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abstract: 'We show that the duality symmetry of the BFKL equation can be interpreted as a symmetry under rotation of the BFKL Kernel in the transverse space from $s$-channel (color dipole model) to $t$-channel (reggeized gluon formulation). We argue that the duality symmetry holds also in the non-forward case due to a very special structure of the non-forward BFKL Kernel, which can be written as a sum of three forward BFKL Kernels. The duality symmetry is established by identifying the dual coordinates with the transverse coordinates of a non-diagonal dipole scattered off the target.'
author:
- |
Alex Prygarin\
[*II. Institute for Theoretical Physics, Hamburg University, Germany*]{}\
title: '**Duality symmetry of BFKL equation: reggeized gluons vs color dipoles.** '
---
Introduction {#sec:intro}
============
The Balistky-Fadin-Kuraev-Lipatov (BFKL) equation [@BFKL] describes the amplitude of scattering at very high center-of-mass energy $\sqrt{s}$ with $|t/s|\ll 1 $, where $t$ is the square of the transferred momentum. The leading order BFKL is obtained by summing terms $(\alpha_s \log s)^n$, where each power of the coupling constant $\alpha_s$ is accompanied by the corresponding power of the logarithm of energy. This kinematic regime is called multiregge kinematics. In the multiregge kinematics, the transverse degrees of freedom fully decouple from the longitudinal ones. This allows to formulate the BFKL equation as evolution in complex time (rapidity) with the integral Kernel operating in the transverse space. The BFKL equation was originally formulated using the fact that $t$-channel gluons reggeize and the production vertices of the $s$-channel gluons factorize in the Regge kinematics. In this picture the BFKL equation describes a compound state of two reggeized gluons. An alternative derivation of the BFKL evolution was proprosed by Mueller [@MUCD] using $s$-channel unitarity for evolution of colorless dipoles in the limit of the large number of colors. The BFKL equation was solved [@BFKLsol] using the conformal invariance of the BFKL Kernel. It was also noticed that the BFKL Kernel has another interesting property called the *duality* symmetry found by Lipatov [@Lipatov:1998as]. This symmetry means that the form of the BFKL equation does not change if the gluon momentum $k$ is replaced by its conjugate coordinate. It was shown that this symmetry can explain the integrability of the BFKL equation. However, it was also suggested that the duality symmetry should hold only for the case of zero momentum transfer for a system of two reggeized gluons.
The objective of the present study is to show that the duality symmetry of the BFKL equation holds also in the non-forward case, though not in an explicit way. We continue the analysis started by the author [@Prygarin:2009tn] and establish the duality symmetry as a symmetry between reggeized gluon formulation and the dipole picture of the BFKL evolution. In particular, we show that the evolution equation for a dipole with different sizes to the left and to the right of the unitarity cut can be written in the form of the BFKL equation in the dual coordinates. The dual momenta coordinates and the conjugate coordinates are not *a priori* related objects, the fact that can identify them is to be understood as a sign for the duality symmetry. However, there seem to be no obvious choice of the Fourier transform (at least of a single variable) that can take one picture to another. This is the reason why we prefer to call this symmetry - the *hidden* duality symmetry. The hidden duality symmetry can also be interpreted as a symmetry under rotation of the BFKL Kernel in the transverse space from $t$-channel (reggeized gluons) to $s$-channel (color dipoles) and back.
Duality symmetry of BFKL equation {#sec:dualityBFKL}
=================================
In this section we explain briefly how the duality symmetry appears in the leading order BFKL equation and show why it is related to the integrability.
The duality symmetry of the system of interacting reggeons in the limit of a large number of colors was first formulated by Lipatov [@Lipatov:1998as]. In the following we briefly outline the major relevant points of this study.
We start with a general description of the BFKL approach and present its formulation in terms of the holomorphic Hamiltonian in the Schödiner like equation.
The BFKL equation describes the behavior of the scattering amplitude in the limit of the center-of-mass energy $\sqrt{s}$ being much larger than the typical transferred momentum $|t/s|\ll 1$ (the Regge kinematics). The leading order BFKL evolution equation is obtained by summing the powers of the parameter $\alpha_s \log s$, where each power of the strong coupling constant $\alpha_s$ is accompanied by the corresponding power of the logarithm of energy. In this picture the BFKL Pomeron appears as a compound state of two reggeized gluons of transverse momenta $\vec{k}$ and $\vec{k}-\vec{q}$ as illustrated in Fig. \[fig:bfkl\]. The color singlet BFKL in the limit of large number of color $N_c$ reads
$$\begin{aligned}
\label{bfkl}
\left(\frac{\partial }{\partial y}-\epsilon(-\vec{k}^2)-\epsilon(-(\vec{k}-\vec{q})^2)\right)\mathcal{F}(\vec{k},\vec{k}-\vec{q})=
\frac{\alpha_s N_c}{2\pi^2} \int d^2 \vec{\chi} \frac{K(\vec{k},\vec{\chi})}{\vec{\chi}^2 (\vec{\chi}-\vec{q})^2}\mathcal{F}(\vec{\chi},\vec{\chi}-\vec{q})\end{aligned}$$
where the gluon reggeization enters the equation through the Regge gluon trajectory $$\begin{aligned}
\label{regge}
\epsilon(-\vec{k}^2)=\frac{\alpha_s N_c}{4\pi^2} \int d^2 \vec{\chi} \frac{-\vec{k}^2}{\vec{\chi}^2(\vec{\chi}-\vec{k})^2}\end{aligned}$$
The real emission part of the Kernel is given by $$\begin{aligned}
\label{kernel}
K(\vec{k},\vec{\chi})=\vec{q}^2-\frac{\vec{k}^2(\vec{\chi}-\vec{q})^2}{(\vec{\chi}-\vec{k})^2}-\frac{\vec{\chi}^2(\vec{k}-\vec{q})^2}{(\vec{\chi}-\vec{k})^2}\end{aligned}$$
![The BFKL evolution equation describes high energy scattering as a compound state of two $t$-channel reggeized gluons, with real $s$-channel gluon emissions crossing the dashed line of the unitarity cut. The effective real production vertices are denoted by the dark blobs and the fact that $t$-channel gluons are reggeized is reflected by crosses. []{data-label="fig:bfkl"}](bfkl){width="1.1in"}
In the leading order BFKL the transverse momenta components decouple from the longitudinal ones (rapidity). Due to this factorization the BFKL Pomeron can be written as a state in the two dimensional transverse space that evolves with rapidity which plays a role of an imaginary time. This fact makes it possible to formulate the color singlet BFKL dynamics in the form of the Schödinger equation for the wave function $f_{m,\tilde{m}}(\vec{\rho}_1,\vec{\rho}_2,...,\vec{\rho}_n;\vec{\rho}_0)$ for a system of $n$-reggeized gluons [@Kwiecinski:1980wb; @Bartels:1980pe; @Lipatov:1990zb], the BFKL equation is obtained for $n=2$. The vectors $\vec{\rho}_k$ are two dimensional coordinates of the reggeized gluons, and $m$ and $\tilde{m}$ are the conformal weights $$\begin{aligned}
m=\frac{1}{2}+i\nu+\frac{n}{2} ,\;\;\; \tilde{m}=\frac{1}{2}+i\nu-\frac{n}{2}\end{aligned}$$ which are expressed in terms of the anomalous dimension $\gamma=1+2i\nu$ and the integer conformal spin $n$. The anomalous dimension and the conformal spin in this context were introduced when solving the BFKL equation in the complex coordinates $$\begin{aligned}
\rho_k=x_k+iy_k, \;\;\; \rho^*_k=x_k-iy_k \end{aligned}$$ using the conformal properties of the BFKL Kernel.
The BFKL wave function $f_{m,\tilde{m}}$ satisfies the Schödinger equation $$\begin{aligned}
E_{m,\tilde{m}}f_{m,\tilde{m}}=H f_{m,\tilde{m}}\end{aligned}$$ with the energy $E_{m,\tilde{m}}$ being proportional to the position of the singularity in the complex angular momentum $j$ plane. In the multicolor limit the Hamiltonian possesses a property of holomorphic separability $$\begin{aligned}
H=\frac{1}{2}\left(h+h^*\right)\end{aligned}$$ where the holomorphic and the anti-holomorphic Hamiltonians $$\begin{aligned}
\label{hamsmall}
h=\sum_{k=1}^n h_{k,k+1}, \;\;\; h^*=\sum_{k=1}^n h^*_{k,k+1}\end{aligned}$$
are expressed through the BFKL operator [@Lipatov:1993qn] $$\begin{aligned}
\label{hholom}
h_{k,k+1}=\log(p_k)+\log(p_{k+1})+\frac{1}{p_k}\log(\rho_{k+1})p_k+\frac{1}{p_{k+1}}\log(\rho_{k+1})p_{k+1}+2\gamma\end{aligned}$$ In Eq. (\[hholom\]) one defines $\rho_{k,k+1}=\rho_k-\rho_{k+1}$, $p_k=i\partial/(\partial \rho_k)$, $p^*_k=i\partial/(\partial \rho^*_k)$ and $\gamma=-\psi(1)$ (the Euler constant). The holomorphic separability of the Hamiltonian means the holomorphic factorization of the wave function $$\begin{aligned}
\label{confbfkl}
f_{m,\tilde{m}}(\vec{\rho}_1,\vec{\rho}_2,...,\vec{\rho}_n;\vec{\rho}_0)=\sum_{r,l} c_{r,l}f^r_{m}(\rho_1,\rho_2,...,\rho_n;\rho_0)
f^l_{\tilde{m}}(\rho^*_1,\rho^*_2,...,\rho^*_n;\rho^*_0)\end{aligned}$$ and the Schödinger equations in the holomorphic and the anti-holomorphic spaces $$\begin{aligned}
\epsilon_m f_m=h f_m, \;\;\; \epsilon_{\tilde{m}}f_{\tilde{m}}=h^* f_{\tilde{m}}, \;\;\;E_{m,\tilde{m}}=\epsilon_m+\epsilon_{\tilde{m}}\end{aligned}$$ The degenerate solutions are accounted for by the coefficients $c_{r,l}$ in Eq. (\[confbfkl\]), which are fixed by the singlevaluedness condition for the wave function in the two dimensional space.
It is interesting to note that the BFKL way function can be normalized in two different ways $$\begin{aligned}
\parallel f\parallel^2_1=\int \prod_{r=1}^n d^2 \rho_r \left| \prod^n_{r=1} \rho^{-1}_{r,r+1} f\right|^2,
\;\;\;
\parallel f\parallel^2_2=\int \prod_{r=1}^n d^2 \rho_r \left| \prod^n_{r=1} p_{r} f\right|^2\end{aligned}$$ This is in an agreement with the hermicity properties of the Hamiltonian, since the transposed Hamiltonian $h^t$ can be obtained by two different similarity transformations [@Lipatov:1993yb] $$\begin{aligned}
\label{norm}
h^t=\prod_{r=1}^n p_r h \prod_{r=1}^n p_r^{-1}=\prod_{r=1}^n \rho_{r,r+1}^{-1} h \prod_{r=1}^n \rho_{r,r+1}\end{aligned}$$
The BFKL Hamiltonian is invariant under cyclic permutations corresponding to the Bose symmetry of the reggeon wave function $i \to i+1 \;(i=1,2...,n)$ in multicolor limit. It was noticed by Lipatov [@Lipatov:1998as] that the Hamiltonian is also invariant under canonical transformation $$\begin{aligned}
\label{change}
\rho_{k-1,k} \to p_{k} \to \rho_{k,k+1}\end{aligned}$$ accompanied by the change of the operator ordering. This property becomes obvious if we rewrite the Hamiltonian Eq. (\[hamsmall\]) in the form of $$\begin{aligned}
h=h_p+h_{\rho}\end{aligned}$$ with $$\begin{aligned}
\label{hp}
h_p=\sum_{k=1}^n \left( \log(p_k) +\frac{1}{2} \sum_{\lambda=\pm1} \rho_{k,k+\lambda} \log(p_k) \rho^{-1}_{k,k+\lambda}+\gamma \right)\end{aligned}$$
and $$\begin{aligned}
\label{hrho}
h_\rho=\sum_{k=1}^n \left( \log(\rho_{k,k+1}) +\frac{1}{2} \sum_{\lambda=\pm1} p^{-1}_{k+(1+\lambda)/2} \log(\rho_{k,k+1}) p_{k+(1+\lambda)/2}+\gamma \right)\end{aligned}$$
The invariance of the BFKL Hamiltonian under the change of the variables Eq. (\[change\]) together with the change of the operator ordering was called the *duality* symmetry. The duality symmetry implies that the BFKL Hamiltonian commutes $[h,A]=0$ with the differential operator $$\begin{aligned}
A=\rho_{12}\rho_{23}...\rho_{n1}p_1p_2...p_n.\end{aligned}$$ or, more generally, there is a family of mutually commuting integrals of motion [@Lipatov:1993yb] $$\begin{aligned}
[q_r,q_s]=0, \;\;\; [q_r,h]=0\end{aligned}$$ and they are given by $$\begin{aligned}
q_r=\sum_{i_1<i_2<...<i_r} \rho_{i_1,i_2}\rho_{i_2,i_3}...\rho_{i_r,i_1}p_{i_1}p_{i_2}... p_{i_r}\end{aligned}$$ The operators $q_r$ build a complete set of the invariants of the transformation. Therefore the Hamiltonian $h$ is their function $$\begin{aligned}
h=h(q_2,q_3,...,q_n)\end{aligned}$$ and a common eigenfunction of $q_r$ is simultaneously a solution to the Schödinger equation. This fact explains why the duality symmetry is related to the integrability of a system of Reggeons in the limit of the large number of colors $N_c$. In the the multicolor limit only nearest neighbor interactions are not suppressed and the BFKL dynamics is similar to that of the Ising spin chain model.
The transformation Eq. (\[change\]) of the holomorphic BFKL Hamiltonian is an unitary transformation only for a vanishing total momentum $$\begin{aligned}
\vec{p}=\sum^n_{r=1}\vec{p}_r\end{aligned}$$ which guarantees the cyclicity of the momenta $p_r$ important for their representation by the difference of coordinates $\rho_{r,r+1}$. For the compound state of two reggeized gluons (usual BFKL case) for $n=2$, this can be achieved only for the zero transferred momentum $\vec{q}=0$. Only in this case one can really identify the dual coordinates $\vec{\rho}_{r,r+1}$ of the momenta $\vec{p}_r$ with their conjugate coordinates. In a more general case these two are not the same object. However, the integrability of the non-forward BFKL suggests that the duality symmetry should be present also in the case of $\vec{q}\neq 0$, but in an *implicit* way. The main objective of the present study is to show that the dipole formulation of the BFKL evolution can provide a suitable framework for studying the duality symmetry of the non-forward BFKL. We show that the evolution equation for the scattering of a non-diagonal dipole coincides with the non-forward BFKL equation in the dual space provided we impose on the dipole scattering amplitude some condition that is dual to the so-called *BFKL condition*. The BFKL condition is a result of the unitarity and the multiregge kinematics used for deriving the leading order BFKL as discussed below. In this formalism the duality symmetry of the non-forward BFKL equation appears in an implicit way due to the fact that we can identify the set of coordinates of the scatterred dipole with a set of dual coordinates of the reggeized gluons momenta. However it looks like that there is no obvious choice of the Fourier transform that can relate the dipole coordinates to the reggeized gluon momenta individually, that is the reason why the duality symmetry is established implicitly. The duality symmetry holds also in the non-forward case because of the special structure of the non-forward BFKL Kernel, which can be viewed as sum of the three forward Kernels. As it was already mentioned the duality symmetry of the forward BFKL can be shown explicitly, which suggests that the sum of the three forward Kernels should also possess this property.
One remark is in order. A system may possess another symmetry with a similar name, called the *dual conformal* symmetry. The dual conformal symmetry is an usual conformal symmetry in dual coordinates ($k_i=x_i-x_{i+1}$) and, generally, is not related to the duality symmetry. The dual conformal symmetry is now successfully implemented in calculating multileg planar amplitudes in SYM $\mathcal{N}=4$ (for an up-to-date discussion see Ref. [@Korchemsky:2009hm] and references wherein), and it was also recently considered in the connection with the BFKL equation [@Gomez:2009bx]. This symmetry is beyond the scope of the present study.
In the next section we write the BFKL equation in the dual coordinates and analyze its structure. We argue that the non-forward BFKL equation can be represented as a three point amplitude, due to the *BFKL condition* associated with the lack of the crossing symmetry in the BFKL approach.
BFKL equation in dual coordinates {#sec:BFKLdualcoordinates}
=================================
In this section we discuss the structure of the BFKL equation and write it in the *dual* coordinates. We show that the *non-forward* BFKL Kernel $q\neq 0$ can written as a sum of three *forward* Kernels, which can be interpreted as two uncut and one cut Kernel (UCU structure). The UCU structure of the BFKL equation is crucial for establishing the duality symmetry also in the non-forward case.
We start with recasting the BFKL equation into a form useful for our discussion.[^1] The direct substitution of Eq. (\[regge\]) and Eq. (\[kernel\]) in Eq. (\[bfkl\]) gives $$\begin{aligned}
\label{bfkl3linesOLD}
\frac{\partial \mathcal{F}(k,k-q) }{\partial y}&=&+
\frac{\alpha_s N_c}{2 \pi^2} \int \frac{d^2 \chi \;k^2}{ \chi^2(\chi-k)^2}\mathcal{F}(\chi,\chi-q)-\frac{\alpha_s N_c}{4 \pi^2} \int \frac{d^2 \chi \;k^2}{ \chi^2(\chi-k)^2}\mathcal{F}(k,k-q) \nonumber \\
&&
+\frac{\alpha_s N_c}{2 \pi^2} \int \frac{d^2 \chi \; (k-q)^2}{ (\chi-q)^2(\chi-k)^2}\mathcal{F}(\chi,\chi-q)-\frac{\alpha_s N_c}{4 \pi^2} \int \frac{d^2 \chi \;(k-q)^2}{ (\chi-q)^2(\chi-k)^2}\mathcal{F}(k,k-q) \nonumber \\
&&
-\frac{\alpha_s N_c}{2 \pi^2} \int \frac{d^2 \chi \; q^2}{ \chi^2(\chi-q)^2}\mathcal{F}(\chi,\chi-q)\end{aligned}$$
For our purpose it is convenient to write the second line of Eq. (\[bfkl3linesOLD\]) in a slightly different form changing the integration variable $\chi \to \chi-q$ $$\begin{aligned}
\label{bfkl3lines}
\frac{\partial \mathcal{F}(k,k-q) }{\partial y}&=&+
\frac{\alpha_s N_c}{2 \pi^2} \int \frac{d^2 \chi \;k^2}{ \chi^2(\chi-k)^2}\mathcal{F}(\chi,\chi-q)-\frac{\alpha_s N_c}{4 \pi^2} \int \frac{d^2 \chi \;k^2}{ \chi^2(\chi-k)^2}\mathcal{F}(k,k-q) \nonumber \\
&&
+\frac{\alpha_s N_c}{2 \pi^2} \int \frac{d^2 \chi \; (k-q)^2}{ \chi^2(\chi-k+q)^2}\mathcal{F}(\chi+q,\chi)-\frac{\alpha_s N_c}{4 \pi^2} \int \frac{d^2 \chi \;(k-q)^2}{ \chi^2(\chi-k+q)^2}\mathcal{F}(k,k-q) \nonumber \\
&&
-\frac{\alpha_s N_c}{2 \pi^2} \int \frac{d^2 \chi \; q^2}{ \chi^2(\chi-q)^2}\mathcal{F}(\chi,\chi-q)\end{aligned}$$
One can see that the Kernel of the non-forward BFKL equation can be written as a sum of the three forward Kernels, where two of them, are the usual forward BFKL Kernels given by the first and the second lines of Eq. (\[bfkl3lines\]), while the third line has only real emission part. This interpretation is better understood from Fig. \[fig:UCUnew\], where the unitarity cut is denoted by the vertical dashed line. The BFKL Kernel that describes the bound states of two reggeized gluons $k$ and $k-q$ can be viewed as sum of the *uncut* forward Kernels for the gluon pair $k$ and $k$ and the gluon pair $k-q$ and $k-q$ ( the first term and the third term on the r.h.s in Fig. \[fig:UCUnew\] ), and the *cut* forward Kernel for the scattering of the pair of fictitious gluons $q$ and $q$ (the second term on the r.h.s in Fig. \[fig:UCUnew\] ). The last contribution seats exactly on the unitarity cut and thus does not possess any virtual contribution. This UCU structure of the non-forward BFKL Kernel plays an important role in showing the duality symmetry of the BFKL equation and in finding its dual in the dipole picture as we show below. To see this we first write Eq. (\[bfkl3lines\]) in the *dual* coordinates properly chosen by making the following observations.
![ The uncut-cut-uncut (UCU) structure of the non-forward BFKL. The non-forward BFKL Kernel can be written as a linear combination of three forward Kernels, two uncut (for two gluon pairs $k$ and $k-q$) and one cut for the gluon pair $q$. The cut Kernel does not possess virtual contributions, which is reflected by the absence of crosses on gluons $q$.[]{data-label="fig:UCUnew"}](UCUnew){width="4.5in"}
The duality symmetry holds also for a case of the multireggeon exchange, and is not limited to the system of two reggeized gluons as in the BFKL equation Eq. (\[bfkl3lines\]). Another important point is that only the upper momenta of the reggeized gluons are to be taken into account. In particular, this means that in the case of the BFKL Pomeron we have only three momenta for the duality symmetry, because the Regge kinematics selects $t$-channel for a propagation of the BFKL state breaking the crossing symmetry. Together with the unitarity condition and the strong ordering of the produced particles (multiregge kinematics) this results into some constraint on the form of the leading order BFKL amplitude, which we call the $BFKL \hspace{0.1cm} condition$. This condition is implicitly written in the LO BFKL as we explain below.
By inspecting the arguments of the BFKL amplitude in Eq. (\[bfkl3lines\]) of both the real and the virtual parts, one can deduce that their difference is always equal to the transferred momentum, namely for $\mathcal{F}(k_i,k_j)$ we have $k_i-k_j=q$. This is a consequence of the use of the $t$-channel unitarity together with a special kinematics in the BFKL approach. We call this condition the $BFKL \hspace{0.1cm} condition$. It suggests to treat the BFKL amplitude as a three point function with external momenta
$$\begin{aligned}
\label{k1k2k3}
k_1=k ;\;\; k_2=q-k; \;\; k_3=-q\end{aligned}$$
At first sight, the BFKL amplitude is four point scattering amplitude with four external (transverse) momenta $k$, $k-q$, $k'$ and $k'-q$, but the BFKL condition removes the necessity in the fourth external momentum leaving three momenta which obey the conservation law. This means that the BFKL amplitude is in fact a function of only two external transverse momenta, i.e. $k$ and $q$ or $k$ and $k-q$. In other words, the duality symmetry deals with only upper gluon momenta or only lower gluon momenta, but never with mixes them. This observation suggests to pick up only three dual coordinates.
For our purposes we define the dual coordinates
$$\begin{aligned}
\label{duals}
k=k_1=x_1-x_2=x_{12}; \;\; q-k=k_2=x_2-x_3=x_{23}; \;\; -q=k_3=x_3-x_1=x_{31}\end{aligned}$$
so that the overall momenta conservation $k_1+k_2+k_3=0$ is automatically satisfied and the BFKL amplitude can be represented as a three point function in the dual space as illustrated in Fig. \[fig:4point3point\]. Using this definition we can write the BFKL equation Eq. (\[bfkl3lines\]) as follows
$$\begin{aligned}
\label{bfkldual}
\frac{\partial \mathcal{F}(x_{12},x_{23}) }{\partial y}&=&+
\frac{\alpha_s N_c}{2 \pi^2} \int \frac{d^2 z \; x_{12}^2}{ z^2(z-x_{12})^2}\left\{\mathcal{F}(z,z+x_{31})-\frac{1}{2}\mathcal{F}(x_{12},-x_{23}) \right\} \nonumber \\
&&
+\frac{\alpha_s N_c}{2 \pi^2} \int \frac{d^2 z \; x_{23}^2}{ z^2(z+x_{23})^2}\left\{\mathcal{F}(z-x_{31},z)-\frac{1}{2}\mathcal{F}(x_{12},-x_{23}) \right\} \nonumber \\
&&
-\frac{\alpha_s N_c}{2 \pi^2} \int \frac{d^2 z \; x_{31}^2}{ z^2(z+x_{31})^2}\mathcal{F}(z,z+x_{31})\end{aligned}$$
![The BFKL condition constraints the representation of the BFKL scattering amplitude as a function of only three dual coordinates instead of four external points. []{data-label="fig:4point3point"}](4point3point){width="6in"}
In the next section we discuss the evolution equation for the non-diagonal dipole scattering and show that it can be written in the form of the BFKL in the dual coordinates Eq. (\[bfkldual\]) by imposing on it the condition dual to the BFKL condition.
Scattering of non-diagonal dipole {#sec:dipole}
=================================
In this section we show that the evolution equation for the scattering of the non-diagonal dipole depicted in Fig. \[fig:dipole12psik\] can be brought to the form of the BFKL equation in the dual space Eq. (\[bfkldual\]). The scattering of the non-diagonal dipole with different coordinates in the amplitude and the conjugate amplitude (to the left and to the right of the unitarity cut) was considered by Levin and the author [@Levin:1900tt] as an auxiliary problem in proving the single inclusive production formula in the dipole formulation. Such a dipole can be constructed if one fixes momentum of the antiquark line and thus keeping it coordinates different to the left and to the right of the unitarity cut, whereas the lower quark line momentum is integrated over resulting in $\delta^{(2)}(\rho_1-\rho_{1'})$. The non-linear evolution equation was derived and solved using the notion of the "generalized optical theorem”. The function $M(12|12')$ for which the equation was derived is an auxiliary function for proving the single inclusive production formula for the dipole model. It has a meaning of the non-diagonal dipole total cross section since for $\rho_2=\rho_{2'}$ it reduces to $M(12|12)=2N(12)$ (the optical theorem in the coordinate space), where $N(12)$ is the BFKL amplitude in the Möbius representation. This property was imposed by the definition of $M(12|12')$, since for $\rho_2=\rho_{2'}$ the non-diagonal dipole takes a form of a usual dipole, which is described by the BFKL equation.
![ The schematic representation of a color dipole, which has different transverse sizes to the left and to the right of the unitarity cut denoted by the vertical dashed line. For our purposes it is enough to consider only difference in the coordinates of the upper (antiquark) line, keeping the the coordinates of the lower (quark) the same. The broken antiquark line illustrates only the fact that the sizes are different. There is no discontinuity in the charge flow etc. []{data-label="fig:dipole12psik"}](dipole12psik){width="2in"}
The evolution equation for the non-diagonal dipole is derived using *real*-*virtual* *non*-*cancellations*, i.e. including the interactions in the final state. The final state interactions fully cancel in the inclusive case, but as far as gluon production is concerned such cancellation does not happen and this fact is crucial for obtaining the closed form of the single gluon production cross section with evolution effects included. For more details about the way it was derived one is referred to Levin and the author [@Levin:1900tt]. Here we only want to discuss the linear version of this evolution equation, its properties and to show that it can be written as a non-forward BFKL in the dual space. This result would mean that there exist a *hidden* *duality symmetry* of the non-forward BFKL, that appears implicitly from our analysis due to the fact that the set of dual coordinates (with dimensions of mass) can be associated with set of the transverse coordinates of the dipoles. This extends the duality symmetry shown by Lipatov for the forward case, to a non-zero momentum transfer, which can potentially explain the integrability of the BFKL equation.
For our purposes we retain only the linear part of the resulting non-linear evolution equation for a non-diagonal dipole scattering [@Levin:1900tt]. It reads
$$\begin{aligned}
\label{evolMA1simple}
\frac{\partial M(12|12')}{\partial y}=\frac{\bar{\alpha_s}}{2\pi}\int d^2\rho_3 \left\{
-\frac{1}{2}\left(\frac{\rho_{13}}{\rho^2_{13}}-\frac{\rho_{23}}{\rho^2_{23}}\right)^2M(12|12')-\frac{1}{2}\left(\frac{\rho_{13}}{\rho^2_{13}}-\frac{\rho_{2'3}}{\rho^2_{2'3}}\right)^2M(12|12')
\right.\end{aligned}$$
$$\begin{aligned}
\left.
+\left(\frac{\rho_{13}}{\rho^2_{13}}-\frac{\rho_{23}}{\rho^2_{23}}\right)\left(\frac{\rho_{13}}{\rho^2_{13}}-\frac{\rho_{2'3}}{\rho^2_{2'3}}\right) \left\{2N(13)+M(32|32')\right\}
-\left(\frac{\rho_{13}}{\rho^2_{13}}-\frac{\rho_{23}}{\rho^2_{23}}\right)\left(\frac{\rho_{23}}{\rho^2_{23}}-\frac{\rho_{2'3}}{\rho^2_{2'3}}\right)
M(13|12')\right. \nonumber
\\
\left.
-\left(\frac{\rho_{2'3}}{\rho^2_{2'3}}-\frac{\rho_{23}}{\rho^2_{23}}\right)\left(\frac{\rho_{13}}{\rho^2_{13}}-\frac{\rho_{2'3}}{\rho^2_{2'3}}\right)
M(12|13)
-\frac{1}{2}\left(\frac{\rho_{23}}{\rho^2_{23}}-\frac{\rho_{2'3}}{\rho^2_{2'3}}\right)^2
M(12|12')
\right\} \nonumber\end{aligned}$$
As it was already mentioned the function $M(12|12')$ is defined such that $M(12|12)=2N(12)$.[^2] This definition follows from the fact that in the simple case of equal dipole sizes $\rho_2=\rho_{2'}$ all necessary *real-virtual cancellations* take place removing all final state interactions, and one deals with the scattering of an usual color dipole described by the BFKL equation in the coordinate space. Indeed, as it easy to see that Eq. (\[evolMA1simple\]) reduces to the BFKL equation for $\rho_2=\rho_{2'}$ (see Ref. [@Levin:1900tt]). This definition and the properties of the initial condition suggested the possible form of the solution to the non-diagonal dipole evolution equation $$\begin{aligned}
\label{optics}
M(12|12')=N(12)+N(12')-N(22')\end{aligned}$$ It was checked by the explicit substitution that this form of the solution keeps also in the non-linear case of the generalized Balitsky-Kovchegov (BK) [@B; @K] equation considered in Ref. [@Levin:1900tt]. The non-linear equation is a generalization of the Balitsky-Kovchegov equation and coincides with it for $\rho_2=\rho_{2'}$ similar to the linear case. Using this form of solution in Eq. (\[evolMA1simple\]) we obtain
$$\begin{aligned}
\label{evolMA1comb}
\frac{\partial (N(12)+N(12')-N(22'))}{\partial y}=\frac{\bar{\alpha_s}}{2\pi}\int
\frac{\rho^2_{12}d^2\rho_3 }{\rho^2_{13}\rho^2_{23}} \left\{N(13)+N(32)-N(12)\right\}
\end{aligned}$$
$$\begin{aligned}
+\frac{\bar{\alpha_s}}{2\pi}\int
\frac{\rho^2_{12'}d^2\rho_3 }{\rho^2_{13}\rho^2_{2'3}} \left\{N(13)+N(32')-N(12')\right\} \nonumber\end{aligned}$$
$$\begin{aligned}
-\frac{\bar{\alpha_s}}{2\pi}\int
\frac{\rho^2_{22'}d^2\rho_3 }{\rho^2_{23}\rho^2_{2'3}} \left\{N(32)+N(32')-N(22')\right\} \nonumber\end{aligned}$$
which is just a linear combination of three BFKL equations for initial dipoles with coordinates $12$, $12'$ and $22'$. This reminds the *uncut-cut-uncut* (UCU) structure of the BFKL equation in the momentum space mentioned in the previous section. It is worthwhile mentioning that generalized BK equation also has the UCU structure, which fully corresponds to the picture drawn by Ciafaloni, Marchesini and Veneziano deriving the Cut Reggeon Calculus [@Ciafaloni:1975jy; @Ciafaloni:1975fh]. They found that the Pomeron can be described as a linear combination of three propagating states, which correspond to one cut and two uncut Pomerons $\phi^++\phi^--\phi^c$. The reggeon field $\phi^+$ stands for the Pomeron to the left of the unitarity cut, $\phi^-$ for the Pomeron to the right of the unitarity cut and $\phi^c$ represents the Pomeron living on the cut. The introduction of the triple Pomeron splitting vertex (“fan“ diagrams) preserves this structure, while the Pomeron loops break it explicitly. The same result was also obtained by Levin and the author [@Levin:2007yv] using generating functional approach to the analysis of the multiparticle states in the dipole model based on Abramovski-Gribov-Kancheli cutting rules [@Abramovsky:1973fm].
Our goal is to show that the evolution equation for the non-diagonal dipole reproduces the non-forward BFKL equation in the dual coordinates. It is not difficult to see that with the help of the solution Eq. (\[optics\]) we can recast Eq. (\[evolMA1comb\]) into form of
$$\begin{aligned}
\label{evolMnew1}
\frac{\partial M(12|12')}{\partial y}&=&\frac{\bar{\alpha_s}}{2\pi}\int
\frac{\rho^2_{12}d^2\rho_3 }{\rho^2_{13}\rho^2_{23}} \left\{M(32|32')-\frac{1}{2}M(12|12')+M(32|22')-\frac{1}{2}M(12|22')\right\}
\\&+&
\frac{\bar{\alpha_s}}{2\pi}\int
\frac{\rho^2_{12'}d^2\rho_3 }{\rho^2_{13}\rho^2_{2'3}} \left\{M(32|32')-\frac{1}{2}M(12|12')+M(32'|22')-\frac{1}{2}M(12'|22')\right\}\nonumber
\\&-&
\frac{\bar{\alpha_s}}{2\pi}\int
\frac{\rho^2_{22'}d^2\rho_3 }{\rho^2_{23}\rho^2_{2'3}} M(32|32') \nonumber\end{aligned}$$
We immediately notice that Eq. (\[evolMnew1\]) is very similar to Eq. (\[bfkldual\]) except the last two terms in brackets of the first two lines. This is despite the fact that the functions $M(12|12')$ is defined in a very much different way than the BFKL amplitude. The main difference is that the BFKL amplitude $\mathcal{F}(k,k-q)$ accounts for the requirement of the BFKL condition. In particular, this means that for $\mathcal{F}(k,k-q)=\mathcal{F}(k_1,k_2)$ the arguments should satisfy $$\begin{aligned}
\label{unit}
k_1-k_2=q\end{aligned}$$
which is translated in terms of the dual coordinates Eq. (\[duals\]) as the function $\mathcal{F}(z_1,z_2)$ should satisfy $z_1-z_2=-x_{31}$. It is now possible to identify the dual coordinates of Eq. (\[duals\]) with the transverse dipole coordinates as follows
$$\begin{aligned}
\label{associate}
\rho_{12}=x_{12}; \;\; \rho_{12'}=-x_{23}; \;\; \rho_{22'}=x_{31}\end{aligned}$$
The dual of the BFKL condition in Eq. (\[unit\]) for $M(ij|ik)$ reads $$\begin{aligned}
\label{unitdual}
\rho_{ij}-\rho_{ik}=-\rho_{22'}\end{aligned}$$ Imposing the dual of the BFKL condition Eq. (\[unitdual\]) on the evolution equation for the non-diagonal dipole removes undesired terms in Eq. (\[evolMnew1\]) and we are left with
$$\begin{aligned}
\label{dipoledualOLD}
\frac{\partial \tilde{M}(\rho_{12}|\rho_{12'})}{\partial y}&=&\frac{\bar{\alpha_s}}{2\pi}\int
\frac{\rho^2_{12}d^2\rho_3 }{\rho^2_{13}\rho^2_{23}} \left\{\tilde{M}(\rho_{32}|\rho_{32'})-\frac{1}{2}\tilde{M}(\rho_{12}|\rho_{12'})\right\}
\\&+&
\frac{\bar{\alpha_s}}{2\pi}\int
\frac{\rho^2_{12'}d^2\rho_3 }{\rho^2_{13}\rho^2_{2'3}}\left\{\tilde{M}(\rho_{32}|\rho_{32'})-\frac{1}{2}\tilde{M}(\rho_{12}|\rho_{12'})\right\} \nonumber
\\&-&
\frac{\bar{\alpha_s}}{2\pi}\int
\frac{\rho^2_{22'}d^2\rho_3 }{\rho^2_{23}\rho^2_{2'3}} \tilde{M}(\rho_{32}|\rho_{32'}) \nonumber\end{aligned}$$
Recasting Eq. (\[dipoledualOLD\]) in a more transparent form we get
$$\begin{aligned}
\label{dipoledualOLD}
\frac{\partial \tilde{M}(\rho_{12}|\rho_{12'})}{\partial y}&=&\frac{\bar{\alpha_s}}{2\pi}\int
\frac{\rho^2_{12}d^2\rho_{32} }{\rho^2_{32}(\rho_{32}-\rho_{12})^2} \left\{\tilde{M}(\rho_{32}|\rho_{32}+\rho_{22'})-\frac{1}{2}\tilde{M}(\rho_{12}|\rho_{12'})\right\}
\\&+&
\frac{\bar{\alpha_s}}{2\pi}\int
\frac{\rho^2_{12'}d^2\rho_{32'} }{\rho^2_{32'}(\rho_{32'}-\rho_{12'})^2}\left\{\tilde{M}(\rho_{32'}-\rho_{22'}|\rho_{32'})-\frac{1}{2}\tilde{M}(\rho_{12}|\rho_{12'})\right\} \nonumber
\\&-&
\frac{\bar{\alpha_s}}{2\pi}\int
\frac{\rho^2_{22'}d^2\rho_{32} }{\rho^2_{32}(\rho_{32}+\rho_{22'})^2} \tilde{M}(\rho_{32}|\rho_{32}+\rho_{22'}) \nonumber\end{aligned}$$
which is identical to the non-forward BFKL equation in the dual space Eq. (\[bfkldual\]) provided we identify the dipole coordinates and the dual coordinates as in Eq. (\[associate\]). It is not surprising that the equation for $M(12|12')$ includes more terms than the BFKL for $\mathcal{F}(k,k-q)$, since $M(12|12')$ was defined without any additional constraint except to reproduce dipole BFKL for $\rho_2=\rho_{2'}$, in contrast to the BFKL amplitude.
By construction of the dipole model the coordinates $\rho_{ij}$ are conjugate to the momenta $k_i$ of the reggeized gluons. The fact that we can identify the dual momenta coordinates of Eq. (\[duals\]) with the dipole coordinates indicates that the duality symmetry is preserved also in the non-forward case. However, there seems to be no obvious way to introduce the Fourier transform that connects them and thus the duality symmetry is *hidden*.
In our discussion we ignored the issue of the initial condition and the impact parameter $b_{12}=(\rho_1+\rho_2)/2$ dependence. These two are related to each other since the impact parameter defines a reference point that connects the evolution to the target. Any fixed reference point breaks explicitly the translational symmetry and thus the impact parameter cannot be related to the set of the dual coordinates Eq. (\[duals\]). For a similar reason we do not consider the lower momenta $k'$ and $k'-q$ of the amputated BFKL amplitude shown in Fig. \[fig:4point3point\], more accurately, we do not consider simultaneously the upper and the lower momenta. We assign the evolution to the upper momenta, while the lower momenta enter through the initial condition (we could do vice versa). Any attempt to include the initial condition to duality picture would contradict the lack of the impact parameter dependence in the dipole picture, but as have already pointed out the $b$-dependence is incompatible with the requirement of the translational invariance.
![The duality symmetry can be interpreted as a symmetry under rotation of the BFKL Kernel in the transverse space from $s$-channel (color dipoles) to $t$-channel (reggeized gluons). The unitarity cut is denoted by a dashed vertical line.[]{data-label="fig:rot"}](rot){width="4in"}
The hidden duality symmetry is related only to the pure evolution, without any reference to the initial condition. As it was anticipated in Ref. [@Prygarin:2009tn], the duality symmetry is the symmetry under rotation of the BFKL Kernel in the transverse space from $s$-channel to $t$-channel and back as illustrated in Fig. \[fig:rot\]. This rotation is, in fact, a rotation between the reggeized gluon formulation of the BFKL evolution and the dipole picture. The connection between the two pictures is certainly not complete without matching the initial condition. The proper matching is formulated as follows. At the first stage, one makes a suitable choice of the dual coordinates, then the physical picture is changed by rotating the Kernel of the evolution equation in the *transverse* space and the function is given the proper interpretation (either reggeized gluon or dipole scattering amplitude). Finally, at the second stage, the initial conditions are chosen in accordance to the physical picture. The second stage is obviously has nothing to do with duality symmetry property of the BFKL evolution. This point seems to be not so much important in the case of the linear evolution considered here, but it becomes crucial for clearifying the meaning of the duality symmetry of the Balitsky-Kovchegov equation.
Conclusion {#sec:concl}
==========
We discussed the duality symmetry of the LO BFKL equation. The duality symmetry of the BFKL equation was formulated by Lipatov [@Lipatov:1998as] for a system of $n$ reggeized gluons, and in the case of the color singlet BFKL equation ($n=2$) the duality symmetry was shown to hold only in the forward ($q=0$) case. In the present study we argue that the duality symmetry is valid also in the non-forward case, though in an *implicit* way. The *hidden* duality symmetry is established by identifying the dual coordinates (with dimension of mass) of the BFKL in the momentum space with the transverse sizes of a non-diagonal dipole scattered off the target. The evolution equation for the non-diagonal dipole having different sizes to the right and to the left of the unitarity cut was derived by Levin and the author [@Levin:1900tt]. Its analytical solution was also found, and it is a linear combination of three amplitudes of usual dipoles. This structure is similar to the structure of the non-forward BFKL, which can be also decomposed in three pieces each corresponding to forward BFKL. Two of the pieces can be viewed as uncut BFKL, while one piece does not have virtual contribution and is interpreted as a cut BFKL. The uncut-cut-uncut (UCU) structure of the BFKL Kernel uncovered in the present study is consistent with the picture drawn by Ciafaloni, Marchesini and Veneziano [@Ciafaloni:1975fh; @Ciafaloni:1975jy] in Cut Reggeon Calculus, where the Pomeron is represented by three fields, which denote two uncut and one cut Pomerons.
We argue that the duality symmetry can be viewed as a symmetry under rotation of the BFKL Kernel in the transverse space from $s$-channel (color dipoles) to $t$-channel (reggeized gluons) and back as illustrated in Fig. \[fig:rot\].
Acknowledgments {#acknowledgments .unnumbered}
---------------
We are deeply indebted to J. Bartels, V. Fadin, G. Korchemsky, J. Kotanski, L. Levin, L. Lipatov and L. Motyka for very helpful discussions. This study was supported by the Minerva Postdoctoral Fellowship of the Max Planck Society.
[99]{}
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G. P. Korchemsky and E. Sokatchev, arXiv:0906.1737 \[hep-th\].
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[^1]: From now on we deal only with two dimensional transverse momenta and omit the vector sign to make the presentation clear.
[^2]: Here indices of the argument stand for the transverse coordinates of the quark $\rho_1$ and the antiquark $\rho_{2}$ ($\rho_{2'}$) lines and not for only the dipole size $\rho_{12}=\rho_1-\rho_2$ in contrast to the common notation.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'Prethermalization, by introducing emergent quasiconserved observables, plays a crucial role in protecting periodically driven (Floquet) many-body phases over exponentially long time, while the ultimate fate of such quasiconserved operators can signal thermalization to infinite temperature. To elucidate the properties of prethermal quasiconservation in many-body Floquet systems, here we systematically analyze infinite temperature correlations between observables. We numerically show that the late-time behavior of the autocorrelations unambiguously distinguishes quasiconserved observables from non-conserved ones, allowing to single out a set of linearly-independent quasiconserved observables. By investigating two Floquet spin models, we identify two different mechanism underlying the quasiconservation law. First, we numerically verify energy quasiconservation when the driving frequency is large, so that the system dynamics is approximately described by a static prethermal Hamiltonian. More interestingly, under moderate driving frequency, another quasiconserved observable can still persist if the Floquet driving contains a large global rotation. We show theoretically how to calculate this conserved observable and provide numerical verification. Having systematically identified all quasiconserved observables, we can finally investigate their behavior in the infinite-time limit and thermodynamic limit, using autocorrelations obtained from both numerical simulation and experiments in solid state nuclear magnetic resonance systems.'
author:
- Chao Yin
- 'Pai Peng (彭湃)'
- Xiaoyang Huang
- Chandrasekhar Ramanathan
- Paola Cappellaro
bibliography:
- 'Biblio.bib'
title: Prethermal quasiconserved observables in Floquet quantum systems
---
[UTF8]{}[gbsn]{}
[^1]
Introduction
============
Controlling quantum systems using a periodic (Floquet) drive has emerged as a powerful tool in the field of condensed matter physics and quantum information science. It has been used to realize Hamiltonians that are not accessible in a static system, such as modifying the tunneling and coupling rates [@Eckardt05; @Tsuji11; @Mentink15; @Kitamura16; @Mikhaylovskiy15; @Gorg18], inducing non-trivial topological structures [@Lindner11; @Wang13s; @Oka09; @Gu11; @Grushin14; @FoaTorres14; @Rudner13; @Jiang11l; @Kundu13; @Kitagawa10; @Else17b], creating synthetic gauge fields [@Goldman14; @Bukov15; @Bukov16; @Struck12; @Aidelsburger13] and spin-orbit couplings [@Struck14]. On a quantum computer, Floquet engineering also enables universal quantum simulation via Trotter-Suzuki scheme [@trotter59; @Lloyd96; @liu19x; @Jotzu14; @Aidelsburger15; @Kokail19; @Childs2018]. Floquet systems also possess interesting dynamical phenomena ranging from discrete time crystalline phase [@Choi17n; @Zhang17n; @Moessner17; @Luitz19; @Machado19x] to dynamical localization [@Dunlap86; @Fishman82], dynamical phase transitions [@Bastidas12; @Bastidas12a] and coherent destruction of tunneling [@Grossmann91; @Grossmann92; @Grifoni98].
While the connection to an effective time-independent Hamiltonian is appealing, the active drive leads to energy absorption by the Floquet many-body system, which is then expected to heat up to infinite temperature. The heating is detrimental to any quantum application, as no local quantum information is retained and all interesting phenomena mentioned above disappear [@Lazarides14; @DAlessio14; @Kim14]. It has been shown theoretically [@Abanin17; @Abanin15; @Kuwahara16; @Abanin17b; @Else17] and experimentally [@Peng2019; @Antonio2020] that even when the system heats up, the thermalization time can be exponentially long in the drive parameters (typically the frequency of a rapid drive). Then, a long-lived *prethermal* quasi-equilibrium is established, that allows exploiting the engineered Floquet Hamiltonian for quantum simulation [@Heyl19; @DAlessio13; @Sieberer19]. The emergent symmetries and conserved observables in the prethermal state distinguish it from the fully thermalized state, and underpin the existence of novel Floquet phases [@Else17; @Luitz19; @Machado19x]. Even more surprisingly, some numerical studies have shown that the emergent conserved observables might not display thermalizing behavior even in the infinite-time limit [@Heyl19; @Sieberer19; @Prosen99; @DAlessio13]. Many-body localization [@Abanin16; @Lazarides15; @Ponte15; @Zhang17n; @Zhang16b; @Po16; @Bordia17; @Khemani16], dynamic localization [@Heyl19; @Sieberer19; @Ji18], and some fine-tuned driving protocols [@Prosen98; @Prosen99; @DAlessio13] provide a way to escape the thermalization fate, which could also be absent in finite-size systems. Indeed, distinguishing the long-lived prethermal state from an eventual thermal state is challenging. Numerical studies are bound to finite-size (and often small) systems, while experiments can only probe finite times, before the external environment induces thermal relaxation.
Here we tackle this problem by a numerical and experimental study of two Floquet models in spin chains, namely the kicked dipolar model (KDM) and the alternating dipolar model (ADM). While most studies on spin chain dynamics have focused on evolution of pure states, here we propose to study Floquet prethermalization using infinite temperature correlations. This metric provides information about quasisconserved observables across the whole spectrum and serves as a direct measurable quantity in nuclear magnetic resonance (NMR) experiments. In Sec. \[sec:conserv\] we show that the existence of long-lived quasiconserved observables can be unambiguously identified using late-time behavior of the correlations, based on which we provide a method to systematically search for all linearly-independent local quasiconserved quantities. Then we provide both numerical and analytical tools to investigate such prethermal conserved observables and their origins. We first show that the prethermal Hamiltonian $H_{pre}$ obtained from the Magnus expansion under rapid drive yields a quasiconserved observable in each model in Sec. \[sec:Hpre\]. We further show in Sec. \[sec:Dpre\] that when the driving Hamiltonian contains a large global rotation, the Floquet propagator can induce an additional conserved observable, as shown by going beyond the usual Magnus expansion. With all the quasiconserved observables at hand, we investigate in Sec. \[sec:limit\] whether they exist in the thermodynamic limit and infinite-time limit, by looking at the dependence of autocorrelations on system size (numerically) and on time (experimentally). Both methods indicate quasiconserved observables vanish and the system thermalizes to infinite temperature.
Quasiconserved observables {#sec:conserv}
==========================
Hamiltonians and Correlations
-----------------------------
![\[fig:dyn\] Typical dynamics of $\langle {\mathcal{O}}(t){{\mathcal{O}}}' \rangle$ in a Floquet spin chain. Here we choose KDM and ${\mathcal{O}}={\mathcal{O}}'$. (a-c) $J\tau=0.5$, (d-f) $J\tau=2$. (a,d) ${\mathcal{O}}=X$, (b,e) ${\mathcal{O}}=Y$, (c,f) ${\mathcal{O}}=Z$. Different colors correspond to different system size $L$, as shown in the legend. ](dyn2.pdf){width="0.98\linewidth"}
In this paper we use the Trotter-Suzuki scheme for the driving protocol, where the time-dependent Hamiltonian is piecewise constant in one driving period. However, our results are general for any form of periodic driving. The evolution of the system we study is given by the unitary propagator in one period $U_F=e^{-iH_2\tau}e^{-iH_1\tau}$, where in each period we consider the system to be under the Hamiltonian $H_1$ for a time $\tau$, and then under $H_2$ for another duration $\tau$. Motivated by NMR experiments, we consider two models of an $L$-site spin-1/2 chain: the kicked dipolar model (KDM), where $H_1^{(K)}\!=\!JD_y$, $H_2^{(K)}\!=\!hZ$, and the alternating dipolar model (ADM), with $H_1^{(A)}\!=\!JD_y$ and $H_2^{(A)}\!=\!JD_x$. Here $D_\alpha=\sum_{j<k} \frac{1}{2}\left(3S_\alpha^j S_\alpha^k - \vec{S}_j\cdot\vec{S}_k \right)/|j-k|^3$ is the dipolar interaction operator in an arbitrary direction set by $\alpha$ $(\alpha=x,y,z)$, where $S_\alpha^j$ are spin-1/2 operators of the $j$-th spin $(j=1,\cdots,L)$ and $\vec{S}_j=(S_x^j,S_y^j,S_z^j)^T$. As shown in Ref. [@Machado19x], the $1/r^3$ interaction is sufficiently short range in 1D to yield no qualitative difference with respect to the nearest-neighbor interaction, thus for simplicity in numerical and analytical studies we only keep the nearest-neighbor interaction unless explicitly mentioned. $Z=\sum_j S^j_z$ is the collective magnetization operator along z-axis, and below we will also use $X=\sum_j S^j_x, Y=\sum_j S^j_y$. $J$ and $h$ are the strength of the dipolar interaction and the collective z-field respectively, and we fix $h=J$ throughout the paper. In numerics we assume periodic boundary conditions.
To investigate quasi-conservation properties we use infinite-temperature correlations as our metric, $\langle {\mathcal{O}}(t){{\mathcal{O}}}' \rangle_{\beta=0} \equiv \text{Tr}[ U_t{\mathcal{O}}U_t^\dagger{{\mathcal{O}}}' ] /{\left( \| {\mathcal{O}}\|\|{{\mathcal{O}}}'\| \right)}$, where $U_t$ is the unitary evolution during time $t$, ${{\mathcal{O}}}$ and ${{\mathcal{O}}}'$ are observables, and the norm is defined as $\|{\mathcal{O}}\|\equiv \sqrt{\text{Tr}{\mathcal{O}}^2}$. In the following we drop the subscript $\beta=0$ for simplicity.
Figure \[fig:dyn\] shows numerical simulations of some exemplary correlations, the magnetization along three axes ${\mathcal{O}}={{\mathcal{O}}}'=Z,X,Y$ in KDM (the qualitative behavior is general for other observables and models.) The autocorrelations of $X$ and $Y$ display oscillations around $0$ and damping, which originate from the z-field and the dipolar interaction, respectively. Instead, $\langle Z(t)Z \rangle$ exhibits a more interesting behavior. For small $J\tau$, it quickly equilibrates at a nonzero value independent of $L$, and it remains constant afterwards. For relatively large $J\tau$, there is a slow decay of $\langle Z(t)Z \rangle$ toward a final value that decreases with increasing $L$. We thus expect the final value to be zero in the thermodynamic limit, corresponding to an infinite-temperature final state. Indeed, the observable $Z$ displays the defining characteristics of what we deem a quasiconserved observable in the prethermal regime: the autocorrelation of a quasiconserved observable is nonzero in the prethermal regime, but goes to zero in the fully thermalized state. In simulations, autocorrelations of quasiconserved observables still have nonzero value at infinite time due to the small system size (e.g. $\langle Z(t)Z\rangle$ in Fig. \[fig:dyn\]), while for non-conserved observables autocorrelations are zero (e.g. $\langle X(t)X\rangle$ in Fig. \[fig:dyn\]). These distince behaviors serve as a direct metric to identify quasiconserved observables. As any observable that overlaps with a quasiconserved observable would have non-zero infinite-time autocorrelation, we want to find a linearly independent, orthogonal set of *eigen-*quasiconserved observables.
Eigen-quasiconserved Observables
--------------------------------
We design a systematic procedure to search for the set of eigen-quasiconserved observables, $\{\mathcal E_\mu\}$ starting from the infinite-time correlations $\langle {\mathcal{O}}(\infty){\mathcal{O}}' \rangle\equiv\lim_{T\to\infty}(1/T)\int_0^T\langle {\mathcal{O}}(t){\mathcal{O}}' \rangle dt$. We note that eigenvectors $\{E_\mu\}$ of the Floquet (super)propagator $\hat U_{F}$ form an orthogonal vector basis for the space of operators (here $\hat U[\mathcal O]=U{\mathcal{O}}U^\dagger$.), $|\langle E_j(\infty)E_k\rangle| \propto \delta_{jk}$, that we can call “eigen-observables”.
However, this operator basis is in general highly non-local, and thus not practical. We then want to find a small, local set of observables that approximate the exact eigen-observables, and have non-zero eigenvalues, that is, are quasiconserved. We start from a basis set $\{{\mathcal{O}}_{(\alpha)}\}$ of Hermitian observables that are translationally invariant sums of local operators: $${\mathcal{O}}_{(\alpha)} = \sum_j S^j_{\alpha_1}S^{j+1}_{\alpha_2}\cdots S^{j+r-1}_{\alpha_r}.$$ Here $(\alpha)\equiv(\alpha_1, \cdots, \alpha_r)$ with $\alpha_{k}\in\{x,y,z,0\}$, where $S_0^j$ denotes the identity matrix operating on the $j$-th spin. By imposing $\alpha_1,\alpha_r\neq0$, we say ${\mathcal{O}}_{(\alpha)}$ is of range $r$: each term in $\mathcal O_{(\alpha)}$ acts non-trivially on at most $r$ neighboring spins. Since the number of operators is exponentially large in system size, we restrict our search to the operator subspace spanned by ${\mathcal{O}}_{(\alpha)}$ whose range $r\le r_c$, which are local and thus experimentally relevant. Starting from an orthonormal operator basis $\{{{\mathcal{O}}}_\mu\}$ of this subspace (with $\langle{{\mathcal{O}}}_\mu{{\mathcal{O}}}_\nu\rangle=\delta_{\mu\nu}$) we construct a matrix from all pair correlations, $\Lambda_{\mu\nu}=\langle{{\mathcal{O}}}_\mu(\infty){{\mathcal{O}}}_\nu\rangle$. The matrix $\Lambda$ is the projection of the infinite-time propagator $\hat U_F(t\to\infty)$ onto the $r_c$-local subspace. The diagonalization of $\Lambda$ yields the local eigen-observables $\mathcal E_k$, and eigenvalues $\lambda_k$, satisfying $\langle\mathcal{E}_k(\infty)\mathcal{E}_l\rangle=\lambda_k\delta_{kl}$. Note that since $\Lambda$ is not ensured to be unitary, its eigenvalues do not have unit amplitude, $\lambda_k\leq 1$. We note that the larger the $\lambda_k$, the better $\mathcal E_k$ approximates an exactly conserved observable. [The correlations $\langle {\mathcal{O}}(\infty){{\mathcal{O}}'} \rangle$ between any two observables whose locality is bounded by $r_c$ can be directly derived by decomposing the observables onto the $\mathcal E_\mu$ basis $$\label{eq:decomp}
\langle {\mathcal{O}}(\infty){{\mathcal{O}}}' \rangle = \sum_\mu \lambda_\mu \langle{\mathcal{O}}\mathcal E_\mu\rangle \langle\mathcal E_\mu{{\mathcal{O}}'}\rangle.$$ ]{}
![\[brute\] By considering the matrix $\Lambda$ obtained for each $J\tau$ Trotter step, we calculate three largest eigenvalues as a function of $J\tau$ for KDM (a) and ADM (b). Curve color represents different eigenvalues and curve style represents different system sizes. From the eigenvalues and their dependence on system size, we see there are two eigen-quasiconserved observables in KDM while only one in ADM. ](all_KDM3.pdf "fig:"){width="0.48\linewidth"} ![\[brute\] By considering the matrix $\Lambda$ obtained for each $J\tau$ Trotter step, we calculate three largest eigenvalues as a function of $J\tau$ for KDM (a) and ADM (b). Curve color represents different eigenvalues and curve style represents different system sizes. From the eigenvalues and their dependence on system size, we see there are two eigen-quasiconserved observables in KDM while only one in ADM. ](all_ADM2.pdf "fig:"){width="0.48\linewidth"}
We apply this systematic procedure to the two models under consideration. The infinite time limit $\mathcal{O}(\infty)$ is taken by considering the diagonal ensemble of $\mathcal{O}$ (that is, keeping only the diagonal matrix elements of $\mathcal{O}$ in the Floquet energy eigenbasis), which gives the same result as averaging $\mathcal{O}$ over long time. The results for $r_c=3$ are shown in Fig. \[brute\]. At large Trotter steps, $\tau$, most eigenvalues go to zero. The upward trends of the eigenvalues when $J\tau=h\tau\to\pi$ (most pronounced for the largest eigenvalue) is due to the fact that $[e^{-iH_1^{(K)}\tau},e^{-iH_2^{(K)}\tau}]=0$ at $J\tau=h\tau\to\pi$, making the system equivalent to a time-independent system. Even for small Trotter steps, most eigenvalues are already small, and decrease when increasing system size. However, a few eigenvalues are large, and show little dependence on system size. This last group comprises the eigenvalues associated with the eigen-quasiconserved observables that govern the nontrivial dynamics at long times.
Based on these results, we find that there are two eigen-quasiconserved observables for KDM, $\mathcal E^{(K)}_1,\mathcal E^{(K)}_2$, and one for ADM, $\mathcal E^{(A)}_1$. In both models, $\mathcal E_1$ is close to their average Hamiltonian $\overline H=H_1+H_2$ (blue curves in Fig. \[brute\]), while $\mathcal E^{(K)}_2$ for KDM is close to $D_z$ \[red curves in Fig. \[brute\](a)\]. We can thus more carefully analyze these quasiconserved observables and describe analytically their origin in the limit of small $\tau$ in the next section. Even so, we remark that there is an interesting regime at intermediate $\tau$ , where $\mathcal E^{(K)}_1,\mathcal E^{(K)}_2$ are well conserved, since $\lambda^{(K)}_1,\lambda^{(K)}_2$ are still large, but they deviate from their static ($\tau\to0$) counterparts. This indicates that the quasiconserved observables truly arise from the Floquet dynamics, and are not simply a remnant of the approximated, static Hamiltonian.
Analytical Derivation of Conserved Observables
==============================================
Prethermal Hamiltonian {#sec:Hpre}
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It is intuitive to expect that a quasiconserved observable might emerge from energy conservation. Indeed, one can always regard the Floquet evolution as arising from an effective static Hamiltonian by setting $U_F=e^{-i\tau H_F}$ for some Hermitian operator $H_F$. However, in general this Hamiltonian is highly non-local and thus it is not associated to a local quasi-conserved observable. Still, when the driving frequency is large compared to local energy scales (here $J,h$), the stroboscopic dynamics is given by a time-independent local prethermal Hamiltonian $H_{pre}$ plus a small correction $\delta H(t)$ [@Kuwahara16; @Abanin17], which may be nonlocal. It is this prethermal Hamiltonian $H_{pre}$ that can be associated with a local quasiconserved observable. $H_{pre}$ can be obtained from the Floquet-Magnus expansion [@Magnus54; @Blanes09] truncated at an optimal order $m^*$: $$\label{eq:Hpre}
H_{pre}=\sum_{m=0}^{m^*}\tau^m \Omega_m,$$ where the zeroth order term is the average Hamiltonian $\Omega_0\!=\!\overline H\!=\!1/\tau\int_{0}^\tau\! H(t)dt$ and higher order terms $\Omega_m$ involve $m$ nested commutators. Then, for spin chains with [nearest-neighbor]{} couplings the range of $\Omega_m$ grows linearly with $m$.
The truncation $m^*$ is crucial not only to keep the prethermal Hamiltonian local, but also because the series in Eq. \[eq:Hpre\] diverges for a generic many-body system [@Kuwahara16]. The time-dependent correction $\delta H$ is however exponentially small in $1/J\tau$, leading to an exponentially long time $t_{pre}$ for the system to heat up. Thus, for $t<t_{pre}$, the system effectively prethermalizes to the state $e^{-\beta H_{pre}}$ where $\beta$ is determined by the initial state energy, making $H_{pre}$ an eigen-quasiconserved observable. Although one should investigate the prethermalization process by studying the dynamics of an infinitely large system at long times approaching infinity, numerically we can only tackle small system sizes, so we take a different approach – we set the time to infinity, and study how the observable correlations change when increasing system size. The validity of this approach relies on the fact that for a system size $L<m^*$ the term $\delta H$ does not appear in the expansion, making ${\mathcal{O}}_1= H_{pre}$ exactly conserved even at infinite time for sufficiently small $\tau$. From a physics point of view, this means that the energy $2\pi\hbar/\tau$ is larger than the many-body bandwidth ($\sim JL$), and thus the system cannot absorb energy from the drive if it is faster than $1/JL$. Since the zeroth order term of $H_{pre}$ is $\overline H$, the autocorrelation of $H_{pre}$ provides a bound for that of $\overline H$, leading to bounded Trotter error in the Trotter-Suzuki scheme [@Heyl19].
As further verification, we calculate numerically the Floquet Magnus expansion, Eq. (\[eq:Hpre\]), up to $m=10$ and evaluate not only the convergence of the expansion, but also operator conservation. For the first metric, we plot $\|\Omega_m\|$ in Fig. \[Hid\_theory\](a) and (d) for the two models studied. We find that, up to the computationally accessible order, the norm of $\Omega_m$ decays exponentially, indicating that $H_{pre}$ converges when $\tau$ is small. From the slopes in Fig. \[Hid\_theory\](a) and (d), we get radii of convergence $J\tau\approx 3$ for both models. Still, the expansion convergence does not guarantee the resulting $H_{pre}$ is a quasiconserved observable. In Fig. \[Hid\_theory\](b) and (e), we compute the long-time infidelity ($1-\langle H_{pre}(\infty) H_{pre} \rangle$) by truncating the expansion in Eq. (\[eq:Hpre\]) at increasing orders. When $J\tau$ is small, the autocorrelation exponentially approaches 1 with increasing order, suggesting that the optimal truncation order $m^*$ should be larger than our largest accessible order here, or even absent in the system size we study. Instead, for larger $J\tau$, the correlation stops converging at some order; for even larger $J\tau$ ($J\tau=1$ for example) the correlation is almost zero for all orders. Therefore, even within the radius of convergence $J\tau\approx3$, $H_{pre}$ from Eq. \[eq:Hpre\] may fail to be quasiconserved. We plot the infinite-time correlation $\langle H_{pre}(\infty) H_{pre} \rangle$ versus $J\tau$ in Fig. \[Hid\_theory\](c) and (f) and show how it changes with system size (here $H_{pre}$ is evaluated to $7^\mathrm{th}$ order). The drop of $\langle H_{pre}(\infty) H_{pre} \rangle$ with increasing system size is evident for $J\tau\gtrsim1.2$ in both models, suggesting that for the system size we explore the effective Hamiltonian picture fails in the above parameter space. Note that in the $L\to\infty$ limit the correlations are expected to be zero for any $\tau>0$ as will be discussed in Sec. \[sec:limit\].
Here we present experimental study of the prethermal Hamiltonian of ADM (that of KDM can be found in Ref. [@Peng2019]). The experimental system is a single crystal of fluorapatite (FAp) [@VanderLugt64]. We study the dynamics of $^{19}$F spin-$1/2$ using NMR techniques. Although the sample is 3D, $^{19}$F form quasi-1D structure because the interaction within the chain is $\sim$40 times larger than the interaction between different chains [@Cappellaro07l; @Zhang09; @Ramanathan11]. Average chain length is estimated to be larger than 50 and the coherence time of the $^{19}F$ spins is $T_1\approx0.8s$. The sample is placed in 7 T magnetic field thus $^{19}F$ spins interact with each other via the secular dipolar interaction $H=J_0D_z$ with $J_0=-29.7$ krad/s (we define $z$ as the magnetic field direction). While the corresponding 1D, nearest-neighbor XXZ Hamiltonian is integrable [@Alcaraz87; @Sklyanin88; @Wang16a], the Hamiltonian we consider can lead to diffusive [@Sodickson95; @Zhang98l] and chaotic behavior [@Jyoti17x] in 3D. In the presence of a transverse field, the system is known to show a quantum phase transition [@Isidori11]. We use 16 RF pulse to engineer the natural Hamiltonian into $H_1^{(A)}=JD_y$ and $H_2^{(A)}=JD_x$ with tunable $J$. Similar Hamiltonian engineering technique has been used in Ref. [@Wei18; @Wei19; @Peng2019]. In order to faithfully quantify the Floquet heating rate, experimental imperfections have to be kept constant. We fix $\tau$ and change $J$ when varying the unitless Floquet period $J\tau$ so that decoherence and pulse errors remain the same. The initial state is a high-temperature thermal state with small thermal polarization in the magnetic field direction $\rho(0)\approx(\mathbb{1}-\epsilon Z)/2^L$ with $\epsilon\approx 10^{-5}$, and the observable is collective magnetization long x-axis ${\mathcal{O}}=X$. As the identity part does not change under unitary evolution and does not contribute to signal, it is convenient to consider only the deviation from identity $\delta\rho(0)=Z$. Therefore, the NMR signal is $\mathrm{Tr}[D_z(t)D_z]\equiv\langle\delta\rho(t){\mathcal{O}}\rangle$. To study the autocorrelation of $H_{pre}=\overline{H}+O(\tau)$ in ADM, we use the Jeener-Broekaert pulse pair to evolve the $\delta\rho$ and ${\mathcal{O}}$ into $D_z\propto\overline{H}^{(A)}$ [@Jeener67].
Figure \[HidA\](a) shows the autocorrelation of $D_z$, which decays when more Floquet periods are applied. The decay rate is shown in Fig. \[HidA\](b) and can be fitted into an exponential function in $1/(J\tau)$ on top of a constant background. The background decay is a result of experimental imperfections (see SM for more discussion). In Fig. \[HidA\](c) we normalize correlations at each $n$ by that acquired under the fastest driving to cancel the background decay. The overall drop of the curve when increasing $n$ is an indicator of Floquet heating. For given $n$, the normalized correlation decreases when increasing $J\tau$, because $H_{pre}=\overline{H}+O(J\tau)$ thus $\overline{H}$ that we measure has less overlap with the true quasiconserved observable $H_{pre}$ for larger $J\tau$.
Emergent dipolar order {#sec:Dpre}
----------------------
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To search for additional conserved observables in KDM we develop a method inspired by the existence of discrete time-translation symmetry-protected phases in prethermal Floquet systems [@Else17]. Similar results have been obtained for the static Hamiltonian $\overline H=hZ+JD_y$ associated with the (zero-order) KDM. For this model, it has been shown that the polarization $Z$ is quasiconserved, and does not reach its thermal equilibrium value until a time exponentially long in $h/J$ [@Else17; @Abanin17b; @Wei19], even if according to ETH the system should thermalize.
Since the average Hamiltonian picture breaks down when increasing $\tau$ and we see from Fig. \[brute\](a) that the other observable is conserved for even larger $\tau$, we must go beyond the static case, and work directly in the Floquet system. This kind of system was first studied in [@Else17], where they further focused on the case $h=\pi$ to identify a prethermal Floquet time crystal. Here we generalize their analysis to obtain the novel quasiconserved observable for any $h$.
We transform the Floquet operator by going to a rotated frame as $$\label{eq:pre0}
e^{S} e^{-i hZ\tau}e^{-i H_1\tau} e^{-S} = e^{-i h Z\tau} e^{-i\tau (JD+\delta H)},$$ and demand $[Z,D]=0$. By appropriately choosing $S,D$, it will be shown that $\delta H$ is exponentially small in $ \min[O(\frac{h}{J}), O(\frac{1}{h\tau})]$ [@Peng2019footnote]. Therefore, for small $\tau$ and large enough ratio $h/J\gtrsim0.5$ [@Wei19], the operator $D$ approximately commutes with the Floquet unitary in the rotated frame, making $D_{pre}=e^{-S}De^S$ a prethermal quasiconserved observable in the original frame. We emphasize that the right-hand side of Eq. \[eq:pre0\] still describes a Floquet system, therefore we derived the quasiconservation without first transforming to a static Hamiltonian. Note that $Z_{pre}=e^{-S}Ze^S$ is quasiconserved in the same sense as $D_{pre}$. However, whereas $D_{pre}$, is orthogonal to $H_{pre}$ to zeroth order, $Z_{pre}\approx H_{pre}-D_{pre}$ and it cannot thus be considered an eigen-quasiconserved observable.
Now we give the details of finding the desired $S,D$. We first write the transformation Eq. \[eq:pre0\] in an equivalent form $$\label{eq:pre}
e^{i \epsilon hZ\tau} e^S e^{-i \epsilon hZ\tau} e^{-i \epsilon^2 H_1\tau} e^{-S} = e^{-i\tau (D+\delta H)},$$ Here we assume that $J/h$ and $h\tau$ are small parameters whose magnitude are of the same order marked by $\epsilon$, and do perturbation in $\epsilon \ll 1$. After expanding the operators, $D=\epsilon D_1+\epsilon^2 D_2+\cdots, S = \epsilon S_1+\epsilon^2 S_2+\cdots$, one can collect terms that are of order $\epsilon^{j}$ on both sides of Eq. \[eq:pre\], and get a series of equations indexed by $j$. In practice we do not calculate exponentials directly but use the Magnus expansion of the left-hand side. The $j$-th order is given by $$\label{eq:pre_j}
-i\tau D_j = \left[S_{j-1},-ihZ\tau\right] + h_j,$$ where $h_j$ only contains $-ihZ\tau$, $-iH_1\tau$ and $S_{j'}$ with $j'<j-1$. The first few orders can be written explicitly, $$\begin{aligned}
&h_1=0,\nonumber \\
&h_2=-iH_1\tau,\\
&h_3=[S_1,h_2]+\frac{ih\tau}2([S_1,[S_1,-Z]]+[Z,[ihZ\tau,S_1]]),\nonumber\end{aligned}$$ while higher orders can be found recursively. Assuming all orders $S_{j'}$ with $j'<j-1$ are known (which is trivially true for $j=2$), we determine $S_{j-1}$ from Eq. \[eq:pre\_j\] by requiring $\left[S_{j-1},-ihZ\tau\right]$ to cancel the terms in $h_j$ that do not commute with $Z$. Similar to the prethermal Hamiltonian Eq. \[eq:Hpre\], the expansion in $\epsilon$ generally diverges and should be truncated at some order, leading to the exponentially small nonlocal residual $\delta H$, see, e.g. Ref. [@Abanin17; @Else17].
Here we explain in detail how to obtain $S_{j-1}$ from Eq. \[eq:pre\_j\] by taking advantage of the special structure of the field operator $Z$. We first decompose $h_j = \sum_{q=0,\pm1,\cdots} h_{jq}$ such that $[Z, h_{jq}] = q h_{jq}$ ($h_{jq}$ are called the $q$-th quantum coherence of $Z$ [@Wei18; @Munowitz75; @Garttner18]). This decomposition is only possible when the dominant part of the Hamiltonian has equally spaced eigenvalues, such as for the collective rotation $H_2^{(K)}=JZ$ in our case. Equation \[eq:pre\_j\] is then satisfied by setting $-i \tau D_j = h_{j0}$ and $S_{j-1} = i\sum_{q\neq 0} h_{jq}/(hq\tau)$. We note that $S$ is a sufficiently local operator, $r(S_j)=j$, for KDM with nearest-neighbor interaction.
When $\tau$ is small, the $S_j$ operators are dominated by the $(J/h)^{j}$ term. Therefore, in the $\tau\to0$ limit, the quasiconserved observable found here for the Floquet model reduces to the prethermal quasiconserved observable of the static Hamiltonian $\overline H^{(K)}$ [@Wei19; @Else17], where the expansion is a series of $J/h$ and $\delta \tilde{H}\approx \exp(-O(h/J))$. In this regime, $D_{pre}=- \frac{1}{2}D_z +O((J/h)^2)$, and the expansion converges for $h/J \gtrsim 0.5$ (up to truncation at exponentially large order) as shown in Ref. [@Wei19] (Note that here we used $h/J=1$). Instead, for relatively larger $h\tau$, the $S_j$ operators are dominated by $(h\tau)^j$ and $\delta \tilde{H}\approx \exp(-O(1/h\tau))$, in agreement with the exponentially slow Floquet heating.
We numerically evaluate the convergence properties of $D_{pre}$ in the KDM \[Fig. \[Dpre\_theory\](a)\], using the metrics discussed in the previous section, convergence of the order-by-order expansion terms and infinite-time autocorrelation. We find that the series converges up to order 7 in the $h\tau$ regime we are interested in. The infinite-time autocorrelation is close to $1$ at small $\tau$, as shown in Fig. \[Dpre\_theory\](b) and (c), confirming that the local truncation of $D_{pre}$ (as obtained by the first few orders) gives rise to quasiconserved observable $\mathcal E_2^{(K)}$. Comparing these results to the prethermal Hamiltonian shown in Fig. \[Hid\_theory\](b) and (c), we find that (i) the normalized autocorrelation of $D_{pre}$ converges to 1 in a larger parameter range ($J\tau\lesssim 1.6$ for $D_{pre}$ and $J\tau\lesssim 1$ for $H_{pre}$), (ii) the autocorrelation shows a significant drop at $J\tau\gtrsim1.8$ for $D_{pre}$ and $J\tau\gtrsim1.2$ for $H_{pre}$, with a steeper drop when $L$ is increased from 8 to 12. Both facts suggest that $D_{pre}$ is more robust than $H_{pre}$, in agreement with the experimental results presented in Ref. [@Peng2019]. This provides evidence that it is possible to realize novel Floquet phases beyond the effective Hamiltonian picture.
Toward infinite temperature: experimental and numerical signatures {#sec:limit}
==================================================================
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Although it is generally believed that Floquet many-body systems should heat up to infinite temperature, some numerical works [@Heyl19; @Sieberer19; @Prosen99; @DAlessio13] have found signs of non-thermal behavior in various models. Here we provide evidence of thermalization in the long-time and thermodynamic limit, using numerics and experiments in a NMR quantum simulator [@Peng2019; @Wei18; @Wei19], respectively. In simulations, we can access the infinite-time limit using exact diagonalization, but only for small system sizes. Conversely, the system size in NMR experiments is large enough to achieve the thermodynamic limit, but the evolution time cannot be too long due to hardware limitation. Still, by looking at the dynamics for increasingly longer times (experimentally) and larger system sizes (numerically), we can extract insight on the final fate of the Floquet systems.
The experimental system is a single crystal of fluorapatite (FAp) [@VanderLugt64]. We study the dynamics of $^{19}$F spin-$1/2$ using NMR techniques. Although the sample is 3D, $^{19}$F form quasi-1D structure because the interaction within the chain is $\sim$40 times larger than the interaction between different chains [@Cappellaro07l; @Zhang09; @Ramanathan11]. Average chain length is estimated to be $>50$ and the coherence time of the $^{19}F$ spins is $T_1\approx0.8s$. The sample is placed in 7 T magnetic field where the Zeeman interaction dominates, thus reducing the $^{19}F$ spins interaction to the secular dipolar Hamiltonian $H=J_0D_z$ with $J_0=-29.7$ krad/s (we define $z$ as the magnetic field direction). While the corresponding 1D, nearest-neighbor XXZ Hamiltonian is integrable [@Alcaraz87; @Sklyanin88; @Wang16a], the experimental $1/r^3$ Hamiltonian can lead to diffusive [@Sodickson95; @Zhang98l] and chaotic behavior [@Jyoti17x] in 3D. In the presence of a transverse field, the system is known to show a quantum phase transition [@Isidori11]. We use 16 RF pulses [@Wei18; @Wei19; @Peng2019; @Sanchez20] to engineer the natural Hamiltonian into $H_1^{(A)}=JD_y$ and $H_2^{(A)}=JD_x$ with tunable $J$. This enables varying the Floquet steps by tuning $J$, while keeping $\tau$ fixed. Then, experimental imperfections such as decoherence and pulse errors remain the same, and we can faithfully quantify the Floquet heating rate. The initial state is a high-temperature thermal state with small thermal polarization in the magnetic field direction $\rho(0)\approx(\mathbb{1}-\epsilon Z)/2^L$ with $\epsilon\approx 10^{-5}$, and the observable is the collective magnetization along x-axis ${\mathcal{O}}=X$. As the identity part does not change under unitary evolution and does not contribute to signal, it is convenient to consider only the deviation from the identity $\delta\rho(0)=Z$, which can be rotated to a desired observable ${\mathcal{O}}'$. Therefore, the NMR signal is equivalent to an infinite-temperature correlation $\mathrm{Tr}[\delta\rho(t)X]\to \langle{\mathcal{O}}'(t){\mathcal{O}}\rangle_{\beta=0}$.
We experimentally study the heating rates of the quasiconserved observables and their scaling with Floquet period, to reveal the prethermal phase and investigate the eventual heating to infinite temperature. In Fig. (\[fig:ADMexp\]) we show results for ADM (the two quasiconserved observable in KDM show similar behavior as reported elsewhere [@Peng2019].) To study the autocorrelation of $H_{pre}=\overline{H}+O(\tau)$ in ADM, we measure the average Hamiltonian $\overline H^{(A)}\!=\!JD_y\!+\!JD_x\!=\!-JD_z$, since the higher order terms in Eq. \[eq:Hpre\] are not accessible. We use the Jeener-Broekaert pulse pair [@Jeener67] to evolve the initial state $\delta\rho$ and experimental observable $X$ into $D_z\propto\overline{H}^{(A)}$. Because of the difference $H_{pre}-\overline H$, we still expect an initial transient, over a time $\sim \|H_{pre}\|^{-1}$, where the average Hamiltonian thermalizes to the prethermal Hamiltonian. When more Floquet periods are applied, the autocorrelation of $D_z$ slowly decays from its prethermal value.
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The decay rate in the prethermalization regime is shown in Fig. \[fig:ADMexp\](b), and can be fitted to an exponential function in $1/(J\tau)$ on top of a constant background decay (which is due to experimental imperfections, see SM [@SOM] for more details.) By normalizing the data to the data collected under the fastest drive ($J\tau=0.35$), the background decay is cancelled, and the resulting dynamics only arises from the coherent evolution, as shown in Fig. \[fig:ADMexp\](c). For given $n$, the normalized correlation decreases when increasing $J\tau$, because $H_{pre}=\overline{H}+O(J\tau)$ thus $\overline{H}$ that we measure has less overlap with the true quasiconserved observable $H_{pre}$ for larger $J\tau$. The overall drop of the curves when increasing $n$ is instead an indicator of Floquet heating.
To better quantify the final thermalization process, we define a critical value $J_c$ such that when $J\tau>J_c\tau$ the system is thermalized, at a given number $n$ of periods in the thermodynamic limit, or for a system size $L$ at infinite time. Studying the scaling of $J_c$ as a function of $n$ (experimentally) and $L$ (numerically) provides hints on the long-time, thermodynamic limits.
We numerically obtain the autocorrelations $\langle \mathcal O(\infty) \mathcal O\rangle$ as a function of $J\tau$, using exact diagonalization. In Fig. \[fig:tscaling\](a) we show simulation results for $\mathcal O=\overline H^{(K)}, D_z$ for KDM and $\mathcal O=\overline H^{(A)}$ for ADM. (Here we explicitly consider the exact dipolar interaction instead of truncating to nearest neighbors.) Note that both observables in KDM show a non-monotonic behavior. They appear to be quasiconserved until $J\tau=1$; the decrease in overlap is however interrupted by a revival at $J\tau=1.6$. This is because $\overline{H}^{(K)}$ and $D_z$ are approximation of $H_{pre}$ and $D_{pre}$ to leading order. Thus $\overline{H}^{(K)}$ ($D_z$) still has a small overlap with $D_{pre}$ ($H_{pre}$), giving rise to a second plateau at $J\tau\approx 1.6$ ($J\tau\approx 1$). The experimentally measured autocorrelations of quasiconserved observables in KDM can be find in [@Peng2019]. For both experiments and simulations we then find $J_c\tau$ from the point where the curves drop below a threshold value of 0.5 (any other reasonable choice would not qualitatively change the results). We linearly interpolate between data points to get $J_c\tau$ for every quasiconserved observable and plot the $J_c\tau$ in Fig. \[fig:tscaling\](b) and (c). The decrease of numerically calculated $J_c\tau$ with $L$ in Fig. \[fig:tscaling\](b) indicates that even the correlations of quasiconserved observables decay to zero as the system thermalizes to infinite temperature, suggesting this non-thermalizing behavior should not persist to the thermodynamic limit. Similar result is also observed from experimentally measured $J_c\tau$ as shown in Fig. \[fig:tscaling\](c) [^2]. Note that although $J_c\tau$ for $\langle \overline H^{(K)}(n)\overline H^{(K)}\rangle$ shows only a moderate dependence on $n$ \[Fig. \[fig:tscaling\](c)\], its decay is still larger than experimental uncertainties.
Conclusion {#sec:conclusion}
==========
As Floquet driving is a promising avenue for quantum simulation, it is crucial to evaluate its robustness, the existence of a long-lived prethermal phase, and the eventual thermalization to infinite temperature. Investigating Floquet heating, which breaks the prethermal regime, is particularly challenging, not only because of inherent limitations in numerical and experimental studies, but also because of the challenge to properly identifying all quasiconserved observables in the complex, many-body driven dynamics.
Here we tackle both these issues by combining analytical, numerical and experimental tools. First, we provide a systematic strategy to find local, eigen-quasiconserved observables in the prethermal regime using infinite-temperature correlations. By systematically searching over local operators, we find that counter-intuitive quasiconserved observables might emerge, as we identify two eigen-quasiconserved observables: the first, not surprisingly is associate with energy, $H_{pre}$, under sufficient fast drive; in addition, we find another quasiconserved observable, $D_{pre}$, for the KDM in the presence of a large driving field.
We then use numerical and experimental evidence to obtain insight into the inaccessible thermodynamic limit and long-time regime, to show that autocorrelations of quasiconserved observables indeed decrease toward zero due to Floquet heating, suggesting the Floquet system approaches the infinite temperature state.
Our results not only provide a metric to study thermalization in driven quantum systems, but also open intriguing perspectives into the existence of quasiconserved observables other than the energy. It is an open question when they emerge and how they interact with each other. A better understanding of quasiconserved observables would benefit understanding of heating in closed driven systems, and designing robust protocol to slow down thermalization.
Authors would like to thank H. Zhou, W.-J Zhang and Z. Li for discussion. This work was supported in part by the National Science Foundation under Grants No. PHY1734011, No. PHY1915218, and No. OIA-1921199.
Supplemental Material {#supplemental-material .unnumbered}
=====================
Experimental background decay rate as a function of $J\tau$ {#app:background}
===========================================================
In the main text we measured the Floquet heating for a periodic, Hamiltonian switching scheme. While it would be easy to change the period by increasing the time between switches, this would lead to experiments performed with different total times or a different number of control operations. In turns, this can introduce variable amount of decoherence and relaxation effects, and of control errors. Instead, we kept the time for one Floquet period constant and used Hamiltonian engineering to vary the Hamiltonian strength in order to vary the Floquet driving frequency.
One of the assumptions in our work is that the background decay rate does not change much with driving frequency (compared to the change in Floquet heating rate). In this section, we provide experimental evidence for this assertion. When changing driving frequency, we are changing (i) the effective strength $J$ of the engineered dipolar interaction $JD_y$ and (ii) the kicking angle in the kicked dipolar model by a phase shift (see \[app:Ham\]). As phase shift angles are usually very accurately implemented in NMR experiments, we focus on the engineered dipolar interaction, which is obtained by Floquet engineering itself, as explained in \[app:Ham\]. To quantify how good is the engineered $JD_y$, we measure $\langle Y(n)Y\rangle$ and $\langle D_y(n)D_y\rangle$ under the engineered Hamiltonian $JD_y$, without kicking field nor direction alternation, as shown in Fig. \[fig:bgDR\].
![\[fig:bgDR\] Decay rate of $\langle Y(n)Y\rangle$ (blue) and $\langle D_y(n)D_y\rangle$ (green) under engineered dipolar Hamiltonian $JD_y$ as a function of $J\tau$. [The range of $J\tau$ studied was obtained by varying the scaling $u$ (see SM [@SOM]) from 0.098 to 0.646, while keeping fixed $\tau=120\mu s$. In the inset, we compare the background decay rates with the Floquet decay rates (dashed lines).]{} ](bgDR4.pdf){width="0.98\columnwidth"}
Note that the maximum difference between the decay rate of $\langle D_y(n)D_y\rangle$ over the range of $J\tau$ considered is $\sim 0.003$, much smaller than the Floquet heating rate in the main text. A quantitative analysis is challenging because the specific form of error terms is unknown, and $JD_y$ is an interacting Hamiltonian thus error accumulation is intractable. Here we use some simple arguments to argue that variations in the background decay with $J\tau$ have little to no influence on our results. First, we note that while in the main text we are interested in the decay of the autocorrelation of $H_{pre}$ and $D_{pre}$, here with $H=JDy$ we can only discuss the decay of $D_y$ and $Y$, since other observables that are not conserved display very fast decay which is not informative. For example, in the main text we measure $D_z$, which thermalizes even under the ideal $D_y$ and thus we cannot distinguish thermalization from decay due to experimental imperfections in the engineered dipolar Hamiltonian $D_y$. Still, as $D_z$ and $D_y$ overlap, if the background decay of $D_z$ had a significant change with $J\tau$, it would be reflected in $D_y$, which is not observed. Therefore, we expect the change in the background decay rate for $\langle D_z(n) D_z\rangle$ to be small as well. Here we can only probe the background decay rate of $Y$, while in the main text we are interested in the longitudinal magnetization, $Z$, that appears in $\langle \overline H^{(K)}(n)\overline H^{(K)}\rangle$ \[see Fig. \[fig:tscaling\](c)\]. The transverse magnetization decay rate is, however, a upper bound for $Z$, since in NMR experiments $Z$ is usually more robust against errors than $Y$ due to the large magnetic field in z-axis that suppresses decoherence and experimental errors that do not conserve the total Zeeman energy (we note that we typically do not explicitly write the Zeeman energy in the Hamiltonians as we work in the rotating frame). Even if the variation in the background decay for $Z$ were as large as what observed for $Y$ in these experiments ($\sim0.009$), it would still be still small compared with Floquet (see inset of Fig. \[fig:bgDR\]). In addition, in the kicked dipolar model, we can consider $ D_y$ as being subjected to rotations along $Z$ that further cancel out the error terms in the engineered $JD_y$ that do not conserve $Z$. As a result, the decay rate of $Y$ due to the engineered $D_y$ is larger, by about a factor of 2, than the baseline decay of $\langle \overline H^{(K)}(n)\overline H^{(K)}\rangle$ in the kicked dipolar model (they are 0.254 and 0.123, respectively, in the fastest driving case $J\tau=0.35$).
Experimental System, Control and Data Analysis
==============================================
Experimental System {#app:exp}
-------------------
The system used in the experiment was a single crystal of fluorapatite (FAp). Fluorapatite is a hexagonal mineral with space group $P6_3/m$, with the $^{19}$F spin-1/2 nuclei forming linear chains along the $c$-axis. Each fluorine spin in the chain is surrounded by three $^{31}$P spin-1/2 nuclei. We used a natural crystal, from which we cut a sample of approximate dimensions 3 mm$\times$3 mm$\times$2 mm. The sample is placed at room temperature inside an NMR superconducting magnet producing a uniform $B=7$ T field. The total Hamiltonian of the system is given by $$H_\mathrm{tot}=\omega_F \sum_k S_z^k+\omega_P \sum_\kappa s_z^\kappa+H_{F}+H_P+H_{FP}
\label{eq:Hamtot}$$ The first two terms represent the Zeeman interactions of the F($S$) and P($s$) spins, respectively, with frequencies $\omega_F=\gamma_FB\approx (2\pi)282.37$ MHz and $\omega_P=\gamma_PB=(2\pi)121.51$ MHz, where $\gamma_{F/P}$ are the gyromagnetic ratios. The other three terms represent the natural magnetic dipole-dipole interaction among the spins, given generally by $$H_\mathrm{dip}=\sum_{j<k}\frac{\hbar\gamma_j\gamma_k}{|\vec r_{jk}|^3}\left[\vec S_j\cdot\vec S_k-\frac{3\vec S_j\cdot\vec r_{jk}\,\vec S_k\cdot\vec r_{jk}}{|\vec r_{jk}|^2}\right],$$ where $\vec r_{ij}$ is the vector between the $ij$ spin pair. Because of the much larger Zeeman interaction, we can truncate the dipolar Hamiltonian to its energy-conserving part (secular Hamiltonian). We then obtain the homonuclear Hamiltonians $$\begin{aligned}
H_F&=\frac{1}{2}\sum_{j<k}J^F_{jk}(2 S_z^j S_z^{k}- S_x^j S_x^{k}- S_y^j S_y^{k}) \\ H_P&=\frac{1}{2}\sum_{\lambda<\kappa}J^P_{\kappa\lambda}(2s_z^\lambda s_z^{\kappa}-s_x^\lambda s_x^{\kappa}-s_y^\lambda s_y^{\kappa})
\end{aligned}$$ and the heteronuclear interaction between the $F$ and $P$ spins, $$H_{FP}=\sum_{k,\kappa} J^{FP}_{k,\kappa}S_z^ks_z^\kappa,$$ with $J_{jk}=\hbar\gamma_j\gamma_k\frac{1-3\cos(\theta_{jk})^2}{|\vec r_{jk}|^3}$, where $\theta_{jk}$ is the angle between the vector $\vec r_{jk}$ and the magnetic field $z$-axis. The maximum values of the couplings (for the closest spins) are given respectively by $J^F=-32.76$ krad s$^{-1}$, $J^P=1.20$ krad s$^{-1}$ and $J^{FP}=6.12$ krad s$^{-1}$.
![**A** Fluorapatite crystal structure, showing the Fluorine and Phosphorus spins in the unit cell. **B** NMR scheme for the generation and detection of MQC. In the inset (**C**) an exemplary pulse sequence for the generation of the $H_\mathrm{dipy}$. Note that thanks to the ability of inverting the sign of the Hamiltonian, the scheme amounts to measuring out-of-time order correlations. []{data-label="fig:mqcd"}](FApDQseq2.pdf){width="0.98\linewidth"}
The dynamics of this complex many-body system can be mapped to a much simpler, quasi-1D system. First, we note that when the crystal is oriented with its $c$-axis parallel to the external magnetic field the coupling of fluorine spins to the closest off-chain fluorine spin is $\approx40$ times weaker, while in-chain, next-nearest neighbor couplings are $8$ times weaker. Previous studies on these crystals have indeed observed dynamics consistent with spin chain models, and the system has been proposed as solid-state realizations of quantum wires [@Cappellaro07l; @Cappellaro11; @Ramanathan11]. This approximation of the experimental system to a 1D, short-range system, although not perfect has been shown to reliably describe experiments for relevant time-scales [@RufeilFiori09b; @Zhang09]. The approximation breaks down at longer times, with a convergence of various effects: long-range in-chain and cross-chain couplings, as well as pulse errors in the sequences used for Hamiltonian engineering. In addition, the system also undergoes spin relaxation, although on a much longer time-scale ($T_1=0.8~$s for our sample).
Error analysis {#app:err}
--------------
In experiments, we want to measure the correlation $\langle\delta\rho(t)\mathcal{O}\rangle$, where $\delta\rho(t)= U(t)\delta\rho(0)U(t)$ is the nontrivial part of the density matrix evolved under a pulse-control sequence for a time $t$. Instead of just performing a single measurement after the sequence, we continuously monitor the free evolution of $\delta\rho(t)$ under the natural Hamiltonian $H_\mathrm{dip}$, from $t$ to $t+t_{\textrm{FID}}$. The measured signal is called in NMR free induction decay (FID) and a typical FID trace is shown in Fig. \[fig:FID\]). This signal trace allows us to extract not only the amplitude of the correlation (from the first data point) but also its uncertainty. We take the standard deviation of the last 20 data points in the FID as the uncertainty of the $\langle\delta\rho(t)\mathcal{O}\rangle$. This uncertainty is used with linear error propagation to obtain the error bars of all the quantities analyzed in the main text.
![\[fig:FID\] An example of FID. 128 data points are taken in total. The first data point gives $\langle(\delta\rho(t)\mathcal{O}\rangle$ and the standard deviation of the last 20 points gives the uncertainty of $\langle(\delta\rho(t)\mathcal{O}\rangle$. ](FIDsignal.pdf){width="80mm"}
Hamiltonian Engineering {#app:Ham}
-----------------------
In the main text we focused on the Floquet heating (Trotter error) for a periodic alternating scheme, switching between two Hamiltonians. In order to avoid longer times and/or different numbers of control operations when changing the Trotter step (Floquet period), we engineered Hamiltonians of variable strengths. Then, the Hamiltonians themselves are obtained stroboscopically by applying periodic rf pulse trains to the natural dipolar Hamiltonian that describes the system, and are thus themselves Floquet Hamiltonians. Since we only varied the sequences, but not the Floquet period, this step does not contribute to the behavior described in the main text, as we further investigate in \[app:background\].
We used Average Hamiltonian Theory (AHT [@Haeberlen68]) as the basis for our Hamiltonian engineering method, to design the control sequences and determine the approximation errors. The dynamics is induced by the total Hamiltonian $H=H_\text{dip}+H_\text{rf}$, where $H_\text{dip}=\frac{1}{2}\sum_{j<k}J_{jk}(2 S_z^j S_z^{k}-S_x^j S_x^{k}-S_y^j S_y^{k})+\sum_j h_j S_z^j$ is the system Hamiltonian, and $H_\text{rf}(t)$ is the external Hamiltonian due to the rf-pulses. The density matrix $\rho$ evolves under the total Hamiltonian according to $\dot\rho=-i[H,\rho]$. We study the dynamics into a convenient interaction frame, defined by $\rho'={U_\text{rf}}^{\dagger}\rho U_\text{rf}$, where $U_\text{rf}(t)=\mathcal{T}\exp[-i\int_0^t H_\text{rf}(t') dt']$ and $\mathcal{T}$ is the time ordering operator. In this *toggling* frame, $\rho'$ evolves according to $\dot{\rho}'=-i[H(t),\rho']$, where $H(t)={U_\text{rf}}^{\dagger}H_\text{dip} U_\text{rf}$. Since $U_\text{rf}$ is periodic, $H(t)$ is also periodic with the same period $\tau$, and gives rise to the Floquet Hamiltonian, $H_F$, as as $U(\tau)=\exp[-i H_F \tau]$. Note that if the pulse sequence satisfies the condition $U_\text{rf}(\tau)=1$, the dynamics of $\rho$ and $\rho'$ are identical when the system is viewed stroboscopically, i.e., at integer multiples of $\tau$, where the toggling frame coincides with the (rotating) lab frame.
We devised control sequences to engineer a scale-down, rotated version of the dipolar Hamiltonian [@Wei18; @Wei19]. We usually look for control sequences that would engineer the desired Hamiltonian up to second order in the Magnus-Floquet expansion. Then, to engineer the interaction $D_y$, we use a 16-pulse sequence. The basic building block is given by a 4-pulse sequence [@Kaur12; @Yen83] originally developed to study MQC. We denote a generic 4-pulse sequence as $P(\tau_1,{\bf n}_1,\tau_2,{\bf n}_2,\tau_3,{\bf n}_3,\tau_4,{\bf n}_4,\tau_5)$, where ${\bf n}_j$ represents the direction of the $j$-th $\pi/2$ pulse, and $\tau_j$’s the delays interleaving the pulses. In our experiments, the $\pi/2$ pulses have a width $t_w$ of typically 1 $\mu$s. $\tau_j$ starts and/or ends at the midpoints of the pulses (see also Fig. \[fig:mqcd\]). In this notation, our forward 16-pulse sequence can be expressed as
$$\begin{gathered}
P(\tau_1,{\bf x},\tau_2,{\bf y},2\tau_1,{\bf y},\tau_2,{\bf x},\tau_1)P(\tau_1,{\bf x},\tau_2,{\bf y},2\tau_1,{\bf y},\tau_2,{\bf x},\tau_1)P(\tau_1,{\bf \overline{x}},\tau_2,{\bf \overline{y}},2\tau_1,{\bf \overline{y}},\tau_2,{\bf \overline{x}},\tau_1)P(\tau_1,{\bf \overline{x}},\tau_2,{\bf \overline{y}},2\tau_1,{\bf \overline{y}},\tau_2,{\bf \overline{x}},\tau_1)\end{gathered}$$
and the backward sequence as $$\begin{gathered}
P(\tau_3,{\bf y},\tau_3,{\bf x},2\tau_4,{\bf x},\tau_3,{\bf y},\tau_3)P(\tau_3,{\bf y},\tau_3,{\bf x},2\tau_4,{\bf x},\tau_3,{\bf y},\tau_3)P(\tau_3,{\bf \overline{y}},\tau_3,{\bf \overline{x}},2\tau_4,{\bf \overline{x}},\tau_3,{\bf \overline{y}},\tau_3)P(\tau_3,{\bf \overline{y}},\tau_3,{\bf \overline{x}},2\tau_4,{\bf \overline{x}},\tau_3,{\bf \overline{y}},\tau_3)\end{gathered}$$
where $\{{\bf \overline{x}},{\bf \overline{y}}\}\equiv \{{\bf -x},{\bf -y}\}$. The delays are given by $$\begin{gathered}
\begin{aligned}
\tau_1&=\tau_0(1-u), \quad
\tau_2=\tau_0(1+2u), \\
\tau_3&=\tau_0(1+u), \quad
\tau_4=\tau_0(1-2u),
\end{aligned}\end{gathered}$$ where $\tau_0$ is 5 $\mu$s in this paper. The cycle time $t_c$, defined as the total time of the sequence, is given by $\tau=24\tau_0$. $u$ is a dimensionless adjustable parameter, and is restricted such that none of the inter-pulse spacings becomes negative. To the zeroth order Magnus expansion, the above sequence realizes Hamiltonian $uJ_0D_y$ and $uJ_0=J$.
[^1]: C.Y. and P.P. contributed equally to this work.
[^2]: We note that discrepancies in the value of $J_c\tau$ and order of curves in Fig. \[fig:tscaling\](a) and (c) are to be expected, because although $J_c\tau$ approaches zero when $L\to\infty$ and $n\to \infty$, the convergence speed depends on the path to that limit.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'It has recently been shown that the maximal kinematical invariance group of polytropic fluids, for smooth subsonic flows, is the semidirect product of $SL(2,R)$ and the static Galilei group $G$. This result purports to offer a theoretical explanation for an intriguing similarity, that was recently observed, between a supernova explosion and a plasma implosion. In this paper we extend this result to discuss the symmetries of discontinuous flows, which further validates the explanation by taking into account shock waves, which are the driving force behind both the explosion and implosion. This is accomplished by constructing a new set of Rankine-Hugoniot conditions, which follow from Noether’s conservation laws. The new set is dual to the standard Rankine-Hugoniot conditions and is related to them through the $SL(2, R)$ transformations. The entropy condition, that the shock needs to satisfy for physical reasons, is also seen to remain invariant under the transformations.'
---
MIT-CTP-3519
[Symmetries of Discontinuous Flows and the Dual Rankine-Hugoniot Conditions in Fluid Dynamics]{}
[Oliver Jahn[^1]]{}
[Center for Theoretical Physics]{}
[Massachusetts Institute of Technology]{}
[Cambridge, MA 02139-4307]{}
[U.S.A.]{}
[V. V. Sreedhar[^2]]{}
[Department of Physics]{}
[Indian Institute of Technology]{}
[Kanpur, 208016]{}
[India]{}
[Amitabh Virmani[^3]]{}
[Department of Physics]{}
[University of California]{}
[Santa Barbara, CA 93106-9530]{}
[U.S.A.]{}
Introduction
============
It has recently been observed that the density profiles of a supernova explosion and an inertial confinement plasma implosion[@sn1; @sn2; @sn3] are strikingly similar. An empirical basis for this intriguing duality between explosion and implosion was given by Drury and Mendonça [@drury] who pointed out that Euler’s equations of fluid dynamics, which describe both the systems, are form–invariant under a set of nonlinear coordinate transformations [*viz.*]{} $\vec x \rightarrow \vec x/t, ~~~t
\rightarrow -1/ t$. The minus sign in the time transformation maps an explosion to an implosion and the inversion allows large time scales to be mapped to small time scales and [*vice versa*]{}. These transformations suggest that the maximal kinematical invariance group $\mathcal{G}$ of fluid dynamics is larger than the standard Galilei group. It is now known that this larger group is a twelve-parameter semidirect product, $\mathcal{G} = SL(2,R)\wedge G$ [@sreedhar; @jackiw], where $G$ is the nine-parameter, connected, static Galilei group: $$\vec x \rightarrow {\bf R} \vec x + \vec v t + \vec a, \qquad t \rightarrow t$$ and $SL(2,R)$ is the group consisting of the transformations: $$t \rightarrow \frac{\alpha t + \beta}{\gamma t + \delta} \;,
\qquad \vec x \rightarrow \frac{\vec x}{\gamma t + \delta}
\qquad\text{with}\qquad \alpha \delta - \beta \gamma = 1 \;.$$ Physically, the three-parameter $SL(2,R)$ group consists of time translations, scale transformations, and a one-parameter set of time-dependent scale transformations called expansions. The transformations proposed by Drury and Mendonça are a special case of the $SL(2, R)$ transformations with $(\alpha,
~\beta,~\gamma,~\delta) = (0,~-1,~1,~0)$. The $SL(2,R)$ part of ${\cal G}$ is therefore important for a better understanding of the explosion–implosion map.
It should be pointed out that the naive expectation of using time-reversal invariance, to explain the similarity between explosion and implosion, is untenable here since the length and time scales involved in the two systems are drastically different. Invoking scaling arguments is not of much help since, although a composition of time-reversal and suitable scalings leaves the equations of fluid mechanics and the Reynolds number invariant, it has the property of reversing the direction of time’s arrow and thereby violates the second law of thermodynamics. As a consequence, when applied to a shock wave, such transformations violate entropy conditions that define the physicality of the shock. As is well-known, however, both the supernova explosion and the plasma implosion are driven by the formation and propagation of a shock wave. It is therefore important to examine whether the $SL(2, R)$ symmetry, that purports to explain the observed duality, respects the physicality of the shock wave. With this in view, we extend the study of [@sreedhar] – in which the explanation of explosion-implosion duality based on the symmetry group ${\cal G}$ was restricted to smooth, subsonic flows – to examine shock waves. It will be shown that shock wave solutions are consistent with the symmetries of the maximal kinematical invariance group ${\cal G}$ in the following precise sense.
A shock in a fluid is described mathematically by the well-known Rankine-Hugoniot jump conditions [@rich]. So the natural question to ask is: What happens to these conditions under the action of the $SL(2, R)$ group? This question is best answered not in the framework of the partial differential equations of fluid dynamics, but by reverting back to their so-called primitive form [*i.e.*]{} expressing them as conservation laws. The conservation laws are completely equivalent to the partial differential equations for smooth flows, but produce the Rankine-Hugoniot jump conditions for discontinuous flows in a natural and well-defined manner. The connection with $SL(2, R)$ is made by appealing to Noether’s theorem which asserts that corresponding to every continuous symmetry, there exists a conserved charge.
Anticipating that the $SL(2,R)$ transformations will mix the conservation laws corresponding to various symmetries, we construct the Noether charges corresponding to them and the boost transformations, in addition to the well-known ones for rotations, space and time translations. We then use the attendant conservation laws to establish a new set of jump conditions. It turns out that the new conditions are identically satisfied if the standard conditions for mass, momentum and energy conservation are satisfied. Although seemingly redundant because of this reason, the new set holds independently; following, as it does, from the symmetries of the fluid equations. In fact, these conditions are useful to prove the form-invariance of the Rankine-Hugoniot conditions under the $SL(2,R)$ transformations. Thus, to each physical system governed by the fluid dynamics equations two independent, but physically equivalent, sets of jump conditions can be associated, the two being related by $SL(2,R)$ transformations. We conclude that the $SL(2,R)$ transformations map the Rankine-Hugoniot conditions of the explosion to the dual Rankine-Hugoniot conditions of the implosion and [*vice versa*]{}. Further, by specialising to the Drury-Mendonça transformations, $\vec x \rightarrow \vec x/t, ~~~t
\rightarrow -1/ t$, we show that the jump conditions for boosts and expansions, along with the continuity equation for mass conservation, provide an independent, albeit equivalent, description of the shock. They may be viewed either as the dual of the standard Rankine-Hugoniot conditions, or, in the language of passive coordinate transformations, as the standard Rankine-Hugoniot conditions in the dual coordinate system corresponding to the choice $(\alpha, \beta, \gamma, \delta ) = (0, -1, 1, 0)$. Similar dual conditions exist for each choice of the $SL(2, R)$ parameters.
It is well-known that Rankine-Hugoniot conditions describe not only shocks, but other discontinuities like slip and contact discontinuities, for example. Therefore, the map between the dual sets of Rankine-Hugoniot conditions would be relevant to explosion-implosion duality only if both the sets refer to shocks. In other words, only those Rankine-Hugoniot conditions that describe a shock and only those $SL(2, R)$ transformations which map a shock to a shock are of interest for explosion-implosion duality. Moreover, Rankine-Hugoniot conditions say nothing about the physicality of the shock – this information is contained in additional inequalities for its entropy that the shock needs to satisfy. Physical shocks are distinguished from others because their entropy always increases across the shock front. We verify explicitly that this requirement is unaffected by the $SL(2, R)$ transformations.
The physicality of the map between explosion and implosion may also be established in the following subtle manner: Although the notions of viscosity and heat conduction lose their meaning in the immediate vicinity of the shock, because the changes in all the quantities they depend on are so great, they do play an important role in the formation and maintenance of a shock discontinuity[@courant]. In particular, the positivity of the coefficients of viscosity and heat conduction guarantees that the shock satisfies the appropriate entropy conditions[@courant; @landau]. Hence, Euler’s equations ought to be considered as a special case of a more general set of fluid equations with vanishingly small viscosity. Requiring the sign of the viscosity to remain unchanged under the transformations establishes the physicality. The Navier-Stokes equations – which are the obvious choice for including viscosity – are not invariant under the full $SL(2,R)$ part of ${\cal G}$, but only under the standard Galilean transformations. However, a more general set of fluid equations with viscosity fields transforming appropriately under the $SL(2, R)$ transformations has a maximal kinematical invariance group given by ${\cal G}$ [@sreedhar]. Hence we use these equations for our purpose of examining the behaviour of non–vanishing viscosity under the $SL(2, R)$ transformations. Similar arguments apply for heat conduction, but it does not bring in any new qualitative features and hence is omitted from further discussion.
Symmetries of Fluid Dynamics
============================
In this section we briefly recapitulate the results of [@sreedhar]. The general fluid equations in $n$-dimensional space are [@landau] $$\begin{aligned}
\label{e1}
D \rho &=& - \rho \vec{\nabla} \cdot \vec{u}
\\\label{e2}
\rho D \vec{u} &=& -\vec{\nabla} p + \vec{V}
\\\label{e3}
D \varepsilon &=& - (\varepsilon + p) \vec{\nabla} \cdot \vec{u} \end{aligned}$$ where $$D = \frac{\partial}{\partial t} + \vec{u} \cdot \vec{\nabla}$$ and $$V_i = \nabla_j\left(\eta(\nabla_j u_i + \nabla_i u_j - \frac{2}{n} \delta_{ij} \nabla_k u_k)\right) + \nabla_i(\zeta \nabla_k u_k)$$ In the above equations $\rho, \vec{u},p,\varepsilon$ stand for the density, velocity, pressure and energy density of the fluid respectively and $\eta, \zeta$ are the bulk and shear viscosity fields.
The above differential equations are usually augmented by an algebraic condition called the polytropic equation of state which relates the pressure to the energy density as $$\label{eos}
p = (\gamma_0 - 1) \varepsilon$$ where $\gamma_0$ is a constant called the polytropic exponent. As shown in [@sreedhar], the maximal invariance group of the above set of equations is ${\cal G} = SL(2, R)\wedge G$, provided the polytropic exponent takes the standard value for an ideal, nonrelativistic fluid [*viz.*]{} $\gamma_0 = 1 +
\frac{2}{n}$. For this value, the fluid equations are invariant under the following transformations [@sreedhar]:
### Connected, static Galilei transformations: {#connected-static-galilei-transformations .unnumbered}
Let $g$ denote a general element of this sub-group then $$g: \quad t' = t \;,\qquad \vec x' = {\bf R} \vec x + \vec v t + \vec a$$ with ${\bf R}$ an orthogonal matrix. Under the action of $g$, the fields $\rho$ and $\vec u$ transform as $$\label{field1}
\rho' = \rho \quad\mbox{and}\quad
\quad
\vec u ' = \vec u + \vec v$$
### SL(2,R) transformations: {#sl2r-transformations .unnumbered}
Let $\sigma$ denote a general element of the $SL(2,R)$ part of $\mathcal{G}$ then $$\label{cor2}
\sigma : \quad t' = \frac{\alpha t + \beta}{\gamma t + \delta} \;,\quad
\vec x' = \frac{\vec x}{\gamma t + \delta}
\quad\text{where}\quad \alpha \delta - \beta \gamma = 1$$ Under the action of $\sigma$, the fields transform as $$\begin{aligned}
\label{field2}
\rho' = (\gamma t + \delta)^n \rho\quad\mbox{and}\quad
\vec u ' = (\gamma t + \delta)\vec u - \gamma\vec x \end{aligned}$$ For both $g$ and $\sigma$, the transformations of $\varepsilon$ and $p$ can be worked out once the transformation properties of $\rho$ are known since $$\label{rel}
\varepsilon= \chi\rho^{\gamma_0}\;, \qquad
p=(\gamma_0-1)\varepsilon\;,$$ with the field $\chi$ – related to entropy – transforming like a scalar. The transformation properties of the viscosity fields are similar to the density $\rho$ $$\eta' = \bigl(\gamma t+ \delta\bigr)^n\eta \quad\mbox{and}\quad
\zeta' = \bigl(\gamma t+ \delta\bigr)^n\zeta$$ The above results were derived in [@sreedhar] by requiring the invariance of the Action for the simple case of an inviscid and isentropic fluid. The symmetry of the equations followed by subsequently relaxing the simplifications to arrive at the general fluid equations. It should be noted that the requirement of the invariance of the Action is sufficient, but not necessary, for the form invariance of the equations that follow from it. Any transformation that leaves the Action invariant upto a multiplicative factor produces equations of motion which have the same form. If this is taken into account, the condition $\alpha\delta - \beta\gamma
= 1$ is no longer required and $SL(2, R)$ gets replaced by $GL(2, R)$ in the maximal invariance group of the general fluid equations. However, it is sufficient for our purposes to concentrate on the variational symmetries of the fluid equations and for this purpose, $\mathcal{G} = SL(2,R)\wedge G$. Finally, for the sake of completeness, it should also be pointed out that the $SL(2,R)$ condition is invariant under the following discrete symmetries $(\alpha,\beta, \gamma,\delta)~~\rightarrow~~ (\alpha,-\beta,-\gamma,\delta),
~~\rightarrow~~ (-\alpha,\beta,\gamma,-\delta), ~~\rightarrow~~ (-\alpha,
-\beta,-\gamma, -\delta)$ of the $SL(2,R)$ parameters.
Conservation Laws
=================
In this section we construct the conservation laws corresponding to the symmetries outlined in the previous section. In order to do this, it is useful to revert back to the Action formalism and obtain the results for the subclass of inviscid, isentropic and irrotational flows. The corresponding expressions for a general fluid can then be worked out along the lines of [@sreedhar].
For inviscid, isentropic and irrotational flows, the Lagrangian density is given by $${\cal L} = \rho\bigl(\dot\phi - {1\over 2}(\vec\nabla\phi)^2\bigr) -
\rho^{\gamma_0}$$ where $\vec\nabla\phi$ stands for the curl-free part of the velocity vector field $\vec u$. Let $\mu = 0,1,2,3$ and $x^\mu$ be a four-vector under the transformations of the previous section [*i.e.*]{} $x^i$ with $i = 1,2,3$ are the components of $\vec x$ and $x^0 = t$. Let the infinitesimal variations in the coordinates and fields be defined as $$\delta x^\mu = {x^\mu}' - x^\mu \quad\mbox{and}\quad \delta\phi (x) = \phi'(x')
- \phi (x)$$ Then the variations for translations, rotations, boosts, dilatations, and expansions respectively are given by $$\delta x^\mu = a^\mu,~~ \delta x^i = \omega^{ij}x^j,~~ \delta x^i = v^it,~~
\delta x^i = \lambda x^i,~~ \delta t = 2\lambda t~~\mbox{and}~~
\delta x^\mu = -\mu tx^\mu$$ where the parameters $\lambda,~a^0,~\mu$ are expressible in terms of the $SL(2,R)$ parameters $\alpha,~\beta,~\gamma,~\delta$. The field variation is given by $$\delta\phi = \Lambda = {[\gamma(x + a) -\delta v]^2\over 2\gamma(\gamma t
+\delta )}$$ The variation in $\rho$ is not important since no derivatives of $\rho$ appear in the Lagrangian density. Using these results we find, by a straightforward application of Noether’s theorem [@goldstein], that the following quantities, integrated over all space, are constants of motion: $${\hbox{\bf Temporal Translations}}:\qquad
H = {\rho\over 2}(\vec\nabla\phi )^2 + \rho^{\gamma_0}$$ $${\hbox{\bf Spatial Translations}}:\qquad
\vec P = \rho\vec\nabla\phi$$ $${\hbox{\bf Rotations}}:\qquad
\vec L = \vec P \times \vec x$$ $${\hbox{\bf Boosts}}:\qquad
\vec K = \vec P t - \rho\vec x$$ $${\hbox{\bf Dilatations}}:\qquad
D = -2t H + \vec x \cdot \vec P$$ $${\hbox{\bf Expansions}}:\qquad
A = t^2 H - t \vec x\cdot\vec P
+ {\rho\over 2}{\vec x}^2$$ The conditions of irrotationality and isentropicity can be relaxed easily and one sees that Euler’s equations $$\dot\rho = -\vec\nabla\cdot(\rho\vec u)$$ $$\rho {\dot{\vec u}} = -\rho(\vec u\cdot\vec\nabla)\vec u - \vec \nabla p$$ $$\dot\varepsilon = -\vec\nabla\cdot(\varepsilon\vec u) - p \vec\nabla
\cdot\vec u$$ can be expressed in the form of conservation laws, $${\partial\over\partial t}\rho = - {\partial\over\partial x_j} (\rho u_j)$$ $${\partial\over\partial t}(\rho u_i) = - {\partial\over\partial x_j} (\rho
u_iu_j +\delta_{ij}p)$$ $${\partial\over\partial t}({1\over2}\rho\vec u^2 +\varepsilon ) = -
{\partial\over\partial x_j} [({1\over 2}\rho\vec u^2 + \varepsilon + p)u_j]$$ for mass and the translation generators found above. These can be reexpressed succinctly as follows: $$\partial_\mu J^\mu_{(\rho)} = 0,\quad \partial_\mu J^\mu_{(\vec P)} = 0,
\quad\mbox{and}\quad \partial_\mu J^\mu_{(H)} = 0$$ The zeroeth components of the above currents, namely $\rho$, $\rho\vec u$, and ${1\over 2}\rho\vec u^2 +\varepsilon$, give the charge densities which, when integrated over all space, give the conserved charges. As is well-known, these are merely statements of mass, momentum flux, and total energy conservation. The corresponding current densities are $$J^j_\rho = \rho u_j$$ $$J^j_{P_i} = \rho u_iu_j + \delta_{ij}p$$ $$J^j_{H} = ({1\over 2}\rho\vec u^2 + \varepsilon + p)u_j$$ The conservation laws corresponding to rotations, boosts, dilatations and expansions can be stated similarly $$\partial_\mu J^\mu_{(\vec L)} = 0,\quad \partial_\mu J^\mu_{(\vec K)} = 0,
\quad \partial_\mu J^\mu_{({D})},
\quad\mbox{and}\quad \partial_\mu J^\mu_{({A})} = 0$$ The charge densities are shown in (19) -(22) respectively, and the corresponding currents are $$\vec J_{L_i} = \epsilon_{ikl}x_k\vec J_{P_l}$$ $$\vec J_{K_i} = t\vec J_{P_i} - x_i\vec J_\rho$$ $$\vec J_{D} = x_i\vec J_{P_i} - 2t\vec J_{H}$$ $$\vec J_{A} = {1\over 2}\vec x^2\vec J_\rho - tx_i\vec J_{P_i} +
t^2\vec J_{H}$$ It may be mentioned that the above results are not surprising in the light of [@ajp], where corresponding results for a free, nonrelativistic, point particle were obtained through a discussion that essentially parallels the above. The noteworthy linear relations between the currents will, however, play a crucial role in this paper when we consider flows with discontinuities.
Discontinuous Flows and Jump Conditions
=======================================
As long as the flows are smooth, [*i.e.*]{} the functions $\rho, \vec{u}, p,
\varepsilon \in \mathbf{C}^1$ in their dependence on $\vec x$ and $t$, the systems (23 – 25) and (26 – 28) are equivalent. However, real flows are not always smooth and can develop discontinuities as time elapses. Such flows are described by weak solutions of differential equations [@rich]. A weak solution is generally piecewise smooth. The smooth parts satisfy the differential equation in the usual, or strong, form, but that does not generally suffice to determine the course of motion for initial data, and the equation must be supplemented by jump conditions. The resulting jump conditions are most clearly derived from the conservation laws.
By definition any, possibly non-smooth, function $J^\mu(\vec{x},t)$ that satisfies $$\label{weak} \int \partial_\mu w(\vec x,t) \, J^\mu(\vec x,t)
\,{\mathrm{d}}^3x\, {\mathrm{d}}t = 0$$ for all test functions $w(\vec{x},t)$ is said to be a weak solution of the differential equation $\partial_\mu J^\mu=0$.
We now use the above definition to obtain the jump conditions associated with the system of conservation laws derived in the last section. Suppose $J^\mu(\vec x,t)$ has a jump discontinuity across a hyper-surface $\mathcal{S}$ in ${\vec x},t$ space, while otherwise being continuously differentiable in some neighbourhood $\mathcal{N}$ of $\mathcal{S}$ (see Fig.1).
Let $w(\vec x,t)$ be a test function with support in $\mathcal{N}$. Let $\mathcal{R}$ be the part of the support of $w(\vec x,t)$ that lies on *one side* of $\mathcal{S}$, say the right. Then, by Gauss’s theorem $$\label{tough}
\int_{\mathcal{R}} \partial_\mu w \,J^\mu \,{\mathrm{d}}^3 x\,{\mathrm{d}}t
+ \int_{\mathcal{R}} w \,\partial_\mu J^\mu \,{\mathrm{d}}^3 x\,{\mathrm{d}}t
= \int_{\mathcal{R}} \partial_\mu (w J^\mu) \,{\mathrm{d}}^3 x\,{\mathrm{d}}t
= \int_{\mathcal{S}} w n_\mu J^\mu \,{\mathrm{d}}^3 \mathcal{S}$$ since $w(\vec x,t) = 0$ on the boundary of $\mathcal{R}$ except on $\mathcal{S}$. The second integral in the above equation is zero, because the conservation law holds in the strong sense in the interior of $\mathcal{R}$. Here, $n(\vec x,t)$ is the outward normal vector to the hypersurface $\mathcal{S}$. Therefore, if we integrate similarly over the left part of the support of $w$, add the result and make use of (\[weak\]), we find that: $$0 = \int_{\mathcal{S}} w n_\mu \Delta J^\mu {\mathrm{d}}\mathcal{S}$$ where $\Delta f$ denotes the difference of the two limiting values of a function $f$ on the two sides of the hypersurface $\mathcal{S}$ [*i.e.*]{} the jump of the function. This result follows because the vector $n_\mu$, which by convention points outwards, flips its sign on the left side of the support. Since $w$ is an arbitrary test function, the above equation implies the jump condition $$\label{master}
n_\mu \Delta J^\mu = 0 \quad\mbox{on } \mathcal{S} \;.$$ Applying (\[master\]) to the conservation laws (29) for $J^\mu_{(\rho)}$, $J^\mu_{(\vec P)}$ and $J^\mu_{(H)}$, we obtain $$\begin{aligned}
0 &= n_\mu \Delta J_{(\rho)}^\mu \;, \label{RH1} \\
0 &= n_\mu \Delta J_{(\vec P)}^\mu \;, \label{RH2}\\
0 &= n_\mu \Delta J_{(H)}^\mu \label{RH3}\end{aligned}$$ From here the standard Rankine-Hugoniot conditions can be derived in their usual form [@rich]. Similarly, one can apply (\[master\]) to the conservation laws (33) for $J^\mu_{(\vec L)}$, $J^\mu_{(\vec K)}$, $J^\mu_{(D)}$, and $J^\mu_{(A)}$ to obtain a new set of jump conditions: $$\begin{aligned}
0 &= n_\mu \Delta J^\mu_{(\vec L)} \\
0 &= n_\mu \Delta J_{(\vec K)}^\mu \\
0 &= n_\mu \Delta J_{(D)}^\mu \\
0 &= n_\mu \Delta J_{(A)}^\mu .\end{aligned}$$ Since the coordinates $\vec x$ and $t$ are continuous on $\mathcal{S}$, these conditions are all identically satisfied because of the jump conditions for mass, momentum and energy conservation, in (42-44) – a fact that can be easily verified using (19-22) and (34-37).
The Dual Rankine-Hugoniot Conditions
====================================
We have seen that the new set of jump conditions associated with rotations, boosts, dilatations, and expansions, follow from the jump conditions associated with mass, momentum and energy. This suggests that the Rankine-Hugoniot conditions are invariant under the full kinematical invariance group of smooth flows, including the $SL(2,R)$ part.
To see this explicitly, we consider the transformation properties of the conserved currents under $SL(2,R)$. Let us begin by considering the simplest of these, namely the time-component of $J^\mu_{(\rho )},~i.e.~\rho$. From equation (10), now with $n = 3$, $$\begin{aligned}
\label{density}
\rho' = (\gamma t + \delta)^3 \rho \end{aligned}$$ The $(\gamma t + \delta)^3$ factor is cancelled by an identical factor coming from the change of variables when we perform an integration over all space. Moreover, the transformation does not mix $\rho$ with any other current. Thus, $\rho$ transforms under the singlet representation of $SL(2, R)$ as a scalar density. Let us now consider the transformation of the time-component of $J^\mu_{({\vec P})},~i.e.~\vec P = \rho\vec u$. From (10) it now follows, after a little algebra, that $$\begin{aligned}
\label{momentum}
\vec P' = \rho'\vec u' = (\gamma t + \delta)^3(\delta \vec P + \gamma\vec K)\end{aligned}$$ Thus the transformation of the spatial translation generator mixes it with the boost generator together with which, it forms a doublet representation of $SL(2, R)$, with the prefactor $(\gamma t + \delta)^3$ now making it a vector density. The latter fact is, in fact, generic to the time-components of all the currents. Likewise, we may consider the generator of time translations, namely the Hamiltonian, and it follows that $$\begin{aligned}
\label{Hamiltonian}
\ H' = (\gamma t + \delta)^3(\gamma^2 A - \delta\gamma D + \delta^2 H)\end{aligned}$$ Thus the transformation of the time translation generator mixes it with the generator of dilatations and expansions, the three of them form the triplet (or adjoint) representation. The transformation properties of the rest of the currents can be similarly worked out and the results summarised as follows: If the (abstract) symmetry generators $T_r$ transform as $$T'_r \equiv \sigma^{-1} T_r \sigma =
\sum_s M_{r s}(\sigma) T_s \;,$$ where the matrix $M(\sigma)$ is determined by the group structure of $SL(2,R)\wedge G$, then the corresponding currents transform as $$J^{\mu\prime}_r(x') = \det\left(\frac{\partial x}{\partial x'}\right)
\frac{\partial x^{\mu\prime}}{\partial x^\nu}
\sum_s M_{r s}(\sigma) J^\nu_s(x) \;.$$ Assembling the currents in a column, $$J^\mu = \left(\begin{matrix} J^\mu_{(\rho )} \\ J^\mu_{(\vec K)} \\
J^\mu_{(\vec P)} \\ J^\mu_{(A)} \\ J^\mu_{(D)} \\ J^\mu_{(H)}
\end{matrix}\right)$$ one has for the transformation matrix, $$M = \left(\begin{matrix}
1 & 0 & 0 & 0 & 0 & 0 \\
0 & \alpha & \beta & 0 & 0 & 0 \\
0 & \gamma & \delta & 0 & 0 & 0 \\
0 & 0 & 0 & \alpha & -\alpha\beta & \beta^2 \\
0 & 0 & 0 & -2\alpha\gamma & (\beta\gamma + \alpha\delta) & -2\beta\delta \\
0 & 0 & 0 & \gamma^2 & -\gamma\delta & \delta^2
\end{matrix}\right)$$ Using $\alpha\delta - \beta\gamma = 1$, and the fact that the determinant of a block diagonal matrix is the product of the determinants of the blocks, it is easy to check that the matrix $M$ has unit determinant. As already pointed out, the fact that the currents transform like vector densities is reflected in the temporal components picking up a multiplicative factor $(\gamma t + \delta)^3$. The spatial components follow the example $$J_{(\rho)}^{i\prime} = (\gamma t+\delta)^{n+1} J_{(\rho)}^i
- \gamma x^i (\gamma t+\delta)^n J_{(\rho)}^0$$ with the same $SL(2, R)$ transformations defined by the matrix $M$.
The dual Rankine-Hugoniot conditions are now easily obtained. The normal vector $n_\mu$ appearing in the jump condition (\[master\]) transforms like a covector, $$n_\mu' \propto \frac{\partial x^{\nu}}{\partial x^{\mu\prime}} n_\nu \;,$$ so the transformed jump condition for $J_r$ is $$n_\mu' \Delta J^{\mu\prime}_r \propto
\det\left(\frac{\partial x}{\partial x'}\right)
\sum_s M_{r s}(\sigma) n_\mu \Delta J^\mu_s(x)
= 0 \quad\mbox{on } \mathcal{S} \;.$$ Since the determinant is smooth across the surface $\mathcal{S}$, the factor in front of the sum can be omitted. The transformed jump condition is therefore a linear combination of the original jump conditions. In particular, the conditions for $J_{(\rho)}$, $J_{(\vec P)}$ and $J_{(H)}$ (the Rankine-Hugoniot conditions) become linear combinations of the jump conditions for $J_{(\rho)}$, $J_{(\vec P)}$, $J_{(\vec K)}$, $J_{(H)}$, $J_{(D)}$ and $J_{(A)}$, $$\begin{aligned}
n_\mu' \Delta J^{\mu\prime}_{(\rho)} &\propto n_\mu \Delta J^\mu_{(\rho)} \;, \\
n_\mu' \Delta J^{\mu\prime}_{(\vec P)}
&\propto n_\mu ( \gamma \Delta J^\mu_{(\vec K)}
+ \delta \Delta J^\mu_{(\vec P)} ) \;, \\
n_\mu' \Delta J^{\mu\prime}_{(H)}
&\propto n_\mu ( \delta^2 \Delta J^\mu_{(H)} - \gamma\delta \Delta J^\mu_{(D)}
+ \gamma^2 \Delta J^\mu_{(A)} ) \;.\end{aligned}$$ The standard Rankine-Hugoniot conditions (42-44), in conjunction with the new set of jump conditions (45-48), then imply that the right hand side of the above equations is identically zero [*i.e.*]{} the Rankine-Hugoniot conditions are form–invariant. In particular, this holds for the Drury-Mendonça transformation $t\to-1/t$, $\vec x\to\vec x/t$ used to relate the explosion and implosion problems. For this, $(\alpha,\beta, \gamma,
\delta) = (0, -1, 1, 0)$ and it follows that $$\begin{aligned}
0 &= n_\mu \Delta J^\mu_{(\rho)} \;, \label{dualRH1} \\
0 &= n_\mu \Delta ( x_i J_{(\rho)}^\mu - t J_{(P_i)}^\mu ) \;, \label{dualRH2}\\
0 &= n_\mu \Delta ( -t^2 J_{(H)}^\mu + t x_i J_{(P_i)}^\mu - \tfrac12 x^2
\label{dualRH3} J_{(\rho )}^\mu ) \;.\end{aligned}$$ where we have substituted the explicit expressions for the currents $J^\mu_{
(\vec K)}$ and $J^\mu_{(A)}$. The conditions (62-24) are the dual Rankine-Hugoniot conditions. If an explosion is described by the standard Rankine-Hugoniot conditions, the corresponding implosion, obtained by a Drury-Mendonça transformation, is described by the dual Rankine-Hugoniot conditions (62-64).
Since the coordinates $\vec x$ and $t$ are continuous on $\mathcal{S}$, and crucially because the relations between the currents are linear, the conditions (62-64) are equivalent to the jump conditions obtained from mass, momentum and energy conservation, in (42-44). In fact, these two sets of equations imply, and are implied by, each other. In conclusion, the dual set of jump conditions associated with mass, boosts and expansions, is completely equivalent to the usual Rankine-Hugoniot conditions and may be used for an independent description of the shock.
The Entropy Condition
=====================
For a polytropic gas, by choosing $\varepsilon = \chi \rho^{\gamma_0}$, we can rewrite Eq. (5) as $$D\chi = 0$$ In [@sreedhar] we defined an isentropic flow to be one for which $\chi =~constant$. For a general flow, it followed that $\chi$ transforms like a scalar. For a polytropic gas, it is also well-known [@courant] that $\chi$ is related to the specific entropy (entropy per unit mass), $S$ as follows: $$S-S_0 = C_v{\hbox{log}}\bigl[\chi (\rho V)^{\gamma_0}\bigr]$$ where $C_v = R/(\gamma_0 - 1)$, $R$ being the universal gas constant divided by the molecular weight of the particular gas, $V$ the volume and $S_0$ an appropriate constant. It is obvious from this equation that as a particle of the medium moves about, the specific entropy at the moving particle remains constant under an $SL(2, R)$ transformation. Hence, under an $SL(2, R)$ transformation, a physical shock gets mapped to a physical shock.
We now require the positivity of viscosity to be preserved under an $SL(2, R)$ transformation – a requirement that guarantees that the shock respects the entropy condition. As already pointed out (see eq. (12)), in three-dimensional space, the viscosity fields transform as follows: $$\eta' = \bigl(\gamma t+ \delta\bigr)^3\eta \quad\mbox{and}\quad
\zeta' = \bigl(\gamma t+ \delta\bigr)^3\zeta$$ Thus the transformation properties of the viscosity fields are similar to $\rho$ [*i.e.,*]{} they transform like scalar densities. Hence, if we integrate the viscosity field over all space, to get the viscosity, it is an invariant under the $SL(2, R)$ transformations. Likewise, the specific viscosity (viscosity per unit mass), is an invariant. It follows that the positivity of the viscosity is maintained without any additional restrictions on the $SL(2, R)$ parameters.
Conclusions
===========
In this paper, we extended the analysis of [@sreedhar] to discuss the symmetries of discontinuous flows in fluid dynamics. The maximal kinematical invariance group of an ideal, polytropic fluid is ${\cal G}=SL(2,R)\wedge G$, not just for smooth, but for discontinuous flows also. This is made manifest by writing the fluid equations in their conservation law form. New conservation laws follow from a direct application of Noether’s theorem, enabling us to construct a dual set of Rankine-Hugoniot shock conditions. The $SL(2, R)$ transformations map the standard Rankine-Hugoniot shock conditions to the dual ones and [*vice versa*]{}. These transformations also respect the entropy conditions that physical shocks need to satisfy. Hence we conclude that, under these transformations, an explosion gets mapped to an implosion, thus offering a theoretical explanation for the intriguing observations of [@sn1; @sn2; @sn3].
Acknowledgements {#acknowledgements .unnumbered}
================
We thank L. O’C Drury, Pravir Dutt and S.G. Rajeev for discussions. VVS thanks A. J. Niemi for his hospitality in Uppsala, Sweden, and the Center for Dynamical Processes and Structure Formation (CDP), Uppsala University, for financial support.
[99]{} B. Remington, “Supernova Hydrodynamics Up Close,” Science and Technology Review Jan’/Feb’ 2000, Lawrence Livermore National Laboratory; `http://www.llnl.gov/str` I. Hachisu [*et al.*]{}, “Rayleigh-Taylor instabilities and mixing in the helium star models for Type Ib/Ic supernovae,” Astrophysical Journal [**368**]{} (1991) L27–30; H. Sakagami and K. Nishihara, “Rayleigh-Taylor instability on the pusher-fuel contact surface of stagnating targets,” Physics of Fluids [**B 2**]{} (1990) 2715–2730. L. O’C Drury and J. T. Medonça, “Explosion Implosion Duality and the Laboratory Simulation of Astrophysical Systems”, Physics of Plasma **7** (2000) 5148-5152 L. O’Raifeartaigh and V. V. Sreedhar, “The maximal kinematical invariance group of fluid dynamics and explosion-implosion duality,” Annals Phys.293:215-227,2001 hep-th/0007199.
In a different context, this group was first discovered by R. Jackiw, Phys. Today [**25**]{} (1972) 23; U. Niederer, Helvetica Physica Acta, [**45**]{} (1972) 802; C. R. Hagen, Phys. Rev [**D5**]{} (1972) 377. For related work in fluid mechanics see, M. Hassaine and P. A. Horvathy, Ann. of Phys. [**282**]{} (2000) 218; A. M. Grundland and L.Lalague, Can. J. Phy. [**72**]{} (1994) 362, Can. J. Phys. [**73**]{}, (1995) 463. For a review see R. Jackiw, physics/0010042.
Robert D. Richtmyer, Principles of Advanced Mathematical Physics, Vol. 1 1978 Springer-Verlag, See section 2.9, 17.1 –17.6
R. Courant and K. O. Friedrichs, Supersonic Flow and Shock Waves, 1948 Interscience Publishers, Inc., New York.
L. D. Landau and E. M. Lifshitz, Fluid Mechanics, Pergamon Press (1959).
Herbert Goldstein, Classical Mechanics, Second Edition 1980, Addison Wesley, See section 12-7.
O. Jahn and V. V. Sreedhar, Am. J. Phys. 69, 1039 (2001) math-ph/0102011
[^1]: [email protected]
[^2]: [email protected]
[^3]: [email protected]
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We discuss correlated lattice models with a time-dependent potential across a barrier and show how to implement a Josephson-junction-like behavior. The pairing occurs by a correlation effect enhanced by the symmetry of the system. In order to produce the effect we need a mild distortion which causes avoided crossings in the many-body spectrum. The Josephson-like response involves a quasi-adiabatic evolution in the time-dependent field. Besides, we observe an inverse-Josephson (Shapiro) current by applying an AC bias; a supercurrent in the absence of electromotive force can also be excited. The qualitative arguments are supported by explicit exact solutions in prototype 5-atom clusters with on-site repulsion. These basic units are then combined in ring-shaped systems, where one of the units sits at a higher potential and works as a barrier. In this case the solution is found by mapping the low-energy Hamiltonian into an effective anisotropic Heisenberg chain. Once again, we present evidence for a superconducting flux quantization, i.e. a Josephson-junction-like behavior suggesting the build-up of an effective order parameter already in few-electron systems. Some general implications for the quantum theory of transport are also briefly discussed, stressing the nontrivial occurrence of asymptotic current oscillations for long times in the presence of bound states.'
address: |
$^1$ INFN, Laboratori Nazionali di Frascati, P.O. Box 13, 00044 Frascati, Italy.\
$^2$ Dipartimento di Fisica, Università di Roma Tor Vergata, Via della Ricerca Scientifica, Roma, Italy\
$^3$ Dipartimento di Scienze Fisiche, Università di Roma Tre, Via della Vasca Navale 84, 00146 Roma, Italy\
$^4$ Instituto de Estructura de la Materia. Consejo Superior de Investigaciones Cient[í]{}ficas. Serrano 123, 28006 Madrid. Spain
author:
- 'Stefano Bellucci$^1$ Michele Cini$^{1,2}$, Pasquale Onorato$^{1,3}$, and Enrico Perfetto$^{1,4}$\'
title: Correlated Nanoscopic Josephson Junctions
---
ł Ł ¶ § c
Introduction
============
The Josephson effect is one of the clearest fingerprints from which one can diagnose superconductivity. A DC electromotive force (e.m.f.) $V$ across a thin insulating layer in a superconductor-insulator-superconductor junction produces[@josephson] an AC current response. The Josephson current is given by the well known expression[@grosso] \[Josephson\] I=I\_[J]{}( t+\_[0]{}),where $I_{J}$ and $\g_{0}$ are constants. In textbooks this result is explained in terms of the *order parameter* $\psi$ at the junction; within the insulator one has (z)=\_[left]{}e\^[-z]{}+\_[right]{}e\^[(z-s)]{}, where $s$ is the barrier width, $\a$ is a characteristic inverse length and $z$ is the normal coordinate; $\psi_{left}$ and $\psi_{right}$ are complex constants. The quantum mechanical current $I$ is computed by the Ginzburg-Landau formula, that is, by using $\psi$ as a single-particle wave function. Thus, $I$ is proportional to $\sin\theta$, where $\theta$ is the phase difference between $\psi_{left}$ and $\psi_{right}$. Letting $\theta(t)=\frac{2e
V}{\hbar}t+\g_{0}$ we obtain (\[Josephson\]).
The electrons cannot come into play in this picture, since the order parameter is a macroscopic concept. A very enlightening model argument which is microscopic in spirit is the one due to Ferrel and Prange[@ferrel]. Let $H_{T}$ denote the electron tunneling Hamiltonian between the left and right superconductors. The junction with $N-\n$ pairs on the left and $\n$ on the right (both numbers being huge) is mapped to a tight-binding chain described by an effective Hamiltonian $H_{F}$ that describes pair hopping. For $H_{T}\rightarrow 0$ the energy $E_{0}$ of the junction does not depend on $\n$. So, the state is defined by the amplitude $\phi_{\n}$ obtained from
H\_[F]{}\_=E\_[0]{} \_ +F(\_[+1]{}+\_[-1]{}), where the pair hopping matrix element arising to the second-order in $H_{T}$ reads
F=\_[i]{} and the sum is over the intermediate states $i$ of energy $E_{i}$ with an electron in the barrier. The eigenfunctions of $H_{F}$ are plane waves in $\n$ space, with eigenvalues (k)=E\_[0]{}+2Fk.A current $$\label{ferrel} I=\frac{4 e F}{\hbar}\sin k$$ is associated to the group velocity $\frac{1}{\hbar}\frac{\de
E}{\de k}$ in the fictitious chain and to the real current across the unbiased junction. A wave packet centered at $\bar{k}$ in the fictitious chain would be accelerated by |[k]{} = 2 e V and this inserted into (\[ferrel\]) reproduces (\[Josephson\]) again, seen from another angle. The description by Ferrel and Prange is still aimed at a macroscopic system with huge pair numbers enabling us to use the concept of a group velocity.
The question that arises is: what happens in small systems, when the fictitious chain is very short and the group velocity has no meaning? The Ferrel - Prange description must be modified, however pairs will continue to hop as bound units, and the AC response to a DC bias is a qualitative feature which looks well suited to test any model for superconductivity. Being a qualitative feature, it is hard to see how it can depend on size. Moreover, the order parameter which is well defined only in the thermodynamic limit must be supplemented by a microscopic description valid in finite systems; there, such concepts like long-range order do not apply but clear signatures like lack of dissipation must still distinguish superconducting from normal currents. The very nature of these signatures deserves investigation.
In this paper we wish to look for precursors to the Josephson behavior far from the thermodynamic limit; small cluster studies are motivated by conceptual as well as practical reasons. Indeed, the growing technological interest in nano-systems justifies the identification and search. We focus on repulsive Hubbard-like models that we want to test for correlation-induced superconductivity; this is relevant to the search for possible non-conventional mechanisms, that have been considered by several workers e.g. in the context of high-T$_{C}$ materials. The choice of the geometry has been prompted by previous studies[@topicalreview] in the framework of the $W=0$ theory; this develops the role of symmetry in inducing pairing from repulsive interactions. Besides, we also found the conditions leading to superconducting flux quantization in mesoscopic systems. We are therefore interested in testing this mechanism in time-dependent conditions.
Below we use a Gedanken experiment for understanding the problem. Developing this idea, one meets the difficulty that small systems always give AC response to a DC bias for quantum mechanical reasons; that is, one finds oscillating currents analogous to Rabi oscillations. However, the frequency of the oscillations is proportional to the charge, so a sharp effect of pairing remains, i.e. supercurrents are signaled by doubled frequencies. This is a conceptual problem and we shall be in position to add more to try to clarify this point in the conclusions.
There are further reasons of interest in this kind of problems. Transient current response are now actively studied, and within the TDLDA the asymptotic behavior of currents yields the static current-voltage characteristic[@stefanucci04]. Here we shall find a class of systems where, owing to correlation induced bound states, the asymptotic behavior is [*oscillatory.*]{} There is a clear need to develop new techniques, capable of tackling such situations.
Here we outline the plan of this paper. In Section \[w0\] we summarize the theory of $W=0$ pairing in Hubbard clusters. Section \[unosolo\] presents our one-unit model. Here we discuss the simplest case, namely a 5-site Hubbard cluster with square symmetry. For short, we shall refer to it as the CuO$_{4}$ unit; this is known to yield pairing and flux quantization, and here we show that it is also a prototype system for dynamic phenomena. One must perturb the square symmetry, in order to mimic a barrier in such a tiny system. The current excited by a constant electro-moving force (emf) is oscillatory with a frequency which is exactly the same as the Josephson frequency in Eq. (\[Josephson\]); moreover we shall observe Shapiro spikes (inverse Josephson effect) which represent a DC response to an AC bias at certain frequencies. A constant current (with some periodic ripple) flows in the absence of an applied potential difference, which is typical of superconducting systems. All these effects disappear when pairing fails, e.g. if the repulsion $U$ is removed. In Section \[many\] we shall extend the investigation of Josephson-like currents to large systems that are rings composed of any number of Cu$O_{4}$ units, up to the thermodynamic limit in principle. For small inter-unit hopping, the many-electron system can be exactly mapped into a Heisenberg-Ising-like chain. Systems hosting many pairs are described. An effective barrier is simulated by lifting the one-body levels of one of the Cu$O_{4}$ clusters; the superconducting flux quantization is again obtained and time-dependent solutions with a constant emf are also presented. Finally, we present our conclusions in Section \[Conclusion\].
W=0 pairing {#w0}
===========
In the context of High-T$_{C}$ superconductivity, several approaches based on the Hubbard Hamiltonian have appeared in the literature. The quantity (N)=E(N)+E(N-2)-2E(N-1), where $E(N)$ is the ground state energy of the system with $N$ fermions is used to evaluate the effective interaction between particles in the interacting ground state. A negative $\D$ means that an effective attraction is developing from the repulsive interactions. We refer to Ref. for the general theory based on the fact that highly-symmetric clusters possess 2-particle singlet eigenstates which do not feel the on-site repulsion $W$; these are called $W=0$ pairs. The persistence of the “$W=0$” character in the interacting case is discussed in detail there, with the flux quantization properties and their group theory analysis. The CuO$_{4}$ cluster with Hamiltonian H\_[0]{}= \_[p]{}\_[,i=2]{}\^[5]{}p\_[i,]{}\^p\_[i,]{}+ t\_[pd]{} \_[i]{}( d\^\_p\_[, i]{}+ p\_[, i]{}\^d\_)+ U(n\^[(d)]{}\_ n\^[(d)]{}\_+\_[i]{}n\^[(p)]{}\_[, i ]{}n\^[(p)]{}\_[, i]{}), is the simplest one to give the effect; here, $p^{\dagger}_{\sigma, i}$ is the creation operator for a fermion onto the Oxygen $i=2,..,5,$ $d$ destroys a fermion on the central site 1; besides, $\e_{p}$ is the O energy level, $t_{pd}$ is the p-d hopping matrix element, and $U$ is the Hubbard repulsion. In a wide range of parameters, this model yields $\D<0$ for $N=4.$
The magnetic properties of the CuO$_{4}$ cluster will be discussed below in more detail.
\[unit\]
(-8,-30)(8,-20) (-8,-30)(8,-20) (0,-20)
(-3,-5)
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(3,-5)
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\[bicornotondo\]
The one-unit model {#unosolo}
==================
For simplicity, Cu$O_{4}$ clusters are natural units to build more interesting systems. We start from the simple 5-atom unit and then show how we can combine many units to build large systems in Sect. \[many\]. In order to have closed circuits where flux tubes can be inserted, we consider the time-dependent Hamiltonian (see Figure 1)
H\_[tot]{} = H\_[0]{}+H\_(t),where
H\_(t)= \_[i]{}\_[i,i+1]{} p\_[i,]{}\^p\_[i+1,]{} + h.c.
Henceforth $t_{pd}$ will be our energy unit; we could dispense from introducing $\e_{p},$ but in this section $\e_{p}=3.5$ to make the flux quantization pattern more pronounced. There are various cases to consider.
Symmetric cluster
-----------------
Letting $\tilde{t}_{i,i+1}=t_{Ox}$ we have the symmetric cluster of Figure 1 a). Moreover we insert flux tubes (see Figure 1 a)) by setting $\tilde{t}_{i,i+1}=t_{Ox}e^{i \gamma (t)}$ with $\gamma =
\frac{2\pi \phi}{\phi_{0}} ,$ where $\phi $ is the flux in each tube; the flux does not affect the $t_{pd}$ bonds by symmetry. We observe a superconducting flux quantization due to the $W=0$ pairs as shown in Figure 2, solid line. A typical double-minimum pattern is found with sharp maxima due to the level crossing of paired states with different symmetry. This patter is obtained with $t_{Ox}=-0.05$; if the absolute value of the hopping $t_{Ox}$ is increased too much, the coupling to the flux becomes excessive compared to the pairing energy and the system turns paramagnetic[@topicalreview]. There are two minima at $\phi={\phi_{0}\over 2}$ and $\phi=\phi_{0};$ in both minima, there is pairing ($\Delta <0$). This pattern disappears together with pairing for $U=0.$
For a time-independent flux, the $\tilde{t}$ bonds carry a screening current, since $\hat{j} \propto - {\de H \over \de \phi}$[@kohn]. $ \phi=
\phi(t)$ is equivalent to applying an emf across each O-O bond. In the absence of interactions, the current response to a constant emf across e.g. the 2-3 bond is periodic due to the finite system (see below). If the flux varies slowly, the system will follow the ground state curve adiabatically up to the crossing point. However since each level has conserved quantum numbers, the system cannot adiabatically follow the ground state with slowly increasing $\phi.$ As a result, the adiabatic current is periodic with the same period as in a normal system (e.g. the cluster with $U=0$).
Static properties of the mildly distorted cluster
-------------------------------------------------
A more interesting behavior is obtained by breaking the square symmetry, as shown in Figure 1b). A superconducting pattern is still visible (Figure 2, dashed line). Here, $\tilde{t}_{i,i+1}=t_{ox}$ is a constant for all bonds, except for the bond 2-3, where we take $\tilde{t}_{2,3}={t_{ox}\over
2} e^{i\g(t)}.$ No flux is present in the other O-O-Cu triangles. In Figure 3, dashed line, we see that the superconducting double minimum pattern is still there but the level crossing is avoided opening a gap in the 4-fermion spectrum. However, the repulsion is needed to produce pairing and flux quantization. The dotted line shows the drastic effect of setting $U=0$: the system becomes paramagnetic and normal.
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(0,0)
\[scossadyn\]
Emf and frequency of currents
-----------------------------
In the paired situation of Figure 2, dashed line, we will consider the time-dependent case, $\phi/\phi_{0} \rightarrow Vt$, where $V$ plays the role of an electromotive force, and $t$ in units of $\frac{\hbar}{t_{pd}}$. An adiabatic behavior is now achieved when the time dependence is slow compared to the gap. We are using the Gauss system, hence \_[0]{}=and the Faraday law reads V=-.
Therefore, we introduce the frequency $\h$ by setting =-.
Equation (\[Josephson\]) becomes =(2t)for superconductor junctions, while the 2 is missing for normal systems.
This estimate agrees with the results of Figure 3. The solid line is the current through the 2-3 bond obtained by integrating the Schrödinger equation. The initial conditions are $\phi=0$ with the system in the ground state with $N$=4. The dotted line, reported for comparison, is the adiabatic response $j_{ad}(t)=c\frac{\de
E(4)}{\de \phi(t)}.$ The two curves agree fairly well. Since $t_{Ox}\ll 1,$ the main contribution to the current response comes from the 123 plaquette, which responds with frequency $\h$ for normal systems and $2\h$ for superconducting behavior. Some current contribution involves other plaquettes as well, perhaps mixing different frequencies. The agreement with the adiabatic approximation confirms the present analysis and will be used in Sect.IV . The normal response would be periodic with half the frequency.
(-30,-.1)(300,.1)
(0,0)(220,0) (0,-.09)(0,.09) (-5,-.00)[0]{} (100,-.005)(100,.005)(100,.0155)[100]{} (200,-.005)(200,.005)(200,-.015)[200]{}(220,-.002)[t]{} (20,.05)[$j(t)$]{} (-5,-.05)(5,-.05)(-20,-.05)[-.05]{} (-5,.05)(5,.05)(-20,.05)[.05]{} \[[[0., 0]{}, [2., -0.00339753]{}, [4., -0.00691894]{}, [6., -0.0103962]{}, [8., -0.0137411]{}, [10., -0.0171257]{}, [12., -0.0205779]{}, [14., -0.0239042]{}, [16., -0.0271086]{}, [18., -0.0303438]{}, [20., -0.0335343]{}, [22., -0.0365201]{}, [24., -0.0393614]{}, [26., -0.042131]{}, [28., -0.0446921]{}, [30., -0.0469448]{}, [32., -0.0489428]{}, [34., -0.0506688]{}, [36., -0.0519953]{}, [38., -0.0528676]{}, [40., -0.0532935]{}, [42., -0.0532319]{}, [44., -0.0526219]{}, [46.,-0.0514222]{}, [48., -0.0496312]{}, [50., -0.0472926]{}, [52., -0.0444153]{}, [54., -0.0409684]{}, [56., -0.0370504]{}, [58., -0.0328378]{}, [60., -0.0283548]{}, [62., -0.0236344]{}, [64., -0.0189498]{}, [66., -0.0145099]{}, [68., -0.0102919]{}, [70., -0.00645534]{}, [72., -0.00334274]{}, [74., -0.000995043]{}, [76., 0.000660623]{}, [78., 0.00138565]{}, [80., 0.0010407]{}, [82., -0.000137375]{}, [84., -0.00208146]{}, [86., -0.0049123]{}, [88., -0.00837457]{}, [90., -0.012096]{}, [92., -0.0160462]{}, [94., -0.0200579]{}, [96., -0.023637]{}, [98., -0.0265761]{}, [100., -0.0288549]{}, [102., -0.0301104]{}, [104., -0.0300441]{}, [106., -0.0287835]{}, [108., -0.0263032]{}, [110., -0.022374]{}, [112., -0.0171807]{}, [114., -0.0110685]{}, [116., -0.00404562]{}, [118., 0.00375753]{}, [120., 0.0118364]{}, [122., 0.0199012]{}, [124., 0.0278897]{}, [126., 0.0354504]{}, [128., 0.0421876]{}, [130., 0.0480498]{}, [132., 0.0529852]{}, [134., 0.0567652]{}, [136., 0.0593559]{}, [138., 0.0609155]{}, [140., 0.0614922]{}, [142., 0.0611443]{}, [144., 0.0600891]{}, [146., 0.0585274]{}, [148., 0.0566046]{}, [150., 0.0545216]{}, [152., 0.0524576]{}, [154., 0.0505249]{}, [156., 0.0488609]{}, [158., 0.0475654]{}, [160., 0.0466234]{}, [162., 0.0460493]{}, [164., 0.0458844]{}, [166., 0.046033]{}, [168., 0.0463657]{}, [170., 0.0468801]{}, [172., 0.0475216]{}, [174., 0.0480965]{}, [176., 0.0485386]{}, [178., 0.0488999]{}, [180., 0.049077]{}, [182., 0.0489324]{}, [184., 0.0485548]{}, [186., 0.0480335]{}, [188., 0.0472482]{}, [190., 0.0461872]{}, [192., 0.0450442]{}, [194., 0.0438533]{}, [196., 0.0424827]{}, [198., 0.0410148]{}, [200., 0.0396242]{}]{} \] \[[[0, 0]{}, [2, -0.00703588]{}, [4, -0.0138933]{}, [6, -0.0204247]{}, [8, -0.0265082]{}, [10, -0.0320627]{}, [12, -0.0370449]{}, [14, -0.0414509]{}, [16, -0.0452972]{}, [18, -0.0486157]{}, [20, -0.0514473]{}, [22, -0.0538359]{}, [24, -0.0558241]{}, [26, -0.0574515]{}, [28, -0.0587536]{}, [30, -0.0597617]{}, [32, -0.060503]{}, [34, -0.0610001]{}, [36, -0.0612724]{}, [38, -0.0613363]{}, [40, -0.0612056]{}, [42, -0.0608921]{}, [44, -0.0604054]{}, [46, -0.0597542]{}, [48, -0.0589457]{}, [50, -0.0579865]{}, [52, -0.0568822]{}, [54, -0.0556383]{}, [56, -0.0542595]{}, [58, -0.0527507]{}, [60, -0.0511165]{}, [62, -0.0493612]{}, [64, -0.0474896]{}, [66, -0.0455063]{}, [68, -0.043416]{}, [70, -0.0412236]{}, [72, -0.0389342]{}, [74, -0.036553]{}, [76, -0.0340857]{}, [78, -0.0315377]{}, [80, -0.0289151]{}, [82, -0.0262238]{}, [84, -0.0234702]{}, [86, -0.0206607]{}, [88, -0.0178019]{}, [90, -0.0149005]{}, [92, -0.0119635]{}, [94, -0.00899782]{}, [96, -0.00601057]{}, [98, -0.0030089]{}, [100, 0]{}, [102, 0.0030089]{}, [104, 0.00601057]{}, [106, 0.00899782]{}, [108, 0.0119635]{}, [110, 0.0149005]{}, [112, 0.0178019]{}, [114, 0.0206607]{}, [116, 0.0234702]{}, [118, 0.0262238]{}, [120, 0.0289151]{}, [122, 0.0315377]{}, [124, 0.0340857]{}, [126, 0.036553]{}, [128, 0.0389342]{}, [130, 0.0412236]{}, [132, 0.043416]{}, [134, 0.0455063]{}, [136, 0.0474896]{}, [138, 0.0493612]{}, [140, 0.0511165]{}, [142, 0.0527507]{}, [144, 0.0542595]{}, [146, 0.0556383]{}, [148, 0.0568822]{}, [150, 0.0579865]{}, [152, 0.0589457]{}, [154, 0.0597542]{}, [156, 0.0604054]{}, [158, 0.0608921]{}, [160, 0.0612056]{}, [162, 0.0613363]{}, [164, 0.0612724]{}, [166, 0.0610001]{}, [168, 0.060503]{}, [170, 0.0597617]{}, [172, 0.0587536]{}, [174, 0.0574515]{}, [176, 0.0558241]{}, [178, 0.0538359]{}, [180, 0.0514473]{}, [182, 0.0486157]{}, [184, 0.0452972]{}, [186, 0.0414509]{}, [188, 0.0370449]{}, [190, 0.0320627]{}, [192, 0.0265082]{}, [194, 0.0204247]{}, [196, 0.0138933]{}, [198, 0.00703588]{}, [200, 0]{}]{} \]
(0,0)
\[scossadynuzero\]
The system becomes normal for $U=0$, a fact which is apparent from the magnetic response (dotted line of Figure 2). The current response through the 2-3 bond also become normal, as we can see from the halved frequency of the current response in Figure 4. Note that the adiabatic result (dashed) does not agree very well with the response obtained by integrating the Schrödinger equation (solid). In fact, the gap $\sim t_{Ox}$ is missing in the spectrum.
Supercurrent flowing without bias
---------------------------------
In a macroscopic junction one can excite a supercurrent that will last forever if the exciting bias is removed, unless there are dissipative elements elsewhere in the circuit. This is also borne out in the paired situation of Figure 2, dashed line; we will consider the time-dependent case, $\phi/\phi_{0} \rightarrow Vt$, where $V$ plays the role of an emf of finite duration. In Figure 5 we plot the current flowing across the 2-3 bond (barrier) when the emf lasts from $t=0$ to $t=20$. At $t=0$, $\phi={\pi\over 2}.$ A constant supercurrent is superimposed on the oscillatory response.
\[supercurrent\]
(-50,-10)(350,14)
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(0,0) (0,0)(307,0)(317,1)[t]{}(-10,0)[0]{} (0,-.5)(0,13)(44,12)[$1000\times
j(t)$]{} (100,-1)(100,1)(97,-2)[100]{} (200,-1)(200,1)(203,-2)[200]{} (-10,10)(10,10)(-16,10)[10]{}(-10,20)(10,20)
Shapiro-like effect
-------------------
Still in the paired situation of Figure 2, dashed line, we also consider the time-dependent case of an oscillating field. In superconducting junctions there are spikes in the current-voltage characteristics when the voltage has a constant component and an oscillating one at radio-frequency $\w_{r}$ of amplitude $V_{r}$ (Shapiro effect): V\_[tot]{}=V+V\_[r]{}(\_[r]{}t).Then Eq.(\[Josephson\]) reads: I(t)=I\_[J]{}.In particular one finds a zero-voltage spike, because at $V=0$ the resulting $I(t)$ has a DC component. We can mimic such a behavior with our model. In Figure 6 we plot the current response through the 2-3 bond to a flux (t)=\_[0]{}+2(\_[r]{}t)with $\g_{0}={\pi \over 2},$ $\h_{r}=\frac{\pi}{200}$ and $\w_{r}={\pi
\over 10}.$ We observe the typical Shapiro-like DC current with a ripple on it.
(-50,-2)(350,34)
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(0,0) (0,0)(307,0)(317,1)[$t$]{} (0,-.5)(0,23)(44,22)[$1000\times
j(t)$]{} (100,-1)(100,1)(100,-1)[100]{} (200,-1)(200,1)(200,-1)[200]{} (-10,10)(10,10)(-16,10)[10]{}(-10,20)(10,20)(-16,20)[$20$]{}
\[shapiro\]
Comment on the one-unit model
-----------------------------
It appears from the above that the 5-atom cluster with suitable parametrization can simulate all the characteristic signatures of Josephson junctions, as if it already contained a seed of the order parameter. This is indeed surprising, but we show below that combining the small units we can go smoothly to large systems still keeping the same characteristic behavior; simplicity stems from the fact that if there are many pairs they move coherently.
The many-units model {#many}
====================
In This Section we study the time-dependent propagation of many $W=0$ pairs along rings of CuO$_{4}$ Hubbard clusters in presence of a barrier.
Our system is made of $L-1$ CuO$_{4}$ units plus a “barrier” standing between the 1-st and the $L-1$-th units. The barrier itself is taken to be a CuO$_{4}$ cluster, but with very high on-site energies, in order to make its occupancy only virtual.
The $L-1$ units are connected each other by the usual inter-cluster hopping Hamiltonian $H_{\tau}$, with real hopping integral $\tau$, while the hopping between the barrier and the 1-st and the $L-1$-th units is assumed to be complex and time-dependent: $\tau ' e^{i
\gamma (t)}$, where the phase $\gamma (t)$ will be discussed later. By means of such a choice we have in mind that in transport gedankenexperiments the potential drops only through the barrier.
According to the notation of our previous work[@sfq], the model reads: H\_[tot]{} = H\_[0]{}+H\_+H\_[barrier]{}+H\_[’]{}(t) with $$H_{0}=\sum_{\alpha =1}^{L-1} [ t_{pd} \sum_{i\sigma}( d^{\dagger}_{\alpha \sigma}p_{\alpha, i\sigma}+
p_{\alpha, i\sigma}^{\dagger}d_{\alpha \sigma})+ U(n^{(d)}_{\alpha
\uparrow} n^{(d)}_{\alpha \downarrow}+\sum_{i}n^{(p)}_{\alpha, i
\uparrow}n^{(p)}_{\alpha, i\downarrow})],$$ where, $p^{\dagger}_{\alpha, i\sigma}$ is the creation operator for a hole onto the Oxygen $i=1,..,4$ of the $\alpha$-th cell and so on. In this section $\e_{p}=0.$
$H_{\tau}$ is an inter-cell hopping Hamiltonian which allows a fermion in the $i$-th Oxygen site of the $\alpha$-th unit to move towards the $i$-th Oxygen site of the $\alpha \pm 1$-th unit (without involving the $L$-th unit, which is the barrier) with hopping integral $\tau$: $$H_{\tau}=\sum_{\alpha=1}^{L-2}\sum_{i\sigma} \tau\;
p_{\alpha,i\sigma}^{\dagger}p_{\alpha +1,i\sigma}\; + h.c. .$$ Finally we model the barrier with the following Hamiltonian: H\_[barrier]{}=t\_[pd]{} \_[i]{}( d\^\_[L ]{}p\_[L, i]{}+ p\_[L, i]{}\^d\_[L ]{})+ U(n\^[(d)]{}\_[L ]{} n\^[(d)]{}\_[L ]{}+\_[i]{}n\^[(p)]{}\_[L, i ]{}n\^[(p)]{}\_[L, i]{}) + WN\_[tot]{}, where $N_{tot}$ is the number of particles in the $L$-th cluster and $W$ is an extra (large) energy felt by each particle in the barrier, which can be provided, for instance, by a gate voltage. The barrier is linked to the rest of the ring by a complex hopping Hamiltonian: H\_[’]{}(t)=’ e\^[i (t)]{} \_[i]{} (p\_[L-1,i]{}\^p\_[L,i]{} + p\_[L,i]{}\^p\_[1,i]{}) + h.c. ’ e\^[i (t)]{} H\_[J]{} +h.c.
In the static case we set (with the same notation of [@sfq]) = ,where $\phi$ is the total flux through the ring and $\Lambda=2$ as long as the flux is associated with two bonds. In the time-dependent case, we will set $\phi/\phi_{0} \rightarrow Vt$, where $V$ plays the role of an electromotive force, as in the previous Section.
Static Case
-----------
Let us begin with the static case. As usual we introduce $H_{\tau}+H_{\tau'}$ perturbatively, in the same spirit of our previuos work[@sfq]. When $U/t\simeq 5$, $\tau=\tau'=0$ and the number of particles is appropriate and even, each CuO$_{4}$ cluster is populated by 2 or 4 particles (since the occupations with 3 particles have a gap of the order of $|\Delta|$ ), while the barrier is totally empty, since the extra energy even for a single occupancy is very large. Now, if $|\tau|,|\tau'|<<|\Delta|$, the perturbations are operative only by the second order. In particular $H_{\tau}$ provides an effective hopping $\tau_{eff}\sim \tau^{2}/|\Delta|$ for bound $W=0$ pairs, which can propagate in the 1,2,...,$L-1$ clusters as hard-core bosons. On the other hand $H_{\tau'}$ acts differently. Indeed the occupancy of barrier is highly unfavoured in the ground state because of the large energy $W$. As a consequence the $W=0$ pair can jump from the 1-st cluster to the $L-1$-th cluster an viceversa only at the fourth order in $H_{\tau'}$. In this case the effective hopping through the barrier is $ J e^{i \frac{4\pi \phi}{ \phi_{0}}
} $ with J= -(’)\^[4]{}C , where = E(3)+E(2)+E(1)+W (with $E(N)$ the ground state energy of the CuO$_{4}$ with $N$ particles), and C is a order 1 costant given by
$$\begin{array}{c}
C=\langle \Psi_{L-1}(4)| \otimes \langle \Psi_{L}(0) |\otimes
\langle \Psi_{1}(2) | \ H_{J} |
\Psi_{1}(2) \rangle \otimes | \Psi_{L}(1) \rangle\otimes |\Psi_{L-1}(3)
\rangle \times
\\
\langle \Psi_{L-1}(3)| \otimes \langle \Psi_{L}(1) |\otimes \langle
\Psi_{1}(2) | \ H_{J} |
\Psi_{1}(3) \rangle \otimes | \Psi_{L}(0) \rangle\otimes |\Psi_{1}(3) \rangle \times
\\
\langle \Psi_{L-1}(3)| \otimes \langle \Psi_{L}(0) |\otimes \langle
\Psi_{1}(3) | \ H_{J} |
\Psi_{1}(3) \rangle \otimes | \Psi_{L}(1) \rangle\otimes |\Psi_{L-1}(2) \rangle \times
\\
\langle \Psi_{L-1}(2)| \otimes \langle \Psi_{L}(1) |\otimes \langle
\Psi_{1}(3) | \ H_{J} |
\Psi_{1}(2) \rangle \otimes | \Psi_{L}(0) \rangle\otimes |\Psi_{1}(4) \rangle ,
\end{array}$$
where $|\Psi_{\alpha}(N) \rangle$ is the ground state of the $\alpha$-th cluster with $N$ particles.
The resulting low-energy effective hamiltonian is then equivalent to a 1-dimensional spin $1/2$ chain with ${\cal L} =L-1$ sites, given by (see Ref. [@bic2] for the case of $\tau' =0$): H\_[eff]{}()=-2\_[eff]{} \_[=1]{}\^[[L]{}-1]{} S\^[z]{}\_S\^[z]{}\_[+ 1]{}+ \_[eff]{} \_[=1]{}\^[[L]{}-1]{} (S\^[+]{}\_S\^[-]{}\_[+ 1]{}+h.c.) -\_[eff]{}S\^[z]{}\_[[L]{}]{}S\^[z]{}\_[ 1]{}+ (J e\^[i ]{}S\^[+]{}\_[[L]{}]{}S\^[-]{}\_[1]{}+h.c.).
We observe that the barrier has been “integrated out” and the resulting model has the 1-st and the ${\cal L}$-th sites linked by $J$.
It is worth to note that this model, because of the the presence of two different parametres $\tau_{eff}$ and $J$, is not solvable by the Bethe ansatz, therefore we must resort numerical methods. We solved $H_{eff}(\phi)$ with ${\cal L}$ up to 12 and $S^{z}_{tot}$ up to 3. In this case, in the original fermionic model there were $L=13$ CuO$_{4}$ clusters (12 + barrier) and $6$ $W=0$ pairs, that is $12
\times 2 + 6\times 2=36$ fermionic particles. In other words the number of $W=0$ pairs is $2 \times S^{z}_{tot}$, which is a conserved quantity.
In figure we show the ground state energy $E(\phi)$ as a function of the magnetic flux for ${\cal L}=12$ and $2\times
S^{z}_{tot}=1,2,3,4,5,6$. It is remarkable that whatever is the number of $W=0$ pairs in the original systems (1,2,3,4,5,6), the peridicity of $E(\phi)$ does not change and shows the same superconducting flux quantization. This is a very interesting feature, suggesting that all the $W=0$ pairs may behave “coherently”, providing the same macroscopic response, no matter of their number. It is also clear that the oscillations of $E(\phi)$ get more and more pronounced as the number of pairs in the ground state become of the same order of number of the lattice sites. This is an indication that such a behavior should survive in the thermodynamic limit.
Finally we observe that the trend of $E(\phi)$ by incresing the number of particles is deeply different with respect to the one observed in the repulsive Hubbard ring. In the latter case[@yufowler] [@kusmartsev], there are many minima between $\phi=0$ and $\phi=\phi_{0}$, depending on the ratio $Nt/LU$, where $N$ is the number of particles and $L$ is the number of lattice sites. On the other hand, in the case under consideration it seems that the superconducting nature of the charge carriers “freezes” the period oscillations to be $\phi_{0}/2$.
As a last remark, we notice that $E(\phi)$ is a smooth function of $\phi$. This means that no level crossing occurs between $\phi=0$ and $\phi=\phi_{0}$. This is a direct consequence of the breaking of the translational symmetry induced by the presence of the barrier. As discussed in the previous Section, this condition will be important in the time-dependent case, where, at least in the adiabatic approximation, the system at $t=0$ (supposed to be in the ground state at $\phi=0$) will be able to evolve continuosly to the static ground state at $\phi=\phi_{0}/2$. This condition is necessary in order to have a time-dependent superconducting response.
Time-Dependent Case
-------------------
Now we consider a time-dependent perturbation, with the effective hopping through the barrier given by $
J e^{i 4\pi Vt }.
$ In this case we rely on the adiabatic approximation, such that, nevertheless of the time-dependence, the low energy Hamiltonian is the same as $H_{eff}$, with $\phi/\phi_{0} \rightarrow Vt$. The correctness of this approximation has been checked directly in the previous Section, in the time-dependent analysis of the single CuO$_{4}$ cluster. There, the exact time-dependent solution of the whole original electronic problem was achieved and compared to the approximated one.
The exact time-dependent solution of the Schrödinger equation associated to $H_{eff}(t)$ is much harder than the static solution. Therefore we only consider the case ${\cal L}=4$ and $2\times
S^{z}_{tot}=2$. Anyway, in the original fermionic model, it means $L=5$ CuO$_{4}$ clusters (4+barrier) and 2 $W=0$ pairs, that is $12$ particles. As initial condition at $t=0$ we take the gruond state of $H_{eff}(\phi)$ at $\phi =0$.
Our aim is to evaluate the time-dependent current $I(t)$ through the barrier, i.e. through the bond governed by $J e^{i \frac{4\pi Vt}{
\phi_{0}} }$, according to I(t)=J (t) | e\^[i 4Vt ]{}S\^[+]{}\_[[L]{}]{}S\^[-]{}\_[1]{}- e\^[-i 4Vt ]{}S\^[-]{}\_[[L]{}]{}S\^[+]{}\_[1]{} | (t) .
The exact numerical solution shows that $I(t)$ is a peridic function of $t$ with period $1/(2V)$. In figure we plot $I(t)$ as a function of $Vt$ in two different unitary intervals: (0,1) and (48,49).
We stress that the time-dependent response of the many-units model have the same Josephson-like behaviour of the single-unit model.
Conclusions {#Conclusion}
===========
We have shown how to model a Josephson-like junction using microscopic ingredients like electrons hopping in an atomic lattice, a tunneling barrier, and on-site correlation. The barrier consists of a bond characterized by a weak time-dependent hopping, where the emf is applied. The current response is easier to calculate and analyze when there is emf on a single bond. We stress that the Hubbard interaction $U$ leads to a current response in quantitative agreement with Eq. (\[Josephson\]). This happens in circumstances that we understand, namely: i) the formation of $W=0$ pairs, ii) a well developed double minimum in the energy-flux plot, iii) a soft distortion of the symmetry opening a small gap (due to avoided crossing) in the many-body spectrum, iv) a small enough emf $V$, in order to allow for adiabatic response. When the last condition is realized, calculations based on the adiabatic approximation are in good agreement with the full solutions; this allows us to extend the analysis to fairly large systems. In modeling rings of Hubbard clusters, the barrier was formed by a CuO$_{4}$ unit with high one-electron energy; in order to deal with the low-energy physics, we derived an effective Hamiltonian where the barrier is integrated out and represented by an effective, weakened, complex bond, similar to the barrier in the above distorted CuO$_{4}$ case.
In the single-unit model, the superconducting AC current has the same frequency ${2e V\over \hbar}$ as one would expect to observe in a macroscopic junction; in the latter case, however, the normal response is a DC current. In our cluster calculations, on the other hand, the current response of the normal systems is periodic with half the Josephson frequency; this happens with one-particle systems and with 4-body ones with $U=0$, when no pairing occurs. This is due to a size effect that we wish to comment upon here.
As shown in Ref. (), in the non-interacting case (which is clearly normal) the transient current across a tunnel barrier between infinite leads tends to a constant for long times; the asymptotic $t\rightarrow \infty$ behavior yields the current-voltage characteristics calculated by several people by time-independent approaches [@landauer][@caroli][@feuchtwang][@meir]. This has recently been proven as a theorem by Stefanucci and Almbladh[@stefanucci04] within the Time-Dependent Density Functional Theory in the Local Density Approximation. In order to understand the difference, we recall that the infinite junctions considered in ([@cini80]) never reach equilibrium, and the steady current that holds in the $t\ra \inf$ limit is just a never-ending transient. On the contrary, the oscillating behavior of the small cluster follows almost adiabatically the instantaneous ground state. Another difference is that the normal system is unable to trap the magnetic flux: when the field penetrates the sample our modeling should be less appropriate than in superconductors. So we can expect that the normal current oscillation at half frequency, that we observe in the small cluster, should acquire a longer and longer transient behaviour with increasing sample size, while the field penetration should superimpose different oscillation frequencies. Instead, the superconducting pairs should keep their frequency in line with equation (\[Josephson\]) right to the thermodynamic limit.
According to our interpretation, the Josephson effect is essentially another facet of the superconducting flux quantization. The necessary distortion is always present in macroscopic systems, where in addition a large energy separates the states of different pair symmetry. Unbiased supercurrents and Shapiro spikes are also obtained by the same models that mimic the AC Josephson behaviour.
In the many-units model, the CuO$_{4}$ clusters are combined in large systems linked by weak hopping integrals; the solution is found by mapping the problem into a spin 1/2 chain. In the time-independent case, we presented evidence that the flux quantization pattern is unmodified when several bound pairs propagate together, except that the energy barriers separating the minima grow large. In the time-dependent case we solved Schrödinger equation numerically, and again we got a Josephson-junction-like behavior.
Finally, it is remarkable that even with one pair, that can be on the left or on the right, there is enough uncertainty on the pair number to create some ghost order parameter.
Acknowledgements {#acknowledgements .unnumbered}
================
E. P. was supported by INFN grant 10068.
[99]{}
B. D. Josephson, Phys. Letters [**1**]{}, 251 (1962) G. Grosso and G. Pastori Parravicini, Solid State Physics, Academic Press, San Diego (2000). J. G. Bednorz and K. A. Müller, Z. Phys. [**B 64**]{}, 189 (1986). R. A. Ferrel and R. E. Prange, Phys. Rev. Letters [**10**]{}, 479 (1963) A. Balzarotti, M. Cini, E.Perfetto and G.Stefanucci, J. Phys.: Condens. Matter 16 (2004) R1387-R1422. W. Kohn, Phys. Rev. [**133**]{} A171 (1964). G. Stefanucci and C.-O. Almbladh, Phys. Rev. [**B69**]{} 195318 (2004) A. Callegari, M. Cini, E. Perfetto and G. Stefanucci, Eur. Phys. J.B [**34**]{}, 455 (2003). N. Yu and M. Fowler, Phys. Rev.[**B45**]{} 11795 (1992). F. V. Kusmartsev, Phys. Rev.[**B52**]{} 14445 (1995). M. Cini, Phys. Rev.[**B22**]{} 5587 (1980). R. Landauer, IBM J. Res. Dev. [**1**]{}, 233 (1957) C. Caroli, R. Combescot, P. Nozi$\acute{e}$res and D. Saint-James, J. Phys. C [**4**]{}, 916 (1971) T. E. Feuchtwang, Phys. Rev. [**B12**]{}, 3979 (1975) Y. Meir and N. S. Windgreen, Phys. Rev. Lett. [**68**]{}, 2512 (1992) A. Callegari, M. Cini, E. Perfetto and G. Stefanucci, Eur. Phys. J. B [**34**]{}, 455 (2003).
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: '[We analyze the effects of noise on the permutation entropy of dynamical systems. We take as numerical examples the logistic map and the Rössler system. Upon varying the noise strengthfaster, we find a transition from an almost-deterministic regime, where the permutation entropy grows slower than linearly with the pattern dimension, to a noise-dominated regime, where the permutation entropy grows faster than linearly with the pattern dimension. We perform the same analysis on experimental time-series by considering the stochastic spiking output of a semiconductor laser with optical feedback. Because of the experimental conditions, the dynamics is found to be always in the noise-dominated regime. Nevertheless, the analysis allows to detect regularities of the underlying dynamics. By comparing the results of these three different examples, we discuss the possibility of determining from a time series whether the underlying dynamics is dominated by noise or not.]{}'
author:
- 'C. Quintero-Quiroz[^1]'
- Simone Pigolotti
- 'M. C. Torrent'
- Cristina Masoller
title: Numerical and experimental study of the effects of noise on the permutation entropy
---
[*Keywords*]{}: Time series analysis, Entropy, Ordinal patterns, Permutation entropy, Stochastic systems, Symbolic analysis.
Introduction
============
Ordinal analysis is a method of time series analysis that consists of computing the probabilities of *ordinal patterns*, which are defined according to the ordering of $D$ consecutive values in the series [@bp2002]. The entropy of these probabilities, referred to as [ *permutation entropy*]{}, is a tool to detect possible regularities in the time series. In recent years, ordinal patterns and permutation entropy have been widely used to investigate complex dynamical systems [@zanin; @special_issue]. They have been employed in the attempt to distinguish noise from chaos [@amigo2006order; @rosso_prl_2007; @amigo2008combinatorial; @aragoneses_2013], to detect noise-induced order [@rosso_pre_2009], serial correlations [@tiana_pra_2010] and dependencies between two or more time series [@groth_pre_2005; @canovas2011using; @bahraminasab2008direction; @matilla2011spatial; @matilla2008non; @saco_physicaa_2010; @barreiro_chaos_2011], among many other examples. Applications to experimental time series analysis include classification and discrimination of dynamical states in normal and epileptic EEG [@Nicolett-2012; @veisi2007fast; @li2007predictability; @bruzzo2008permutation] and detection of heart rate variability under different physiological and pathological conditions [@parlitz2012classifying; @frank2006permutation; @berg2010comparison].
Given this growing interest, it is relevant to understand the relation between the permutation entropy and other complexity measures. In particular, a well-established way to characterize the production of information of a dynamical system is the Kolmogorov-Sinai entropy $h_{ks}$, see e.g. [@beck; @compl]. To compute $h_{ks}$, the time series is discretized by partitioning the phase space into regions and assigning a symbol to each region. Then one computes the probabilities of [*blocks*]{}, which are vectors of $D$ consecutive symbols (more details in the following sections). The entropy of the block probabilities is the [*block entropy*]{}. The Kolmogorov-Sinai entropy is finally obtained as the rate of growth, for $D\rightarrow \infty$ and in the limit of a very refined partition, of the block entropy.
Similarly to $h_{ks}$, one can introduce a [*permutation entropy rate*]{} as the rate of growth for $D\rightarrow \infty$ of the permutation entropy. Both the permutation entropy and the Kolmogorov-Sinai entropy measure the “asymptotic” information rate of representations of the time series, the former with ordinal patterns (based in the relative order of consecutive values) and the latter with blocks (based on a partition of the phase space). The permutation entropy rate and $h_{ks}$ are not only conceptually related: for piecewise monotone interval maps on the real line, they were shown to be equal [@KBP_2002]. This result has been later extended to a broad class of dynamics [@amigo_2005; @amigo_2012]. This equivalence is non-trivial considering, for example, that the number of total possible ordinal patterns grows with $D$ as $D!$, while the number of blocks grows as $Q^D$ where $Q$ is the total number of symbols. The two quantities can be equal only thanks to the large number of forbidden ordinal patterns, strongly limiting the growth of the permutation entropy as $D$ is increased.
These mathematical results clarify that, under general hypotheses, permutation and block entropies share the same asymptotic behavior. However, due to difficulties in reaching the asymptotic regime, this equivalence can be of little use in many practical cases. For example, it has been noted [@KBP_2002] that the rate of convergence of the permutation entropy to the Kolmogorov-Sinai entropy is extremely slow even for one-dimensional maps, while on the contrary, block entropies converge very quickly, see e.g. [@beck].
Comparing the two analyses becomes even more problematic for high-dimensional and/or noisy dynamics, such as typical experimental time-series. Consider for example the extreme case of a time series dominated by noise, in which all symbols are equally probable and temporal correlations are absent. In this case, the block entropy of length $D$ is equal to $D\ln(Q)$, where $Q$ is the total number of symbols, while the permutation entropy with patterns of length $D$ is equal to $\ln(D!)\sim D\ln D$. This means that the block entropy is linear in $D$, with a slope $\ln(Q)$, explicitly dependent on the chosen partition, which diverges only in the limit of a very refined partition, $Q\rightarrow \infty$. In contrast, the permutation entropy grows more than linearly, so that their asymptotic slope is infinite. In both cases, the result is an infinite entropy rate. However, to discover it, in the first case one needs to construct a very refined partition. In the second case, one needs to reach large values of $D$ to appreciate that the slope increases logarithmically. Both these tasks can be very difficult when analyzing a finite time series due to statistical limitations.
Our goal is to get a better understanding of how noise influences the permutation entropy. To this aim, we analyze simulated and experimental time series. We mostly focus on permutation entropy as the effect of noise on block entropies is fairly well understood, see e.g. [@cencini_2000; @falcioni_2003]. We first analyze time series generated from the logistic map and from Poincaré sections of the three-dimensional Rössler system. We conclude with an experimental example of output intensity data recorded from a semiconductor laser with optical feedback.
Methods
=======
Numerical data
--------------
We consider two dynamical systems: the one-dimensional logistic map and the three-dimensional Rössler system. In both cases, we study the effect of adding to the dynamical equations a Gaussian white noise, $\xi_t $ with $\langle\xi_t\rangle=0$ and temporal correlation $\langle\xi_t \xi_{t'}\rangle=\delta_{t,t'}$. We considered also the case of observational noise (not shown), where the dynamics is deterministic but the noise affects the observation, obtaining very similar results.
### Logistic map
$$x_{t+1} = 4 x_{t} (1 - x_{t}) + \alpha \xi_t,
\label{log}$$
where $x_t$ is the state of the system at iteration $t$ and $\alpha$ is the noise strength. In order to constrain the variable $x_t$ in the interval $[0,1]$, the values of $\xi_t$ that would lead to $x_{t + 1}>1$ or $x_{t+1}<0$ are simply discarded and redrawn. Thus, the noise $\xi_t$ is temporally uncorrelated, but not purely Gaussian due to this truncation effect. To investigate the variation of the permutation entropy with the noise strength, we computed the permutation entropy, for each value of $\alpha$, from time series of length $N = 12 \times
10^{7}$. We have also studied other nonlinear one-dimensional maps (Tent, Bernoulli and Quadratic) and obtained very similar results to those of the logistic map (results not shown).
### Rössler system
The Rössler equations read $$\begin{aligned}
\dot{X} &=& - Y - Z + \alpha \xi(t) , \nonumber\\
\dot{Y} &=& X + aY \label{eq1}\\
\dot{Z} &=& b+Z(X-c)\nonumber\end{aligned}$$ where $\{X,Y,Z\}$ are the states of the system at time $t$, $\alpha$ is the noise strength and $\{a,b,c\}$ are the local parameters set at {$0.1, \,0.1, \,18.0$}, respectively.
In order to apply the symbolic methods (ordinal patterns or blocks) we need to discretize the dynamics. Instead of employing temporal sampling [@demicco], we introduce a Poincaré section [@review] at $ X =
0 $, and analyze the time intervals between consecutive crossings of the Poincaré plane. For each value of $\alpha$, the permutation entropy is computed from time-series of $N=12\times10^{7}$ data points.
Experimental data
-----------------
Experimental data was recorded from the output intensity of a semiconductor laser with optical feedback operating in the low-frequency fluctuations (LFFs) regime. In this regime, the laser intensity displays sudden and apparently random dropouts, followed by gradual recoveries. This spiking dynamics has received considerable attention because the intensity dropouts are induced by stochastic effects and deterministic nonlinearities. The optical feedback introduces a delay which renders the system in principle infinite dimensional. Therefore, the laser in the LFF regime generates complex fluctuations that, because of the stochastic and high-dimensional nature of the underlying dynamics, are suitable to be investigated by means of complexity measures such as the permutation and block entropies.
The experimental setup is the same as in [@taciano_optics_express_2015] and uses a 650 nm AlGaInP semiconductor laser (SONY SLD1137VS) with optical feedback. The feedback was given through a mirror placed 70 cm apart from the laser cavity, with a round trip of 4.7 ns. The feedback was controlled using a neutral density filter that can adjusts the light intensity injected into the laser. The laser has a solitary threshold current of $I_{th} = 28.4$ mA. The temperature and current of the laser were stabilized using a combi controller Thorlabs ITC501 with an accuracy of $0.01$ C and $0.01$ mA, respectively. The current used during the experiment was $I = 29.3$ mA and the temperature was set at $T = 17$ C. The neutral density filter was adjusted so that the threshold reduction due to feedback was about 7%. The signal was captured using a photo detector (Thorlabs DET210) connected to a FEMTO HSA-Y-2-40 amplifier and registered with a 1 GHz digital oscilloscope (Agilent Infiniium DSO9104A) with $0.2$ ns of sampling. The intensity time series were acquired from the oscilloscope by a LabVIEW program that uses a threshold to detect the times when the intensity drops, and calculates the time intervals between successive threshold crossings (in the following, referred to as inter-dropout-intervals, IDIs). We recorded in this way time series of more than $10^5$ consecutive IDIs.
Methods of analysis
-------------------
We compare two different methods to transform a time-series, $x(t)=
\{x(1)$, $ x(2)\dots x(N)\}$, into a sequence of symbols, $s(t)$: ordinal patterns and blocks.
In both cases, one needs to choose a [*dimension*]{} $D$ for defining vectors made up of consecutive entries of the time series, i.e. $\{x(i),
x(i+1), \dots, x(i+D-1)\}$. Ordinal patterns and blocks differ by the way in which the entries of these vectors are transformed into symbols. Ordinal patterns classify them according to the ranking (from the largest to the smallest value) of the $D$ entries in the vectors. The total number of ordinal patterns of length $D$ is then equal to the number of permutations, $D!$. For example, with $D = 2$ there are two ordinal patterns: $x(t_i) > x(t_{i+1})$ corresponding to the ordinal pattern ‘$01$’ and $x(t_i) < x(t_{i+1})$ corresponding to the ordinal pattern ‘$10$’.
For blocks, the phase space is first divided into $Q$ regions, associating a symbol to each region. Blocks represent all vectors in which each value of the time series correspond to the same symbol. For example, let us consider the time series $x (t) = \{0.1,0.6,0.7,0.3\}$, and partition the phase space into the two regions $[0, 0.5)$ and $[0.5,
1]$, associating to them the symbols $0$ and $1$ respectively. With $D=2$, the blocks associated to the time series are $\{01,11,10\}$.
Then, the permutation entropy [@bp2002] or the block entropy [@beck; @compl] are simply the entropy of the frequency $p_i$ of the different patterns in the time series $$H_D = -\sum_i^M p_i\ln p_i,$$ where $M$ is the number of possible patterns: for permutation entropy, $M=D!$, while for block entropy, $M=Q^D$.
Results
=======
![Permutation entropy ($H$) as a function of the size of the ordinal pattern ($D$) and the noise strength ($\alpha$) for data generated from the Logistic map. (a) $H$ vs $D$ for $\alpha=
1\times10^{-4}$ (stars), $\alpha= 2\times10^{-2}$ (triangles), $\alpha= 5\times10^{-2}$ (inverted triangles), $\alpha=0.1$ (circles), $\alpha=$1 (pentagons) and $H_{max}=\ln D!$ (solid line). (b) $H$ versus $\alpha$ for $D=2$ (stars), $D=3$ (triangles), $D=4$ (inverted triangles), $D=5$ (circles), $D=6$ (pentagons), $D=7$ (squares) and $D=8$ (diamonds).[]{data-label="fig1"}](fig1-a "fig:"){width="49.50000%"} ![Permutation entropy ($H$) as a function of the size of the ordinal pattern ($D$) and the noise strength ($\alpha$) for data generated from the Logistic map. (a) $H$ vs $D$ for $\alpha=
1\times10^{-4}$ (stars), $\alpha= 2\times10^{-2}$ (triangles), $\alpha= 5\times10^{-2}$ (inverted triangles), $\alpha=0.1$ (circles), $\alpha=$1 (pentagons) and $H_{max}=\ln D!$ (solid line). (b) $H$ versus $\alpha$ for $D=2$ (stars), $D=3$ (triangles), $D=4$ (inverted triangles), $D=5$ (circles), $D=6$ (pentagons), $D=7$ (squares) and $D=8$ (diamonds).[]{data-label="fig1"}](fig1-b "fig:"){width="49.50000%"}
Figure \[fig1\]a displays the permutation entropy, $H$, vs the dimension of the ordinal patterns, $D$, computed from time series of the logistic map at different noise strengths. It can be observed that $H$ increases monotonically with $D$, regardless of the noise strength. As the noise increases, $H$ approaches its maximum value, corresponding to equally probable ordinal patterns, $H_{max}=\ln D!$ (solid black line). Note that at ${\alpha = 1}$ (pentagons) the values of $H$ is already very close to $H_{max}$. Figure \[fig1\]b displays $H$ as a function of the noise strength $\alpha$. A clear transition from low-noise to high-noise can be observed, for a value of the noise strength approximately independent of $D$. The difference between the values of the entropies at low and high noise becomes more pronounced as $D$ increases.
To further investigate this transition, Fig. (\[fig2\]a) displays the difference $H_D-H_{D-1}$ as a function of $D$, for various values of noise strength. As before, we indicate with a thin black line the noise-dominated limit in which all patterns are equiprobable, $H_D -
H_{D-1} = (\ln D! - \ln (D-1)!)$. In the opposite limit of almost-deterministic, as $D$ grows the expected value of $H_D-H_{D-1}$ is the Kolmogorov-Sinai entropy [@KBP_2002], which for a one-dimensional chaotic map is equal to the Lyapunov exponent $\lambda$. In the case of logistic map for a local parameter set at $4$ one has $\lambda =\ln2$, indicated by the thick black line. As shown in detail in Fig. (\[fig2\]c), we identify three possibilities:
- a almost-deterministic regime in which $H_D-H_{D-1}$ decreases for large $D$,
- a noise-dominated regime in which $H_D-H_{D-1}$ increases for large $D$,
- an intermediate regime in which $H_D-H_{D-1}$ remains nearly constant with $D$.
![Comparison of the entropy computed from ordinal patterns, and the entropy computed from the blocks, for the Logistic map. The difference $H_D-H_{D-1}$ is plotted vs. the dimension of the ordinal patterns (a,c) and of the blocks (b,d) for various values of noise strength \[the noise strengths are as in Fig. (\[fig1\]a)\]. In panels (a) and (b) the solid lines indicate the asymptotic values for low noise (thick) and high noise (thin). Panel (c) and (d) display a detail of (a) and (b).[]{data-label="fig2"}](fig2){width="80.00000%"}
In principle, this qualitative feature of the permutation entropy can be applied to experimental time series to assess whether the dynamics is dominated by noise or by the deterministic dynamics.
We remind that this distinction can not be done for the block entropy, as in this case $H_D-H_{D-1}$ is necessarily a decreasing function of $D$ (see e.g. [@shannon48; @1056823]). This fundamental difference between permutation entropy and block entropy can be appreciated by comparing the left and right panels of Fig. (\[fig2\]).
For the analysis of the Rossler data, we considered the Poincaré map $X=0$, shown in Fig. (\[fig3\]a), and analyzed the sequence of time-intervals between consecutive crossings. Figure (\[fig3\]b) shows the difference $H_D-H_{D-1}$ vs $D$, for different values of $\alpha$. The solid line indicates the expected value if all ordinal patterns were equally probable, $H_D-H_{D-1} =\ln D!-\ln (D-1)!$. Because of the high level of stochasticity, we calculate the confidence interval that is consistent with the null hypothesis of equally probable ordinal patterns: in Fig. (\[fig3\]b) the gray region represents the expected value $\pm 3 \sigma$, where $\sigma$ is the standard deviation calculated for a hundred surrogated (shuffle) time-series.
![(a) Rössler attractor and Poincaré section in $X = 0$. (b) Permutation entropy difference, $H_D-H_{D-1}$, vs the dimension of the ordinal patterns, $D$, for noise strength $\alpha
= [0$ (star), $0.8 $ (triangle), $ 1.6 $ (inverted triangle), $2.4$ (circle), $ 3.2 $ (pentagon), $ 4$ (square)$]$. The gray region indicates the values of $H_D-H_{D-1}$ that are consistent with equally probable ordinal patterns (see text for details). For the smallest value of alpha, $H_D-H_{D-1}$ shows a non-monotonic behavior, while for higher values of the noise strength, $H_D-H_{D-1}$ grows monotonically with $D$.[]{data-label="fig3"}](fig3-a "fig:"){width="49.50000%"} ![(a) Rössler attractor and Poincaré section in $X = 0$. (b) Permutation entropy difference, $H_D-H_{D-1}$, vs the dimension of the ordinal patterns, $D$, for noise strength $\alpha
= [0$ (star), $0.8 $ (triangle), $ 1.6 $ (inverted triangle), $2.4$ (circle), $ 3.2 $ (pentagon), $ 4$ (square)$]$. The gray region indicates the values of $H_D-H_{D-1}$ that are consistent with equally probable ordinal patterns (see text for details). For the smallest value of alpha, $H_D-H_{D-1}$ shows a non-monotonic behavior, while for higher values of the noise strength, $H_D-H_{D-1}$ grows monotonically with $D$.[]{data-label="fig3"}](fig3-b "fig:"){width="49.50000%"}
Before testing the method in experimental data we want to investigate how the choice of the Poincaré section influences the results. We consider a Poincaré section in the plane $ Z = \beta $, as shown in Fig. (\[fig4\]a), and varying $\beta$ in the range $[0.05-26.7]$, for a fixed value of $\alpha = 0$. In this case, to discretize the time series, we analyze the time values when the trajectory intersects the Poincaré section and $Z$ grows.
Figure (\[fig4\]b) displays the difference $H_D-H_{D-1}$ vs. $D$, for different values of $\beta$. We can see that the difference $H_D -
H_{D-1}$ increases with $\beta$. This is due to the fact that, as $\beta$ is increased, consecutive values in the time-series become increasingly uncorrelated, similarly to when increasing the noise strength. On the contrary, for the minimum value of $\beta$, the variation of $H_D -
H_{D-1}$ with $D$ is resemblant to the behavior under almost-deterministic observed in Fig. (\[fig3\]b).
![(a) Rössler attractor and Poincaré section placed in $ z
= \beta $. (b) Permutation entropy difference, $H_D-H_{D-1}$, vs the dimension of the ordinal patterns, $D$, for $\beta=0.05$ (stars), $\beta= 6.7$ (triangles), $\beta=13.4$ (inverted triangles), $\beta= 20.0$(circles), $\beta= 26.7$ (pentagons). The behavior is qualitatively similar to the one observed in Fig. (\[fig3\]b).[]{data-label="fig4"}](fig4-a "fig:"){width="49.50000%"} ![(a) Rössler attractor and Poincaré section placed in $ z
= \beta $. (b) Permutation entropy difference, $H_D-H_{D-1}$, vs the dimension of the ordinal patterns, $D$, for $\beta=0.05$ (stars), $\beta= 6.7$ (triangles), $\beta=13.4$ (inverted triangles), $\beta= 20.0$(circles), $\beta= 26.7$ (pentagons). The behavior is qualitatively similar to the one observed in Fig. (\[fig3\]b).[]{data-label="fig4"}](fig4-b "fig:"){width="49.50000%"}
Finally, we analyze experimental data from the laser output intensity, displayed in Fig. (\[fig5\]a). To discretize the data we consider the thresholds indicated with horizontal lines in Fig. (\[fig5\]a), and analyzed the time intervals between consecutive threshold-crossings [@tiana_pra_2010; @aragoneses_2013; @taciano_optics_express_2015]. Figure (\[fig5\]b) displays the difference $H_D-H_{D-1}$ vs. $D$, for different thresholds. Note that $H_D-H_{D-1}$ varies with the threshold in a similar way as in Fig. (\[fig4\]b): as the threshold decreases, correlations between consecutive dropouts are lost.
For all the thresholds, $H_D-H_{D-1}$ grows monotonically with $D$. The reason is that the empirical time series is very noisy and the “almost-deterministic” regime is not seen, not even for the highest threshold. Nevertheless, the values of $H_D-H_{D-1}$ lie outside the gray region that indicates values consistent with equally probable ordinal patterns. This reveals that the sequence of intensity dropouts are not completely uncorrelated, and thus, this method can determine regularities also in very noisy data.
![(a) Experimentally recorded time-series for the output intensity of a semiconductor laser, which operates in the low-frequency fluctuations (LFFs) regime, induced by self time-delayed optical feedback. The horizontal lines indicate the thresholds used to detect the dropout times. (b) Permutation entropy difference, $H_D-H_{D-1}$, vs the dimension of the ordinal patterns, for different thresholds: $-0.5$ (stars), $-2$ (inverted triangles) and $-4$ (pentagons).[]{data-label="fig5"}](fig5-a "fig:"){width="49.50000%"} ![(a) Experimentally recorded time-series for the output intensity of a semiconductor laser, which operates in the low-frequency fluctuations (LFFs) regime, induced by self time-delayed optical feedback. The horizontal lines indicate the thresholds used to detect the dropout times. (b) Permutation entropy difference, $H_D-H_{D-1}$, vs the dimension of the ordinal patterns, for different thresholds: $-0.5$ (stars), $-2$ (inverted triangles) and $-4$ (pentagons).[]{data-label="fig5"}](fig5-b "fig:"){width="49.50000%"}
Conclusion
==========
We have studied the influence of noise in the permutation entropy of dynamical systems, considering both, simulated data and experimental data. In the simulated data, when increasing the noise strength, a transition between a almost-deterministic regime and a noise-dominated regime was clearly observed. The noise value at which this transition occurs is roughly independent of the size $D$ of the ordinal pattern.
In the almost-deterministic regime, the permutation entropy grows almost linearly or sub-linearly with $D$. This behavior is qualitatively similar to that of the block entropy. However, to observe a quantitative equivalence it is often needed to analyze extremely long time series, which can be computationally unfeasible even for relatively simple dynamical systems. In the noise-dominated regime, the growth is faster than linear, i.e. the differences $H_D-H_{D-1}$ increase with $D$. In principle, this fact can be used to determine whether the dynamics is in a noise-dominated or a almost-deterministic regime from an experimental time series where the noise strength can not be externally tuned. However, care must be taken in interpreting the results, as extracting a one-dimensional time series from a purely deterministic high-dimensional time series via a Poincaré map can lead to ordinal patterns which look effectively noisy, as we demonstrated in the example of the Rössler systems. This fact reflects the well-known difficulties of distinguishing deterministic dynamics from noise when dealing with high-dimensional systems [@cencini_2000].
Acknowledge
===========
This work was supported by grants EOARD FA9550-14-1-0359, Ministerio de Ciencia e Innovacion, Spain and FEDER, FIS2012-37655-C02-01 and the Marie Curie Initial Training Network NETT, FP7-PEOPLE-2011-ITN 289146.
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[^1]: [[email protected]]([email protected])
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---
abstract: 'We study a continuous-time financial market with continuous price processes under model uncertainty, modeled via a family $\mathcal{P}$ of possible physical measures. A robust notion ${\rm NA}_{1}(\mathcal{P})$ of no-arbitrage of the first kind is introduced; it postulates that a nonnegative, nonvanishing claim cannot be superhedged for free by using simple trading strategies. Our first main result is a version of the fundamental theorem of asset pricing: ${\rm NA}_{1}(\mathcal{P})$ holds if and only if every $P\in\mathcal{P}$ admits a martingale measure which is equivalent up to a certain lifetime. The second main result provides the existence of optimal superhedging strategies for general contingent claims and a representation of the superhedging price in terms of martingale measures.'
author:
- 'Sara Biagini [^1]'
- 'Bruno Bouchard [^2]'
- 'Constantinos Kardaras [^3]'
- 'Marcel Nutz [^4]'
title: Robust Fundamental Theorem for Continuous Processes
---
*Keywords* Fundamental Theorem of Asset Pricing; Arbitrage of the First Kind; Superhedging duality; Nondominated Model
*AMS 2010 Subject Classification* 91B25; 60G44; 93E20
Introduction
============
We consider a financial market where stocks are traded in continuous time. The (discounted) stock price process $S$ is assumed to be continuous, but its distribution in the sense of a stochastic model is not necessarily known. Rather, the market is considered under a family ${\mathcal{P}}$ of probability measures: each $P\in{\mathcal{P}}$ is understood as a possible model for the real-world dynamics of $S$. Two fundamental questions are studied in this context: the absence of arbitrage and its relation to linear pricing rules (fundamental theorem of asset pricing), and the range of arbitrage-free prices (superhedging theorem).
We introduce a robust notion of market viability, called no-arbitrage of the first kind and denoted ${{\rm NA}_{1}(\mathcal{P})}$. Given a contingent claim $f\geq 0$ at maturity $T$, let $v^{{\rm simp}}(f)$ be the minimal initial capital necessary to superhedge $f$ simultaneously under all models $P\in{\mathcal{P}}$, $$v^{{\rm simp}}(f) { := }\inf\big\{x:\,\exists\, H \mbox{ with } x+ H{\stackrel{\mbox{\tiny$\bullet$}}{}}S_{T} \geq f\;P{\mbox{-a.s.}}\mbox{ for all }P\in{\mathcal{P}}\big\}.$$ In the above, we allow only simple trading strategies $H$, so that there are no limitations related to defining the stochastic integral $H{\stackrel{\mbox{\tiny$\bullet$}}{}}S$—no semimartingale assumption is made. Our condition ${{\rm NA}_{1}(\mathcal{P})}$ then postulates that $$v^{{\rm simp}}(f)=0\quad \mbox{implies} \quad f=0\:\: P{\mbox{-a.s.}}\mbox{ for all }P\in{\mathcal{P}}.$$ To state the same in reverse, the price $v^{{\rm simp}}(f)$ should be strictly positive if $P \{f>0 \} >0$ holds for some $P\in{\mathcal{P}}$. This condition corresponds to [@Kardaras.10 Definition 1.1] when ${\mathcal{P}}$ is a singleton; it will turn out to be a notion of market viability that is well suited for model uncertainty in continuous time.
The main goal of the fundamental theorem is to deduce the existence of martingale measures, or linear pricing rules, from the absence of arbitrage opportunities. In the classical case [@DalangMortonWillinger.90; @DelbaenSchachermayer.94], this measure is equivalent to the physical measure $P$. In the case of model uncertainty in a discrete-time market, the fundamental theorem of [@BouchardNutz.13] yields a family ${\mathcal{Q}}$ of martingale measures such that each $P\in{\mathcal{P}}$ is dominated by a martingale measure; the families ${\mathcal{P}}$ and ${\mathcal{Q}}$ are equivalent in the sense that they have the same polar sets. In the present setting with continuous processes, we find a result which is stronger in the sense that each $P$ admits an equivalent martingale measure $Q$. On the other hand, equivalence needs to be defined in a weaker way: it is necessary to allow for a loss of mass in our martingale deflators; thus, the measures $Q$ may allocate mass outside the support of ${\mathcal{P}}$. As a result, the equivalence of measures holds only up to a random time $\zeta$, and so does the martingale property. More precisely, we suppose that our model is set on a canonical space $\Omega$ of paths which are continuous before possibly jumping to a cemetery state, and $\zeta$ is the time of this jump. This “lifetime” is infinite and thus invisible under all $P\in{\mathcal{P}}$, but may be finite under some $Q\in{\mathcal{Q}}$. With these notions in place, our version of the fundamental theorem then states that ${{\rm NA}_{1}(\mathcal{P})}$ holds if and only if for every $P\in{\mathcal{P}}$ there exists a local martingale measure $Q$ such that $Q$ and $P$ are equivalent prior to $\zeta$. See Definition \[defn:ELMM\] and Theorem \[thm: FTAP\] for the precise statements.
A related setting is considered in [@DolinskySoner.12] where $S$ is the canonical process in the space of continuous paths. Roughly speaking, the market model considered there corresponds to declaring all paths to be possible for the stock price, or including all measures in ${\mathcal{P}}$. There is, then, no necessity for a definition of arbitrage; in some sense, the absence of the latter is implicit in the fact that all paths are possible. Nevertheless, the duality result stated in [@DolinskySoner.12] implies a conclusion in the direction of the fundamental theorem; namely, it follows that there must exist at least one martingale measure under the conditions of that result. A similar result on Skorokhod space is reported in [@DolinskySoner.14]. We also refer to [@DavisHobson.07] for a discussion of different notions of arbitrage in the context of traded options. For versions of the robust fundamental theorem for discrete-time frictionless markets, see [@AcciaioBeiglbockPenknerSchachermayer.12; @BouchardNutz.13; @BurzoniFrittelliMaggis.14; @Riedel.11]; for discrete-time markets with transaction costs, see [@BayraktarZhang.13; @BayraktarZhangZhou.13; @BouchardNutz.14trans; @DolinskySoner.13].
The second main result of the present paper is a superhedging theorem in our setting. Assume that ${{\rm NA}_{1}(\mathcal{P})}$ holds and let $f\geq0$ be a contingent claim, measurable at time $T$. Then, we establish the duality $$\begin{gathered}
\sup_{Q\in{\mathcal{Q}}} E^Q[f{\mathbf{1}}_{\zeta>T}] \\
=\inf\big\{x:\,\exists\, H \mbox{ with } x+ H{\stackrel{\mbox{\tiny$\bullet$}}{}}S_{T} \geq f\;P{\mbox{-a.s.}}\mbox{ for all }P\in{\mathcal{P}}\big\};\end{gathered}$$ moreover, we construct an optimal superhedging strategy $H$—naturally, this necessitates continuous trading. See Theorem \[th:duality\] for the precise statement.
The line of argument in the proof is similar to [@Nutz.14] where it is assumed that ${\mathcal{P}}$ consists of martingale measures in the first place. In the present case, the martingale property holds only prior to $\zeta$ which necessitates a number of additional considerations. Generally speaking, the superhedging theorem is fairly well studied in the situation where ${\mathcal{P}}$ consists of martingale measures; cf. [@BouchardMoreauNutz.12; @DenisMartini.06; @FernholzKaratzas.11; @NeufeldNutz.12; @NutzSoner.10; @NutzZhang.12; @Peng.10; @PossamaiRoyerTouzi.13; @SonerTouziZhang.2010rep; @SonerTouziZhang.2010dual], among others, or when all paths are considered possible for the stock and options are also traded; see, e.g., [@CoxObloj.11; @DavisHobson.07; @DolinskySoner.12; @DolinskySoner.14; @GalichonHenryLabordereTouzi.11; @Hobson.98]. We also refer to [@AcciaioBeiglbockPenknerSchachermayer.12; @BayraktarZhangZhou.13; @BouchardNutz.13; @DolinskySoner.13; @Nutz.13] for discrete-time markets. Finally, in the forthcoming independent work [@CheriditoKupperTangpi.14], absence of a duality gap will be established by functional analytic methods in a market more general than ours, under a condition that is stronger than ${{\rm NA}_{1}(\mathcal{P})}$.
The remainder of this paper is organized as follows. The setup is detailed in Section \[sec: set up\], where we also define ${{\rm NA}_{1}(\mathcal{P})}$. In Section \[sec : FTAP\], we discuss our version of the fundamental theorem of asset pricing. Section \[sec: dyna prog and local mart\] provides some technical results on prior-to-$\zeta$ equivalent martingale measures; these are used in Section \[sec: superhedging duality\], where we study the superhedging theorem. Finally, the Appendix collects auxiliary results on Föllmer’s exit measure and the particular path space that are used in the body of this paper.
Setup {#sec: set up}
=====
Measurable Space and Model Uncertainty {#subsec: prelims}
--------------------------------------
We first construct the underlying measurable space $(\Omega,{\mathcal{F}})$ used throughout the paper. Let $E$ be a Polish space and let $d_E$ be a complete metric consistent with the topology of $E$. Adjoining an isolated “cemetery” state $\triangle$, we shall work with $\bar E:=E\cup \{\triangle\}$. It is easy to see that $\bar E$ is again a Polish space under the metric $$d_{\bar E}(x,y):= 1\wedge d_{E}(x,y) {\mathbf{1}}_{\{\triangle \notin\{x,y\}\}}+{\mathbf{1}}_{\{\triangle \in\{x,y\}\}\cap \{x\ne y\}},\quad x,y\in\bar E.$$ We then define $\Omega$ to be the space of all paths $\omega : {\mathbb{R}}_{+} \to \bar E$ which start at a given point $x_{*}\in E$, are càdlàg on $[0,\zeta(\omega))$ and constant on $[\zeta(\omega),\infty)$, where $$\zeta(\omega):=\inf\{t\geq0: \,\omega_{t}=\triangle\}$$ is the “lifetime” of $\omega$. The function $\zeta$ takes values in $(0,\infty]$ since $x_{*}\in E$ and the paths are right-continuous. It is shown in Lemma \[le:OmegaPolish\] in the Appendix that $\Omega$ carries a natural Polish topology.
We denote by $B=(B_{t})_{t\in{\mathbb{R}}_{+}}$ the canonical process, defined by $B_{t}(\omega)=\omega_{t}$, and by ${\mathbb{F}}=({\mathcal{F}}_{t})_{t\in{\mathbb{R}}_{+}}$ its the natural filtration, ${\mathcal{F}}_{t}=\sigma(B_{s},\,s\leq t)$, and finally ${\mathcal{F}}=\sigma(B_{s},\,s\in{\mathbb{R}}_{+})$. The set of ${\mathbb{F}}$-stopping times is denoted by ${\mathcal{T}}$. The minimal right-continuous filtration containing ${\mathbb{F}}$ is denoted by ${\mathbb{F}}_+ = ({\mathcal{F}}_{t+})_{{{t \in {\mathbb R}_+}}}$, while ${\mathcal{T}}_+$ is the set of all ${\mathbb{F}}_+$-stopping times. With these notions in place, we observe that $ \{ \zeta \leq t \} = \{B(t) = \triangle \} \in {\mathcal{F}}_t$ for all ${{t \in {\mathbb R}_+}}$ and hence that $\zeta \in {\mathcal{T}}$.
To represent model uncertainty, we shall work with a (nonempty) family ${\mathcal{P}}$ of probability measures on $(\Omega,{\mathcal{F}})$, rather than a single measure. Each element $P\in{\mathcal{P}}$ is interpreted as a possible model for the real-world dynamics; no domination assumption is made. We say that a property holds ${\mathcal{P}}$-quasi-surely (or ${\mathcal{P}}$-q.s.) if it holds $P$-a.s. for all $P\in {\mathcal{P}}$. We shall assume throughout that $$\zeta = \infty \quad { {\mathcal{ P}}}{\mbox{-q.s.}}$$ Thus, the cemetery state is actually invisible under the real-world models; its role will be to absorb the residual mass of certain martingale measures.
Given a $\sigma$-field ${\mathcal{G}}\subseteq {\mathcal{F}}$, we denote by ${\mathbf{L}^0_+}({\mathcal{G}})$ the set of all $[0, \infty]$-valued, ${\mathcal{G}}$-measurable random variables that are ${ {\mathcal{ P}}}$-q.s. finite.
Trading and Arbitrage
---------------------
The tradable assets are modeled by an ${\mathbb{R}}^{d}$-valued, ${\mathbb{F}}$-adapted and right-continuous process $S : {\mathbb{R}}_{+}\times\Omega \to {\mathbb R}^d$ such that $$\mbox{the paths of $S$ are ${ {\mathcal{ P}}}$-q.s.\ continuous.}$$ No other assumption is made on $S$ at this stage; in particular, no semimartingale property is assumed. However, structural properties will follow later as a consequence of our no-arbitrage condition.
A simple predictable[^5] strategy is a process $H = \sum_{i = 1}^n h_i {\mathbf{1}}_{{]\kern-0.15em] \tau_{i-1}, \tau_i ]\kern-0.15em]}}$, where $h_i = (h_i^j)_{j\le d}$ is ${\mathcal{F}}_{\tau_{i-1}+}$-measurable for all $i\le n$, and $(\tau_i)_{i\le n}$ is a nondecreasing ${\mathcal{T}}_+$-valued sequence with $\tau_0 = 0$. Given an initial capital $x \in {\mathbb R}_{+}$ and a simple predictable strategy $H$, we define the associated wealth process $$X^{x, H} = x + H{\stackrel{\mbox{\tiny$\bullet$}}{}}S = x + \sum_{i=1}^n \sum_{j=1}^d h_i^j {\left(S^j_{\tau_i \wedge \cdot} - S^j_{\tau_{i-1} \wedge \cdot}\right)}.$$ Moreover, we define ${\mathcal{H}}^{\rm simp}(x)$ as the class of all simple predictable processes $H$ such that $X^{x, H}$ remains nonnegative ${ {\mathcal{ P}}}$-q.s. (The superscript “${\rm simp}$” acts as a mnemonic for “simple” in what follows.) Given $T \in {\mathbb R}_+$ and $f \in {\mathbf{L}^0_+}({\mathcal{F}}_T)$, let $$v^{{\rm simp}}(T, f) { := }\inf {\left\{ x \in {\mathbb R}_+ : \, \exists H \in {{\mathcal{H}}^{\rm simp}}(x) \text{ with } X^{x, H}_T \geq f \: { {\mathcal{ P}}}\text{-q.s.}\right\}}$$ be the superhedging price of the claim $f$ over the class of simple strategies. We can then introduce our notion of no-arbitrage of the first kind, stating that the superhedging price is null if and only if the claim is null ${\mathcal{P}}$-q.s.
\[def: NA1(Pc)\] We say that ${{\rm NA}_{1}(\mathcal{P})}$ holds if $$\forall T \in {\mathbb R}_+ \text{ and } f \in {\mathbf{L}^0_+}({\mathcal{F}}_T), \quad v^{{\rm simp}}(T, f) = 0 \;\Longrightarrow\; f = 0 \; { {\mathcal{ P}}}\text{-q.s.}$$
This condition coincides with [@Kardaras.10 Definition 1.1] when ${\mathcal{P}}$ is a singleton.
Fundamental Theorem of Asset Pricing {#sec : FTAP}
====================================
In order to state our version of the fundamental theorem of asset pricing, we first need to introduce the concept of prior-to-$\zeta$ equivalence.
\[def : prior to zeta abs cont and equiv\] Given two measures $P$ and $Q$ on $(\Omega, {\mathcal{F}})$, we say that $Q$ is *prior-to-$\zeta$ absolutely continuous* with respect to $P$, if $Q\ll P$ holds on the space ${\left({\left\{t < \zeta\right\}}, {\mathcal{F}}_t \cap {\left\{t < \zeta\right\}}\right)}$ for all ${{t \in {\mathbb R}_+}}$. This relation is denoted by $Q {\ll_{\zeta}}P$. If $Q {\ll_{\zeta}}P$ and $P {\ll_{\zeta}}Q$, we say that $P$ and $Q$ are *prior-to-$\zeta$ equivalent* and denote this fact by $Q {\sim_{\zeta}}P$.
In this definition, equivalence is used in the sense of unnormalized measures. Namely, even if the measures are probabilities on $(\Omega,{\mathcal{F}})$, they need not be probabilities on ${\left({\left\{t < \zeta\right\}}, {\mathcal{F}}_t \cap {\left\{t < \zeta\right\}}\right)}$, and $Q {\sim_{\zeta}}P$ does not mean that $P (A) = 1$ implies $Q (A) = 1$, even if $A\in {\mathcal{F}}_t \cap {\left\{t < \zeta\right\}}$. A second remark is that local (on ${\mathcal{F}}_t$, for all ${{t \in {\mathbb R}_+}}$) equivalence of two probabilities trivially implies prior-to-$\zeta$ equivalence, but the converse fails. The following simple example demonstrates these phenomena.
Suppose that $E$ is a singleton. Then, ${\mathbb{F}}$ is the smallest filtration that makes $\zeta$ a stopping time and ${\mathcal{F}}_t \cap {\left\{t < \zeta\right\}} = {\left\{\emptyset, {\left\{t < \zeta\right\}}\right\}}$ holds for all ${{t \in {\mathbb R}_+}}$. It follows that prior-to-$\zeta$ equivalence for any two probabilities $P$ and $Q$ on $(\Omega, {\mathcal{F}})$ is tantamount to checking that $P\{\zeta > t\} > 0$ if and only if $ Q\{\zeta > t\} > 0$, for all ${{t \in {\mathbb R}_+}}$. On $(\Omega, {\mathcal{F}})$, one can prescribe probabilities endowing any given law to $\zeta$; letting $P$ be such that $P\{\zeta < \infty\} = 0$ and $Q$ be such that $Q\{\zeta > t\} = \exp(-t)$ for ${{t \in {\mathbb R}_+}}$, it follows that $P$ is a probability on ${\left({\left\{t < \zeta\right\}}, {\mathcal{F}}_t \cap {\left\{t < \zeta\right\}}\right)}$ for all $t \in (0, \infty)$, while $Q$ is a strict sub-probability. Note also that the probabilities $P$ and $Q$ fail to be equivalent on ${\mathcal{F}}_t$ whenever $t \in (0, \infty)$; indeed, $P\{\zeta \leq t\} = 0$ and $Q\{\zeta \leq t\} > 0$ hold for all $t \in (0, \infty)$.
We refer to Section \[sec: follmer measure\] for further discussions on prior-to-$\zeta$ equivalence and proceed with the relevant concept of a local martingale measure.
\[defn:ELMM\] Fix $P \in { {\mathcal{ P}}}$. A probability $Q$ on $(\Omega, {\mathcal{F}})$ is a *prior-to-$\zeta$ equivalent local martingale measure corresponding to $P$* if $Q {\sim_{\zeta}}P$ and there exists a nondecreasing sequence $({\tau}_n)_{{{n \in {\mathbb N}}}}\subset {\mathcal{T}}_+$ such that
1. $\tau_n < \zeta$ for all ${{n \in {\mathbb N}}}$ and ${\lim_{n \to \infty}}\tau_n = \zeta$ hold $Q$-a.s.,
2. $(S_{t\wedge \tau_n })_{{{t \in {\mathbb R}_+}}}$ is an $({\mathbb{F}}_+, Q)$-martingale for all ${{n \in {\mathbb N}}}$.
The class of all such probabilities $Q$ will be denoted by ${\mathcal{Q}}^P$.
What follows is the main result of this section, the fundamental theorem of asset pricing. In the present incarnation, it states that the condition ${{\rm NA}_{1}(\mathcal{P})}$ of Definition \[def: NA1(Pc)\] holds if and only if we can find (at least) one prior-to-$\zeta$ equivalent local martingale measure for each possible model $P\in {\mathcal{P}}$.
\[thm: FTAP\] Condition $
{{\rm NA}_{1}(\mathcal{P})}$ holds if and only if ${\mathcal{Q}}^P \neq \emptyset$ for all $P \in { {\mathcal{ P}}}$.
We emphasize that this result necessitates the continuity of $S$; it is to be compared to the discrete-time case of [@BouchardNutz.13]. The following is a direct consequence of the theorem, but will actually be established in the course of its proof.
\[co:FTAP\] Let ${{\rm NA}_{1}(\mathcal{P})}$ hold. Then $S$ is a semimartingale under each $P\in {\mathcal{P}}$.
To be precise, we should indicate a filtration in the above statement. In fact, the $P$-semimartingale property holds equivalently in any of the filtrations ${\mathbb{F}}$, ${\mathbb{F}}_{+}$ or ${\mathbb{F}}_{+}^{P}$ (the $P$-augmentation of ${\mathbb{F}}_{+}$), or more generally in any intermediate filtration ${\mathbb{F}}\subset {\mathbb{G}}\subset {\mathbb{F}}_{+}^{P}$; see, e.g., [@NeufeldNutz.13a Proposition 2.2]. We shall use this fact in Section \[sec: superhedging duality\].
*Step 1.* We first prove the easy implication; that is, we assume that ${\mathcal{Q}}^P \neq \emptyset$ for all $P\in{\mathcal{P}}$. Fix $T \in {\mathbb R}_+$ and $f \in {\mathbf{L}^0_+}({\mathcal{F}}_T)$ with $v^{{\rm simp}}(T, f) = 0$. Moreover, let $P\in{\mathcal{P}}$ be arbitrary but fixed; we need to show that $f=0$ $P$-a.s.
Indeed, let ${\mathcal{X}}^{{\rm simp}}$ be the class of all processes of the form $X^{x, H}$ for $x \in {\mathbb R}_+$ and $H \in {\mathcal{H}}^{\rm simp}(x)$. By assumption, there exists some $Q\in {\mathcal{Q}}^{P}$. Let $(\tau_n)_{{{n \in {\mathbb N}}}}$ be the localizing sequence appearing in Definition \[defn:ELMM\]. Since the stopped process $S_{\cdot \wedge \tau_n}$ is a $Q$-martingale, it follows that $X_{\cdot \wedge \tau_n}$ is a local $Q$-martingale for all $X \in {\mathcal{X}}^{{\rm simp}}$ and ${{n \in {\mathbb N}}}$. A straightforward argument then shows that $X {\mathbf{1}}_{[\![0, \zeta[\![}$ is a $Q$-supermartingale for all $X \in{\mathcal{X}}^{{\rm simp}}$.
Let $X^n\in {\mathcal{X}}^{{\rm simp}}$ be such that $X_0^n = 1/n$ and $X^n_T \geq f$ ${ {\mathcal{ P}}}$-q.s., then the above supermartingale property yields that $$E^Q [f {\mathbf{1}}_{T < \zeta}] \leq E^Q[X^n_T {\mathbf{1}}_{T < \zeta}] \leq E^Q[X^n_0 ] = 1/n,\quad n\geq1.$$ Therefore, $E^Q [f {\mathbf{1}}_{T < \zeta}] = 0$ which implies that $Q \{f > 0, T < \zeta\} = 0$. Since $Q {\sim_{\zeta}}P$ and $\zeta=\infty$ $P$-a.s., it follows that $P \{f > 0\} = 0$. This completes the proof of the “if” implication in Theorem \[thm: FTAP\].
*Step 2.* The converse implication will be established through a third equivalent condition. To this end, consider ${{\rm NA}_{1}}(P):={{\rm NA}_{1}}(\{P\})$ for a fixed $P\in{\mathcal{P}}$; that is, the condition that $$\forall T \in {\mathbb R}_+ \text{ and } f \in {\mathbf{L}^0_+}({\mathcal{F}}_T), \quad v^{{\rm simp},P}(T, f) = 0 \;\Longrightarrow\; f = 0 \; P \text{-a.s.},$$ where $$v^{{\rm simp},P}(T, f) = \inf \big\{ x \in {\mathbb R}_+ : \, \exists H \in {\mathcal{H}}^{{\rm simp},P}(x) \text{ with } X^{x, H}_T \geq f \, P \text{-a.s.}\big\}$$ and ${\mathcal{H}}^{{\rm simp},P}(x)$ is the class of all simple predictable processes $H$ such that $X^{x, H}$ is nonnegative $P$-a.s. We claim that $$\label{eq: na_1_easy}
\text{${{\rm NA}_{1}(\mathcal{P})}$ holds~~~~if and only if~~~~${{\rm NA}_{1}}(P)$ holds for all $P \in { {\mathcal{ P}}}$.}$$ Indeed, the observation that ${\mathcal{H}}^{\rm simp}(x) \subseteq {\mathcal{H}}^{{\rm simp},P}(x)$ shows that the validity of ${{\rm NA}_{1}}(P)$ for all $P \in { {\mathcal{ P}}}$ implies ${{\rm NA}_{1}(\mathcal{P})}$. To see the converse, suppose that there exists $P \in { {\mathcal{ P}}}$ such that ${{\rm NA}_{1}}(P)$ fails. Then, there are $T \in {\mathbb R}_+$ and $g \in {\mathbf{L}^0_+}({\mathcal{F}}_T)$ such that $v^{{\rm simp},P}(T, g) = 0$ and $P\{ g > 0\}>0$. That is, for any ${{n \in {\mathbb N}}}$ there exists $H^n \in {\mathcal{H}}^{{\rm simp},P} (1/n)$ such that $X^{1/n, H^n}_T \geq g$ $P$-a.s. Define $$\tau^n = \inf \big \{ {{t \in {\mathbb R}_+}}:\, X^{1/n, H^n}_t < 0 \big \}\in {\mathcal{T}}_+,\quad G^{n} = H^n {\mathbf{1}}_{{]\kern-0.15em] 0, \tau^n ]\kern-0.15em]}}.$$ Then $\tau^n \in {\mathcal{T}}_+$ as the paths of $S$ are right-continuous and thus $G^{n}$ is a simple predictable strategy. Since $\tau^{n}=\infty$ $P$-a.s., we have $G^{n} = H^n$ $P$-a.s.; in particular, $G^{n}$ still satisfies $X^{1/n, G^n}_T \geq g$ $P$-a.s. In addition, the definition of $\tau^{n}$ guarantees that $X^{1/n, G^n}$ is nonnegative ${\mathcal{P}}$-q.s.—the continuity of $S$ is crucial in this step. Consider $$f := \inf_{{{n \in {\mathbb N}}}} X^{1/n, G^n}_T \in {\mathbf{L}^0_+}({\mathcal{F}}_T)$$ and note that $v^{{\rm simp}}(T, f) = 0$ holds by definition. Moreover, we have $f \geq g$ $P$-a.s. and thus $P\{f > 0\} > 0$, contradicting ${{\rm NA}_{1}(\mathcal{P})}$. Therefore, has been established.
*Step 3.* In view of , it remains to show that ${{\rm NA}_{1}}(P)$ implies ${\mathcal{Q}}^{P}\neq\emptyset$, for arbitrary but fixed $P\in{\mathcal{P}}$. Thus, we are essentially in the realm of classical stochastic analysis and finance; in particular, we may use the tools in the Appendix as well as [@Kardaras.10; @Kardaras.13].
Define ${\mathcal{X}}^{{\rm simp},P}$ as the class of all processes of the form $X^{x, H}$ for $x \in {\mathbb R}_+$ and $H \in {\mathcal{H}}^{{\rm simp},P}(x)$. The set $\{X \in {\mathcal{X}}^{{\rm simp},P}:\, X_0 = 1 \}$ has the essential properties of [@Kardaras.13 Definition 1.1] needed to conclude that ${\mathcal{X}}^{{\rm simp},P}$ consists of $P$-semimartingales, see [@Kardaras.13 Theorem 1.3], and that (the immediate extension of) condition ${{\rm NA}_{1}}(P)$ is also valid for the closure ${\mathcal{X}}^{P}$ of ${\mathcal{X}}^{{\rm simp},P}$ in the $P$-semimartingale topology; see [@Kardaras.13 Remark 1.10]. In particular, a standard localization and integration argument (using local boundedness of $S$ under $P$) shows that $S$ is itself a $P$-semimartingale.
The set ${\mathcal{X}}^{P}$ coincides with the class of all $P$-a.s. nonnegative stochastic integrals of $S$ under $P$, using general predictable and $S$-integrable integrands. This is seen by using density (in the semimartingale topology) of simple stochastic integrals with respect to general stochastic integrals, as well as a stopping argument which again uses that $S$ has continuous paths $P$-a.s. As a result, using condition ${{\rm NA}_{1}}(P)$ for ${\mathcal{X}}^{P}$, we infer the existence of a strictly positive $({\mathbb{F}}_+, P)$-local martingale $Y$ with $Y_0 = 1$ such that $Y S$ is an $({\mathbb{F}}_+, P)$-local martingale; cf. [@Kardaras.10 Theorem 4]. We can now use Theorem \[thm: foelmeasure\_1\] in the Appendix to construct a probability $Q {\sim_{\zeta}}P$ such that $Y$ is the prior-to-$\zeta$ density of $Q$ with respect to $P$. Using the facts that $Y S$ is an $({\mathbb{F}}_+, P)$-local martingale, $\zeta$ is foretellable under $Q$ (for the latter, see Definition \[defn: foretellable\] and Theorem \[thm: foelmeasure\_1\] in the Appendix) and Remark \[rem:localization\_under\_both\], we can construct the required ${\mathcal{T}}_+$-valued sequence $(\tau_n)_{{{n \in {\mathbb N}}}}$ such that $S_{\cdot \wedge \tau_n}$ is an $({\mathbb{F}}_+, Q)$-martingale for all ${{n \in {\mathbb N}}}$. The last fact translates to $Q \in {\mathcal{Q}}^P$ and concludes the proof.
Dynamic Programming Properties of Prior-to-$\zeta$ Supermartingale Measures {#sec: dyna prog and local mart}
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For our proof of the superhedging theorem in Section \[sec: superhedging duality\], it will be crucial to know that the set of (super-)martingale measures satisfies certain dynamic programming properties. In this section, we impose assumptions on the set ${\mathcal{P}}$ which is the primary object of our model, and show how these properties are inherited by the corresponding set of supermartingale measures.
Additional Assumptions and Notation {#subsec: add condi and notations}
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From now on, we assume that the Polish space $E$ is a topological vector space and that the paths $\omega\in\Omega$ start at the point $x_{*}=0\in E$.
For $x,y\in \bar E$, we use the convention $x+y=\triangle$ if $x=\triangle$ or $y=\triangle$. Let $t\geq0$. Given $\omega,\tilde\omega\in\Omega$, we set $$(\omega\otimes_{t}\tilde \omega)_{s}=\omega_{s}{\mathbf{1}}_{[0,t)}(s)+ (\omega_{t}+\tilde \omega_{s-t}){\mathbf{1}}_{[t,\infty)}(s).$$ Given also a process $Z$, we define $$Z_{s}^{t,\omega}(\tilde \omega):=Z_{t+s}(\omega\otimes_{t}\tilde \omega), \quad s\ge 0;$$ note that a shift in the time variable is part of our definition. We view a random variable $\xi$ as a process which is constant in time, so that $$\xi^{t,\omega}(\tilde \omega):=\xi(\omega\otimes_{t}\tilde \omega).$$ We denote by ${\mathfrak{P}}(\Omega)$ the collection of all probability measures on $\Omega$, equipped with the topology of weak convergence. Given a probability $R \in {\mathfrak{P}}(\Omega)$, we define $R^{t,\omega}$ by $$R^{t,\omega}(A)=R_t^\omega\{\omega \otimes_{t} \tilde \omega: \tilde \omega \in A\}, \quad A\in {\mathcal{F}},$$ where $R_t^\omega$ is a regular conditional distribution of $R$ given ${\mathcal{F}}_{t}$ satisfying $$R_t^\omega \{\omega'\in \Omega: \omega'=\omega\mbox{ on } [0,t]\} =1, \quad \omega\in \Omega.$$ The existence of $R_t^\omega$ is guaranteed by the fact that ${\mathcal{F}}_{t}$ is countably generated; cf. Lemma \[le:OmegaPolish\] and [@StroockVaradhan.79 Theorem 1.1.8 and p.34]. It then follows that $$\label{eq:conditioning}
E^{R^{t,\omega}}[\xi^{t,\omega}]=E^{R^{\omega}_{t}}[\xi]=E^{R}[\xi|{\mathcal{F}}_{t}](\omega)\quad\mbox{for $R$-a.e.\ $\omega\in\Omega$}.$$
We shall assume that our set ${\mathcal{P}}$ admits a family of $(t,\omega)$-conditional models. More precisely, we start with a family $\{{\mathcal{P}}_{t}(\omega):\, t\in{\mathbb{R}}_{+},\,\omega \in \Omega\}$ of subsets of ${\mathfrak{P}}(\Omega)$ which is adapted in the sense that ${\mathcal{P}}_{t}(\omega)={\mathcal{P}}_{t}(\tilde\omega)$ if $\omega|_{[0,t]}=\tilde\omega|_{[0,t]}$. In particular, ${\mathcal{P}}_{0}={\mathcal{P}}_{0}(\omega)$ is independent of $\omega$. We impose the following structural conditions—compare with [@NeufeldNutz.12; @NutzVanHandel.12] in the case $\zeta\equiv \infty$.
\[def : analytic and stable family\] An adapted family $\{{\mathcal{R}}_{t}(\omega):\, t\in{\mathbb{R}}_{+},\,\omega \in \Omega\}$ of subsets of ${\mathfrak{P}}(\Omega)$ is *analytic and stable prior to $\zeta$* if the following hold for all $ t\ge s\ge0$, $\bar \omega \in \Omega$ and $ R\in {\mathcal{R}}_{s}(\bar \omega)$.
1. $\{(R',\omega): \omega \in \Omega, R'\in {\mathcal{R}}_{t}(\omega)\}\subset {\mathfrak{P}}(\Omega)\times \Omega$ is analytic[^6].
2. $R^{t-s,\omega} \in {\mathcal{R}}_{t}(\bar \omega \otimes_{s} \omega)$ for $R$-a.e. $\omega \in \{\zeta^{s,\bar \omega}>t\}$.
3. If $\nu:\Omega\mapsto {\mathfrak{P}}(\Omega)$ is an ${\mathcal{F}}_{t-s}$-measurable kernel and $\nu(\omega)\in {\mathcal{R}}_{t}(\bar \omega\otimes_{s} \omega)$ for $R$-a.e. $\omega \in \{\zeta^{s,\bar \omega}>t\}$, then the measure defined by $$\bar R(A):=\iint ({\mathbf{1}}_{A})^{t-s,\omega}(\omega')\,\nu^{R}(d\omega';\omega)\,R(d\omega),\quad A\in {\mathcal{F}},$$ $$\mbox{where}\quad \nu^{R}(\omega):=\nu(\omega){\mathbf{1}}_{\{\zeta^{s,\bar \omega}>t\}}(\omega) + R^{t-s,\omega}{\mathbf{1}}_{\{\zeta^{s,\bar \omega}\leq t\}}(\omega),$$ belongs to ${\mathcal{R}}_{s}(\bar \omega)$.
Condition (A1) is of technical nature; it will be used for measurable selection arguments. Conditions (A2) and (A3) are natural consistency conditions, stating that the family is stable under “conditioning” and “pasting.”
\[ass : Pc(t,omega)\] We have ${\mathcal{P}}={\mathcal{P}}_{0}$ for a family $\{{\mathcal{P}}_{t}(\omega):\, t\in{\mathbb{R}}_{+},\,\omega \in \Omega\}$ which is analytic and stable prior to $\zeta$. Moreover, $S^{t,\omega}$ is ${\mathcal{P}}_{t}(\omega)$-q.s. continuous prior to $\zeta^{t,\omega}-t$, for all $t\in{\mathbb{R}}_{+}$ and $\omega \in \Omega$.
A canonical example of such a set ${\mathcal{P}}$ is the collection of all laws $P$ of Itô semimartingales $\int_0^\cdot \alpha_u \,du + \int_0^\cdot \sigma_u \,dW_u$, each one situated on its own probability space with a Brownian motion $W$, drift rate $\alpha$ valued in a given measurable set $A\subset {\mathbb{R}}^{d}$, and volatility $\sigma$ such that $\sigma\sigma^{\top}$ is valued in a given measurable set $\Sigma$ of positive definite $d\times d$ matrices. In this case, we can take ${\mathcal{P}}_{t}(\omega)={\mathcal{P}}$ for all $(t,\omega)$ because the sets $A$ and $\Sigma$ are constant; cf. [@NeufeldNutz.13b]. The continuity condition is clearly satisfied for the canonical choice $S=B$ and then ${{\rm NA}_{1}}({\mathcal{P}})$ holds, for instance, when $A$ and $\Sigma$ are compact.
Prior-to-$\zeta$ Supermartingale Measures {#sec: analytic and stable}
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For technical reasons, it will be convenient to work with supermartingale (rather than local martingale) measures in what follows. The purpose of this section is to define a specific family of supermartingale measures satisfying the conditions of Definition \[def : analytic and stable family\]; it will be used to construct the optimal strategy in the superhedging theorem (Theorem \[th:duality\]). We first need to define a conditional notion of prior-to-$\zeta$ absolute continuity.
Let $(t,\omega)\in {\mathbb{R}}_{+}\times\Omega$ and $P,Q\in {\mathfrak{P}}(\Omega)$. We write $Q{\ll_{\zeta^{t,\omega}}}P$ (with some abuse of notation) if $$Q\ll P \quad\mbox{on}\quad {\mathcal{F}}_{s}\cap \{s<\zeta^{t,\omega}-t\},\quad s\in{\mathbb{R}}_{+}.$$
We also need to consider wealth processes conditioned by $(t,\omega)\in{\mathbb{R}}_{+}\times \Omega$. More precisely, let $$\label{eq:defXsimple}
{\mathcal{X}}_{t}^{\rm simp}(\omega) := \big\{1 + (H{\stackrel{\mbox{\tiny$\bullet$}}{}}S^{t,\omega})^{\tau^{n}_{H,S^{t,\omega}}}:\,H\in{\mathcal{H}}^{\rm simp} , \, n\in{\mathbb{N}}\big\},$$ where ${\mathcal{H}}^{\rm simp}$ is the set of all simple predictable processes and $$\tau^{n}_{H,S^{t,\omega}}:=\inf\big\{ s\ge 0 : (H{\stackrel{\mbox{\tiny$\bullet$}}{}}S^{t,\omega})_{s}\notin (-1,n) \big\}.$$ Here the stopping at $-1$ corresponds to the nonnegativity of the wealth process, whereas the stopping at $n$ is merely for technical convenience. The point in this specific definition of ${\mathcal{X}}_{t}^{\rm simp}(\omega)$ is to have a tractable dependence on $\omega$; in this respect, we note that the set ${\mathcal{H}}^{\rm simp}$ is independent of $\omega$.
\[def: super martin abs measures\] Let $(t,\omega)\in {\mathbb{R}}_{+}\times\Omega$ and $P\in {\mathfrak{P}}(\Omega)$. We introduce the sets $$\Pa{\zeta^{t,\omega}}{P}=\{Q\in {\mathfrak{P}}(\Omega):\, Q {\ll_{\zeta^{t,\omega}}}P\},$$ $${\mathcal{Y}}_{t}(\omega)=\big\{ Q\in{\mathfrak{P}}(\Omega):\, X{\mathbf{1}}_{[\![0,\zeta^{t,\omega}-t[\![} \mbox{ is a $Q$-supermartingale $\forall\; X\in{\mathcal{X}}_{t}^{\rm simp}(\omega)$}\big\},$$ $${\mathcal{Q}}_{t}(\omega,P)=\Pa{\zeta^{t,\omega}}{P}\cap {\mathcal{Y}}_{t}(\omega),$$ $${\mathcal{Q}}_{t}(\omega)=\bigcup_{P\in {\mathcal{P}}_{t}(\omega)} {\mathcal{Q}}_{t}(\omega,P).$$ The elements of ${\mathcal{Q}}_{t}(\omega)$ are called *prior-to-$\zeta$ absolutely continuous supermartingale measures* given $(t,\omega)$.
We observe that the family $\{ {\mathcal{Q}}_{t}(\omega):\, t\in{\mathbb{R}}_{+},\,\omega \in \Omega\}$ is adapted. Furthermore, we recall from Theorem \[thm: FTAP\] that ${\mathcal{Q}}_{0}\neq\emptyset$ under ${{\rm NA}_{1}(\mathcal{P})}$. In the rest of this subsection, we show that the family $\{ {\mathcal{Q}}_{t}(\omega)\}$ inherits from $\{ {\mathcal{P}}_{t}(\omega)\}$ the properties of Definition \[def : analytic and stable family\].
\[pr:QsatisfiesCondA\] The family $\{ {\mathcal{Q}}_{t}(\omega)\}$ satisfies [(A1)–(A3)]{}.
The proof is split into the subsequent lemmas. For ease of reference, we first state the following standard result.
\[lem: map omega Q to E is Borel\] Let $A$ be a Borel space and let $(a,\omega)\in A\times \Omega \mapsto \xi(a,\omega)\in {\mathbb{R}}_{+}$ be Borel-measurable. Then, $
(a,R)\in A\times {\mathfrak{P}}(\Omega)\mapsto E^{R}[\xi(a,\cdot)]
$ is Borel-measurable.
See, e.g., Step 1 in the proof of [@NutzVanHandel.12 Theorem 2.3].
\[le:superSimple\] There exist a countable set $\tilde {\mathcal{H}}\subset{\mathcal{H}}^{\rm simp}$ and a countable set $\tilde{\mathcal{T}}\subset {\mathcal{T}}$ of bounded stopping times with the following property:
Given $(t,\omega)\in{\mathbb{R}}_{+}\times \Omega$ and $Q\in{\mathfrak{P}}(\Omega)$ such that $S^{t,\omega}$ is $Q$-a.s. continuous prior to $\zeta^{t,\omega}-t$, we have equivalence between
1. $X{\mathbf{1}}_{[\![0,\zeta^{t,\omega}-t[\![}$ is a $Q$-supermartingale for all $X\in{\mathcal{X}}_{t}^{\rm simp}(\omega)$,
2. $X{\mathbf{1}}_{[\![0,\zeta^{t,\omega}-t[\![}$ is a $Q$-supermartingale for all $X\in\tilde{\mathcal{X}}_{t}(\omega)$,
3. $E^{Q}[X_{\sigma}{\mathbf{1}}_{\sigma < \zeta^{t,\omega}-t}] \geq E^{Q}[X_{\tau}{\mathbf{1}}_{\tau < \zeta^{t,\omega}-t}]$ for $X\in\tilde{\mathcal{X}}_{t}(\omega)$ and $\sigma\leq \tau$ in $\tilde{\mathcal{T}}$,
where $\tilde{\mathcal{X}}_{t}(\omega)$ is defined like but using only integrands $H\in\tilde {\mathcal{H}}$. Moreover, if $S^{t,\omega}{\mathbf{1}}_{[\![0,\zeta^{t,\omega}-t[\![}$ is a semimartingale under $Q$, the above are equivalent to
1. $X{\mathbf{1}}_{[\![0,\zeta^{t,\omega}-t[\![}$ is a $Q$-supermartingale for all $X\in{\mathcal{X}}_{t}(\omega)$,
where ${\mathcal{X}}_{t}(\omega)$ is defined like but using arbitrary predictable integrands.
For each $s\geq 0$, let $\tilde{\mathcal{F}}_{s}$ be a countable algebra generating ${\mathcal{F}}_{s}$; cf. Lemma \[le:OmegaPolish\]. Let $\tilde{\mathcal{T}}$ be the set of all stopping times $$\tau=\sum_{j=1}^{n} t_{j}{\mathbf{1}}_{A_{j}},$$ where $n\in {\mathbb{N}}$, $t_{j}\in{\mathbb{Q}}_{+}$ and $A_{j}\in \tilde{\mathcal{F}}_{t_{j}}$. Moreover, let $\tilde {\mathcal{H}}\subset{\mathcal{H}}^{\rm simp}$ be the set of all processes $$H= \sum_{j=0}^{n} \alpha_{j}{\mathbf{1}}_{]t_{j}, t_{j+1}]},$$ where $n\in {\mathbb{N}}$, $0=t_{0}\leq t_{1}\leq \cdots \leq t_{n}\in{\mathbb{Q}}_{+}$ and each random variable $\alpha_{j}$ is of the form $$\alpha_{j}= \sum_{i=0}^{n} a_{j}^{i}{\mathbf{1}}_{A_{j}^{i}}$$ for some $a_{j}^{i}\in{\mathbb{Q}}^{d}$ and $A_{j}^{i}\in\tilde{\mathcal{F}}_{t_{j}}$.
It is clear that (i)$\Rightarrow$(ii)$\Rightarrow$(iii). To see that (iii) implies (i), fix $Q\in{\mathfrak{P}}(\Omega)$ and $X\in{\mathcal{X}}_{t}^{\rm simp}(\omega)$. We first observe that it suffices to show that
1. $E^{Q}[X_{\sigma}{\mathbf{1}}_{\sigma < \zeta^{t,\omega}-t}] \geq E^{Q}[X_{\tau}{\mathbf{1}}_{\tau < \zeta^{t,\omega}-t}]$ for all $X\in{\mathcal{X}}_{t}^{\rm simp}(\omega)$ and all $\sigma\leq \tau$ in $\tilde{\mathcal{T}}$.
Indeed, since $\tilde{\mathcal{T}}$ contains all stopping times of the form $\tau=u{\mathbf{1}}_{A}+v{\mathbf{1}}_{A^{c}}$ and $\sigma=u$, where $u\leq v\in {\mathbb{Q}}_{+}$ and $A\in\tilde{\mathcal{F}}_{u}$, it readily follows that (i’) implies the supermartingale property of $X{\mathbf{1}}_{[\![0,\zeta^{t,\omega}-t[\![}$ at rational times, and then the supermartingale property on ${\mathbb{R}}_{+}$ follows by right-continuity.
To show that (iii) implies (i’), fix $\sigma\leq\tau$ and let $T\in{\mathbb{R}}_{+}$ be such that $\tau\leq T$. The claim will follow by passing to suitable limits in the inequality $$\label{eq:ineqLimits}
E^{Q}[X_{\sigma}{\mathbf{1}}_{\sigma < \zeta^{t,\omega}-t}] \geq E^{Q}[X_{\tau}{\mathbf{1}}_{\tau < \zeta^{t,\omega}-t}];$$ we confine ourselves to a sketch of the proof. Let $X\in{\mathcal{X}}_{t}^{\rm simp}(\omega)$ be given and recall that $S^{t,\omega}$ is ($Q$-a.s.) continuous prior to $\zeta^{t,\omega}-t$.
Using a stopping argument and monotone convergence, we may reduce to the case where $\bar{X}:=X{\mathbf{1}}_{[\![0,\zeta^{t,\omega}-t[\![}$ is uniformly bounded. Then, using dominated convergence and another stopping argument, we may reduce to the case where $\bar{X}$ is also uniformly bounded away from zero prior to $\zeta^{t,\omega}-t$. Using standard arguments we can find a sequence $(H^{k})$ of simple predictable integrands with deterministic jump times such that $X^{k}:=1+H^{k}{\stackrel{\mbox{\tiny$\bullet$}}{}}S^{t,\omega} \to X$ uniformly on $[\![0,\zeta^{t,\omega}-t[\![$ in $Q$-probability. Using that $X$ is bounded and bounded away from zero, it follows that $$\bar{X}^{k}:=1+(H^{k}{\stackrel{\mbox{\tiny$\bullet$}}{}}S^{t,\omega})^{\tau^{n}_{H^{k},S^{t,\omega}}}{\mathbf{1}}_{[\![0,\zeta^{t,\omega}-t[\![} \to \bar{X}$$ uniformly on $[0,T]$ in $Q$-probability, for a sufficiently large $n\in{\mathbb{N}}$. After an additional approximation, we may obtain the same property with $H^{k}\in\tilde{\mathcal{H}}$, and we may show using dominated convergence that the validity of for each $\bar{X}^{k}$ implies the validity for $\bar{X}$.
If $S^{t,\omega}$ is a semimartingale under $Q$, one shows that (iii) implies (iv) by using similar arguments as well as standard results about stochastic integrals, in particular [@Protter.05 Theorems II.21 and IV.2].
\[lem: A1 holds\] The family $\{ {\mathcal{Q}}_{t}(\omega)\}$ satisfies [(A1)]{}.
Fix $t\geq0$. It suffices to show that the set $$\Gamma:=\{(\omega,P,Q) :\omega\in \Omega,\; P\in {\mathcal{P}}_{t}(\omega),\; Q\in {\mathcal{Q}}_{t}(\omega,P)\} \subset \Omega\times{\mathfrak{P}}(\Omega)\times {\mathfrak{P}}(\Omega)$$ is analytic. Indeed, once this is established, the graph of ${\mathcal{Q}}_{t}(\cdot)$ is analytic as a projection of $\Gamma$; that is, [(A1)]{} is satisfied.
As a first step, we show that $$\label{eq:analyticity2}
\operatorname{graph}(\Pa{\zeta^{t,\cdot}}{\cdot}):=\{(\omega,P,Q ):\, \omega \in \Omega, \;P\in {\mathfrak{P}}(\Omega),\;Q\in \Pa{\zeta^{t,\omega}}{P}\}\quad\mbox{is Borel}$$ and in particular analytic. Indeed, it follows from Lemma \[lem : PXa countable representation\] that $$\Pa{\zeta^{t,\omega}}{P}=\bigcap_{q\in {\mathbb{Q}}_{+}} \Pa{\zeta^{t,\omega}}{P,q},$$ where $$\Pa{\zeta^{t,\omega}}{P,q} := \big\{Q\in{\mathfrak{P}}(\Omega):\, Q\ll P \mbox{ on }{\mathcal{F}}_{q} \cap \{q<\zeta^{t,\omega}-t\}\big\}.$$ Hence, it suffices to show that $$\{(\omega, P,Q)\in \Omega\times{\mathfrak{P}}(\Omega)\times {\mathfrak{P}}(\Omega):\;\, Q\in \Pa{\zeta^{t,\omega}}{P,q}\}$$ is Borel for fixed $q$. Since ${\mathcal{F}}_{q}$ is countably generated, cf. Lemma \[le:OmegaPolish\], a standard argument (see [@DellacherieMeyer.78 Theorem V.58, p.52] and the subsequent remarks) shows that we can construct a Borel function $D_{q}: \Omega \times {\mathfrak{P}}(\Omega)\times {\mathfrak{P}}(\Omega)\to {\mathbb{R}}$ such that $D_{q}(\cdot ,Q,P)$ is a version of the Radon–Nikodym derivative of the absolutely continuous part of $Q$ with respect to $P$ on $ {\mathcal{F}}_{q}$. Then, $Q\in \Pa{\zeta^{t,\omega}}{P,q}$ if and only if $E^{P}[D_{q}( Q,P){\mathbf{1}}_{q<\zeta^{t,\omega}-t}]=Q\{q<\zeta^{t,\omega}-t\}$. Using the fact that $$(\omega,P,Q)\mapsto E^{P}[D_{q}(Q,P){\mathbf{1}}_{q<\zeta^{t,\omega}-t}]-Q\{q<\zeta^{t,\omega}-t\}$$ is Borel by Lemmas \[lem: map omega Q to E is Borel\] and \[le:OmegaPolish\], we conclude that holds.
Let $\sigma,\tau\in{\mathcal{T}}$ and let $X\in{\mathcal{X}}_{t}^{\rm simp}(\omega)$; recall that $X=X^{H}$ is of the form . Then the map $$(\omega,Q)\in \Omega\times {\mathfrak{P}}(\Omega)\mapsto \psi^{H,\sigma,\tau}(\omega,Q):= E^{Q}[X_{\tau}{\mathbf{1}}_{\tau < \zeta^{t,\omega}-t}] - E^{Q}[X_{\sigma}{\mathbf{1}}_{\sigma < \zeta^{t,\omega}-t}]$$ is Borel as a consequence of Lemma \[lem: map omega Q to E is Borel\]. If $(\omega,Q)$ are such that $S^{t,\omega}$ is $Q$-a.s. continuous prior to $\zeta^{t,\omega}-t$, Lemma \[le:superSimple\] shows that $$Q\in {\mathcal{Y}}_{t}(\omega) \quad \mbox{if and only if}\quad \psi^{H,\sigma,\tau}(\omega,Q)\leq0 \;\; \forall H\in\tilde {\mathcal{H}}, \, \sigma\leq\tau\in\tilde{\mathcal{T}}.$$ Using the obvious embeddings of $\operatorname{graph}({\mathcal{P}}_{t})$ and $\operatorname{graph}({\mathcal{Y}}_{t})$ into $\Omega\times{\mathfrak{P}}(\Omega)\times {\mathfrak{P}}(\Omega)$, it follows that $$\begin{aligned}
\Gamma & = \operatorname{graph}({\mathcal{P}}_{t}) \cap \operatorname{graph}(\Pa{\zeta^{t,\cdot}}{\cdot}) \cap \operatorname{graph}({\mathcal{Y}}_{t}) \\
& = \operatorname{graph}({\mathcal{P}}_{t}) \cap \operatorname{graph}(\Pa{\zeta^{t,\cdot}}{\cdot}) \cap \bigcap_{H\in\tilde {\mathcal{H}}, \, \sigma\leq\tau\in\tilde{\mathcal{T}}} \{\psi^{H,\sigma,\tau}\leq 0\}.\end{aligned}$$ Here we have used that if $(\omega,P,Q)$ belong to the first intersection, then $S^{t,\omega}$ is $P$-a.s. and hence $Q$-a.s. continuous prior to $\zeta^{t,\omega}-t$; cf. Assumption \[ass : Pc(t,omega)\]. The above representation shows that $\Gamma$ is analytic as a countable intersection of analytic sets.
The family $\{ {\mathcal{Q}}_{t}(\omega)\}$ satisfies [(A2)]{}.
For simplicity of notation, we state the proof for $s=0$; the extension to the general case is immediate. Fix $Q\in {\mathcal{Q}}_{0}$; then $Q\in {\mathcal{Q}}_{0}(P)$ for some $P\in {\mathcal{P}}$. We shall show that $$ Q^{t,\omega}\in \Pa{\zeta^{t,\omega}}{P^{t,\omega}}\cap {\mathcal{Y}}_{t}(\omega) \quad\mbox{for $Q$-a.e.\ $\omega\in\{\zeta>t\}$};$$ this will imply the lemma because $P^{t,\omega}\in{\mathcal{P}}_{t}(\omega)$ holds for $P$-a.e. $\omega\in\Omega$, cf. Assumption \[ass : Pc(t,omega)\], and thus for $Q$-a.e. $\omega\in\{\zeta>t\}$ as $Q\in {\mathcal{Q}}_{0}(P)$.
Let $Y$ be the prior-to-$\zeta$ density process of $Q$ with respect to $P$ (see Remark \[rem : density process constructed from Q\] for details on this notion) and set $$\tilde Y={\mathbf{1}}_{[0,t)}+(Y/Y_{t}){\mathbf{1}}_{[t,\infty)},$$ where we use the convention $0/0=0$. We first establish that given $s\geq0$, we have $Q^{t,\omega}\ll P^{t,\omega}$ on ${\mathcal{F}}_{s}\cap \{\zeta^{t,\omega}-t>s\}$ and in fact $$dQ^{t,\omega}=\tilde Y_{s}^{t,\omega} dP^{t,\omega}\quad\mbox{on}\quad {\mathcal{F}}_{s}\cap \{\zeta^{t,\omega}-t>s\}$$ for $Q$-a.e. $\omega\in\{\zeta>t\}$. Indeed, let $g\geq0$ be an ${\mathcal{F}}_{s}$-measurable random variable; then there exists an ${\mathcal{F}}_{s+t}$-measurable random variable $\bar g$ such that $\bar g^{t,\omega}=g$. Recalling , we have for $Q$-a.e. $\omega\in\{\zeta>t\}$ that $$\begin{aligned}
E^{Q^{t,\omega}}[g{\mathbf{1}}_{\zeta^{t,\omega}-t>s}]
&=E^{Q}[\bar g{\mathbf{1}}_{\zeta>s+t}|{\mathcal{F}}_{t}](\omega)\\
&=E^{P}[(Y_{s+t}/Y_{t})\bar g{\mathbf{1}}_{\zeta>s+t}|{\mathcal{F}}_{t}](\omega)\\
&=E^{P}[\tilde Y_{s+t}\bar g{\mathbf{1}}_{\zeta>s+t}|{\mathcal{F}}_{t}](\omega)\\
&=E^{P^{t,\omega}}[\tilde Y^{t,\omega}_{s}g{\mathbf{1}}_{\zeta^{t,\omega}-t>s}].
\end{aligned}$$ We have shown in particular that $Q^{t,\omega}\ll P^{t,\omega}$ on ${\mathcal{F}}_{s}\cap \{\zeta^{t,\omega}-t>s\}$ for all $s\in{\mathbb{Q}}_{+}$ holds for $Q$-a.e. $\omega\in\{\zeta>t\}$, which by Lemma \[lem : PXa countable representation\] implies that $$ Q^{t,\omega}\in \Pa{\zeta^{t,\omega}}{P^{t,\omega}} \quad \mbox{for $Q$-a.e.\ $\omega\in \{\zeta>t\}$.}$$ It remains to prove that $$\label{eq:A2proofAim1}
Q^{t,\omega}\in {\mathcal{Y}}_{t}(\omega)\quad \mbox{for $Q$-a.e.\ $\omega\in \{\zeta>t\}$.}$$ Let $X\in {\mathcal{X}}_{t}^{\rm simp}(\omega)$, then we observe that $X=\bar{X}^{t,\omega}$ for some $\bar{X}\in {\mathcal{X}}^{\rm simp}_{0}$. Moreover, let $\sigma\in{\mathcal{T}}$ be bounded, then $\sigma=\bar\sigma^{t,\omega}-t$ for some bounded $\bar\sigma\in{\mathcal{T}}$ satisfying $\bar\sigma\geq t$ (both $\bar{X}$ and $\bar\sigma$ do not depend on $\omega$). We have $X_{\sigma}=(\bar{X}^{t,\omega})_{\bar\sigma^{t,\omega}-t}=(\bar{X}_{\bar\sigma})^{t,\omega}$ (where $\bar{X}_{\bar\sigma}$ is considered as a random variable) and thus $$E^{Q^{t,\omega}}[X_{\sigma}{\mathbf{1}}_{\zeta^{t,\omega}-t>\sigma}]=E^{Q^{t,\omega}}[(\bar{X}_{\bar\sigma})^{t,\omega}{\mathbf{1}}_{\zeta^{t,\omega}>\bar\sigma^{t,\omega}}] = E^{Q}[\bar{X}_{\bar\sigma} {\mathbf{1}}_{\zeta>\bar\sigma}|{\mathcal{F}}_{ t }](\omega)$$ for $Q$-a.e. $\omega\in \{\zeta>t\}$. If $\tau\geq\sigma \in{\mathcal{T}}$ is bounded and $\bar\tau\geq \bar\sigma $ has the obvious meaning, we deduce from the supermartingale property of $Q\in{\mathcal{Y}}_{0}$ that $$\begin{aligned}
E^{Q^{t,\omega}}[X_{\sigma}{\mathbf{1}}_{\zeta^{t,\omega}-t>\sigma}]
&= E^{Q}[\bar{X}_{\bar\sigma} {\mathbf{1}}_{\zeta>\bar\sigma}|{\mathcal{F}}_{t}](\omega) \\
& \geq E^{Q}[\bar{X}_{\bar\tau} {\mathbf{1}}_{\zeta>\bar\tau}|{\mathcal{F}}_{t}](\omega) \\
&= E^{Q^{t,\omega}}[X_{\tau}{\mathbf{1}}_{\zeta^{t,\omega}-t>\tau}]\end{aligned}$$ for $Q$-a.e. $\omega\in \{\zeta>t\}$. Now Lemma \[le:superSimple\] implies and the proof is complete.
The family $\{ {\mathcal{Q}}_{t}(\omega)\}$ satisfies [(A3)]{}.
Again, we state the argument for the case $s=0$. Let $Q\in {\mathcal{Q}}_{0}$; then $Q\in {\mathcal{Q}}_{0}(P)$ for some $P\in {\mathcal{P}}={\mathcal{P}}_{0}$. Moreover, let $t\ge 0$ and let $\nu$ be an ${\mathcal{F}}_{t}$-measurable kernel such that $\nu(\omega)\in {\mathcal{Q}}_{t}( \omega)$ for $Q$-a.e. $\omega \in \{\zeta>t\}$. Using Assumption \[ass : Pc(t,omega)\] and the measurability results established in the proof of Lemma \[lem: A1 holds\], it follows that the set $$\{(\omega,P',Q') :\omega\in \Omega,\; P'\in {\mathcal{P}}_{t}( \omega),\; Q' =\nu(\omega),\;Q'\in {\mathcal{Q}}_{t}( \omega,P')\}$$ is analytic. Let ${\mathcal{F}}^{*}_{t}$ be the universal completion of ${\mathcal{F}}_t$. Applying the measurable selection theorem, cf. [@BertsekasShreve.78 Proposition 7.49], we can find an ${\mathcal{F}}^{*}_{t}$-measurable kernel $\mu'$ such that $\mu'(\omega)\in {\mathcal{P}}_{t}(\omega)$ and $\nu(\omega)\in {\mathcal{Q}}_{t}( \omega,\mu'(\omega))$ for all $\omega\in\{\zeta>t\}$ outside the ${\mathcal{F}}^{*}_{t}$-measurable $Q$-nullset $$N':=\{\nu\notin{\mathcal{Q}}_{t}\}\cap \{\zeta>t\}$$ and, e.g., $\mu'(\omega)=P^{t,\omega}$ for $\omega\in N'$. We can then find an ${\mathcal{F}}_{t}$-measurable kernel $\mu$ and a $P$-nullset $N$ such that $\mu(\omega)=\mu'(\omega)$ for all $\omega\notin N$; cf. [@BertsekasShreve.78 Lemma 7.27]. Using Assumption \[ass : Pc(t,omega)\] and $Q\ll_{\zeta}P$, we have $$\begin{aligned}
\mu(\omega)&\in {\mathcal{P}}_{t}(\omega) &\mbox{for $P$-a.e.\ $\omega\in \{\zeta>t\}$;}\nonumber\\
\nu(\omega)&\in {\mathcal{Q}}_{t}( \omega, \mu(\omega)) &\mbox{for $Q$-a.e.\ $\omega\in \{\zeta>t\}$.}\label{eq: nu in Q of mu}
\end{aligned}$$ By Assumption \[ass : Pc(t,omega)\], the measure $$\bar P(A):=\iint ({\mathbf{1}}_{A})^{t,\omega}(\omega')\,\mu^{P}(d\omega';\omega)\,P(d\omega),\quad A\in {\mathcal{F}}$$ is an element of ${\mathcal{P}}$; cf. Definition \[def : analytic and stable family\] for the notation. Set $$\bar Q(A):=\iint ({\mathbf{1}}_{A})^{t,\omega}(\omega')\,\nu^{Q}(d\omega';\omega)\,Q(d\omega),\quad A\in {\mathcal{F}}.$$ Next, we show that $\bar Q \ll_{\zeta} \bar P$; i.e., that $$\bar Q \ll \bar P \quad \mbox{on}\quad {\mathcal{F}}_{s}\cap\{s<\zeta\}, \quad s\geq0.$$ This is clear for $s\leq t$ since $\bar Q = Q \ll_{\zeta} P = \bar P$ on ${\mathcal{F}}_{t}$. Let $s>t$ and let $A\in{\mathcal{F}}_{s}$ be such that $\bar P(A\cap \{s<\zeta\})=0$. Then $$\mu(\omega)\{(A\cap \{s<\zeta\})^{t,\omega}\}=\bar P^{t,\omega}\{(A\cap \{s<\zeta\})^{t,\omega}\}=0$$ and thus $$\bar Q^{t,\omega}\{(A\cap \{s<\zeta\})^{t,\omega}\}=\nu(\omega)\{(A\cap \{s<\zeta\})^{t,\omega}\}=0$$ for $Q$-a.e. $\omega\in \{\zeta>t\}$, by . It follows that $$\bar Q(A\cap \{s<\zeta\})=E^{Q}\big[E^{\bar Q}[{\mathbf{1}}_{A\cap \{s<\zeta\}}|{\mathcal{F}}_{t}]\big]=0$$ as desired.
To see that $\bar Q\in{\mathcal{Y}}_{0}$, let $X\in\tilde{\mathcal{X}}_{0}$ (recall the notation from Lemma \[le:superSimple\]); then $X{\mathbf{1}}_{[\![0,\zeta[\![}$ is a $Q$-supermartingale. Moreover, noting that $X^{t,\omega}$ is an element of the scaled space $X_{t}(\omega){\mathcal{X}}_{t}^{\rm simp}(\omega)$, we have that $X^{t,\omega}{\mathbf{1}}_{[\![0,\zeta^{t,\omega}-t[\![}$ is a $\nu(\omega)$-supermartingale for all $\omega$ such that $\nu(\omega)\in{\mathcal{Q}}_{t}(\omega)$. Using Fubini’s theorem, it then follows that $X{\mathbf{1}}_{[\![0,\zeta[\![}$ is a $\bar Q$-supermartingale as desired.
We have shown that $\bar Q\in \Pa{\zeta }{\bar P}\cap {\mathcal{Y}}_{0}\subset {\mathcal{Q}}_{0}$ and the proof is complete.
Superhedging Duality {#sec: superhedging duality}
====================
In this section, we provide a superhedging duality and the existence of an optimal strategy. To this end, we require an enlargement of the set of admissible strategies, allowing for continuous trading. We first introduce the filtration ${\mathbb{G}}= ({\mathcal{G}}_t)_{t\geq 0}$, where $${\mathcal{G}}_t:= {\mathcal{F}}^{*}_{t}\vee \mathcal{N}^{{\mathcal{P}}};$$ here ${\mathcal{F}}^{*}_{t}$ is the universal completion of ${\mathcal{F}}_t$ and $\mathcal{N}^{{\mathcal{P}}}$ is the collection of sets which are $({\mathcal{F}},P)$-null for all $P\in{\mathcal{P}}$. Moreover, Assumption \[ass : Pc(t,omega)\] is in force throughout this section.
Let ${{\rm NA}_{1}(\mathcal{P})}$ hold, then Corollary \[co:FTAP\] implies the $({\mathbb{G}},P)$-semimartingale property of $S$ for each $P\in {\mathcal{P}}$. Therefore, we may introduce the class ${\cal L}({\mathcal{P}})$ of all predictable processes on $(\Omega, {\mathbb{G}})$ that are $S$-integrable under every $P \in { {\mathcal{ P}}}$. Given $H\in {\cal L}({\mathcal{P}})$ and $P\in{\mathcal{P}}$, we can construct the usual stochastic integral $H{\stackrel{\mbox{\tiny$\bullet$}}{}}S$ under $P$ (the dependence on $P$ is suppressed in the notation—but see also [@Nutz.11int]). For $x \in {\mathbb R}_+$, we denote by ${\mathcal{H}}(x)$ the collection of all $H \in {\cal L}({\mathcal{P}})$ such that $x + H {\stackrel{\mbox{\tiny$\bullet$}}{}}S$ remains $P$-a.s. nonnegative for all $P \in { {\mathcal{ P}}}$.
To be consistent with the classical literature, the following superhedging theorem is stated with the set $${\mathcal{Q}}:=\bigcup_{P\in {\mathcal{P}}} {\mathcal{Q}}^{P}$$ of prior-to-$\zeta$ local martingale measures; cf. Definition \[defn:ELMM\]. The subsequent Lemma \[le:sameSupremum\] provides an equivalent version with the set ${\mathcal{Q}}_{0}$ of supermartingale measures.
\[th:duality\] Let ${{\rm NA}_{1}(\mathcal{P})}$ hold, let $T\in {\mathbb{R}}_{+}$ and let $f:\Omega\to [0,\infty]$ be an upper semianalytic [^7], ${\mathcal{G}}_T$-measurable function with $\sup_{Q\in{\mathcal{Q}}} E^Q[f{\mathbf{1}}_{\zeta>T}]<\infty$. Then $$\begin{gathered}
\sup_{Q\in{\mathcal{Q}}} E^Q[f{\mathbf{1}}_{\zeta>T}] \\
=\min\big\{x:\,\exists\, H\in {\mathcal{H}}(x)\mbox{ with } x+ (H{\stackrel{\mbox{\tiny$\bullet$}}{}}S)_{T} \geq f\;P{\mbox{-a.s.}}\mbox{ for all }P\in{\mathcal{P}}\big\}.
\end{gathered}$$
In order to prove this theorem, we first show that ${\mathcal{Q}}$ can equivalently be replaced by ${\mathcal{Q}}_0$ in its statement.
\[le:sameSupremum\] Let ${{\rm NA}_{1}(\mathcal{P})}$ hold, let $T\in {\mathbb{R}}_{+}$ and let $f:\Omega\to [0,\infty]$ be a ${\mathcal{G}}_{ T }$-measurable function. Then $$\sup_{Q\in{\mathcal{Q}}} E^Q[f{\mathbf{1}}_{\zeta>T}] = \sup_{Q\in{\mathcal{Q}}_{0}} E^Q[f{\mathbf{1}}_{\zeta>T}].$$
Since ${\mathcal{Q}}\subseteq {\mathcal{Q}}_{0}$, we only have one non-trivial inequality to prove. Fix $Q_0 \in {\mathcal{Q}}_{0}$, and let $P \in { {\mathcal{ P}}}$ be such that $Q_0\ll_{\zeta} P$. By Remark \[rem : density process constructed from Q\] in the Appendix, one can construct a càdlàg adapted process $Y^0\geq0$ which is the prior-to-$\zeta$ density of $Q_{0}$ with respect to $P$. Then, the same arguments as in [@LarsenZitkovic.07 Proposition 3.2] show that one may write $Y^0 = Y D$, where $D$ is an ${\mathbb{F}}_+$-predictable nonincreasing process with $D_{0}=1$ and $Y$ is a $P$-a.s. strictly positive càdlàg $({\mathbb{F}}_+, P)$-local martingale such that $Y S$ is also an $({\mathbb{F}}_+, P)$-local martingale. Applying Theorem \[thm: foelmeasure\_1\] from the Appendix, we construct $Q {\sim_{\zeta}}P$ whose prior-to-$\zeta$ density with respect to $P$ is $Y$. Clearly, $$E^{Q}[f{\mathbf{1}}_{\zeta>T}]=E^{P}[Y_{T}f] \ge E^{P}[Y^0_{T}f] = E^{Q_{0}}[f{\mathbf{1}}_{\zeta>T}]$$ since $f\ge 0$. It remains to show that $Q \in {\mathcal{Q}}^{P}$, which follows in a straightforward way from the fact that $Y S$ is an $({\mathbb{F}}_+, P)$-local martingale and that $\zeta$ is foretellable under $Q$; see Definition \[defn: foretellable\] and Theorem \[thm: foelmeasure\_1\] in the Appendix.
The remainder of this section is devoted to the proof of Theorem \[th:duality\]. In the course of this proof, $T>0$ is fixed and $f$ satisfies the assumptions stated in the theorem. We will use Lemma \[le:sameSupremum\] without further mention. To simplify the notation, we may assume that $$S=S{\mathbf{1}}_{[\![0,\zeta[\![}$$ and moreover we set $$g:=f{\mathbf{1}}_{\zeta>T};$$ note that $g$ is upper semianalytic like $f$.
We begin by proving the easy inequality of the theorem. Let $x\in{\mathbb{R}}$ and suppose there exists $H\in{\mathcal{H}}(x)$ such that $x+ H{\stackrel{\mbox{\tiny$\bullet$}}{}}S_{T}\geq g$ $P$-a.s. for all $P\in{\mathcal{P}}$. Fix $Q\in{\mathcal{Q}}$; then there exists $P\in{\mathcal{P}}$ such that $Q\sim_{\zeta}P$. Remark \[rem:foretell\_and\_pred\] from the Appendix shows that $\zeta$ is a predictable stopping time in the $Q$-augmentation ${\mathbb{G}}^{Q}_{+}$ of ${\mathbb{G}}_{+}$. It follows that $H':=H {\mathbf{1}}_{[\![0,\zeta[\![}$ is predictable in ${\mathbb{G}}^{Q}_{+}$, and thus $x+H'{\stackrel{\mbox{\tiny$\bullet$}}{}}S$ is a nonnegative local martingale under $Q$; in particular, a $Q$-supermartingale. Using that $g=0$ on $\{\zeta\leq T\}$, we see that $x+ H'{\stackrel{\mbox{\tiny$\bullet$}}{}}S_{T}\geq g$ $Q$-a.s., and now taking expectations yields $x\geq E^Q[g]$. Since $Q\in{\mathcal{Q}}$ was arbitrary, the inequality “$\geq$” of the theorem follows.
To complete the proof of the theorem, we shall construct in the remainder of this section a strategy $H$ satisfying $$\label{eq:aimH}
\sup_{Q\in{\mathcal{Q}}} E^{Q}[g] + H{\stackrel{\mbox{\tiny$\bullet$}}{}}S_{T} \geq g \quad P \mbox{-a.s.} \quad \mbox{for all} \quad P \in {\mathcal{P}}.$$ Given $t\geq0$ and an upper semianalytic function $h\geq 0$ on $\Omega$, we define $${\mathcal{E}}_t(h)(\omega):=\sup_{Q\in{\mathcal{Q}}_{t}(\omega)} E^Q[h^{t,\omega}],\quad\omega\in\Omega.$$ Moreover, we denote ${\mathbb{F}}^{*}=({\mathcal{F}}^{*}_{t})_{t\in {\mathbb{R}}_{+}}$.
\[le:Esupermart\] The process $\{{\mathcal{E}}_{t}(g)\}_{t\in[0,T]}$ is a $(Q,{\mathbb{F}}^{*})$-supermartingale for all $Q\in{\mathcal{Q}}_{0}$, and in particular for all $Q\in{\mathcal{Q}}$.
Let $s\leq t$. In view of Proposition \[pr:QsatisfiesCondA\] and Lemma \[le:OmegaPolish\], we may adapt the proof of [@NutzVanHandel.12 Theorem 2.3] to establish that ${\mathcal{E}}_t(g)$ is ${\mathcal{F}}_t^*$-measurable and upper semianalytic, that $${\mathcal{E}}_s(g{\mathbf{1}}_{\zeta>t})(\omega) = {\mathcal{E}}_s({\mathcal{E}}_t(g){\mathbf{1}}_{\zeta>t})(\omega)\quad\mbox{for all}\quad \omega\in\Omega,$$ and that $${\mathcal{E}}_s(g{\mathbf{1}}_{\zeta>t}) = \mathop{\operatorname{ess\, sup}^Q}_{Q'\in {\mathcal{Q}}^{Q}_{s}} E^{Q'}[{\mathcal{E}}_t(g){\mathbf{1}}_{\zeta>t}|{\mathcal{F}}_s]\quad Q{\mbox{-a.s.}}\quad\mbox{for all}\quad Q\in{\mathcal{Q}},$$ where ${\mathcal{Q}}^{Q}_{s}=\{Q'\in {\mathcal{Q}}:\, Q'=Q \mbox{ on } {\mathcal{F}}_s\}$. Since $\{\zeta>T\} \subseteq \{\zeta>t\}$ for $t\leq T$, we have $g{\mathbf{1}}_{\zeta>t}=f{\mathbf{1}}_{\zeta>T}{\mathbf{1}}_{\zeta>t}=f{\mathbf{1}}_{\zeta>T}=g$. Hence, the above simplifies to $$\label{eq:DPP}
{\mathcal{E}}_s(g) = {\mathcal{E}}_s({\mathcal{E}}_t(g)),\quad s\leq t\leq T$$ and $$\label{eq:esssupDPP}
{\mathcal{E}}_s(g) = \mathop{\operatorname{ess\, sup}^Q}_{Q'\in {\mathcal{Q}}^{Q}_{s}} E^{Q'}[{\mathcal{E}}_t(g)|{\mathcal{F}}_s]\quad Q{\mbox{-a.s.}}\quad\mbox{for all}\quad Q\in{\mathcal{Q}},\quad s\leq t\leq T.$$ Our assumption that ${\mathcal{E}}_{0}(g)<\infty$ and applied with $s=0$ yield that $\sup_{Q\in{\mathcal{Q}}} E^{Q}[{\mathcal{E}}_{t}(g)]<\infty$ for all $t$; in particular, ${\mathcal{E}}_{t}(g)$ is integrable under all $Q\in{\mathcal{Q}}$. Moreover, yields that $${\mathcal{E}}_s(g) \geq E^{Q}[{\mathcal{E}}_t(g)|{\mathcal{F}}_s]=E^{Q}[{\mathcal{E}}_t(g)|{\mathcal{F}}^{*}_s]\quad Q{\mbox{-a.s.}}\quad\mbox{for all}\quad \!Q\in{\mathcal{Q}},\quad s\leq t\leq T,$$ which is the desired supermartingale property.
\[le:Zworks\] Define $$Z'_{t}:= \limsup_{r\downarrow t,\, r\in {\mathbb{Q}}} {\mathcal{E}}_{r}(g)\quad\mbox{for}\quad t<T \quad\mbox{and}\quad Z'_{T}:= {\mathcal{E}}_{T}(g),$$ let $N$ be the set of all $\omega\in\Omega$ such that $Z'(\omega)$ is not càdlàg, and $$Z:=Z'{\mathbf{1}}_{N^{c}}.$$ Then $(Z_{t})_{t\in[0,T]}$ is a càdlàg, ${\mathbb{G}}_{+}$-adapted process which is a $Q$-supermartingale for all $Q\in{\mathcal{Q}}$. Moreover, $$\label{eq:aimZ}
Z_{0} \leq \sup_{Q\in{\mathcal{Q}}} E^{Q}[g] \quad \mbox{and} \quad Z_{T}=g \quad P\mbox{-a.s.} \quad \mbox{for all} \quad P \in {\mathcal{P}}.$$
Recall Lemma \[le:Esupermart\]. The modification theorem for supermartingales [@DellacherieMeyer.82 Theorem VI.2] yields that $N$ is ${\mathcal{Q}}$-polar, the limit superior in its definition is actually a limit outside a ${\mathcal{Q}}$-polar set, and moreover that $Z'$ is a $({\mathbb{G}}_+,Q)$-supermartingale for all $Q\in{\mathcal{Q}}$.
To see that $N\in{\mathcal{N}}^{{\mathcal{P}}}$, we fix an arbitrary $P\in{\mathcal{P}}$ and show that $N$ is $P$-null. Indeed, we may decompose $N$ as $$N = (N \cap \{\zeta \leq T\}) \cup (N \cap \{\zeta > T\}).$$ The first set is $P$-null because $\{\zeta <\infty\}$ was assumed to be ${\mathcal{P}}$-polar. We know that there exists $Q\in{\mathcal{Q}}$ such that $P\sim_{\zeta} Q$. Since $N$ is $Q$-null relative to ${\mathcal{F}}^{*}_{T}$, there exists an ${\mathcal{F}}_{T}$-measurable $Q$-nullset $N^{Q}$ such that $N\subseteq N^{Q}$. Now $P\sim_{\zeta} Q$ implies that $ N^{Q} \cap \{\zeta > T\}$ is $P$-null, and then so is $N \cap \{\zeta > T\}$. As a result, we have $N\in{\mathcal{N}}^{{\mathcal{P}}}$ and in particular $N\in{\mathcal{G}}_{0}$. This implies that $Z:=Z'{\mathbf{1}}_{N^{c}}$ is still a $({\mathbb{G}}_+,Q)$-supermartingale for all $Q\in{\mathcal{Q}}$, while in addition all paths of $Z$ are càdlàg. Moreover, for any $P\in{\mathcal{P}}$, it follows from ${\mathcal{G}}_{T}={\mathcal{F}}_{T}$ $P$-a.s. and that $Z_{T}=Z'_{T}={\mathcal{E}}_{T}(g)=g$ $P$-a.s.
It remains to show the first part of . Since $Z_0$ is ${\mathcal{G}}_{0+}$-measurable, ${\mathcal{G}}_{0+}$ is equal to ${\mathcal{F}}_{0+}$ up to $P$-nullsets for any $P\in{\mathcal{P}}$, and any $P\in{\mathcal{P}}$ is dominated on ${\mathcal{F}}_{0+}$ by some $Q\in{\mathcal{Q}}$, it suffices to show that $$Z_0 \leq \sup_{Q'\in{\mathcal{Q}}} E^{Q'}[g]\equiv {\mathcal{E}}_0(g) \quad Q\mbox{-a.s.}$$ for all $Q\in{\mathcal{Q}}$. The proof of this fact is similar to the proof of [@Nutz.14 Inequality (3.3)]. Namely, it follows from Lemma \[le:Esupermart\] and the construction of $Z$ that $$\sup_{Q'\in{\mathcal{Q}}}E^{Q'}[Z_0]\leq \sup_{Q'\in{\mathcal{Q}}} E^{Q'}[g].$$ Then, one shows that $\sup_{Q'\in{\mathcal{Q}}}E^{Q'}[Z_0]$ dominates the $Q$-essential supremum of $Z_{0}$ for any $Q\in{\mathcal{Q}}$ by verifying that ${\mathcal{Q}}$ is stable under ${\mathcal{F}}_{0+}$-measurable, equivalent changes of measure—see Theorem \[thm: foelmeasure\_1\]. We omit the details.
\[le:classicalOptDecomp\] Let $Q\in{\mathcal{Q}}$. Then there exists a ${\mathbb{G}}_{+}^{Q}$-predictable process $H^{Q}$ which is $S$-integrable under $Q$ such that $$Z- H^{Q} {\stackrel{\mbox{\tiny$\bullet$}}{}}S\quad \mbox{is nonincreasing $Q$-a.s.\ on $[\![0,\zeta[\![\cap [\![0,T]\!]$.}$$
Let $\sigma_{n}$ be an announcing sequence for $\zeta$ associated with $Q$ and set $\tau_{n}:=\sigma_{n}\wedge T$. Let $Q'$ be a probability on ${\mathcal{F}}_{T}$ which is equivalent to $Q$ and such that $S^{\tau_{n}}$ is a $Q'$-local martingale; we show that $Z^{\tau_{n}}$ is a $Q'$-supermartingale. Indeed, let $Y'=(Y'_{t})_{t\in[0,T]}$ be the density process of $Q'$ with respect to $Q$ and the filtration ${\mathbb{G}}_{+}^{Q}$, a strictly positive $Q$-martingale with unit expectation. Define $$Y''_{t} := Y'_{t\wedge \tau_{n}},\quad t\geq 0;$$ then $Y''$ is the density process of a probability $Q''$ with respect to $Q$ and it is elementary to verify that $Q''\in{\mathcal{Q}}$. Thus, $Z$ is a $Q''$-supermartingale by Lemma \[le:Zworks\]. As $Q''=Q'$ on ${\mathcal{G}}_{\tau_{n}+}$, it follows that $Z^{\tau_{n}}$ is a $Q'$-supermartingale as desired. As a result, we may apply the classical optional decomposition theorem (see [@FollmerKabanov.98]) to obtain an integrand $H^{Q,n}$ such that $$Z^{\tau_{n}}- H^{Q,n}{\stackrel{\mbox{\tiny$\bullet$}}{}}S^{\tau_{n}}\quad \mbox{is nonincreasing $Q$-a.s.}$$ The result follows by a passage to the limit $n\to\infty$.
We can now construct $H$ as in by arguments similar to the proof of [@Nutz.14 Theorem 2.4]. To this end, we recall that $S=S{\mathbf{1}}_{[\![0,\zeta[\![}$. Moreover, as we will be working in the filtration ${\mathbb{G}}$ and ${\mathcal{N}}^{{\mathcal{P}}}\subset{\mathcal{G}}_{0}$, we may assume without loss of generality that all paths of $S$ are continuous prior to $\zeta$.
The $(d+1)$-dimensional process $(S,Z)$ is essentially a ${\mathbb{G}}_{+}$-semimartingale under all $Q\in{\mathcal{Q}}$; that is, modulo the fact that $S$ may fail to have a left limit at $\zeta$. Following the construction of [@NeufeldNutz.13a Proposition 6.6][^8], there exists a ${\mathbb{G}}_{+}$-predictable (and hence ${\mathbb{G}}$-predictable) process $C^{(S,Z)}$ with values in ${\mathbb{S}}^{d+1}_{+}$ (the set of nonnegative definite symmetric matrices), having ${\mathcal{Q}}$-q.s. continuous and nondecreasing paths prior to $\zeta$, and which coincides $Q$-a.s. with ${\langle (S,Z)^{c} \rangle}^{Q}$ under each $Q\in{\mathcal{Q}}$, prior to $\zeta$. Here ${\langle (S,Z)^{c} \rangle}^{Q}$ denotes the usual second characteristic of $(S,Z)$ under $Q$; i.e., the quadratic covariation process of the continuous local martingale part of $(S,Z)$.
Let $C^{S}$ be the $d\times d$ submatrix corresponding to $S$ and let $C^{SZ}$ be the $d$-dimensional vector corresponding to the quadratic covariation of $S$ and $Z$. Let $A_{t}:=\operatorname{tr}C^{S}_{t}$ be the trace of $C^{S}$; then, prior to $\zeta$, $C^{S}\ll A$ ${\mathcal{Q}}$-q.s. and $C^{SZ}\ll A$ ${\mathcal{Q}}$-q.s. (i.e., absolute continuity holds outside a polar set). Thus, we have $dC^{S} = c^{S}dA$ ${\mathcal{Q}}$-q.s. and $dC^{SZ} = c^{SZ}dA$ ${\mathcal{Q}}$-q.s. for the derivatives defined by $$c^{S}_{t}:= \tilde{c}^{S}_{t} {\mathbf{1}}_{\{\tilde{c}^{S}_{t}\in {\mathbb{S}}^{d}_{+}\}},
\quad \tilde{c}^{S}_{t} := \limsup_{n\to\infty} \frac{C^{S}_{t} - C^{S}_{(t-1/n)\vee 0}} {A_{t} - A_{(t-1/n)\vee 0}}$$ and $$c^{SZ}_{t}:= \tilde{c}^{SZ}_{t} {\mathbf{1}}_{\{\tilde{c}^{SZ}_{t}\in {\mathbb{R}}^{d}\}},
\quad \tilde{c}^{SZ}_{t} := \limsup_{n\to\infty} \frac{C^{SZ}_{t} - C^{SZ}_{(t-1/n)\vee 0}} {A_{t} - A_{(t-1/n)\vee 0}},$$ where all operations are componentwise and $0/0:=0$. Let $(c^{S})^{\oplus}$ be the Moore–Penrose pseudoinverse of $c^{S}$ and define the ${\mathbb{G}}$-predictable process $$H:= \begin{cases}
c^{SZ} (c^{S})^{\oplus} &\mbox{on }[\![0,\zeta[\![\cap [\![0,T]\!], \\
0 & \mbox{otherwise};
\end{cases}$$ we show that $H$ satisfies .
Fix $Q\in{\mathcal{Q}}$. By Lemma \[le:classicalOptDecomp\], there exist an $S$-integrable process $H^{Q}$ and a nondecreasing process $K^{Q}$ such that $$\label{eq:usualOptDecomp}
Z=Z_{0}+ H^{Q}{\stackrel{\mbox{\tiny$\bullet$}}{}}S - K^{Q}\quad \mbox{$Q$-a.s.\ on $[\![0,\zeta[\![\cap [\![0,T]\!]$.}$$ It follows that $$d{\langle S,Z \rangle}=H^{Q} d{\langle S \rangle} \quad Q{\mbox{-a.s.}},$$ or equivalently $$c^{SZ}=H^{Q} c^{S} \quad Q\times dA\mbox{-a.e.}$$ By Itô’s isometry, this implies that $H$ is $S$-integrable under $Q$ and $$H{\stackrel{\mbox{\tiny$\bullet$}}{}}S = H^{Q}{\stackrel{\mbox{\tiny$\bullet$}}{}}S \quad \mbox{$Q$-a.s.\ on $[\![0,\zeta[\![\cap [\![0,T]\!]$.}$$ Now implies that $$Z-Z_{0}- H{\stackrel{\mbox{\tiny$\bullet$}}{}}S \quad \mbox{is nonincreasing and nonpositive $Q$-a.s.\ on $[\![0,\zeta[\![\cap [\![0,T]\!]$.}$$ Noting that $$Z_{t}{\mathbf{1}}_{\zeta \leq t} = {\mathcal{E}}_{t+}(f{\mathbf{1}}_{\zeta > T}){\mathbf{1}}_{\zeta \leq t} = {\mathcal{E}}_{t+}(f{\mathbf{1}}_{\zeta > T}{\mathbf{1}}_{\zeta \leq t})={\mathcal{E}}_{t+}(0)=0\quad Q{\mbox{-a.s.}},$$ we see that $Z=0$ on $[\![0,T]\!] \setminus [\![0,\zeta[\![$. Since $H$ also vanishes on that set, we conclude that $$Z-Z_{0}- H{\stackrel{\mbox{\tiny$\bullet$}}{}}S\quad \mbox{is nonincreasing and nonpositive $Q$-a.s.\ on $[\![0,T]\!]$.}$$ In particular, $Z_{0}+H{\stackrel{\mbox{\tiny$\bullet$}}{}}S \geq0$ $Q$-a.s. As $Q\in{\mathcal{Q}}$ was arbitrary, it easily follows that $Z_{0}+H{\stackrel{\mbox{\tiny$\bullet$}}{}}S \geq0$ $P$-a.s. and that $$Z-Z_{0}- H{\stackrel{\mbox{\tiny$\bullet$}}{}}S\quad \mbox{is nonincreasing $P$-a.s.\ on $[\![0,T]\!]$}$$ for all $P\in{\mathcal{P}}$. Thus, we have $$\sup_{Q\in{\mathcal{Q}}} E^{Q}[g] + H{\stackrel{\mbox{\tiny$\bullet$}}{}}S_{T} \geq Z_{0} +H{\stackrel{\mbox{\tiny$\bullet$}}{}}S_{T} \geq Z_{T} = g \quad P\mbox{-a.s.} \quad \mbox{for all} \quad P \in {\mathcal{P}}$$ and $H\in{\mathcal{H}}(x)$ for $x=\sup_{Q\in{\mathcal{Q}}} E^{Q}[g]$. This completes the proof of and thus of Theorem \[th:duality\].
Appendix {#sec : Appendix}
========
Notions from Measure Theory {#sec:measureTheory}
---------------------------
Given a measurable space $(\Omega,{\mathcal{A}})$, let ${\mathfrak{P}}(\Omega)$ the set of all probability measures on ${\mathcal{A}}$. The *universal completion* of ${\mathcal{A}}$ is the $\sigma$-field $\cap_{P\in{\mathfrak{P}}(\Omega)} {\mathcal{A}}^P$, where ${\mathcal{A}}^P$ denotes the $P$-completion of ${\mathcal{A}}$. When $\Omega$ is a topological space with Borel $\sigma$-field ${\mathcal{B}}(\Omega)$, we endow ${\mathfrak{P}}(\Omega)$ with the topology of weak convergence. Suppose that $\Omega$ is Polish, then ${\mathfrak{P}}(\Omega)$ is Polish as well. A subset $A\subset\Omega$ is called *analytic* if it is the image of a Borel subset of another Polish space under a Borel-measurable mapping. Analytic sets are stable under countable union and intersection, and under forward and inverse images of Borel functions. Any Borel set is analytic, and any analytic set is universally measurable. A function $f: \Omega\to [-\infty,\infty]$ is *upper semianalytic* if $\{f\geq c\}$ is analytic for every $c\in{\mathbb{R}}$. In particular, any Borel function is upper semianalytic. We refer to [@BertsekasShreve.78 Chapter 7] for these results and further background.
Föllmer’s Exit Measure {#sec: follmer measure}
----------------------
Important references on Föllmer’s exit measure are [@Foellmer.72] and [@Meyer.72]; see also [@PerkowskiRuf.13] and the references therein for recent developments. The first result of this section provides an alternative, seemingly stronger characterization of the notion of prior-to-$\zeta$ absolute continuity—compare with Definition \[def : prior to zeta abs cont and equiv\].
\[lem : PXa countable representation\] Let $\xi$ be a random time and $P,Q\in{\mathfrak{P}}(\Omega)$. Then $$\label{eq: prior to zeta equiv}
P(A\cap \{\tau<\xi\})=0\quad\Rightarrow \quad Q(A\cap \{\tau<\xi\})=0\quad\forall\; \tau\in {\mathcal{T}}_{+},\;A\in {\mathcal{F}}_{\tau+}$$ holds if and only if $$\label{eq: prior to zeta equiv coutable}
P(A\cap \{q<\xi\})=0\quad\Rightarrow \quad Q(A\cap \{q<\xi\})=0\quad\forall\; q\in {\mathbb{Q}}_{+},\;A\in {\mathcal{F}}_{q}.$$
It is clear that implies . For the converse, we first note that it suffices to check for ${\mathbb{F}}$-stopping times taking finitely many values in ${\mathbb{Q}}_{+}\cup\{\infty\}$. Indeed, let $\tau \in {\mathcal{T}}_{+}$ be given; then $$\tau_{n}:=\inf\big\{(k+1)2^{-n}:\, 0\leq k\leq n2^{n},\, \tau\leq k2^{-n}\big\}$$ (where $\inf\emptyset=\infty$) is a sequence of such stopping times and $\tau_{n}\downarrow \tau$. Now $A\cap \{\tau_{n}<\xi\}$ increases to $A\cap \{\tau<\xi\}$ for $A\in {\mathcal{F}}_{\tau+}\subset {\mathcal{F}}_{\tau_{n}}$; therefore, if is valid for each $\tau_{n}$, then $ P(A\cap \{\tau<\xi\})=0$ implies $ P(A\cap \{\tau_{n}<\xi\})=0$ which in turn implies $ Q(A\cap \{\tau_{n}<\xi\})=0$ and thus $Q(A\cap \{\tau<\xi\})=0$ by monotone convergence.
Any ${\mathbb{F}}$-stopping time $\tau$ with finitely many values in ${\mathbb{Q}}_{+}\cup\{\infty\}$ is of the form $\tau=\sum_{i=1}^{n} t_{i}{\mathbf{1}}_{A_{i}}$, where $n\in {\mathbb{N}}$, $t_{i}\in{\mathbb{Q}}_{+}\cup\{\infty\}$ and $A_{i}\in {\mathcal{F}}_{t_{i}}$ are disjoint. Hence, $$R(A\cap \{\tau<\xi\})=\sum_{i=1}^{n} R\big(A\cap \{\tau\le t_{i}\}\cap A_{i}\cap \{t_{i}<\xi\}\big),\quad R\in \{P,Q\}$$ and it follows that implies .
\[rem:localization\_under\_both\] Let $Q {\sim_{\zeta}}P$. It is a consequence of Lemma \[lem : PXa countable representation\] that $Q$ and $P$ are equivalent on ${\mathcal{F}}_{\tau+} \cap {\left\{\tau < \zeta\right\}}$ for any $ {\tau} \in {\mathcal{T}}$. Suppose now that $(\tau_n)_{{{n \in {\mathbb N}}}}$ is a nondecreasing ${\mathcal{T}}$-valued sequence such that $\tau { := }{\lim_{n \to \infty}}\tau_n \geq \zeta$ holds in the $Q$-a.s. sense. Since ${\left\{\tau < \zeta\right\}} \in {\mathcal{F}}_{\tau+} \cap {\left\{\tau < \zeta\right\}}$ has zero $Q$-measure, we conclude that $P\{\tau < \zeta\} = 0$, i.e., that $\tau \geq \zeta$ also holds in the $P$-a.s. sense. In particular, if $\zeta = \infty$ $P$-a.s., it follows that $\tau = \infty$ $P$-a.s.
\[rem : density process constructed from Q\] Let $P$ and $Q$ be two probability measures on $(\Omega, {\mathcal{F}})$ with $Q {\ll_{\zeta}}P$ and $\zeta = \infty$ $P$-a.s. By utilizing appropriate versions of the Radon–Nikodym theorem and a càdlàg modification procedure, one may establish the existence of a $P$-a.s. nonnegative càdlàg adapted process $Y$ such that $$\label{eq: measure_change}
Q (A_\tau \cap {\left\{\tau < \zeta\right\}}) = E^P {\left[Y_\tau {\mathbf{1}}_{A_\tau} {\mathbf{1}}_{\tau < \zeta}\right]} \quad \mbox{for all}\quad \tau \in {\mathcal{T}}_+ \text{ and } A_\tau \in {\mathcal{F}}_{\tau+}.$$ The above process $Y$ will be called the *prior-to-$\zeta$ density process of $Q$ with respect to $P$*. It is strictly positive under $P$ when $Q{\sim_{\zeta}}P$. Note that uniquely specifies $Q$, since the class of sets $ A_T \cap \{ T < \zeta \} , T \in {\mathbb R}_+ , A_T \in {\mathcal{F}}_T $ generates ${\mathcal{F}}_{\zeta-} = {\mathcal{F}}$ and is also a $\pi$-system. Therefore, the specification of the prior-to-$\zeta$ density process of $Q$ with respect to $P$ is uniquely defined up to $P$-evanescent sets.
Suppose that $Q {\sim_{\zeta}}P$ and $Y$ is the prior-to-$\zeta$ density process of $Q$ with respect to $P$. In particular, since $Q$ and $P$ are equivalent on ${\mathcal{F}}_{0+}$ and $\zeta > 0$, gives $E^P[Y_0] = 1$. Furthermore, for $0 \leq s < t<\infty$ and $A_s \in {\mathcal{F}}_{s+}$, note that $$E^P [Y_t {\mathbf{1}}_{A_s}] = Q (A_s \cap {\left\{t < \zeta\right\}}) \leq Q (A_s \cap {\left\{s < \zeta\right\}}) = E^P [Y_s {\mathbf{1}}_{A_s}],$$ which implies that $Y$ is an $({\mathbb{F}}_+, P)$-supermartingale.
Theorem \[thm: foelmeasure\_1\] that follows, essentially due to Föllmer in [@Foellmer.72], is a converse to the previous observation: starting with a probability $P$ and a candidate density process $Y$, a probability $Q$ is constructed that has $Y$ as a prior-to-$\zeta$ density with respect to $Q$. The statement requires the following notion.
\[defn: foretellable\] We say that $\zeta$ is *foretellable* under a probability $Q$ if there exists a ${\mathcal{T}}_+$-valued sequence $(\tau_n)_{{{n \in {\mathbb N}}}}$ such that $Q\{\tau_n < \zeta\} = 1$ for all $n$ and $Q\{{\lim_{n \to \infty}}\tau_n = \zeta\} = 1$.
It is clear that the above sequence of stopping times can be chosen to be nondecreasing. Also, note that foretellability of $\zeta$ does *not* remain invariant under prior-to-$\zeta$ equivalent probability changes.
\[rem:foretell\_and\_pred\] By [@HeWangYan.92 Theorem 4.16], $\zeta$ is foretellable under $Q$ if and only if $\zeta$ is $Q$-a.s. equal to a predictable stopping time on $(\Omega, \, {\mathbb{F}}_+)$.
\[thm: foelmeasure\_1\] Let $Y$ be a strictly positive $({\mathbb{F}}_+, P)$-supermartingale with $E^P {\left[Y_0\right]} = 1$. Then, there exists $Q {\sim_{\zeta}}P$ such that $Y$ is the prior-to-$\zeta$ density process of $Q$ with respect to $P$. Furthermore, if $Y$ is actually an $({\mathbb{F}}_+, P)$-local martingale, $\zeta$ is foretellable under $Q$.
Recall that for $\xi \in {\mathcal{T}}_+$, the $\sigma$-field ${\mathcal{F}}_{\xi-}$ is generated by the collection ${\left\{A_s \cap {\left\{s < \xi\right\}}:\, s \geq 0,\, A_s \in {\mathcal{F}}_s\right\}}$. With this definition in place, we observe that ${\mathcal{F}}= {\mathcal{F}}_{\zeta-}$, because $B_t$ is ${\mathcal{F}}_{\zeta-}$-measurable for all $t \geq 0$. Indeed, Borel subsets of $E \cup \{ \triangle \}$ are of the form $A$ or $A \cup \{ \triangle \}$, where $A \in {\mathcal{B}}(E)$, and for any such $A$, we have $\{ B_t \in A \} = \{ B_t \in A \} \cap \{ t < \zeta \} \in {\mathcal{F}}_{\zeta -}$ and ${\left\{B_t \in A \cup \{ \triangle \}\right\}} = (\{ B_t \in A \} \cap \{ t < \zeta \}) \cup \{ \zeta \leq t \} \in {\mathcal{F}}_{\zeta -}$.
By [@PerkowskiRuf.13 Section 4.2], one can construct $\xi \in {\mathcal{T}}_+$ with $P\{\xi < \infty\} = 0$ and a probability $Q^0$ on $(\Omega, {\mathcal{F}}_{\xi-})$, such that $$Q^0 (A_\tau \cap {\left\{\tau < \xi\right\}}) = E^P {\left[Y_\tau {\mathbf{1}}_{A_\tau} {\mathbf{1}}_{\tau < \xi}\right]}$$ holds for all $\tau \in {\mathcal{T}}_+$ and $A_\tau \in {\mathcal{F}}_{\tau+}$. In particular, $Q^{0}\{\xi >0\}=E^{P}[Y_{0}]=1$. Since $A_\tau \cap {\left\{\tau < \xi\wedge\zeta\right\}} \in {\mathcal{F}}_{\tau+}$ for all $A_\tau \in {\mathcal{F}}_{\tau+}$, the above formula also holds for $\xi':=(\xi\wedge \zeta) {\mathbf{1}}_{\xi > 0} + \zeta {\mathbf{1}}_{\xi = 0}$. Thus, we may assume that $\xi \in {\mathcal{T}}_+$ satisfies $0 < \xi \leq \zeta$ and $P\{\xi = \zeta\} = 1$, and that $Q^0 (A_\tau \cap {\left\{\tau < \xi\right\}}) = E^P {\left[Y_\tau {\mathbf{1}}_{A_\tau} {\mathbf{1}}_{\tau < \xi}\right]}$ holds for all $\tau \in {\mathcal{T}}_+$ and $A_\tau \in {\mathcal{F}}_{\tau+}$. We shall extend $Q^0$ to a probability $Q$ on ${\mathcal{F}}= {\mathcal{F}}_{\zeta-}$ such that $Q\{\xi = \zeta\} = 1$ holds; this will immediately establish . Define a map $\psi: \Omega \rightarrow \Omega$ as follows: for $\omega \in \Omega$, set $\psi_{t}(\omega) = \omega_{t}$ when $t < \xi (\omega)$ and $\psi_{t}(\omega) =\triangle$ when $\xi (\omega) \leq t$. Since ${\mathcal{F}}$ is generated by the coordinate projections and $$\{ \psi \in \Lambda \} = ( \{\omega :\, \omega_{t} \in \Lambda \cap E \} \cap \{ t < \xi \} ) \cup \{t \geq \xi \} \in {\mathcal{F}}_{\xi-}$$ holds for all ${{t \in {\mathbb R}_+}}$ and Borel subsets $\Lambda$ of $\bar{E}=E\cup \{ \triangle\}$, it follows that $\psi$ is $({\mathcal{F}}_{\xi-} / {\mathcal{F}})$-measurable. By construction, $\zeta \circ \psi = \xi$. We claim that $\xi \le \xi\circ \psi$ holds as well. Indeed, since $\xi\wedge t$ is ${\mathcal{F}}_{t-}$-measurable for all ${{t \in {\mathbb R}_+}}$, [@DellacherieMeyer.78 Theorem 96, Chapter IV] implies that $\xi \wedge t = (\xi \wedge t)\circ k_{t}$, where $k$ is the killing operator defined via $k_{t}(\omega) = \omega{\mathbf{1}}_{[0,t)} + \triangle {\mathbf{1}}_{[t,\infty)}$ for $\omega \in \Omega$. Since $\xi(\omega)\wedge t= \xi\circ k_{t}(\omega) \wedge t$ holds for all $(\omega, t) \in \Omega \times {\mathbb R}_+$, plugging in $t = \xi(\omega)$ gives $$\xi(\omega)= \xi\circ k_{\xi(\omega)}(\omega) \wedge \xi(\omega)=\xi\circ \psi (\omega) \wedge \xi(\omega), \quad \omega \in \Omega,$$ where we have used that $k_{\xi(\omega)}(\omega) = \psi(\omega)$ holds for all $\omega \in \Omega$. Therefore, $\xi \le \xi\circ \psi$. The last inequality, combined with $\xi \leq \zeta$ and $\zeta \circ \psi = \xi$, gives $\zeta \circ \psi = \xi \circ \psi$. Define $Q$ on ${\mathcal{F}}$ via $Q(A )= Q^0(\psi^{-1}(A))$ for all $A \in {\mathcal{F}}$. By construction, $Q$ is an extension of $Q^0$, and follows since $$Q\{\xi < \zeta\} = Q^0\{\xi(\psi) < \zeta(\psi)\} = Q^0(\emptyset) = 0.$$
Finally, if $Y$ is an $({\mathbb{F}}_+, P)$-local martingale, let $(\tau_n)$ be a localizing sequence and call $\tau { := }{\lim_{n \to \infty}}\tau_n$. Note that $\tau = \infty = \zeta$ holds in the $P$-a.s. sense. By Remark \[rem:localization\_under\_both\], $\tau\geq \zeta$ holds in the $Q$-a.s. sense. Furthermore, from , we obtain $Q\{\tau_n < \tau\} = E_P [Y_{\tau_n}] = 1$ for all ${{n \in {\mathbb N}}}$. Therefore, $\zeta$ is foretellable under $Q$.
On the Path Space $\Omega$
--------------------------
The goal of this section is to show that $\Omega$ carries a natural Polish topology, which is required for the measurable selection arguments in Sections \[sec: dyna prog and local mart\] and \[sec: superhedging duality\]. To the best of our knowledge, this result is not contained in the previous literature—only the Lusin property is mentioned; see, e.g., [@Meyer.72].
Let ${\mathbb{D}}={\mathbb{D}}_{x_{*}}([0,\infty);E)$ be the usual Skorokhod space of $E$-valued càdlàg paths on $[0,\infty)$ starting at the point $x_{*}\in E$ and let $\delta_{\infty}$ be its usual metric, rendering ${\mathbb{D}}$ a Polish space. We may think of a path $\omega\in\Omega$ as consisting of a path $\tilde\omega\in{\mathbb{D}}$ and a lifetime $z\in(0,\infty]$; in this context, it is useful to equip $(0,\infty]$ with the complete metric $d_{(0,\infty]}(z,z'):=|z^{-1}-z'^{-1}|$, where $\infty^{-1}:=0$. More precisely, given $z\in(0,\infty]$, let $$e_{z}(t):=\begin{cases}
t&\mbox{if }z=\infty,\\
z(1-e^{-t})&\mbox{if }z<\infty.
\end{cases}$$ We note that $e_{z}: [0,\infty)\to[0,z)$ is a monotone bijection; thus, precomposition with $e_{z}$ turns a path $\omega\in\Omega$ with lifetime $z=\zeta(\omega)$ into an element of ${\mathbb{D}}$. As a result, we can define $$\delta_{\Omega}(\omega,\omega'):= d_{(0,\infty]}\big(\zeta(\omega),\zeta(\omega')\big) + \delta_{\infty}\big(\omega\circ e_{\zeta(\omega)}, \omega'\circ e_{\zeta(\omega')}\big), \quad \omega,\omega'\in\Omega.$$
\[le:OmegaPolish\] The space $(\Omega,\delta_{\Omega})$ is Polish and its Borel $\sigma$-field coincides with ${\mathcal{F}}$. Moreover, ${\mathcal{F}}_{\tau}=\sigma(B_{t\wedge\tau},\,t\in{\mathbb{R}}_{+})$ for any ${\mathbb{F}}$-stopping time $\tau$; in particular, ${\mathcal{F}}_{\tau}$ is countably generated.
It is clear that $\delta_{\Omega}$ defines a metric on $\Omega$. Moreover, the mapping $$\Omega \to {\mathbb{D}}\times (0,\infty],\quad \omega \mapsto \big(\omega \circ e_{\zeta(\omega)}, \zeta(\omega)\big)$$ admits the inverse $${\mathbb{D}}\times (0,\infty] \to \Omega,\quad (\tilde\omega,z) \mapsto (\tilde \omega \circ e_{z}^{-1}) \,{\mathbf{1}}_{[0,z)} + \triangle \,{\mathbf{1}}_{[z,\infty)}.$$ By the definition of $\delta_{\Omega}$, these mappings constitute an isometric isomorphism between $\Omega$ and ${\mathbb{D}}\times (0,\infty]$; in particular, $\Omega$ is Polish like ${\mathbb{D}}\times (0,\infty]$. Let ${\mathcal{B}}(\Omega)$ be the Borel $\sigma$-field on $\Omega$. To prove that ${\mathcal{F}}\subset{\mathcal{B}}(\Omega)$, it suffices to show that the evaluation $B_{t}: \omega\mapsto\omega_{t}$ is Borel-measurable for any fixed $t\geq0$. To this end, note that the functions $$\omega\mapsto \zeta(\omega)\in (0,\infty], \quad \omega\mapsto \omega\circ e_{\zeta(\omega)}\in {\mathbb{D}},\quad \omega\mapsto e^{-1}_{\zeta(\omega)}(t)\in[0,\infty)$$ are continuous on $\Omega$. Let $\tilde B$ be the canonical process on ${\mathbb{D}}$ and recall that $(t,\tilde\omega)\mapsto \tilde B_{t}(\tilde\omega)$ is jointly Borel-measurable. It then follows that $$\omega\mapsto B_{t}(\omega)= \tilde{B}_{e^{-1}_{\zeta(\omega)}(t)}\big(\omega\circ e_{\zeta(\omega)}\big) \,{\mathbf{1}}_{[0,\zeta(\omega))}(t) + \triangle \,{\mathbf{1}}_{[\zeta(\omega),\infty)}(t)$$ is Borel-measurable as well.
To prove the reverse inclusion ${\mathcal{B}}(\Omega)\subset{\mathcal{F}}$, it suffices to show that any continuous function $f:\Omega\to{\mathbb{R}}$ is ${\mathcal{F}}$-measurable. Indeed, the maps $$\omega\mapsto \zeta(\omega)\in (0,\infty], \quad \omega\mapsto \omega\circ e_{\zeta(\omega)}\in {\mathbb{D}}$$ are clearly ${\mathcal{F}}$-measurable. Moreover, any function $f$ on $\Omega$ induces a unique function $\tilde f$ on ${\mathbb{D}}\times (0,\infty]$ satisfying $$f(\omega) = \tilde f\big(\omega\circ e_{\zeta(\omega)}, \zeta(\omega)\big),\quad \omega\in\Omega.$$ If $f$ is continuous, it follows that $\tilde f$ is continuous and hence the composition $\omega\mapsto f(\omega)=\tilde f(\omega\circ e_{\zeta(\omega)}, \zeta(\omega))$ is ${\mathcal{F}}$-measurable. This completes the proof that ${\mathcal{F}}={\mathcal{B}}(\Omega)$.
The last claim follows from the fact that $\bar E$ is Polish and standard arguments; see [@StroockVaradhan.79 Lemma 1.3.3 and Exercise 1.5.6].
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[^1]: Department of Economics and Management, University of Pisa, Pisa, [email protected]
[^2]: CEREMADE, Université Paris Dauphine and CREST-ENSAE, [email protected]. Research supported by ANR Liquirisk and Investissements d’Avenir (ANR-11-IDEX-0003/Labex Ecodec/ANR-11-LABX-0047).
[^3]: Department of Statistics, London School of Economics and Political Science, London, [email protected].
[^4]: Departments of Statistics and Mathematics, Columbia University, New York, [email protected]. Research supported by NSF Grants DMS-1208985 and DMS-1512900. The authors would like to thank the Associate Editor and the anonymous referees for their constructive comments.
[^5]: We define simple predictable strategies with respect to the filtration ${\mathbb{F}}_+$; however, we recall that the class of predictable processes on $(\Omega, {\mathbb{F}})$ coincides with the class of predictable processes on $(\Omega, {\mathbb{F}}_+)$. The symbol ${]\kern-0.15em] \tau_{i-1}, \tau_i ]\kern-0.15em]}$ denotes the stochastic interval.
[^6]: The definition of an analytic set is recalled in Section \[sec:measureTheory\] of the Appendix.
[^7]: The definition of an upper semianalytic function is recalled in Section \[sec:measureTheory\] of the Appendix. In particular, any Borel function is upper semianalytic.
[^8]: That proposition does not use the separability assumptions on the filtration that are imposed for the main results of [@NeufeldNutz.13a].
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We will give a complete description of $\mathcal{I}$, the set of invertible quasicontinuous functions on the unit circle. After doing this, we will then classify the path-connected components of $\mathcal{I}$ and show that $\mathcal{I}$ has uncountably many path-connected components. We will then use the above classifications to characterize $\mathcal{F}$, the set of Fredholm operators of the C$^*$-algebra generated by the Toeplitz operators $T_\phi$ with quasicontinuous symbols $\phi$. Then we will classify the path-connected components of $\mathcal{F}$ and show that $\mathcal{F}$ also has uncountably many path-connected components.'
address: |
244 Mathematics Building\
University at Buffalo\
Buffalo, NY 14260
author:
- Adam Orenstein
title: 'Fredholm operators in the Toeplitz Algebra $\mathcal{I}(QC)$'
---
Introduction
============
Let $\mathbb{T}$ be the unit circle with the Lebesgue measure. Let $C(\mathbb{T})$ be the set of all continuous functions on $\mathbb{T}$. For any $n\in\mathbb{Z}$, let $\chi_n:\mathbb{T}\rightarrow\mathbb{T}$ be defined by $$\chi_n(t)=e^{int}.$$ For $p=1,2,\infty,$ let $H^p(\mathbb{T})$ be defined by $$H^p(\mathbb{T})=\left\{f\in L^p(\mathbb{T}): \int_0^{2\pi}f(t)\chi_n(t)dt=0 \text{ for all } n>0\right\}.$$ The spaces $H^p(\mathbb{T})$ are called the Hardy spaces on $\mathbb{T}$. It is shown in [@Doug page 147] as proposition 6.36 that $H^\infty+C(\mathbb{T})$ is a Banach subalgebra of $L^\infty\left(\mathbb{T}\right)$. Let $QC$ be the set defined by $$QC=\left[H^\infty+C(\mathbb{T})\right]\bigcap \overline{\left[H^\infty+C(\mathbb{T})\right]}$$ where $\overline{H^\infty+C(\mathbb{T})}=\{\overline{f}:f\in H^\infty+C(\mathbb{T})\}$ [@Doug page 157]. A function in $QC$ is called quasicontinuous. It is easy to show that $QC$ is a commutative Banach subalgebra of $L^\infty(\mathbb{T})$.
Let $\mathfrak{L}(H^2)$ be the set of all bounded linear operators on $H^2$. Denote the operator norm on $\mathfrak{L}(H^2)$ by $\|\cdot\|$. For any $f\in L^\infty\left(\mathbb{T}\right)$, let $T_f$ be the operator on $H^2$ defined by $$T_f(g)=P(fg) \text{ for all }g\in H^2$$ where $P$ is the orthogonal projection of $L^2(\mathbb{T})$ onto $H^2$. $T_f$ is called the Toeplitz operator on $H^2$ with symbol $f$. Since $f\in L^\infty(\mathbb{T})$, $T_f\in\mathfrak{L}(H^2)$ and $\|T_f\|=\|f\|_\infty$ [@Doug page 160]. For any subset $S$ of $L^\infty(\mathbb{T})$, let $\mathcal{I}(S)$ be the smallest closed subalgebra of $\mathfrak{L}(H^2)$ containing $\{T_f:f\in S\}$. $\mathcal{I}(S)$ is called a Toeplitz algebra.
Toeplitz algebras and Toeplitz operators on $H^2$ with quasicontinuous symbols have been studied by many different people in the literature. In particular the path-connected components of the Fredholm operators in $\mathcal{I}(C(\mathbb{T}))$ have been completely determined. The purpose of this paper is to do the same for $\mathcal{I}(QC)$. In doing this, we will have an example of a Toeplitz algebra whose Fredholm operators have uncountably many path-connected components. Until this paper, no such example has been published.
Preliminaries
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More Notation {#notat}
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Here we will establish some more notation. Let $\mathbb{D}$ be the open unit disk in $\mathbb{C}$. For any $D\subseteq \mathbb{T}$, let $|D|$ denote the Lebesgue measure of $D$ on $\mathbb{T}$. Let $C_{\mathbb{R}}(\mathbb{T})$ be the subset of $C(\mathbb{T})$ consisting of all real-valued functions, $L_{\mathbb{R}}^\infty(\mathbb{T})$ be the subset of $L^\infty(\mathbb{T})$ consisting of all real-valued functions and $QC_\mathbb{R}$ denote the set of all real-valued functions in $QC$. We will also use the following conventions: for any two sets $Q$ and $B$ and for any function $g$, let $QB=\{cb:c\in Q, b\in B\}$, $gQ=\{g\}Q$, $Q\setminus B=\{q\in Q: q\notin B\}$, $Q+B=\{q+b:q\in Q, b\in B\}$ and $\exp(Q)=\{\exp(q):q\in Q\}$.
Important Functions
-------------------
Now we will define some important functions that we will use. Let $w\in L^1(\mathbb{T})$. Then following [@krantz page 56 and page 81] and [@zhuOp page 255] respectively, we define $\widetilde{w}:\mathbb{T}\rightarrow\mathbb{C}$ and $\widehat{w}:\mathbb{D}\rightarrow\mathbb{C}$ by $$\label{hilDef}\widetilde{w}(\theta)=\sum_{n\in\mathbb{Z}}(-i{\operatorname{sgn}}(n))a_w(n)\chi_n(\theta)$$ where $$\label{coeff}a_w(n)=\frac{1}{2\pi}\int_0^{2\pi} w(t)\chi_{-n}(t)dt \hspace{.2cm} \text{ for all } n\in\mathbb{Z},$$ and $$\label{poiDef}\widehat{w}(z)=\frac{1}{2\pi}\int_0^{2\pi} w(t)\text{Re}\left[\frac{e^{it}+z}{e^{it}-z}\right]dt.$$ We will call $\widetilde{w}$ the Hilbert Transform of $w$ and $\widehat{w}$ the Poisson transform of $w$.
Another important function we will occasionally use is $\mathcal{C}(w):\mathbb{D}\cup \mathbb{T}\rightarrow \mathbb{C}$ defined by $$\label{conjDef}\begin{split}&\mathcal{C}(w)(z)=\frac{1}{2\pi}\int_0^{2\pi} w(t)\text{Im}\left[\frac{e^{it}+z}{e^{it}-z}\right]dt \text{ for all } z\in \mathbb{D}\\& \mathcal{C}(w)(e^{i\theta})=\lim_{r\nearrow1}\mathcal{C}(w)(re^{i\theta}) \text{ for all } \theta\in[0,2\pi)\end{split}$$ [@zhuOp page 256]. As in [@zhuOp page 256], we will call $\mathcal{C}(w)$ the conjugation operator of $w$ and the mapping $w \mapsto \mathcal{C}(w)$ the conjugation operator.
Important Spaces {#imSpaces}
----------------
In this section, we will define the spaces of functions that we will be working with. For any $f\in L^1(\mathbb{T})$ and any subarc $I$ of $\mathbb{T}$, let $$f_I=\frac{1}{|I|}\int_I f(\theta)d\theta.$$ For any $f\in L^2(\mathbb{T})$, define $\|f\|_{BMO}$ by $$\|f\|_{BMO}= \sup\left\{\left[\frac{1}{|I|}\int_I\left|f(\theta)-f_I\right|^2d\theta\right]^{\frac{1}{2}}:I \text{ is any subarc of }\mathbb{T}\right\}.$$ Let $BMO$ and $VMO$ be defined by $$BMO=\left\{f\in L^2(\mathbb{T}):\|f\|_{BMO}<\infty\right\}$$ and $$\label{VMOdef}VMO=\left\{f\in BMO:\lim_{|I|\rightarrow 0}\left[\frac{1}{|I|}\int_I\left|f(\theta)-f_I\right|^2d\theta\right]=0\right\}.$$ We say $BMO$ is the space of all functions in $L^2(\mathbb{T})$ that have bounded mean oscillation on $\mathbb{T}$ and $VMO$ is the space of all functions in $BMO$ that have vanishing mean oscillation on $\mathbb{T}$. Here our definitions of $BMO$ and $VMO$ are taken from [@zhuOp page 266 and page 275]. We will also denote the set of all real-valued functions in $VMO$ by $VMO_{\mathbb{R}}$. In fact $$\label{quasi} QC=VMO\cap L^\infty\left(\mathbb{T}\right)$$ [@Garn page 377].
Note that some people use the conjugation operator instead of the Hilbert transform when working with $BMO$ and $VMO$. However for every $f\in L^1\left(\mathbb{T}\right)$, Lemma 1.2 [@Garn page 103] and Theorem 1.6.11 [@krantz page 87] together yield $$\label{conjHil} -\widetilde{f}=\mathcal{C}(f) \text{ [a.e] on } \mathbb{T}.$$
Moreover as proved in [@zhuOp page 277], $$VMO=C(\mathbb{T})+\mathcal{C}\left(C(\mathbb{T})\right).$$ It is also easy to see that $f\in L^1(\mathbb{T})$ and real-valued implies $\widetilde{f}$ is real-valued. Then by and some straightforward calculations, $$\label{vmoReal} VMO=C(\mathbb{T})+\widetilde{C(\mathbb{T})} \text{ and }VMO_\mathbb{R}=C_\mathbb{R}(\mathbb{T})+\widetilde{C_\mathbb{R}(\mathbb{T})}.$$
Fredholm Index {#SectInd}
==============
Let $\mathfrak{LC}(H^2)$ be the set of all compact operators on $H^2$ and $\mathfrak{F}(H^2)$ be the set of all Fredholm operators on $H^2$. For any $A\in\mathfrak{L}(H^2)$ let $A^*$ be the adjoint of $A$. Let $j:\mathfrak{F}(H^2)\rightarrow\mathbb{Z}$ be defined by$$j(A)=\dim(\ker(A))-\dim(\ker(A^*)).$$ Then $j$ is surjective, continuous with respect to the operator norm and for any $A,B\in \mathfrak{F}(H^2)$ and $K\in\mathfrak{LC}(H^2)$, we have $j(AB)=j(A)+j(B)$ and $j(A+K)=j(A)$ [@Doug page 123].
Let $$(H^\infty+C(\mathbb{T}))^{-1}=\{f\in H^\infty+C(\mathbb{T}):f^{-1}\in H^\infty+C(\mathbb{T})\}.$$ Following [@Doug page 169] and [@sarason], we will work with the function ${\operatorname{ind}}:(H^\infty+C(\mathbb{T}))^{-1}\rightarrow \mathbb{Z}$ defined by $${\operatorname{ind}}(f)=n\left(\hat{f}_r\left(e^{i\theta}\right),0\right)$$ where $\hat{f}_r\left(e^{i\theta}\right)=\widehat{f}(re^{i\theta})$ for all $\theta\in[0,2\pi)$ and $n\left(\hat{f}_r(e^{i\theta}),0\right)$ is the winding number of $\hat{f}_r\left(e^{i\theta}\right)$ about 0 for all $r>0$ such that $r_0<r<1$ where $r_0=r_0(f)>0$ is as in Theorem 4 [@sarason]. By both Corollary 7.34 [@Doug page 168] and Theorem 7.36 [@Doug page 169], $$\label{index} j(T_f)=-{\operatorname{ind}}(f) \text{ for all } f\in (H^\infty+C(\mathbb{T}))^{-1}.$$ It follows that ${\operatorname{ind}}$ is continuous with respect to the norm $\|\cdot\|_\infty$. We will prove below that ${\operatorname{ind}}(fg)={\operatorname{ind}}(f)+{\operatorname{ind}}(g)$ for all $f,g\in (H^\infty+C(\mathbb{T})^{-1}$.
Main Results {#motivQC}
============
Recall above that $\mathfrak{I}(QC)$ be the $C^*$-algebra generated by $\{T_\phi:\phi\in QC\}$ and $$\mathcal{I}=\{f\in QC:f^{-1}\in QC\}.$$ and $\mathcal{F}$ be defined by $$\mathcal{F}=\mathfrak{F}(H^2)\cap\mathfrak{I}(QC).$$ In this paper we will completely classify $\mathcal{I}$ and $\mathcal{F}$, and to give a complete description of the path-connected components of $\mathcal{I}$ and of $\mathcal{F}$. More specifically, we will prove the following three theorems:
\[classI\] $$\mathcal{I}=\{\chi_n\}_{n\in\mathbb{Z}}\exp\left(QC_\mathbb{R}\right)\exp\left(iVMO_{\mathbb{R}}\right).$$
\[mainI\] For any $h\in VMO_{\mathbb{R}}$, let $$\label{pPath} P_h=\{\exp(ig): g\in QC_{\mathbb{R}}+\{h\}\}$$ and let $$\label{uniPath} Q_h=\exp(QC_{\mathbb{R}})P_h.$$ Then the path-connected components of $\mathcal{I}$ are $\{\chi_kQ_g\}_{\left\{\begin{subarray}{l} k\in\mathbb{Z} \text{ and either }\\ g\in VMO_{\mathbb{R}}\setminus L_\mathbb{R}^\infty \text{ or }\\ g=0\end{subarray}\right\}}$.
\[mainF\] With the same notation from Theorem \[mainI\], let $$\label{fredPath} {_m}V_h=\{T_f+K\in \mathcal{F}:f\in\chi_{-m}Q_{h}, K\in\mathfrak{LC}(H^2)\}.$$ Then the path-connected components of $\mathcal{F}$ are $\{{_m}V_h\}_{\left\{\begin{subarray}{l} m\in\mathbb{Z} \text{ and either } \\ h\in VMO_{\mathbb{R}}\setminus L_\mathbb{R}^\infty \text{ or }\\ h=0\end{subarray}\right\}}$.
In addition to the above theorems, we will also prove the following:
\[uncountable\] $\mathcal{I}$ and $\mathcal{F}$ both have uncountably many path-connected components.
Classification of $\mathcal{I}$
===============================
Important lemmas {#Prelem}
----------------
First we will show ${\operatorname{ind}}(fg)={\operatorname{ind}}(f)+{\operatorname{ind}}(g)$ for all $f,g\in (H^\infty+C(\mathbb{T}))^{-1}$. To prove this, we will need the following two propositions found in [@Doug page 159 and page 164].
\[commToe\] If $\phi\in L^\infty$ and $\psi,\overline{\theta}\in H^\infty$, then $T_{\phi}T_{\psi}=T_{\phi\psi}$ and $T_{\theta}T_\phi=T_{\theta\phi}$.
\[ToeCommCompact\] If $\phi\in C(\mathbb{T})$ and $\psi\in L^\infty$, then $T_{\phi}T_{\psi}-T_{\phi\psi},T_{\psi}T_{\phi}-T_{\psi\phi}\in\mathfrak{LC}(H^2)$.
\[indSum\] For any $f,g\in (H^\infty+C(\mathbb{T}))^{-1}$, ${\operatorname{ind}}(fg)={\operatorname{ind}}(f)+{\operatorname{ind}}(g)$.
Let $f,g\in (H^\infty+C(\mathbb{T}))^{-1}$. So we may write $f=h_f+p_f$ and $g=h_g+p_g$ where $h_f,h_g\in H^\infty$ and $p_f,p_g\in C(\mathbb{T})$. Then by Proposition \[commToe\] and Proposition \[ToeCommCompact\], there exist $L_1,L_2\in\mathfrak{LC}(H^2)$ such that $$\begin{split}T_fT_g&=T_{h_f}T_{h_g}+T_{h_f}T_{p_g}+T_{p_f}T_{h_g}+T_{p_f}T_{p_g}\\&=T_{h_fh_g}+T_{h_fp_g}+L_1+T_{p_fh_g}+T_{p_fp_g}+L_2\\&= T_{h_fh_g+h_fp_g+p_fh_g+p_fp_g}+L_1+L_2\\&=T_{fg}+L_1+L_2.\end{split}$$ Thus $T_fT_g=T_{fg}+J$ for some $J\in\mathfrak{LC}(H^2)$. Then by Section \[SectInd\] $j(T_{fg})=j(T_fT_g)=j(T_f)+j(T_g)$. Therefore by , ${\operatorname{ind}}(fg)={\operatorname{ind}}(f)+{\operatorname{ind}}(g)$.
We will also need the following lemmas in order to prove Theorem \[classI\]. Some of these results will also be used to prove Theorem \[mainI\] and Theorem \[mainF\]. The first lemma will be used many times.
\[exp\] Let $\mathfrak{B}$ be a Banach algebra, $G$ be the collection of all the invertible elements in $\mathfrak{B}$ and $G_0$ be the connected component in $G$ which contains the identity. If $\mathfrak{B}$ is commutative then $G_0=\exp(\mathfrak{B})$.
From now on for convenience, we will denote the path-connected component in $\mathcal{I}$ containing 1 by $G_0$. As we will see, being able to calculate $\widetilde{\widetilde{f}}$ for any $f\in H^\infty+C(\mathbb{T})$ is necessary. The next lemma shows us how to do this.
\[DblHil\] For every $f\in L^2(\mathbb{T})$, $\widetilde{\widetilde{f}}=-f+a_f(0)$.
Let $f\in L^2(\mathbb{T})$. Then by Theorem 1.6.16 [@krantz page 87], $f=\sum_{n\in\mathbb{Z}} a_f(n)\chi_n$ \[a.e\] on $\mathbb{T}$. Also from and Parseval’s identity, $\|\widetilde{f}\|_2^2 = \sum_{n=-\infty}^\infty |-i{\operatorname{sgn}}(n)a_f(n)|^2 \leq \sum_{n=-\infty}^\infty |a_f(n)|^2=\|f\|_2^2$ where $\|\cdot\|_2$ is the norm on $L^2(\mathbb{T})$. It follows that the Hilbert Transform is both bounded and linear on $L^2(\mathbb{T})$. Hence $$\begin{split}\widetilde{\widetilde{f}}=\sum_{n\in\mathbb{Z}}(-i{\operatorname{sgn}}(n))^2a_f(n)\chi_n&=-\sum_{n\in\mathbb{Z}}({\operatorname{sgn}}(n))^2a_f(n)\chi_n\\& =-\sum_{n\in\mathbb{Z}}a_f(n)\chi_n +a_f(0).\end{split}$$ Therefore $\widetilde{\widetilde{f}}=-f+a_f(0)$.
An immediate consequence of Lemma \[DblHil\] and is $$\label{doublevmo} \widetilde{VMO}\subseteq VMO \text{ and } \widetilde{VMO_\mathbb{R}}\subseteq VMO_\mathbb{R}.$$
The next lemma will be crucial in classifying $\mathcal{I}$.
\[I\] Let $\phi\in QC$. Then $\phi\in \mathcal{I} \text{ if and only if } \phi, \overline{\phi}\in (H^\infty+C(\mathbb{T}))^{-1}$.
$(\Rightarrow)$ Suppose $\phi\in \mathcal{I}$. Then $\phi g=1$ for some $g\in QC$. Thus $\overline{\phi}\overline{g}=1$. Since $QC$ is self-adjoint $\overline{\phi},\overline{g}\in QC$. Therefore since $QC\subseteq H^\infty+C(\mathbb{T})$ we have $\phi, \overline{\phi}\in (H^\infty+C(\mathbb{T}))^{-1}$.
$(\Leftarrow)$ Assume $\phi, \overline{\phi}\in (H^\infty+C(\mathbb{T}))^{-1}$. Then for some functions $g,h\in H^\infty+C(\mathbb{T})$, $\phi g=\overline{\phi}h=1$. This means $\overline{\phi}\overline{g}=\overline{\phi}h$. Thus $\overline{g}=h$ and $g\in \left[H^\infty+C(\mathbb{T})\right]\bigcap \overline{\left[H^\infty+C(\mathbb{T})\right]}$. So $g\in QC$. Therefore $\phi\in \mathcal{I}$.
From the lemma, we can see that to fully classify $\mathcal{I}$, we must first classify $(H^\infty+C(\mathbb{T}))^{-1}$. To do this, we will need Lemma \[inHardy\] and the following Theorem from [@sarason].
\[uniModInv\] Let $n\in\mathbb{Z}$ and $w$ be a unimodular function in $(H^\infty+C(\mathbb{T}))^{-1}$ satisfying ${\operatorname{ind}}(w)=n$. Then for some $u,v\in C_{\mathbb{R}}(\mathbb{T})$, $w=\chi_n\exp(i(u+\mathcal{C}(v)))$.
\[inHardy\] Let $w\in L_{\mathbb{R}}^\infty$. Then $\exp(w-i\widetilde{w})$ is invertible in $H^\infty$.
Let $G:\mathbb{D}\rightarrow\mathbb{C}$ be defined by $$G(re^{it})=\exp\left(\widehat{w}(re^{it})+i\mathcal{C}\left(w\right)(re^{it})\right).$$ Then from Theorem 11.32 [@Rud page 249] and Theorem 17.16 [@Rud page 343], $G$ is analytic on $\mathbb{D}$ and $\lim_{r\nearrow 1}G(re^{it})$ exists for almost every $e^{it}\in\mathbb{T}$. Let $G^*:\mathbb{T}\rightarrow \mathbb{C}$ be defined by $$G^*(e^{it})=\left\{
\begin{array}{cc}
\lim_{r\nearrow 1}G(re^{it})& \mbox{ if } \lim_{r\nearrow 1}G(re^{it}) \mbox{ exists}\\
0 & \mbox{otherwise}
\end{array}\right.$$ Then by Theorem 11.23 [@Rud page 244], and above $$G^*=\exp(w-i\widetilde{w}) \text{ almost everywhere on }\mathbb{T}.$$ It follows that $G^*\in L^\infty(\mathbb{T})$. Moreover by the proof of Theorem 11.32 [@Rud page 249], $G=\widehat{G^*}$. Thus by Lemma 6.43 [@Doug page 150-151] $$G(re^{i\theta})=\sum_{n\in\mathbb{Z}}^\infty a_{G^*}(n)r^{|n|}\chi_n \text{ for all }z\in\mathbb{D}.$$ Since $G$ is analytic on $\mathbb{D}$, we must have $a_{G^*}(n)=0$ for all $n<0$. Hence $G^*\in H^\infty$. Since $w\in L_{\mathbb{R}}^\infty$ was arbitrary, the above argument also implies $\exp(-w+i\widetilde{w})\in H^\infty$. Therefore $\exp(w-i\widetilde{w})$ is invertible in $H^\infty$.
Now we will classify $(H^\infty+C(\mathbb{T}))^{-1}$.
\[saras\] Let $f\in L^\infty(\mathbb{T})$. Then $$f\in (H^\infty+C(\mathbb{T}))^{-1} \text{ if and only if } f=\chi_n\exp(ig)\exp(w-i\widetilde{w})$$ for some $n\in\mathbb{Z}$, $w\in L_\mathbb{R}^\infty$ and $g\in VMO_{\mathbb{R}}$.
Let $g\in VMO_\mathbb{R}$, $n\in\mathbb{Z}$ and $w\in L_\mathbb{R}^\infty$. Then by $g=h-\widetilde{u}$ for some $u,h\in C_\mathbb{R}(\mathbb{T})$. Thus $\exp(ig)=\exp(ih)\exp(u-i\widetilde{u})\exp(-u)$ Then from Lemma \[inHardy\], $\exp(w-i\widetilde{w})\exp(ih)\exp(u-i\widetilde{u})\exp(-u)\in (H^\infty+C(\mathbb{T}))^{-1}$. Hence $\chi_n\exp(ig)\exp(w-i\widetilde{w})\in (H^\infty+C(\mathbb{T}))^{-1}$.
Assume $f\in (H^\infty+C(\mathbb{T}))^{-1}$. It follows from the continuous functional calculus [@zhuAl page 62] that $\ln|f|\in L_\mathbb{R}^\infty$ and thus by Lemma \[inHardy\] $\exp\left[\ln|f|-i\left(\widetilde{\ln|f|}\right)\right]\in (H^\infty+\mathcal{C}(\mathbb{T}))^{-1}$. It is also true that\
$\left|\exp\left[\ln|f|-i\left(\widetilde{\ln|f|}\right)\right]\right|=|f|$. Hence by Theorem \[uniModInv\], and , $f\exp\left[-\ln|f|+i\left(\widetilde{\ln|f|}\right)\right]=\chi_n\exp(ig)$ for some $n\in\mathbb{Z}$ and $g\in VMO_{\mathbb{R}}$. Therefore $f=\chi_n\exp(w-i\widetilde{w})\exp(ig)$ for some $n\in\mathbb{Z}$, $w\in L_\mathbb{R}^\infty$ and $g\in VMO_{\mathbb{R}}$.
We will now prove that the only integer valued functions in $VMO$ are the constant functions. Recall that for any measurable function $g:\mathbb{T}\rightarrow\mathbb{C}$, the essential range of $g$ is the set $R_g$ defined by $$R_g=\{\lambda\in\mathbb{C}:\left|\{e^{it}\in\mathbb{T}:|g(e^{it})-\lambda|<\epsilon\}\right|>0 \text{ for every }\epsilon>0\}.$$
\[essRan\] If $g\in VMO_\mathbb{R}$, then $R_g$ is a connected subset of $\mathbb{R}$.
From the definition of $R_g$, we can see that $g(\mathbb{T})\subseteq \mathbb{R}$ implies $R_g\subseteq \mathbb{R}$. Thus it suffices to show if $a,b\in R_g$ with $a<b$ and $a<c<b$, then $c\in R_g$.
Assume $c\notin R_g$. Then for some $\gamma>0$, $|g^{-1}(c-\gamma,c+\gamma)|=0$. Choose $0<\epsilon\leq \gamma$ so that $a< c-\epsilon$ and $c+\epsilon<b$. Let $A=g^{-1}\left((-\infty,c-\epsilon]\right)$ and $B=g^{-1}\left([c+\epsilon,\infty)\right)$. Clearly $B\subseteq \mathbb{T}\setminus A$. Then $a,b\in R_g$ implies $|A|>0$ and $|\mathbb{T}\setminus A|>0$. Let $\omega_A:\mathbb{T}\rightarrow\{0,1\}$ be defined by
$$\omega_A(x)=
\left\{
\begin{array}{cc}
1 & \mbox{ if } x\in A\\
0 & \mbox{if } x\notin A\\
\end{array}
\right.$$
and let $I$ be any subarc of $\mathbb{T}$. Since $\frac{1}{|I|}\int_{I}\omega_A=\frac{|I\cap A|}{|I|}$, we have $$\begin{split}&\frac{1}{|I|}\int_{I}\left|\omega_A-\frac{|I\cap A|}{|I|}\right|^2=\\&=\frac{1}{|I|}\int_{I\cap A}\left|\omega_A-\frac{|I\cap A|}{|I|}\right|^2+\frac{1}{|I|}\int_{I\cap (\mathbb{T}\setminus A)}\left|\omega_A-\frac{|I\cap A|}{|I|}\right|^2\\&=\frac{1}{|I|}\int_{I\cap A}\left|1-\frac{|I\cap A|}{|I|}\right|^2+\frac{1}{|I|}\int_{I\cap (\mathbb{T}\setminus A)}\left|0-\frac{|I\cap A|}{|I|}\right|^2\\&=\frac{|I\cap A||I\cap (\mathbb{T}\setminus A)|\left(|I\cap A|+|I\cap (\mathbb{T}\setminus A)|\right)}{|I|^3}\\&=\frac{|I\cap A||I\cap (\mathbb{T}\setminus A)|}{|I|^2}\\&=\frac{|I\cap A||I\cap B|}{|I|^2}\end{split}$$ where the latter equality follows from the fact that $|\mathbb{T}\setminus A|=|B|+|g^{-1}\left((c-\epsilon,c+\epsilon)\right)|=|B|$. By proposition 5.3 [@XiaVmo page 461],\
$\omega_A\notin VMO_\mathbb{R}$. So for some subarc $J$ of $\mathbb{T}$, $$\lim_{|J|\rightarrow0}\frac{|J\cap A||J\cap B|}{|J|^2}\neq0.$$
On the other hand direct calculation yields $$\frac{1}{|I|}\int_I|g-g_I|^2=\frac{1}{2|I|^2}\int_I\int_I|g(t)-g(x)|^2dxdt$$ for any subarc $I$ of $\mathbb{T}$ [@zhuOp page 276]. Thus $$\begin{split}\lim_{|I|\rightarrow0}\frac{1}{|I|}\int_I|g-g_I|^2&\geq \lim_{|I|\rightarrow0}\frac{1}{2|I|^2}\int_{I\cap A}\int_{I\cap B}|g(t)-g(x)|^2dxdt\\&\geq \lim_{|I|\rightarrow0}\frac{(2\epsilon)^2|I\cap A||I\cap B|}{2|I|^2}.\end{split}$$ So since $g\in VMO_\mathbb{R}$, $\lim_{|J|\rightarrow0}\frac{4\epsilon^2|J\cap A||J\cap B|}{2|J|^2}=0$ and which implies $$\lim_{|J|\rightarrow0}\frac{|J\cap A||J\cap B|}{|J|^2}=0.$$ This is a contradiction by the above statements. Therefore $c\in R_g$ and $R_g$ is connected.
Hence we get the following corollary.
\[Z\] If $f\in VMO_{\mathbb{R}}$ and integer valued, then $f$ is constant.
Proof of Theorem \[classI\]
---------------------------
By Lemma \[exp\], Lemma \[I\] and Proposition \[saras\], $$\{\chi_n\}_n\exp\left(QC_\mathbb{R}\right)\exp\left(iVMO_{\mathbb{R}}\right)\subseteq \mathcal{I}.$$
Let $\phi\in\mathcal{I}$. Then by Lemma \[I\], $\phi,\overline{\phi}\in (H^\infty+C(\mathbb{T}))^{-1}$. So by Proposition \[saras\], $$\phi=\chi_n\exp(w-i\widetilde{w})\exp(ig)$$ for some $n\in\mathbb{Z}$, $w\in L_\mathbb{R}^\infty$ and $g\in VMO_{\mathbb{R}}$. This yields $$\exp(2w)=\phi\overline{\phi}\in \mathcal{I}.$$ This implies $0\notin \sigma\left(\exp(2w)\right)$ where $\sigma\left(\exp(2w)\right)$ is the spectrum of $\exp(2w)$ as an element of $QC$. Then by Corollary 9.5 [@zhuAl page 57] and the fact that $\sigma\left(\exp(2w)\right)$ is compact, $\sigma\left(\exp(2w)\right)\subseteq [c,\|\exp(2w)\|_\infty]$ for some $c>0$. Then by the continuous functional calculus [@zhuAl page 62], $\frac{1}{2}\ln(\exp(2w))\in QC$. It follows that $w\in QC_\mathbb{R}$. Then by and , $\widetilde{w}\in VMO_{\mathbb{R}}$. This means $$\phi=\chi_n\exp(w)\exp(ih)$$ where $h=\widetilde{w}+g$. Hence $$\mathcal{I}\subseteq \{\chi_n\}_n\exp\left(QC_\mathbb{R}\right)\exp\left(iVMO_{\mathbb{R}}\right).$$ Therefore Theorem \[classI\] holds.
Corollary to Theorem \[classI\]
-------------------------------
For any $k\in\mathbb{Z}$, let $$I_k=\{\phi\in \mathcal{I}:{\operatorname{ind}}(\phi)=k\}.$$ Then by Theorem \[classI\] and the next lemma, we can completely classify $I_k$.
\[ind\] For any $g\in VMO_{\mathbb{R}}$, $w\in L_\mathbb{R}^\infty$ and $n\in\mathbb{Z}$, $${\operatorname{ind}}(\chi_n)=n, \ {\operatorname{ind}}(\exp(ig))=0\text{ and } {\operatorname{ind}}(\exp(w-i\widetilde{w}))=0.$$
Let $g\in VMO_{\mathbb{R}}$, $w\in L_\mathbb{R}^\infty$ and $n\in\mathbb{Z}$. By Lemma 6.4.3 [@Doug page 151-152], $\widehat{1}=1$ and $\widehat{\chi_n}(z)=z^n$ for all $z\in\mathbb{D}$ and all $n\in\mathbb{Z}$. This yields $$\label{indexConst}{\operatorname{ind}}(\chi_n)=n \text{ for every }n\in\mathbb{Z}.$$
Since $w\in L_\mathbb{R}^\infty$, $\exp(w-i\widetilde{w})$ is invertible in $H^\infty$ by Lemma \[inHardy\]. Moreover with $f=\exp(w-i\widetilde{w})$, $T_f\in \mathfrak{F}(H^2)$ by Lemma \[fredI\] below and $T_f$ is invertible by Proposition \[commToe\]. It follows by Corollary 7.25 [@Doug page 165] that $j(T_f)=0$. Hence by , $$\label{indL}{\operatorname{ind}}(\exp(w-i\widetilde{w}))=0 \text{ for every } w\in L_\mathbb{R}^\infty.$$
By , $\exp(ig)=\exp(i(u-\widetilde{v}))=\exp(iu)\exp(-i\widetilde{v})$ for some $u,v\in C_\mathbb{R}(\mathbb{T})$. Now by Proposition \[saras\] and the fact that $L^\infty$ is a commutative Banach Algebra with respect to function multiplication, the function $F:[0,1]\rightarrow (H^\infty+C(\mathbb{T}))^{-1}$ defined by $F(\lambda)=\exp(-\lambda v)$ is a path in $(H^\infty+C(\mathbb{T}))^{-1}$ between 1 and $\exp(-v)$. It follows that ${\operatorname{ind}}(\exp(-v))={\operatorname{ind}}(1)=0$. Then since $\exp(-i\widetilde{v})=\exp(v-i\widetilde{v})\exp(-v)$, implies ${\operatorname{ind}}(\exp(-i\widetilde{v}))=0$. Similarly the function $G:[0,1]\rightarrow (H^\infty+C(\mathbb{T}))^{-1}$ defined by $G(\lambda)=\exp(i\lambda u)$ is a path in $(H^\infty+C(\mathbb{T}))^{-1}$ between $\exp(iu)$ and 1. Hence ${\operatorname{ind}}(\exp(iu))=0$. So by Lemma \[indSum\] $$\label{indexVMO}{\operatorname{ind}}(\exp(ig))=0.$$ Therefore ${\operatorname{ind}}(\chi_n)=n$ and ${\operatorname{ind}}(\exp(ig))={\operatorname{ind}}(\exp(w-i\widetilde{w}))=0.$
\[classIk\] $$\label{kComp}\begin{split}
& I_0 = \exp\left(QC_\mathbb{R}\right)\exp\left(iVMO_{\mathbb{R}}\right) \\& \text{and } I_k=\chi_kI_0 \ \text{ for all } k\in\mathbb{Z}\end{split}$$
Let $\chi_n\exp(ig)\exp(w)\in\mathcal{I}$ where $g\in VMO_\mathbb{R}$, $w\in QC_\mathbb{R}$ and $n\in\mathbb{Z}$. Clearly $\exp(ig)\exp(w)=\exp(i(g+\widetilde{w}))\exp(w-i\widetilde{w})$. Then for any $k\in\mathbb{Z}$, Lemma \[ind\] yields $${\operatorname{ind}}(\chi_n\exp(ig)\exp(w))=k \text{ if and only if } n=k.$$ Therefore for any $n,k\in\mathbb{Z}$, $\chi_n\exp(ig)\exp(w)\in I_k$ if and only if $n=k$ and holds.
It is implied from that in order to fully understand the sets $\{I_k\}_{k\in\mathbb{Z}}$, it suffices to study $I_0$. This is the approach we will use to classify the path-connected components of $\mathcal{I}$.
Path-connected components
=========================
We will also need the following lemma in order to classify the path-connected components of $\mathcal{I}$ and of $\mathcal{F}$. The lemma follows from the definition of the path-connected components of a topological space [@MunTop page 160].
\[equiv\] Let $X$ be a topological space and $\{A_\alpha\}_{\alpha\in \Omega}$ be a collection of subspaces of $X$. Then $\{A_\alpha\}_{\alpha\in \Omega}$ are the path-connected components of $X$ if all of the following conditions hold:
1. $X=\bigcup_{\alpha\in \Omega}A_\alpha$.
2. $\{A_\alpha\}_{\alpha\in \Omega}$ are pairwise disjoint.
3. $\{A_\alpha\}_{\alpha\in \Omega}$ are all path-connected in $X$.
4. Any nonempty path-connected subspace of $X$ intersects only one $A_\alpha$ from $\{A_\alpha\}_{\alpha\in \Omega}$.
Special Case $I_0$
------------------
### First the Unimodular functions
Let $U_0$ be the set of all unimodular functions in $I_0$. Then by Corollary \[classIk\], $$\label{unimDef}U_0=\exp\left(iVMO_{\mathbb{R}}\right).$$ We will first prove $\{P_g\}_{\left\{\begin{subarray}{l} g\in VMO_{\mathbb{R}}\setminus L_\mathbb{R}^\infty \text{ or }\\ g=0\end{subarray}\right\}}$ are the path-connected components of $U_0$ where for any $g\in VMO_\mathbb{R}$, $P_g$ is defined by .
\[pathCond\] Let $\exp(ig), \exp(ih) \in U_0$ where $g,h\in VMO_{\mathbb{R}}$. Then $\exp(ig)$ is path-connected to $\exp(ih)$ in $U_0$ $\text{ if and only if } g-h\in QC_{\mathbb{R}}$.
Let $\exp(ig)$ and $\exp(ih)$ be as above. Clearly $\exp(ig)$ is path-connected to $\exp(ih)$ in $U_0$ if and only if $\exp(i(g-h))$ is path-connected to 1 in $U_0$. So it suffices to show $\exp(i(g-h))$ is path-connected to 1 in $U_0$ if and only if $g-h\in QC_{\mathbb{R}}$.
$(\Rightarrow)$ Assume $\exp(i(g-h))$ is path-connected to 1 in $U_0$. Recall $G_0$ is the connected component of $\mathcal{I}$ containing 1. Then $\exp(i(g-h))\in G_0$ and Lemma \[exp\] implies $$\exp(i(g-h))=\exp(u+iv)$$ for some $u+iv\in QC$ where $u$ and $v$ are real-valued. It follows that $$1=\exp(u)\mbox{ and } \exp(i(g-h))=\exp(iv).$$ Thus we have $$\label{uniEq}u=0 \mbox{ and } g-h=v+2\pi\sigma \text{ for some } \sigma:\mathbb{T}\rightarrow\mathbb{Z}.$$ Since $u+iv\in QC$, direct calculation and imply $v\in QC_{\mathbb{R}}$. Hence by , $\sigma\in VMO$. Then by Corollary \[Z\], $\sigma$ is constant. Therefore $g-h\in QC_{\mathbb{R}}$.
$(\Leftarrow)$ Now assume $g-h\in QC_{\mathbb{R}}$. Since $QC$ is a commutative Banach Algebra, the function $G:[0,1]\rightarrow U_0$ defined by $G(\lambda)=\exp(i\lambda(g-h))$ is continuous on $[0,1]$. It follows that $G$ is a path between $\exp(i(g-h))$ and 1 in $U_0$. Therefore $\exp(i(g-h))$ is path-connected to 1 in $U_0$.
\[pathComp\] The path-connected components of $U_0$ are\
$\{P_g\}_{\left\{\begin{subarray}{l} g\in VMO_{\mathbb{R}}\setminus L_\mathbb{R}^\infty \text{ or }\\ g=0\end{subarray}\right\}}$
From Lemma \[pathCond\], $$P_g \text{ is path-connected in }U_0\text{ for all } g\in VMO_{\mathbb{R}}.$$ Let $\exp(ih)\in P_w\cap P_q$ with $h\in VMO_{\mathbb{R}}$, $P_w,P_q\in \{P_g\}_{\left\{\begin{subarray}{l} g\in VMO_{\mathbb{R}}\setminus L_\mathbb{R}^\infty \text{ or }\\ g=0\end{subarray}\right\}}$. Then by , $h-w, h-q\in QC_{\mathbb{R}}$. Hence $w-q\in QC_{\mathbb{R}}$. This means $P_w=P_q$. It follows that $\{P_g\}_{\left\{\begin{subarray}{l} g\in VMO_{\mathbb{R}}\setminus L_\mathbb{R}^\infty \text{ or }\\ g=0\end{subarray}\right\}}$ is pairwise disjoint.
By and $P_g\subseteq U_0 \ \text{ for all } g \in VMO_{\mathbb{R}}$. This yields $P_0\cup \bigcup_{g\in VMO_{\mathbb{R}}\setminus L_\mathbb{R}^\infty} P_g\subseteq U_0$. Let $\exp(ih)\in U_0$. Then by definition of $U_0$, $h\in VMO_\mathbb{R}$. If $h\in L_\mathbb{R}^\infty$, then by , $h\in QC_\mathbb{R}$. Thus $\exp(ih)\in P_0$. If $h\notin L_\mathbb{R}^\infty$, then $\exp(ih)\in P_h\subseteq\bigcup_{g\in VMO_{\mathbb{R}}\setminus L_\mathbb{R}^\infty} P_g $. So $U_0\subseteq P_0\cup\bigcup_{g\in VMO_{\mathbb{R}}\setminus L_\mathbb{R}^\infty} P_g$. Thus $$U_0=P_0\cup \bigcup_{g\in VMO_{\mathbb{R}}\setminus L_\mathbb{R}^\infty} P_g.$$
Let $V$ be any nonempty path-connected subspace of $U_0$ such that $V\cap P_h\neq\emptyset$ and $V\cap P_w\neq\emptyset$ for some $P_w,P_h\in\{P_g\}_{\left\{\begin{subarray}{l} g\in VMO_{\mathbb{R}}\setminus L_\mathbb{R}^\infty \text{ or }\\ g=0\end{subarray}\right\}}$. Then since $\exp(ih)\in P_h$ and $\exp(iw)\in P_w$, $\exp(ih)$ is path-connected to $\exp(iw)$ in $U_0$. So by Lemma \[pathCond\] $w-h\in QC_\mathbb{R}$ and $P_h=P_w$. Therefore by Lemma \[equiv\] $\{P_g\}_{\left\{\begin{subarray}{l} g\in VMO_{\mathbb{R}}\setminus L_\mathbb{R}^\infty \text{ or }\\ g=0\end{subarray}\right\}}$ are the path-connected components of $U_0$.
### Path-connected components in $I_0$
We can now completely classify the path-connected components of $I_0$.
\[connIo\] The path-connected components of $I_0$ are $\{Q_g\}_{\left\{\begin{subarray}{l} g\in VMO_{\mathbb{R}}\setminus L_\mathbb{R}^\infty \text{ or }\\ g=0\end{subarray}\right\}}$ where $Q_g$ is defined by for any $g\in VMO_\mathbb{R}$.
Let $x,y\in QC_\mathbb{R}$. Since $QC$ is a Banach Algebra, the function $F:[0,1]\rightarrow \exp(QC_\mathbb{R})$ defined by $F(\lambda)=\exp(x+\lambda(y-x))$ is a path between $\exp(x)$ and $\exp(y)$. Thus $\exp(QC_\mathbb{R})$ is a path-connected subset of $\mathcal{I}$. So by Lemma \[pathComp\] and Corollary \[classIk\], $$Q_g \text{ is path-connected in } I_0\text{ for all } g\in VMO_{\mathbb{R}}.$$
Let $\exp(w)\exp(ih)\in Q_q\cap Q_y$, where $w\in QC_{\mathbb{R}}$, $h\in VMO_\mathbb{R}$, and $Q_q,Q_y\in \{Q_g\}_{\left\{\begin{subarray}{l} g\in VMO_{\mathbb{R}}\setminus L_\mathbb{R}^\infty \text{ or }\\ g=0\end{subarray}\right\}}$. Then for some $u\in QC_\mathbb{R}$ and $v\in VMO_\mathbb{R}$ with $\exp(iv)\in P_q$, $\exp(w)\exp(ih)=\exp(u)\exp(iv)$. Then $w=u$ and $h-v=2\pi\eta$ for some integer valued function $\eta$ on $\mathbb{T}$. By Corollary \[Z\], $\eta$ is constant and $\exp(ih)\in P_q$. Similarly $\exp(ih)\in P_y$. Then by Lemma \[pathComp\], $P_q=P_y$. This means $\{Q_g\}_{\left\{\begin{subarray}{l} g\in VMO_{\mathbb{R}}\setminus L_\mathbb{R}^\infty \text{ or }\\ g=0\end{subarray}\right\}}$ is pairwise disjoint.
By Corollary \[classIk\] and Lemma \[pathComp\], $$I_0=\exp(QC_\mathbb{R})U_0=\exp(QC_\mathbb{R})P_0\cup \bigcup_{g\in VMO_{\mathbb{R}}\setminus L_\mathbb{R}^\infty} \exp(QC_\mathbb{R})P_g.$$ Hence by , $$I_0=Q_0\cup\bigcup_{g\in VMO_{\mathbb{R}}\setminus L_\mathbb{R}^\infty}Q_g.$$
Let $V$ be any nonempty path-connected subspace of $I_0$ satisfying $V\cap Q_h\neq\emptyset$ and $V\cap Q_w\neq\emptyset$ for some $Q_h,Q_w\in\{Q_g\}_{\left\{\begin{subarray}{l} g\in VMO_{\mathbb{R}}\setminus L_\mathbb{R}^\infty \text{ or }\\ g=0\end{subarray}\right\}}$. Then since $\exp(ih)\in Q_h$ and $\exp(iw)\in Q_w$, $\exp(ih)$ is path-connected to $\exp(iw)$ in $I_0$ by some path $F$. This means $\frac{F}{|F|}$ is a path in $U_0$ between $\exp(iw)$ and $\exp(ih)$. So by Lemma \[pathComp\], $P_h= P_w$. Thus $Q_h=Q_w$. Therefore by Lemma \[equiv\] $\{Q_g\}_{\left\{\begin{subarray}{l} g\in VMO_{\mathbb{R}}\setminus L_\mathbb{R}^\infty \text{ or }\\ g=0\end{subarray}\right\}}$ are the path-connected components of $I_0$.
Case of $I_k$ with $k\neq0$
---------------------------
Now that we have classified the path-connected components of $I_0$, we can classify the path-connected components of $I_k$ for any $k\in\mathbb{Z}\setminus\{0\}$. This part of the classification is very easy due to the classification of $I_0$ and Proposition \[homIndex\] below.
\[homIndex\] For any $k\in\mathbb{Z}$, $I_k$ is homeomorphic to $I_0$.
Let $k\in\mathbb{Z}$. Let $\Gamma_k:I_0\rightarrow I_k$ and $\Theta_k:I_k\rightarrow I_0$ be defined by $$\Gamma_k(\phi)=\chi_k\phi \mbox{ and } \Theta_k(\psi)=\chi_{-k}\psi.$$ Both $\Gamma_k$ and $\Theta_k$ are well-defined by Corollary \[classIk\]. Since $QC$ is a Banach Algebra, $\Gamma_k$ and $\Theta_k$ are both continuous on their domains. Moreover since $\left(\chi_k\right)^{-1}=\chi_{-k}$ with respect to multiplication, we have $\Gamma_k^{-1}=\Theta_k$. Therefore $\Gamma_k$ is a homeomorphism and $I_k$ is homeomorphic to $I_0$.
\[connIk\] For any $k\in\mathbb{Z}$, the path-connected components of $I_k$ are $\{\chi_kQ_g\}_{\left\{\begin{subarray}{l}g\in VMO_{\mathbb{R}}\setminus L_\mathbb{R}^\infty \text{ or }\\ g=0\end{subarray}\right\}}$.
Let $\Gamma_k$ be as in Proposition \[homIndex\]. Then $$\Gamma_k(Q_g)=\chi_kQ_g \text{ for all } g\in VMO_{\mathbb{R}}.$$ Therefore a straightforward calculation using $\Gamma_k$ and $\Theta_k$ along with Theorem \[connIo\] , Proposition \[homIndex\] and Lemma \[equiv\], $\{\chi_kQ_g\}_{\left\{\begin{subarray}{l} g\in VMO_{\mathbb{R}}\setminus L_\mathbb{R}^\infty \text{ or }\\ g=0\end{subarray}\right\}}$ are the path-connected components of $I_k$.
Proof of Theorem \[mainI\]
==========================
By Theorem \[connIk\], $$\{\chi_kQ_g\}_{\left\{\begin{subarray}{l} k\in\mathbb{Z} \text{ and either }\\ g\in VMO_{\mathbb{R}}\setminus L_\mathbb{R}^\infty \text{ or }\\ g=0\end{subarray}\right\}} \text{ is a collection of path-connected sets in } \mathcal{I}$$ and $$\mathcal{I}=\bigcup_{\left\{\begin{subarray}{l} k\in\mathbb{Z} \text{ and either } \\ g\in VMO_{\mathbb{R}}\setminus L_\mathbb{R}^\infty \text{ or }g=0\end{subarray}\right\}}\chi_kQ_g.$$
Let $\chi_p\exp(w)\exp(ih)\in \chi_nQ_q\cap \chi_mQ_y$ for some $w\in QC_\mathbb{R}$, $h\in VMO_\mathbb{R}$, $p\in\mathbb{Z}$ and $\chi_mQ_y,\chi_nQ_q\in\{\chi_kQ_g\}_{\left\{\begin{subarray}{l} k\in\mathbb{Z} \text{ and either }\\ g\in VMO_{\mathbb{R}}\setminus L_\mathbb{R}^\infty \text{ or }\\ g=0\end{subarray}\right\}}$. Then by by Theorem \[connIk\], $\chi_p\exp(w)\exp(ih)\in I_n\cap I_m$. It follows that ${\operatorname{ind}}(\chi_p\exp(w)\exp(ih))=m$ and ${\operatorname{ind}}(\chi_p\exp(w)\exp(ih))=n$. So by Lemma \[ind\] $m=n=p$ and by Theorem \[connIk\], $\chi_nQ_g=\chi_mQ_y$. That is $\{\chi_kQ_g\}_{\left\{\begin{subarray}{l} k\in\mathbb{Z} \text{ and either }\\ g\in VMO_{\mathbb{R}}\setminus L_\mathbb{R}^\infty \text{ or }\\ g=0\end{subarray}\right\}}$ is pairwise disjoint.
Let $W$ be a nonempty path-connected subspace of $\mathcal{I}$ so that $W\cap \chi_nQ_q\neq\emptyset$ and $W\cap \chi_mQ_y\neq\emptyset$ for some $\chi_mQ_y,\chi_nQ_q\in\{\chi_kQ_g\}_{\left\{\begin{subarray}{l} k\in\mathbb{Z} \text{ and either }\\ g\in VMO_{\mathbb{R}}\setminus L_\mathbb{R}^\infty \text{ or }\\ g=0\end{subarray}\right\}}$. Then since $\chi_n\exp(iq)\in \chi_nQ_q$ and $\chi_m\exp(iy)\in \chi_mQ_y$, $\chi_n\exp(iq)$ is path-connected to $\chi_m\exp(iy)$ in $\mathcal{I}$. Let $F:[0,1]\rightarrow\mathcal{I}$ be the path. Since ${\operatorname{ind}}$ is continuous on $\mathcal{I}$, we must have $m=n={\operatorname{ind}}(F(\lambda))$ for all $\lambda\in[0,1]$. Hence $\chi_n\exp(iq)$ is path-connected to $\chi_n\exp(iy)$ by $F$ in $I_n$. Then from Theorem \[connIk\], $Q_y=Q_q$. Therefore $\chi_mQ_y=\chi_nQ_q$ and by Lemma \[equiv\], $\{\chi_kQ_g\}_{\left\{\begin{subarray}{l} k\in\mathbb{Z} \text{ and either }\\ g\in VMO_{\mathbb{R}}\setminus L_\mathbb{R}^\infty \text{ or }\\ g=0\end{subarray}\right\}}$ are the path-connected components of $\mathcal{I}$.
Uncountably many path-connected components of $\mathcal{I}$ {#uncount}
===========================================================
In this section we will prove $\mathcal{I}$ has uncountably many path-connected components. To do this we will construct an explicit example of a function $H$ such that $H\in VMO_\mathbb{R}\setminus L_\mathbb{R}^\infty$. We will also use $H$ later on to show $\mathcal{F}$ has uncountably many path-connected components. To construct $H$, we will need two following theorems from [@TrigSer page 182] and [@SeriesFourier page 101] respectively .
\[trigThm\] Suppose $a_v\geq a_{v+1}$ and $a_v\rightarrow0$. Then a necessary and sufficient condition for the uniform convergence of $\sum_{v=1}^\infty a_v\sin(vx)$ is $va_v\rightarrow 0$.
\[fourSeri\] If the coefficients $a_n$ and $b_n$ of the series $$\frac{a_0}{2}+\sum_{n=1}^\infty a_n\cos(nx), \sum_{n=1}^\infty b_n\sin(nx)$$ are positive and decrease monotonically to zero as $n\rightarrow\infty$, then both series converge uniformly on any interval $[a,b]$ which does not contain points of the form $x=2k\pi$ $(k=0,\pm 1\pm 2,\ldots)$.
Now we will construct $H$.
\[Ex\] There exists a function $H\in VMO_\mathbb{R}$ such that $H\notin L_\mathbb{R}^\infty$.
For each $N\geq 2$ let $g_N:\mathbb{R}\rightarrow\mathbb{R}$ be defined by $g_N(x)=\sum_{k=2}^N \frac{\sin(kx)}{k\ln(k)}$. By Theorem \[trigThm\], $\{g_N\}_{N=2}^\infty$ converges uniformly on $\mathbb{R}$. Let $g:\mathbb{R}\rightarrow \mathbb{R}$ be defined by $g(x) =\sum_{k=2}^\infty \frac{\sin(kx)}{k\ln(k)}$. Then $g\in C_\mathbb{R}(\mathbb{T})$ and by , $\widetilde{g}\in VMO_\mathbb{R}$.
Direct calculation shows $\widetilde{g_N}(x)=-\sum_{k=2}^N \frac{\cos(kx)}{k\ln(k)}$ for all $x\in\mathbb{R}$. Recall that by the proof of Lemma \[DblHil\], the Hilbert transform is bounded on $L^2(\mathbb{T})$ with respect to the $L^2(\mathbb{T})$ norm $\|\cdot\|_2$. So by the preceding paragraph, $\lim_{N\rightarrow\infty}\|\widetilde{g_N}-\widetilde{g}\|_2 =0$. Also by Theorem \[fourSeri\], $-\sum_{k=2}^\infty \frac{\cos(kx)}{k\ln(k)}$ converges uniformly on each closed subinterval of $\mathbb{R}\setminus \{2k\pi\}_{k\in\mathbb{Z}}$. Hence $\widetilde{g}(x)=-\sum_{k=2}^\infty \frac{\cos(kx)}{k\ln(k)}$ for almost every $x\in\mathbb{R}$. Note that since $\sum_{k=2}^\infty \frac{1}{k\ln(k)}=\infty$, $\{\widetilde{g_N}(0)\}_{N=2}^\infty$ is not bounded. This means $\{\widetilde{g_N}\}_{n=2}^\infty$ is not uniformly bounded in $N$ and $x$.
Let $H:\mathbb{R}\rightarrow\mathbb{R}$ be defined by $$H(x) = \left\{
\begin{array}{ll}
-\sum_{k=2}^\infty \frac{\cos(kx)}{k\ln(k)}& \mbox{ if }\ \widetilde{g}(x)=-\sum_{k=2}^\infty \frac{\cos(kx)}{k\ln(k)}\\
0 & \mbox{otherwise}
\end{array}
\right.$$
Then $H\in VMO_\mathbb{R}$. Suppose $H\in L_\mathbb{R}^\infty$. Let $\sigma_N:\mathbb{T}\rightarrow \mathbb{C}$ be defined by $\sigma_N(x)=\frac{1}{2(N+1)\pi}\int_0^{2\pi} H(x-t)\left(\frac{\sin\left(\frac{(N+1)t}{2}\right)}{\sin\left(\frac{t}{2}\right)}\right)^2dt$. Then $\left|\sigma_N(x)\right|\leq \|H\|_\infty$ for every $N\geq 2$ and $x\in\mathbb{T}$ [@krantz page 63]. Moreover by [@krantz page 60] $\sigma_N(x)=\frac{1}{n+1}\sum_{j=2}^n \widetilde{g_N}(x)$. Furthermore a straightforward induction argument shows $\widetilde{g_N}(x)-\sigma_N(x)=\frac{1}{N+1}\sum_{j=2}^N \frac{\cos(jx)}{\ln(j)}$. Hence for some $M_1>0$, $|\widetilde{g_N}(x)-\sigma_N(x)|\leq M_1$ for all $N\geq 2$ and all $x\in\mathbb{T}$. It follows that $\{\widetilde{g_N}(x)\}_{N=2}^\infty$ is uniformly bounded in $x$ and $N$, which is a contradiction by the above statements. So $H\notin L_\mathbb{R}^\infty$. Therefore $H\in VMO_\mathbb{R}\setminus L_\mathbb{R}^\infty$.
\[uncountI\] $\mathcal{I}$ has uncountably many path-connected components.
Let $H$ be as in Proposition \[Ex\]. Then by , $H\notin QC_{\mathbb{R}}$. By , and Corollary \[Z\], $$Q_g=Q_q \text{ if and only if } g-q\in QC_{\mathbb{R}}$$ for any $g,q\in VMO_{\mathbb{R}}$. Hence for all $\beta, \sigma\in\mathbb{R}$, $$\beta \neq\sigma \text{ implies }Q_{\beta H}\neq Q_{\sigma H}.$$ Thus $\{Q_{\beta H}\}_{\beta\in\mathbb{R}}$ is an uncountable collection of path-connected components in $I_0$ and $I_0$ has uncountably many path-connected components. Therefore by Theorem \[mainI\], $\mathcal{I}$ has uncountably many path-connected components.
In fact by Theorem \[connIk\] and Theorem \[uncountI\], $I_k$ has uncountable many path-connected components for all $k\in\mathbb{Z}$.
Classification of $\mathcal{F}$ and the Path-connected components of $\mathcal{F}$ {#fred}
==================================================================================
Here we will classify $\mathcal{F}$ and the path-connected components of $\mathcal{F}$. As we will see, the classification of the path-connected components of $\mathcal{F}$ is based on the classification of the path-connected components of $\mathcal{I}$. So one can really say that Theorem \[mainF\] is a corollary to Theorem \[mainI\]. We will also show that $\mathcal{F}$ has uncountably many path-connected components.
For this section, we will use the following lemma which holds by proposition 7.12 [@Doug page 161] and the proof of proposition 7.11 [@Doug page 161].
\[compactToe\] Let $f\in L^\infty(\mathbb{T})$ and $K\in\mathfrak{LC}(H^2)$. Then $\|T_f+K\|\geq\|T_f\|$.
The above implies that the only compact Toeplitz operator on $H^2$ is the zero operator. That is, $T_f\in\mathfrak{LC}(H^2)$ if and only if $f=0$.
Classification of $\mathcal{F}$
===============================
Recall from above that $\mathfrak{I}(QC)$ is the $C^*$-subalgebra of $\mathfrak{L}(H^2)$ generated by $\{T_f\}_{f\in QC}$. The next lemma is well-known in the literature.
\[ToeQC\] $\mathfrak{I}(QC)=\{T_f+K:f\in QC, K\in\mathfrak{LC}(H^2)\}$
We will also need the following lemma, which is Corollary 7.34 [@Doug page 168].
\[fredI\] Let $f\in H^\infty+C(\mathbb{T})$. Then $T_f\in\mathfrak{F}(H^2)$ if and only if $f\in (H^\infty+C(\mathbb{T}))^{-1}$.
We can now prove the following Theorem which describes $\mathcal{F}$ completely.
\[F\] $\mathcal{F}=\{T_f+K:f\in \mathcal{I}, K\in \mathfrak{LC}(H^2)\}$.
By Lemma \[ToeQC\], $\mathcal{F}=\mathfrak{F}(H^2)\cap\{T_f+K:f\in QC, K\in\mathfrak{LC}(H^2)\}$. Also by proposition 5.15 [@Doug page 113], $T_f+K\in\mathfrak{F}(H^2)$ if and only if $T_f\in\mathfrak{F}(H^2)$ for any $K\in\mathfrak{LC}(H^2)$ and any $f\in L^\infty\left(\mathbb{T}\right)$. So it suffices to show $T_f\in\mathcal{F}$ if and only if $f\in \mathcal{I}$.
$(\Leftarrow)$ If $f\in \mathcal{I}$ then $f\in (H^\infty+C(\mathbb{T}))^{-1}$. So by Lemma \[fredI\], $T_f\in \mathfrak{F}(H^2)$. Then by definition of $\mathcal{F}$ and Lemma \[ToeQC\], $T_f\in \mathcal{F}$.
$(\Rightarrow)$ Suppose $T_f\in \mathcal{F}$. Then $T_f\in\mathfrak{F}(H^2)$ and $T_f=T_g+K$ for some $g\in QC$ and $K\in\mathfrak{LC}(H^2)$. Then $T_{f-g}\in\mathfrak{LC}(H^2)$ which means $f=g$ by Lemma \[compactToe\]. So $f\in QC$. Also, by proposition 5.15 [@Doug page 115], $T_{\overline{f}}\in\mathfrak{F}(H^2)$. Then by Lemma \[fredI\], $f, \overline{f}\in (H^\infty+C(\mathbb{T}))^{-1}$. Therefore by Lemma \[I\], $f\in \mathcal{I}$.
Path-connected components of the elements of $\mathcal{F}$ of index k
=====================================================================
For any $k\in\mathbb{Z}$, let $$E_k=\{T_f+K\in \mathcal{F}: j(T_f+K)=k\}.$$ By Section \[SectInd\], $$j(T_f+K)=j(T_f)=-{\operatorname{ind}}(f) \text{ for all } f\in \mathcal{I} \text{ and for all } K\in\mathfrak{LC}(H^2).$$ Hence $$\label{firstDefEk}E_k=\{T_f+K\in \mathcal{F}: f\in I_{-k}, K\in\mathfrak{LC}(H^2)\}.$$
Let $\zeta:L^\infty\left(\mathbb{T}\right)\rightarrow \mathfrak{L}(H^2)$ be the map defined by $\zeta(f)=T_f$. Clearly $\zeta$ is well-defined. Moreover by proposition 7.4 [@Doug page 159] and Corollary 7.8 [@Doug page 160], $\zeta$ is \*-linear and an isometry. Let $\xi=\zeta|_{\mathcal{I}}$. Notice $\xi$ is continuous on $\mathcal{I}$. We will now use $\xi$ and Theorem \[connIk\] to classify the path-connected components of $E_k$. Recall for each $g\in VMO_{\mathbb{R}}$ and $k\in\mathbb{Z}$, ${_k}V_g$ is defined by .
\[pathEk\] The path-connected components of $E_k$ are $$\{_kV_g\}_{\left\{\begin{subarray}{l} g\in VMO_{\mathbb{R}}\setminus L_\mathbb{R}^\infty \text{ or } \\ g=0\end{subarray}\right\}}.$$
As mentioned in the proof of Lemma \[ToeQC\], $\mathfrak{LC}(H^2)$ is a two-sided \*-ideal in $\mathfrak{L}(H^2)$. This implies $\mathfrak{LC}(H^2)$ is convex in $\mathfrak{L}(H^2)$. Also for any $g\in VMO_{\mathbb{R}}$ and any $k\in\mathbb{Z}$, $$\label{xi}{_k}V_g=\xi(\chi_{-k}Q_g)+\mathfrak{LC}(H^2).$$ It then follows from Theorem \[connIk\] that $${_k}V_g \text{ is path-connected in } E_k \text{ for all } g\in VMO_{\mathbb{R}}.$$ By Theorem \[connIk\] and , we can see that $$E_k = \bigcup_{\left\{\begin{subarray}{l} g\in VMO_{\mathbb{R}}\setminus L_\mathbb{R}^\infty \text{ or } \\ g=0\end{subarray}\right\}} {_k}V_g.$$
Let $T_{f_0}+K_0\in {_k}V_q\cap {_k}V_h$ for some $f_0\in I_{-k}$, $K_0\in\mathfrak{LC}(H^2)$ and ${_k}V_q, {_k}V_h\in\{_kV_g\}_{\left\{\begin{subarray}{l} g\in VMO_{\mathbb{R}}\setminus L_\mathbb{R}^\infty \text{ or } \\ g=0\end{subarray}\right\}}$. Then for some $f_1\in\chi_{-k}Q_h$ and $K_1\in\mathfrak{LC}(H^2)$, $T_{f_0}+K_0=T_{f_1}+K_1$. Thus by Lemma \[compactToe\], $f_0=f_1$ and $f_0\in\chi_{-k}Q_h$. Similarly, $f_0\in\chi_{-k}Q_q$. Then by Theorem \[connIk\], ${_k}V_q= {_k}V_h$. Hence $\{_kV_g\}_{\left\{\begin{subarray}{l} g\in VMO_{\mathbb{R}}\setminus L_\mathbb{R}^\infty \text{ or } \\ g=0\end{subarray}\right\}}$ is pairwise disjoint.
Let $W$ be a nonempty path-connected subspace of $E_k$ with ${_k}V_h\cap W\neq\emptyset$ and ${_k}V_q\cap W\neq\emptyset$ for some ${_k}V_h,{_k}V_q\in\{_kV_g\}_{\left\{\begin{subarray}{l} g\in VMO_{\mathbb{R}}\setminus L_\mathbb{R}^\infty \text{ or } \\ g=0\end{subarray}\right\}}$. Let $f_h=\chi_{-k}\exp(ih)$ and $f_q=\chi_{-k}\exp(iq)$. Then since $T_{f_h}\in {_k}V_h$ and $T_{f_q}\in {_k}V_q$, $T_{f_h}$ is path-connected to $T_{f_q}$ in $E_k$. Let $H:[0,1]\rightarrow E_k$ be the path between $T_{f_q}$ and $T_{f_h}$. For each $\lambda\in[0,1]$, let $T_{f_\lambda}+K_\lambda=H(\lambda)$ where $f_0=f_q$, $f_1=f_h$, $K_\lambda\in \mathfrak{LC}(H^2)$ for each $\lambda\in (0,1)$ and $K_0,K_1$ both equal the zero operator. Then by Lemma \[compactToe\], $$\|f_{\lambda}\|_\infty=\|T_{f_\lambda}\|\leq\|H(\lambda)\| \text{ for all } \lambda\in [0,1].$$ It follows that $f_q$ is path-connected to $f_h$ in $I_{-k}$. Then from Theorem \[connIk\], $\chi_{-k}Q_h=\chi_{-k}Q_q$. Hence ${_k}V_q= {_k}V_h$ and by Lemma \[equiv\], $\{_kV_g\}_{\left\{\begin{subarray}{l} g\in VMO_{\mathbb{R}}\setminus L_\mathbb{R}^\infty \text{ or } \\ g=0\end{subarray}\right\}}$ are the path-connected components of $E_k$.
Proof of Theorem \[mainF\]
==========================
By Theorem \[pathEk\], $${_k}V_g \text{ is path-connected in } \mathcal{F} \text{ for all } g\in VMO_{\mathbb{R}} \text{ and for all } k\in\mathbb{Z}.$$ By Theorem \[F\], and the fact that $\mathcal{I}=\bigcup_{k\in\mathbb{Z}}I_k$, $\mathcal{F}=\bigcup_{k\in\mathbb{Z}} E_k$. Thus another application of Theorem \[pathEk\] yields $$\mathcal{F}=\bigcup_{\left\{\begin{subarray}{l} k\in\mathbb{Z} \text{ and either } \\ g\in VMO_{\mathbb{R}}\setminus L_\mathbb{R}^\infty \text{ or }\\ g=0 \end{subarray}\right\}} {_k}V_g .$$
Suppose for some $f\in\mathcal{I}$ and $K\in\mathfrak{LC}(H^2)$, $T_f+K\in {_k}V_y\cap{_m}V_h$ for some ${_k}V_y,{_m}V_h\in\{{_k}V_g\}_{\left\{\begin{subarray}{l} k\in\mathbb{Z} \text{ and either } \\ g\in VMO_{\mathbb{R}}\setminus L_\mathbb{R}^\infty \text{ or } \\ g=0\end{subarray}\right\}}$. Then for some $q\in \chi_{-k}Q_y$ and $L\in\mathfrak{LC}(H^2)$ we have $T_f+K=T_q+L$. Then by Lemma \[compactToe\] $f=q$ and $f\in \chi_{-k}Q_y$. Similarly $f\in \chi_{-m}Q_h$. Then by Theorem \[mainI\], $\chi_{-k}Q_y=\chi_{-m}Q_h$. It follows that ${_k}V_y={_m}V_h$ and $\{{_k}V_g\}_{\left\{\begin{subarray}{l} k\in\mathbb{Z} \text{ and either } \\ g\in VMO_{\mathbb{R}}\setminus L_\mathbb{R}^\infty \text{ or } \\ g=0\end{subarray}\right\}}$ is pairwise-disjoint.
Suppose $W$ is a nonempty path-connected subspace of $\mathcal{F}$ such that $W\cap {_k}V_y\neq \emptyset$ and $W\cap {_m}V_h\neq\emptyset$ for some ${_k}V_y,{_m}V_h\in\{{_k}V_g\}_{\left\{\begin{subarray}{l} k\in\mathbb{Z} \text{ and either } \\ g\in VMO_{\mathbb{R}}\setminus L_\mathbb{R}^\infty \text{ or } \\ g=0\end{subarray}\right\}}.$ Let $f_y\in Q_y$ and $f_h\in Q_h$. Then since $T_{\chi_{-k}f_y}+K\in {_k}V_y$ and $T_{\chi_{-m}f_h}+K\in{_m}V_h$ where $K\in\mathfrak{LC}(H^2)$, $T_{\chi_{-k}f_y}+K$ is path-connected to $T_{\chi_{-m}f_h}+K$ in $\mathcal{F}$. Then since $j$ is continuous on $\mathcal{F}$ and integer valued, $j(T_{\chi_{-k}f_y}+K)=j(T_{\chi_{-m}f_h}+K)$. That is $m=k$. Then by Theorem \[pathEk\], ${_k}V_y= {_m}V_h$. Therefore by Lemma \[equiv\], $\{{_k}V_h\}_{\left\{\begin{subarray}{l} k\in\mathbb{Z} \text{ and either } \\ g\in VMO_{\mathbb{R}}\setminus L_\mathbb{R}^\infty \text{ or } \\ g=0\end{subarray}\right\}}$ are the path-connected components of $\mathcal{F}$.
Uncountably many path-connected components of $\mathcal{F}$
===========================================================
\[uncountF\] $\mathcal{F}$ has uncountably many path-connected components.
Let $k\in\mathbb{Z}$ and assume for some $h,g\in VMO_\mathbb{R}$, ${_k}V_{h}={_k}V_{g}$. Then by Lemma \[compactToe\], $\chi_{-k}Q_{h}=\chi_{-k}Q_{g}$. Let $H$ be as in Proposition \[Ex\]. It follows from the proof of Theorem \[uncountI\] that $$\beta\neq\gamma \text{ implies } {_k}V_{\beta H}\neq{_k}V_{\gamma H}$$ for all $\beta,\gamma\in\mathbb{R}$. Hence $\{{_k}V_g\}_{\left\{\begin{subarray}{l}g\in VMO_{\mathbb{R}}\setminus L_\mathbb{R}^\infty \text{ or } \\ g=0\end{subarray}\right\}}$ is uncountable. So $E_k$ has uncountably many path-connected components. Therefore $\mathcal{F}$ has uncountably many path-connected components.
In fact since $k$ in the proof of Corollary \[uncountF\] was arbitrary, $E_k$ has uncountable many path-connected components for all $k\in\mathbb{Z}$.
Acknowledgment {#acknowledgment .unnumbered}
--------------
The author would like to thank his dissertation committee for their help in writing this article: Dr. Jingbo Xia, Dr. Lewis Coburn and Dr. Ching Chou.
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|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'Multipartite quantum entanglement serves as a resource for spatially separated parties performing distributed quantum information processing. Any multipartite entangled state can be generated from appropriately distributed bipartite entangled states by local operations and classical communication (LOCC), and in this sense, any distributed process based on shared multipartite entanglement and LOCC is simulatable by using only bipartite entangled states and LOCC. We show here that this reduction scenario does not hold when there exists a limitation on the size of the local quantum system of each party. Under such a limitation, we prove that there exists a set of multipartite quantum states such that these states in the set *cannot* be prepared from any distribution of bipartite entanglement while the states *can* be prepared from a common resource state exhibiting multipartite entanglement. We also show that temporal uses of bipartite quantum communication resources within a limitation of local system sizes are sufficient for preparing this common resource state exhibiting multipartite entanglement, yet there also exist other states exhibiting multipartite entanglement which cannot be prepared even in this setting. Hence, when the local quantum system sizes are limited, multipartite entanglement is an indispensable resource without which certain processes still cannot be accomplished.'
author:
- Hayata Yamasaki
- Alexander Pirker
- Mio Murao
- Wolfgang Dür
- Barbara Kraus
title: |
Multipartite entanglement outperforming bipartite entanglement\
under limited quantum system sizes
---
Introduction
============
Multipartite quantum entanglement ubiquitously appears in many-body quantum systems in condensed matter physics [@A2] and quantum gravity [@R4], and also serves as a resource for multiparty tasks in distributed quantum information processing such as measurement-based quantum computation [@R1; @R2; @R3], distributed sensing [@K1; @E4], and quantum networking [@P3]. Such a distributed setting is also considered as a promising candidate for realizing large-scale quantum computation due to technological limitations on the number of low-noise qubits which can be stored in a single quantum device. In a distributed setting where spatially separable parties can freely perform local operations and classical communication (LOCC), any multipartite entangled state can be prepared by LOCC from initially distributed bipartite entangled states among the parties, using quantum teleportation [@B4]. In this regard, even if multipartite entanglement is used for a task, initially sharing bipartite entangled states is sufficient, and hence, it would be natural to doubt whether multipartite entanglement is necessary for performing tasks by LOCC.
![The task of system-size-limited quantum state transformation, where the parties transform a common resource state represented by blue circles by LOCC into an arbitrary state in a given target set $\left\{\Ket{\psi_0},\Ket{\psi_1},\ldots\right\}$ represented by red circles. To differentiate the capabilities of common resource states exhibiting multipartite entanglement at the top and those consisting only of bipartite entanglement at the bottom, where each connected pair of blue circles represents a bipartite entangled state, we consider the static setting where each party’s local system size for storing the common resource state is limited. We also consider the dynamic setting where the parties have to prepare a common resource state within these limitations by performing quantum communication, in addition to storing the common resource state. The difference in the capabilities arises in terms of achievability of this task.[]{data-label="fig:intro"}](intro2.pdf){width="0.79\linewidth"}
In this paper, we show *nontrivial* examples demonstrating the difference between entangled resource states consisting only of bipartite entanglement and those exhibiting multipartite entanglement. This difference arises when there exists a limitation on the size of each party’s local quantum system, that is, the dimension of the Hilbert space representing the local quantum system. Our comparison between bipartite and multipartite entanglement, motivated by technological limitations on the number of qubits which can be stored in one quantum device, differs from the comparison in the context of quantum key distribution [@E3; @P4] as we consider the cost of LOCC to be negligible. The difference can also be observed in a trivial example of qubits as follows. Consider three parties $A$, $B$, and $C$ sharing two Bell states ${\left(\frac{1}{\sqrt{2}}\left(\Ket{00}+\Ket{11}\right)\right)}^{\otimes 2}$, one of which is between $A$ and $B$, and the other of which is between $B$ and $C$. These two Bell states as a whole are regarded as a state consisting of bipartite entangled states. In this case, once these two Bell states are given to the parties, the parties can transform the two Bell states by LOCC into any three-qubit state shared among $A$, $B$, and $C$, such as the Greenberger-Horne-Zeilinger (GHZ) state $\Ket{\textup{GHZ}}\coloneqq\frac{1}{\sqrt{2}}\left(\Ket{000}+\Ket{111}\right)$ and the $W$ state $\Ket{W}\coloneqq\frac{1}{\sqrt{3}}\left(\Ket{100}+\Ket{010}+\Ket{001}\right)$, which we regard as states exhibiting multipartite entanglement. However, if each party’s local system size is limited to one qubit, the parties cannot store any state consisting of a collection of bipartite entangled states to obtain $\Ket{\textup{GHZ}}$ and $\Ket{W}$ by LOCC while the parties can still store these states exhibiting multipartite entanglement as a resource for performing some task by LOCC.
Apart from the above trivial example of qubits, this paper aims to demonstrate the difference even in cases where the size of local systems of some parties is not limited to one qubit. Given an entangled state transformable into another entangled state, the former state can be considered to have more capability as a resource than the latter state. If such a resource state having more capability is shared among parties, the parties may transform the shared resource state by LOCC into a suitable form for performing a given task. This paradigm yields a *common resource state* [@S3; @G3] transformable into any state in a given set, that is, a resource state having more capability than any state in the set, such as the two Bell states for the set $\left\{\Ket{\textup{GHZ}},\Ket{W}\right\}$. We call this set of states the *target set*. Similarly, Ref. [@H3] also introduces common resource states in terms of state convertibility by stochastic LOCC.
As illustrated in Fig. \[fig:intro\], we consider two settings of state preparation tasks for differentiating capabilities of common resource states consisting only of a collection of bipartite entangled states and those exhibiting multipartite entanglement. We call the tasks *system-size-limited quantum state preparation*, where one of the two settings is called a *static* setting, and the other is called a *dynamic* setting. In the static setting, we analyze each party’s local system size for storing a common resource state for a given target set. For a given target set of states of a multipartite system in general, there may not exist any common resource state in the multipartite system itself transformable by LOCC into all the states in the set. In particular, given a multipartite system where each local dimension is $d$, almost no LOCC transformation among pure states of the system is possible [@V2; @S1; @S2; @H2; @G2; @S4]. This fact implies that, in general, a common resource state for a set of multipartite states may be a state of a higher-dimensional system than that for the set itself. If there is a limitation on each party’s local system size, it may not be possible for the parties to store an entangled state of a higher-dimensional system serving as a common resource state. Despite the efforts to understand properties of multipartite entanglement [@H1; @P1; @E1; @E2; @W1; @B2], general quantitative conditions of the smallest system size for common resource states have not yet been established. In this paper, we provide nontrivial instances where, within a given limitation on local system sizes, the preparation of a state in a given target set is *not* achievable by any common resource state consisting of a collection of bipartite entangled states but it is achievable by a common resource state exhibiting multipartite entanglement. These examples show the difference in the capabilities between these two types of common resource states.
As for the dynamic setting, in addition to considering a limitation on local system sizes for storing a common resource state, the parties are also required preparation of the common resource state within this limitation by performing quantum communication. Some of the common resource states exhibiting multipartite entanglement analyzed in the static setting can be prepared within the limitation using quantum communication. Hence, temporal uses of bipartite quantum communication resources are still sufficient for preparing such common resource states. In contrast, we also show other examples of states exhibiting multipartite entanglement which can be stored but cannot be prepared within a limitation on local system sizes.
The rest of this paper is structured as follows. In Sec. \[sec:common\_resource\_state\], we recall the definition of a common resource state for a given target set. In Sec. \[sec:def\], we introduce the tasks of system-size-limited quantum state preparation in the static setting and the dynamic setting. We analyze system-size-limited quantum state preparation in the static setting in Sec. \[sec:analysis\] and also analyze the dynamic setting in Sec. \[sec:analysis2\]. Our conclusion is given in Sec. \[sec:conclusion\].
\[sec:common\_resource\_state\]Definition of common resource states
===================================================================
We begin with recalling the definition of a common resource state for a given set of states under LOCC [@S3; @G3]. In the following, superscripts of an operator or a ket indicate the Hilbert space on which the operator acts or to which the ket belongs. Note that although we only consider the cases of pure states, generalization to mixed states is straightforward.
To define a target set, consider a quantum system for the states in a target set shared among $N$ parties denoted by $v_1,\ldots,v_N$. The corresponding Hilbert space is denoted by $\mathcal{H}\coloneqq\bigotimes_{k=1}^{N}\mathcal{H}^{v_k}$, where each party $v_k$’s system is represented by a Hilbert space $\mathcal{H}^{v_k}$. Let $S$ denote a set of states of $\mathcal{H}$ to be prepared from a common resource state, which we call the *target set*. Note that the target set $S$ can be either finite or infinite.
For a given target set $S$ on $\mathcal{H}$, a common resource state [@S3; @G3] is defined as follows. The total system for a common resource state is denoted by $\overline{\mathcal{H}}\coloneqq\bigotimes_{k=1}^{N}\overline{\mathcal{H}}^{v_k}$, where $\overline{\mathcal{H}}^{v_k}$ for each party $v_k$ denotes the Hilbert space corresponding to $v_k$. This total system $\overline{\mathcal{H}}$ includes $\mathcal{H}$ for a target set as a subspace, that is, $\overline{\mathcal{H}}^{v_k}\supset\mathcal{H}^{v_k}$ for each party $v_k$. A state $\Ket\phi\in \overline{\mathcal{H}}$ is called a *common resource state* for the target set $S$ under LOCC if for any state $\Ket{\psi}\in S$ there exists an LOCC protocol which transforms $\Ket\phi$ into $\Ket{\psi}$ deterministically and exactly. Regarding a formal definition of LOCC, we refer to Ref. [@C2] and the references therein. Note that the common resource state $\Ket\phi$ for $S$ on $\mathcal{H}$ may only exist in a higher-dimensional Hilbert space $\overline{\mathcal{H}}$ than that for $S$ itself, that is, $\dim\overline{\mathcal{H}}^{v_k}\geqq\dim\mathcal{H}^{v_k}$ for each $v_k$.
To compare bipartite and multipartite entanglement, we introduce the notion of a common resource state consisting of a collection of bipartite entangled states and that exhibiting multipartite entanglement. In the following, common resource states are assumed to be fully entangled, that is, entangled with respect to any bipartition of the parties. Consider a collection of bipartite entangled states distributed among the parties $v_1,\ldots,v_N$. The distribution of the bipartite entangled states can be represented by a graph $G=(V,E)$, where each vertex in the set $V=\left\{v_1,\ldots,v_N\right\}$ represents a party, and each edge $e=\left\{v_k,v_{k'}\right\}\in E$ a bipartite entangled state $\Ket{\phi_e}^e$ shared between two parties $v_k$ and $v_{k'}$. A common resource state $\Ket{\phi}$ for a target set $S$ is called a state *consisting of bipartite entanglement* if there exists a graph $G=(V,E)$ such that $\Ket\phi$ is locally unitarily equivalent to a state in the form $\bigotimes_{e\in E}\Ket{\phi_e}^e$. Otherwise, $\Ket{\phi}$ is called a state *exhibiting multipartite entanglement*.
For any target set $S$, we can always obtain a common resource state consisting of bipartite entanglement using quantum teleportation [@B4] or a more efficient protocol proposed in Ref. [@Y1]. This common resource state consists of maximally entangled states distributed among the parties $v_1,\ldots,v_N$ according to a tree $T=(V,E)$, which is a graph including no cycle as a subgraph. The common resource state can be written as $\bigotimes_{e\in E}\Ket{\Phi_{M_e}^+}^e$, where for each edge $e=\{v_k,v_{k^\prime}\}\in E$, $\Ket{\Phi_{M_e}^+}^e\coloneqq\frac{1}{\sqrt{M_e}}\sum_{l=0}^{M_e -1}\Ket{l}^{v_k}\otimes\Ket{l}^{v_{k^\prime}}$ is a maximally entangled state of Schmidt rank $M_e$ shared between $v_k$ and $v_{k'}$. For any tree $T=(V,E)$, if an edge $e\in E$ is deleted, $T$ is divided into two disjoint trees, whose vertices are represented by disjoint sets $V_e$ and $\overline{V}_e$ satisfying $V=V_e\cup \overline{V}_e$. For any $\Ket{\psi}\in S$ on $\mathcal{H}=\bigotimes_{k=1}^{N}\mathcal{H}^{v_k}$, we let $R_e\left(\Ket{\psi}\right)$ denote the Schmidt rank of $\Ket{\psi}$ with respect to the bipartition $\bigotimes_{v_k\in V_e}\mathcal{H}^{v_k}$ and $\bigotimes_{v_k\in \overline{V}_e}\mathcal{H}^{v_k}$ of $\mathcal{H}=\bigotimes_{k=1}^{N}\mathcal{H}^{v_k}$. Given any $\Ket{\psi}\in S$ and a tree $T=(V,E)$, Ref. [@Y1] provides the necessary and sufficient condition for the resource state $\bigotimes_{e\in E}\Ket{\Phi_{M_e}^+}^e$ being transformable into $\Ket{\psi}$ by LOCC. This transformation is achievable if and only if the Schmidt rank of each bipartite maximally entangled state of the resource state is not smaller than the Schmidt rank of $\Ket{\psi}$ with respect to the corresponding bipartition, that is, for each $e\in E$, $$\label{eq:bipartite}
M_e\geqq R_e\left(\Ket{\psi}\right).$$ To obtain a common resource state consisting of bipartite entanglement for $S$, it is sufficient to ensure that the condition of the Schmidt ranks given in Inequality is fulfilled for all the states in $S$.
![A simple example of a graph representing a graph state and a quantum circuit representing a class of states parameterized by $\alpha$ which can be deterministically prepared using this graph state. Given a graph state $\Ket{\Phi}^{v_1,v_2,v_3}$ as illustrated on the left, by performing the unitary $\exp\left(\textup{i}\alpha X^{v_1}\right)$ parameterized by $\alpha$ and a measurement in the $Z$ basis $\left\{\Ket{0},\Ket{1}\right\}$ on the qubit represented by the black vertex $v_1$, followed by local unitary corrections on the white vertices $v_2$ and $v_3$ conditioned by the measurement outcome, we can deterministically obtain a two-qubit state $\Ket{\psi\left(\alpha\right)}$ defined in Eq. represented by $v_2$ and $v_3$. The state $\Ket{\psi\left(\alpha\right)}$ can also be represented as the output of the quantum circuit on the right, where a two-qubit gate $\exp\left(\textup{i}\alpha Z^{v_2}\otimes Z^{v_3}\right)$ parameterized by $\alpha$ is applied to $\Ket{+}^{v_2}\otimes\Ket{+}^{v_3}$.[]{data-label="fig:correspondence"}](correspondence2.pdf){width="3.4in"}
As a common resource state exhibiting multipartite entanglement, we can use a class of graph states proposed in Ref. [@S3]. A graph state [@H4; @H5] is a multiqubit entangled state characterized by a graph $G=(V,E)$. Note that, while graphs in this paper also represent distribution of bipartite entanglement as explained above, a graph state is a different concept, which is a state exhibiting multipartite entanglement obtained for a graph $G=(V,E)$ as follows: first, for each vertex $v_k\in V$, a qubit labeled $v_k$ is initialized as $$\Ket{+}^{v_k}\coloneqq\frac{1}{\sqrt{2}}\left(\Ket{0}^{v_k}+\Ket{1}^{v_k}\right),$$ and then, for each edge $e=\left\{v_k,v_{k^\prime}\right\}\in E$, the controlled-$Z$ gate $$\label{eq:cz}
\begin{split}
&CZ^{v_k,v_{k^\prime}}\\
&\coloneqq{\left(\Ket{00}\Bra{00}+\Ket{01}\Bra{01}+\Ket{10}\Bra{10}-\Ket{11}\Bra{11}\right)}^{v_k,v_{k^\prime}}
\end{split}$$ is applied to two qubits labeled as $v_k$ and $v_{k^\prime}$. Reference [@S3] proposes an LOCC protocol for preparing any pure state of an arbitrary number of qubits by performing sequential projective measurements and local unitary corrections on a particular type of graph states. (See also measurement-based quantum computation [@R1; @R2; @R3].) To see how this protocol works, consider the three-vertex graph shown in Fig. \[fig:correspondence\] as a simple example. The graph state $\Ket{\Phi}^{v_1,v_2,v_3}$ represented by this graph is invariant under a local unitary transformation $X^{v_1}\otimes Z^{v_2}\otimes Z^{v_3}$, that is, $$X^{v_1}\otimes Z^{v_2}\otimes Z^{v_3}\Ket{\Phi}^{v_1,v_2,v_3}=\Ket{\Phi}^{v_1,v_2,v_3},$$ where $X$ and $Z$ are the Pauli operators. Thus, if the unitary operator $\exp\left(\textup{i}\alpha X^{v_1}\right)$ parameterized by $\alpha$ is performed on qubit $v_1$, the action is equivalent to $$\begin{aligned}
&\exp\left(\textup{i}\alpha X^{v_1}\right)\otimes\openone^{v_2}\otimes\openone^{v_3}\Ket{\Phi}^{v_1,v_2,v_3}\\
&=\openone^{v_1}\otimes\exp\left(\textup{i}\alpha Z^{v_2}\otimes Z^{v_3}\right) \Ket{\Phi}^{v_1,v_2,v_3},\end{aligned}$$ which can be shown using the Taylor series of the exponential function. Then, it is straightforward to verify that, performing $\exp\left(\textup{i}\alpha X^{v_1}\right)$ and a measurement in $Z$ basis $\left\{\Ket{0},\Ket{1}\right\}$ on the qubit $v_1$, we obtain a state of two qubits $v_2$ and $v_3$ which can be deterministically transformed by local unitary corrections $\openone^{v_2}\otimes\openone^{v_3}$ or $Z^{v_2}\otimes Z^{v_3}$ conditioned by the measurement outcome $\Ket{0}$ or $\Ket{1}$, respectively, into $$\label{eq:psi_alpha}
\Ket{\psi\left(\alpha\right)}^{v_2,v_3}\coloneqq\exp\left(\textup{i}\alpha Z^{v_2}\otimes Z^{v_3}\right)\left(\Ket{+}^{v_2}\otimes\Ket{+}^{v_3}\right).$$ In the same way, it is shown in Ref. [@S3] that any quantum circuit consisting of one-qubit Clifford gates and multiqubit gates $\exp\left(\textup{i}\alpha Z\otimes Z\otimes\cdots\otimes Z\right)$ parameterized by $\alpha$ can be implemented by performing sequential projective measurements and local unitary corrections on a particular graph state corresponding to the quantum circuit. In addition, it is shown that any pure state of an arbitrary number of qubits is locally unitarily equivalent to a pure state generated by a quantum circuit consisting of these types of gates. Using this argument, we can obtain a graph state serving as a common resource state exhibiting multipartite entanglement for a given target set.
We remark that the above graph states for common resource states require at least one auxiliary qubit per parameter describing local unitary equivalence classes of $n$-qubit states. Since the number of parameters of states increases exponentially with respect to $n$, the size of the required graph states is also exponentially large. At the same time, if we consider the target set $S$ to be a smaller subset of the $n$ qubits, the number of the parameters describing the states in $S$ can be decreased. In this case, we may construct another class of graph states which serve as common resource states for the smaller subset $S$, which require a smaller number of auxiliary qubits than the original graph states serving as common resource states for the set of arbitrary $n$-qubit states. We will use this observation in the subsequent section where the task of system-size-limited quantum state preparation is introduced.
\[sec:def\]Definition of system-size-limited quantum state preparation
======================================================================
We introduce the tasks of system-size-limited quantum state preparation. We consider a scenario in which a multipartite system is distributed among spatially separated parties $v_1,\ldots,v_N$, and each party’s local system size is limited. The system-size-limited quantum state preparation for a given target set $S$ is a task for the parties to transform a shared common resource state into an arbitrary state $\Ket{\psi}\in S$ by performing local operations on a limited-size quantum system and classical communication. To compare multipartite and bipartite resources, we analyze system-size-limited quantum state preparation in two settings—the *static* setting and the *dynamic* setting. In this section, we first describe the definition of local operations on a limited-size quantum system and then define system-size-limited quantum state preparation in the static setting and the dynamic setting.
To clarify the meaning of local operations on a limited-size quantum system, we assume that each party $v_k\in\left\{v_1,\ldots,v_N\right\}$ has a quantum system corresponding to a Hilbert space $\overline{\mathcal{H}}^{v_k}$ of dimension $$d^{\left(v_k\right)}\coloneqq\dim\overline{\mathcal{H}}^{v_k}.$$ The configuration of system sizes for all the parties is denoted by a tuple $$\boldsymbol{d}=\left(d^{\left(v_1\right)},\ldots,d^{\left(v_N\right)}\right).$$ As explained in Sec. \[sec:common\_resource\_state\], the target set $S$ is given from a subspace $\mathcal{H}$ of the total system $\overline{\mathcal{H}}$. Each party $v_k$ can perform any unitary and any measurement on the system $\overline{\mathcal{H}}^{v_k}$ but is *not* allowed to add an auxiliary system to increase the dimension of $\overline{\mathcal{H}}^{v_k}$. Measurements are represented by quantum instruments, and while an indirect measurement may require an auxiliary working quantum system, the protocols in this paper use only projective measurements [^1]. Classical information processing and classical communication without using a quantum system can be freely performed. For a given configuration of system sizes specified by $\boldsymbol{d}$, we assume in both the static setting and the dynamic setting that the parties can perform local operations on a limited-size quantum system in the above sense and classical communication. We refer to this restricted LOCC as *LOCC within the configuration $\boldsymbol{d}$*.
In the dynamic setting, we also allow any two parties $v_k$ and $v_{k^\prime}$ to perform quantum communication. When $v_k$ sends a state of a $d$-dimensional system to $v_{k^\prime}$ by quantum communication, $v_k$ has to initially store the state to be sent in a $d$-dimensional subsystem of $\overline{\mathcal{H}}^{v_k}$, and $v_{k^\prime}$ has to initialize a $d$-dimensional subsystem of $\overline{\mathcal{H}}^{v_{k^\prime}}$ as a fixed state $\Ket{0}$ so that $v_{k^\prime}$ receives the state using this subsystem. After each quantum communication, the $d$-dimensional subsystem of $\overline{\mathcal{H}}^{v_k}$ is initialized as a fixed state $\Ket{0}$ so that $v_k$ can reuse this subsystem. Each quantum communication from one party to another party is called one *round* of quantum communication. If a protocol includes multiple rounds of quantum communication, the multiple rounds of quantum communication are performed sequentially. Quantum communication between the parties is allowed only if it is stated explicitly.
The system-size-limited quantum state preparation in the static setting for a configuration $\boldsymbol{d}$ of system sizes and a target set $S$ is a task for $N$ parties to achieve the following:
1. A common resource state $\Ket\phi\in\overline{\mathcal{H}}$ for $S$ is given to the parties;
2. A particular target state $\Ket{\psi}\in S$ is chosen from the target set $S$, and all the parameters of $\Ket{\psi}$ given to all the parties. Then the parties perform LOCC within the configuration $\boldsymbol{d}$ to transform the common resource state $\Ket\phi$ into this target state $\Ket\psi$.
Our analysis concerns properties of the common resource state $\Ket\phi$ for achieving a system-size-limited quantum state preparation, that is, whether the task is achievable or not when the common resource state $\Ket\phi$ is a state consisting of bipartite entanglement or a state exhibiting multipartite entanglement.
In a similar way, we define system-size-limited quantum state preparation in the dynamic setting as follows. The system-size-limited quantum state preparation in the dynamic setting for a configuration $\boldsymbol{d}$ of system sizes and a target set $S$ is a task for $N$ parties to achieve the following:
1. A common resource state $\Ket\phi\in\overline{\mathcal{H}}$ for $S$ is prepared by the parties using quantum communication in addition to LOCC within the configuration $\boldsymbol{d}$;
2. A particular target state $\Ket{\psi}\in S$ is chosen from the target set $S$, and all the parameters of $\Ket{\psi}$ given to all the parties. Then the parties perform LOCC within the configuration $\boldsymbol{d}$ to transform the common resource state $\Ket\phi$ into this target state $\Ket\psi$.
In this dynamic setting, $\Ket\phi$ can be a state exhibiting multipartite entanglement as long as $\Ket\phi$ is deterministically prepared by finitely many rounds of quantum communication. Note that while we differentiate the capabilities of common resource states consisting of bipartite entanglement and those exhibiting multipartite entanglement in the static setting, common resource states in the dynamic setting are expected to have an intermediate capability, since only temporal uses of bipartite quantum communication resources are allowed in the dynamic setting for preparing the common resource states.
In the following, we provide nontrivial examples differentiating between bipartite and multipartite entanglement in the static setting in Sec. \[sec:analysis\]. Also, other examples in the dynamic setting are provided in Sec. \[sec:analysis2\] for differentiating the capability of the common resource states in the dynamic setting from that of the common resource states consisting of bipartite entanglement and exhibiting multipartite entanglement in the static setting.
\[sec:analysis\]System-size-limited quantum state preparation in the static setting
===================================================================================
In this section, we analyze system-size-limited quantum state preparation in the static setting. We show the existence of a system-size-limited quantum state preparation which is achievable by a common resource state exhibiting multipartite entanglement but not by any common resource state consisting of bipartite entanglement.
To show such a nontrivial example, consider eight parties $v_1,\ldots,v_8$. The configuration $\boldsymbol{d}_0=\left(d_0^{\left(v_1\right)},\ldots, d_0^{\left(v_8\right)}\right)$ of the quantum system sizes are given as follows: $$\label{eq:d}
\begin{split}
d_0^{\left(v_k\right)}&=\dim\overline{\mathcal{H}}^{v_k} = 4,\; \dim\mathcal{H}^{v_k} = 2, \;\forall v_k\in\{v_1,\ldots,v_7\};\\
d_0^{\left(v_8\right)}&=\dim\overline{\mathcal{H}}^{v_8} =\dim\mathcal{H}^{v_8} = 2.
\end{split}$$ For each $v_k\in\left\{v_1,\ldots,v_7\right\}$, we regard the four-dimensional system $\overline{\mathcal{H}}^{v_k}$ as two qubits, where one is for the target set denoted by $\mathcal{H}^{v_k}$ and the other is an auxiliary qubit denoted by $\mathcal{H}_\textup{a}^{v_k}$, that is, $\overline{\mathcal{H}}^{v_k}=\mathcal{H}^{v_k}\otimes\mathcal{H}_\textup{a}^{v_k}$.
We define a target set $S_0$ on $\mathcal{H}=\bigotimes_{k=1}^{N}\mathcal{H}^{v_k}$ as the set of all the possible output states of a quantum circuit illustrated in Fig. \[fig:target\_set\]. This circuit consists of seven two-qubit gates $\exp\left(\textup{i}\alpha_i Z\otimes Z\right)$ parameterized by $\alpha_i\in\left\{\alpha_1,\ldots,\alpha_7\right\}$, where $0\leqq\alpha_i < 2\pi$ for each $\alpha_i$. The tuple of the seven parameters is denoted by $$\boldsymbol\alpha\coloneqq\left(\alpha_1,\ldots,\alpha_7\right).$$ As input to the circuit, we consider an eight-qubit product state $\Ket{+}^{\otimes 8}\in\mathcal{H}$. The target set $S_0$ consists of the eight-qubit output states of the circuit parameterized by $\boldsymbol\alpha$, that is, $$\label{eq:s}
S_0\coloneqq\left\{\Ket{\psi\left(\boldsymbol\alpha\right)}\in\mathcal{H}:\boldsymbol\alpha=\left(\alpha_1,\ldots,\alpha_7\right)\right\},$$ where each qubit is placed at one of the parties, as illustrated in Fig. \[fig:target\_set\]. For example, given the parameters $$\boldsymbol\alpha_0\coloneqq\left(0,0,0,0,0,0,0\right),$$ each gate in the circuit reduces to the identity and hence $\Ket{\psi\left(\boldsymbol\alpha_0\right)}=\Ket{+}^{\otimes 8}\in S_0$ is a product state. In contrast, given the parameters $$\boldsymbol\alpha_{\frac{\pi}{4}}\coloneqq\left(\frac{\pi}{4},\frac{\pi}{4},\frac{\pi}{4},\frac{\pi}{4},\frac{\pi}{4},\frac{\pi}{4},\frac{\pi}{4}\right),$$ $\Ket{\psi\left(\boldsymbol\alpha_\frac{\pi}{4}\right)}\in S_0$ is a fully entangled state, since each gate $\exp\left(\textup{i}\frac{\pi}{4} Z\otimes Z\right)$ entangles $\Ket{+}\otimes\Ket{+}$.
Using the configuration $\boldsymbol{d}_0$ and the target set $S_0$ defined above, we present the following two propositions on the system-size-limited quantum state preparation for $\boldsymbol{d}_0$ and $S_0$. Proposition \[thm:multipartite\] shows the feasibility of the system-size-limited quantum state preparation using a common resource state exhibiting multipartite entanglement while Proposition \[thm:bipartite\] is a no-go theorem for any common resource state consisting of bipartite entanglement.
![A quantum circuit generating all the states in the target set $S_0\coloneqq\left\{\Ket{\psi\left(\boldsymbol{\alpha}\right)}\right\}$ for the system-size-limited quantum state preparation in Propositions \[thm:multipartite\] and \[thm:bipartite\], where $\boldsymbol{\alpha}=\left(\alpha_1,\ldots,\alpha_7\right)$ is a tuple of parameters. The wires of the circuit starting from the input $\Ket{+}^{v_1},\ldots,\Ket{+}^{v_8}$ represent qubits held by the parties $v_1,\ldots,v_8$, respectively. The circuit consists of seven two-qubit gates $\exp\left(\textup{i}\alpha_i Z\otimes Z\right)$ parameterized by $\alpha_i\in\left\{\alpha_1,\ldots,\alpha_7\right\}$.[]{data-label="fig:target_set"}](eight_qubit2.pdf){width="3.4in"}
![A graph representing a $15$-qubit graph state $\Ket{\Phi_\textup{res}}$ used as a common resource state exhibiting multipartite entanglement in Proposition \[thm:multipartite\]. Each of the parties $v_k\in\left\{v_1,\ldots,v_7\right\}$ holds two qubits $\mathcal{H}^{v_k}\otimes\mathcal{H}_\textup{a}^{v_k}$, while party $v_8$ holds one qubit $\mathcal{H}^{v_8}$. Eight of the $15$ qubits $\mathcal{H}^{v_1},\ldots,\mathcal{H}^{v_8}$ represented by white vertices are qubits which can be prepared in any state $\Ket{\psi\left(\boldsymbol{\alpha}\right)}$ in the target set $S_0$. The other seven $\mathcal{H}_\textup{a}^{v_1},\ldots,\mathcal{H}_\textup{a}^{v_7}$ represented by black vertices are auxiliary qubits to be measured. To obtain $\Ket{\psi\left(\boldsymbol{\alpha}\right)}\in S_0$ parameterized by $\boldsymbol\alpha=\left(\alpha_1,\ldots\alpha_7\right)$, each party $v_i\in\left\{v_1,\ldots,v_7\right\}$ performs the following protocol in order. First, a unitary $\exp\left(\textup{i}\alpha_i X\right)$ parameterized by $\alpha_i$ is performed on $\mathcal{H}_\textup{a}^{v_i}$. Then, the qubit $\mathcal{H}_\textup{a}^{v_i}$ is measured in the $Z$ basis $\left\{\Ket{0},\Ket{1}\right\}$, and depending on the outcome, a unitary correction is applied to the remaining qubits. Using this protocol, the parties can deterministically transform $\Ket{\Phi_\textup{res}}$ into $\Ket{\psi\left(\boldsymbol{\alpha}\right)}\in S_0$ for any $\boldsymbol\alpha$.[]{data-label="fig:tree"}](tree5.pdf){width="3.4in"}
\[thm:multipartite\] *Multipartite entanglement in a system-size-limited quantum state preparation in the static setting.* The system-size-limited quantum state preparation in the static setting for the configuration $\boldsymbol{d}_0$ defined in Eq. and the target set $S_0$ defined in Eq. is achievable using a common resource state exhibiting multipartite entanglement.
\[thm:bipartite\] *Bipartite entanglement in a system-size-limited quantum state preparation in the static setting.* The system-size-limited quantum state preparation in the static setting for the configuration $\boldsymbol{d}_0$ defined in Eq. and the target set $S_0$ defined in Eq. is *not* achievable using any common resource state consisting only of bipartite entanglement.
In the following, we prove Propositions \[thm:multipartite\] and \[thm:bipartite\]. Note that while shallower quantum circuits having a similar structure to the circuit in Fig. \[fig:target\_set\] are not sufficient for differentiating between multipartite and bipartite entanglement, this example might not be the simplest, and further sets of states with the same properties will also be given after the proofs.
We provide a common resource state exhibiting multipartite entanglement for the target set $S_0$, which is the $15$-qubit graph state $\Ket{\Phi_\textup{res}}$ illustrated in Fig. \[fig:tree\] held by the parties $v_1,\ldots,v_8$. In the same way as explained in Sec. \[sec:common\_resource\_state\], given the graph state $\Ket{\Phi_\textup{res}}$ in Fig. \[fig:tree\], for each $i\in\left\{1,\ldots,7\right\}$, performing $\exp\left(\textup{i}\alpha_i X^{v_i}\right)$ parameterized by $\alpha_i$ and a measurement in the $Z$ basis $\left\{\Ket{0},\Ket{1}\right\}$ on the qubit represented by $\mathcal{H}_\textup{a}^{v_i}$, followed by local unitary corrections on other qubits conditioned by the measurement outcome, the parties can obtain $\Ket{\psi\left(\boldsymbol\alpha\right)}\in S_0$ deterministically for any parameters $\boldsymbol\alpha=\left(\alpha_1,\ldots,\alpha_7\right)$.
We derive a necessary condition for preparing the state $\Ket{\psi\left(\boldsymbol\alpha_{\frac{\pi}{4}}\right)}\in S_0$ from a resource state consisting of bipartite entanglement by LOCC within the configuration $\boldsymbol d_0$. Observe that the state $\Ket{\psi\left(\boldsymbol\alpha_{\frac{\pi}{4}}\right)}$ is fully entangled, that is, entangled with respect to any bipartition of the eight qubits. To prepare a fully entangled state, the resource state at party $v_8$ has to be entangled with some other parties. As $\dim \overline{\mathcal{H}}^{v_8}=2$, the party $v_8$ can store only one qubit of a bipartite resource state entangled with another party, which we label as $u_7\in\{v_1,\ldots,v_7\}$. The quantum system $\overline{\mathcal{H}}^{u_7}$ at $u_7$ is decomposed into $\overline{\mathcal{H}}^{u_7}=\mathcal{H}^{u_7}_{\{u_7,v_8\}}\otimes\mathcal{H}^{u_7}_\textup{r}$ where $\mathcal{H}^{u_7}_{\{u_7,v_8\}}$ is a system of more than one dimension for the bipartite entangled resource state shared with $v_8$, and $\mathcal{H}^{u_7}_\textup{r}$ the remaining quantum system. It is necessary that $$\label{eq:dim}
\begin{split}
&\dim\mathcal{H}^{u_7}_{\{u_7,v_8\}}=2,\\
&\dim\mathcal{H}^{u_7}_\textup{r}=2,
\end{split}$$ which can be shown by contradiction as follows. Assume that $\dim\mathcal{H}^{u_7}_{\{u_7,v_8\}}>2$. Then we have $\dim\mathcal{H}^{u_7}_\textup{r}<2$, and the resource state shared between the parties $u_7$ and $v_8$ cannot be entangled with any of the other parties. This contradicts the assumption that a fully entangled state can be prepared, and Eq. is shown. As $\dim\mathcal{H}^{u_7}_\textup{r}=2$, the party $u_7$ can store another single qubit of a bipartite resource state entangled with a party other than $v_8$, which we label as $u_6\in\{v_1,\ldots,v_7\}\setminus\{u_7\}$. By iterating the above argument, any resource state consisting of bipartite entanglement for preparing a fully entangled state by LOCC within the configuration $\boldsymbol{d}_0$ is required to be seven two-qubit entangled states shared between $u_1$–$u_2$, $\ldots$, $u_6$–$u_7$, and $u_7$–$v_8$, respectively, where $$\label{eq:perm}
\begin{split}
(u_1,\ldots,u_7)~\text{is a permutation of}~(v_1,\ldots,v_7).
\end{split}$$ Note that although $u_1$ uses only one qubit in this case, the remaining system of $u_1$, which is two dimensional, cannot be used for sharing an entangled state with the other parties since there is no dimension left in the quantum systems of the other parties. Therefore, the distribution of the two-qubit entangled states is represented by a line-topology graph, as illustrated in Fig. \[fig:permutation\]. Note that this line-topology graph is a tree. Since the target set $S_0$ includes a fully entangled state $\Ket{\psi\left(\boldsymbol\alpha_\frac{\pi}{4}\right)}$, it is necessary that any common resource state consisting of bipartite entanglement for $S_0$ within the configuration $\boldsymbol d_0$ is a state consisting of seven two-qubit entangled states represented by the line-topology tree as shown in Fig. \[fig:permutation\].
![A line-topology tree representing a resource state consisting of bipartite entanglement to prepare a fully entangled state within the configuration $\boldsymbol{d}_0$. Since the target set $S_0$ includes a fully entangled state $\Ket{\psi\left(\boldsymbol\alpha_\frac{\pi}{4}\right)}$, the common resource states consisting of bipartite entanglement for $S_0$ have to be represented by the line-topology tree in the figure, which leads to a contradiction with the condition given in Inequality as shown in the main text.[]{data-label="fig:permutation"}](permutation.pdf){width="3.4in"}
We prove that the state $\Ket{\psi(\boldsymbol\alpha_\frac{\pi}{4})}$ cannot be prepared from any such resource state. Since any two-qubit entangled state can be obtained by LOCC from a Bell state $\frac{1}{\sqrt{2}}\left(\Ket{00}+\Ket{11}\right)$, it suffices to consider resource states consisting of seven Bell states represented by the line-topology tree. Thus, the condition given in Inequality implies that the state $\Ket{\psi\left(\boldsymbol\alpha_\frac{\pi}{4}\right)}$ can be prepared from resource states consisting of seven Bell states represented by a line-topology tree if and only if $$R_e\left(\Ket{\psi\left(\boldsymbol\alpha_\frac{\pi}{4}\right)}\right)\leqq 2$$ for any edge $e$ of the line-topology tree. In other words, the Schmidt rank of $\Ket{\psi\left(\boldsymbol\alpha_\frac{\pi}{4}\right)}$ with respect to each edge of the line-topology tree needs to be smaller than or equal to $2$. However, the explicit calculation of $R_e\left(\Ket{\psi(\boldsymbol\alpha_\frac{\pi}{4})}\right)$ for all the edges $e$ of all the $7!=5040$ different trees obtained from the permutations of $v_1,\ldots,v_7$ in Eq. shows that, for any of the permutations, there exists an edge $e$ such that $$\label{eq:r_e}
R_e\left(\Ket{\psi\left(\boldsymbol\alpha_\frac{\pi}{4}\right)}\right)>2.$$ For details, see Appendix \[sec:calculation\]. The calculation of $R_e\left(\Ket{\psi\left(\boldsymbol\alpha_\frac{\pi}{4}\right)}\right)$ implies that the state $\Ket{\psi(\boldsymbol\alpha_\frac{\pi}{4})}$ cannot be prepared from any resource state consisting of the seven Bell states. Therefore, we conclude that there exists no common resource state consisting of bipartite entanglement for the target set $S_0$ within the configuration $\boldsymbol d_0$.
We remark that, regarding the required system sizes for storing a common resource state consisting of bipartite entanglement, the above argument for Proposition \[thm:bipartite\] based on the Schmidt ranks of a common resource state consisting of bipartite entanglement can also apply to more general target sets. In particular, we analyze in Appendix \[sec:general\] a target set $S$ of $2m$-qudit states, where the size of each qudit is $d$, and each state in $S$ has maximal Schmidt ranks with respect to any bipartition between $m$ qudits and the other $m$ qudits. Note that random weighted graph states or random pure states fulfill this condition, for which the reduced states have almost maximum entropy for any bipartition [@C4]. We show in Appendix \[sec:general\] that for any resource state consisting of bipartite entanglement to obtain such a state by LOCC, or even by stochastic LOCC, there has to be at least one party for which the local quantum system size for storing this resource state needs to be almost quadratically larger than $d$, that is, greater than or equal to $d^{2-\frac{1}{m}}$. We also note that for some special configurations of local system sizes, these differences between bipartite and multipartite entanglement do not arise, especially if $\dim\overline{\mathcal{H}}^{v_1}\geqq\prod_{k=2}^{N}\dim\overline{\mathcal{H}}^{v_k}$ [@R5].
\[sec:analysis2\]System-size-limited quantum state preparation in the dynamic setting
=====================================================================================
In this section, we analyze the difference in system-size-limited quantum state preparation between the static setting and the dynamic setting. Before analyzing multipartite cases, we first discuss a simpler bipartite case to clarify the difference between the static setting and the dynamic setting. Consider two parties $v_1$ and $v_2$, where each party has two qubits; that is, the configuration $\left(d^{\left(v_1\right)},d^{\left(v_2\right)}\right)$ is given by $$\begin{aligned}
d^{\left(v_1\right)}&=\dim\overline{\mathcal{H}}^{v_1} = 4,\\
d^{\left(v_2\right)}&=\dim\overline{\mathcal{H}}^{v_1} = 4.\end{aligned}$$ In this case, these two parties can store an entangled resource state of Schmidt rank $4$ in the static setting. However, in the dynamic setting, the parties can prepare an entangled resource state of Schmidt rank at most $2$, which is shown as follows. Consider any shared state $\Ket{\phi}^{v_1,v_2}$ after the last round of quantum communication for preparing $\Ket{\phi}^{v_1,v_2}$, where we assume that the direction of the quantum communication in the last round is from $v_1$ to $v_2$ without loss of generality. Since the quantum communication sends out at least one qubit from $v_1$, the rank of $v_1$’s reduced state for $\Ket{\phi}^{v_1,v_2}$ is at most $2$; that is, the Schmidt rank of $\Ket{\phi}^{v_1,v_2}$ is at most two. Since the Schmidt rank is monotonically nonincreasing by LOCC [@L], $v_1$ and $v_2$ after the last round of quantum communication cannot prepare an entangled resource state of Schmidt rank more than $2$, which yields the conclusion. Although this two-party example is trivial, we also show nontrivial cases of more than two parties in the following.
We present the following two propositions. Proposition \[prp:bipartite\_dynamic\] shows that the common resource states available in the dynamic setting can still have more capability than any common resource state consisting of bipartite entanglement in the static setting, similarly to the common resource states exhibiting multipartite entanglement in the static setting. In contrast, Proposition \[prp:multipartite\_dynamic\] shows the existence of common resource states which cannot be prepared in the dynamic setting by the parties within a limitation of local system sizes, while the common resource states can still be stored within the limitation in the static setting. This implies that the common resource states in the dynamic setting have in this case less capability than a common resource state exhibiting multipartite entanglement in the static setting.
{width="7.0in"}
\[prp:bipartite\_dynamic\] *A common resource state in the dynamic setting having more capability than any common resource state consisting of bipartite entanglement.* The state $\Ket{\Phi_\textup{res}}$ in the proof of Proposition \[thm:multipartite\] and in Fig. \[fig:tree\] can be used as a common resource state for achieving the system-size-limited quantum state preparation in the dynamic setting for the configuration $\boldsymbol{d}_0$ defined in Eq. and the target set $S_0$ defined in Eq. , while the system-size-limited quantum state preparation in the static setting for $\boldsymbol{d}_0$ and $S_0$ cannot be achieved by any common resource state consisting of bipartite entanglement due to Proposition \[thm:bipartite\].
\[prp:multipartite\_dynamic\] *Common resource states exhibiting multipartite entanglement which cannot be prepared in the dynamic setting.* Consider four parties $v_1$, $v_2$, $v_3$, and $v_4$. Given a configuration $\boldsymbol{d}_1=\left(d_1^{\left(v_1\right)},d_1^{\left(v_2\right)},d_1^{\left(v_3\right)},d_1^{\left(v_4\right)}\right)$, where $$\begin{aligned}
d_1^{\left(v_1\right)}&=\dim\overline{\mathcal{H}}^{v_1} = 4,\\
d_1^{\left(v_k\right)}&=\dim\overline{\mathcal{H}}^{v_k} = 2, \; \forall v_k\in\left\{v_2,v_3,v_4\right\},
\end{aligned}$$ any fully entangled common resource state $\Ket{\phi}\in\overline{\mathcal{H}}$ whose Schmidt rank with respect to the bipartition between $v_1$ and $v_2 v_3 v_4$ is more than $2$ cannot be prepared in the dynamic setting, although there exists such a common resource state which can be stored in the static setting.
First, we prove Proposition \[prp:bipartite\_dynamic\] as follows.
We show that the common resource state $\Ket{\Phi_\textup{res}}$ in the proof of Proposition \[thm:multipartite\] and in Fig. \[fig:tree\] can be prepared by the parties using quantum communication in addition to LOCC within the configuration $\boldsymbol{d}_0$. The protocol for preparing $\Ket{\Phi_\textup{res}}$ is represented by a quantum circuit illustrated in Fig. \[fig:multipartite\]. In this circuit, the parties repeatedly perform $CZ$ gates defined in Eq. to entangle qubits initialized as $\Ket{+}$, distribute one qubit of the entangled state by quantum communication, and perform a $CZ$ gate again to entangle the remaining part of the entangled state with another qubit initialized as $\Ket{+}$. After this protocol, the state $\Ket{\Phi_\textup{res}}$ is shared among the parties $v_1,\ldots,v_8$.
Next, we prove Proposition \[prp:multipartite\_dynamic\] in a similar way to the example given at the beginning of this section.
Consider any fully entangled state $\Ket{\phi}^{v_1,v_2,v_3,v_4}$ shared among $v_1$, $v_2$, $v_3$, and $v_4$ after the last round of quantum communication for preparing $\Ket{\phi}^{v_1,v_2,v_3,v_4}$. The direction of the quantum communication in the last round is either of the following three possibilities:
1. From $v_1$ to $v_k$ where $k\in\left\{2,3,4\right\}$;
2. From $v_k$ to $v_{k^\prime}$ where $k,k^\prime\in\left\{2,3,4\right\}$ and $k\neq k^\prime$;
3. From $v_k$ to $v_1$ where $k\in\left\{2,3,4\right\}$.
Since $\Ket{\phi}^{v_1,v_2,v_3,v_4}$ is fully entangled, we exclude the latter two possibilities 2 and 3, which lead to a product state between $v_k$ and the others. Regarding possibility 1, after sending at least one qubit from $v_1$ to $v_k$, the rank of $v_1$’s reduced state for $\Ket{\phi}^{v_1,v_2,v_3,v_4}$ is at most $2$; that is, the Schmidt rank of $\Ket{\phi}^{v_1,v_2,v_3,v_4}$ with respect to the bipartition between $v_1$ and $v_2 v_3 v_4$ is at most $2$. Since the Schmidt rank is monotonically nonincreasing by LOCC [@L], the parties after the last round of quantum communication cannot prepare any common resource state whose Schmidt rank with respect to the bipartition between $v_1$ and $v_2 v_3 v_4$ is more than $2$, which yields the conclusion.
Note that under the limitation in Proposition \[prp:multipartite\_dynamic\], the parties can prepare any state whose Schmidt rank with respect to the bipartition between $v_1$ and $v_2 v_3 v_4$ is not more than $2$. This is because $v_1$’s reduced state can be represented by one qubit in this case, and hence, the parties can perform quantum communication to bring arbitrary two qubits to $v_1$ to perform any two-qubit gates. We also remark that, while we assume in our analysis that quantum communication is performed sequentially, one can also consider simultaneous quantum communication between two parties, which is considered as a swap operation between the two. However, this simultaneous quantum communication yields a trivial result, since the parties under the limitation in Proposition \[prp:multipartite\_dynamic\] can prepare any state $\Ket{\Phi}\in\overline{\mathcal{H}}$ using swap operations for letting $v_1$ perform arbitrary two-qubit gates.
\[sec:conclusion\]Conclusion
============================
We introduced and analyzed the task of system-size-limited quantum state preparation for comparing multipartite and bipartite entanglement from the viewpoint of state convertibility by local operations and classical communication (LOCC). In contrast to previous studies on the LOCC convertibility between multipartite pure states of the *same*-dimensional systems [@V2; @S1; @S2; @H2; @G2; @S4; @T; @T2], we analyzed LOCC transformation from a common resource state [@S3; @G3] of a *higher*-dimensional Hilbert space into a set of states of a *lower*-dimensional Hilbert space.
Introducing a limitation on the size of the local system of each party, we analyzed the capabilities of common resource states exhibiting multipartite entanglement and those consisting of bipartite entanglement. By showing a nontrivial example, we differentiate the capabilities of these common resource states in terms of achievability of the system-size-limited quantum state preparations for the same target set in the static setting where a common resource state has to be stored within a given limitation of local system sizes. In addition to this static setting, we considered the dynamic setting where the parties may use a common resource state exhibiting multipartite entanglement, but this common resource state has to be prepared by temporal uses of bipartite quantum communication resources within the limitation of local system sizes. We also provided nontrivial examples implying that common resource states in the dynamic setting have an intermediate capability between the common resource states exhibiting multipartite entanglement and those consisting of bipartite entanglement.
Our results provide examples implying that multipartite entanglement outperforms bipartite entanglement when limitations on the local system sizes matter in both the static setting and the dynamic setting. Further research will be needed to establish more general connections between the system sizes for common resource states and properties differentiating multipartite and bipartite entanglement.
\[sec:calculation\]How to calculate the ranks in Inequality
============================================================
We show that the Schmidt rank $R_e\left(\Ket{\psi(\boldsymbol\alpha_\frac{\pi}{4})}\right)$ in Inequality can be exactly calculated with the help of a computer program. Although computers cannot calculate irrational numbers exactly, we can reduce the Schmidt rank $R_e\left(\Ket{\psi(\boldsymbol\alpha_\frac{\pi}{4})}\right)$ of a vector $\Ket{\psi(\boldsymbol\alpha_\frac{\pi}{4})}$ with irrational elements to that of a vector only with integer elements. To remove irrational coefficients for normalization of the state $\Ket{+}$ and the gates $\exp\left(\textup{i}\frac{\pi}{4}Z\otimes Z\right)$, we substitute $\Ket{+}$ and $\exp\left(\textup{i}\frac{\pi}{4}Z\otimes Z\right)$ in the circuit in Fig. \[fig:target\_set\] with $\sqrt{2}\Ket{+}$ and $\sqrt{2}\exp\left(\textup{i}\frac{\pi}{4}Z\otimes Z\right)$, respectively. The resulting vector $$\Ket{\tilde\psi\left(\boldsymbol\alpha_\frac{\pi}{4}\right)}\coloneqq 2^{\frac{15}{2}}\Ket{\psi\left(\boldsymbol\alpha_\frac{\pi}{4}\right)}$$ has the same Schmidt ranks as $\Ket{\psi\left(\boldsymbol\alpha_\frac{\pi}{4}\right)}$ for any bipartition, and all the elements of $\Ket{\tilde\psi\left(\boldsymbol\alpha_\frac{\pi}{4}\right)}$ are complex numbers whose real and imaginary parts are both integers by construction. Therefore, we can exactly calculate Schmidt ranks of $\Ket{\psi\left(\boldsymbol\alpha_\frac{\pi}{4}\right)}$ by calculating those of $\Ket{\tilde\psi\left(\boldsymbol\alpha_\frac{\pi}{4}\right)}$ by computer.
\[sec:general\]Requirement for common resource states consisting of bipartite entanglement for preparing states having maximal Schmidt ranks
============================================================================================================================================
We show the following proposition.
*Requirement for resource states consisting of bipartite entanglement for preparing a multipartite entangled state having maximal Schmidt ranks.* Consider a $2m$-qudit state $\Ket{\psi}\in\mathcal{H}\coloneqq{\left(\mathbb{C}^d\right)}^{\otimes 2m}$ of local system size $d$ which has maximal Schmidt rank with respect to bipartite cuts between any $m$ qudits and the other $m$ qudits; that is, for any such bipartite cut, the Schmidt rank is $d^m$. If $2m$ parties $v_1,\ldots,v_{2m}$ prepare $\Ket{\psi}$ by LOCC from any resource state consisting only of bipartite entanglement, then there has to exist at least one party $v\in V\coloneqq\left\{v_1,\ldots,v_{2m}\right\}$ for which the local system size $\dim\overline{\mathcal{H}}^{v}$ for storing this resource state is almost quadratically larger, that is, $$\label{eq:full_schmidt_rank}
\max_{v_k\in V} \left\{\dim\overline{\mathcal{H}}^{v_k}\right\} \geqq d^{2- \frac{1}{m}}.$$
Since any bipartite state can be obtained from a maximally entangled state, it suffices to evaluate $\dim\overline{\mathcal{H}}^{v_k}$ for storing a resource state consisting of bipartite maximally entangled states distributed according to the complete graph $K=\left(V,E\right)$, that is, the fully-connected graph for the $2m$ parties. We let $M_e\in\left\{1,2,\ldots\right\}$ denote the Schmidt rank of the maximally entangled state for each edge $e\in E$.
We first derive a lower bound of the total system size for storing $\bigotimes_{e\in E}\Ket{\Phi_{M_e}^+}^e$, that is, $\prod_{e\in E}{\left(M_e\right)}^2$. Consider an edge cut $C$ [@B7] of $K$ between any $m$ vertices and the other $m$ vertices. Since the Schmidt rank is monotonically nonincreasing under LOCC [@L], it is necessary that, for any $C$, $$\label{eq:schmidt_rank_condition}
\prod_{e\in C}M_e\geqq d^m.$$ Considering Inequality for all the ${{2m\choose m}}/2$ possible choices of $C$ between any $m$ vertices and the other $m$ vertices, and taking the products of the right- and left-hand sides of these inequalities, we obtain $$\prod_{C} \prod_{e\in C} M_e\geqq d^{m\frac{{2m\choose m}}{2}}.$$ Since $M_e$ for each $e \in E$ appears ${2m-2\choose m-1}$ times in the product on the left-hand side, the last inequality can be written as $$\prod_{C} \prod_{e\in C} M_e =\prod_{e\in E}{\left(M_e\right)}^{{2m-2}\choose{m-1}}\geqq d^{m\frac{{2m\choose m}}{2}}.$$ Therefore, a lower bound of the total system size is $$\prod_{e\in E}{\left(M_e\right)}^2\geqq d^{2\left(2m-1\right)}.$$
Since the total system size for storing $\bigotimes_{e\in E}\Ket{\Phi_{M_e}^+}^e$ is written as $$\dim\overline{\mathcal{H}}=\prod_{v_k\in V}\dim\overline{\mathcal{H}}^{v_k},$$ we have $$\prod_{v_k\in V}\dim\overline{\mathcal{H}}^{v_k}\geqq\prod_{e\in E}{\left(M_e\right)}^2\geqq d^{2\left(2m-1\right)}.$$ Therefore, we obtain $$\max_{v_k\in V} \left\{\dim\overline{\mathcal{H}}^{v_k}\right\}\geqq{\left(\prod_{v_k\in V}\dim\overline{\mathcal{H}}^{v_k}\right)}^{\frac{1}{2m}}\geqq d^{2-\frac{1}{m}},$$ which yields the conclusion.
Note that the lower bound of local system sizes in Inequality is almost sufficient for fulfilling the condition of the Schmidt ranks by storing a symmetric distribution of maximally entangled states shared between all pairs of the parties. In this case, since each party shares maximally entangled states with the other $2m-1$ parties, the maximally entangled state corresponding to each $e\in E$ satisfies $M_e=\left\lceil d^\frac{1}{m}\right\rceil$, and the local system size for each $v\in V$ is ${\left\lceil d^\frac{1}{m}\right\rceil}^{2m-1}$, where $\lceil{}\cdots{}\rceil$ is the ceiling function.
[10]{} L. Amico, R. Fazio, A. Osterloh, and V. Vedral, Rev. Mod. Phys. **80**, 517 (2008). M. Rangamani and T. Takayanagi, Lect. Notes Phys. **931**, 1 (2017). H. J. Briegel and R. Raussendorf, Phys. Rev. Lett. **86** 910 (2001). R. Raussendorf and H. J. Briegel, Phys. Rev. Lett. **86**, 5188 (2001). R. Raussendorf, D. E. Browne, and H. J. Briegel, Phys. Rev. A **68**, 022312 (2003). P. K[ó]{}m[á]{}r, E. M. Kessler, M. Bishof, L. Jiang, A. S. S[ø]{}rensen, J. Ye, and M. D. Lukin, Nat. Phys. **10**, 582 (2014). Z. Eldredge, M. Foss-Feig, J. A. Gross, S. L. Rolston, and A. V. Gorshkov, Phys. Rev. A 97, 042337 (2018). A. Pirker, J. Wallnöfer, and W. Dür, New J. Phys. **20**, 053054 (2018). C. H. Bennett, G. Brassard, C. Crepeau, R. Jozsa, A. Peres, and W. K. Wootters, Phys. Rev. Lett. **70**, 1895 (1993). M. Epping, H. Kampermann, C. Macchiavello, and D. Bru[ß]{}, New J. Phys. **19**, 093012 (2017). M. Pivoluska, M. Huber, and M. Malik, Phys. Rev. A **97**, 032312 (2018). C. Spee, J. I. de Vicente, and B. Kraus, Phys. Rev. A **88**, 010305(R) (2013). C. Guo, E. Chitambar, and R. Duan, arXiv:1601.06220. M. Hebenstreit, M. Gachechiladze, O. Gühne, and B. Kraus, Phys. Rev. A **97**, 032330 (2018). J. I. de Vicente, C. Spee, and B. Kraus, Phys. Rev. Lett. **111**, 110502 (2013). C. Spee, J. I. de Vicente, and B. Kraus, in *Quantum \[Un\]Speakables II*, edited by R. Bertlmann and A. Zeilinger (Springer, 2017), pp. 365–380. C. Spee, J. I. de Vicente, and B. Kraus, J. Math. Phys. **57**, 052201 (2016). M. Hebenstreit, C. Spee, and B. Kraus, Phys. Rev. A **93**, 012339 (2016). G. Gour, B. Kraus, and N. R. Wallach, J. Math. Phys. **58**, 092204 (2017). D. Sauerwein, N. R. Wallach, G. Gour, and B. Kraus, Phys. Rev. X **8**, 031020 (2018). R. Horodecki, P. Horodecki, M. Horodecki, and K. Horodecki, Rev. Mod. Phys. **81**, 865 (2009). M. B. Plenio and S. Virmani, Quantum Inf. Comput. 7, 1 (2007). J. Eisert and D. Gross, in *Lectures on Quantum Information*, edited by D. Bru[ß]{} and G. Leuchs (Wiley, Weinheim, 2007), Chap. 13. C. Eltschka and J. Siewert, J. Phys. A **47**, 424005 (2014). M. Walter, D. Gross, and J. Eisert, arXiv:1612.02437. I. Bengtsson and K. Życzkowski, *Geometry of Quantum States: An Introduction to Quantum Entanglement* (Cambridge University Press, New York, 2017) Chap. 17. See *e.g.* M. J. Donald, M. Horodecki, and O. Rudolph, J. Math. Phys., **43**, 4252 (2002); E. Chitambar, W. Cui, and H.-K. Lo, Phys. Rev. Lett. **108**, 240504 (2012); E. Chitambar, D. Leung, L. Mancinska, M. Ozols, and A. Winter, Commun. Math. Phys. **328**, 1, 303 (2014). H. Yamasaki, A. Soeda, and M. Murao, Phys. Rev. A **96**, 032330 (2017). M. Hein, J. Eisert, and H. J. Briegel, Phys. Rev. A **69**, 062311 (2004). M. Hein, W. Dür, J. Eisert, R. Raussendorf, M. Van den Nest, and H. J. Briegel, in *Quantum Computers, Algorithms and Chaos* (IOS Press, Amsterdam, 2006), pp. 115–218. H.-K. Lo and S. Popescu, Phys. Rev. A **63**, 022301 (2001). J. Calsamiglia, L. Hartmann, W. Dür, and H. J. Briegel, Phys. Rev. Lett. **95**, 180502 (2005); D. N. Page, Phys. Rev. Lett. **71**, 1291 (1993); P. Hayden, D. W. Leung, A. Winter, Commun. Math. Phys. **265**, 1, 95, (2006). R. Duan and Y. Shi, Quantum Inf. Comput. **10**, 0925 (2010). S. Turgut, Y. Gül, and N. K. Pak, Phys. Rev. A **81**, 012317 (2010). S. Kintas and S. Turgut, J. Math. Phys. **51**, 092202 (2010). J. A. Bondy and U. S. R. Murty, *Graph Theory*, Volume 244 of Graduate Texts in Mathematics (Springer, London, 2008).
[^1]: For the completeness of the definition, we may allow each party $v_k$ to implement an indirect measurement using a projective measurement and one auxiliary working qubit in addition to the system $\overline{\mathcal{H}}^{v_k}$ itself. This auxiliary working qubit has to be traced out after each measurement. The use of only one auxiliary working qubit is sufficient for implementing any indirect measurement according to \[E. Andersson and D. K. L. Oi, Phys. Rev. A **77**, 052104 (2008).\].
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'Quantitative predictions of the Li intercalation voltage and of the electronic properties of rechargeable battery cathode materials are a substantial challenge for first-principles theory due to the possibility of (1) strong correlations associated with localized transition metal $d$ electrons and (2) significant van der Waals (vdW) interactions in layered systems, both of which are not accurately captured by standard approximations to density functional theory (DFT). Here, we perform a systematic benchmark of electronic structure methods based on the widely-used generalized gradient approximation of Perdew, Burke, and Ernzerhof (PBE) and the new strongly constrained and appropriately normed (SCAN) meta-generalized gradient approximation for battery cathode materials. Studying layered Li$_x$TiS$_2$, Li$_x$NiO$_2$, and Li$_x$CoO$_2$, olivine Li$_x$FePO$_4$, and spinel Li$_x$Mn$_2$O$_4$, we compute the voltage, crystal structure, and electronic structure with and without extensions to incorporate on-site Hubbard interactions and vdW interactions. Within pure DFT (i.e., without corrections for on-site Hubbard interactions), SCAN is a significant improvement over PBE for describing cathode materials, decreasing the mean absolute voltage error by more than 50%. Although explicit vdW interactions are not critical and in cases even detrimental when applied in conjunction with SCAN, Hubbard $U$ corrections are still in general necessary to achieve reasonable agreement with experiment. We show that no single method considered here can accurately describe the voltage and overall structural, electronic, and magnetic properties (i.e., errors no more than 5% for voltage, volume, band gap, and magnetic moments) of battery cathode materials, motivating a strong need for improved electronic structure approaches for such systems.'
author:
- 'Eric B. Isaacs'
- Shane Patel
- Chris Wolverton
bibliography:
- 'scan\_battery.bib'
title: 'Prediction of Li intercalation voltages in rechargeable battery cathode materials: effects of exchange-correlation functional, van der Waals interactions, and Hubbard $U$'
---
Introduction {#sec:intro}
============
Li-ion rechargeable battery cathodes, which are typically composed of transition metal oxides, represent a challenging testbed for first-principles theory. Prediction of the Li intercalation voltage is of particular importance given it is a fundamental battery property that helps determine the battery power and closely relates to the (de)lithiation mechanism. The average Li intercalation voltage $V$ relates to the difference in Li chemical potential between the cathode and anode. For a battery cathode whose Li content changes changes from $x_1$ to $x_2>x_1$, $V$ relative to Li metal is given by $eV=E_{\mathrm{Li}} + \frac{E(x_1)-E(x_2)}{(x_2-x_1)},$ where $e$ is the elementary charge, $E_{\mathrm{Li}}$ is the energy of Li metal, and $E(x)$ is the energy of the cathode material with Li concentration $x$ [@aydinol_ab_1997; @wolverton_cation_1998]. For example, for Li$_x$CoO$_2$, over the full range of Li ($x_1=0$ for CoO$_2$, $x_2=1$ for LiCoO$_2$), $V$ is given by $eV = E_{\mathrm{Li}}+E(0)-E(1)$. In principle, Gibbs free energies should be used in the previous expressions; we ignore pressure-volume and entropic contributions typically small compared to the magnitude of $V$ [@reynier_entropy_2004]. Since $V$ is a function of the total energies of phases whose electron distributions differ starkly (leading to fewer opportunities for error cancellation), it is a useful observable to serve as a stringent benchmark of *ab initio* thermodynamics approaches. Density functional theory (DFT) [@hohenberg_inhomogeneous_1964; @kohn_self-consistent_1965], within the generalized gradient approximation (GGA) [@perdew_generalized_1996] in particular, has become the *de facto* standard for electronic structure calculations of solids. Despite its many successes, DFT struggles to capture the composition-dependent energetics necessary to describe the intercalation voltage and compositional phase stability of battery cathode materials. For example, in the case of olivine Li$_x$FePO$_4$ [@padhi_phospho-olivines_1997; @padhi_effect_1997], DFT in the GGA substantially underestimates the experimental $V$ (by $\sim$20%) and fails to qualitatively capture the experimentally-observed phase separation for intermediate Li concentrations [@zhou_phase_2004; @zhou_first-principles_2004]. In order to address this challenge, a wide variety of electronic structure approaches, including DFT with different exchange-correlation functionals [@aydinol_ab_1997; @wolverton_cation_1998; @wolverton_first-principles_1998; @wang_computational_2016; @chakraborty_predicting_2018], hybrid functionals [@chevrier_hybrid_2010; @seo_calibrating_2015], van der Waals (vdW) functionals [@aykol_van_2015], DFT plus on-site Hubbard $U$ (DFT+$U$) [@zhou_first-principles_2004; @bacq_impact_2004; @ong_voltage_2011; @aykol_local_2014; @shishkin_self-consistent_2016; @isaacs_compositional_2017], DFT+$U$+$V$ (where $V$ is an inter-site interaction) [@cococcioni_energetics_2019], DFT plus dynamical mean-field theory [@isaacs_compositional_2019], and diffusion quantum Monte Carlo [@saritas_charge_2020] have been applied to battery cathode voltages. It remains an open question what interactions and level of theory beyond DFT in the GGA are needed to adequately describe such materials.
Among these works, we mention two recent developments pertaining to cathode voltage prediction. The first is the work of Aykol *et al.*, who found that vdW interactions, employed in conjunction with DFT+$U$ and the widely-used GGA of Perdew, Burke, and Ernzerhof (PBE), are necessary to accurately describe the voltage for layered Li$_x$CoO$_2$ [@aykol_van_2015]. Second, also studying layered cathode materials, Chakraborty *et al.* found that the new strongly constrained and appropriately normed (SCAN) DFT functional considerably improves the voltage prediction even without Hubbard $U$ or explicit vdW corrections [@chakraborty_predicting_2018].
In this work, we perform a systematic study of the average Li intercalation voltage of five classic cathode materials: layered Li$_x$TiS$_2$, olivine Li$_x$FePO$_4$, layered Li$_x$NiO$_2$, spinel Li$_x$Mn$_2$O$_4$, and layered Li$_x$CoO$_2$. For the five cathode materials, we investigate the impact on $V$ of (1) the new SCAN exchange-correlation functional, (2) the use of density functionals explicitly considering vdW interactions, and (3) on-site Hubbard $U$ within DFT+$U$. In all cases, the calculations are critically compared to experiments to assess accuracy. We also consider other quantities (volumes, electronic band gaps, and magnetic moments) in order to provide a more complete picture of the accuracy of the methods.
Within pure DFT [^1], we find that SCAN is a significant improvement for describing battery cathode materials, decreasing the mean-absolute-error for $V$ by more than 50%, as compared to PBE [@perdew_generalized_1996], from 0.67 V to 0.30 V. In some cases (e.g., Li$_x$FePO$_4$ and Li$_x$Mn$_2$O$_4$) Hubbard $U$ corrections are still necessary to achieve reasonable agreement (i.e., within 5%) with experiment. Therefore, given the improvement of SCAN over PBE, quantitative voltage predictions within pure DFT are closer but still currently out of reach. When applied in combination with SCAN, Hubbard $U$ within DFT+$U$ can lead to worsened predictions in some cases (i.e., Li$_x$TiS$_2$ and Li$_x$CoO$_2$). In other words, within DFT+$U$, Hubbard $U$ added to SCAN does not universally help or hurt the predictions. In the one case in which SCAN itself provides a sufficient prediction of $V$ (i.e., Li$_x$NiO$_2$), we find it is still possible to achieve a better overall electronic structure description using DFT+$U$ calculations based on PBE rather than SCAN. Therefore, calculations based on SCAN should not necessarily be considered universally better than those based on PBE when considering both the energetics and overall electronic structure. Using PBE, adding vdW interactions provide appreciably improved $V$ predictions, even for non-layered cathode materials lacking a clear van der Waals gap when delithiated. In the majority of the cases, when Hubbard $U$ is also considered with PBE, the experimental $V$ can be achieved with or without vdW interactions (for different values of $U$). Therefore, it is not clear that missing vdW interactions are a significant source of PBE’s well-known voltage underprediction in general. We find that such vdW interactions are significantly less important in terms of $V$ when applied in conjunction with SCAN, which already contains some intermediate-range vdW interactions. Overall, despite the significantly improved $V$ predictions of SCAN as compared to PBE within pure DFT, we illustrate that no single method considered here can generally describe the voltage and overall electronic structure of battery cathode materials, motivating a strong need for improved electronic structure approaches for such systems.
Summary of electronic structure approaches tested {#sec:methodology}
=================================================
Exchange-correlation functional
-------------------------------
The key ingredient to DFT is the exchange-correlation functional $E_{xc}$, which encapsulates all the interaction effects beyond the single-particle kinetic energy and mean-field (Hartree) Coulomb energy [@martin_electronic_2008]. $E_{xc}$ in the local density approximation (LDA) depends solely on the electron density $\rho$ and is parametrized to exactly describe the homogeneous electron gas (jellium) [@ceperley_ground_1980]. In GGA, $E_{xc}$ depends on $\nabla\rho$ in addition to $\rho$, which allows for the satisfaction of additional constraints such as the correct behavior in the slowly- and rapidly-varying density limits [@perdew_generalized_1996]. A higher level of theory is the meta-GGA, in which $E_{xc}$ exhibits an additional dependence on the orbital kinetic energy density $$\tau = \sum_i \frac{1}{2}
|\nabla\psi_i|^2,$$ where $\psi_i$ is the $i$th occupied Kohn-Sham wavefunction, corresponding to functional that is implicitly nonlocal in $\rho$.
The strongly constrained and appropriately normed (SCAN) functional [@sun_strongly_2015] is a new meta-GGA, which satisfies 17 known constraints of the exact $E_{xc}$ and has shown significant promise in the description of solids [@sun_strongly_2015; @sun_accurate_2016; @tran_rungs_2016; @kitchaev_energetics_2016; @zhang_comparative_2017]. Just as PBE is built on top of LDA (reproducing the LDA result for jellium), SCAN is built on top of PBE and exhibits the same behavior as PBE for slowly-varying densities in the metallic bonding regime of $\tau$. We note that the SCAN functional implicitly contains some “intermediate-range” vdW interactions [@sun_strongly_2015], and it also can be incorporated in methods containing explicit vdW interactions [@peng_versatile_2016], as discussed below.
Very relevant to the prediction of battery cathode voltages is that SCAN has been shown to yield significant improvement to formation energy predictions as compared to PBE for strongly-bound compounds like oxides [@isaacs_performance_2018; @zhang_efficient_2018]. Indeed, for a few layered cathode materials, very recent work has suggested that SCAN achieves more accurate $V$ prediction compared to PBE [@chakraborty_predicting_2018; @isaacs_compositional_2019]. In particular, based on calculations of layered Li$_x$NiO$_2$, Li$_x$CoO$_2$, and Li$_x$MnO$_2$ with PBE, PBE+$U$, and SCAN, Chakraborty *et al.* found that SCAN performs better than PBE and PBE+$U$ for the $V$ profiles. Based on the $V$ behavior, as well as predicted lattice parameters, densities of states, and $\rho$ (as compared to that from the PBE0 hybrid functional), they concluded that SCAN without Hubbard $U$ exhibits good overall performance for layered cathode materials. Whether such trends hold more generally (e.g., for non-layered cathodes) is an open question addressed by this work.
Explicit van der Waals interactions
-----------------------------------
The lack of nonlocal correlation effects needed to capture vdW interactions is a well-documented limitation of standard DFT [@grimme_density_2011; @klimes_perspective_2012; @berland_van_2015; @grimme_dispersion-corrected_2016; @hermann_first-principles_2017; @stohr_theory_2019]. In order to address this limitation, first-principles vdW density functionals have been developed. In such functionals, a nonlocal correlation energy term (explicitly nonlocal in $\rho$) of the form $$E_c^{nl}=\frac{1}{2}\int\int \rho(r) \phi(r, r')
\rho(r') d^3rd^3r'$$ is incorporated in $E_{xc}$ [@dion_van_2004; @roman-perez_efficient_2009]. Here, $\phi(r, r')$ is the kernel, which is typically based on approximations to the frequency-dependent polarizability. For example, Dion *et al.* devised a kernel based on a plasmon pole approximation to the dielectric function $\epsilon$ and a second-order expansion of the polarization $S=1-\epsilon^{-1}$ [@dion_van_2004; @grimme_dispersion-corrected_2016], such that the kernel is a function of $\rho$ and $\nabla\rho$ at spatial coordinates $r$ and $r'$ as well as $|r-r'|$.
Aykol *et al.* recently tested a variety of methodologies incorporating vdW interactions (including first-principles and semiempirical approaches) on Li$_x$CoO$_2$ [@aykol_van_2015]. The first-principles opt-type vdW density functionals [@klimes_chemical_2010; @klimes_van_2011], such as optPBE-vdW, were found to yield the most accurate $V$ predictions and correspond to a significant improvement over standard density functionals lacking vdW interactions. This opens up the question of how important vdW interactions are to describe battery cathode materials in general (i.e., beyond Li$_x$CoO$_2$), which we address in this work.
We focus on optPBE-vdW in this work [@klimes_chemical_2010; @klimes_van_2011]. optPBE-vdW combines a linear combination of the exchange forms of PBE and the related RPBE [@hammer_improved_1999], LDA local correlation, and the kernel of Ref. . In optPBE-vdW, the fraction of PBE-like [@perdew_generalized_1996] and RPBE-like [@hammer_improved_1999] exchange and the two parameters employed in both such forms have been optimized (hence the “opt”) to minimize interaction energy errors for the Set 22 (S22) quantum chemistry benchmark [@jurecka_benchmark_2006]. When we refer to adding vdW interactions to PBE in this work, we are referring to the optPBE-vdW method. Although this is not strictly accurate as the difference between PBE and optPBE-vdW is not additive, we do so for convenience and since optPBE-vdW is closely connected to PBE. We also consider the SCAN plus revised Vydrov-Van Voorhis 2010 (SCAN+rVV10) vdW functional [@vydrov_nonlocal_2010; @sabatini_nonlocal_2013; @peng_versatile_2016], which corresponds to a different choice of kernel with one of its two parameters fit to best reproduce the Ar dimer binding curve from coupled cluster singles, doubles, and perturbative triples \[CCSD(T)\] quantum chemistry calculations. SCAN+rVV10 is explored in this work for purely practical reasons as it is currently the only vdW functional implemented in conjunction with the SCAN functional in the Vienna *ab initio* simulation package (<span style="font-variant:small-caps;">vasp</span>). In this work, we also refer to SCAN+rVV10 as SCAN+vdW for convenience.
On-site Hubbard $U$ corrections
-------------------------------
In an attempt to correct for the deficiencies of DFT (using common approximations like the GGA), the DFT+$U$ approach [@himmetoglu_hubbard-corrected_2014] has become a widely used method to describe cathode materials. In this methodology, DFT is augmented with an on-site Hubbard interaction $U$ (solved within static mean-field theory) related to strong electronic correlations in a chosen subspace of localized orbitals (defined via transition metal $d$ orbital projectors in this work). In this methodology, the energy depends on the on-site density matrix for the transition metal $d$ orbitals in addition to $\rho$. In particular, using the simplified rotationally-invariant formalism of Dudarev *et al.* [@dudarev_electron-energy-loss_1998] and the fully localized limit (FLL) double counting [@anisimov_density-functional_1993], the DFT+$U$ energy can be written as $$E_{DFT+U} = E_{DFT}[\rho^s] + \frac{1}{2}U\sum_{\tau,
m, s}n_m^{\tau s}(1-n_m^{\tau s}),\label{eq:dftu}$$ where $\rho^s$ is the spin-density, $E_{DFT}[\rho^s]$ is the (spin-dependent) DFT energy and $n_m^{\tau s}$ is the $m$th eigenvalue of the density matrix corresponding to transition metal site $\tau$ and spin projection $s$. Written this way, it is visible that the effect of DFT+$U$ is to penalize non-integer occupancy of the localized orbitals.
DFT+$U$ has been shown to help alleviate the voltage underestimation of DFT in the GGA [@zhou_first-principles_2004], and it has become a standard tool to describe cathode materials. However, recent evidence suggests it may lead to considerable problems. In particular, for Li$_x$CoO$_2$, DFT+$U$ yields spurious gaps and charge ordering, as well as overestimated Li order-disorder temperatures [@isaacs_compositional_2017]. The ability of DFT+$U$ to accurately describe cathode materials in general is an open question we aim to address in this work. We note that DFT+$U$ is a static approximation to the more accurate DFT plus dynamical mean-field theory (DFT+DMFT), in which the local correlation problem is solved exactly rather than via the Hartree-Fock approximation of DFT+$U$. Recent DFT+DMFT calculations found a significantly different $V$ prediction for Li$_x$CoO$_2$ as compared to DFT+$U$, suggesting dynamical correlations neglected by DFT+$U$ but captured by DFT+DMFT may also be important in battery cathode materials [@isaacs_compositional_2019]. However, due to the large computational cost to solve the quantum impurity problem in DFT+DMFT, we do not explore the role of dynamical correlations in this work.
Computational Details {#sec:compdetails}
=====================
Spin-dependent density functional theory calculations using the projector augmented wave (PAW) method [@blochl_projector_1994; @kresse_ultrasoft_1999] and a 520 eV plane wave kinetic energy cutoff are performed using <span style="font-variant:small-caps;">vasp</span> [@kresse_ab_1994; @kresse_ab_1993; @kresse_efficient_1996; @kresse_efficiency_1996]. We use the Perdew–Burke–Ernzerhof (PBE) GGA [@perdew_generalized_1996] and the strongly constrained and appropriately normed (SCAN) [@sun_strongly_2015] meta-GGA to the exchange-correlation functional. The impact of vdW interactions is assessed via calculations with the optPBE-vdW functional [@klimes_chemical_2010; @klimes_van_2011] and the SCAN+rVV10 functional [@peng_versatile_2016]. On-site Hubbard $U$ is included for the transition metal $d$ states using the rotationally-invariant DFT+$U$ approach [@liechtenstein_density-functional_1995; @himmetoglu_hubbard-corrected_2014]. We use the recommended <span style="font-variant:small-caps;">vasp</span> 5.2 PBE PAW potentials for all calculations [@vasp_paw]. Uniform $k$-meshes are chosen with $\ge 8,000/N_{\mathrm{atoms}}$ $k$-points, where $N_{\mathrm{atoms}}$ is the number of atoms in the unit cell. The ionic forces and total energy are converged to 10$^{-2}$ eV/Å and 10$^{-6}$ eV, respectively. We employ 0.1 eV 1st-order Methfessel-Paxton smearing [@methfessel_high-precision_1989] for structural relaxations and the tetrahedron method with Blöchl corrections [@blochl_improved_1994] for static runs.
{width="\linewidth"}
Results and Discussion {#sec:results}
======================
Crystal structures and nominal electronic configurations
--------------------------------------------------------
We begin by briefly discussing the structures and electronic configuration of the cathode materials considered. The crystal structures and nominal transition metal (TM) $d$ electron configurations for the five cathode materials are illustrated in Fig. \[fig:crystal\_structures\]. All the compounds considered have octahedral coordination of TM by oxygen. Although the octahedra are often distorted, we still refer to the lowest energy three $d$ levels as $t_{2g}$ and the highest energy two $d$ levels as $e_g$ for simplicity, where $t_{2g}$ and $e_g$ are the irreducible representations of $d$ orbitals in perfect octahedral symmetry.
Li$_x$TiS$_2$, Li$_x$NiO$_2$, and Li$_x$CoO$_2$ are layered materials with alternating layers of Li and edge-sharing TM–oxygen octahedra. TiS$_2$ ($t_{2g}^0e_{g}^0$), LiTiS$_2$ ($t_{2g}^1e_{g}^0$), and CoO$_2$ ($t_{2g}^5e_{g}^0$) are in the hexagonal $P\bar{3}m1$ structure (O1 structure) [@chianelli_structure_1975; @dahn_structure_1980; @amatucci_coo2_1996]. NiO$_2$ and LiCoO$_2$ (both $t_{2g}^6e_{g}^0$) are considered in the rhombohedral $R\bar{3}m$ structure (O3 structure) [@johnston_preparation_1958; @orman_cobaltiii_1984; @seguin_structural_1999; @croguennec_nio2_2000]. We model LiNiO$_2$ ($t_{2g}^6e_{g}^1$) with the monoclinic $Pm$ structure [@cao_local_2009], which captures the Jahn-Teller distortion. For the Li$_x$CoO$_2$ case, in addition to computing $V$ for the full $0 < x < 1$ range, we also compute $V$ for $x<\frac{1}{2}$ and $x>\frac{1}{2}$. To do so, we consider Li$_{1/2}$CoO$_2$ in the known monoclinic $P2/m$ structure, which corresponds to an in-plane Li/vacancy ordering in a unit cell twice as large as the primitive rhombohedral cell [@takahashi_single-crystal_2007].
In contrast to the other materials, Li$_x$FePO$_4$ and Li$_x$Mn$_2$O$_4$ do not exhibit layered crystal structures. Olivine FePO$_4$ ($t_{2g}^3e_{g}^2$) and LiFePO$_4$ ($t_{2g}^4e_{g}^2$) crystallize in an orthorhombic $Pnma$ structure consisting of (1) one-dimensional channels of Li and (2) layers of corner sharing Fe–oxygen octahedra connected via phosphate groups [@santoro_antiferromagnetism_1967; @rousse_magnetic_2003]. To model Li$_x$Mn$_2$O$_4$, we consider Mn$_2$O$_4$ ($t_{2g}^3e_{g}^0$) in the ideal spinel-like $Fd\bar{3}m$ structure ($\lambda$-MnO$_2$), which consists of a diamond sublattice of Li and a three-dimensional network of edge-sharing Mn–oxygen octahedra [@hunter_preparation_1981; @mosbah_phases_1983]. In order to capture possible Jahn-Teller effects, we model LiMn$_2$O$_4$ (nominally in the $t_{2g}^{3.5}e_{g}^0$ configuration) with the symmetry-broken ferromagnetic monoclinic $C2/c$ structure from Ref. . Ferromagnetic ordering is considered for all magnetic compounds except Li$_x$FePO$_4$, which exhibits antiferromagnetic ordering [@rousse_magnetic_2003].
Pure DFT
--------
![Average intercalation voltage for $0 < x < 1$ within PBE, optPBE-vdW, SCAN, and SCAN+vdW for DFT (i.e., $U=0$). The solid black horizontal lines indicate the experimental voltage. Mean absolute error (MAE) values for the 5 cathode materials are indicated in the legend.\[fig:dft\_voltages\]](voltages.pdf){width="\linewidth"}
Figure \[fig:dft\_voltages\] shows the average intercalation voltages over the full Li concentration range ($0 < x < 1$) within pure DFT ($U=0$). As has been shown previously [@zhou_first-principles_2004], PBE systematically and substantially underpredicts $V$, yielding a mean absolute error (MAE) of 0.67 V. SCAN represents a significant improvement over PBE in terms of the predicted $V$, reducing the MAE by over 50% to 0.30 V. However, some errors are still unacceptably large (e.g., 15% error for Li$_x$Mn$_2$O$_4$). In this sense, quantitative $V$ predictions within DFT are closer to being achieved but are still currently out of reach. For most of the cathode materials, SCAN still underpredicts the experimental values despite the appreciable increase in $V$ with respect to PBE values. There are two exceptions to this trend: (1) Li$_x$NiO$_2$, for which the SCAN prediction (3.8 V) is nearly identical to the experimental value (3.9 V) and (2) Li$_x$CoO$_2$, for which the SCAN prediction (4.5 V) is appreciably larger than experiment (4.2 V). The increase in $V$ of SCAN with respect to PBE is highly system dependent: while this increase is 1.0 V for Li$_x$CoO$_2$, it is a mere 0.2 V for Li$_x$Mn$_2$O$_4$.
![Volumes in Å$^3$ per formula unit within pure DFT for (a) Li$_x$TiS$_2$, (b) Li$_x$FePO$_4$, (c) Li$_x$NiO$_2$, (d) Li$_x$Mn$_2$O$_4$, and (e) Li$_x$CoO$_2$ as a function of Li concentration for the various methods considered in this work. The panels are labeled by the transition metal. Experimental values are shown as black horizontal lines [@whittingham_lithium_1975; @mizushima_lixcoo2_1980; @hunter_preparation_1981; @hirano_relationship_1995; @amatucci_coo2_1996; @rousse_magnetic_2003; @takahashi_single-crystal_2007; @cao_local_2009].[]{data-label="fig:dft_volume"}](dft_volumes.pdf){width="1.0\linewidth"}
Explicit vdW interactions also generally yield an increase in predicted $V$, though of a smaller magnitude. For example, adding vdW interactions to PBE (i.e., optPBE-vdW) reduces the MAE from 0.67 V to 0.42 V. Here, the voltage increases are less system-dependent: similar increases of 0.1–0.3 V (4–10%) for optPBE-vdW with respect to PBE are found for all five cathode materials. The $V$ enhancement is not generally smaller for non-layered materials: $V$ increases by 10% for Li$_x$FePO$_4$, for example. In contrast, adding vdW interactions to SCAN (i.e., SCAN+vdW) does not appreciably increase predicted $V$ and an MAE of 0.27 V (negligibly smaller than the 0.30 V value for SCAN) is obtained. We believe this behavior stems from the construction of SCAN+vdW since a parameter in the rVV10 form is fit specifically for SCAN, which already intrinsically contains some intermediate-range vdW interactions. Based on the predicted $V$ behavior, we find that explicit vdW methods are not critical when applied in conjunction with SCAN for battery cathode materials. This suggests that the vdW interactions intrinsic to SCAN are likely sufficient to describe vdW interactions in this class of materials. We note that Chakraborty *et al.* reached a similar conclusion using a distinct dispersion-corrected DFT approach [@chakraborty_predicting_2018].
The predicted volume is another observable with which we can benchmark different computational methods. As shown in Fig. \[fig:dft\_volume\], we find SCAN+vdW leads to worsened volume predictions compared to SCAN for all the systems considered. This suggests that the explicit vdW interactions contained within SCAN+vdW may be not only unnecessary, but even harmful to the description of battery cathode materials. This behavior is in contrast to that of optPBE-vdW, which generally improves the volume predictions as compared to PBE.
We note that the impact of the explicit vdW interactions on $V$ is not primarily structural in nature. For example, freezing to PBE ground state structures, the SCAN $V$ value for Li$_x$CoO$_2$ changes by only 24 meV relative to the value calculated using the SCAN ground state structures. We find similar behavior for the other cathode materials. For example, for Li$_x$TiS$_2$, the computed $V$ changes by at most 0.1 V for the case with structures relaxed with vdW interactions and that with structures relaxed without vdW interactions, for all the functionals considered.
![Average Li$_x$CoO$_2$ intercalation voltage for $0 < x <
1/2$ (left horizontal lines) and $1/2 < x < 1$ (right horizontal lines) within PBE, optPBE-vdW, SCAN, and SCAN+vdW for DFT (i.e., $U=0$). The solid black horizontal lines indicate the corresponding experimental values.\[fig:dft\_half\_voltages\]](half_voltages.pdf){width="\linewidth"}
For Li$_x$CoO$_2$, we also consider the separate “half voltages,” i.e., the distinct voltage averages for $0 < x < 1/2$ and $1/2 < x <
1$, shown in Fig. \[fig:dft\_half\_voltages\] for DFT. In Fig. \[fig:dft\_half\_voltages\], one can observe the same main trends discussed above for the Li$_x$CoO$_2$ $V$ over the full range of $x$: (1) SCAN significantly increases the voltage, exceeding experiment and (2) incorporating explicit vdW interactions moderately enhances the voltages when added to PBE, but negligibly when added to SCAN. We focus on the “voltage gap” at $x=1/2$, i.e., the difference between the voltage average of $1/2 < x < 1$ and that of $0 < x < 1/2$. Such a voltage gap $\Delta V$ is a measure of the formation energy of a stable (on the convex hull) phase of intermediate Li concentration with respect to the $x=0$ and $x=1$ endmembers, which can be written as $-x(1-x)e\Delta V$ [@aykol_local_2014]. Therefore, the $x=1/2$ voltage gap of Li$_x$CoO$_2$ is a convenient benchmark for compositional phase stability. The voltage gap predicted by PBE ($-0.9$ V) is significantly larger in magnitude than the experimental value ($-0.4$ V). SCAN predicts an improved, but still too large (in magnitude) voltage gap of $-0.7$ V. This is consistent with the conclusion that SCAN provides an improved, though still imperfect, description of the energetics of battery cathode materials. vdW interactions also improve the predicted $\Delta V$, yielding values of $-0.8$ V for optPBE-vdW and $-0.5$ V for SCAN+vdW.
DFT+$U$
-------
{width="\linewidth"}
Figure \[fig:dftu\_voltages\] shows the DFT+$U$ average intercalation voltages over the full Li concentration range. We first comment on the general impact of $U$ on the intercalation voltages. Although an increase in $V$ has typically been found with increasing $U$ for battery cathode materials (shown here as well as in previous works [@zhou_first-principles_2004; @zhou_phase_2004; @aykol_local_2014; @aykol_van_2015; @isaacs_compositional_2017; @cococcioni_energetics_2019]), we also find the opposite behavior in the small-$U$ limit in some of the cases (e.g., Li$_x$TiS$_2$). An increase in $V$ with $U$ necessarily stems from the larger energy penalty on the $x=0$ endmember than the $x=1$ endmember, since $eV$ is proportional to $E(0) - E(1).$ Similarly, a decrease in $V$ with $U$ corresponds to a greater energetic penalty on $x=1$ than $x=0.$
Nominal electron counting corresponding to completely filled or completely empty states (as suggested by most of the level diagrams in Fig. \[fig:crystal\_structures\]) is insufficient by itself to explain these trends, as the energy penalty from DFT+$U$ (using the FLL double counting) exactly vanishes in such a fully localized limit, as can be seen in Eq. \[eq:dftu\]. However, knowledge of the electron counting in conjunction with the overall electronic structure can be used to explain the observed trends. For example, for Li$_x$CoO$_2$, $U$ penalizes metallic $x=0$ more than the band insulator $x=1$, which has closer-to-integer $d$ orbital occupations [@isaacs_compositional_2017] due to electron counting (as well as to the increased ionicity stemming from Li). Therefore, $V$ increases with $U$. The reverse situation occurs for Li$_x$TiS$_2$: here, $x=0$ is the band insulator and $x=1$ has a partially-filled $t_{2g}$ shell, corresponding to a metal. This explains the decrease in $V$ in the small-$U$ limit (the increase at larger $U$ is discussed later). Analogously, a negative $\partial V/\partial U$ in the small-$U$ limit is also found for Li$_x$Mn$_2$O$_4$ using PBE and optPBE-vdW since within these levels of theory Mn$_2$O$_4$ is insulating and LiMn$_2$O$_4$ is metallic for small $U$. The voltage increases with $U$ for Li$_x$FePO$_4$ and Li$_x$NiO$_2$, though the origin of the increases is different than the Li$_x$CoO$_2$ case. For Li$_x$FePO$_4$, whose endmembers are both magnetic insulators, it is the enhanced covalency of the $x=0$ endmember [@isaacs_compositional_2017] that gives it a larger energy penalty. And for Li$_x$NiO$_2$, despite the nominally partially-filled $e_g$ shell for $x=1$, the increased ordering from the Jahn-Teller distortion allows the $x=1$ endmember to be less affected by $U$ than the $x=0$ endmember.
{width="1.0\linewidth"}
Due to the diversity of behavior observed, we now discuss the specific results (including the behavior with different levels of theory in comparison with experiment) for each material separately. Afterwards, we present a general synthesis of the results.
We begin with Li$_x$TiS$_2$. As discussed above, small values of $U$ for Li$_x$TiS$_2$ serve to decrease the predicted $V$, which already underestimates the experimental value within pure DFT. We also note that, for the majority of the pure DFT levels of theory considered here, $U$ also hurts the volume prediction (shown in Fig. \[fig:dftu\_volume\]) for Li$_x$TiS$_2$. Therefore, in the case of Li$_x$TiS$_2$, adding Hubbard $U$ serves to hurt the description. Consistent with this finding, we note that the past work of Chevrier *et al.* avoided the use of Hubbard $U$ for Ti-based compounds [@chevrier_hybrid_2010].
Here, we comment on the discontinuous behavior at larger $U$. A metal-insulator transition for LiTiS$_2$ at larger $U$ decreases its energy penalty relative to TiS$_2$, leading to a change in sign in $\partial V/\partial U.$ The insulating behavior is spurious as LiTiS$_2$ is actually metallic in experiment [@klipstein_transport_1987]. This raises the question of whether this predicted insulating state for LiTiS$_2$ for larger $U$ corresponds to (1) an intrinsic failure of DFT+$U$ or simply (2) the use of an unphysically-large $U$ parameter. Using a self-consistent linear response approach and PBE, Shishkin and Sato computed $U$ of 5.5 eV for LiTiS$_2$ (using same the PAW potential for Ti we employ, which treats the $4s$ semicore states as valence states) [@shishkin_self-consistent_2016]. Since Shishkin and Sato found LiTiS$_2$ to still be metallic at this $U$ value [@shishkin_self-consistent_2016], their results suggest the second case above (too-large $U$); however, we note that this may be a borderline case as we find insulating LiTiS$_2$ for $U=6$ eV.
In conjunction with the pure DFT results discussed above, we find that the application of SCAN (as opposed to PBE) and vdW interactions significantly improves the voltage prediction for Li$_x$TiS$_2$, while $U$ hurts the description. We note that SCAN+vdW, which exhibits the best agreement with the experimental $V$ (error of only 0.1 V), has worse volume predictions than those of SCAN for Li$_x$TiS$_2$ (for LiTiS$_2$ in particular), as shown in Fig. \[fig:dft\_volume\].
For Li$_x$FePO$_4$, the pure DFT approaches are insufficient to quantitatively describe the voltage. However, the predicted voltage increases roughly linearly with $U$ in all cases, enabling agreement with experiment using DFT+$U$. For PBE, the optimal $U$ value to achieve agreement with experiment is 4.2 V, in agreement with previous work [@zhou_phase_2004; @zhou_first-principles_2004; @isaacs_compositional_2017]. This value also agrees well with the overall magnitude of the first-principles $U$ values for the $x=0$ (4.9 V) and $x=1$ (3.7 V) endmembers computed from first principles (with PBE) via the linear response approach [@zhou_first-principles_2004]. Since adding vdW interactions to PBE (i.e., optPBE-vdW) provides a roughly rigid increase in the predicted $V$, of around 0.3 V, the optimal $U$ value to achieve agreement with the experimental $V$ for optPBE-vdW is 2.0 eV, substantially lower than the PBE case. SCAN+$U$ and SCAN+vdW+$U$ yield essentially identical $V$ predictions, consistent with the intrinsic vdW interactions in SCAN. For such methods, the predicted $V$ matches experiment for $U=3.0$ eV, also significantly lower than the PBE case.
Although not computed here, it would be interesting to assess whether the first-principles $U$ values based on optPBE-vdW and/or SCAN(+vdW) would also be appreciably lower than those of PBE, leading to the same consistency observed for PBE in terms of the first-principles $U$ and $U$ fit to experimental $V$. We note that the optimal $U=3.0$ eV for SCAN(+vdW), in terms of $V$, agrees well with $U$ values found to reproduce the FeO/Fe$_2$O$_3$ (2.9 eV) and FeO/Fe$_3$O$_4$ (3.3 eV) experimental oxidation reaction energies in a recent SCAN+$U$ work by Sai Gautam and Carter [@sai_gautam_evaluating_2018].
The volume behavior for Li$_x$FePO$_4$ is shown in Fig. \[fig:dftu\_volume\]. SCAN+$U$ yields the best volume prediction for Li$_x$FePO$_4$ among all the methods considered in this work, though some underestimation of the LiFePO$_4$ volume persists. The band gap and local Fe magnetic moment behaviors for Li$_x$FePO$_4$ are shown in the Supplemental Material [^2]. The application of $U$ to SCAN also significantly improves the predicted LiFePO$_4$ band gap, though the FePO$_4$ gap (already in good agreement with experiment for $U=0$) becomes overestimated. A similar effect is found in terms of the local Fe magnetic moment, with overestimation (underestimation) for FePO$_4$ (LiFePO$_4$). Ultimately, while $U$ can be chosen to yield agreement with the experimental $V$ using DFT+$U$ based on any of the pure DFT methodologies considered here, we find that SCAN+$U$ using $U$ of $\sim 3$ eV provides the best (although still imperfect) overall description of Li$_x$FePO$_4$ when also taking into account the volume, band gap, and local magnetic moments.
For Li$_x$NiO$_2$, the pure DFT prediction using PBE significantly underestimates experiment, but agreement can be reached for $U$ of $\sim 6$ eV, which is close in value to the PBE first-principles computed endmember $U$ values [@zhou_first-principles_2004]. The behavior is similar for optPBE-vdW+$U$, whose $V$ predictions are $\sim 0.2$ eV larger than those of PBE+$U$. The behavior for SCAN is quite distinct. Here, the SCAN-predicted $V$ already exhibits excellent agreement (within $\sim 0.1$ V) with experiment even without Hubbard $U$. Therefore, the application of $U$ in this case pushes $V$ to far too large values. This is also true for SCAN+vdW+$U$, which exhibits a small, roughly constant $\sim 0.1$ V increase in $V$ with respect to SCAN+$U$.
Although one can achieve a satisfactory quantitative $V$ prediction using PBE/optPBE-vdW with $U$ ($\sim 6$ eV) or SCAN(+vdW) without $U$, the volume prediction (shown in Fig. \[fig:dftu\_volume\]) suggests such approaches are not equivalent in their overall description. SCAN and SCAN+vdW provide worsened volume predictions as compared to PBE+$U$. Although optPBE-vdW+$U$ yields a similar LiNiO$_2$ volume as PBE+$U$, its volume prediction for NiO$_2$ is significantly worse than PBE+$U$. We note additionally that PBE+$U$ yields an accurate band gap prediction for LiNiO$_2$, as shown in the Supplemental Material. Overall, despite the excellent $V$ prediction using SCAN(+vdW), we find PBE+$U$ provides the overall best description of Li$_x$NiO$_2$.
The Li$_x$Mn$_2$O$_4$ case is similar to that of Li$_x$FePO$_4$ in that a quantitatively accurate $V$ prediction can be achieved using calculations based on any of the pure DFT methodologies considered here, but only using Hubbard $U$. $U$ values of 5.7 eV (reasonably close in value to the first-principles computed endmember $U$ values [@zhou_first-principles_2004]) and 4.4 eV are needed to achieve agreement with the experimental $V$ for PBE and optPBE-vdW, respectively. The SCAN+$U$ and SCAN+vdW+$U$ voltages agree with experiment for the significantly smaller value of $U=2.6$ eV. This value is in good agreement with the $U$ values found to reproduce the MnO/Mn$_2$O$_3$ (2.9 eV) and Mn$_2$O$_3$/MnO$_2$ (2.5 eV) experimental oxidation reaction energies in the work of Sai Gautam and Carter [@sai_gautam_evaluating_2018].
Here, as in the Li$_x$FePO$_4$ case, the SCAN+vdW $V$ result is nearly identical to that of SCAN. This suggests that the energetic impact of the vdW interactions in SCAN+vdW beyond those already contained within SCAN itself is especially small for the non-layered cathode materials. In contrast, optPBE-vdW+$U$ yields a substantially larger $V$ prediction than PBE+$U$. Based on the volume data shown in Fig. \[fig:dftu\_volume\] and band gap data shown in the Supplemental Material, we find that DFT+$U$ calculations based on SCAN exhibit a better overall description than those based on PBE for Li$_x$Mn$_2$O$_4$. In particular, SCAN+$U$ and SCAN+vdW+$U$ do not exhibit the significant volume overestimation of PBE+$U$ and optPBE-vdW+$U$ for appreciable $U$. In addition, the LiMn$_2$O$_4$ band gap is underestimated by SCAN+$U$ and SCAN+vdW+$U$ by a much smaller degree than PBE+$U$ and optPBE-vdW+$U$.
![Li$_x$CoO$_2$ intercalation voltage for (a) $x<1/2$ and (b) $x>1/2$ within PBE, optPBE-vdW, SCAN, and SCAN+vdW for DFT+$U$. The solid black horizontal lines indicate the experimental voltage.\[fig:dftu\_half\_voltages\]](half_voltages_u.pdf){width="\linewidth"}
The Li$_x$CoO$_2$ voltage is significantly underestimated within pure DFT using PBE (by 0.8 V). The significant increase in the predicted $V$ when adding $U$ to PBE, which is dampened via a spurious metal-insulator transition for CoO$_2$, is still insufficient to achieve agreement with the experimental $V$ [@aykol_local_2014; @aykol_van_2015; @isaacs_compositional_2017]. As was previously shown [@aykol_van_2015], adding vdW interactions via optPBE-vdW+$U$ further enhances $V$ with respect to PBE+$U$ and enables agreement with experiment. Therefore, it was suggested [@aykol_van_2015] that such nonlocal correlation effects were associated with the $V$ underprediction within PBE+$U$ for Li$_x$CoO$_2$ (and possibly other transition metal oxides). We reproduce the previous result here and find that the optPBE-vdW+$U$ voltage agrees with experiment for $U=4.4$ eV.
SCAN provides a drastically different $V$ prediction for Li$_x$CoO$_2$ [@chakraborty_predicting_2018; @isaacs_compositional_2019], moderately *over*estimating (by 0.3 V) the experimental voltage. Since $U$ serves to increase $V$ in this case, the SCAN+$U$ voltage predictions for Li$_x$CoO$_2$ become even further from experiment. As observed in many of the cases discussed above, adding vdW interactions to SCAN+$U$ (SCAN+vdW+$U$) has a relatively modest impact as compared to the difference between optPBE-vdW+$U$ and PBE+$U$. SCAN+vdW+$U$ provides $V$ predictions for Li$_x$CoO$_2$ no more than $0.1$ V larger (further from experiment) than SCAN+$U$.
Similar behavior is found in terms of the half voltages for Li$_x$CoO$_2$, shown in Fig. \[fig:dftu\_half\_voltages\]: (1) For PBE, $U$ enhances the half voltages, but not enough to reach experimental values, (2) optPBE-vdW+$U$ provides a substantial increase over PBE+$U$ and enables agreement with experiment (for $U$ close to 3 eV), and (3) $U$ (vdW interactions) generally tends to significantly (moderately) enhance the already-too-large voltages of SCAN. We note that, despite $U$ further overestimating the voltage magnitudes when applied to SCAN(+vdW), it does lead to an improved voltage gap at $x=1/2$.
Although optPBE-vdW+$U$ achieves agreement with the experimental voltage (overall the full and half $x$ ranges), it may not provide an accurate overall description of Li$_x$CoO$_2$. As shown in the Supplemental Material, although it exhibits an accurate prediction of the LiCoO$_2$ band gap, optPBE-vdW+$U$ exhibits the same spurious orderings as PBE+$U$: CoO$_2$ gap opening and large magnetic moment of PBE+$U$, as well as Li$_{1/2}$CoO$_2$ charge ordering and gap opening.
We discuss two possible alternatives to optPBE-vdW+$U$ for best describing Li$_x$CoO$_2$. The first alternative is to use pure SCAN. Despite modest voltage overestimation (e.g., 0.3 V for $0 < x < 1$), SCAN does not exhibit any of the spurious gap opening or charge ordering discussed above. It also exhibits a very accurate LiCoO$_2$ band gap, Li$_{1/2}$CoO$_2$ magnetic moment, and reasonably accurate volume predictions, as shown in Fig. \[fig:dft\_volume\] and the Supplemental Material. In addition, although the overall voltage magnitudes are moderately overestimated, the $x=1/2$ voltage gap (related to the $x=1/2$ formation energy) agrees decently well with experiment, as discussed in the previous section. The second alternative is to use DFT+DMFT (to which DFT+$U$ is a static approximation ignoring dynamical correlations) in conjunction with SCAN, as very recent work using non-charge-self-consistent DFT+DMFT [@isaacs_compositional_2019] found that dynamical correlations (1) are large and $x$-dependent in Li$_x$CoO$_2$, (2) help eliminate the spurious gaps and charge ordering of DFT+$U$, and (3) reduce the predicted $V$ such that the SCAN+DMFT voltage is likely to agree well with experiment. Further work to assess which of these alternatives (or another) is optimal to accurately describe Li$_x$CoO$_2$ will be important future work.
Finally, we summarize our overall findings regarding describing battery cathode materials within DFT+$U$. As discussed in the previous section, within pure DFT, it is clear that (1) SCAN is superior to PBE and (2) adding additional vdW interactions beyond those intrinsic to SCAN is not essential and is in some cases detrimental. With DFT+$U$, the results are less clear cut.
In the case of Li$_x$TiS$_2$, adding Hubbard $U$ generally yields no improvement over the corresponding pure DFT $V$ results (which are only modestly underestimated with SCAN and SCAN+vdW), if one takes into account the spurious LiTiS$_2$ metal-insulator transition predicted by DFT+$U$ occurring for sufficiently-large $U$. In contrast, Hubbard $U$ is essential to achieve a voltage prediction in agreement with experiment for Li$_x$FePO$_4$ and LiMn$_2$O$_4$. Therefore, the new SCAN functional does not eliminate the need for Hubbard $U$ corrections. In fact, we find SCAN+$U$ provides the best description of these two cathode materials. Therefore, it is not true that SCAN eliminates the need for $U$ for battery cathode materials in general, in contrast to this conclusion reached by Chakraborty *et al.* in their study of layered systems [@chakraborty_predicting_2018]. Although SCAN provides an excellent voltage prediction for Li$_x$NiO$_2$, an improved description can be achieved via DFT+$U$ calculations based on PBE. Therefore, we find that calculations based on SCAN should not be universally considered superior to those based on PBE. Finally, in the case of Li$_x$CoO$_2$, none of the methods considered here gives a sufficient description of both the voltage and electronic structure, though SCAN arguably fares the best.
Taking all these results into account, despite the improved performance obtained via pure DFT and DFT+$U$ calculations based on SCAN for certain cases, we find that no single method can sufficiently accurately describe the voltage and overall structural, electronic, and magnetic properties (i.e., yielding errors no more than 5% for voltage, volume, band gap, and magnetic moments) of the battery cathode materials considered here. Our results strongly motivate the need for improved electronic structure approaches for such systems.
Conclusions {#sec:conclusions}
===========
Despite the great need for an accurate and computationally inexpensive approach to characterize and design battery cathode materials, such a method still remains out of reach at present. Within pure DFT, SCAN is a significant improvement over PBE for describing cathode materials, though appreciable errors remain. Methods incorporating explicit vdW interactions are not critical and in cases even detrimental when applied in conjunction with SCAN, which already intrinsically contains some intermediate-range vdW interactions.
Hubbard $U$ corrections considered within DFT+$U$ are essential to achieve an accurate voltage prediction in some cases (e.g., Li$_x$FePO$_4$ and Li$_x$Mn$_2$O$_4$) and detrimental in others (e.g., Li$_x$TiS$_2$). Although we find SCAN+$U$ provides the best description for Li$_x$FePO$_4$ and Li$_x$Mn$_2$O$_4$, we find PBE+$U$ gives the best description for Li$_x$NiO$_2$, suggesting DFT+$U$ calculations based on SCAN should not be considered universally superior to those based on PBE. No method here is completely satisfactory to describe Li$_x$CoO$_2$, though the SCAN description perhaps has the fewest deficiencies. Our results motivate the need to develop improved electronic structure descriptions that can accurately describe the thermodynamics and electronic structure of battery cathode materials.
We acknowledge support from Toyota Research Institute through the Accelerated Materials Design and Discovery program (development of software tools for automating electronic structure calculations) and the Center for Electrochemical Energy Science (CEES), an Energy Frontier Research Center funded by the U.S. Department of Energy, Office of the Science, Basic Energy Science under Award No. DE-AC02-06CH11357 (voltage calculations). Computational resources were provided by the National Energy Research Scientific Computing Center (U.S. Department of Energy Contract DE-AC02-05CH11231) and the Extreme Science and Engineering Discovery Environment (National Science Foundation Contract ACI-1548562).
[^1]: We use “pure DFT” to refer to all the methodologies studied in this work without Hubbard $U$ corrections, such that the total energy is purely a functional of the density (even if only implicitly). This includes SCAN and the vdW functionals.
[^2]: See Supplemental Material for additional calculation results (magnetic moment, band gap, and density of states) and details on the experimental voltage data.
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abstract: 'We study a generalization of the Set Cover problem called the *Partial Set Cover* in the context of geometric set systems. The input to this problem is a set system $({X}, {\mathcal{S}})$, where ${X}$ is a set of elements and ${\mathcal{S}}$ is a collection of subsets of ${X}$, and an integer $k \le |{X}|$. The goal is to cover at least $k$ elements of ${X}$ by using a minimum-weight collection of sets from ${\mathcal{S}}$. The main result of this article is an rounding scheme which shows that the integrality gap of the Partial Set Cover is at most a constant times that of the Set Cover for a certain projection of the set system $({X}, {\mathcal{S}})$. As a corollary of this result, we get improved approximation guarantees for the Partial Set Cover problem for a large class of geometric set systems.'
author:
- Tanmay Inamdar
- Kasturi Varadarajan
bibliography:
- 'wpgsc.bib'
title: On Partial Covering For Geometric Set Systems
---
Introduction
============
In the Set Cover () problem, the input is a set system $({X}, {\mathcal{S}})$, where ${X}$ is a set of $n$ elements, and ${\mathcal{S}}$ is a collection of subsets of ${X}$ . The goal is to find a minimum size collection ${\mathcal{S}}' \subseteq {\mathcal{S}}$ that *covers* ${X}$, i.e., the union of the sets in ${\mathcal{S}}'$ contains the elements of ${X}$. In the weighted version, each set $S_i \in {\mathcal{S}}$ has a non-negative weight $w_i$ associated with it, and we seek to minimize the weight of ${\mathcal{S}}'$. A simple greedy algorithm finds a solution that is guaranteed to be within $O(\log n)$ factor from the optimal, and it is not possible to do better in general, under certain standard complexity theoretic assumptions [@Feige1998].
The question of whether we can improve the $O(\log n)$ bound has been extensively studied for geometric set systems. We focus on three important classes – covering, hitting, and art gallery problems. In the Geometric Set Cover problem, ${X}$ typically consists of points in ${\mathbb{R}}^d$, and ${\mathcal{S}}$ contains sets induced by a certain class of geometric objects via containment. For example, each set in ${\mathcal{S}}$ might be the subset of ${X}$ contained in a hypercube. Some of the well-studied examples include covering points by disks in plane, fat triangles, etc. In the Geometric Hitting Set problem, ${X}$ is a set of geometric objects, and each set in ${\mathcal{S}}$ is the subset consisting of all objects in ${X}$ that are pierced by some point. In an example of the art gallery problem, ${X}$ consists of a set of points in a simple polygon, and each set in ${\mathcal{S}}$ is the subset consisting of all points in ${X}$ that can be seen by some vertex of the polygon [@KingKirk]. Thus, the set system here is defined by visibility.
For many such geometric set systems, it is possible to obtain approximation guarantees better than $O(\log n)$. We survey two of the main approaches to obtain such guarantees. The first and the most successful approach is based on the Linear Program () and its connection to ${\varepsilon}$-nets. For completeness, we state the standard for the weighted case. $$\begin{aligned}
{3}
\text{minimize} \displaystyle&\sum\limits_{S_i \in {\mathcal{S}}} w_{i}x_{i} & \nonumber\\
\text{subject to} \displaystyle&\sum\limits_{i:e_{j} \in S_{i}} x_{i} \geq 1, \quad & \quad e_j \in {X}\\
\displaystyle &\qquad \quad x_i \ge 0, & \quad S_i \in {\mathcal{S}}\end{aligned}$$ For the unweighted case, @Even2005 showed that if for a certain set system, $O{\left(\frac{1}{{\varepsilon}} \cdot g{\left(\frac{1}{{\varepsilon}}\right)}\right)}$ size ${\varepsilon}$-nets exist, then the integrality gap of the is $O(g(OPT))$, where $OPT$ is the size of the optimal solution. This result is constructive, in that an efficient algorithm for constructing ${\varepsilon}$-nets also yields an efficient algorithm for obtaining an $O(g(OPT))$ approximation. (A similar result was obtained earlier by @BronnimannG1995, without using LP machinery). It is fairly well-known ([@Clarkson1987; @HausslerV1987]) that for a large class of geometric set systems, ${\varepsilon}$-nets of size $O{\left(\frac{1}{{\varepsilon}} \log{\left(\frac{1}{{\varepsilon}}\right)}\right)}$ can be computed efficiently, which implies $O(\log(OPT))$ approximation for the set cover problem on the corresponding geometric set system. @ClarksonV2007 showed that if the *union complexity* of any set of $n$ objects is $O(n\cdot h(n))$, then ${\varepsilon}$-nets of size $O{\left(\frac{1}{{\varepsilon}} \cdot h{\left(\frac{1}{{\varepsilon}}\right)}\right)}$ exist. @AronovES2010 gave a tighter bound of $O(\frac{1}{{\varepsilon}} \cdot \log h(\frac{1}{{\varepsilon}}))$ on the size of ${\varepsilon}$-nets for the objects of union complexity $O(n\cdot h(n))$ (see also [@VaradarajanUnion2009]). Some of these results were extended to the weighted case in [@VaradarajanWGSC2010; @ChanGKS12] by a technique called *quasi-uniform sampling*. We summarize some of these ${\varepsilon}$-net based results for the set cover problem for geometric set systems in the accompanying table.
${X}$ Geometric objects inducing ${\mathcal{S}}$ Integrality Gap of
-------------------------------- -------------------------------------------- ----------------------
Disks (via containment) $O(1)$
Fat triangles (containment) $O(\log \log^* n)$
Unit cubes (containment) $O(1)$
Halfspaces (containment) $O(1)$
Rectangles in ${\mathbb{R}}^3$ Points (via piercing) $O(\log \log n)$
Points on 1.5D terrain Points on terrain (via visibility) $O(1)$ [@ElbTerrain]
: LP-based approximation ratios for . See [@ClarksonV2007; @AronovES2010; @VaradarajanWGSC2010; @EASFat; @ChanGKS12] for the references establishing these bounds. Except for piercing rectangles in ${\mathbb{R}}^3$ by points, these bounds hold for the weighted . For these problems, we obtain analogous results for weighted .
Another approach for tackling for geometric set systems is by combinatorial algorithms. The dominant paradigm from this class is the simple Local Search algorithm. The effectiveness of Local Search was first demonstrated by @MustafaR2010, who gave the first PTAS for covering points by disks in plane. There have been a series of results that build on their work, culminating in @GovindarajanRRB2016, who show that Local Search yields a PTAS for for a fairly general class of objects such as pseudodisks and non-piercing regions in plane. @KrohnTerrain gave a PTAS for the terrain guarding problem, where the geometric set system is defined by visibility. Another common strategy, called the *shifting strategy*, was introduced by @HochbaumM1985. They give a PTAS for covering points by unit balls in ${\mathbb{R}}^d$; however in this case the set ${\mathcal{S}}$ consists of *all* unit balls in ${\mathbb{R}}^d$. @ChanPiercing gave a PTAS for piercing a set of fat objects in ${\mathbb{R}}^d$ using a minimum number of points from ${\mathbb{R}}^d$.
Now we turn to the Partial Set Cover () problem. The input to is the same as that to the , along with an additional integer parameter $k \le |{X}|$. Here the goal is to cover at least $k$ elements from ${X}$ while minimizing the size (or weight) of the solution ${\mathcal{S}}' \subseteq S$. It is easy to see that is a generalization of , and hence it is at least as hard as . We note here that another classical problem that is related to both of these problems is the so-called Maximum Coverage () problem. In this problem, we have an upper bound on the number of sets that can be chosen in the solution, and the goal is to cover the maximum number of elements. It is a simple exercise to see that an exact algorithm for the unweighted can be used to solve exactly, and vice versa. However the reductions are not approximation-preserving. In particular, the greedy algorithm achieves $1 - 1/e$ approximation guarantee for — which is essentially the best possible — whereas it is -hard to approximate within $o(\log n)$ factor in general. We refer the reader to [@KhullerMN1999] for a generalization of and a survey of results.
For , the greedy algorithm is shown to be an $O(\log \Delta)$ approximation in [@Slavik1997], where $\Delta$ is the size of the largest set in ${\mathcal{S}}$. @Gandhi2004 give a primal-dual based algorithm which achieves an approximation guarantee of $f$, where $f$ is the maximum frequency of any element in the sets. A special case of is the Partial Vertex Cover () problem, where we need to pick a minimum size (or weight) subset of vertices that covers at least $k$ edges of the graph. @BshoutyL1998 describe a $2$-approximation based on rounding for . Improvements for some special classes of graphs are described in @Gandhi2004. See also [@Mestre2009; @KonemannPS2011] for more recent results on , , and related problems.
While for various geometric set systems has been studied extensively, there are relatively fewer works studying in the geometric setting. @Gandhi2004 give a PTAS for a geometric version of where ${\mathcal{S}}$ consists of *all* unit disks in the plane. They provide a dynamic program on top of the standard shifting strategy of @HochbaumM1985, thus adapting it for . Using a similar technique, @GlasserRS2008 give a PTAS for a generalization of partial geometric covering, under a certain assumption on the *density* of the given disks. @ChanN2015 give a PTAS for covering points by unit squares in the plane.
#### Our Results and Techniques {#our-results-and-techniques .unnumbered}
Suppose that we are given a instance $({X}, {\mathcal{S}}, k)$. For any set of elements ${X}_1 \subseteq {X}$, let ${\mathcal{S}}_{{X}_1} := \{S \cap {X}_1 \mid S \in {\mathcal{S}}\}$ denote the projected set system. Suppose also that for any projected instance $({X}_1, {\mathcal{S}}_{|{X}_1})$, (where ${X}_1 \subseteq {X}$) and a corresponding feasible solution $\sigma_1$, we can round $\sigma_1$ to a feasible integral solution with cost at most $\beta$ times that of $\sigma_1$. That is, we suppose that we can efficiently compute a $\beta$-approximation for the ${\textsf{SC}\xspace}$ instance $({X}_1, {\mathcal{S}}_{|{X}_1})$ by solving the natural LP relaxation and rounding it. Then, we show that we can round the natural to an integral solution to within a $2\beta + 2$ factor. By the previous discussion about existence of such rounding algorithms for for a large class of geometric objects (cf. ), we get the same guarantees for the corresponding instances as well (up to a constant factor). For clarity, we describe a sample of these applications.
1. Suppose we are given a set $P$ of $n$ points and a set ${\mathcal{T}}$ of [*fat*]{} triangles in the plane and a positive weight for each triangle in ${\mathcal{T}}$. We wish choose a subset ${\mathcal{T}}' \subseteq
{\mathcal{T}}$ of triangles that covers $P$, and minimize the weight of ${\mathcal{T}}'$, defined to be the sum of the weights of the triangles in it. This is a special case of weighted obtained by setting ${X}= P$, and adding the set $T \cap P$ to ${\mathcal{S}}$ for each triangle in $T \in {\mathcal{T}}$, with the same weight. There is an $O(\log \log^* n)$ approximation for this problem based on rounding [@EASFat; @ChanGKS12]. We obtain the same approximation guarantee for the partial covering version, where we want a minimum weight subset of ${\mathcal{T}}$ covering any $k$ of the points in $P$.
2. Suppose we are given a set ${\mathcal{R}}$ of $n$ axis-parallel rectangles and a set $P$ of points in ${\mathbb{R}}^3$, and we wish to find a minimum cardinality subset of $P$ that hits (or pierces) ${\mathcal{R}}$. This a special case of obtained by setting ${X}= {\mathcal{R}}$, and adding the set $\{R \in {\mathcal{R}}\ | \ p \in R\}$ to ${\mathcal{S}}$ for each point $p \in P$. There is an $O(\log \log n)$ approximation for this problem based on rounding [@AronovES2010]. Thus, we obtain the same approximation guarantee for the partial version, where we want a minimum cardinality subset of $P$ piercing any $k$ of the rectangles in ${\mathcal{R}}$.
3. Suppose we have a 1.5D terrain (i.e.,an $x$-monotone polygonal chain in ${\mathbb{R}}^2$), a set $P$ of points and a set $G$ of $n$ points, called guards, on the terrain along with a positive weight for each guard in $G$. The goal is to choose a subset $G' \subset G$ such that each point in $P$ is seen by some guard in $G$, and minimize the weight of $G'$. Two points $p$ and $g$ on the terrain see each other if the line segment connecting them does not contain a point below the terrain. This is a special case of obtained by setting ${X}= P$, and adding the set $\{p \in P \ | \ g \mbox{ sees } p\}$ to ${\mathcal{S}}$ for each guard $g \in G$. There is an $O(1)$ approximation guarantee for this problem based on rounding [@ElbTerrain]. Thus, we obtain an $O(1)$ approximation for the partial version, where we want a minimum weight subset of $G$ that sees any $k$ of the points in $P$.
Our algorithm for rounding a solution to the natural corresponding to partial cover instance $({X}, {\mathcal{S}}, k)$ proceeds as follows. Let ${X}_1$ be the elements that are covered by the LP solution to an extent of at least $1/2$. By scaling the solution by a factor of $2$, we get a feasible solution to the corresponding to $({X}_1, {\mathcal{S}}_{|X_1})$, which we round using the -based $\beta$-approximation algorithm. For the set ${X}\setminus {X}_1$, the solution provides a total fractional coverage of at least $k - |{X}_1|$. Crucially, each element of ${X}\setminus {X}_1$ is [*shallow*]{} in that it is covered to an extent of at most $1/2$. We use this observation to round the solution to an integer solution, of at most twice the cost, that covers at least $k - |{X}_1|$ points of ${X}\setminus {X}_1$. This rounding step and its analysis are inspired by the rounding scheme of [@BshoutyL1998], however there are certain subtleties in adapting it to the problem. To the best of our knowledge, this connection between the and was not observed before.
The rest of this article is organized as follows. In , we describe the standard formulation for the problem, and give an integrality gap example. We describe how to circumvent this integrality gap by preprocessing the input in . Finally, in , we describe and analyze the main rounding algorithm.
Preliminaries {#sec:prelim}
=============
**Formulation** We use the following Integer Programming formulation of (left). Here, for each element $e_j \in {X}$, the variable $z_j$ denotes whether it is one of the $k$ elements that are chosen by the solution. For each such chosen element $e_j$, the first constraint ensures that at least one set containing it must be chosen. The second constraint ensures that at least $k$ elements are covered by the solution. We relax the integrality \[constr:integral-z,constr:integral-x\], and formulate it as a Linear Program (right).
$$\begin{aligned}
{3}
\text{minimize} \displaystyle&\sum\limits_{S_i \in {\mathcal{S}}} w_{i}x_{i} & \nonumber\\
\text{subject to} \displaystyle&\sum\limits_{i:e_{j} \in S_{i}} x_{i} \geq z_j, \quad &e_j \in {X}\nonumber\\
\displaystyle&\sum_{e_j \in {X}}z_j \ge k, & \nonumber\\
\displaystyle &z_j \in \{0, 1\}, & e_j \in {X}\label[constr]{constr:integral-z}\\
\displaystyle &x_i \in \{0, 1\}, & S_i \in {\mathcal{S}}\label[constr]{constr:integral-x}
\end{aligned}$$ Integer Program
$$\begin{aligned}
{3}
\text{minimize} \displaystyle&\sum\limits_{S_i \in {\mathcal{S}}} w_{i}x_{i} & \label[constr]{constr:objective}\\
\text{subject to} \displaystyle&\sum\limits_{i:e_{j} \in S_{i}} x_{i} \geq z_j, \quad &e_j \in {X}\label[constr]{constr:coverage}\\
\displaystyle&\sum_{e_j \in {X}}z_j \ge k, & \label[constr]{constr:k-cover}\\
\displaystyle &z_j \in [0, 1], & e_j \in {X}\label[constr]{constr:fractional-z}\\
\displaystyle &x_i \in [0, 1], & S_i \in {\mathcal{S}}\label[constr]{constr:fractional-x}
\end{aligned}$$ Linear Program
\
Since is a special case of where $k = n$, the corresponding can be obtained by setting $k$ appropriately in . However, in this case, the can be further simplified as described earlier. We denote the cost of a solution $\sigma = (x, z)$, for the instance $({X}, {\mathcal{S}})$, as $cost(\sigma) := \sum_{S_i \in {\mathcal{S}}} w_i x_i$, and the cost of an solution is defined in exactly the same way. Also, for any collection of sets ${\mathcal{S}}' \subseteq {\mathcal{S}}$, we define $w(S') := \sum_{S_i \in {\mathcal{S}}'} w_i$. Finally, for a instance $({X}, {\mathcal{S}}, k)$, let $OPT({X}, {\mathcal{S}}, k)$ denote the cost of an optimal solution for that instance.
Unlike , the integrality gap of can be as large as $O(n)$, even for the unweighted case.
**Integrality Gap**. Consider the set system $({X}, {\mathcal{S}})$, where ${X}= \{e_1, {\ldots}, e_n\}$, and ${\mathcal{S}}= \{S_1\}$, where $S_1 = X$. Here, $k = 1$, so at least one element has to be covered. The size of the optimal solution is $1$, because the only set $S_1$ has to be chosen. However, consider the following fractional solution $\sigma = (x, z)$, where $z_j = \frac{1}{n}$ for all $e_j \in {X}$, and $x_1 = \frac{1}{n}$, which has the cost of $\frac{1}{n}$. This shows the integrality gap of $n$.
However, @Gandhi2004 show that after “guessing” the heaviest set in the optimal solution, the integrality gap of the corresponding to the residual instance is at most $f$, where $f$ is the maximum frequency of any element in the set system (this also follows from the modification to the rounding algorithm of [@BshoutyL1998], as commented earlier). In this article, we show that after guessing the heaviest set in the optimal solution, the residual instance has integrality gap at most $2\beta+2$, where $\beta$ is the integrality gap of the for some projection of the same set system.
Preprocessing {#sec:preproc}
=============
Let $({X}', {\mathcal{S}}', k')$ be the original instance. To circumvent the integrality gap, we preprocess the given instance to “guess” the heaviest set in the optimal solution, and solve the residual instance as in [@BshoutyL1998; @Gandhi2004] – see . Let us renumber the sets ${\mathcal{S}}' = \{S_1, \ldots, S_m\}$, such that $w_1 \le w_2 \le \ldots \le w_m$. For each $S_i \in {\mathcal{S}}'$, let ${\mathcal{S}}_i = \{S_1, S_2, \ldots, S_{i-1}\}$, and ${X}_i = {X}' \setminus S_i$. We find the approximate solution $\Sigma_i$ for this residual instance $({X}_i, {\mathcal{S}}_i, k_i)$ with coverage requirement $k_i = k-|S_i|$, if it is feasible (i.e. $\left|\bigcup_{S \in {\mathcal{S}}_i} S \cap X_i \right| \ge k_i$). We return $\Sigma = \arg\min_{S_i \in {\mathcal{S}}'} w(\Sigma_i \cup \{S_i\})$ over all $S_i$ such that the residual instance $({X}_i, {\mathcal{S}}_i, k_i)$ is feasible.
Sort and renumber the sets in ${\mathcal{S}}' = \{S_1, {\ldots}, S_m \}$ such that $w_1 \le \ldots \le w_m$. ${\mathcal{S}}_i \gets \{S_1, \ldots, S_{i-1}\}$ ${X}_i \gets {X}' \setminus S_i$ $k_i \gets k' - |S_i|$ $\Sigma_i \gets $ approximate solution to $({X}_i, {\mathcal{S}}_i, k_i)$ $\Sigma_i \gets \perp$ $\arg\min_{S_i \in {\mathcal{S}}': \Sigma_i \neq \perp} w(\Sigma_i \cup \{S_i\})$
Let $\Sigma^*$ be the optimal partial cover for the instance $({X}', {\mathcal{S}}', k')$, and let $S_p$ be the heaviest set in $\Sigma^*$. Let $\Sigma_p$ be the approximate solution to $({X}_p, {\mathcal{S}}_p, k_p)$ returned by the Rounding Algorithm of , and $\Sigma'$ be the solution returned by . Then,
1. $OPT({X}', {\mathcal{S}}', k') = OPT({X}_p, {\mathcal{S}}_p, k_p) + w_p$
2. $w(\Sigma') \le w(\Sigma_p \cup \{S_p\}) \le (2\beta + 2) \cdot OPT({X}', {\mathcal{S}}', k')$
Since the optimal solution $\Sigma^*$ contains $S_p$, $\Sigma^*_p := \Sigma^* \setminus \{S_p\}$ covers at least $k' - |S_p| = k_p$ elements from ${X}' \setminus S_p$. Therefore, $\Sigma^*_p$ is feasible for $({X}_p, {\mathcal{S}}_p, k_p)$. If we show that $w(\Sigma^*_p) = OPT({X}_p, {\mathcal{S}}_p, k_p)$, the first part follows. Assume for contradiction that there is a set $\Sigma'_p \subseteq {\mathcal{S}}_p$ such that $w(\Sigma'_p) = OPT({X}_p, {\mathcal{S}}_p, k_p) < w(\Sigma^*_p)$. However, $\Sigma'_p$ covers at least $k_p = k' - |S_p|$ elements from ${X}' \setminus S_p$. So $\Sigma'_p \cup \{S_p\}$ covers at least $k'$ elements from ${X}'$, and has weight $w(\Sigma_p') + w_p < w(\Sigma^*_p) + w_p = w(\Sigma^*)$, which is a contradiction.
From , we have an approximate solution $\Sigma_p$ to the instance $({X}_p, {\mathcal{S}}_p, k_p)$ such that $w(\Sigma_p) \le (2\beta + 2) \cdot OPT({X}_p, {\mathcal{S}}_p, k_p) + B$, where $B = w_p$ is the weight of the heaviest set in the optimal solution. Now returns a solution whose cost is at most $w(\Sigma_p \cup \{S_p\}) \le (2\beta + 2) \cdot OPT({X}_p, {\mathcal{S}}_p, k_p) + w_p + w_p \le (2\beta+2) \cdot (OPT({X}_p, {\mathcal{S}}_p, k_p) + w_p) \le (2 \beta+2) \cdot OPT({X}', {\mathcal{S}}', k').$ We use the result from part 1 in the final inequality.
We summarize our main result in the following theorem, which follows easily from .
Let $({X}', {\mathcal{S}}')$ be a set system, such that we can round a feasible for any projected set system $({X}_1, {\mathcal{S}}'_{|{X}_1})$ to within $\beta$ factor, where ${X}_1 \subseteq {X}'$. Then, we can find a $(2\beta+2)$-factor approximation for the partial set cover instance $({X}', {\mathcal{S}}', k')$, where $1 \le k' \le n$.
Rounding Algorithm {#sec:rounding}
==================
Let $({X}', {\mathcal{S}}', k')$ be the original instance. To circumvent the integrality gap, we preprocess the given instance to “guess” the heaviest set in the optimal solution as follows. Let us renumber the sets ${\mathcal{S}}' = \{S_1, \ldots, S_m\}$, such that $w_1 \le w_2 \le \ldots \le w_m$. For each $S_i \in {\mathcal{S}}'$, let ${\mathcal{S}}_i = \{S_1, S_2, \ldots, S_{i-1}\}$, and ${X}_i = {X}' \setminus S_i$. We find the approximate solution $\Sigma_i$ for this residual instance $({X}_i, {\mathcal{S}}_i, k_i)$ with coverage requirement $k_i = k-|S_i|$, if it is feasible. We return $\Sigma = \arg\min_{S_i \in {\mathcal{S}}'} w(\Sigma_i \cup S_i)$ over all $S_i$ such that the residual instance $({X}_i, {\mathcal{S}}_i, k_i)$ is feasible. It is well-known ([[ cite Gandhi et al]{}]{}) that if we have an algorithm that returns a solution $\Sigma_i$ whose cost is $c_1$ times that of the optimal partial cover cost $OPT({X}_i, {\mathcal{S}}_i, k_i)$ for the residual instance, then the cost of $\Sigma$ is at most $c_1$ times that of the cost of the optimal solution to the original instance $OPT({X}', {\mathcal{S}}', k')$. We state it in the following theorem.
\[thm:guess-residual\] For any $S_i \in {\mathcal{S}}'$, let $\Sigma_i \subseteq {\mathcal{S}}_i$ be a solution to $({X}_i, {\mathcal{S}}_i, k_i)$ such that $w(\Sigma_i) \le c_1 \cdot OPT({X}_i, {\mathcal{S}}_i, k_i)$. Then $w(\Sigma) \le c_1 \cdot OPT({X}', {\mathcal{S}}', k')$, where $\Sigma$ and $({X}_i, {\mathcal{S}}_i, k_i)$ are as described above.
Suppose that we have guessed the maximum weight set $S_p \in {\mathcal{S}}'$ in the optimal solution for the original instance $({X}', {\mathcal{S}}', k')$, as described in the previous section. Thus, we now have the residual instance $({X}_p, {\mathcal{S}}_p, k_p)$, where ${X}_p = ({X}' \setminus S_p), {\mathcal{S}}_p = \{S_1, S_2, {\ldots}, S_{p-1}\}$, and $k_p = k' - |S_p|$. We solve the corresponding to the instance $({X}_p, {\mathcal{S}}_p, k_p)$ to obtain an optimal solution $\sigma^* = (x, z)$. In the following, we describe a polynomial time algorithm to round on this instance.
Let $0 < \alpha \le 1/2$ be a parameter (finally we will set $\alpha = 1/2$). Let ${X}_1 = \{e_j \in {X}_p \mid \sum_{i: e_j \in S_i} x_i \ge \alpha\}$ be the set of elements that are covered to an extent of at least $\alpha$ by the solution.
We create an instance $\sigma_1$ of a feasible set cover for the instance $({X}_1, {{\mathcal{S}}_p}_{|{X}_1})$ as follows. For all sets $S_i \in {\mathcal{S}}_p$, we set $x'_i = \min\{\frac{x_i}{\alpha}, 1\}$. Note that cost of this fractional solution is at most $\frac{1}{\alpha}$ times that of $\sigma^*$. Also, note that $\sigma_1$ is feasible for the because for any element $e_j \in {X}_1$, we have that $$\sum_{i: e_j \in S_i} x'_i = \sum_{i: e_j \in S_i} \min\left\{1, \frac{x_i}{\alpha}\right\} \ge \min\bigg\{ 1, \frac{1}{\alpha} \sum_{i:e_j \in S_i} x_i\bigg\} \ge 1$$ Suppose that there exists an efficient rounding procedure to round a feasible solution $\sigma_1$, for the instance $({X}_1, {{\mathcal{S}}_p}_{|{X}_1})$ to a solution with weight at most $\beta \cdot cost(\sigma_1)$. In the remainder of this section, we describe an algorithm () for rounding $\sigma^* = (x, z)$ into a solution that (1) covers at least $k_p - |{X}_1|$ elements from ${X}_p \setminus {X}_1$, and (2) has cost at most $\frac{1}{\alpha} \cdot cost(\sigma^*) + B$, where $B$ is the weight of the heaviest set in ${\mathcal{S}}_p$. Combining the two solutions thus acquired, we get the following theorem.
\[thm:rounding-theorem\] There exists a rounding algorithm to round a partial cover corresponding to $({X}_p, {\mathcal{S}}_p, k_p)$, which returns a solution $\Sigma_p$ such that $w(\Sigma_p) \le (2\beta+2) \cdot OPT({X}_p, {\mathcal{S}}_p, k_p) + B$, where $B$ is the weight of the heaviest set in ${\mathcal{S}}_p$.
Let $\Sigma_p = \Sigma_{p1} \cup \Sigma_{p2}$, where $\Sigma_{p1}$ is the solution obtained by rounding $\sigma_1$, and $\Sigma_{p2} = \Sigma \cup {\mathcal{S}}_e$ is the solution returned by . By assumption, $\Sigma_{p1}$ covers ${X}_1$, and $\Sigma_{p2}$ covers at least $k_p - |{X}_1|$ elements from ${X}_p \setminus {X}_1$ by . Therefore, $\Sigma_p$ covers at least $k_p$ elements from ${X}_p$.
By assumption, we have that $w(\Sigma_{p1}) \le \beta \cdot cost(\sigma_1) \le \frac{\beta}{\alpha} cost(\sigma^*)$. Also, from , we have that $w(\Sigma_{p2}) \le \frac{1}{\alpha } cost(\sigma^*) + B$. We get the claimed result by combining previous two inequalities, setting $\alpha = 1/2$, and noting that $cost(\sigma^*) \le OPT({X}_p, {\mathcal{S}}_p, k_p)$.
\[lem:set-cover-rounding\] Let $\mathcal{F} = ({X}_1, {\mathcal{S}}_{{X}_1})$, $\sigma_1$ be the set cover instance, and the corresponding created as above. If ${\mathcal{S}}_{{X}_1}$ is known to be induced by a specific class of objects, then there exists an algorithm to round the to obtain a collection $\Sigma_{{X}_1} \subseteq {\mathcal{S}}_{{X}_1}$ such that $w(\Sigma_{{X}_1}) \le \beta \cdot cost(\sigma_1) \le \frac{\beta}{\alpha} \cdot cost(\sigma^*)$, where $\beta = \beta(\mathcal{F}, n)$ is a function depends on the set system.
Let ${\mathcal{S}}_1 = \{S_i \in {\mathcal{S}}_p \mid x_i \ge \alpha\}$ be the sets that are opened to more than $\alpha$. Note that without loss of generality, we can assume that $\cup_{S_i \in {\mathcal{S}}_1} S_i \subseteq {X}_1$. If $|{X}_1| \ge k_p$, we are done. Otherwise, let ${X}\gets {X}_p \setminus {X}_1$, ${\mathcal{S}}\gets {\mathcal{S}}_p \setminus {\mathcal{S}}_1$, and $k \gets k_p - |{X}_1|$. Let $\sigma = (x, z)$ be the solution $\sigma^*$ restricted to the instance $({X}, {\mathcal{S}}, k)$, that is, $x = (x_i \mid S_i \in {\mathcal{S}}), z = (z_j \mid e_j \in {X})$. We show how to round $\sigma$ on the instance $({X}, {\mathcal{S}}, k)$ to find a collection of sets that covers at least $k$ elements from ${X}$. In the following lemma, we show that the solution $\sigma$ is feasible for the instance $({X}, {\mathcal{S}}, k)$.
The solution $\sigma = (x, z)$ is feasible for the instance $({X}, {\mathcal{S}}, k)$. Furthermore, $cost(\sigma) \le cost(\sigma^*)$.
Note that $x_i$ and $z_j$ values are unchanged from the optimal solution $\sigma^*$, therefore the \[constr:fractional-x,constr:fractional-z\] are satisfied.
Note that by definition, for any element $e_j \in {X}$, $e_j \not\in \cup_{S_{i'} \in {\mathcal{S}}_1} S_{i'}$, and $e_j \not\in {X}_1$. Therefore, by , we have that $\displaystyle \sum_{i:e_j \in S_i} x_j = \sum_{i:e_j \in S_i, S_i \in {\mathcal{S}}} x_j \ge z_j$.
As for , note that $$\begin{aligned}
{1}
\sum_{e_j \in {X}_p} z_j &\ge k_p \tag{By feasibility of optimal solution $\sigma^*$}
\\\implies \sum_{e_j \in {X}} z_j &\ge k_p - \sum_{e_j \in {X}_1} z_j \tag{${X}= {X}_p \setminus {X}_1$}
\\\implies \sum_{e_j \in {X}} z_j &\ge k_p - |{X}_1| \tag{$z_j \le 1$ for $e_j \in {X}_1$ by feasibility}
\\\implies \sum_{e_j \in {X}} z_j &\ge k \tag{$k = k_p - |{X}_1|$}
\end{aligned}$$
Finally, note that $cost(\sigma) = \sum_{S_i \in {\mathcal{S}}} w_i x_i \le \sum_{S_i \in {\mathcal{S}}_p} w_i x_i = cost(\sigma^*)$, because ${\mathcal{S}}\subseteq {\mathcal{S}}_p$, and the $x_i$ values are unchanged.
Algorithm for Rounding Shallow Elements
---------------------------------------
Now we describe . Recall that we now have a residual instance $({X}, {\mathcal{S}}, k)$, and a feasible solution $\sigma = (x, z)$, such that i) For all sets $S_i \in {\mathcal{S}}$, $x_i \le \alpha$, and ii) For all elements $e_j \in {X}, \sum_{i: e_j \in S_i} x_i \le \alpha$. The goal of the algorithm is to find a solution from ${\mathcal{S}}$, such that it covers at least $k$ elements, and its cost is not too large as compared to the cost of the solution.
To this end, we pair up sets in each iteration, and try to increase the $x_i$ value of one of the sets at the expense of other. We maintain the feasibility and cost of the solution in this process. This is carefully ensured in the procedure .
Initially, we start with copies of the elements and the sets – ${X}_c$ and ${\mathcal{S}}_c$ respectively. We remove a set $S_i$ from ${\mathcal{S}}_c$ if its $x_i$ value becomes $0$ or $\alpha$. In the latter case when an $x_i$ value reaches $\alpha$, we remove all the elements it contains from ${X}_c$. The sets which are removed from ${\mathcal{S}}_c$ by virtue of their $x_i$ value becoming $\alpha$ form the output of the rounding algorithm.
In each iteration, we select a set $S_a \in {\mathcal{S}}_c$ arbitrarily, and pair it up with another arbitrary set $S_b \in {\mathcal{S}}_c$, and pair it up with $S_a$. Depending on the “cost-effectiveness” of the sets $S_a$ and $S_b$ – $\frac{|{X}_c \cap S_a|}{w_a}$ and $\frac{|{X}_c \cap S_b|}{w_b}$ respectively – we call ($S_a, S_b, w, \sigma, {{X}_c}, {{\mathcal{S}}_c}$), or ($S_b, S_a, w, \sigma, {{X}_c}, {{\mathcal{S}}_c}$). In the former case, $x_a$ is increased at the expense of $x_b$, and in the latter case $x_b$ is increased at the expense of $x_a$.
Notice that if we paired up sets $S_a$ and $S_b$ arbitrarily and rounded using , the feasibility of the may not be maintained. In particular, we cannot ensure that for all elements $e_j \in {X}_c$, $z_j \le 1$. To this end, once we choose a set $S_a \in {\mathcal{S}}_c$ arbitrarily, we fix it to be one of the paired sets until it is removed from ${\mathcal{S}}_c$ (because of its $x_a$ value becoming $0$ or $\alpha$), or it is the only remaining set in ${\mathcal{S}}_c$. By doing this, we maintain the following two invariants:
1. Let ${X}_o = \{e_j \in {X}_c \mid z_j \ge \alpha \}$. During the execution of while loop of , there exists a set $S_a \in {\mathcal{S}}_c$ such that ${X}_o \subseteq S_c$.
2. For all sets $S_i \in {\mathcal{S}}_c \setminus \{S_a\}$, the $x_i$ values are unchanged, unless and until the set $S_i$ is paired up with $S_a$.
In , we show that these invariants imply that the constraint is maintained.
The invariants are trivially true before the start of the while loop.
We have an solution $\sigma$ for the instance $({X}, {\mathcal{S}}, k)$. Note that for any $S_i \in {\mathcal{S}}$, $x_i < \alpha$, and for any $e_j \in {X}, \alpha > \sum_{i: e_j \in S_i} x_i \ge z_j$, i.e. each element is *shallow*. We now describe , which rounds $\sigma$ to an integral solution to the instance $({X}, {\mathcal{S}}, k)$. At the beginning of , we initialize ${\mathcal{S}}_c$, the collection of “unresolved” sets, to be ${\mathcal{S}}$; and ${X}_c$, the set of “uncovered” elements, to be ${X}$.
At the heart of the rounding algorithm is the procedure , which takes input two sets $S_1, S_2 \in {\mathcal{S}}_c$, and rounds the corresponding variables $x_1, x_2$ such that either $x_1$ is increased to $\alpha$, or $x_2$ is decreased to $0$ (cf. part 3). A set is removed from ${\mathcal{S}}_c$ if either of these conditions is met. In addition, if $x_i$ reaches $\alpha$, then the set $S_i$ is added to $\Sigma$, which is a part of the output, and all the elements in $S_i$ are added to the set $\Xi$. At a high level, the goal of is to resolve all of the sets in either way, while maintaining the cost and the feasibility of the .
$\Sigma \gets \emptyset$, $\Xi \gets \emptyset$ ${{X}_c}\gets {X}$, ${{\mathcal{S}}_c}\gets {\mathcal{S}}$ \[lin:outer-whileloop\] $S_a \gets$ an arbitrary set from ${{\mathcal{S}}_c}$. \[lin:Sa-chosen\] \[lin:inner-whileloop\] $S_b \gets$ an arbitrary set from ${{\mathcal{S}}_c}\setminus \{S_a\}$. \[lin:Sb-chosen\]
\[lin:sets-compared\] $(x_a, x_b, z) \gets $ ${{\mathcal{S}}_c}\gets {{\mathcal{S}}_c}\setminus \{S_b\}$ $\Xi \gets \Xi \cup S_a$, ${{X}_c}\gets {{X}_c}\setminus S_a$. $\Sigma \gets \Sigma \cup \{S_a\}, {{\mathcal{S}}_c}\gets {{\mathcal{S}}_c}\setminus \{S_a\}$ $(x_b, x_a, z) \gets $ ${{\mathcal{S}}_c}\gets {{\mathcal{S}}_c}\setminus \{S_a\}$ $S_a \gets S_b$ \[lin:Sa-chosen-2\] $\Xi \gets \Xi \cup S_b$, ${{X}_c}\gets {{X}_c}\setminus S_b$. $\Sigma \gets \Sigma \cup \{S_b\}, {{\mathcal{S}}_c}\gets {{\mathcal{S}}_c}\setminus \{S_b\}$ \[lin:end-outerwhile\] ${\mathcal{S}}_e \gets {\mathcal{S}}_c$ \[lin:last-set\] $\Sigma \cup {\mathcal{S}}_e$
------------------------------------------------------------------------
\[fn:round2sets\] $\delta \gets \min\{\alpha - x_1, \frac{w_2}{w_1} \cdot x_2\}$ $x_1 \gets x_1 + \delta$ $x_2 \gets x_2 - \frac{w_1}{w_2} \cdot \delta$ For all elements $e_j \in {{X}_c}$, update $z_j \gets \sum_{i: e_j \in S_i} x_i$ $(x_1, x_2, z)$
Given the procedure , we choose the pairs of sets to be rounded carefully. In , we pick a set $S_a \in {\mathcal{S}}_c$ arbitrarily, and then we pair it up with another set $S_b \in {\mathcal{S}}_c$ chosen arbitrarily in . To ensure that the constraint is maintained, we carefully determine whether to increase $x_a$ and decrease $x_b$ in , or vice versa. Thinking of $\frac{|{X}_c \cap S_a|}{w_a}$, and $\frac{|{X}_c \cap S_b|}{w_b}$ as the “cost-effectiveness” of the sets $S_a$ and $S_b$ respectively, we increase $x_a$ at the expense of $x_b$, if $S_a$ is more cost-effective than $S_b$ or vice versa.
Notice that if we paired up sets $S_a$ and $S_b$ arbitrarily and rounded using , the feasibility of the may not be maintained. In particular, we cannot ensure that for all elements $e_j \in {X}_c$, $z_j \le 1$. To this end, we maintain the following two invariants:
1. Let ${X}_o = \{e_j \in {X}_c \mid z_j \ge \alpha \}$. During the execution of while loop of , the elements of ${X}_o$ are contained in the set $S_a \in {\mathcal{S}}_c$, that is chosen in or .
2. Fix any set $S_i \in {\mathcal{S}}_c \setminus \{S_a\}$. The $x_i$ value is unchanged since the beginning of the algorithm until the beginning of the current iteration of while loop of ; the $x_i$ value can change in the current iteration only if $S_i$ is paired up with $S_a$.
In , we show that these invariants imply that is maintained.
The invariants are trivially true before the start of the while loop. Let $S_a \in {{\mathcal{S}}_c}$ be a set chosen in , or . During the while loop, we maintain the invariants by pairing up the $S_a$ with other arbitrary sets $S_b$, until $S_a$ is removed from ${\mathcal{S}}_c$ in one of the two ways; or until it is the last set remaining. It is easy to see that is maintained – we argue about in the subsequent paragraphs. From the second invariant, we have that if the change in the $z_j$ value for any element $e_j \in {X}_c$ is positive, then it due to the change in the $x_a$ value corresponding to $S_a$ (recall that when the $x_i$ value of a set $S_i \in {\mathcal{S}}_c$ increases to $\alpha$, all the elements contained in it are removed from ${X}_c$).
Now we describe in detail how the first invariant is being maintained in the course of the algorithm. Consider the first case, i.e. in , we increase $x_a$ and decrease $x_b$. If after this, $x_b$ becomes $0$, then we remove $S_b$ from ${\mathcal{S}}_c$. If, on the other hand, $x_a$ increases to $\alpha$, then all the elements in ${X}_c \cap S_a$ are covered to an extent of at least $\alpha$, and so we remove $S_a$ from ${\mathcal{S}}_c$ and $S_a \cap {X}_c$ from ${X}_c$. In the first case, the set ${X}_o$ continues to be a subset of $S_a$, while in the second case, it becomes empty. Thus, is maintained automatically in both cases.
In the second case, in , $x_a$ is decreased and $x_b$ is increased. This case is a bit more complicated, because $z_j$ values of elements $e_j \in S_b$ are being increased by virtue of increase in $x_b$. Therefore, we need to explicitly maintain . If $x_b$ reaches $\alpha$, then $S_b$ is removed from ${\mathcal{S}}_c$ and all the elements covered by $S_b$ are removed from ${X}_c$ (and thus the invariant is maintained). On the other hand, if $x_a$ reaches $0$, then the net change in the $z_j$ values for the elements $e_j \in S_a \setminus S_b$ is non-positive – this follows from , as the $x_i$ values of the sets in ${\mathcal{S}}_c \setminus \{S_a, S_b\}$ are unchanged, and $x_a$ is now zero. Therefore, the set ${X}_o \cap (S_a \setminus S_b)$ becomes empty. However, ${X}_o \cap S_b$ may be non-empty because of the increase in $x_b$. Therefore, we rename $S_b$ as $S_a$, and continue pairing it up with other sets. Notice that we have maintained although the set $S_a$ has changed.
From the above discussion, we have the following result.
\[cl:z-excess-set\] Throughout the execution of the while loop of , are maintained.
Finally, if at the end of while loop of , we set ${\mathcal{S}}_e$ to be ${\mathcal{S}}_c$, and add it to our solution. Note that at this point, ${\mathcal{S}}_c$ can be empty, or it may contain one set. We show that in either case, the resulting solution $\Sigma \cup {\mathcal{S}}_e$ covers at least $k$ elements.
Analysis
--------
In this section, we analyze the behavior of . In the following lemma, we show that in each iteration, we make progress towards rounding while maintaining the cost of the solution.
Let $\sigma = (x, z), \sigma' = (x', z')$ be the solutions just before and after the execution of $(S_1, S_2, w, \sigma, {{X}_c}, {{\mathcal{S}}_c})$ for some sets $S_1, S_2 \in {{\mathcal{S}}_c}$ in some iteration of the algorithm, such that $\sigma$ is a feasible solution to the . Then,
1. $cost(\sigma) = cost(\sigma')$.
2. $\sum_{e_j \in {{X}_c}} z'_j \ge \sum_{e_j \in {{X}_c}} z_j$.
3. Either $x_1' = \alpha$ or $x_2' = 0$ (or both).
<!-- -->
1. Note that the $x_i$ variables corresponding to all the sets $S_i \notin \{S_1, S_2\}$ remain unchanged. The net change in the cost of the solution is $$w_1 \cdot (x_1' - x_1) + w_2 \cdot (x_2' - x_2) = w_1 \cdot \delta - w_2 \cdot {\left(\frac{w_1}{w_2} \cdot \delta\right)} = 0.$$
2. Let $A = S_1 \cap {{X}_c}$, and $B = S_2 \cap {{X}_c}$. $z'_j = z_j$ for all elements $e_j \not\in A \cup B$, i.e. $z_j$ values are modified only for the elements $e_j \in A \cup B$.
For $|A|$ elements $e_j \in A$, $z_j$ value is increased by $\delta$ by virtue of increase in $x_1$. Similarly, for $|B|$ elements $e_{j'} \in B$, $z_{j'}$ value is decreased by $\frac{w_1}{w_2} \cdot \delta$. However by assumption, we have that $\frac{|A|}{w_1} \ge \frac{|B|}{w_2}$. Therefore, the net change in the sum of $z_j$ values is $$|A| \cdot \delta - |B| \cdot {\left(\frac{w_1}{w_2} \cdot \delta \right)} \ge |A| \cdot \delta - {\left(\frac{|A|}{w_1} \cdot w_1\right)} \cdot \delta \ge 0.$$
3. The value of $\delta$ is chosen such that $\delta = \min \{ \alpha - x_1, \frac{w_2}{w_1} \cdot x_2\}$. If $\delta = \alpha - x_1 \le \frac{w_2}{w_1} \cdot x_2$, then $x_1' = x_1 - (\alpha - x_1) = \alpha$, and $x_2' = x_2 - \frac{w_1}{w_2} \cdot (\alpha - x_1) \ge x_2 - x_2 = 0$. In the other case when $\delta = \frac{w_2}{w_1} \cdot x_2 < (\alpha - x_1)$, we have that $x_1' = x_1 + \frac{w_2}{w_1} \cdot x_2 < x_1 + (\alpha - x_1) = \alpha$, and $x_2' = x_2 - \frac{w_2}{w_1} \cdot \frac{w_1}{w_2} \cdot x_2 = 0$.
Note that (in particular Part 2 of ) alone is not sufficient to show the feasibility of the after an execution of —we also have to show that $z_j' \le 1$. This is slightly involved, and is shown in with the help of .
runs in polynomial time.
In each iteration of the inner while loop , is called on some two sets $S_1, S_2 \in {{\mathcal{S}}_c}$, and as such from , either $x_1' = \alpha$ or $x_2' = 0$. Therefore, at least one of the sets is removed from ${{\mathcal{S}}_c}$ in each iteration. Therefore, there are at most $O(|{\mathcal{S}}|)$ iterations of the inner while loop. It is easy to see that each execution of takes $O(|{\mathcal{S}}| \cdot |{X}|)$ time.
\[cl:z-excess-set\] Consider the set ${X}_o = \{e_j \in {{X}_c}\mid z_j > \alpha\}$ at any point during the execution of . There exists a set $S_a \in {\mathcal{S}}$ chosen in or , such that ${X}_o \subseteq S_a$.
Consider the first time when an arbitrary set $S_a$ is chosen in . Since $S_a$ is the first set arbitrarily chosen in , the claim is true at the start.
Assume that the claim is true at the start of an iteration of the while loop of . We show that the claim remains true at the end of the iteration. If for the set $S_b$ chosen in , we have that $\frac{|{{X}_c}\cap S_a|}{w_a} \ge \frac{|{{X}_c}\cap S_b|}{w_b}$, then in the execution of , $x_a$ is increased and $x_b$ is decreased. By , we have that either $x_a = \alpha$ or $x_b = 0$. In the first case, $S_a$ is removed from ${{\mathcal{S}}_c}$ at the end of iteration, and the set ${X}_o$ becomes empty at the end of iteration. In the second case, $z_j$ values are increased only for the elements belonging to the set $S_a$. Therefore, ${X}_o \subseteq S_a$ using the inductive hypothesis, and the reason from the previous sentence.
Consider the second case, i.e. $\frac{|{{X}_c}\cap S_a|}{w_a} < \frac{|{{X}_c}\cap S_b|}{w_b}$, and in the execution of , $x_a$ is decreased and $x_b$ is increased. Again, we have that either $x_a = 0$ or $x_b = \alpha$. In the second case, the elements from $S_b \cap {X}_o \subseteq S_a$ are removed from ${{X}_c}$, and hence from ${X}_o$. For the remaining elements $e_j \in {X}_o \setminus S_b$, the $z_j$ value is decreased, and the claim is true. In the second case, $x_a = 0$, and so $S_a$ is removed from ${\mathcal{S}}$. Using the inductive hypothesis, the increase in the $z_j$ values for any element $e_j \in {X}_o$ is caused by the increase in the $x_a$ value (by the cases of previous paragraph). However, this increase is at most $\alpha$, because of the condition of the while loop. Therefore, all the elements in $e_j \in {X}_o \setminus S_a$ are removed by the removal of $S_a$ from ${{\mathcal{S}}_c}$ – i.e. after the removal of $S_a$ from ${\mathcal{S}}$, we have that ${X}_o \subseteq S_b$. However, in this case $S_b$ is renamed to be $S_a$ in , and again the statement is true.
In the following Lemma, we show that is being maintained by the algorithm. This, when combined with , shows that we maintain the feasibility of the at all times.
During the execution of , for any element $e_j \in {{X}_c}$, we have that $z_j \le 2 \alpha$. By the choice of range of $\alpha$, the feasibility of the is maintained.
At the beginning of the algorithm, we have that $z_j \le \alpha$ for all elements $e_j \in {X}_c = {X}$. Now at any point in the while loop, consider the set ${X}_o = \{e_j \in {{X}_c}\mid z_j > \alpha\}$ as defined earlier. For any element $e_j \in {{X}_c}\setminus {X}_o$, the condition is already met, therefore we need to argue only for the elements in ${X}_o$. We know by that there exists a set $S_a \in {\mathcal{S}}_c$ such that ${X}_o \subseteq S_a$.
By , the $x_i$ values of all sets $S_i \in {\mathcal{S}}_c \setminus \{S_a\}$ are unchanged, and therefore for all elements $e_j \in {X}_c$, the net change to the $z_j$ variable is positive only by the virtue of increase in the $x_a$ value. However, the net increase in the $x_a$ value is at most $\alpha$ because a set $S_i$ is removed from ${\mathcal{S}}_c$ as soon as its $x_i$ value reaches $\alpha$. Accounting for the initial $z_j$ value which is at most $\alpha$, we conclude that $z_j \le 2 \alpha$. Consider the set ${X}_o = \{e_j \in {{X}_c}\mid z_j > \alpha\}$ as defined in . Note that for any element $e_j \in {{X}_c}\setminus {X}_o$, the condition is already met, therefore we need to argue only for the elements in ${X}_o$. However, we know that ${X}_o \subseteq S_a$ for some set $S_a \in {{\mathcal{S}}_c}$ by .
The increase in the $z_j$ value of any element $e_j \in {X}_o$ caused by the increase in $x_a$ is at most $\alpha$, and therefore, the increased $z_j$ value because of contribution of $S_a$ is upper bounded by $2\alpha$. Therefore, it suffices to show that no other sets contribute to this increase.
Consider the case when the current $S_a$ (say $S_1$) was assigned in . That is, in the previous iteration, $S_1$ was chosen to be $S_b$, and some other $S_a$ (say $S_2$) had its $x_2$ value decreased to $0$. In this case, for the elements $e_j \in {{X}_c}\cap S_2$, the overall increase in the $z_j$ value due to $S_2$ is non-positive. Therefore, if the set ${X}_o$ is not empty at the end of last iteration, then the increase in the $z_j$ values for the elements $e_j \in {X}_o$ is the result of increase in $x_1$, i.e. $x_a$.
Using the argument from the proof of \[cl:z-excess-set\], note that if for some set $S_b$, $x_b$ is increased to $\alpha$ during an execution of , then all elements $e_j \in S_b \cap {X}_o$ are removed. On the other hand, if $x_a = 0$ after the execution of , then we use the argument from the previous paragraph (where $S_1 = S_a$ and $S_2 = S_b$), and the claim holds.
Note that after the end of while loop (), we must have $|{\mathcal{S}}_c| \le 1$. That is in , we either let ${\mathcal{S}}_e \gets {\mathcal{S}}_c = \emptyset$, or ${\mathcal{S}}_e \gets {\mathcal{S}}_c = \{S_i\}$ for some set $S_i \in {\mathcal{S}}$.
To state the following claim, we introduce the following notation. Let $\sigma' = (x', z')$ be the solution at the end of . Let ${\mathcal{S}}_r = {\mathcal{S}}\setminus \Sigma$, where $\Sigma$ is the collection at the end of the while loop of , and let ${X}_r = {X}\setminus \Xi$. Note that any element $e_j \in {X}_r$ is contained only in the sets of ${\mathcal{S}}_r$. Finally, let $Z_r = \sum_{e_j \in {X}_r} z'_j$.
\[cl:last-set\] If ${\mathcal{S}}_e \neq \emptyset$, then at least $Z_r$ elements are covered by ${\mathcal{S}}_e$.
By assumption, we have that ${\mathcal{S}}_e \neq \emptyset$, i.e. ${\mathcal{S}}_e = \{S_i\}$ for some $S_i \in {\mathcal{S}}$. For each $S_l \in {\mathcal{S}}_r$ with $l \neq i$, we have that $x_l = 0$, again by the condition of the outer while loop. Since is made tight for all elements in each execution of , for any element $e_{j'} \in {X}_r$ but $e_{j'} \not\in S_i$, we have that $z_{j'} = 0$. On the other hand, for elements $e_j \in {X}_r \cap S_i$, we have that $z'_j = x'_i \le \alpha$. If the number of such elements is $p$, then we have that $Z_r \le \alpha \cdot p$. The lemma follows since choosing $S_i$ covers all of these $p$ elements, and $p \ge Z_r/\alpha \ge Z_r.$
In the following lemma, we show that produces a feasible solution.
The solution $\Sigma \cup {\mathcal{S}}_e$ returned by covers at least $k$ elements.
There are two cases – ${\mathcal{S}}_e = \emptyset$, or ${\mathcal{S}}_e = \{S_i\}$ for some $S_i \in {\mathcal{S}}$. In the first case, all elements in ${X}_r$ are uncovered, and for all such elements, $e_{j'} \in {X}_r$, we have that $z_{j'} = 0$. In this case, it is trivially true that the number of elements of ${X}_r$ covered by ${\mathcal{S}}_e$ is $Z_r$ $(= 0)$. In the second case, the same follows from . Therefore, in both cases we have that, $$\begin{aligned}
\text{Number of elements covered} &\ge |\Xi| + Z_r
\\&\ge \sum_{e_j \in \Xi} z'_j + \sum_{e_j \in {X}_c} z'_i \tag{By \Cref{lem:z-feasibility} and $z_j' \le 1$}
\\&= \sum_{e_j \in {X}} z'_j
\\&\ge \sum_{e_j \in {X}} z_j \tag{\Cref{lem:weight-maintained}, Part 2}
\\&\ge k \tag{By \Cref{lem:feasibility} and \Cref{constr:k-cover}}
\end{aligned}$$ Recall that $z_j$ refers to the $z$-value of an element $e_j$ in the optimal LP solution $\sigma$, at the beginning of the algorithm.
Let $\Sigma \cup {\mathcal{S}}_e$ be the solution returned by , and let $B$ be the weight of the heaviest set in $S$. Then,
1. $w(\Sigma) \le \frac{1}{\alpha} \sum_{S_i \in \Sigma} w_i x'_i$
2. $w({\mathcal{S}}_e) \le B$
3. $w(\Sigma \cup S_e) \le \frac{1}{\alpha} \sum_{S_i \in \Sigma} w_i x'_i + B$
For the first part, note that a set $S_i$ is added to $\Sigma$ only if $x'_i \ge \alpha$. For the second part, note that ${\mathcal{S}}_e$ contains at most one set $S_i \in {\mathcal{S}}$. By definition, weight of any set in $S_i$ is bounded by $B$, the maximum weight of any set in ${\mathcal{S}}$. The third part follows from the first and the second parts.
From and , we conclude that $\Sigma \cup {\mathcal{S}}_e$ is the solution that covers at least $k = k_p - |{X}_1|$ elements from ${X}_p \setminus {X}_1$, and whose cost is at most $\frac{1}{\alpha} \sum_{S_i \in \Sigma} w_i x'_i + B$.
Generalization of
==================
Consider the following generalization of problem, where the elements $e_j \in {X}'$ have profits $p_j \ge 0$ associated with them. Now the goal is to choose a minimum-weight collection $\Sigma \subseteq {\mathcal{S}}'$ such that the total profit of elements covered by the sets of $\Sigma$ is at least $K$, where $0 \le K \le \sum_{e_j \in {X}} p_j$ is provided as an input. Note that setting $p_j = 1$ for all elements we get the original problem. This generalization has been considered in [@KonemannPS2011].
It is easy to modify our algorithm that for , such that it returns a $2\beta + 2$ approximate solution for this generalization as well. We briefly describe the modifications required. Firstly, we modify of to incorporate the profits as follows: $$\sum_{e_j \in {X}} z_j \cdot p_j \ge K$$ The preprocessing and the rounding algorithms work with the straightforward modifications required to handle the profits. One significant change is in the rounding algorithm (). We compare the “cost-effectiveness” of the two sets $S_a, S_b$ in for the as $\frac{|S_a \cap {X}_c|}{w_a} \ge \frac{|S_b \cap {X}_c|}{w_b}$. For handling the profits of the elements, we replace this with the following condition – $\frac{P_a}{w_a} \ge \frac{P_b}{w_b}$, where $P_a := \sum_{e_j \in S_a \cap {X}_c} p_j$, and $P_b := \sum_{e_j \in S_b \cap {X}_c} p_j$. With similar straightforward modifications, the analysis of goes through with the same guarantee on the cost of the solution. We remark here that despite the profits, the approximation ratio only depends on that of the standard , which is oblivious to the profits.
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{
"pile_set_name": "ArXiv"
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---
abstract: 'We observed the HH24-26 region in the L1630 Orion molecular cloud complex with the X-ray observatory ASCA in the 0.5$-$10 keV band. X-ray emission was detected from the T Tauri star SSV61 and from the region where the Class I protostars SSV63E and SSV63W are located (hereafter SSV63E+W). The spectra of both SSV63E+W and SSV61 are well explained by an optically thin thermal plasma model. The spectrum of the T Tauri star SSV61 has a low temperature of $kT=0.9$ (0.7$-$1.2) keV and a moderate absorption of $N_{\rm{H}}=1.3$ (0.9$-$1.7) $\times10^{22}$ cm$^{-2}$, while that of the protostar SSV63E+W has a high temperature of $kT=5.0$ (3.3$-$7.9) keV and a heavy absorption of $N_{\rm{H}}=1.5$ (1.2$-$1.8) $\times10^{23}$ cm$^{-2}$. The X-ray light curve of SSV63E+W showed a flare during the observation. The peak flux reached about 9 times that of the quiescent flux. The temperature and the absorption column density do not change conspicuously during the flare. The 0.5$-$10 keV luminosity of SSV63E+W was about $1\times10^{32}$ erg s$^{-1}$ in the quiescent state. The present detection of hard X-rays from SSV63E+W is remarkable, because this is the first X-ray detection of a protostar in Orion.'
author:
- 'H. Ozawa, F. Nagase, Y. Ueda, T. Dotani, M. Ishida'
title: |
Detection of hard X-rays from a Class I protostar\
in the HH24-26 region in the Orion Molecular Cloud
---
Introduction
============
Young stellar objects (YSOs) are divided into 4 classes by the spectral-energy distribution at infrared to millimeter wave lengths (André & Montmerle 1994): Class 0 and Class I protostars, Class II sources (classical T Tauri stars, CTTS) and Class III sources (weak-line T Tauri stars, WTTS). The Class 0 protostars have massive envelopes which collapse towards the central region. Their age is estimated to be $\sim$10$^4$ yr. The class I protostars have circumsteller disks and dilute dust envelopes. Their age is estimated to be $\sim$10$^5$ yr. The Einstein and ROSAT observatories detected soft X-ray emission from both CTTS and WTTS, whose luminosities range from 10$^{28}$ erg s$^{-1}$ to 10$^{30}$ erg s$^{-1}$ (Montmerle et al. 1993, Neuhäuser 1997). Recently, ROSAT and ASCA discovered X-ray emission from Class I protostars (Casanova et al. 1995, Koyama et al. 1996, Grosso et al. 1997, Kamata et al. 1997). Since X-ray emission has been detected from only a dozen protostars (Carkner et al. 1998), it is important to increase the sample size of X-ray emitting protostars to investigate the origin of their X-ray emission.
The HH24-26 region is located in the L1630 Orion molecular cloud complex at a distance of about 460 pc (Chini et al. 1993). This region is an active star-forming region containing several YSOs, Class 0 sources (HH24MMS, HH25MMS), Class I sources (SSV63, SSV59) and a Class II source (SSV61) (Bontemps et al. 1995). SSV63 and SSV61 were detected in the 2.2 $\mu$ mapping survey (Strom et al. 1976) and later SSV63 was resolved into four near-infrared sources, SSV63E, SSV63W, SSV63NE-1, and SSV63NE-2 (Moneti & Reipurth 1995). Both SSV63E and SSV63W show flat or rising spectra in the near- and mid-infrared band (Zealy et al. 1992), and hence belong to Class I. SSV63NE-1 and SSV63NE-2 are unclassified diffuse sources, and may be reflection nebulae (Moneti & Reipurth 1995). A complex of Herbig-Haro jets (HH24) was detected around SSV63 by optical observations and some of the jets are probably associated with SSV63 itself (Jones et al. 1987, Mundt et al. 1991). In this paper, we report the first detection of X-ray emission from the infrared protostar SSV63.
Observations
============
We observed the HH24-26 region with the ASCA observatory (Tanaka et al. 1994) for a net exposure time of about 30 ks on 1998 October 2. The center coordinates of the pointing position were R.A.(J2000)$=5^{\rm{h}} 46^{\rm{m}} 27.6^{\rm{s}}$ and Decl.(J2000)$=-0^{\circ} 7' 58.7''$. ASCA has four identical X-ray telescopes (XRT), which achieve large effective area at energies up to 10 keV (Serlemitsos et al. 1995). Two Solid state Imaging Spectrometers (SIS0, SIS1) and two Gas Imaging Spectrometers (GIS2, GIS3) are located at each focus of an XRT. Each SIS has four CCD chips, which have an 11’ $\times$ 11’ field of view each. These instruments cover the energy range from 0.4 keV to 10 keV (Burke et al. 1991) with an energy resolution of $\sim$160 eV (FWHM) at 6 keV and $\sim$100 eV (FWHM) at 1.5 keV at the time of the present observation. The GIS has a field of view of 40’ diameter and covers the energy range from 0.7 keV to 10 keV with an energy resolution of 0.5 keV (FWHM) at 5.9 keV (Ohashi et al. 1996). In this observation, the SIS was operated in the 2-CCD mode, while the GIS was in the normal pulse-height mode with nominal bit assignments.
Results
=======
X-ray image of HH24-26 region
-----------------------------
Figure \[image\] shows the SIS soft band (0.7$-$2.0 keV) and hard band (2.0$-$8.0 keV) images of the HH24-26 region averaged over the whole observation. In the figure, the SIS0 and SIS1 images are superimposed and smoothed by a gaussian function with $\sigma$ of 0.2’. The symbols with numbers from 1 to 8 in Figure \[image\] correspond to the sources cited in the caption. Clearly, we detected two X-ray sources. The southern source is conspicuous only in the soft band (0.7$-$2.0 keV) image, whereas the northern source is predominant in the hard band (2.0$-$8.0 keV) image. Both detections are consistent with point sources within the angular resolution of the XRT (half power diameter about 3’). In order to derive accurate positions and count rates of the detected sources, we performed 2-dimensional fitting to the SIS image using the point spread function of the XRT (Ueda 1996). The positions determined with ASCA are ($5^{\rm{h}}46^{\rm{m}}7.0^{\rm{s}}$, $-0^{\circ}10'7''$) and ($5^{\rm{h}}46^{\rm{m}}7.3^{\rm{s}}$, $-0^{\circ}12'4''$), with 1 $\sigma$ error of 14”. Based on the positional coincidence, the southern source is identified with the T Tauri star SSV61. The northern source is identified with the protostars SSV63E/SSV63W, but the ASCA image resolution does not allow us to resolve X-ray emission of the two (hereafter we designate the emission as from SSV63E+W). The count rates in the 0.7$-$7.0 keV band of SSV61 and SSV63E+W are 1.4 (1.3$-$1.6) $\times 10^{-2}$ counts s$^{-1}$ (the errors in parentheses indicate a 90% confidence region) and 8.2 (7.0$-$9.4) $\times 10^{-3}$ counts s$^{-1}$ , respectively. We were not able to detect any evidence of X-ray emission from the Class I protostar SSV59 or the Class 0 protostars, HH24MMS and HH25MMS. We derived upper limits (2 $\sigma$) of count rates of these protostars, 2 $\times 10^{-4}$ counts s$^{-1}$, 8 $\times 10^{-4}$ counts s$^{-1}$, and 5 $\times 10^{-4}$ counts s$^{-1}$, respectively for SSV59, HH24MMS, and HH25MMS.
Light curve
-----------
Figure 2 shows background-subtracted light curves in the hard (3.0$-$8.0 keV) and the soft (0.7$-$2 keV) X-ray bands. In the figure, all the data from the four sensors (SIS0, SIS1, GIS2, GIS3) are combined to obtain the best statistics. The data were taken from regions of 6’ radius for the GIS and 4’ radius for the SIS, centered at the position of SSV63E+W. The circle shown in Figure \[image\] indicates the region where the SIS light curve was derived. We extracted the background which is constructed from the whole FOV of each sensor where point sources were excluded, correcting for the positional dependence of the background intensity (Ueda 1996). Although the region contains both SSV61 and SSV63E+W, the hard band light curve should correspond to that of SSV63E+W and the soft band light curve to that of SSV61, as expected from the image analysis in section 3.1. We discarded the data in the 2.0$-$3.0 keV band from the hard band light curve because X-rays both from SSV63E+W and SSV61 contribute comparably in the 2.0$-$3.0 keV band, as shown in Figure \[spec\] of section 3.3.
As can be seen from the top panel of Figure \[lc\], SSV63E+W showed flare-like time variability. On the other hand, the soft band light curve (bottom panel of Figure \[lc\]) showed little intensity variability, suggesting that the T Tauri star SSV61 was relatively stable during the ASCA observation. The flare from SSV63E+W is characterized by a slow rise followed by an exponential decay. We fitted the light curve with a model consisting of a linear rise and an exponential decay to a constant level of the form, $$F(t)=\cases{\frac{N}{\tau_{r}}(t-t_{p}+\tau_r)+C, & $(t < t_{p})$, \cr
N exp\{\frac{-(t-t_{p})}{\tau_{d}}\}+C, & $(t \ge t_{p}).$ }$$ This model has five free parameters, flare peak time $t_{p}$, rise time $\tau_{r}$, a constant level $C$, the normalization of the exponential function $N$, and e-folding decay time $\tau_{d}$. The solid curve in Figure 2 shows the best-fit model. From the fit, we derived a peak flux ($N+C$) of 0.18 (0.16$-$0.19) cts s$^{-1}$, $C$ = 0.02 (0.016$-$0.026) cts s$^{-1}$, $\tau_{r}$ = 1.7 (1.2$-$2.2) $\times 10^{4}$ s, and $\tau_d$ = 1.2 (0.9$-$1.5) $\times 10^{4}$ s.
Energy spectrum
---------------
The upper and lower panels in Figure \[spec\] show background-subtracted spectra taken from the interval A (hereafter termed the “flare state”) and the interval B (the “quiescent state”) indicated in Figure \[lc\], respectively. Since the separation between SSV63E+W and SSV61 ($\sim$2’) is not large enough compared with the PSF of the XRT of ASCA, it is difficult to obtain spectra from SSV63E+W and SSV61 without contamination from the other even if we extract the spectra in a small region around each source. Hence, we accumulated photons in the same region as used in section 3.2 to obtain better statistics. The background spectrum is also extracted from the same region as used in section 3.2. Since the region includes both the positions of SSV63E+W and SSV61, the spectra will contain contributions from both the sources. Both of the spectra in Figure \[spec\] show a double peak feature, suggesting the existence of two (hard and soft) components. Combined with the results from the image analysis in section 3.1, we assume that the soft and the hard components correspond to SSV61 and SSV63E+W, respectively. In fact, the soft component does not show a significant change between the two states, while the hard component is enhanced during the flare state. This confirms that the hard component in the spectrum really corresponds to the emission from the Class I protostar SSV63E+W.
In the hard component during the flare state, an emission line feature is clearly seen at 6.6 (6.5$-$6.7) keV, which is consistent with the K$\alpha$ line from He-like iron ions. This suggests that the hard component during the flare state originates from an optically thin thermal plasma. On the other hand, the origin of the soft component from SSV61 is also considered to be thin thermal coronal emission, which is typical for T Tauri stars (Montmerle et al. 1993, Neuhäuther 1997). Accordingly, we fit the combined GIS and SIS spectra with a model consisting of two thermal components with different absorption column densities, in which the high and low temperature components represent the contribution from SSV63E+W and from SSV61, respectively. This model of two thermal components was also adopted for the quiescent spectrum, utilizing the thin thermal spectral model calculated by Raymond and Smith (1977; hereafter the “RS model”). Since the statistics of the spectrum are not good enough to determine the abundance of the soft component, we fixed this at 0.5 solar for SSV61 because the metal abundances of late-type stars are roughly 0.5 solar (Carkner et al. 1996). For the hard component, we used an Auxiliary Response Function (ARF) constructed for a point source located in the center of the region selected (i.e., SSV63E+W), while for the soft component, we used an ARF constructed for a point source located at the position of SSV61.
Using this “two-component” RS model, we obtained acceptable fits for both of the flare-state and quiescent state spectra. The temperature and the absorption column density did not change significantly between the flare state and the quiescent state for either the hard or soft components. Hence, to make the tightest constraints on these parameters, we extracted spectra averaged over the whole observation, and fit them with the same model. We again obtained acceptable fits with parameters listed in Table \[tbl-2\]. We derived plasma temperatures $kT=$ 0.9 (0.7$-$1.2) and 5.0 (3.3$-$7.9) keV, time-averaged emission measures of 5.3 (3.2$-$9.0) $\times$ 10$^{54}$ cm$^{-3}$ and 14 (9.5$-$25) $\times$ 10$^{54}$ cm$^{-3}$, and absorption column densities $N_{\rm H} =$1.3 (0.9$-$1.7) $\times$ 10$^{22}$ cm$^{-2}$ and 1.5 (1.2$-$1.8) $\times$ 10$^{23}$ cm$^{-2}$, respectively for the low and high temperature components. We obtained an abundance of 0.45 (0.28$-$0.69) solar for the high temperature component, SSV63E+W. Finally, to determine the spectral parameters of SSV63E+W separately during the flare state and the quiescent state, we repeated the fit by fixing the parameters of the soft component at the best-fit values obtained from the whole observation. The results are also listed in Table \[tbl-2\].
Discussion
==========
Since X-ray emission from Class 0 protostars has not been detected yet, it is interesting whether Class 0 protostars in the HH24-26 region, HH24MMS and HH25MMS, emit X-rays or not. The present ASCA observation, however, failed to detect X-ray emission from either HH24MMS or HH25MMS. We estimate upper limits of X-ray luminosity as 9 $\times$ 10$^{30}$ erg s$^{-1}$ for HH24MMS, 7 $\times$ 10$^{30}$ erg s$^{-1}$ for HH25MMS, and 4 $\times$ 10$^{30}$ erg s$^{-1}$ for the Class I protostar SSV59, assuming a RS model with $kT=$ 5 keV, $N_{\rm{H}}=$ 1 $\times$ 10$^{23}$ cm$^{-2}$, and a metal abundance of 0.5 solar. We note that these estimates depend on the spectral parameters assumed.
The most remarkable result here is the first detection of X-ray emission from a Class I protostar SSV63E+W in Orion. The significant features of the X-ray emission are : (1) a large X-ray luminosity of about 1 $\times$ 10$^{32}$ erg s$^{-1}$ in the 0.5$-$10 keV band and a high temperature of $kT$ = 5.2 ( $>$ 2.3) keV during the quiescent state, (2) an X-ray flare with total energy release of $5 \times 10^{36}$ erg with an elevated temperature of $kT \sim 6$ keV, and (3) a large absorption column density of $N_{\rm{H}}=$ 1.1 (1.0$-$1.3) $\times$ 10$^{23}$ cm$^{-2}$.
SSV63E+W has a high temperature plasma even during the quiescent state, which is comparable to that during the flare state. Such a high temperature is not observed in TTSs during the quiescent state. A cluster of Class I protostars in the R CrA molecular cloud also showed a high temperature of $kT \sim 7$ keV during the quiescent state (Koyama et al. 1996). Thus, some of the Class I sources have high temperature plasmas of $\sim$ 6 keV. In order to know the mechanism of the quiescent X-ray emission, we compare the L$_{X}$/L$_{bol}$ ratio of SSV63E+W with those of T Tauri stars. The bolometric luminosity of SSV63E is estimated to be 22.4 L$_{\odot}$ (Berrilli et al. 1989, Zealey et al. 1992). Although that of SSV63W is highly uncertain, assuming the X-ray emission of SSV63E+W comes from only SSV63E, we obtained L$_{X}$/L$_{bol}$ ratio of 1.2 $\times$ 10$^{-3}$ in the quiescent state. Since the typical L$_{X}$/L$_{bol}$ ratio of T Tauri stars is 5 $\times$ 10$^{-4}$ and the maximum is 3 $\times$ 10$^{-3}$ (Preibisch et al. 1998), our value of the L$_{X}$/L$_{bol}$ ratio of SSV63E+W is comparable to that of T Tauri stars. Thus the X-ray emission mechanism from protostars during the quiescent state has to produce higher temperature plasmas of $\sim$ 6 keV, but not larger L$_{X}$/L$_{bol}$ ratios than those of T Tauri stars. The enhanced disk-magnetosphere interaction (Shu et al. 1997) may provide such high temperature plasmas in protostars.
The hard ($kT \sim 6$ keV) X-ray flare of SSV63E+W can probably be explained by strong magnetic activity, by analogy with the X-ray flares of TTSs. Such a hard X-ray flare was also observed from a protostar in the R Cr A molecular clouds (Koyama et al. 1996). Hayashi et al. (1996) suggested a model for hard X-ray flares in protostars, based on a closed magnetic loop connecting the central star and disk. This model can explain the high temperature of X-ray emitting protostars during the flare state. From the decay time analysis (van den Oord & Mewe 1989, Montmerle 1990) of the observed flare, we estimated the electron density $n_e$ $\sim 1 \times$ 10 $^{11}$ cm$^{-3}$, the volume of the emission region $V \approx E.M./n_e^2$ $\sim 2 \times 10^{33}$ cm$^{3}$, the typical length of the region $d = V^{\frac{1}{3}}$ $\sim 1 \times 10^{11}$ cm, and the magnetic field strength $B$ $\sim$ 200 gauss for the flare region in SSV63E+W. These parameters constrain the model of X-ray flare of the protostar.
The authors are grateful to Dr. Hugh Hudson and the referee, Dr. Thierry Montmerle for their stimulating discussions. H.O. achnowledges the award of Research Fellowships of the Japan Society for the Promotion of Science for Young Scientists.
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[ccccc]{}
$kT$ (keV) & 5.0 (3.3$-$7.9) & 6.3 (3.9$-$11.6) & 5.2 ($>$2.3) & 0.9 (0.7$-$1.2) $N_{\rm{H}}$ (10$^{22}$ cm$^{-2}$) & 15 (12$-$18) & 14 (12$-$17) & 14 (8.7$-$23) & 1.3 (0.9$-$1.7) abundance & 0.45 (0.28$-$0.69) & 0.55 (0.33$-$0.96) & 0.34 (0.0$-$2.2) & 0.5 (fixed) $E.M. $ (10$^{54}$ cm$^{-3}$) & 14 (9.5$-$25) & 21 (14$-$35) & 7.4 (3.5$-$2.7) & 5.3 (3.2$-$9.0) $L\rm{x}$ (10$^{31}$ erg s$^{-1}$) & 22 & 35 & 11 & 6.2 $\chi_\nu^2$/d.o.f. & 68.5/69 & 54.7/56 & 39.7/37 & 68.5/69
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{
"pile_set_name": "ArXiv"
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---
abstract: 'A key objective in multi-view learning is to model the information common to multiple parallel views of a class of objects/events to improve downstream learning tasks. In this context, two open research questions remain: How can we model hundreds of views per event? Can we learn robust multi-view embeddings without any knowledge of how these views are acquired? We present a neural method based on multi-view correlation to capture the information shared across a large number of views by subsampling them in a view-agnostic manner during training. To provide an upper bound on the number of views to subsample for a given embedding dimension, we analyze the error of the bootstrapped multi-view correlation objective using matrix concentration theory. Our experiments on spoken word recognition, 3D object classification and pose-invariant face recognition demonstrate the robustness of view bootstrapping to model a large number of views. Results underscore the applicability of our method for a view-agnostic learning setting.'
bibliography:
- 'references.bib'
---
Introduction {#intro}
============
Across many application domains, we often rely on data collected from multiple views of a target object/event to learn a reliable and comprehensive representation. This group of (machine) learning problems is referred to as multi-view learning. A distinguishing feature of this paradigm is that the different views of a given instance share an association or a *correspondence* that can be exploited to build more informed models of the observed event [@xu2013survey]. Much like the process by which humans learn by reconciling different views of information that may appear conflicting [@klemen2012current], data from different views contain both contrasting and complementary knowledge that can be used to offer robust learning solutions.
We define a *view* as data that is sampled from observing an object/event at different states or with different instruments to capture its various presentations. For example, a person’s face photographed at different angles or audio, language and visuals in an movie. The objective of multi-view learning is to learn vector representations (embeddings/features) that are discriminative of the underlying events by explicitly factoring in/out the shared correspondence between the many views. These embeddings can provide robust features for downstream tasks such as classification and clustering, e.g., text-to-image retrieval [@dorfer2018end] and bilingual word embeddings [@wang2015deep]. They can also be used in an unsupervised fashion to uncover the inherent structure in such data, e.g., learning common components from brain signals across individuals [@parra2018correlated].
Multi-view learning solutions have explored various ways to model the *correspondence* between multiple views to fuse the knowledge across them. They can be broadly categorized into (1) subspace alignment methods, (2) generative models and (3) fusion-based methods [@li2018survey]. The present work can be classified as subspace-alignment, which deals with learning projections between two or more views to maximize the similarity. Most existing subspace-alignment methods learn multi-view representations by estimating at least one distinct projection matrix per view, often assuming that the view information for the probing sample is available at training/testing. Considering the sheer scale of multi-view problems–amount of data and number of views–two critical questions arise: how can we model hundreds of views of an event, and can we learn the multi-view representations effectively in a view-agnostic fashion?
In this paper, we build upon the work by Somandepalli et. al., where a *multi-view correlation* objective (mv-corr) was proposed to learn shared subspaces across multiple views. Data from different views is transformed using identical neural networks (*sub-networks*) to obtain view-invariant embeddings discriminative of the underlying event. We advance this framework along two directions: First, we explore view bootstrapping during training to be able to incorporate a large number of views. We provide a theoretical analysis for the bootstrapped mv-corr objective and derive an upper bound for the number of views to subsample with respect to the embedding dimension. This result is significant because it allows us to determine the number of sub-networks to use in the mv-corr framework. Second, we conduct several experiments to benchmark the performance of view-bootstrapping for downstream learning tasks and highlight its applicability for modeling a large number of views in a view-agnostic fashion. In practice, this framework only needs to know that the sample of views considered at each training iteration have a *correspondence*. That is, the multiple views are obtained from observing the same underlying event. A natural example for this setting is audio recordings from multiple microphones distributed in a conference room. In this example, we can use the timestamps to construct a correspondence. This method can also be used for applications such as pose-invariant face recognition in a semi-supervised setting. We do not need the pose information (view-agnostic) or the total number of classes during training. All we need to know is that the different face images are of the same person.
Related Work {#background}
============
Subspace alignment for more than two views
------------------------------------------
Widely used correlation-based methods include canonical correlation analysis (CCA) [@hotelling1992relations] and its deep learning versions [@andrew2013deep; @dumpala2018sentiment] that can learn non-linear transformations of the two views to derive maximally correlated subspaces. Several metric-learning based methods were proposed to extend CCA for multiple views by learning a view-specific or view-invariant feature space by transforming data. For example, generalized CCA [@horst1961generalized; @benton2017deep] and multi-view CCA [@chaudhuri2009multi]. Their applications include audio-visual speaker clustering and phoneme classification from speech and articulatory information.
In a supervised setting, a discriminative multi-view subspace can be obtained by treating labels as an additional view. Prominent examples of this idea include generalized multi-view analysis (GMA, @sharma2012generalized), partial least squares regression based methods [@cai2013regularized] and multi-view discriminant analysis (MvDA, @kan2015multi). They were effectively used for applications such as image captioning and pose-invariant face recognition. However the generalizability of these methods to hundreds of views remains to be explored.
View-agnostic multi-view learning
---------------------------------
The subspace methods discussed thus far assume that the view information is readily available during training and testing. For instance, GMA and MvDA estimate a within-class scatter matrix specific to each view. In practice, view information may not be available for the probe data (e.g., pose of a face during testing). A promising direction to address this problem was proposed by Ding and Fu . To eliminate the need for view information of the probe sample, a low-rank subspace representation was used to bridge the view-specific and view-invariant representations. Here, a single projection matrix per view was used which would scale linearly with increasing number of views.
Domain adaptation in a multi-view paradigm
------------------------------------------
A recent survey by @ding2018robust presents a unified learning framework mapping out the similarities between multi-view learning and domain adaptation. Typical domain adaptation methods seek domain-invariant representations which are akin to view-invariant representations if we treat different domains as views. The benefit of the multi-view paradigm in this context is that the variabilities associated with multiple views can be *washed out* to obtain discriminative representations of the underlying classes.
This formulation is particularly useful in the domain of speech/audio processing for applications such as wake-word recognition [@kepuska2009novel]. Here we need to recognize a keyword (e.g., “Alexa”, “OK Google”, “Siri”) no matter who says it or where it is said (i.e., the specific background acoustic conditions). Speaker verification methods based on joint factor analysis [@dehak2009support] and total variability modeling [@dehak2011front] have explored the ideas of factoring out the speaker-dependent factors and speaker-independent factors to obtain robust speaker representations in the context of domain adaption. Recently, @somandepalli2019multiview showed that a more robust speech representation can be obtained by explicitly modeling multiple utterances of a word as corresponding views.
Views vs. Modalities {#sec:spiel}
--------------------
Following ideas proposed in the review by @ding2018robust, we delineate two kinds of allied but distinct learning problems: *multi-view* and *multi-modal*. In related work of this domain (See surveys by @zhao207multi [@ding2018robust; @li2018survey]), the two terms are used interchangeably. We however distinguish the two concepts to facilitate modeling and analysis. Multiple views of an event can be modeled as samples drawn from identically distributed random processes, e.g., a person’s face at different poses. However, the individual modalities in multi-modal data need not arise from identically distributed processes, e.g., person’s identity from their voice, speech and pose.
In this work, we focus on multi-view problems, specifically to learn embeddings that capture the shared information across the views. The premise that multiple views can be modeled as samples from identically distributed processes not only facilitates the theoretical analysis of the mv-corr objective, but also helps us to formulate domain adaptation problems in a multi-view paradigm; particularly, for applications that need to scale for hundreds of views (e.g., speaker-invariant word recognition). While it should be noted that the methods explored in this work may not be directly applied to multi-modal problems where we are generally interested to capture both modality-specific and modality-shared representations, the theory developed in this work can be extended to other methods such as GMA [@sharma2012generalized] and multi-view deep network [@kan2016multi] for the broader class of multi-modal problems.
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Proposed Approach {#methods}
=================
We first review the multi-view correlation (mv-corr) objective developed by Somandepalli et. al., . Next, we consider practical aspects for using this objective in a deep learning framework followed by view-bootstrapping. Then, we develop a theoretical analysis to understand the error of the bootstrapped mv-corr objective.
Multi-view correlation (mv-corr)
--------------------------------
Consider $N$ samples of $d$-dimensional features sampled by observing an object/event from $M$ different views. Let $\*X_l \in \mathbb{R}^{d\times N}: l=1,..., M$, be the data matrix for the $l$-th view with columns as mean-zero features. We can use the same feature dimension $d$ across all views because we assume that that the multiple views are sampled from identical distributions (See Sec. \[sec:spiel\]). We describe the mv-corr objective in the context of CCA. The premise of applying CCA-like approaches to multi-view learning is that the *inherent variability associated with a semantic class is uncorrelated across multiple views* to represent the signal shared across the views. For $M=2$, CCA finds projections of same dimensions $\*v_1\text{ and }\*v_2$ in the direction that maximizes the correlation between them. Formally, $$\begin{aligned}
\label{eqn:cca}
(\*v_1^*, \*v_2^*)
= \operatorname*{argmax}_{\*v_1,\*v_2 \in {\mathbb{R}}^{d}}\frac{\*v_1^{{\top}}\*{\Sigma}_{12}\*v_2}{\sqrt{\*v_1^{{\top}}\*\Sigma_{11}\*v_1\*v_2^{{\top}}\*\Sigma_{22}\*v_2}}\end{aligned}$$ where $\*\Sigma_{12}$ is the cross-covariance and $\*\Sigma_{11}\,,\*\Sigma_{22}$ are the covariance terms for the two views. To extend the CCA formulation for more than two views, we consider the sum of all pairwise covariance terms. That is, find a projection matrix or a *multi-view shared subspace* $\*W\in{\mathbb{R}}^{k\times d}$ that maximizes the ratio of the sum of between-view over within-view covariances in the projected space: $$\begin{aligned}
\label{eqn:mcca}
\*W^*
= \operatorname*{argmax}_{\*W}\frac{\*W^{{\top}}\big(\*X_1\*X_2^{{\top}} +\dotsc+\*X_{M-1}\*X_{M}^{{\top}}\big)\*W}{\*W^{{\top}}\big(\*X_1\*X_1^{{\top}} +\dotsc+\*X_{M}\*X_{M}^{{\top}}\big)\*W}\end{aligned}$$ We refer to the numerator and denominator covariance sums in Eq. \[eqn:mcca\] as *between-view covariance* $\*R_b$ and *within-view covariance* $\*R_w$ which are sums of $M(M-1)$ and $M$ covariance terms, respectively. Because we assume feature columns in $\*X_l$ to be mean-zero, we estimate the covariance matrices as a cross product without loss of generality. We now define a multi-view correlation $\*\Lambda$ as the normalized ratio of between- and within-view covariance matrix: $$\begin{aligned}
\label{eqn:mv-corr}
\*\Lambda = \max_{\*W}\frac{1}{M-1}\frac{\*W^{{\top}}\*R_b\*W}{\*W^{{\top}}\*R_w\*W}\end{aligned}$$ here, the common scaling factor $M(N-1)$ in the covariance estimates are omitted from the ratio.
A version of this *ratio of covariances* has been considered in several related multi-view learning methods. One of the earliest works by @hotelling1992relations presented a similar formulation for scalars, also referred to as multi-set CCA by some works (e.g., @parra2018correlated). Notice that this ratio is similar to the use of between-class and within-class scatter matrices in linear discriminant analysis (LDA, @fisher1936use) and more recently in multi-view methods such as GMA and MvDA. Another version of this ratio known as the intraclass correlation coefficient [@bartko1966intraclass] has been extensively used to quantify test-retest repeatability of clinical measures (e.g., @somandepalli2015short).
The primary difference of mv-corr formulation from these methods is that it does not consider the class information explicitly while estimating the covariance matrices. All we need to know is that the subset of $M$ views *correspond* to the same object/event. Additionally we consider the sum of covariances for all pairs of views, eliminating the need for view-specific transformation which enables us to learn the shared subspace $\*W$ in a view-agnostic manner. On the downside, we only capture the shared representation across multiple views and discard view-specific information which may be of interest for some multi-modal applications.
Implementation and practical considerations
-------------------------------------------
Using ideas similar to the deep variants of CCA [@andrew2013deep] and LDA [@dorfer2015deep], we can use deep neural networks (DNN) to learn non-linear transformations of the multi-view data to obtain (possibly) low-dimensional representations. In Eq. \[eqn:mv-corr\], the solution $\*W$ jointly diagonalizes the two covariances $\*R_b$ and $\*R_w$ because $\*W$ is their common eigenspace. Thus, we use the trace ($\Tr$) form of Eq. \[eqn:mv-corr\] to fashion a loss function, $\rho_M$ for batch optimization in DNN for data from $M$ views. $$\begin{aligned}
\label{eqn:rho-M}
\rho_{M}
= \max_{\*W}\frac{1}{d(M-1)}\frac{\Tr(\*W^{{\top}}\*R_b\*W)}{\Tr(\*W^{{\top}}\*R_w\*W)} \end{aligned}$$
The DNN framework for mv-corr consists of one network per view $l$, referred to as $l^{\text{th}}$ *sub-network* denoted by $f_l$. The architecture of the sub-network is the same for multiple views and the weights are not shared across the sub-networks for any layer. The output from the top-most layer of each sub-network is passed to a fully-connected layer of $d$ neurons. Let $\*H_l=f_l(\*X_l)\in{\mathbb{R}}^{d\times N}$ be the activations from this last layer where $N$ is now the batch size. Thus, for each batch we estimate the between- and within-view covariances $\*R_b$ and $\*R_w$ using $\*H_l\,,l=1,\dotsc,M$ to compute the loss in Eq. \[eqn:rho-M\]. The subspace $\*W$ is obtained by solving the generalized eigenvalue (GEV) problem using Cholesky decomposition.
**Total view covariance**: For a large number of views $M$, estimating $\*R_b$ in each batch is expensive as it is $\mathcal{O}(M^2)$. We instead compute a total-view covariance term $\*R_t$ which only involves estimating a single covariance for the sum of all views and is $\mathcal{O}(M)$, and then estimate $\*R_b=\*R_t-\*R_w$. See Supplementary (Suppl.) methods S1 for the proof. $$\begin{aligned}
\label{eqn:rt}
\*R_t = \*R_b + \*R_w = \frac{1}{M}\bigg(\sum_{l=1}^{M}\*X_l\bigg)\bigg(\sum_{l=1}^{M}\*X_l\bigg)^{{\top}}\end{aligned}$$ **Choosing batch size**: A sample size of $\mathcal{O}(d\log d)$ is sufficient to approximate the sample covariance matrix of a general distribution in ${\mathbb{R}}^d$ [@vershynin2010introduction]. Thus we choose a batch size of $N=\text{ceil}(d\log d)$ for a network with $d$-dimensional embeddings. In our experiments, choosing $N<d\log d$ was detrimental to model convergence.
**Regularize $\*R_w$**: Maximizing $\rho_M$ (Eq. \[eqn:rho-M\]) corresponds to maximizing the mean of eigenvalues of $\*R_w^{-1}\*R_b$. Estimating $\*R_w$ with rank deficient $\*H_l$ may lead to spuriously high $\rho$. One solution is to truncate the eigenspace $\*W$. However, this will reduce the number of directions of separability in the data. To retain the full dimensionality of the covariance matrix, we use “shrinkage" regularization [@ledoit2000well] for $\*R_w$ with a parameter $\nu=0.2$ and normalized trace parameter $\bar{\lambda}=\Tr(\*R_w)$ as $\Tilde{\*R}_w = (1-\nu)\*R_w + \nu\bar{\lambda}\*I_d/d$ **Loss function is bounded**: The objective $\rho_M$ is the average of $d$ eigenvalues obtained by solving GEV. We can analytically show that this objective is bounded above by 1 (See Suppl. methods S2). Thus, during training, we minimize the loss $1-\rho_M$ to avoid trivial solutions.
**Inference**: Maximizing $\rho$ leads to maximally correlated embeddings. Thus, during inference we only need to extract embeddings from one of the sub-networks. The proposed loss ensures that the different embeddings are maximally correlated (See Suppl. methods simulations S3).
View bootstrapping
------------------
Modeling a large number of views would require many sub-networks which is not practical for hundreds of views. To address this issue, we propose view bootstrapping. The schematic of the overall method is shown in Figure \[fig:schematic\]. Here, we construct a network with $m$ sub-networks and sample with replacement a small number of views $m\ll M$ to model data with $M$ views. During training, we do not keep track of views being sampled for specific sub-networks which ensures that the model is view-agnostic. The bootstrapped objective can be written as: $$\label{eq:stochastic}
\rho^{*} = \mathbb{E}_{m\sim\mathcal{U}(1,M)}\rho_m \approx \rho_M$$ The intuition behind our stochastic extension lies in law of large numbers applied to the covariance matrices in Eq. \[eqn:rho-M\]. Let $\*R_{\{b,w\}}$ now denote the covariances estimated from $m$ views. Asymptotically, with a large $M$ and as $m\rightarrow M$, we have ${\mathbb{E}}{\*R_b^{(m)}}\rightarrow \*\Sigma_b$ and ${\mathbb{E}}{\*R_w^{(m)}}\rightarrow\*\Sigma_w$ where $\*\Sigma_b\text{ and }\*\Sigma_w$ are the between- and within-view covariance estimated for all $M$ views. In practice, the number of available view samples is finite and the total number of views possible is often unknown. Thus, we analyze the error of the estimate $\rho_m$ with respect to $\rho^*=d^{-1}\Tr(\*W^{{\top}}\*\Sigma_b\*W)/\Tr(\*W^{{\top}}\*\Sigma_w\*W)$ in a non-asymptotic setting.
Let $\*X=[\*A^{(1)},\dotsc,\*A^{(N)}]$ be the $m\times d$ matrices of $m$ views sampled from an unknown number of views $M$. Let the rows $\*A_l$ of the view matrices $\*A$ be independent subgaussian vectors in ${\mathbb{R}}^d$ with $\norm{\*A_l}_2=1:l=1,\dotsc,m$. Then for any $t\geq0$, with probability at least $1-2\exp({-ct^2})$, we have
[\_m]{}(1, C\^[\*]{})
*Here, $\rho_m$ and $\rho^*$ are the mv-corr objectives for subsampled views $m$ and the total number of views $M$ respectively. The constant $C$ depends only on the subgaussian norm $K$ of the view space, with $K = \displaystyle\max_{i,l}\norm{\*A_l^{(i)}}_{\psi_2}$*
Here we highlight the main elements of the proof. Please see Suppl. methods, Theorem S6 for the detailed work. Recall that $\*R_b$ and $\*R_w$ now denote covariance matrices for $m$ views. Using properties of trace and spectral norm, we can rewrite the expression of the corresponding $\rho_m$ as: $$\begin{aligned}
\rho_m = \frac{\Tr(\*W^{{\top}}{\*R}_b\*W)}{\Tr(\*W^{{\top}}{\*R}_w\*W)}
= \frac{\<{\*R_b} + \*{\Sigma}_b - \*{\Sigma}_b, \*W\*W^{{\top}}\>}{\<{\*R}_w + \*{\Sigma}_w - \*{\Sigma}_w, \*W\*W^{{\top}}\>}\\
\leq \frac{\<{\*{\Sigma}}_b, \*W\*W^{{\top}}\> + \norm{{\*R}_t - \*{\Sigma}_t} + \norm{{\*R}_w - \*{\Sigma}_w}}{\<{\*{\Sigma}}_w, \*W\*W^{{\top}}\> - \norm{{\*R}_w - \*{\Sigma}_w}}
$$ where $\*{\Sigma}_b$ and $\*{\Sigma}_w$ are the previously defined between- and within-view covariances respectively for $M$ views. From Eq. \[eqn:rt\], recall the result: $\*R_b = \*R_t-\*R_w$. The rest follows through triangular inequalities. Observe that the ratio $\<{\*{\Sigma}}_B, \*W\*W^{{\top}}\>/\<{\*{\Sigma}}_W, \*W\*W^{{\top}}\>$ is the optimal $\rho^*$ estimated from the unknown number of views $M$. Also, the two trace terms are sum of normalized eigenvalues. Thus $\abs{\<{\*{\Sigma}}_b, \*W\*W^{{\top}}\>},\abs{\<{\*{\Sigma}}_W, \*W\*W^{{\top}}\>}\in[1,d]$.
Next, we need to bound the two norms $\delta_t = \norm{\*R_t-\*\Sigma_t}$ and $\delta_w=\norm{\*R_w-\*\Sigma_w}$. In the statement of the theorem, note that the multi-view data matrix $\*X$ was rearranged as $[\*A^{(1)},\dotsc,\*A^{(N)}]$ using the features as rows in the view-matrices $\*A$. Thus, using the identicality assumption of multiple views, we have: [$$\begin{aligned}
\delta_w
&= \norm{\sum_{i=1}^{N}\frac{1}{m}\*A^{(i){\top}}\*A^{(i)} - {\mathbb{E}}\*A^{(i){\top}}\*A^{(i)}} \\\nonumber
&\leq \sum_{i=1}^{N}\norm{\frac{1}{m}\*A^{(i){\top}}\*A^{(i)} - \*{\Sigma}_w^{(i)}}
\leq N\norm{\frac{1}{m}\*A^{\top}\*A - \*\Sigma_w}\end{aligned}$$]{}
The term $\norm{\frac{1}{m}\*A^{{\top}}\*A-\*\Sigma_w}$ has been extensively studied for the case of isotropic distributions i.e., $\*\Sigma_w=\*I$ by @vershynin2010introduction. Here, we obtain a bound for the general case of $\*\Sigma_w$ and show that $\delta_w=\norm{\*R_w-\*\Sigma_w}$ is $\mathcal{O}(d/m)$. Similarly, we can show that $\delta_t=\norm{\*R_t-\*\Sigma_t}\leq m$. The intuition here is that $\*R_t$ is sum of $m$ view vectors, hence it is $\mathcal{O}(m)$. Detailed proofs for $\delta_w$ and $\delta_t$ are provided in Suppl. methods, Lemmas S4 and S5. Using these results and the fact that we always choose an embedding dimension $d$ greater than $m$, we can prove that $\rho_m$ is $\mathcal{O}({m^2}/{d})$.
This result is significant because we can now show that, to obtain $d$-dimensional multi-view embeddings, we only need to subsample $m\leq\sqrt{d}$ number of views from the larger set of views. For example, for a 64-dimensional embedding, we would need to sample at most $8$ views. In other words, the DNN architecture in this case would have $8$ sub-networks. Additionally, the choice of $d$ is important because a small $d$ would only discriminate between classes that are already easily separable in the data. In contrast, a larger $d$ would require a greater $m$ which in turn inflates the number of parameters in the DNN.
Experiments
===========
We conducted experiments with three different datasets to benchmark the performance of our method with respect to the competitive baselines specific to these domains. We chose these datasets to assess the applicability of our method for downstream learning tasks in two distinct multi-class semi-supervised settings: (1) uniform distribution of views per class and (2) variable number of views per class.
3D object classification {#sec:3d-object}
------------------------
We use Princeton ModelNet dataset [@wu20153d] to classify the object type from 2D images acquired at multiple view points. We use the train/test splits for the 40-class subset provided in their website[^1]. Each class has $100$ CAD models ($80$/$20$ for train/test) with 2D images ($100\times100$px) rendered in two settings by @su2015multi: **V-12**: $12$ views by placing virtual cameras at $30$ degree intervals around the consistent upright position of an object and **V-80**: $80$ views rendered by placing $20$ cameras pointed towards the object centroid and rotating at $0, 90,
180, 270$ degrees along the axis passing through the camera and the object centroid.
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### Deep mv-corr Model {#sec:model}
As shown in Figure \[fig:schematic\], we use identical sub-networks to model the data from each view. The number of sub-networks is equal to the number of views subsampled $m$. We use a simple 3-block VGG architecture [@chatfield2014return] as illustrated in the inset in Figure \[fig:schematic\]. To reduce the number of trainable parameters, we use global average pooling after the last layer instead of vectorizing its activations before passing them to a dense layer of $d$ neurons. The embedding layer is constrained to have a unit $l_2$ norm. For all our experiments, we observed that a sigmoid activation for all layers yielded maximum $\rho$ at convergence. The loss $1-\rho$ was minimized using SGD with a learning rate of $0.01$, momentum of $0.9$ and a decay of $1e-6$. To determine model convergence, we applied early stopping criteria (stop training if $1-\rho$ at the end of a training epoch did not decrease by $10^{-3}$ for $5$ consecutive epochs). All models were implemented in TensorFlow[^2] and trained on a GeForce GTX 1080 Ti GPU.
The result in Theorem 3.1 only tells us about the relation between $d$ and $m$ and not their effect on classification accuracy, so we trained models for $m=[2,3,4,5,7,9]$ and $d=[16,32,40,64]$. Note that, during training we only need to know that the $m$ samples per instance in a batch are of the same class, hence the training can be considered semi-supervised. During inference, we just extract embeddings from one of the sub-networks which is randomly chosen. We did not observe significant changes in performance by choosing a different sub-network.
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Ours supervised
-- -------- ------------------------ ------------------------ --
seen 82.9 $\pm$ 0.5 **88.7** $\pm$ **1.2**
unseen 82.1 $\pm$ 0.7 81.5 $\pm$ 0.9
seen 84.2 $\pm$ 0.4 **89.2** $\pm$ **1.4**
unseen **85.7** $\pm$ **1.1** 80.3 $\pm$ 1.5
: Accuracy of clustering for seen and unseen views. SD computed from ten trials. Bold indicates significantly higher acc.[]{data-label="table:3d-object"}
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### Robustness to unseen views
To setup a view-agnostic evaluation, of the $80$ CAD models in the ModelNet training data, we pick $6$ views for **V-12** and $40$ views for **V-80** to create a train split. We create ten such trials by choosing the 50% of the views using a different random seed. View-information was only used to ensure no overlap of views in train/test splits. We then evaluate the performance of our model on the $20$ CAD models in the test-set both for views that were *seen* and *unseen* in training.
As described in Sec. \[sec:model\], we train our models in a semi-supervised fashion. We use k-means algorithm [@scikit-learn] (no. clusters set to $40$) to classify the $40$ classes in the test set. For baselines, we train a fully supervised CNN in a view-agnostic fashion with same architecture as our sub-network. This baseline can be considered as an upper bound of performance as it is fully supervised. First, we examine the clustering accuracy[^3] for different choices of $m$ and $d$ on the test-set of unseen views in **V-12**. As shown in Figure \[fig:3d-acc\], we found that $d=40$ with the number of sub-networks $m=5$ gave the best performance. Consistent with our theory, $m>\sqrt{d}$ did not improve the performance further. The dip in performance for $m\geq7$ in this case maybe due to the limited data for larger networks. Then, we compare the clustering performance of the chosen model on the test set for the views seen and unseen during training, as well as with the supervised baseline. As shown in Table \[table:3d-object\], for our method, there is no significant[^4] difference between accuracy scores for seen and unseen views for the ten trials. The results for the supervised baseline show significantly better performance for seen views compared to that of unseen views. This suggests that our method performs better for views not in training data. Additionally for the **V-80** dataset, our model performs significantly better than the supervised baseline, suggesting the benefit of multi-view modeling in case of a denser view sampling.
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Method Acc. mAP
------------------------------------- ---------- ----------
Loop-view CNN [@jiang2019mlvcnn] 0.94 **0.93**
HyperGraph NN [@feng2019hypergraph] **0.97** -
Factor GAN [@khan2019unsupervised] 0.86 -
MVCNN [@su2015multi] 0.90 0.80
Ours + 3-layer DNN 0.94 0.89
: 3D object recognition and retrieval comparison with other methods. Bold indicates results of the SoA.[]{data-label="table:3d-soa"}
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### Object recognition and retrieval {#sec:supervised}
To evaluate our embeddings in a supervised setup, we train a model as described in sec. \[sec:model\] using $40$ CAD models in the train split. We extract the embeddings for the remaining $40$ CAD models and train a 3-layer fully connected (sigmoid activation) DNN to classify the object category. We use classification accuracy and mean average precision (mAP) to evaluate recognition and retrieval tasks. For baselines, we compare our method with the ModelNet leader-board for **V-12**. We highlight our results in Table \[table:3d-soa\] in the context of state of the art (SoA) performance for this application as well as examples from widely used class of methods such as domain-invariant applications of GAN [@khan2019unsupervised] and multi-view CNN for object recognition [@sun2019multi]. Unlike our method, these methods are fully supervised and are not generally view-agnostic.
Our method performs within 4% points of the SoA for recognition and retrieval tasks (See Table \[table:3d-soa\]). In all experiments, we observed that the bound for maximum number of sub-networks is better in practice than the theoretical bound, i.e. $m\approx d^{2/5}$. Also, the choice of $m$ only varied with $d$ and not the larger set of views $M$ which is a useful property to note for practical settings. The parameter $d$ however needs to be tuned for classification tasks as it depends on intra- and inter-class variabilities which determine the complexity of the downstream task.
Pose-invariant face recognition {#sec:pose}
-------------------------------
Robust face recognition is yet another application where multi-view learning solutions are attractive because we are interested in the shared representation across different presentations of a person’s face. For this task, we use the Multi-PIE face database [@gross2010multi] which includes face images of $337$ subjects in $15$ different poses, $20$ lighting conditions and $6$ expressions across 4 sessions.
In Sec. \[sec:3d-object\], we evaluated our model to classify object categories available for training, but with a focus on the performance of seen vs. unseen views during training. In this experiment, we wish to test the usefulness of our embeddings to recognize faces not seen in training. We use a similar train/test split as in GMA [@sharma2012generalized] of $129$ subjects in $5$ lighting conditions ($1, 4, 7, 12, 17$) common to all four sessions as test data and remaining $120$ subjects in session 01 for training. For performance evaluation, we use 1-NN matching with normalized euclidean distance similarity score as the metric. The gallery consisted of faces images of the $129$ individuals in frontal pose and frontal lighting and the remaining images from all poses and lighting conditions were used as probes. All images were cropped to contain only the face and resized to $100\times100$ pixels. No face alignment was performed.
### Model and baselines
For our model architecture, we first choose $m=2$ sub-networks and examine the mv-corr value at convergence for different embedding dimension $d$. Based on this we pick $d=64$. Following our observations in the object classification task, we choose $m=4$ sub-networks. The sub-network architecture is the same as before (See inset Figure \[fig:schematic\]). We did not explore other architectures because our goal here was to evaluate the use of mv-corr loss and not necessarily the best performing model for a specific task. During training, we sample with replacement, $m$ face images per individual agnostic to the pose or lighting condition. For matching experiments, we extract embeddings from a single randomly chosen sub-network.
For baselines, we train deep CCA (DCCA @andrew2013deep) using its implementation[^5] with the same sub-network architecture as ours. We trained separate DCCA models for five poses: 15, 30, 45, 60 and 75 degrees. While training the two sub-networks in DCCA, we sample face images of subjects across all lighting conditions with a frontal pose for one sub-network and images of specific pose for the second. This matches the testing conditions where we only have frontal pose images in the gallery. During testing we use the pose-specific sub-network to extract embeddings. We also compare our method with two other variants of GMA: GMLDA and GMMFA reported by @sharma2012generalized.
As shown in Table \[table:face-pose\], our model successfully matches at least 90% of the probe images to the frontal faces in the gallery, across all poses. The performance drop across different poses was also minimal compared to a pairwise method such as DCCA which assumes that the pose of a probe image is available in testing conditions. However, the view-agnostic benefit of our method and the Multi-PIE dataset needs to be viewed in the context of the broader research domain of face recognition. Methods such as MvDA [@kan2015multi] which build view-specific transformations have shown nearly 100% face recognition rate on Multi-PIE when the pose information of the probe and gallery images was known. Furthermore, the face images in this dataset were acquired in strictly controlled conditions. While it serves as an effective test-bed for benchmarking, we must consider other sources of noise for robust face recognition besides pose and lighting [@wang2018devil]. Our future work will focus on adapting mv-corr for face recognition in-the-wild.
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Method $15^{\circ}$ $30^{\circ}$ $45^{\circ}$ $60^{\circ}$ $75^{\circ}$ Avg.
-------- -------------- -------------- -------------- -------------- -------------- ----------
GMLDA 92.6 80.9 64.4 32.3 28.4 59.7
GMMFA 92.7 81.1 64.7 32.6 28.6 59.9
DCCA 82.4 79.5 73.2 62.3 51.7 69.8
Ours 95.7 93.1 94.5 92.3 91.1 **93.3**
: 1-NN matching accuracy comparison for pose-invariant face recognition. Bold indicates the best performing model[]{data-label="table:face-pose"}
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Spoken word recognition {#speech}
-----------------------
The multi-view datasets considered in sections \[sec:3d-object\] and \[sec:pose\] for benchmarking our method were acquired in controlled conditions. They also have nearly uniform distribution of number of distinct views per class as well as as uniform number of samples per view. In practical settings, we often have to deal with a variable number of views per class. To study the our framework in this context, we evaluate our method for spoken word recognition using the publicly available Speech Commands Dataset (SCD, @warden2018speech).
### Speech Commands Dataset
SCD includes variable number of one second audio. recordings from over 1800 speakers saying one or more of 30 commands such as “On” and “Off”. The application of mv-corr for spoken-word recognition and text-dependent speaker recognition in SCD was studied by @somandepalli2019multiview. Their results showed improved performance for speaker recognition task compared to the SoA in this domain [@snyder2017deep]. Building upon their work, in this paper, we analyze spoken-word recognition on SCD in a greater detail.
The different speakers saying the same word can be treated as multiple views to obtain discriminative embeddings of the speech commands invariant to the speaker (view). Specifically, we are interested in the performance of our method for the case of variable number of views per class. Thus, we analyze the performance of each word with respect to the number of unique speakers (views) available for that word. We choose $m=4$ sub-networks (See inset Figure \[fig:schematic\] for the architecture) to obtain 64-dimensional embeddings. Of the $1868$ speakers, we use $1000$ speakers for training and the remaining for testing to ensure that we only test on speakers (views) not seen during training. To assess generalizability to unseen classes, we create three folds by including $20$ words for training and the remaining $10$ words for testing. The models are trained in a semi-supervised fashion as described in \[sec:model\]. We use the k-means algorithm to cluster the embeddings for the test splits with the number of clusters set to $10$.
The per-class accuracy from the clustering task is shown in Figure \[fig:scd\]. The average number of speakers across the thirty commands was $400.3\pm52.5$ which underscores the variable number of views per class. We observe a minimal association (Spearman rank correlation = $0.12$) between the number of unique speakers per word and the per-class accuracy scores. However, it is difficult to disambiguate this result from the complexity of the downstream learning task. That is, we may need more views for certain words to account for inter-class variability (similar sounding words e.g., “on” vs. “off” or “tree” vs. “three”) and intra-class variability (e.g., different pronunciations of the word “on”).
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### Domain adversarial learning
Finally, in the context of domain adaptation for experiments with SCD, we compare our multi-view learning method with two recent domain adversarial learning methods: domain adversarial networks (DAN, @ganin2016domain) and cross-gradient training (CrossGrad, @shankar2018generalizing). The central idea of these methods is to achieve domain invariance by training models to perform better at classifying a label than at classifying the domain (view).
\[table:scd\] 0.15in
Method DAN CrossGrad Ours + 2-layer DNN
--------- ------ ----------- --------------------
Acc (%) 77.9 89.7 **92.4**
: Comparison of mv-corr framework with domain adversarial methods
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As described in Sec. \[sec:supervised\], we adapt our embeddings for a supervised setting on a subset of $12$ commands in SCD to compare with the results in @sharma2012generalized. We first train the mv-corr model of four sub-networks using $500$ speakers from the training set. We then obtain 64-dimensional embeddings on the remaining $500$ speakers and train a 2-layer fully connected DNN (sigmoid activation) to classify the $12$ commands, and test on the remaining $868$ speakers. For baselines, we replicate the experiments for DAN and CrossGrad using released code.[^6] We use the same splits of $500$ speakers each for training/development and $868$ speakers for testing. The classification accuracy of our method and that of DAN and CrossGrad is shown in Table \[table:scd\]. We observed a significant improvement[^7] over CrossGrad, suggesting that a multi-view formulation can be effectively used for domain adaptation problems such as in SCD.
Conclusion
==========
In this paper, we explored a neural method based on multi-view correlation (mv-corr) to capture the information shared across large number of views by bootstrapping the data from multiple views during training in a view-agnostic manner. We discussed theoretical guarantees of view bootstrapping as applied to mv-corr and derived an upper bound for the number of views to subsample for a given embedding dimension. Our experiments on 3D object classification and retrieval, pose-invariant face recognition and spoken word recognition showed that our approach performs on par with competitive methods in the respective domains. Our results underscore the applicability of our framework for large-scale practical applications of multi-view data where we may not know how the multiple corresponding views were acquired. In future work, we wish to extend the ideas of view-bootstrapping and related theoretical analysis to the broader class of multi-view learning problems.
Supplementary Methods {#supplementary-methods .unnumbered}
=====================
The following sections provide detailed proofs for propositions, lemmas and the theorem presented in the associated ICML submission. We also provide details of simulation analysis that we conducted to support one of the claims made in the paper.
[l | p[5cm]{} ]{} Section & Link\
\
Table of Notations & \[table:notation\]\
Proposition: Total-view Covariance & \[prop:rt\]\
Proposition: Multi-view correlation objective is bounded above by 1 & \[prop:rho\]\
Simulation Experiments & \[sec:supp-affinity\]\
Lemma: Upper Bound for Bootstrapped Within-View Covariance & \[sec-lemma:rw\]\
Lemma: Upper Bound for Bootstrapped Total-View Covariance & \[sec-lemma:rt\]\
Theorem: Error of the Bootstrapped Multi-view Correlation & \[thm:rho\]\
Notation {#notation .unnumbered}
========
------------------------------------------------------------- ----------------------------------------------------------------------------------
$N$ Number of samples
$M$ Number of views
$d$ Embedding dimension
$m$ Bootstrap view sample size / number of subsampled views
$\*x_i\in{\mathbb{R}}^d$ Embedding/feature vector
$i=1,\dotsc,N$ Index for sample
$l=1,\dotsc,M$ Index for view
$\*X_l=[\*x_1,...,\*x_N] \in {\mathbb{R}}^{d\times N}$ $l^{th}$-view data matrix
$\*X=[\*X_1,\dotsc,\*X_M]$ Multi-view data matrix. Assume mean-zero columns without loss of generality
$\*R_b\in{\mathbb{R}}^{d\times d}$ Sum of between-view covariance matrices for $m$ views: *Between-view covariance*
$\*R_w\in{\mathbb{R}}^{d\times d}$ Sum of within-view covariance matrix : *Within-view covariance*
$\*R_t\in{\mathbb{R}}^{d\times d}$ Total-view covariance matrix
$\*\Sigma_b$ Between-view covariance for $M$ views
$\*\Sigma_w$ Within-view covariance for $M$ views
$\*\Sigma_t$ Total-view covariance for $M$ views
$\*A_l\in{\mathbb{R}}^d$ $d$-dimensional feature row, mean-zero and $\norm{\*A_l}_2=1$
$\*A^{(i)}=[\*A_1;\dotsc;\*A_m]\in{\mathbb{R}}^{m\times d}$ View-matrix from the $i^{th}$ sample for $m$ views with features as rows
$\*X = [\*A^{(1)},\dotsc,\*A^{(N)}]$ Rearranged m-view data matrix
$\*W \in{\mathbb{R}}^{d\times d}$ Shared subspace / Common Eigenspace of $\*R_b$ and $\*R_w$
$\norm{\cdot}_2 \equiv \norm{\cdot}$ Spectral norm
$\norm{\cdot}_{\psi_1}$ Sub-exponential norm
$\norm{\cdot}_{\psi_2}$ Sub-gaussian norm
------------------------------------------------------------- ----------------------------------------------------------------------------------
: Notations used in the proofs and text[]{data-label="table:notation"}
\[tab:TableOfNotationForMyResearch\]
Proposition: Total-view Covariance {#prop:rt}
==================================
Consider the sum of $\*R_b$ and $\*R_w$ which includes $M^2$ terms. Note that we assume $\*X_l:l=1,\dotsc,M$ to have mean-zero columns. Therefore covariance estimation is just the cross-product: $$\begin{aligned}
\label{eq:rt}
\*R_w + \*R_b
&= \frac{1}{M}\sum_{l=1}^{M}{\*X}_{l}({\*X}_{l})^{\top} + \frac{1}{M}\sum_{k=1}^{M}\sum_{l=1,l\neq k}^{M}{\*X}_{l}({\*X}_{k})^{\top} &&\text{[By definition]} \\\nonumber
&= \frac{1}{M}\sum_{l=1}^{M}\sum_{k=1}^{M}{\*X}_{l}({\*X}_{k})^{\top} &&\text{[Summing all terms]}\\\nonumber
&= \frac{1}{M}\bigg(\sum_{l=1}^{M}{\*X}_{l}\bigg)\bigg(\sum_{l=1}^{M}{\*X}_{l}\bigg)^{\top} =\*R_t &&\text{[Total-view covariance]}\\\nonumber\end{aligned}$$ where the total-view matrix is $\sum_{j=1}^{M}\*X_j$. Thus, $\*R_t$ can be easily estimated as the covariance of a single total-view matrix, without having to consider the sum of $M^2-M$ covariance matrices. Note that we excluded the normalization factor $N-1$ in the esimtation of the covariance terms above. This gives us the following useful relation which simplifies many computations in practice. $$\begin{aligned}
\*R_t &= \*R_b + \*R_w\end{aligned}$$
Proposition: Multi-view correlation objective is bounded above by 1 {#prop:rho}
===================================================================
Recall the multi-view correlation objective for $M$ views: $$\begin{aligned}
\label{eqn:rho-M}
\rho_{M}
= \max_{\*W}\frac{1}{d(M-1)}\frac{\Tr(\*W^{{\top}}\*R_b\*W)}{\Tr(\*W^{{\top}}\*R_w\*W)} \end{aligned}$$
It is desirable to have an upper bound for the objective similar to the correlation coefficient metric which is normalized to have a maximum value of 1. Let us begin with the definition of the multi-view correlation matrix: $$\begin{aligned}
\label{eqn:mv-corr}
\*\Lambda = \max_{\*W}\frac{1}{M-1}\frac{\*W^{{\top}}\*R_b\*W}{\*W^{{\top}}\*R_w\*W}\end{aligned}$$ Here, $\*W\in{\mathbb{R}}^{d\times M}\,,M\leq d$
Define a matrix $\*Y_l = \*W^{{\top}}\*X_l\in{\mathbb{R}}^{M\times N}\,,M\leq d$ where the column vectors $\*y\in{\mathbb{R}}^M$ are a low-dimensional projection of the input features $\*X$. The column vector elements are $y_i^l\in{\mathbb{R}}:i=1,\dotsc,N;l=1,\dotsc,M$ with that the ratio in Eq. \[eqn:mv-corr\], ignoring the max operation can be written as:
$$\begin{aligned}
\*\Lambda &= \frac{1}{M-1}\frac{\*W^{{\top}}(\*X_1\*X_2^{{\top}}+\dotsc+\*X_M\*X_{M-1}^{{\top}})\*W}{\*W^{{\top}}(\*X_1\*X_1^{{\top}}+\dotsc+\*X_M\*X_M^{{\top}})\*W} \\
&= \frac{1}{M-1}\frac{(\*Y_1\*Y_2^{{\top}}+\dotsc+\*Y_M\*Y_{M-1}^{{\top}})}{(\*Y_1\*Y_1^{{\top}}+\dotsc+\*Y_M\*Y_M^{{\top}})} \\
&= \frac{1}{M-1}\frac{\sum_i\sum_l\sum_{k\neq l}y_i^ly_i^k}{\sum_i\sum_l (y_i^l)^2}\\
&= \frac{1}{M-1}\frac{r_b}{r_w}\end{aligned}$$
To show that $\rho\leq1$, we can also equivalently prove the following expression is non-negative:
$$\begin{aligned}
0 &\leq (M-1)r_w - r_b
= (M-1)r_w - (r_t - r_w) &&\text{[From total-covariance proposition: Sec.\ref{prop:rt}]} \\
&= Mr_w - r_t
= M\sum_i\sum_l(y_i^l)^2 - \sum_i\big(\sum_l y_i^l\big)^2
:= F\end{aligned}$$
Now, we need to find the $y_i^l$ that minimizes $F$. Therefore, take the gradient of $F$ with respect to $y_i^l$ and check if the curvature is non-negative where the gradient is zero.
$$\begin{aligned}
\pdv{F}{y_i^l}
&= 2My_i^l - 2\sum_j\sum_ly_j^k\sum_l\delta_{ji}^{kl}
= 2My_i^l - 2\sum_ky_j^k\\
\pdv[2]{F}{y_i^l}{y_j^k} &= 2M\delta^{lk}_{ij} - 2\sum_t\delta_{ji}^{jt}
= 2\delta_{ji}(M\delta^{lk}-1) := J\end{aligned}$$
Solving for $\pdv{F}{y}=0$ has a unique solution: $y_i^l = \frac{1}{M}\sum_k y_i^k = \Bar{y}_i^*$. Putting this result back gives $F=0$ at this solution. To show this solution minimizes $F$ and therefore $\rho<1$, we need to show that the Jacobian $J$ in Eq. 5 has only non-negative eigenvalues. Note that there are only $\delta$ variables in Eq. 5. Thus, in a matrix form across all views we have $J=M\*I_M - \*I_M$ yielding non-negative eigenvalues. Hence $\rho\leq1$
Simulation Experiments {#sec:supp-affinity}
======================
In order to show that the output embeddings from the sub-networks are maximally correlated. we need to empirically show that mv-corr is learning highly correlated vector representations. For this, we generate synthetic observations as detailed in [@parra2018correlated] where the number of common signal components across the different views is known. Because the source signal is given, we can also empirically examine the correlation of the shared components with the source signal.
Data generation
---------------
Consider $N$ samples of signal and noise components for $M$ views to be $\*s^l_n\in\mathbb{R}^K$ and $\*b^l_n\in\mathbb{R}^D\, n=1,...,N\,, l=1,...,M\,, K<D$ respectively, both drawn from standard normal distribution. Because our objective is to obtain correlated components across the views, we fixed the same signal component across the $M$ views, i.e, $\*s^l_n \approx \*s_n$, but corrupted with a view-specific noise$\ \*{\eta^l}$. Thus, signals were mapped to the measurement space as$\ \*x_{s,n}^l=\*A_s^l\*s_n + \*{\eta}^l, \*x_{b,n}^l=\*A_b^l\*b^l_n$ and were z-normalized. The multiplicative noise matrices were generated as $\*A_s^l = \*O_s^l\*D_s^l \in \mathbb{R}^{D\times K}$ and $\*A_b^l=\*O_b^l\*D_b^l \in \mathbb{R}^{D\times D}$ The two matrices $\*O_s^l\in\mathbb{R}^{D\times K}\text{ and }\*O_b^l\in\mathbb{R}^{D\times D}$ are composed of orthonormal columns.
The non-zero eigenvalues of the signal and noise covariance matrices were set with$\ \*D_s^l\in\mathbf{R}^{K\times K}$ and $\*D_b^l\in\mathbf{R}^{D\times D}$ by constructing$\ D_{ii}=\exp(d_i), d_i\sim\mathcal{N}(0,1)$. We used different matrices $\*A_s^l$ and $\*A_b^l$ to simulate a case where the different views of the underlying signal are corrupted by different noise. As is the case with many real world datasets, the noise in the measurement signal is further correlated between the views. We simulated this by $\*x_{b,t}^l\gets \alpha\*x_{b,n}^l + (1-\alpha)\*x_{b,n}^l , \alpha \in [0,1]$. Finally the SNR of the measurements is controlled by $\beta$ to generate the multiview data as $\*y^l_n = \beta\*x_{s,n}^l + (1-\beta)\*x_{b,n}^l , \beta \in [0,1]$ resulting in a data matrix of size $\ N\times\ D\times M$ with $N$ samples of $D$-dimensional data from $M$ views. For all our experiments, we generated data with $N=100000, D=1024, K=10, M=4, \beta = 0.7$ and spatial noise correlation $\alpha=0.5$.
Deep mv-corr Model {#deep-mv-corr-model}
------------------
The network consists of 4 sub-networks where each sub-network is composed of 2 fully connected layers of 1024 and 512 nodes which is then fed into an embedding layer with $d=[5, 10, 15, 20, 40, 50, 64, 128]$ neurons. The output embedding dimension was varied in order to examine the affinity of the representations with the source signal. This is important, since in real world applications the number of correlated components is not known apriori. The models were trained as explained in the main paper.
Affinity metrics to measure correlation
---------------------------------------
The benefit of using synthetic data is that we can examine what the network learns when the generative process is known. The affinity measures we use enable us to compare the similarity of the embedding subspaces to that of the source signal. The objective of our simulations is to measure if the correlated signal components are correctly identified from the measurements. Because the components with equal $\rho$ can be produced by arbitrary linear combination of the vectors in the corresponding subspace, we examined the normalized affinity measure between two subspaces as defined in [@soltanolkotabi2014robust] to compare the representations with the source signal. Let $\ \Hat{\*X}_s^l \in \mathbf{R}^{T\times K'}$ be the reconstructed signal or the representation learnt by optimizing eqn. 11 corresponding to the source signal $\ \*X_s^l \in \mathbf{R}^{T\times K}$. The affinity between $\ \Hat{\*X}\text{ and }\*X$ can be estimated using the principal angles$\ \theta^{(\cdot)}$ as: $$\text{aff}(\*X, \Hat{\*X}) = \sqrt{\frac{\cos^2\theta^{(1)} + ... +\cos^2\theta^{(K\wedge K')} }{K\wedge K'}}$$ The cosine of the principal angles $\ \theta$ are the singular values of the matrix $\ \*U^{\top}\*V$ where$\ \*U$ and $\ \*V$ are the orthonormal bases for $\ \*X \text{ and } \Hat{\*X}$ respectively. The affinity is a measure of correlation between subspaces and has been extensively used to compare distance between subspaces in the subspace clustering literature [@soltanolkotabi2014robust]. This measure is low when the principal angles are nearly orthogonal and has a maximum value equal to one when one of the subspaces is contained in the other.
One of the benefits of using this affinity measure is that it allows us to compare two subspaces of different dimensions. We estimate two affinity measures: 1) *reconstruction affinity*, $\ R_a$: average affinity between the reconstructed signal and the source signal across the$\ N$ views and 2) *inter-set affinity*, $\ R_s$: average affinity between the different views of the reconstructed signal. Formally, [ $$\begin{aligned}
R_a = \frac{1}{N}\sum_{l=1}^{N}\text{aff}(\*X_s^l,\Hat{\*X}_s^l) \\
R_s = \frac{2}{N(N-1)}\sum_{l=1}^{N}\sum_{{\substack{k=1 \\ l\neq k}}}^{N}\text{aff}(\Hat{\*X}_s^l,\Hat{\*X}_s^k) \end{aligned}$$]{}
Figure \[fig:aff\] shows the reconstruction affinity measure ($R_a$) and the inter-set affinity measure ($R_s$) for these parameters. Notice that the maximum $R_a$ is achieved for the embedding dimension of 10 (which is the number of correlated components used to generate the data) indicating that the dMCCA retains some notion of the ambient dimension for maximizing correlation between views. The $R_s$ measure consistently decreased with increasing embedding dimension. Because we estimate covariances in the loss function and use SGD with mini-batches for optimization, we also examine the performance with varying batch sizes. As shown in Fig. \[fig:aff\] a mini-batch size greater than 400 gives consistent results. The results from this simulation study suggests that the multi-view embeddings are maximally correlated. Hence during inference we can use any sub-network to extract the embeddings.
![[[Affinity measures for synthetic data. Number of correlated components in the generated data is 10 (boxed)]{}]{}[]{data-label="fig:aff"}](Ra_Rs.pdf){width="80.00000%"}
Lemma: Upper Bound for Bootstrapped Within-View Covariance {#sec-lemma:rw}
==========================================================
\[lemma:view-matrix\] (Subsampled view matrices, approximate isotropy) Let $\*A$ be a $m\times d$ matrix created by subsampling $m$ views from a larger, unknown number of views. The rows $\*A_i$ of the matrix $\*A$ are independent subgaussian random vectors in $\mathbb{R}^d$ and a second moment matrix $\*{\Sigma}=\mathbb{E}\*A_i\otimes\*A_i$. Then for every $t\geq 0$, with probability at least $1-2\exp(-ct^2)$ we have $$\begin{aligned}
\label{eq:view-matrix}
\norm{\frac{1}{m}\*A^{\top}\*A - \*{\Sigma}} \leq \max(\delta, \delta^2) \quad where \quad \delta = C\sqrt{\frac{d}{m}} + \frac{t}{\sqrt{m}}\end{aligned}$$ Here $C, c >0$ depend only on the subgaussian norm $K = \max_i\norm{\*A_i}_{\psi_2}$ of the view space
This is a straight-forward generalization of Theorem 5.39 [@vershynin2010introduction] for non-isotropic spaces. The proof involves *covering argument* which uses a net $\mathcal{N}$ to discretize the compact view space, which is all the vectors $\*z$ in a unit sphere $\mathcal{S}^{d-1}$. Similar to [@vershynin2010introduction], we prove this in three steps:
1. **$\mathcal{N}_{\epsilon}$ Approximation:** Bound the norm $\norm{\*A\*z}_2$ for all $\*z\in\mathbb{R}^d$ s.t. $\norm{\*z}_2=1$ by discretizing the sphere with a 1/4-net.
2. **Concentration:** Fix a vector $\*z$, and derive a tight bound of $\norm{\*A\*z}_2$.
3. **Union bound:** Take a union bound for all the $\*z$ in the net
**Step 1: $\mathcal{N}_{\epsilon}$ Approximation.** From [@vershynin2010introduction], we use the following statement: $$\begin{aligned}
\exists \delta > 0,\quad \norm{\*B^{\top}\*B - \*I} \leq max(\delta, \delta^2) \implies \norm{\*B}_2 \leq 1 + \delta \end{aligned}$$ We evaluate the operator norm in eq. \[eq:view-matrix\] as follows:
&=\
&=
Let $\*D:=\frac{1}{m}\sum_{i=1}^{m}\*A_i\*A_i^{\top} - \frac{1}{m}\Sigma_{i=1}^{m}\mathbb{E}\*A_i\*A_i^{\top}$. Choose a $\epsilon'$-net $\mathcal{N}$ such that $\lvert\mathcal{N}\rvert\leq 9^{d}$ which provides sufficient coverage for the unit sphere $\mathcal{S}^{d-1}$ at $\epsilon'=1/4$. Then, for every $\*z\in\mathcal{N}$ we have (using Lemma 5.4 in [@vershynin2010introduction]),
&\
&\_\
&2\_[z]{}
For some $\epsilon>0$, we want to show that the operator norm of $\*D$ is concentrated as $$\begin{aligned}
\max_{\*z\in\mathcal{N}}\norm{\*z^{\top}\*D\*z} \leq \frac{\epsilon}{2} \text{ where } \epsilon:=\max(\delta, \delta^2)\end{aligned}$$ **Step 2: Concentration.** Fix any vector $\*z\in\mathcal{S}^{d-1}$ and define $Y_i = \*A_i^{\top}\*z - \mathbb{E}\*A_i^{\top}\*z$ where $\*A_i$ are subgaussian random vectors by assumption with $\norm{\*A_i}_{\psi_2}=K$. Thus, $Y_i$ $i=1,\dotsc,m$ are independent subgaussian random variables. The subgaussian norm of $Y_i$ is calculated as, $$\begin{aligned}
\norm{Y_i}_{\psi_2} = \norm{\*A_i^{\top}\*z - \mathbb{E}\*A_i^{\top}\*z }_{\psi_2}
\leq 2\norm{\*A_i^{\top}\*z}_{\psi_2}
\leq 2\norm{\*A_i}_{\psi_2}\norm{\*z}
= 2K\end{aligned}$$ The above relation is an application of triangular and Jensen’s inequalities: $\norm{X-\mathbb{E}X} \leq 2\norm{X}$ with $\abs*{\mathbb{E}X}\leq\mathbb{E}\abs*{X} \leq \abs*{X}$. Similarly, $Y_i^2$ are independent subexponential random variables with the subexponential norm $K_e = \norm{Y_i}_{\psi_1} \leq \norm{Y_i}_{\psi_2}^2 \leq 4K^2$. Finally, by definition of $Y_i$, we have $$\begin{aligned}
\label{eq:sum}
\norm{\*z^{\top}\*D\*z} = \frac{1}{m}\abs*{\Sigma_{i=1}^{m}Y_i^2}\end{aligned}$$ We use the exponential deviation inequality in Corollary 5.17 from [@vershynin2010introduction] to control the summation term in eq. \[eq:sum\] to give: $$\begin{aligned}
P\Big(\norm{\*z^{\top}\*D\*z} \geq \frac{\epsilon}{2}\Big) &=
P\Big(\frac{1}{m}\abs*{\Sigma_{i=1}^{m}Y_i^2} \geq \frac{\epsilon}{2}\Big) \\ \nonumber
&\leq 2\exp\Bigg[-c\min\bigg(\frac{\epsilon^2}{4K_e^2},\frac{\epsilon}{2K_e}\bigg)m\Bigg] \nonumber\end{aligned}$$ Note that $\epsilon:=max(\delta, \delta^2)$. If $\delta\geq1$ then $\epsilon=\delta^2$. Thus, $\min(\epsilon, \epsilon^2) = \delta^2$. Using this and the fact that $K\geq2\norm{Y_i}_{\psi_2}\geq1$, we get $$\begin{aligned}
P\Big(\norm{\*z^{\top}\*D\*z} \geq \frac{\epsilon}{2}) \leq 2\exp\Bigg[-\frac{c_1}{K^4}\delta^2m\Bigg] \leq 2\exp\Bigg[-\frac{c_1}{K^4}(C^2d + t^2)\Bigg]\end{aligned}$$ by substituting $\delta = C\sqrt{\frac{d}{m}} + \frac{t}{\sqrt{m}}$ and using $(a+b)^2\geq a^2+b^2$.
**Step 3: Union Bound.** Using Boole’s inequality to compute the union bound over all the vectors $\*z$ in the net $\mathcal{N}$ with cardinality $\abs*{\mathcal{N}}=9^d$, we get $$\begin{aligned}
P\Bigg\{\max_{\*z\in\mathcal{N}}\norm{\frac{1}{m}\*A^{\top}\*A - \*{\Sigma}} \geq \frac{\epsilon}{2}\Bigg\} \leq 9^d\cdot2\exp\Bigg[-\frac{c_1}{K^4}(C^2d + t^2)\Bigg]\end{aligned}$$ Pick a sufficiently large $C=C_K\geq K^2\sqrt{\log9/c_1}$, then the probability $$\begin{aligned}
P\Bigg\{\max_{\*z\in\mathcal{N}}\norm{\frac{1}{m}\*A^{\top}\*A - \*{\Sigma}} \geq \frac{\epsilon}{2}\Bigg\} &\leq \frac{2}{\exp\Big(d+\frac{c_1t^2}{K^4}\Big)} \\ \nonumber
&\leq 2\exp{\Big(-\frac{c_1t^2}{K^4}\Big)}\end{aligned}$$ Thus with a high probability of at least $1-2\exp{(-ct^2)}$ eq. \[eq:view-matrix\] holds. In other words, the deviation of the subsampled view matrix from the entire view space, in spectral sense is $\mathcal{O}(d/m)$
(Subsampled within-view covariance bound) Let $\*X$ be the $N\times m\times d$ tensor whose elements $\*A\in\mathbb{R}^{m\times d}$ are identically distributed matrices with rows $\*A_i$ representing $m$-views sampled from a larger set of views in $\mathbb{R}^d$. If $\*A_i$ are independent sub-gaussian vectors with second moment $\*{\Sigma}_w$, then for every $t\geq0$, with probability at least $1-2\exp{(-ct^2)}$, we have $$\norm{{\*R}_w - \*{\Sigma_w}} \leq N\frac{C^2d+t^2}{m}
$$ Here ${\*R}_w$ is the sum of within-view covariance matrices for $m$ views and $C>0$ depends only on the sub-gaussian norm $K=\max_i\norm{\*A_i}_{\psi_{2}}$ of the subsampled view space.
\[lemma:rw\]
Let us now consider the rearranged $m$-view subsampled data tensor $\*X\in \mathbb{R}^{N\times m\times d} = [\*A^{(1)}, ..., \*A^{(N)}]$. Let $\*A$ be the $m\times d$ view-specific data sampled identically for $N$ times. Without loss of generality, assume the rows to be zero mean which makes covariance computation simpler. The rows $\*A_i$ are independent sub-gaussian vectors with second moment matrix $\*{\Sigma}=\mathbb{E}\*A^{\top}\*A$. The between-view covariance matrix ${\*R}_w$ for $m$ views can be written as: $$\begin{aligned}
{\*R}_w = \frac{1}{m}\sum_{i=1}^{N}\sum_{j=1}^{m}\*A_j\otimes\*A_j = \sum_{i=1}^{N}\frac{1}{m}\*A^{(i)\top}\*A^{(i)}\end{aligned}$$ The matrix $\*A$ is a sampling of $m$ views from an unknown and larger number of views $M$ for which the $\*R_w$ is constructed. We want to bound the difference between this term and the within-view covariance of the whole space using lemma \[lemma:view-matrix\]:
&=\
&=\
&\_[i=1]{}\^[N]{} &&\
&= N &&\
&N =C + &&\
&= N( )\^2\
&N() &&
Lemma: Upper Bound for Bootstrapped Total-View Covariance {#sec-lemma:rt}
=========================================================
\[lemma:rt\] (Subsampled total-view covariance bound) Let $\*X$ be the $N\times m\times d$ tensor whose elements $\*A\in\mathbb{R}^{m\times d}$ are identically distributed matrices with rows $\*A_i$ representing $m$-views sampled from a larger set of views in $\mathbb{R}^d$. Construct a total-view matrix ${\*X}\in\mathbb{R}^{m\times d}$ by summing entries across all views. Let $\*{\Sigma}_t$ be the second moment of the total-view space. Then, we have $$\norm{{\*R}_t - \*{\Sigma_t}} \leq Nm
$$ Here ${\*R}_t$ is the total-view covariance matrix and $c_2>0$ depends on the range of the total view space $k$ such that $\abs{{\*X}}\leq k$.
Consider the $m$-view subsampled data tensor rearranged with feature vectors as rows to get $\*X\in \mathbb{R}^{N\times m\times d} = [\*A^{(1)}, ..., \*A^{(N)}]$ with rows of $\*A^*$ as $\*A_i$. Without loss of generality, assume the $d$-dimensional rows of $\*A$ to be zero mean which makes estimating covariances simpler. The covariance ${\*R}_t$ of the total view matrix can be written as follows $$\begin{aligned}
{\*R}_t
= \frac{1}{m}\displaystyle\sum_{i=1}^{N}\Big(\sum_{i=1}^m\*A^{(i)}\Big)\Big(\sum_{i=1}^m\*A^{(i)}\Big)^{\top} = \frac{1}{m}\displaystyle\sum_{i=1}^{N}\Big(\sum_{j=1}^{m}\*A_j^{(i)}\Big)\otimes\Big(\sum_{j=1}^{m}\*A_j^{(i)}\Big)\\\nonumber
= \frac{1}{m}\displaystyle\sum_{i=1}^{N}\Big(\sum_{j=1}^{m}\*A_j^{(i)}\Big)\otimes\Big(\sum_{j=1}^{m}\*A_j^{(i)}\Big)=\frac{1}{m}\displaystyle\sum_{i=1}^{N}\*W_i\*W_i^{\top}\nonumber
$$ We want to bound the difference between this subsampled total-view covariance matrix and the second moment of the total-view space. Let ${\*a}^{(i)} = \sum_{j=1}^{m}\*A_j^{(i)}$ for $i=1,\dotsc,N$. The vector ${\*a}^{(i)}$ is the sum-of-views. We use a useful application of Jensen’s inequality here: $\norm{X-\mathbb{E}X} \leq 2\norm{X}$ with $\abs*{\mathbb{E}X}\leq\mathbb{E}\abs*{X} \leq \abs*{X}$
&=\
&\_[i=1]{}\^[N]{} &&\
&= &&\
& &&\
&= m\^2 = Nm&&
Theorem: Error of the Bootstrapped Multi-view Correlation {#thm:rho}
=========================================================
Let $\*X=[\*A^{(1)},\dotsc,\*A^{(N)}]$ be the $m\times d$ matrices of $m$ views sampled from an unknown number of views $M$. Let the rows $\*A_l$ of the view matrices $\*A$ be independent subgaussian vectors in ${\mathbb{R}}^d$ with $\norm{\*A_l}_2=1:l=1,\dotsc,m$. Then for any $t\geq0$, with probability at least $1-2\exp({-ct^2})$, we have
[\_m]{}(1, C\^[\*]{})
*Here, $\rho_m$ and $\rho^*$ are the mv-corr objectives for subsampled views $m$ and the total number of views $M$ respectively. The constant $C$ depends only on the subgaussian norm $K$ of the view space, with $K = \displaystyle\max_{i,l}\norm{\*A_l^{(i)}}_{\psi_2}$*
Starting from the objective defined in the main paper and ignoring the normalization factors, the objective $\rho_m$ for $m$ views can be rewritten as: $$\begin{aligned}
\rho_m = \frac{\text{Tr}(\*W^{\top}{\*R}_B\*W)}{\text{Tr}(\*W^{\top}{\*R}_W\*W)}
= \frac{\<{\*R_B} + \*{\Sigma}_B - \*{\Sigma}_B, \*W\*W^{\top}\>}{\<{\*R}_W + \*{\Sigma}_W - \*{\Sigma}_W, \*W\*W^{\top}\>}\\
\leq \frac{\<{\*{\Sigma}}_B, \*W\*W^{\top}\> + \norm{{\*R}_T - \*{\Sigma}_T} + \norm{{\*R}_W - \*{\Sigma}_W}}{\<{\*{\Sigma}}_W, \*W\*W^{\top}\> - \norm{{\*R}_W - \*{\Sigma}_W}}
$$ where $\*{\Sigma}_b$ and $\*{\Sigma}_w$ are the second moment matrices for the the between-view and within-view covariances respectively. This can be written using cyclical properties of trace function and relation between spectral norm and trace. Additionally note from the previous result that we can use total covariance to simplify the estimation of $\*R_B$. That is, $\*R_B = \*R_T-\*R_W$. The rest follows through triangular inequalities.
Observe that the ratio $\<{\*{\Sigma}}_B, \*W\*W^{\top}\>/\<{\*{\Sigma}}_W, \*W\*W^{\top}\>$ is the optimal $\rho^*$ we are interested to bound the approximation $\rho_m$ from. We can show that $\abs*{\rho}\leq 1$. Additionally the two trace terms are sum of normalized eigen values (each bounded above by 1). Thus $\<{\*{\Sigma}}_B, \*W\*W^{\top}\>\in[1,d]$ and $\<{\*{\Sigma}}_W, \*W\*W^{\top}\>\in[1,d]$. Furthermore, from lemma \[lemma:rw\], we know that the norm term with $\*R_W$ is greater than 1 i.e., $\norm{\*R_T - \*\Sigma_W}\leq C\frac{d}{m} > 1$, because we always choose the embedding size to be greater than the number of views subsampled. With these inequalities. We can loosely bound the above inequality for $\rho_m$ as:
$$\begin{aligned}
\rho_m &\leq \frac{\<{\*{\Sigma}}_B, \*W\*W^{\top}\>}{\<{\*{\Sigma}}_W, \*W\*W^{\top}\>}\frac{ \norm{{\*R}_T - \*{\Sigma}_T} + \norm{{\*R}_W - \*{\Sigma}_W}}{ \norm{{\*R}_W - \*{\Sigma}_W}}\\
&\leq \rho^{*}\frac{ \norm{{\*R}_T - \*{\Sigma}_T} }{ \norm{{\*R}_W - \*{\Sigma}_W}} \leq \rho^*\frac{2Nm}{NC\frac{(\sqrt{d}+t)^2}{m}} &&\text{[From Lemmas \ref{lemma:rw} and \ref{sec-lemma:rt}]}\\
&\leq C'\rho^*\frac{m^2}{d} \approx \mathcal{O}(\frac{m^2}{d}) \\\end{aligned}$$
where $C'$ is a constant term that depends only the subgaussian norm of the $d$-dimensional feature vectors.
[^1]: 3D object dataset and leader-board:\[note1\]
[^2]: TensorFlow 2.0:
[^3]: Clustering accuracy estimated with Kuhn’s Hungarian method\[kuhn\]
[^4]: Significance testing using Mann-Whitney U test at $\alpha=0.05$
[^5]: Deep-CCA code:
[^6]: CrossGrad and DAN code:
[^7]: Permutation test $n=10^5$, $p = 0.008$
|
{
"pile_set_name": "ArXiv"
}
|
---
address:
- |
Department of Mathematics\
MIT\
Cambridge, MA 02139
- |
Department of Mathematics\
Harvard University\
Cambridge, MA 02140
author:
- 'V. Guillemin'
- 'S. Sternberg'
title: Riemann sums over polytopes
---
Introduction {#sec:1}
============
Given a ${\mathcal{C}^\infty}$ function, $f$, on the interval $[0,1]$ let $R_N (f)$ be the Riemann sum
$$\label{eq:1.1}
\frac{1}{N}\sum^N_{i=1} f(t_i) \, , \quad
\frac{i}{N}\leq t_i < \frac{i+1}{N}\, .$$
In freshman calculus one learns that
$$\label{eq:1.2}
R_N (f) = \int^1_0 f(x) \, dx + O \left( \frac{1}{N}\right)\, .$$
What is not perhaps as well known is that if one chooses the $t_i$’s judiciously, i.e., lets $t_i = \frac{i}{N}$ the $O \left(
\frac{1}{N}\right)$ in (\[eq:1.2\]) can be replaced by a much better error term, an asymptotic series: $$\label{eq:1.3}
\frac{1}{2N} \left(f(1) - f(0)\right) + \sum^{\infty}_{k=1} (-1)^{k-1}
\frac{B_k}{(2k)!} \left(f^{\langle 2k-1 \rangle} (1)-
f^{{\langle 2k-1 \rangle}}(0) \right) N^{-2k}$$ in which the $B_k$’s are the Bernoulli numbers. In particular if $f$ is periodic of period 1 the $O \left( \frac{1}{N}\right)$ in (\[eq:1.2\]) is actually an $O (N^{-\infty})$. (For an expository account of this “Euler–Maclaurin formula for Riemann sums” see [@GS].)
In this article we will prove an $n$-dimensional version of this result in which the interval $[0,1]$ gets replaced by a convex polytope. We will give a precise formulation of our result in §\[sec:4\]; however, roughly speaking, it asserts that if $\Delta$ is a simple convex polytope whose vertices lie on the lattice, ${{\mathbb Z}}^n$, and if $f$ is in ${\mathcal{C}^\infty}(\Delta )$ the difference $$\label{eq:1.4}
\int_{\Delta} f(x) \, dx -\frac1{N^n}\sum_{k \in N \Delta \cap {{\mathbb Z}}^n}
f \left( \frac{k}{N}\right)$$ can be expanded in an asymptotic series in $N^{-1}$ in which the coefficients are explicitly computable by recipes resembling (\[eq:1.3\]). Our formula bears a formal resemblance to the generalized Euler–Maclaurin formulas of [@KP], [@KK], [@CS], [@Gu], [@BV] et al., however in these so-called “exact” Euler–Maclaurin formulas the functions involved are polynomials, not as in the case here, arbitrary ${\mathcal{C}^\infty}$ functions. Somewhat closer in spirit to our result is the Euler–Maclaurin formula with remainder of [@KSW] and the Ehrhard theorem for symbols of [@GSW]. (Our result also yields an Ehrhard theorem for symbols, and its relation to the theorem in [@GSW] will be discussed in § \[sec:4\].)
A word about the organization of this paper. In § \[sec:2\] we will review the proof of the Riemann sum version of Euler–Maclaurin, for the interval, $(-\infty , 0]$ and in §\[sec:3\] show how to extend this result to regions in ${{\mathbb R}}^n$ which are defined by systems of $k$ linearly-independent inequalities
$$\label{eq:1.5}
\langle u_i , x \rangle \leq c_i \, , \quad
u_i \in {{\mathbb Z}}^n \, , \quad
c_i \in {{\mathbb Z}}\, .$$
(We will call such regions *$k-wedges$*.)
In §\[sec:4\] we will derive from this result a Euler–Maclaurin formula for Riemann sums over polytopes and in §\[sec:5\] show that our result has an equivalent formulation as an Ehrhard theorem for symbols.
We would like to thank Dan Stroock and Hans Duistermaat for helpful discussions concerning the material in Section \[sec:2\].
Euler–Maclaurin for the interval $(-\infty , 0]$ {#sec:2}
================================================
Let $\tau (s)$ be the Todd function
$$\label{eq:2.1}
\frac{s}{1-e^{-s}} = 1 + \frac{s}{2} + \sum (-1)^{n-1}
B_n \frac{s^{2n}}{(2n)!}\, .$$
In this section we will show that for Schwartz functions, $f \in
S({{\mathbb R}})$ the difference
$$\label{eq:2.2}
\frac{1}{N} \sum^{\infty}_{k=0} f \left( -\frac{k}{N}\right)
- \int^0_{-\infty} f(x) \, dx$$
has an asymptotic expansion:
$$\label{eq:2.3}
\frac{f(0)}{2N} + \sum^{\infty}_{n=1} (-1)^{n-1}
\frac{B_n}{(2n)!} f^{(2n-1)}(0) N^{-2n} \, .$$
In view of (\[eq:2.1\]) this formula can be written more succinctly in the form $$\label{eq:2.4}
\frac{1}{N} \sum^{\infty}_{k=0} f \left( -\frac{k}{N}\right)
\sim \left( \tau \left(\frac{1}{N}\, \frac{\partial}{\partial h}\right)
\int^h_{-\infty} f(x) \, dx \right) (h=0)$$ and it is this version of it which we will prove.
We first of all observe that if $f(x) = e^{\lambda x}$, $\lambda >0$, then
$$\int^{h}_{-\infty} f(x) \, dx = \frac{1}{\lambda} e^{\lambda h}\,.$$
So for $N>2\pi \lambda$ we may apply the infinite order constant coefficient operator $ \tau \left( \frac{1}{N} \, \frac{\partial}{\partial h}\right)$ to this expression: $$\begin{aligned}
\tau \left( \frac{1}{N} \, \frac{\partial}{\partial h}\right)
\int^h_{-\infty} f(x) \, dx
&=& \tau \left( \frac{1}{N} \, \frac{\partial}{\partial h}\right)
\, \frac{e^{\lambda h}}{\lambda}\\[1ex]
&=& \tau \left( \frac{\lambda}{N} \right) \,
\frac{e^{\lambda h}}{\lambda}\\[1ex]
&=& \frac{1}{N} \, \frac{\lambda}{1-e^{-\lambda /N}} \,
\frac{e^{\lambda h}}{\lambda} \\[1ex]
&=& \frac{1}{N} \left( \sum^{\infty}_{k=0}
e^{-\frac{k}{N}\lambda} \right) e^{\lambda h},\end{aligned}$$ all series being convergent. We conclude that $$\label{eq:2.5}
\frac{1}{N} \sum^{\infty}_{k=0} e^{-\frac{k}{N}\lambda}
= \left( \tau \left( \frac{1}{N}\, \frac{\partial}{\partial h}
\right) \int^{h}_{-\infty} e^{\lambda x}\, dx \right)
(h=0)\, .$$ More generally differentiating this identity $n$ times with respect to $\lambda$ we obtain $$\label{eq:2.6}
\frac{1}{N} \sum^{\infty}_{k=0} \left(-\frac{k}{N}\right)^{n}
e^{-\frac{k}{N}\lambda} = \left( \tau \left( \frac{1}{N}
\, \frac{\partial}{\partial h} \right) \int^h_{-\infty}
x^n e^{\lambda x}\, dx \right) (h=0)$$ verifying (\[eq:2.4\]) for the function $x^n e^{\lambda x}$ and hence for the functions of the form $p(x) e^{\lambda x}$ where $p$ is a polynomial. Now let $f$ be a Schwartz function and $p$ a polynomial having the property that $f(x) - p(x) e^{\lambda x}$ vanishes to order $n+2$ at $x=0$. Let
$$\label{eq:2.7}
g(x) =
\begin{cases}
0 \, , & x \geq 0 \\
f(x) -p(x) e^{\lambda k} \, , & x<0 \, .
\end{cases}$$
Then
$$\label{eq:2.8}
\| g^{(i)} (x) \|_1 \leq \infty \hbox{ for } i \leq n+2$$
and by the Poisson summation formula $$\label{eq:2.9}
\sum_{-\infty <k<\infty} g \left(- \frac{k}{N}\right)
= N \sum_{-\infty <k<\infty}\hat{g} (Nk) \, .$$ However, by (\[eq:2.8\]) $$\label{eq:2.10}
| \hat{g} (Nk) | \leq \operatorname{Const.}N^{-n}k^{-2}$$ for $k \neq 0$, and $$\label{eq:2.11}
\hat{g} (0) = \int^0_{-\infty} g (x) \, dx \, .$$
Hence
$$\label{eq:2.12}
\frac{1}{N} \sum^{\infty}_{k=0} g \left( -\frac{k}{N}\right)
= \int^0_{-\infty} g(x) \, dx +O(N^{-n}) \, .$$
This shows that (\[eq:2.4\]) is true for $g$ modulo $O(N^{-n})$ and hence is true for $f$ modulo $O (N^{-\infty})$.
Q.E.D
In §\[sec:3\] we will also need a version of the theorem above for “twisted” Riemann sums. Let $\omega\neq 1$ be a $q$[$^{\scriptstyle \textrm{th}}$]{} root of unity and let
$$\tau_{\omega} (s) = \frac{s}{1-\omega e^{-s}}
= \frac{s}{1-\omega} + \sum_{i>1} b^{\omega}_i s^i \, .$$
For $f \in S({{\mathbb R}})$ we will show that the twisted Riemann sum
$$\label{eq:2.13}
\frac{1}{N} \sum^{\infty}_{k=0} \omega^k f \left( -\frac{k}{N}\right)$$
is asymptotic to the series
$$\label{eq:2.14}
\frac{1}{1-\omega} \frac{f(0)}{N} + \sum_{i>1} b^{\omega}_i
f^{(i)} (0) N^{-i}\, .$$
As above we can rewrite this in the more succinct form
$$\label{eq:2.15}
\frac{1}{N} \sum^{\infty}_{k=0} \omega^k f
\left(-\frac{k}{N}\right) \sim
\left( \tau_{\omega}\left( \frac{1}{N}\,
\frac{\partial}{\partial h}\right) \int^h_{-\infty}
f(x) \, dx \right) (h=0)$$
and we will prove this by essentially the same proof as before: If $f = e^{\lambda x}$ the expression in parentheses is
$$\begin{aligned}
\label{eq:2.16}
\tau_{\omega} \left( \frac{\lambda}{N}\right)
\frac{e^{\lambda h}}{\lambda }
&=& \frac{1}{N}
\left( \frac{\lambda}{1-\omega e^{-\lambda /N}}\right)
\frac{e^{\lambda h}}{\lambda}\\[1ex]\notag
&=& \frac{1}{N} \left( \sum^{-\infty}_{k=0} \omega^k
e^{-k \lambda /N} \right) e^{\lambda h } \, ,\notag\end{aligned}$$
and by setting $h=0$ we see that (\[eq:2.15\]) is valid for $f=e^{\lambda x}$; and by differentiating both sides of (\[eq:2.16\]) by $\left( \frac{d}{d \lambda}\right)^n$ that it’s valid for $x^n e^{\lambda x}$ and hence for $p(x) e^{\lambda
x}$ where $p (x)$ is a polynomial. Thus, as above, we’re reduced to showing that for the function $g$ defined by (\[eq:2.7\]):
$$\label{eq:2.17}
\frac{1}{N}\sum_{-\infty <k<\infty} \omega^k g
\left( \frac{k}{N} \right) =O (N^{-n})\, .$$
For $r=0,1,\ldots,q-1$, let $g_r (x) = g(qx+ \frac{r}{N})$. Then
$$\label{eq:2.18}
\frac{1}{N} \sum_{-\infty < k < \infty} \omega^k g
\left( \frac{k}{N}\right) = \frac{1}{N} \sum^{q-1}_{r=0}
\omega^r \left( \sum_{-\infty < k < \infty} g_r
\left( \frac{k}{N}\right) \right) \, .$$
Since
$$\hat{g}_r (Nk) = \frac{1}{q} \, e^{i\frac{rk}{q}} \hat{g}
\left(\frac{Nk}{q}\right)$$
the Poisson summation formula yields, as before, the estimate
$$\label{eq:2.19}
\sum^{q-1}_{r=0} \omega^r \int^{\infty}_{-\infty}
g_r (x) \, dx + O (N^{-n})\, ,$$
for the right hand side of (\[eq:2.18\]). However,
$$\int^{\infty}_{-\infty} g_r (x) \, dx
= \int^{\infty}_{-\infty} g_0 (x) \, dx$$
and $\sum^{q-1}_{r=0} \omega^r =0$ so the first summand in (\[eq:2.18\]) is zero. Q.E.D.
We will conclude this discussion of one dimensional Euler–Maclaurin formulas by describing analogues of (\[eq:2.4\]) and (\[eq:2.15\]) in which the sum over $-\infty
< k<0$ gets replaced by a sum over $-\infty < k < \infty$. For simplicity assume that $f \in {\mathcal{C}^\infty}_0 ({{\mathbb R}})$. We claim:
$$\label{eq:2.20}
\frac{1}{N} \sum^{\infty}_{k=-\infty} f
\left( \frac{k}{N}\right) = \int^{\infty}_{-\infty}
f(x) \, dx + O (N^{-\infty})$$
and, for $\omega$ a $q$[$^{\scriptstyle \textrm{th}}$]{} root of unity, $\omega\neq 1$, $$\label{eq:2.21}
\sum^{\infty}_{k=-\infty} \omega^k f \left( \frac{k}{N}\right)
= O (N^{-\infty})\, .$$ To prove (\[eq:2.20\]) we first observe that for $c$ a large positive integer, the left and right hand sides of (\[eq:2.20\]) are unchanged if one substitutes the function, $f
(x+c)$, for $f$, so without loss of generality we can assume that $f$ is supported on the interval, $x<0$, in which case (\[eq:2.3\]) is of order $O (N^{-\infty})$ and (\[eq:2.20\]) is a consequence of (\[eq:2.4\]). Similarly if we replace $f(x)$ by $f(x+cq)$, with $c$ a large positive integer, the left and right hand sides of (\[eq:2.21\]) are unchanged; so we can assume that $f$ is supported on the interval $x<0$, and (\[eq:2.21\]) is a consequence of (\[eq:2.15\]).
Euler–Maclaurin for wedges {#sec:3}
==========================
Let ${{\mathbb Z}}^n$ be the integer lattice in ${{\mathbb R}}^n$, $({{\mathbb Z}}^n)^*$ its dual lattice in $({{\mathbb R}}^n)^*$ and $\langle u,x \rangle$ the usual paring of vectors, $x \in {{\mathbb R}}^n$, and $u \in ({{\mathbb R}}^n)^*$. Given $m$ linearly independent vectors, $u_i \in ({{\mathbb R}}^n)^*$ we will call the subset of ${{\mathbb R}}^n$ defined by the inequalities $$\label{eq:3.1}
\langle u_i,x \rangle \leq c_i \qquad i=1,\ldots ,m$$ an *integer $m$-wedge* if the $c_i$’s are integers and the $u_i$’s primitive lattice vectors in $({{\mathbb Z}}^n)^*$. Let $W$ be the set (\[eq:3.1\]) and $U$ the subspace of $({{\mathbb R}}^n)^*$ spanned by the $u_i$’s. We will call $W$ a *regular* integer $m$-wedge if $u_1,\ldots ,u_m$ is a lattice basis of the lattice $U \cap
({{\mathbb Z}}^n)^*$ i.e., if $$\label{eq:3.2}
U \cap ({{\mathbb Z}}^n)^* = {\mbox{\textrm span}}_{{{\mathbb Z}}} \{ u_1,\ldots ,u_m \}\, .$$ We will need below the following criterion for regularity.
\[lem:3.1\] If (\[eq:3.2\]) holds, $u_i,\ldots ,u_m$ can be extended to a lattice basis, $u_1 ,\ldots ,u_n$ of $({{\mathbb Z}}^n)^*$.
Let $u_{m+1},\ldots ,u_n$ be vectors in $({{\mathbb Z}}^n)^*$ whose projections onto the quotient of $({{\mathbb Z}}^n)^*$by $U \cap ({{\mathbb Z}})^*$ are a lattice basis of this quotient.
For an integer $m$-wedge satisfying (\[eq:3.2\]) the $n$-dimensional generalization of Euler–Maclaurin is relatively straightforward.
\[th:3.2\] Let $W_h$ be the subset of ${{\mathbb R}}^n$ defined by the inequalities $$\label{eq:3.3}
\langle u_i,x \rangle \leq c_i + h_i \, , \quad
i=1,\ldots ,m \, .$$
Then, for $f \in {\mathcal{C}^\infty}_0 ({{\mathbb R}}^n)$, $$\label{eq:3.4}
\frac{1}{N^n} \sum_{k \in {{\mathbb Z}}^n \cap NW} f
\left(\frac{k}{N}\right) \sim \left( \tau \left(
\frac{1}{N}\, \frac{\partial}{\partial h}\right)
\int_{W_h} f(x) \, dx \right) (h=0)$$ where $\tau (s_1 ,\ldots ,s_m) = \prod^m_{i=1} \tau (s_i)$.
By Lemma \[lem:3.1\] we can incorporate $u_1,\ldots ,u_m$ in a lattice basis $u_1 ,\ldots ,u_n$ of $({{\mathbb Z}}^n)^*$. Let $\alpha_1,\ldots ,\alpha_n$ be the dual basis of ${{\mathbb Z}}^n$ and let $v = \sum^m_{i=1}c_i\alpha_i$. Then via the map $$\label{eq:3.5}
x \in {{\mathbb R}}^n \to \sum x_i \alpha_i +v$$ one is reduced to proving the theorem for the standard $m$-wedge: $x_1 \leq 0 , \ldots , x_m \leq 0$, i.e., showing that the sum $$\label{eq:3.6}
\frac{1}{N^n} \sum f \left( \frac{k_1}{N},\cdots ,
\frac{k_n}{N}\right)$$ summed over all $(k_1, \ldots ,k_n) \in {{\mathbb Z}}^n$, with $k_i \leq 0$ for $i \leq m$, is equal to the expression $$\label{eq:3.7}
\tau \left( \frac{1}{N}\, \frac{\partial}{\partial h}\right)
\int^{h_1}_{-\infty} \cdots \int^{h_m}_{-\infty}
dx_1 \ldots dx_m \int^{\infty}_{-\infty} \cdots
\int^{\infty}_{-\infty} f(x) \,
dx_{m+1} \cdots dx_{n}\, ,$$ evaluated at $h=0$, modulo $O (N^{-\infty})$. Moreover, since the subalgebra of ${\mathcal{C}^\infty}_0 ({{\mathbb R}}^n)$ generated by the products $$f(x) = f_1(x_1) \ldots f_n (x_n) \, , \quad f_i \in {\mathcal{C}^\infty}_0 ({{\mathbb R}})\, ,$$ is dense in ${\mathcal{C}^\infty}_0 ({{\mathbb R}}^n)$ it suffices to prove the theorem for functions of this form, and hence it suffices to prove the theorem for $n=1$ and $m=0$ or $1$. However, these two cases were dealt with in §\[sec:2\]. (See (\[eq:2.4\]) and (\[eq:2.20\]).)
We will next describe how (\[eq:3.4\]) has to be modified if the condition (\[eq:3.2\]) isn’t satisfied. As above let $u_{m+1},\ldots , u_n$ be vectors in $({{\mathbb Z}}^n)^*$ whose projections onto the quotient of $({{\mathbb Z}}^n)^*$ by $U \cap ({{\mathbb Z}}^n)^*$ are a lattice basis of this quotient lattice. The vectors, $u_1,\ldots ,u_n$ are now no longer a lattice basis of $({{\mathbb Z}}^n)^*$ but they span a sublattice
$$\label{eq:3.8}
{{\mathbb A}}^* = {\mbox{\textrm span}}_{{{\mathbb Z}}} \{ u_1,\ldots, u_n \}$$
of $({{\mathbb Z}}^n)^*$ of rank $n$, so the quotient $$\label{eq:3.9}
\Gamma = ({{\mathbb Z}}^n)^* /{{\mathbb A}}^*$$ is a finite group. Let $\alpha_1,\ldots ,\alpha_n$ be the basis vectors of ${{\mathbb R}}^n$ dual to $u_1,\ldots ,u_n$. Since ${{\mathbb A}}^*$ is a sublattice of $({{\mathbb Z}}^n)^*$ the dual lattice, $$\label{eq:3.10}
{{\mathbb A}}= {\mbox{\textrm span}}_{{{\mathbb Z}}} \{ \alpha_1 ,\ldots , \alpha_n \}\, ,$$ contains ${{\mathbb Z}}^n$ as a sublattice. Moreover, each element, $x \in
{{\mathbb A}}$, defines a character of the group, $\Gamma$, via the pairing $$\label{eq:3.11}
\gamma \in \Gamma \to e^{2\pi i \langle \gamma , x \rangle}$$ and this character is trivial if and only if $x$ is in ${{\mathbb Z}}^n$. By a theorem of Frobenius the average value of a character of a finite group is zero if the character is non-trivial and is one if it is trivial, so we have $$\label{eq:3.12}
\frac{1}{|\Gamma |} \sum e^{2\pi i \langle \gamma ,x \rangle}=
\begin{cases}
1 \hbox{ if } x \in {{\mathbb Z}}^n\\
0 \hbox{ if } x \notin {{\mathbb Z}}^n \, .
\end{cases}$$ For each $\gamma \in \Gamma$ let $$\label{eq:3.13}
\tau_{\gamma} (s_1,\ldots ,s_m)
= \tau_{\omega_1}(s_1) \cdots \tau_{\omega_m} (s_m)$$ where $\omega_k = e^{2\pi i \langle \gamma ,\alpha_{k} \rangle}$. We will generalize Theorem \[th:3.2\] by showing that for integer $m$-wedges which don’t satisfy condition (\[eq:3.2\]) one has
\[th:3.3\] For $f \in {\mathcal{C}^\infty}_0 ({{\mathbb R}}^n)$ $$\label{eq:3.14}
\frac{1}{N^n} \sum_{k \in N W \cap {{\mathbb Z}}^n}
f \left( \frac{k}{N}\right) = \left( \sum_{\gamma \in \Gamma}
\tau_{\gamma} \left( \frac{1}{N}\, \frac{\partial}{\partial h}
\right)
\int_{W_h} f(x) \, dx \right) (h=0) \mod O (N^{-\infty})\, .$$
By (\[eq:3.11\]) the sum on the left coincides with the sum $$\label{eq:3.15}
\frac{1}{|\Gamma |} \sum_{\gamma \in \Gamma} \frac{1}{N^n}
\sum_{x \in {{\mathbb A}}\cap NW} e^{2\pi i \langle \gamma ,x \rangle}
f \left( \frac{x}{N}\right)$$ so it suffices to show that the $\gamma$-[$^{\scriptstyle \textrm{th}}$]{} summand in (\[eq:3.14\]) is equal to the $\gamma$-[$^{\scriptstyle \textrm{th}}$]{} summand in (\[eq:3.15\]). Via the map (\[eq:3.5\]) the $\gamma$-[$^{\scriptstyle \textrm{th}}$]{} summand in (\[eq:3.15\]) becomes $$\label{eq:3.16}
\frac{1}{N^n |\Gamma |} \sum_{k_1 \leq 0 ,\ldots ,k_m \leq 0}
\omega_1^{k_1} \ldots \omega^{k_m}_m \left(
\sum_{k_{m+1, \ldots ,k_n}} g \left( \frac{k}{N}\right)\right)$$ where $g (x_1,\ldots ,x_n) = f (v + x_1\alpha_1 + \cdots +
x_n\alpha_n)$, and the $\gamma$-[$^{\scriptstyle \textrm{th}}$]{} summand in (\[eq:3.14\]) becomes
$$\label{eq:3.17}
\frac{1}{|\Gamma |}\, \tau_{\gamma}
\left( \frac{1}{N}\, \frac{\partial}{\partial h}\right)
\int^{h_1}_{-\infty} \ldots \int^{h_m}_{-\infty}
dx_1 \ldots dx_m \int^{\infty}_{-\infty}\ldots
\int^{\infty}_{-\infty} g(x) dx_{m+1}\ldots dx_k$$
evaluated at $h=0$. (The reason for the factor, $1/|\Gamma |$, is that this is the Jacobian determinant of the mapping (\[eq:3.5\]).) To prove that (\[eq:3.16\]) and (\[eq:3.17\]) are equal mod $O (N^{-\infty})$ it suffices as above to prove this for functions of the form $g=g_1(x) \ldots g_n(x_n)$ with $g_i \in {\mathcal{C}^\infty}_0 ({{\mathbb R}})$ and hence to show, for $i \leq m$ $$\label{eq:3.18}
\frac{1}{N} \sum^{-\infty}_{k_i=0}\omega^{k_i} g_i
\left( \frac{k_i}{N} \right) \sim \left( \tau_{\omega_i}
\left( \frac{1}{N}\, \frac{\partial}{\partial h_i} \right)
\int^{h_i}_{-\infty} g_i (x_i)\, dx_i \right)(h_i=0)$$ and, for $i >m$ $$\label{eq:3.19}
\frac{1}{N} \sum^{\infty}_{-\infty} g_i
\left( \frac{k_i}{N}\right) = \int^{\infty}_{-\infty} g_i
(x_i) \, dx_i + O(N^{-\infty})\, ,$$ and these follow from the identities (\[eq:2.15\]) and (\[eq:2.20\]).
Riemann sums over polytopes {#sec:4}
===========================
Let $\Delta \subseteq {{\mathbb R}}^n$ be an $n$-dimensional polytope whose verticies lie on the lattice ${{\mathbb Z}}^n$. $\Delta$ is said to be a simple polytope if each codimension $k$ face is the intersection of exactly $k$ facets. (It suffices to assume that the vertices of $\Delta$, i.e., the codimension $n$ faces, have this property or, alternatively, that there are exactly $n$ edges of $\Delta$ meeting at each vertex.) If the number of facets is $d$ then $\Delta$ can be defined by a set of $d$ inequalities $$\label{eq:4.1}
\langle u_i,x \rangle \leq c_i$$ where $c_i$ is an integer and $u_i \in ({{\mathbb Z}}^n)^*$ is a primitive lattice vector which is perpendicular to the $i$[$^{\scriptstyle \textrm{th}}$]{} facet and points “ outward” from $\Delta$. By the simplicity assumption each codimension $k$ face of $\Delta$ is the intersection of $k$ facets lying in the hyperplanes $$\label{eq:4.2}
\langle u_i,x \rangle = c_i\, \quad i \in F$$ where $F$ is a $k$ element subset of $\{ 1,\ldots ,d \}$. Let $W_F$ be the $k$-wedge $$\label{eq:4.3}
\langle u_i,x \rangle \leq c_i\, \quad i \in F\, .$$
We will say that $\Delta$ is regular if each of these $k$-wedges is regular. (As above it suffices to assume this for the zero faces, i.e., the vertices of $\Delta$, or alternatively to assume that for every vertex, $v$, the edges of $\Delta$ which intersect at $v$ lie on rays $$v + t \alpha_i \, , \quad 0 \leq t <\infty$$ where $\alpha_1 ,\ldots ,\alpha_n$ is a lattice basis of ${{\mathbb Z}}^n$.)
For regular simple lattice polytopes one has the following Euler–Maclaurin formula.
\[th:4.1\] Let $\Delta_h$ be the polytope $$\label{eq:4.4}
\langle u_i,x \rangle \leq c_i + h_i \quad i=1,\ldots ,d \, .$$ Then, for $f \in {\mathcal{C}^\infty}_0 ({{\mathbb R}}^n)$ $$\label{eq:4.5}
\frac{1}{N^n} \sum_{k \in {{\mathbb Z}}^n \cap N\Delta} f
\left( \frac{k}{N}\right) \sim \left( \tau \left(
\frac{1}{N} \, \frac{\partial}{\partial h}\right)
\int_{\Delta_h} f(x) \, dx \right) (h=0)$$ where $\tau (s_1,\ldots ,s_d) = \tau (s_1) \ldots \tau (s_d)$.
By a partition of unity argument we can assume that ${\mbox{\textrm supp}}f$ is contained in a small neighborhood of the set (\[eq:4.2\]) and doesn’t intersect the hyperplanes, $ \langle u_i,x \rangle =
c_i$, $i \notin F$. Then for $i \notin F$ $$\tau \left( \frac{1}{N} \, \frac{\partial}{\partial h_i}\right)
\int_{\Delta_h} f \, dx = \int_{\Delta_h} f \, dx
+ \frac{1}{2N} \, \frac{\partial}{\partial h_i}
\int_{\Delta_h} f(x) \, dx + \cdots \, .$$
However, by (\[eq:2.4\]) all the terms on the right except the first are integrals of derivatives of $f$ over the hyperplane $ \langle u_i,x \rangle =
c_i +h_i$, and hence for $h_{i}$ small are zero. Thus the left hand side of (\[eq:4.5\]) becomes $$\left( \prod_{i \in F} \tau \left( \frac{1}{N}\,
\frac{\partial}{\partial h_i} \right) \int_{(W_F)_h}
f(x) \, dx \right) (h=0)$$ and the theorem above reduces to Theorem \[th:3.2\]. Q.E.D.
If $\Delta$ is simple but not regular, one gets a slightly more complicated result. To the codimension $k$-face of $\Delta$ defined by (\[eq:4.2\]) attach the subspace $$U_F = {\mbox{\textrm span}}_{{{\mathbb R}}} \{ u_i \, , \quad i \in F \}$$ of $({{\mathbb R}}^n)^*$, the sublattice $${{\mathbb Z}}_{\Gamma} = {\mbox{\textrm span}}_{{{\mathbb Z}}} \{ u_i \, , \quad i \in F \}$$ and the finite group $$\Gamma_F = U_F \cap ({{\mathbb Z}}^n)^* /{{\mathbb Z}}_F \, .$$ This group coincides with the “torsion group” (\[eq:3.9\]) of the wedge $W_F$. Moreover, if $E$ is a subset of $F$, $U_E$ is contained in $U_F$ and ${{\mathbb Z}}_E$ in ${{\mathbb Z}}_F$, so $\Gamma_E$ is contained in $\Gamma_F$. Let $\Gamma^{\sharp}_F$ be the set of points in $\Gamma_F$ which are *not* contained in $\Gamma_E$ for some proper subset, $E$ of $F$.
\[th:4.2\] For $f \in {\mathcal{C}^\infty}_0 ({{\mathbb R}}^n)$ the sum $$\label{eq:4.6}
\frac1{N^n} \sum_{k \in {{\mathbb Z}}^n \cap N \Delta} f\left( \frac{k}{N}\right)$$ is equal $\mod O (N^{-\infty})$ to $$\label{eq:4.7}
\left( \sum_F \sum_{\gamma \in \Gamma^{\sharp}_F} \tau_{\gamma}\left(
\frac{1}{N} \, \frac{\partial}{\partial h} \right)
\int_{\Delta_h} f(x) \, dx \right) (h=0)\, .$$
As above it suffices to prove this for ${\mbox{\textrm supp}}f$ contained in a small neighborhood of the set (\[eq:4.2\]), and not intersecting the hyperplanes, $\langle u_i,x \rangle =c_i$, $i \notin F$. Then as above, the only contribution to the sum (\[eq:4.7\]) is $$\left( \sum_{\gamma \in \Gamma_F} \tau_{\gamma} \left(
\frac{1}{N} \, \frac{\partial}{\partial h} \right)
\int_{(W_F)_h} f(x) \, dx \right) (h=0)$$ and Theorem \[th:4.2\] reduces to Theorem \[th:3.3\].
An Ehrhart theorem for symbols {#sec:5}
==============================
A function, $f \in {\mathcal{C}^\infty}({{\mathbb R}}^n)$ is a *polyhomogeneous symbol* of degree $d$ if, for large values of $x$, it admits an asymptotic expansion $$\label{eq:5.1}
f(x) \sim \sum^{-\infty}_{j=d} f_j (x)$$ whose summands are homogeneous functions $f_j \in {\mathcal{C}^\infty}({{\mathbb R}}^n -
\{ 0 \})$ of degree $j$. Let $f$ be such a function and let $\Delta $ be a simple lattice polytope in ${{\mathbb R}}^n$ containing the origin in its interior. The Ehrhart function of the pair, $f$, $\Delta$, is defined to be the function $$E (f,\Delta , N) = \sum_{k \in N \Delta \cap {{\mathbb Z}}^n} f(k) \, ,
\quad N \in {{\mathbb Z}}_+ \, .$$ In [@GSW] it was shown that $$E (f,\Delta , N) -\int_{N\Delta}fdx$$ had an asymptotic expansion $$\label{eq:5.2}
\sum^{-\infty}_{j=n+d} c_j N^j + c$$ for $N$ large.
The main result of this section is a variant of this result. As above let $\Delta$ be a simple lattice polytope in ${{\mathbb R}}^n$ and let $C_{\Delta}$ be the polyhedral cone consisting of all points, $(x_1,\ldots ,x_n, x_{n+1})$, in ${{\mathbb R}}^{n+1}$ with $x_{n+1} >0$ and $(x_1 ,\ldots ,x_n)/x_{n+1} \in
\Delta$. Then, for $N \in {{\mathbb Z}}_+$, $N \Delta$ is just the slice of $C_{\Delta}$ by the hyperplane, $x_{n+1}=N$. We will prove that if $f \in {\mathcal{C}^\infty}({{\mathbb R}}^{n+1})$ is a homogeneous symbol of degree $d$ the sum $$\label{eq:5.3}
\sum_{k \in N\Delta \cap {{\mathbb Z}}^n} f(k)$$ admits an asymptotic expression of the form (\[eq:5.2\]).
1. This result, albeit very close in spirit to the theorem in [@GSW] cited above, doesn’t, as far as we can see, seem to be a trivial consequence of it.
2. This result has a number of applications to spectral theory on toric varieties which we’ll explore in future publications.
3. As a corollary of this result one gets another variant of the Ehrhart theorem: Let $$\Delta^\sharp :=\{(x_1,\dots, x_{n+1})\in C_\Delta,\ \ x_{n+1}\leq 1\}.$$ This $(n+1)$-dimensional polytope is [*not*]{} in general simple. However a version of theorem described at the beginning of this section is still true, namely $$E(f,\Delta^\sharp,N)\sim \sum_{i=d+n+1}^{-\infty} c_i^\sharp + c^\sharp \log N.$$ as one can see by summing the differences $$E(f,\Delta^\sharp,N)-E(f,\Delta^\sharp,N-1)$$ and noting that each difference is exactly (\[eq:5.3\]). By combining this result with the Danilov “desingularization trick" [@Da] one can extend the Ehrhart theorem to a much larger class of convex lattice polytopes. We will discuss the details elsewhere.
As above let $$f \sim \sum^{-\infty}_{i=d} f_i$$ where $f_i (x_1,\ldots ,x_{n+1})$ is a homogeneous function of degree $i$. Then on the cone, $C_{\Delta}$: $$f_i (x_1 ,\ldots ,x_{n+1}) = x^i_{n+1} \, f_i \left(
\frac{x_1}{x_{n+1}}\, , \cdots \, ,
\frac{x_n}{x_{n+1}} \, , \, 1 \right)$$ so if we set $\tilde{f}_i (x_1,\ldots ,x_n) = f_i (x_1,\ldots
,x_n,1)$ the sum (\[eq:5.3\]) is equal to the sum $$\label{eq:5.4}
N^i \sum_{k \in N \Delta \cap {{\mathbb Z}}^n}
\tilde{f}_i \left( \frac{k}{N}\right)\, .$$ which is $N^{i+n}$ times the Riemann sum $$\label{eq:5.5}
\frac{1}{N^n}\sum_{k \in N \Delta \cap {{\mathbb Z}}^n}
\tilde{f}_i \left( \frac{k}{N}\right)\, .$$ Thus, by Theorem \[th:4.2\], each of these summands admits an asymptotic expansion: $$\sum^{-\infty}_{k=n+i} c_{i,k} N^k$$ and hence so does the sum (\[eq:5.3\]).
[99]{} M. Brion and M. Vergne, *Lattice points in simple polytopes*, Jour. Amer. Math. Soc. **10** (1997), 371–392. Cappell,S. and Shaneson,J., “ Euler-Maclaurin expansions for lattices above dimension one", [*C. R. Acad. Sci. Paris*]{} Ser. I Math. 321 (1995), 885 Ð 890. V. I. Danilov, V.I., *The geometry of toric varieties*, Russ. Math. Surv. **33** (1978) no. 2, 97–154.
Guillemin, V. W. and Stroock, D.W. “Some Riemann sums are better than others" (to appear). Guillemin, V.W., Sternberg, S., and Weitsman, J. “The Ehrhart function for symbols" [*J. Diff. Geom.*]{} (to appear) Guillemin, V.W., *Riemann-Roch for toric orbifolds*, J. Diff. Geom.**45** (1997), 53–73. Kantor, J.M. and Khovanskii, A.G. *Une application du théorème de Riemann-Roch combinatoire au polyn$\hat{o}$me d’Ehrhart des polytopes entiers de $R\sp d$*, C. R. Acad. Sci. Paris Sér. I Math. **317** (1993), no. 5, 501–507. Khovanskii, A.G and Pukhlikov, A.V., *The Riemann-Roch theorem for integrals and sums of quasipolynomials on virtual polytopes*, Algebra and Analysis **4** (1992), 188–216, translation in St. Petersburg Math. J. (1993), no. 4, 789–812. Karshon,Y., Sternberg, S., and Weitsman, J., *Euler-MacLaurin with remainder for a simple integral polytope* Duke Mathematical Journal **130** 2005, 401- 434
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---
abstract: 'We present our ongoing survey of $\sim$1000 GK-giants with the 9.2-m Hobby-Eberly Telescope in search for planets around evolved stars. The stars selected for this survey are brighter than 11 mag and are located in the section of the HR-diagram, which is approximately delimited by the main sequence, the instability strip, and the coronal dividing line. We use the High Resolution Spectrograph to obtain stellar spectra for radial velocity measurements with a 4-6 m s$^{-1}$ precision. So far, the survey has discovered a planetary-mass companion to the K0-giant HD 17092, and it has produced a number of plausible planet candidates around other stars. Together with other similar efforts, our program provides information on planet formation around intermediate mass main sequence-progenitors and it will create the experimental basis with which to study dynamics of planetary systems around evolving stars.'
---
Introduction
============
Precision radial velocity (RV) studies have established more than a decade ago that GK-giant stars exhibit RV variations ranging from days to many hundreds of days (e.g. [@walker89 Walker et al. 1989], [@HC93 Hatzes & Cochran 1993], [@1994ApJ...422..366H Hatzes & Cochran 1994]). Enough observational evidence has been accumulated to identify three distinct sources of this variability, namely stellar pulsations, surface activity and a presence of substellar companions. A possibility to discover planets around post-MS giants, in numbers comparable to the current statistics of planets around MS-dwarfs (e.g. [@2006ApJ...646..505B Butler et al. 2006]), offers a very attractive way to provide the much needed information on planet formation around intermediate mass MS-progenitors ($\geq 1.5\Msun$) and to create a foundation for studies of the dynamics of planetary systems orbiting evolving stars (e.g. [@1998Icar..134..303D Duncan & Lissauer 1998]).
Fourteen planet discoveries around GK-giants have been reported so far ([@2007ApJ...669.1354N Niedzielski et al. 2007], and references therein, , [@2007ApJ...665..785J Johnson et al. 2007]). Locations of giants with planets in the HR diagram are shown in Fig. 1. All detections have been made using the Doppler velocity technique with the RV precision ranging from $\sim$5 to $\sim$25 m s$^{-1}$, exploiting the availability of many narrow absorption features generated in the cool atmospheres of evolved stars. These developments demonstrate that sufficiently large surveys of post-MS giants should soon furnish enough planet detections to meaningfully address the above problems.
Initial analyses based on the currently available statistics (, [@2007ApJ...665..785J Johnson et al. 2007]) suggest that the frequency of massive planets is correlated with stellar mass. Because more massive stars probably have more massive disks, these results appear to support the core accretion scenarios of planet formation ([@1996Icar..124...62P Pollack et al. 1996]). Furthermore, have used the apparent lack of correlation between the frequency of planets around giants and stellar metallicity to argue that this effect may imply a pollution origin of the observed planet frequency - metallicity correlation for main sequence stars ([@2005ApJ...622.1102F Fischer & Valenti 2005]). Finally, for planets around giants, the absence of planets on tight orbits can be explained as the effect of post-MS evolution of their parent stars, but, as discussed by [@2007ApJ...665..785J], other scenarios must also be considered. For example, the observed paucity of small orbital radii can be the result of faster depletion of disks around more massive stars, as suggested by simulations carried out by [@2007ApJ...660..845B] .
In this paper, we describe our contribution to searches for planets around post-MS stars with a survey of $\sim$1000 GK-giants with the 9.2-m Hobby-Eberly Telescope. Our program has already discovered a number of interesting planet candidates, first of which has been recently published by [@2007ApJ...669.1354N].
The survey
==========
Our long-term project to search for planets around evolved stars with the 9.2-m Hobby-Eberly Telescope ([@lwr98 Ramsey et al. 1998]) and its High Resolution Spectrograph ([@tull98 Tull 1998]) has been established in early 2004. The sample of stars we have been monitoring is composed of two groups, approximately equal in numbers. The first one falls in the “clump giant” region of the HR-diagram ([@jim98 Jimenez et al. 1998]), which contains stars of various masses over a range of evolutionary stages. The second group comprises stars, which have recently left the MS and are located $\sim$1.5 mag above it. Generally, as shown in Fig. 1, all our targets, a total of $\sim$1000 GK-giants brighter than $\sim$11 mag, occupy the area in the HR-diagram, which is approximately defined by the MS, the instability strip, and the coronal dividing line (a narrow strip in the HR-diagram marking the transition between stars with steady hot coronae and those with cool chromospheric winds [@1979ApJ...229L..27L Linsky & Haisch 1979]).
The HET observations and data analysis for this survey have been described by [@2007ApJ...669.1354N]. Briefly, we observe with the HET in its queue-scheduling mode and use the HRS at the R=60,000 resolution with the gas cell ($I_2$) inserted in the optical path. In our target selection, we avoid bright objects, which are accessible to smaller telescopes. Consequently, more than 66% of our target stars are fainter than V=8 mag. The observing scheme follows the standard practices implemented in precision radial velocity measurements with the iodine cell (). The spectral data used for RV measurements are extracted from the 17 echelle orders, which cover the 505 to 592 nm range of the $I_2$ cell spectrum. The observing strategy consists of the initial set of measurements of a target star (2-3 exposures, typically 3-6 months apart), to check for any RV variability exceeding a 30-50 m s$^{-1}$ threshold, followed by more frequent observations, if a significant variability is detected. If the RV variability is confirmed, the star becomes part of the high priority list.
Radial velocities are measured by means of the commonly used $I_2$ cell calibration technique ([@2006ApJ...646..505B Butler et al. 2006]). A template spectrum is constructed from a high-resolution Fourier Transform Spectrometer (FTS) $I_2$ spectrum and a high signal-to-noise stellar spectrum measured without the $I_2$ cell. Doppler shifts are derived from the least-squares fits of template spectra to stellar spectra with the imprinted $I_2$ absorption lines. The average radial velocity for each epoch is calculated as a mean value of the independent determinations from the 17 usable echelle orders. The corresponding uncertainties of these measurements are estimated assuming that errors obey the Student’s t-distribution. Typically, they fall in the 4-5 m s$^{-1}$ range at 1$\sigma$-level. Radial velocities are referred to the Solar System barycenter using the algorithm, which is accurate enough given the RV precision limitations that are intrinsic to the evolved stars.
As the intrinsic variability may contribute to the observed RV variations (e.g. [@2005PASP..117..711G Gray 2005]), stellar line profiles are studied in detail in search for any signatures of a rotation induced spot activity. Also the existing photometry databases like Hipparcos, Tycho or Northern Variability Sky Survey ([@2004AJ....127.2436W Wo[ź]{}niak et al. 2004]) are used to study possible integrated light variations that might be interpreted as a result of pulsations. These analyses are reviewed elsewhere (Niedzielski et al. this vol.).
Results
=======
In almost four years of observations, we have obtained more than one RV measurement for $>$600 GK-giant stars with a 4-6 m $s^{-1}$ precision. Adopting a working definition of RV scatter $\le$40 m s$^{-1}$ for a stable (single) red giant, we find that 55 $\%$ of stars in that sample are single, 20 $\%$ are new binaries and 25$\%$ stars possibly have low-mass companions.
We have been currently monitoring more than 30 planetary candidate companion stars and have obtained preliminary orbital solutions for most of them. Each one obviously requires a thorough examination of stellar activity, which includes bisector analysis and a study of H$_\alpha$ variations. The RV curves and the corresponding best-fit orbital models for our first published planet around the K0 giant, HD 17092 ([@2007ApJ...669.1354N Niedzielski et al. 2007]), and for three other examples of the detections that are being prepared for publication, are shown in Fig. 2. The observed RV curves are highly repeatable and their periods are not reproduced in the measured line bisector and photometric variations. Provisional stellar mass estimates using evolutionary tracks indicate a planetary nature of the companions. It is quite clear that star 162 has a third, long-period companion, whose nature will be established in the course of further observations. The star 37 planet has the most compact orbit among the existing detections (a = 0.6 AU), whereas the planet around star 18 may be orbiting the most massive star in the existing sample (5.5 $\Msun$) and has an exceptionally high intrinsic RV noise.
A steadily increased number of stars observed in this survey makes it possible to carry out statistical studies of RV noise properties of GK-giants. Our preliminary results confirm the intrinsic RV jitter of red giants with the maximum of its distribution at about 20 m s$^{-1}$. Furthermore, the RV scatter increases with B-V, easily reaching 100 m s$^{-1}$ for stars later than K5. Clearly, more observations are needed to understand the nature of the scatter, part of which may be contributed by short-period pulsations, which remain unresolved by the sparse sampling of our survey.
Searches for planets around evolved stars are still in their infancy compared to similar programs for solar-type stars, which have been steadily furnishing new planet detections to bring the count up to over 250 at the time of this writing. However, it is the former searches that are now needed to obtain new information on stellar mass and time-dependent aspects of planet formation and evolution that is not accessible through the latter ones. A continuation of the survey described in this proposal, together with other similar programs, is already creating a base of planet detections around GK-giants, which will soon become sufficient to fully address the questions of stellar mass and chemical composition dependence of planet formation for masses $>1 \Msun$ and of the possible fates of planetary systems under the influence of an evolving parent star. This knowledge will improve our understanding of the astrophysics of planetary systems, it will provide an experimental base for theories of the far future of the Solar System and it will broaden our knowledge of the astrophysical aspects of long-term survival of life on Earth and elsewhere, including a possibility of the emergence of life on planets in the expanded habitable zones of red giants ([@2005ApJ...627..974L Lopez et al. 2005]).
AN and AW were supported in part by the Polish Ministry of Science and Higher Education grant 1P03D 007 30. AW also acknowledges a partial support from the NASA Astrobiology Program. The Hobby-Eberly Telescope (HET) is a joint project of the University of Texas at Austin, the Pennsylvania State University, Stanford University, Ludwig-Maximilians-Universität München, and Georg-August-Universität Göttingen. The HET is named in honor of its principal benefactors, William P. Hobby and Robert E. Eberly.
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{
"pile_set_name": "ArXiv"
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---
abstract: 'Without proper control of numerical and methodological errors in theoretical predictions at the per mille level it is not possible to study the effect of input parameters in current hadron-collider measurements at the required precision. We present a new version of the parton-level code that achieves this requirement through its highly-parallelized nature, significant performance improvements and new features. An automatic differential cutoff extrapolation is introduced to assess the cutoff dependence of all results, thus ensuring their reliability and potentially improving fixed-cutoff results by an order of magnitude. The efficient differential study of uncertainties and set differences at , for multiple sets simultaneously, is achieved by exploiting correlations. We use these improvements to study uncertainties and sensitivity at , using 371 set members. The work described here permits studies that were previously prohibitively expensive, and lays the groundwork necessary for a future implementation of calculations with a jet at Born level in .'
author:
- John Campbell
- Tobias Neumann
bibliography:
- 'refs.bib'
title: Precision phenomenology with MCFM
---
=1
Conclusions {#sec:conclusions}
===========
With the onset of key hadron-collider measurements at the per mille level, interpretation of the results – and thus our understanding of nature – should be limited by uncertainties inherent in the theoretical predictions. Even with current higher-order predictions, that in some cases have percent-level scale uncertainties, control of numerical and methodological errors at the per mille level is required to demonstrate their reliability. This then allows for the study of input parameters, and their impact, at the necessary level of precision. Unfortunately practical resource limitations, set by local workstations or even expensive computing clusters, are easily reached by precise calculations at . These limitations are even easier to saturate when including scans over additional parameters in the predictions. More often than not these limitations result in the introduction of errors, or the use of uncontrolled approximations, that may lead to a loss of the required precision. This can have a direct phenomenological impact that, for instance, can decide between the advent of a signal for new physics or the continued success of the .
In this paper we have addressed this issue by demonstrating that control at the per-mille level can be achieved with a new version of the code . A key component of the theoretical prediction is the numerical integration over the available phase space, where any type of technical cutoff or artifact must be able to be controlled below that level of precision. We first ensured that the raw numerical predictions can reach these levels of precision and that their errors are reliably estimated. This has been achieved through our newly implemented fully parallelized + Vegas integration that adaptively selects contributions with the largest uncertainties and is fully resumable through automatically written snapshots. Our approach allows reliable error estimates for precision predictions because it can use a huge number of calls per single integral estimate, i.e. per iteration. We have compared to a naïve parallelization, obtained by combining many independent low-statistics calls, and find that requires statistical analysis methods to obtain trustworthy error estimates. For such a situation we recommend the use of the well-known bootstrap/jackknife technique.
A further consideration is that a number of today’s calculations, including those implemented in , depend on a slicing cutoff to regularize divergences (a jettiness cutoff, ${\ensuremath{\tau_\text{cut}}}$, in this paper). Results can only be obtained as an extrapolation ${\ensuremath{\tau_\text{cut}}}\to0$, otherwise residual finite ${\ensuremath{\tau_\text{cut}}}$ effects enter as a systematic error. To estimate the slicing cutoff uncertainties for a finite ${\ensuremath{\tau_\text{cut}}}$ additional integrations must be performed for a range of ${\ensuremath{\tau_\text{cut}}}$ values and the dependence assessed. This is especially importantly differentially, where the residual dependence can be highly non-uniform and large compared to inclusive results. This extrapolation is extraordinarily computationally expensive since smaller values of ${\ensuremath{\tau_\text{cut}}}$ lead to larger numerical cancellation effects. Reaching either the required precision or a small enough ${\ensuremath{\tau_\text{cut}}}$ for these independent runs is not always possible \cite{}. To address this we have implemented an automatic correlated sampling of multiple ${\ensuremath{\tau_\text{cut}}}$ values within one single integration run. This saves orders of magnitude in resources or, equivalently, improve results with equal resources by orders of magnitude. Furthermore we have implemented a boosted jettiness definition for all processes and included leading power corrections, which in combination lead to further order of magnitude performance improvements.
Taken together, these improvements ensure that the ${\ensuremath{\tau_\text{cut}}}$ dependence can be controlled at the per mille level on small computing clusters. To assess the reliability and give concrete error estimates, we have presented a detailed scheme for estimating the residual ${\ensuremath{\tau_\text{cut}}}$ error of results based on our automatic sampling of additional ${\ensuremath{\tau_\text{cut}}}$ values and their fitting. Since our fully differential fitting is based on the expected behavior in the asymptotic regime, we can reliably exploit it for both improvements and error or reliability estimates. We have shown examples where the differential fit improves results by an order of magnitude and furthermore makes the differential ${\ensuremath{\tau_\text{cut}}}$ dependence uniform. We have also considered the case where the fit is no longer reported to be reliable, illustrating the identification of regions that need to be scrutinized further for a valid error estimate.
To avoid introducing the approximation of using matrix elements for the calculation of uncertainties, our new implementation allows the use of multiple sets and set members simultaneously at . They are computed simultaneously in a correlated way, saving many orders of magnitude of computational resources compared to uncorrelated integrations. We have used these improvements to study cases where lower-order matrix elements are used to approximate full uncertainties and shown that our correlated implementation allows for per mille level comparisons between different sets. Studies that discern the impact of different data sets and methods in the fits become directly tractable at at a high precision. We have demonstrated this feature for all processes in and also for the high-$p_T$ tail of the Higgs boson transverse momentum spectrum at . In addition we have used our code to attempt to reproduce results contained in the benchmarking exercise of Ref. [@Alioli:2016fum] but find that the cuts used there prohibit a comparison at the per-mille level.
Modern calculations at the level of and beyond require the assembly and availability of dozens of components that each represents years of work. It is therefore mandatory that such components can be easily and reliably reused for further studies. provides a repository of such work, and has been used several times as a basis for fixed order and calculations, resummed calculations, as well as implementations of physics beyond the . Its library of amplitudes has found use in dozens of projects and studies. Given this track record, the overhaul of all core components of the code described here is an important step to increase its usability and reliability, and to keep it an important tool for both experimentalists and theorists. In particular, the features and efficiency gains documented here will enable a public distribution of $W^\pm, Z$ and $H$ production processes in association with a jet in the near future.
#### Acknowledgments.
This work was supported by the U.S. Department of Energy under award No. DE-SC0008347. This document was prepared using the resources of the Fermi National Accelerator Laboratory (Fermilab), a U.S. Department of Energy, Office of Science, HEP User Facility. Fermilab is managed by Fermi Research Alliance, LLC (FRA), acting under Contract No. DE-AC02-07CH11359.
Detailed description of new features {#sec:newfeatures-app}
====================================
In this section we present the new and modified features in and describe how to use them on a technical level, complementing the new manual. With all re-implemented and newly implemented components we strive for Fortran 2008 compliance, making explicit use of its features. Following the Fortran standard furthermore allows us to achieve compatibility with not just the GNU compiler. In previous versions of the licensing was unclear, since none was specified. We now license all code under the GNU GPL 3 license[^1].
#### Improved input file mechanism.
We have implemented a new input file mechanism based on the configuration file parser `config_fortran` [@JTeunis]. This INI-like file format no longer depends on a strict ordering of configuration elements, allows easy access to configuration elements through a single global configuration object, and makes it easy to add new configuration options of scalar and array numerical and string types. Using the parser package also allows one to override or specify all configuration options as command line arguments to , for example running like `./mcfm_omp input.ini -general%nproc=200 -general%part=nlo`. This is useful for batch parameter run scripts. Settings can also be overridden with additional input files that specify just a subset of options.
#### New histogramming.
We replaced the previous Fortran77 implementation of histograms, that used routines from 1988 by M. Mangano, with a new suite of routines. The new histogram implementation allows for any number of histograms with any number of bins, each of which is dynamically allocated. Furthermore, everything is also handled in a fully multi-threaded approach with the integration. For each thread temporary histograms are allocated which are then reduced to a single one after each integration iteration, so that no locks (critical regions) are required.
#### New Vegas integration, part-adaptive and resumable.
The previous implementation of the Vegas routine was based on Numerical Recipes code. We have re-implemented Vegas and the surrounding integration routines. All parts of a or calculation are now chosen adaptively based on the largest absolute numerical uncertainty. A precision goal can be set in the input file as well as a $\chi^2/\text{it}$ goal and a precision goal for the warmup run. If the goals for the warmup are not reached, the warmup repeats with twice the number of calls. With the setting `writeintermediate` one can control whether histograms are written in intermediate stages during the integration. Enabling the setting `readin` allows one to resume the integration from any point from a previous run. Snapshots saving the whole integration state are saved automatically. When resuming, the only parameter that the user can safely officially change is the `precisiongoal`. Further tweak configuration options to control the stages of the integration have been introduced, which can provide benefits over the default settings in certain situations.
The section `integration` in the configuration file allows for tweaks in the following way. The precision goal can be adjusted by setting `precisiongoal` to a relative precision that should be reached. Similarly, the settings `warmupprecisiongoal` and `warmupchisqgoal` control the minimum relative precision and $\chi^2/\text{it}$ for the warmup phase of `iterbatchwarmup` (default 5) iterations. If the warmup criterion fails, the number of calls is increased by a factor of two. The calls per iteration get increased by a factor of `callboost` (default 4) after the warmup. From then on the number of calls per iteration is increased by a factor of `itercallmult` (default 1.4) for a total of `iterbatch1` iterations. After these first `iterbatch1` iterations, the increase happens for every `iterbatch2` iterations. The setting `maxcallsperiter` controls the cap for the number of calls per iteration. The number of Vegas grid subdivisions can be controlled with `ndmx` (default 100).
The purpose of these settings is a fine control in certain situations. For example to compute expensive uncertainties, one wants a relatively precise warmup run (where additional sets are not sampled) and as few calls as necessary afterwards: For the plots in this paper we thus chose a relative warmup precision goal of $10\%$, and set `callboost` to $0.25$. This means that the first `iterbatch1` iterations after the warmup run only with a quarter of the calls than during the warmup. This precision is sufficient to compute precise uncertainties, when making use of the strong correlations as in . Any further iterations come in batches of `iterbatch2`, which we set to $1$. It allows for a quick switching to parts of the cross section that have the largest uncertainty. For normal applications one wants to boost the number of calls after the warmup significantly, so a default value of `callboost=4` is chosen.
We provide default settings for the initial number of calls per iteration for all components of a calculation. They can be overridden with the following settings in the `integration` section: `initcallslord`, `initcallsnlovirt`, `initcallsnloreal`, `initcallsnlofrag` for parts of a calculations, `initcallssnlobelow`, `initcallssnloabove` for parts of a based calculation, and `initcallsnnlobelow`, `initcallsnnlovirtabove`, as well as `initcallsnnlorealabove` for the parts of the coefficient.
#### Low discrepancy sequence.
-8.0 and prior relied on a linear congruential generator implementation from Numerical Recipes for the generation of a pseudo-random sequence. With newer versions the implementation of the C++ standard library is used, and with this version of we include an implementation of the Sobol low discrepancy sequence based on the code sobseq [@Vugt2016] with initialization numbers from ref. [@Joe2010]. The Sobol sequence is used by default and can be toggled using the flag `usesobol = .true.` in the `integration` section of the input file. \[sec:integrationuncertainties\]. When running in mode, the number of nodes has to be a power of two for the Sobol sequence, because we use it in a strided manner. Otherwise the code will automatically fall back to using the sequence with seed value `seed` in the integration section of the input file. A `seed` value of $0$ denotes a randomly initialized seed.
#### Fully parallelized OMP+MPI use of LHAPDF.
In previous versions of calls to were forced to access from only a single OMP thread through a lock. This is because the interface was based on the old LHAglue interface, part of . We have written an interface to from scratch based on the new object oriented treatment of s in 6. For each OMP thread we initialize a copy of the used members which can be called fully concurrently. The amount of sets with or without uncertainties is only limited by the available system memory. The memory usage of can then range from roughly 20MB when only one central grid is being used, to $\sim 7.4$ GB when 32 OMP threads fully load all members of the sets `CT14nnlo`, `MMHT2014nnlo68cl`, `ABMP16als118_5_nnlo`, `NNPDF30_nnlo_as_0118`, `NNPDF31_nnlo_as_0118` and `PDF4LHC15_nnlo_30` for uncertainties. The total number of members for these grids is 371, each loaded for every of the 32 OMP threads.
Since each OMP thread allocates its own copy of members and histograms we have no need to introduce any OMP locks. On the other hand the memory usage increases and one runs into being CPU cache or DRAM bandwidth bound earlier. In practice, we find that this is still faster than having OMP locks, which directly decrease the speedup in the spirit of Amdahl’s law. Ideally the library should be improved to allow for thread-safe calls with just one memory allocation.
#### Histograms for additional values of ${\ensuremath{\tau_\text{cut}}}$, $\mu_R,\mu_F$ and multiple s.
When using the automatic scale variation, in addition to the normal histograms, additional histograms with filenames `_scale_XY_` are generated, where `X` is a placeholder for the renormalization scale variation and `Y` for the factorization scale variation. `X` and `Y` can either be `u` for an upwards variation by a factor of two, `d` for a downwards variation by a factor of two, or just `-` if no change of that scale was made. The envelope of maximum and minimum can then easily be obtained.
For the sampling of additional values of ${\ensuremath{\tau_\text{cut}}}$ for and calculations using jettiness subtractions, additional histograms with filenames `_taucut_XXX_` are written. Here `XXX` is a placeholder for the chosen ${\ensuremath{\tau_\text{cut}}}$ values in the optional array `taucutarray`, if specified, or one of the five automatically chosen values. These additional files only contain the *differences* to the nominal choice of ${\ensuremath{\tau_\text{cut}}}$, so that $\Delta\sigma(\tau_\text{cut,nominal}) - \Delta\sigma(\tau_\text{cut,i})$ is stored. If `taucutarray` has not been specified, the automatic choice of additional ${\ensuremath{\tau_\text{cut}}}$ values is enabled based on the default nominal ${\ensuremath{\tau_\text{cut}}}$ for the process or the users choice of the nominal ${\ensuremath{\tau_\text{cut}}}$ value as specified in `taucut`. In addition a file with `_taucutfit_` is generated, which in addition to the fitted corrections and its uncertainty includes columns for the maximum relative integration uncertainty for the additionally sampled ${\ensuremath{\tau_\text{cut}}}$ values and the reduced $\chi^2$ of the fit. With the procedure in \[sec:benchmark\], the fit together with the individual ${\ensuremath{\tau_\text{cut}}}$ histograms allows the user to assess the systematic ${\ensuremath{\tau_\text{cut}}}$ error and possibly improve results.
When multiple sets are chosen, additional files with the names of the sets are generated. In case uncertainties are enabled, the histograms also include the upper and lower bounds of the uncertainties.
#### User cuts, histograms and re-weighting.
Modifying the plotting routines in the files `src/User/nplotter*.f` allows for modification of the pre-defined histograms and addition of any number of arbitrary observables. The routine `gencuts_user` can be adjusted in the file `src/User/gencuts_user.f90` for additional cuts after the jet algorithm has performed the clustering. In the same file the routine `reweight_user` can be modified to include a manual re-weighting for all integral contributions. This can be used to obtain improved uncertainties in, for example, tails of distributions. One example is included in the subdirectory `examples`, where the `reweight_user` function approximately flattens the Higgs transverse momentum distribution, leading to equal relative uncertainties even in the tail at .
Compatibility with the Intel compiler and benchmarks
----------------------------------------------------
Previous versions of were developed using `gfortran` as a compiler. contained code that did not follow a specific Fortran standard, and was only compatible with using `gfortran`. We fixed code that did not compile or work with the recent Intel Fortran compiler `ifort` 19.0.1. This does not mean that we claim to be strictly standards compliant with a specific Fortran version, but we aim to be compliant with Fortran 2008. We now fully support GCC versions newer than $7$ and Intel compilers newer than $19$. There might still be compatibility issues with other Fortran compilers, but we are happy to receive bug reports for any issues regarding compilation, that are not due to a lack of modern Fortran 2008 features. To use the Intel compiler one has to change the USEINTEL flag in the files `Install` and `makefile` to `YES`.
To see whether can make use of potential Intel compiler improvements over the GNU compiler collection (GCC) we benchmarked the double real emission component of Higgs production at . We perform tests on our cluster with Intel Xeon 64-bit X5650 2.67 GHz Westmere CPUs, where two six-core CPUs are run in a dual-socket mode with a total of twelve cores. Similarly, we have an AMD 6128 HE Opteron 2GHz quad-socket eight-core setup, thus each having 32 cores per node.
We benchmark both the Intel and GCC compilers on both the Intel and AMD systems. On the Intel system we use 16 MPI processes each with 12 OMP threads, and on the AMD system we have 8 MPI processes using 32 OMP threads. With this we have the same total clockrate of for each setup. For all benchmarks we find that the scaling is perfect up to this size, that is if we use half the number of MPI or OMP threads we double our run-time.
We first try both the Intel fortran compiler 19.0.1 and GCC 9.1.0 on the Intel system with the highest generic optimization flags `-O3 -xsse4.2` and `-O3 -march=westmere`, respectively. Furthermore, we lower the optimizations to `-O2` each and remove the processor specific optimization flags `-xsse4.2` and `-march=westmere`, respectively. All our benchmark run-times in the following are consistent within $\pm \SI{0.5}{\s}$.
We do not support enabling unsafe math operations with `-ffast-math`, since the code relies on the knowledge of NaN values and checks on those. Such checks would be skipped with the meta flag`-ffast-math` which sets `-ffinite-math-only`.
------------------------------ ------
ifort -O3 -xsse4.2 90s
ifort -O2 -xsse4.2 86s
ifort -O2 90s
ifort -O1 103s
gfortran -O3 -march=westmere 101s
gfortran -O2 -march=westmere 105s
gfortran -O2 105s
gfortran -O1 110s
------------------------------ ------
: Benchmark results on the Intel system with $10\cdot25$M calls distributed over 16 MPI processes, each using 12 OMP threads. The GCC version is 9.1.0 and the Intel Fortran compiler 19.0.1
\[tab:benchintel\]
The benchmark results in \[tab:benchintel\] show that using the Intel compiler, performance benefits of $\simeq
10-20\%$ can be achieved. Our goal here is not to go beyond this and check whether exactly equivalent optimization flags have been used in both cases. Enabling optimizations beyond `-O2` have little impact, but come with a penalty for the Intel compiler and with a slight benefit for gfortran. We also notice that processor specific optimizations play no significant role. This might also be in part due to the fact that does not offer much space for (automatic) vectorization optimizations. To summarize, the default optimization flags of `-O2` should be sufficient in most cases. We do not expect that the conclusions from these benchmarks change for different processes. On the other hand if computing uncertainties, the majority of time is used by and different optimization flags for might play a role then. We performed the same benchmark with an older version of GCC, version 7.1.0 using `-O2` optimizations, and found that the run-times are the same as for the newer version.
Finally, we performed some benchmarks on our AMD setup and found that it is $\simeq 2.5$ times slower for the same total clockrate. Using the Intel compiler for the AMD setup decreased the performance by another $\simeq 30\%$. This is likely due to the fact that the Intel compiler already optimizes for the general Intel architecture.
These benchmarks try to give a general impression and might depend in detail on the process, the number of histograms and whether to compute uncertainties, for example. Especially when computing uncertainties the perfect scaling we tested here might break down since the computation can become memory bound. We discuss this caveat in more detail in \[subsec:performance\].
Remarks on memory bound performance issues {#subsec:performance}
------------------------------------------
To get numerically precise predictions at the per mille level for cross sections, already hundreds of million of calls are necessary. Obtaining uncertainties using those matrix elements significantly increases the computational time. In a simplified view the total computational time composes as $N_\text{calls}*(T + N_{\text{{\scalefont{.9}}PDF}}{}\cdot T_{\text{{\scalefont{.9}}PDF}}{})$, where $T$ is the computational effort for the matrix element piece, and the part is proportional to the time calling the evolution $N_{\text{{\scalefont{.9}}PDF}}{}$ times and code related to performing the convolutions. For tree level matrix element evaluations, usually also $T \lll T_{\text{{\scalefont{.9}}PDF}}{}$ holds, so the computational cost grows linearly with the number of s.
This naive picture breaks down in practice when a lot of s are sampled together with a lot of histograms or histogram bins. The total memory necessary to store all the histogram information grows like $N_{\text{{\scalefont{.9}}PDF}}{} \cdot N_\text{bins} \cdot N_\text{thr.}$, where $N_{\text{{\scalefont{.9}}PDF}}{}$ is the number of members, $N_\text{bins}$ the number of histogram bins summed over all histograms and $N_\text{thr.}$ is the number of threads. The factor $N_\text{thr.}$ enters since we have thread-local storage to avoid locks. The values are stored in double precision, so the total memory used is $N_{\text{{\scalefont{.9}}PDF}}{} \cdot N_\text{bins} \cdot N_\text{thr.} \cdot 8 \text{ bytes}$.
Assuming for example, 300 members, 10 histograms with each 20 bins and 12 threads, this sums up to of memory. For the virtual corrections and pieces, one has to update this amount of memory once for each call. For the real emission matrix elements one has to accumulate all dipole contributions, so this number additionally scales with the number of dipole contributions. All the histogram updates are usually fully vectorized for modern superscalar processors with SSE and/or AVX extensions. But if this used memory is too large and does not easily fit into the CPU core caches anymore, a transfer to and from DRAM happens, which now is the limiting factor and significantly slows down the computation. Because for that reason, one should work with a minimal number of necessary histograms when working with a lot of members. This is especially important for cluster setups that are not optimized towards memory bound applications, non-NUMA systems. For example in our cluster we have relatively old AMD Opteron quad-socket eight-core nodes with little CPU cache, and with above numbers we are already limited in wall-time improvements with using $\sim16$ cores. Then reducing the number of histograms will *significantly* improve the performance. In principle one can reduce the histogram precision to single precision and cut memory transfer and storage in half, while doubling the computational speed. This might lead to problems with accumulated rounding errors though, and we have not investigated this further, since in practice one can sufficiently limit the number of histograms or sets.
Supporting plots for the jackknife-after-bootstrap procedure {#app:bootstrap}
============================================================
In \[fig:bootstrap\_mt\_first\] and \[fig:bootstrap\_mt\_second\] the lower panel displays a visualization of the jackknife after bootstrap technique. Each point represents one of the $N$ data points that is being left out. The dashed lines represent quantiles of the bootstrap distribution. shows that leaving out the single point number $557$ would significantly shrink the percentiles and make the Gaussian distribution symmetric. After removing the outlier \[fig:bootstrap\_mt\_second\] is obtained, where now no single point would significantly modify the bootstrap distribution. For more details we refer to [@boot3].
![Result of applying the bootstrap technique to our data set of about 4500 data points with 10 million calls each. The sample size is not Gaussian due to one significant outlier. []{data-label="fig:bootstrap_mt_first"}](plots/bootstrap_mt_first.pdf){width="\columnwidth"}
![Result of the applying to bootstrap technique to our data set of about 4500 data points with 10 million calls each. The worst four outliers as shown in the jackknife plot in \[fig:bootstrap\_mt\_first\] have been removed. The result is $31559 \pm 13$. []{data-label="fig:bootstrap_mt_second"}](plots/bootstrap_mt_second.pdf){width="\columnwidth"}
[^1]: See <https://www.gnu.org/licenses/gpl-3.0.en.html>.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'Over the past decade, a consensus picture has emerged in which roughly a quarter of the universe consists of dark matter. The observational evidence for the existence of dark matter is reviewed: rotation curves of galaxies, weak lensing measurements, hot gas in clusters, primordial nucleosynthesis and microwave background experiments. In addition, a new line of research on Dark Stars is presented, which suggests that the first stars to exist in the universe were powered by dark matter heating rather than by fusion: the observational possibilities of discovering dark matter in this way are discussed.'
address: 'Michigan Center for Theoretical Physics, University of Michigan, Ann Arbor, MI 48109, USA'
author:
- Katherine Freese
title: Review of Observational Evidence for Dark Matter in the Universe and in upcoming searches for Dark Stars
---
Introduction
============
A standard model of cosmology is emerging (often dubbed the Concordance Model), in which the universe consists of 4% ordinary baryonic matter, $\sim 23$% dark matter, and $\sim 73$% dark energy, with a tiny abundance of relic neutrinos. The baryonic content is well-known, both from element abundances produced in primordial nucleosynthesis roughly 100 seconds after the Big Bang, and from measurements of anisotropies in the cosmic microwave background (CMB). The evidence for the existence of dark matter is overwhelming, and comes from a wide variety of astrophysical measurements.
Dark Matter in Galaxies and Clusters
====================================
The evidence that 95% of the mass of galaxies and clusters is made of some unknown component of Dark matter (DM) comes from (i) rotation curves (out to tens of kpc), (ii) gravitational lensing (out to 200 kpc), and (iii) hot gas in clusters.
Rotation Curves
---------------
In the 1970s, Ford and Rubin [@FordRubin1970] discovered that rotation curves of galaxies are flat. The velocities of objects (stars or gas) orbiting the centers of galaxies, rather than decreasing as a function of the distance from the galactic centers as had been expected, remain constant out to very large radii. Similar observations of flat rotation curves have now been found for all galaxies studied, including our Milky Way. The simplest explanation is that galaxies contain far more mass than can be explained by the bright stellar objects residing in galactic disks. This mass provides the force to speed up the orbits. To explain the data, galaxies must have enormous dark halos made of unknown Òdark matter.Ó Indeed, more than 95% of the mass of galaxies consists of dark matter. This is illustrated in Fig. 1, where the velocity profile of galaxy NGC 6503 is displayed as a function of radial distance from the galactic center. The baryonic matter which accounts for the gas and disk cannot alone explain the galactic rotation curve. However, adding a dark matter halo allows a good fit to data.
The limitations of rotations curves are that one can only look out as far as there is light or neutral hydrogen (21 cm), namely to distances of tens of kpc. Thus one can see the beginnings of DM haloes, but cannot trace where most of the DM is. The lensing experiments discussed in the next section go beyond these limitations.
{width="\textwidth"}
Lensing
-------
Einstein’s theory of General Relativity predicts that mass bends, or lenses, light. This effect can be used to gravitationally ascertain the existence of mass even when it emits no light. Lensing measurements confirm the existence of enormous quantities of dark matter both in galaxies and in clusters of galaxies.
Observations are made of distant bright objects such as galaxies or quasars. As the result of intervening matter, the light from these distant objects is bent towards the regions of large mass. Hence there may be multiple images of the distant objects, or, if these images cannot be individually resolved, the background object may appear brighter. Some of these images may be distorted or sheared. The Sloan Digital Sky Survey used weak lensing (statistical studies of lensed galaxies) to conclude that galaxies, including the Milky Way, are even larger and more massive than previously thought, and require even more dark matter out to great distances (Adelman-McCarthy [*et al.*]{} [@Sloan2005]). Again, the predominance of dark matter in galaxies is observed.
A beautiful example of a strong lens is shown in Figure 2. The panel on the right shows a computer reconstruction of a foreground cluster inferred by lensing observations made by Tyson et al. using the Hubble Space Telescope. This extremely rich cluster contains many galaxies, indicated by the peaks in the figure. In addition to these galaxies, there is clearly a smooth component, which is the dark matter contained in clusters in between the galaxies.
The key success of the lensing of DM to date is the evidence that DM is seen out to much larger distances than could be probed by rotation curves: the DM is seen in galaxies out to 200 kpc from the centers of galaxies, in agreement with N-body simulations. On even larger Mpc scales, there is evidence for DM in filaments (the cosmic web).
{width="\textwidth"}
Hot Gas in Clusters
-------------------
Another piece of gravitational evidence for dark matter is the hot gas in clusters. Figure 3 illustrates the Coma Cluster. The left panel is in the optical, while the right panel is emission in the x-ray (observed by ROSAT)(Briel & Henry [@Coma1997]). \[Note that these two images are not on the same scale.\] The X-ray image indicates the presence of hot gas. The existence of this gas in the cluster can only be explained by a large dark matter component that provides the potential well to hold on to the gas.
{width="50.00000%"} {width="50.00000%"}
Bullet Cluster
--------------
A recent image of the bullet cluster of galaxies (a cluster formed out of a collision of two smaller clusters) taken by the Chandra X-ray observatory shows in pink the baryonic matter; in blue is an image of the dark matter, deduced from gravitational lensing. In the process of the merging of the two smaller clusters, the dark matter has passed through the collision point, while the baryonic matter slowed due to friction and coalesced to a single region at the center of the new cluster. In modified gravity theories without dark matter, it is not likely that such a differentiation of these two components of the matter would take place.
{width="7cm"}
In summary, the evidence is overwhelming for the existence of an unknown component of DM that comprises 95% of the mass in galaxies and clusters.
Cosmic Abundances
=================
The cosmic abundances tell a consistent story in which the preponderance of the mass in the universe consists of an unknown DM component. The Cosmic Microwave Background provides the most powerful measurements of the cosmological parameters; primordial nucleosynthesis restricts the abundance of baryonic matter; Type IA supernovae provided powerful evidence for the acceleration of the universe, possibly explained by dark energy as the major constituent of the cosmic energy density.
The Cosmic Microwave Background
-------------------------------
Further evidence for dark matter comes from measurements on cosmological scales of anisotropies in the CMB (WMAP Collaboration [@wmap2003],[@wmap2008]). The CMB is the remnant radiation from the hot early days of the universe. The photons underwent oscillations that froze in just before decoupling from the baryonic matter at a redshift of 1000. The angular scale and height of the peaks (and troughs) of these oscillations are powerful probes of cosmological parameters, including the total energy density, the baryonic fraction, and the dark matter component. The sound horizon at last scattering provides a ruler stick for the geometry of the universe: if the light travels in a straight line (as would be the case for a flat geometry), then the angular scale of the first Doppler peak was expected to be found at 1 degree; indeed this is found to be correct. Thus the geometry is flat, corresponding to an energy density of the universe of $\sim 10^{-29} {\rm gm/cm}^3$. The height of the second peak implies that 4% of the total is ordinary atoms, while matching all the peaks implies that 23% of the total is DM.
Primordial nucleosynthesis
--------------------------
When the universe was a few hundred seconds old, at a temperature of ten billion degrees, deuterium became stable: $p + n \rightarrow D +
\gamma$. Once deuterium forms, helium and lithium form as well. The formation of heavier elements such as C, N, and O must wait a billion years until stars form, with densities high enough for triple interactions of three helium atoms into a single carbon atom. The predictions from the Big Bang are 25% Helium-4, $10^{-5}$ deuterium, and $10^{-10}$ Li-7 abundance by mass. These predictions exactly match the data as long as atoms are only 4% of the total constituents of the universe.
Dark Energy
-----------
Evidence for the 70% dark energy in the universe comes from observations of distant supernovae (Perlmutter [*et al.*]{} [@sn1999a], Riess [*et al.*]{} [@sn1999b], Riess [*et al.*]{} [@sn2004]). The supernovae are dimmer than expected, as is most easily explained by an accelerating universe. There are two different approaches to the dark energy: (i) a vacuum energy such as a cosmological constant or time-dependent vacuum (Freese [*et al.*]{} [@fafm1987]) may be responsible, or (ii) it is possible that General Relativity is incomplete and that Einstein’s equations need to be modified (Freese & Lewis [@modgenrel2002a], Freese 2005 [@modgenrel2005], Deffayet [*et al.*]{} [@modgenrel2002b], Carroll [*et al.*]{} [@Carroll_etal2004]). Note, however, that this dark energy does not resolve or contribute to the question of dark matter in galaxies, which remains as puzzling (if not more) than twenty years ago. We now have a concordance model of the universe, in which roughly a quarter of its content consists of dark matter.
Dark Matter Candidates
======================
There is a plethora of dark matter candidates. MACHOs, or Massive Compact Halo Objects, are made of ordinary matter in the form of faint stars or stellar remnants; they could also be primordial black holes or mirror matter (Mohaptra & Teplitz [@MohapatraTeplitz1999]). However, there are not enough of these to completely resolve the question. Of the nonbaryonic candidates, the most popular are the WIMPS (Weakly Interacting Massive Particles) and the axions, as these particles have been proposed for other reasons in particle physics. Ordinary massive neutrinos are too light to be cosmologically significant, though sterile neutrinos remain a possibility. Other candidates include primordial black holes, nonthermal WIMPzillas, and Kaluza-Klein particles which arise in higher dimensional theories.
MACHOs
------
MACHO candidates include faint stars, planetary objects (brown dwarfs), and stellar remnants (white dwarfs, neutron stars, and black holes). Microlensing experiments (the MACHO (Alcock [*et al.*]{} [@alcock2000]) and EROS (Ansari [*et al.*]{} [@eros2004]) experiments) as well as a combination of other observational (HST) and theoretical results (Graff & Freese [@graff1996]) have shown that MACHOs less massive than 0.1 $M_\odot$ make an insignificant contribution to the energy density of the Galaxy. However, there is a detection (Alcock [*et al.*]{} [@alcock2000]) of a roughly 20% halo fraction made of $\sim 0.5 M_\odot$ objects which might be made of stellar remnants such as white dwarfs. We found a number of constraints: the progenitors produce observable element abundances (C,N,He), they require an enormous mass budget, the initial mass function must be extremely sharply peaked, and, most important, the progenitors produce observable infrared radiation. Our conclusion from these constraints is that at most 20% of the Galactic Halo can be made of stellar remnants (Freese [*et al.*]{} [@Freese_etal2000], Fields [*et al.*]{} [@ffgwpb], Graff [*et al.*]{} [@ffgwpc]).
Axions {#sec:axions}
------
The good news is that cosmologists don’t need to “invent” new particles. Two candidates already exist in particle physics for other reasons: axions and WIMPs. Axions with masses in the range $10^{-(3-6)}$ eV arise in the Peccei-Quinn solution to the strong-CP problem in the theory of strong interactions. Axion bounds (Asztalos [*et al.*]{} [@rosenberg]) from the ADMX cavity experiment are approaching the remaining parameter range.
WIMPs (Weakly Interacting Massive Particles) {#sec:WIMPs}
--------------------------------------------
WIMPs are also natural dark matter candidates from particle physics. These particles, if present in thermal abundances in the early universe, annihilate with one another so that a predictable number of them remain today. The relic density of these particles comes out to be the right value: $$\Omega_\chi h^2 = (3 \times 10^{-26} {\rm cm}^3/{\rm sec})
/ \langle \sigma v \rangle_{ann}$$ where the annihilation cross section $\langle \sigma v \rangle_{ann} $ of weak interaction strength automatically gives the right answer. This coincidence is known as “the WIMP miracle” and is the reason why WIMPs are taken so seriously as DM candidates. The best WIMP candidate is motivated by Supersymmetry (SUSY): the lightest neutralino in the Minimal Supersymmetric Standard Model. Supersymmetry in particle theory is designed to keep particle masses at the right value. As a consequence, each particle we know has a partner: the photino is the partner of the photon, the squark is the quark’s partner, and the selectron is the partner of the electron. The lightest superysmmetric partner is a good dark matter candidate (see the reviews by Jungman [*et al.*]{}[@Jungman_etal1996], Lewin & Smith [@jkgb], Primack [*et al.*]{}[@jkgc], Bertone [*et al.*]{}[@Bertone_etal2004]).
There are several ways to search for dark WIMPs. SUSY particles may be discovered at the LHC as missing energy in an event. In that case one knows that the particles live long enough to escape the detector, but it will still be unclear whether they are long-lived enough to be the dark matter. Thus complementary astrophysical experiments are needed. In direct detection experiments, the WIMP scatters off of a nucleus in the detector, and a number of experimental signatures of the interaction can be detected (Goodman & Witten [@gw], Drukier [*et al.*]{} [@dfs]). In indirect detection experiments, neutrinos are detected from the Sun or Earth that arise as annihilation products of captured WIMPs; the first papers suggesting this idea were by Silk [*et al.*]{} [@SOS] in the Sun; and by Freese [@Freese1986] as well as Krauss, Srednicki and Wilczek [@Krauss_etal1986] in the Earth. Another way to detect WIMPs is to look for anomalous cosmic rays from the Galactic Halo: WIMPs in the Halo can annihilate with one another to give rise to antiprotons, positrons, or neutrinos (Ellis [*et al.*]{} [@ellis]). In addition, neutrinos, Gamma-rays, and radio waves may be detected as WIMP annihilation products from the Galactic Center (Gondolo & Silk [@gonsilk]). Many talks in this conference will discuss ongoing and planned DM searches.
Dark Stars
==========
The first stars to form in the universe, at redshifts $z \sim 10-50$, may be powered by dark matter annihilation for a significant period of time (Spolyar, Freese, and Gondolo [@SpolyarFreeseGondolo08]). We have dubbed these objects “Dark Stars.”
As discussed in the last section, WIMP dark matter annihilation in the early universe provides the right abundance today to explain the dark matter content of our universe. This same annihilation process will take place at later epochs in the universe wherever the dark matter density is sufficiently high to provide rapid annihilation. The first stars to form in the universe are a natural place to look for significant amounts of dark matter annihilation, because they form at the right place and the right time. They form at high redshifts, when the universe was still substantially denser than it is today, and at the high density centers of dark matter haloes.
The first stars form inside dark matter (DM) haloes of $10^6 M_\odot$ (for reviews see e.g. Ripamonti & Abel [@RipamontiAbel05], Barkana & Loeb [@BarkanaLoeb01], and Bromm & Larson [@BrommLarson03]; see also Yoshida et al. [@Yoshida_etal06].) One star is thought to form inside one such DM halo. The first stars play an important role in reionization, in seeding supermassive black holes, and in beginning the process of production of heavy elements in later generations of stars. It was our idea to ask, what is the effect of the DM on these first stars? We studied the behavior of WIMPs in the first stars. As our canonical values, we take $m_\chi =
100$GeV for the WIMP mass and $\langle \sigma v \rangle_{ann} = 3
\times 10^{-26} {\rm cm^3/sec}$ for the annihilation cross section (motivated above). We find that the annihilation products of the dark matter inside the star can be trapped and deposit enough energy to heat the star and prevent it from further collapse. A new stellar phase results, a Dark Star, powered by DM annihilation as long as there is DM fuel.
Three Criteria for Dark Matter Heating
--------------------------------------
WIMP annihilation produces energy at a rate per unit volume $$Q_{\rm ann} = \langle \sigma v \rangle_{ann} \rho_\chi^2/m_\chi
\linebreak \simeq 10^{-29} {{\rm erg} \over {\rm cm^3/s}} \,\,\, {\langle
\sigma v \rangle \over (3 \times 10^{-26} {\rm cm^3/s})} \left({n \over {\rm
cm^{-3}}}\right)^{1.6} \left({100 {\rm GeV}\over m_\chi}\right)$$ where $\rho_\chi$ is the DM energy density inside the star and $n$ is the stellar hydrogen density. Paper I (Spolyar, Freese, & Gondolo [@SpolyarFreeseGondolo08]) outlined the three key ingredients for Dark Stars: 1) high dark matter densities, 2) the annihilation products get stuck inside the star, and 3) DM heating wins over other cooling or heating mechanisms. These same ingredients are required throughout the evolution of the dark stars, whether during the protostellar phase or during the main sequence phase.
[**First criterion: High Dark Matter density inside the star.**]{} Dark matter annihilation is a powerful energy source in these first stars because the dark matter density is high. To find the DM density profile, we started with an NFW (Navarro, Frenk & White [@NavarroFrenkWhite96]) profile for both DM and gas in the $10^6
M_\odot$ halo. Originally we used adiabatic contraction ($M(r)r$ = constant) (Blumenthal et al. [@Blumenthal_etal85]) and matched onto the baryon density profiles given by Abel, Bryan & Norman [@AbelBryanNorman02] and Gao et al. [@Gao_etal07] to obtain DM profiles; see also Natarajan, Tan, & O’Shea [@NatarajanTanO'Shea08] for a recent discussion. Subsequent to our original work, we have done an exact calculation (which includes radial orbits) (Freese, Gondolo, Sellwood & Spolyar [@FreeseGondoloSellwoodSpolyar08]) and found that our original results were remarkably accurate, to within a factor of two. At later stages, we also consider possible further enhancements due to capture of DM into the star (discussed below).
[**Second Criterion: Dark Matter Annihilation Products get stuck inside the star**]{}. In the early stages of Pop III star formation, when the gas density is low, most of the annihilation energy is radiated away (Ripamonti Mapelli & Ferrara [@RipamontiMapelliFerrara06]). However, as the gas collapses and its density increases, a substantial fraction $f_Q$ of the annihilation energy is deposited into the gas, heating it up at a rate $f_Q Q_{\rm ann}$ per unit volume. While neutrinos escape from the cloud without depositing an appreciable amount of energy, electrons and photons can transmit energy to the core. We have computed estimates of this fraction $f_Q$ as the core becomes more dense. Once $n\sim 10^{11} {\rm cm}^{-3}$ (for 100 GeV WIMPs), e$^-$ and photons are trapped and we can take $f_Q \sim 2/3$.
[**Third Criterion: DM Heating is the dominant heating/cooling mechanism in the star**]{}. We find that, for WIMP mass $m_\chi =
100$GeV (1 GeV), a crucial transition takes place when the gas density reaches $n> 10^{13} {\rm cm}^{-3}$ ($n>10^9 {\rm cm}^{-3}$). Above this density, DM heating dominates over all relevant cooling mechanisms, the most important being H$_2$ cooling (Hollenbach & McKee [@HollenbachMcKee79]).
Figure 5 shows evolutionary tracks of the protostar in the temperature-density phase plane with DM heating included (Yoshida et al. [@Yoshida_etal08]), for two DM particle masses (10 GeV and 100 GeV). Moving to the right on this plot is equivalent to moving forward in time. Once the black dots are reached, DM heating dominates over cooling inside the star, and the Dark Star phase begins. The protostellar core is prevented from cooling and collapsing further. The size of the core at this point is $\sim 17$ A.U. and its mass is $\sim 0.6 M_\odot$ for 100 GeV mass WIMPs. A new type of object is created, a Dark Star supported by DM annihilation rather than fusion.
{width="\textwidth"}
Building up the Mass
--------------------
We have found the stellar structure of the dark stars (hereafter DS) (Freese, Bodenheimer, Spolyar, & Gondolo [@FreeseBodenheimerSpolyarGondolo08]). They accrete mass from the surrounding medium. In our paper we build up the DS mass as it grows from $\sim 1 M_\odot$ to $\sim 1000 M_\odot$. As the mass increases, the DS radius adjusts until the DM heating matches its radiated luminosity. We find polytropic solutions for dark stars in hydrostatic and thermal equilibrium. We build up the DS by accreting $1 M_\odot$ at a time with an accretion rate of $2 \times 10^{-3} M_\odot$/yr, always finding equilibrium solutions. We find that initially the DS are in convective equilibrium; from $(100-400) M_\odot$ there is a transition to radiative; and heavier DS are radiative. As the DS grows, it pulls in more DM, which then annihilates. We continue this process until the DM fuel runs out at $M_{DS} \sim 800 M_\odot$ (for 100 GeV WIMPs). Figure 6 shows the stellar structure. One can see “the power of darkness:” although the DM constitutes a tiny fraction ($<10^{-3}$) of the mass of the DS, it can power the star. The reason is that WIMP annihilation is a very efficient power source: 2/3 of the initial energy of the WIMPs is converted into useful energy for the star, whereas only 1% of baryonic rest mass energy is useful to a star via fusion.
{width="50.00000%"}
Results and Predictions
-----------------------
Our final result (Freese, Bodenheimer, Spolyar, & Gondolo [@FreeseBodenheimerSpolyarGondolo08]), is very large first stars; e.g., for 100 GeV WIMPs, the first stars have $M_{DS} = 800
M_\odot$. Once the DM fuel runs out inside the DS, the star contracts until it reaches $10^8$K and fusion sets in. A possible end result of stellar evolution will be large black holes. The Pair Instability SN (Heger & Woosley [@HegerWoosley02]) that would be produced from 140-260 $M_\odot$ stars (and whose chemical imprint is not seen) would not be as abundant. Indeed this process may help to explain the supermassive black holes that have been found at high redshift ($10^9 M_\odot$ BH at z=6) and are, as yet, unexplained (Li et al. [@Li_etal07]; Pelupessy et al. [@Pelupessy_etal07]).
The stars are very bright, $\sim 10^6 L_\odot$, and relatively cool, (6000-10,000)K (as opposed to standard Pop III stars whose surface temperatures exceed $30,000K$). Reionization during this period is likely to be slowed down, as these stars can heat the surroundings but not ionize them. One can thus hope to find DS and differentiate them from standard Pop III stars.
Later stages: Capture
---------------------
The dark stars will last as long as the DM fuel inside them persists. The original DM inside the stars runs out in about a million years. However, as discussed in the next paragraph, the DM may be replenished by capture, so that the DS can live indefinitely due to DS annihilation. We suspect that the DS will eventually leave their high density homes in the centers of DM haloes, especially once mergers of haloes with other objects takes place, and then the DM fuel will run out. The star will eventually be powered by fusion. Whenever it again encounters a high DM density region, the DS can capture more DM and be born again.
The new source of DM in the first stars is capture of DM particles from the ambient medium. Any DM particle that passes through the DS has some probability of interacting with a nucleus in the star and being captured. The new particle physics ingredient required here is a significant scattering cross section between the WIMPs and nuclei. Whereas the annihilation cross section is fixed by the relic density, the scattering cross section is a somewhat free parameter, set only by bounds from direct detection experiments. Two simultaneous papers (Freese, Spolyar, & Aguirre [@FreeseSpolyarAguirre08], Iocco [@Iocco08]) found the same basic idea: the DM luminosity from captured WIMPs can be larger than fusion for the DS. Two uncertainties exist here: the scattering cross section, and the amount of DM in the ambient medium to capture from. DS studies following the original papers that include capture have assumed (i) the maximal scattering cross sections allowed by experimental bounds and (ii) ambient DM densities that are never depleted. With these assumptions, DS evolution models with DM heating after the onset of fusion have now been studied in several papers (Iocco et al. [@Iocco_etal08], Taoso et al. [@Taoso_etal08], Yoon et al [@Yoon_etal08]).
In short, the first stars to form in the universe may be Dark Stars powered by DM heating rather than by fusion. Our work indicates that they may be very large ($800 M_\odot$ for 100 GeV mass WIMPs). Once DS are found, one can use them as a tool to study the properties of WIMPs.
Conclusion
==========
95% of the mass in galaxies and clusters of galaxies is in the form of an unknown type of dark matter. We know this from rotation curves, from gravitational lensing, and from hot gas in clusters. This assessment of the DM contribution to the global energy density of the universe is consistent with measurements of the cosmic microwave background, primordial nucleosynthesis, supernova, and large scale structure. A consensus picture has emerged, in which the DM contributes 23% of the overall energy density of the universe. Its nature is still unknown. At most 1/5 of the DM in galaxies can be white dwarfs (or other MACHO candidates), but most is likely to be an exotic particle candidate. DM searches for the best motivated candidates, axions and WMPs are ongoing and promising over the next few years. One of the key properties of WIMP candidates is its annihilation cross section, yielding the proper relic density today. As a consequence of this annihilation, the first stars in the universe may provide another avenue to test the DM hypothesis. These stars may be powered by DM annihilation, and one can look for them in upcoming telescopes. The goal is to decipher this unknown dark matter over the next decade.
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{
"pile_set_name": "ArXiv"
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|
---
abstract: 'The parity-time ($\mathcal{PT}$) symmetric structures have exhibited potential applications in developing various robust quantum devices. In an optical trimmer with balanced loss and gain, we analytically study the $\mathcal{PT}$ symmetric phase transition by investigating the spontaneous symmetric breaking. We also illustrate the single-photon transmission behaviors in both of the $\mathcal{PT}$ symmetric and $\mathcal{PT}$ symmetry broken phases. We find (i) the non-periodical dynamics of single-photon transmission in the $\mathcal{PT}$ symmetry broken phase instead of $\mathcal{PT}$ symmetric phase can be regarded as a signature of phase transition; and (ii) it shows unidirectional single-photon transmission behavior in both of the phases but comes from different underlying physical mechanisms. The obtained results may be useful to implement the photonic devices based on coupled-cavity system.'
author:
- 'L. F. [Xue]{}'
- 'Z. R. Gong'
- 'H. B. [Zhu]{}'
- 'Z. H. [Wang]{}'
title: '$\mathcal{PT}$ symmetric phase transition and single-photon transmission in an optical trimmer system'
---
Introduction
============
Since Bender and Boettcher proposed the concept of parity-time ($\mathcal{PT}$) symmetry [@bender1; @bender2], it has attracted a lot of attentions due to its potential applications. Remarkably, the system with $\mathcal{PT}$ symmetry can undergo a phase transition when the parameter that controls the non-Hermiticity surpasses a critical value which is usually called exceptional point (EP) [@bender2; @AM]. Below the EP, all of the eigen-values of the non-Hermit Hamiltonian are real, and some or all of the eigen values become complex beyond the EP.
The broken of $\mathcal{PT}$ symmetry will lead to a lot of interesting phenomena. For example, people have observed the non-reciprocal photonic transmission in the $\mathcal{PT}$ symmetric structure [@long; @xiaomin; @wujh; @songzhi; @HR; @ZL; @LF1; @NB1], and predicted the enhancement of nonlinear interaction due to field localization [@jiahua1; @jiahua2]. Moreover, the unique property of the system with $\mathcal{PT}$ symmetry has potential application in various fields, such as loss-induced or gain-induced transparency [@Guo; @Jing], efficient photon or phonon lasing [@BP; @LF; @HH; @HJ; @BH], ultralow-threshold optical chaos [@CT; @xinyou] as well as quantum metrology [@jingzhang].
On the other hand, the coupled-cavity system is widely used to coherently control the photon transfer. In the coupled-cavity-array with infinite length, the defect can be introduced to construct a singe-photon switcher [@sun1; @wang0], router [@sun2] and frequency converter [@wang]. Moreover, within the capacity of current experiments, a lot of attentions have been paid on the optical dimmer (also called optical molecule [@MB; @YP], which is composed of two coupled cavities), such as the coherent polariton [@xiao] and state transfer [@state]. Furthermore, motivated by the simulation of photosynthesis harvest system in recent years, there are also studies about the photon [@ultrastrong] and thermal transport [@thermal] in the trimmer structure.
In this paper, we focus on an optical trimmer with $\mathcal{PT}$ symmetry. Here, our scheme is composed of an active gain cavity and a passive loss cavity, which simultaneously couple with a third cavity without loss and gain, to form a coupled-cavity-array as shown in Fig. \[trimmer\]. With balanced gain and loss, which are described phenomenally in this paper, we show a $\mathcal{PT}$ symmetric phase transition in the non-Hermitian system. In the $\mathcal{PT}$ symmetric phase, all of the eigen-values of the non-Hermitian Hamiltonian are real and the corresponding eigen-states show the same $\mathcal{PT}$ symmetric character as that of the Hamiltonian. On the contrary, in the $\mathcal{PT}$ symmetry broken phase, the complex eigen-values emerge, and the $\mathcal{PT}$ symmetry of the eigen-states disappear. In this sense, the system will undergo a spontaneous symmetric breaking as the phase transition occurs. In addition, the $\mathcal{PT}$ symmetric phase transition is accompanied by the field localization. That is, the photon shows an equal weight distribution between the passive and active cavity in $\mathcal{PT}$ symmetric phase and localized at the active cavity in $\mathcal{PT}$ symmetry broken phase. Compared with the phase transition in optical molecule or dimmer [@long; @xiaomin; @jiahua1; @jiahua2], the critical coupling strength in optical trimmer is much smaller, and is therefore easier for experimental realization.
In such a system, we study the unidirectional single-photon transmission in both of the $\mathcal{PT}$ symmetric and $\mathcal{PT}$ symmetry broken phases. In the $\mathcal{PT}$ symmetric phase, it shows a periodical oscillation, and the unidirectional transmission results from the breaking of time reversal symmetry. In the $\mathcal{PT}$ broken symmetric phase, where the gain compensates the loss [@long], the single-photon transfer exhibits a non-periodical feature and the unidirectional transmission stems mainly from the field localization.
The rest of the paper is organized as follows. In Sec. \[model\], we present a $\mathcal{PT}$ symmetry model by an optical trimmer with balanced loss and gain and discuss the $\mathcal{PT}$ symmetric phase transition. In Sec. \[transmission\], we illustrate the unidirectional single-photon transmission in both of the $\mathcal{PT}$ symmetric phase and $\mathcal{PT}$ symmetry broken phase. At last, we give a brief conclusion in Sec. \[conclusion\].
Model and $\mathcal{PT}$ phase transition {#model}
=========================================
As shown in Fig. \[trimmer\], our model consists of an array of three single-mode cavities [@ultrastrong], where a passive and an active cavity (labelled by “$-1$" and “$1$" respectively) simultaneously couple to the third cavity (labelled by “$0$") without loss and gain. By describing the gain and loss in our scheme phenomenologically, the Hamiltonian is written as $$\begin{aligned}
H&=&(\omega_{-1}-i\gamma_{-1})a_{-1}^{\dagger}a_{-1}+\omega_{0}a_{0}^{\dagger}a_{0}
+(\omega_{1}+i\gamma_1)a_{1}^{\dagger}a_{1}\nonumber\\&&+
J(a_{-1}^{\dagger}a_0+a_0^{\dagger}a_{1}+a_{0}^{\dagger}a_{-1}+a_1^{\dagger}a_{0}),
\label{PTH}\end{aligned}$$ where $a_l (l=-1,0,1)$ is the annihilation operator for the $l$th cavity with resonant frequency $\omega_l$. $J$ is the photon-tunneling strength between the two nearest cavities, which can be adjusted by changing the distance of them. In addition, we use $\gamma_{-1}(>0)$ to denote the decay of the passive cavity and $\gamma_1(>0)$ to denote the gain of active cavity. Hereafter, we consider the case that $\omega_{-1}=\omega_0=\omega_1=\omega$ and $\gamma_{-1}=\gamma_{1}=\gamma$, therefore the Hamiltonian satisfies a $\mathcal{PT}$ symmetry, that is $[H, \mathcal{PT}]=0$. Here $\mathcal{P}$ represents the mirror reflection $1\leftrightarrow-1$ and $\mathcal{T}$ denotes the time reversal $i\leftrightarrow-i$ [@bender2].
![(Color online) Schematic illustration of $\mathcal{PT}$ symmetric optical trimmer. In this setup, the passive cavity $-1$ and active cavity $1$ simultaneously couple to the central cavity $0$, which is without loss or/and gain.[]{data-label="trimmer"}](trimmer){width="8cm"}
To deeply investigate the $\mathcal{PT}$ phase transition in our system, we write the Hamiltonian in the form of $$H=\left(\begin{array}{ccc}
a_{1}^{\dagger} & a_{2}^{\dagger} & a_{3}^{\dagger}\end{array}\right)\mathcal{H}\left(\begin{array}{c}
a_{1}\\
a_{2}\\
a_{3}
\end{array}\right),$$ where $$\mathcal{H}=\left(\begin{array}{ccc}
\omega-i\gamma & J & 0\\
J & \omega & J\\
0 & J & \omega+i\gamma
\end{array}\right).
\label{mh}$$ Solving the secular equation ${\rm det}(\mathcal{H}-EI)=0$, where $I$ is a $3\times3$ identity matric, we will obtain the eigenvalues and the corresponding eigenstates of the Hamiltonian $\mathcal{H}$, yielding $$\begin{aligned}
E_0&=&\omega,\,\,|E_0\rangle=\frac{1}{\sqrt{2+(\gamma/J)^2}}(-1,-i\frac{\gamma}{J},1),\\
E_{\pm}&=&\omega\pm\sqrt{2J^2-\gamma^2},\,\,|E_{\pm}\rangle=\frac{1}{\mathcal{N_{\pm}}}(a_{\pm},b_{\pm},1).\end{aligned}$$ where $\mathcal{N}_0$ and $\mathcal{N}_{\pm}$ are the normalized constants and we have defined $$\begin{aligned}
a_{\pm}&:=&\frac{J^2-\gamma^2\mp i\gamma\sqrt{2J^2-\gamma^2}}{J^2},\\
b_{\pm}&:=&\frac{-i\gamma\pm\sqrt{2J^2-\gamma^2}}{J}.\end{aligned}$$
We note that the eigenvalue $E_0$ is always real and is independent of $J$ and $\gamma$, and the eigen-state satisfies the $\mathcal{PT}$ symmetry, that is $\mathcal{PT}|E_0\rangle=e^{i\pi}|E_0\rangle$. However, the other paired eigenvalues $E_{\pm}$ are dependent not only on $\omega$, but also on $\gamma$ and $J$.
To obtain a purely real spectrum, we need a strong inter-cavity coupling strength, i.e., $J>\gamma/\sqrt{2}$. It then satisfies $\mathcal{N}_+=\mathcal{N}_-=2$ and the eigen-state $|E_\pm\rangle$ can be simplified as $$|E_\pm\rangle=\frac{1}{2}(e^{\mp i\theta_1},2e^{\mp i\theta_2},1),$$ where $\theta_1:=\arctan[\gamma\sqrt{2J^2-\gamma^2}/(J^2-\gamma^2)]$ and $\theta_2:=\arctan[\sqrt{2J^2-\gamma^2}/\gamma]$. A simple calculation tells us $$\mathcal{PT}|E_\pm\rangle=e^{\pm i\theta_1}|E_\pm\rangle,$$ which implies that the wave functions $|E_\pm\rangle$ are transformation invariant under the $\mathcal{PT}$ operation (except for a global phase). On the other hand, for the situation of $J<\gamma/\sqrt{2}$, the imaginary parts of $E_{\pm}$ emerge and $E_{\pm}=\omega\pm i\sqrt{\gamma^2 -2J^2}$. Meanwhile, we can not find a global phase $\phi$ to satisfy $\mathcal{PT}|E_\pm\rangle=e^{\pm i\phi}|E_\pm\rangle$. In other words, when the system undergoes a spontaneous symmetry breaking as the inter-cavity coupling crosses the EP $J=\gamma/\sqrt{2}$. In this sense, we name the phase in the regime $J>\gamma/\sqrt{2}$ as the $\mathcal{PT}$ symmetric phase and that for $J<\gamma/\sqrt{2}$ as the $\mathcal{PT}$ symmetry broken phase.
In Figs. \[eigs\] (a) and (b) , we give the real and imaginary parts of $E_{\pm}$ respectively. In the $\mathcal{PT}$ symmetry broken phase ($J<\gamma/\sqrt{2}$), the small coupling strength protects the gained energy flowing from the active cavity to the passive one, and the long lifetime supermode $E_+$ is localized at the active cavity as shown in Fig. \[eigs\] (c) , where we plot $|a_+|$ as a function of the coupling strength $J$. On contrary, in the $\mathcal{PT}$ symmetric phase ($J>\gamma/\sqrt{2}$) , the gained energy is transferred to the passive cavity quickly, and the photon yields an equal weight distribution in the passive and active cavities, that is $|a_\pm|\equiv1$.
![(Color online) The real parts of $E_{\pm}$ (a), imaginary parts of $E_{\pm}$ (b), and $|a_+|$ as a function of the coupling strength $J$. We have chosen $\omega=5\gamma$, and all of the other parameters are in units of $\gamma=1$.[]{data-label="eigs"}](eigsys){width="8cm"}
We point out that here, the similar $\mathcal{PT}$ symmetric phase transition also occurrs in optical dimmer which consists of two cavities with balanced loss and gain (see Refs. [@long; @xiaomin] and the references therein). The differences stems in the following two aspects: On the one hand, in our optical trimmer system, there exist a single real energy level $E_0$, and the corresponding wave function is always invariant under the $\mathcal{PT}$ operation, independent of whether the phase transition occurs. On the other hand, the EP of optical trimmer system is $J=\gamma/\sqrt{2}$ instead of $J=\gamma$ in optical dimmer [@long; @xiaomin; @jiahua1; @jiahua2], that is, a smaller coupling strength is needed, which is more easily to be realized in experiments.
single-photon transmission {#transmission}
==========================
To show the effect of $\mathcal{PT}$ symmetric phase transition on the dynamics of the system, we in this section consider the different behaviors of single-photon transmission when it is excited in the passive or active cavity initially, in both of the $\mathcal{PT}$ symmetric and $\mathcal{PT}$ symmetry broken phases.
![(Color online) The illustration of dynamics of the system when the single photon is initially excited in the passive (a) and active (b) cavities. The parameters are set as $\omega=5\gamma, J=5\gamma$, and all parameters are in units of $\gamma=1$. Under these parameters, the system is in the $\mathcal{PT}$ symmetric phase.[]{data-label="ss"}](sps "fig:"){width="8cm"} ![(Color online) The illustration of dynamics of the system when the single photon is initially excited in the passive (a) and active (b) cavities. The parameters are set as $\omega=5\gamma, J=5\gamma$, and all parameters are in units of $\gamma=1$. Under these parameters, the system is in the $\mathcal{PT}$ symmetric phase.[]{data-label="ss"}](sas "fig:"){width="8cm"}
Since we only consider the transmission of single photon in this paper, the wave function at arbitrary time $t$ can be assumed as $$|\psi(t)\rangle=\alpha(t)|1;0;0\rangle+\beta(t)|0;1;0\rangle+\xi(t)|0;0;1\rangle,$$ where $|m;n;q\rangle:=|m\rangle_{-1}\otimes|n\rangle_{0}\otimes|q\rangle_{1}$, and $|n\rangle_{i}(i=-1,0,1)$ represents that the $i$th cavity is in the Fock state $|n\rangle$.
Governed by the Hamiltonian in Eq. (\[PTH\]), the dynamics of the system is determined by the Schoedinger equation $i\partial_t|\psi\rangle=H|\psi\rangle$, which gives
$$\begin{aligned}
i\frac{d}{dt}\alpha(t)&=&(\omega-i\gamma)\alpha(t)+J\beta(t),\\
i\frac{d}{dt}\beta(t)&=&\omega\beta(t)+J[\alpha(t)+\xi(t)],\\
i\frac{d}{dt}\xi(t)&=&(\omega+i\gamma)\xi(t)+J\beta(t).\end{aligned}$$
\[cof\]
Single-photon transmission in $\mathcal{PT}$ symmetric phase
------------------------------------------------------------
We now study the behavior of the single-photon transmission in the $\mathcal{PT}$ symmetric phase, that is $J>\gamma/\sqrt{2}$. Firstly, we consider the situation that the photon is excited in the passive cavity initially, that is $\alpha(0)=1,\beta(0)=\xi(0)=0$, we can obtain explicitly the probability amplitudes for finding a photon in the three cavities as $$\begin{aligned}
\alpha_s(t)&=&\frac{2J^2e^{-i\omega t}}{\Delta^2}\cos^2(\frac{\Delta t+\phi_1}{2}),\nonumber \\ \\
\beta_s(t)&=&\frac{2 iJ^2e^{-i\omega t}}{\Delta^2}\sin(\frac{\Delta t}{2})\sin(\frac{\Delta t-\phi_2}{2}),\\
\xi_s(t)&=&-\frac{2J^2e^{-i\omega t}}{\Delta^2}\sin^2(\frac{\Delta t}{2}),\end{aligned}$$ where $\Delta=\sqrt{2J^2-\gamma^2},\phi_1=\arctan[\Delta \gamma/(J^2-\gamma^2)],\phi_2=2\arctan(\Delta/\gamma)$. Secondly, when the single photon is initially excited in the active cavity, that is $\alpha(0)=\beta(0)=0,\xi(0)=1$, the solution of Eqs. (\[cof\]) are obtained as $$\begin{aligned}
\alpha'_s(t)&=&-\frac{2J^2e^{-i\omega t}}{\Delta^2}\sin^2(\frac{\Delta t}{2})
, \\
\beta'_s(t)&=&\frac{2 iJ^2e^{-i\omega t}}{\Delta^2}\sin(\frac{\Delta t}{2})\sin(\frac{\Delta t+\phi_2}{2}),\\
\xi'_s(t)&=&
\frac{2J^2e^{-i\omega t}}{\Delta^2}\cos^2(\frac{\Delta t-\phi_1}{2}).\end{aligned}$$
In Figs. \[ss\] (a) and (b), we plot the corresponding probabilities $|h_s(t)|^2$ and $|h'_s(t)|^2$ $(h=\alpha,\beta,\xi)$ as functions of evolution time $t$. As shown in the figure, when the system is in the $\mathcal{PT}$ symmetric phase, the dynamics shows regular periodical oscillations. If the single photon is excited in the passive cavity, i.e., $\alpha(0)=1$, the decay makes $|\alpha_s(t)|^2$ directly decrease to zero and then the revival occurs. However, if it is excited in the active cavity, i.e., $\xi(0)=1$, with the assistance of the gain effect, $|\xi'_s(t)|^2$ firstly reaches its maximal value $[2J^2/(2J^2-\gamma^2)]^2$, which is obviously larger than $1$ and then oscillates between the maximal value and zero. As for the central cavity, we can also observe that $\beta_s(t)\neq\beta'_s(t)$. In this sense, our system exhibits a unidirectional phenomenon in the single-photon level even in the $\mathcal{PT}$ symmetric phase. Furthermore, we find that the initial phases $\phi_1$ and $\phi_2$ are $\gamma$ dependent and $\phi_1(-\gamma)=-\phi_1(\gamma),\phi_2(-\gamma)=-\phi_2(\gamma)$. As a result, by regarding the amplitudes as functions of $t$ and $\gamma$, we will reach the relationship $\alpha_s(t,-\gamma)=\xi'_s(t,\gamma),\beta_s(t,-\gamma)=\beta'_s(t,\gamma)$ and $\xi_s(t,-\gamma)=\alpha'_s(t,\gamma)$. Meanwhile, it is obvious from Eq. (\[mh\]) that $\mathcal{H}(i\leftrightarrow-i)=\mathcal{H}(\gamma\leftrightarrow-\gamma)$, therefore, the single-photon unidirectional transmission in $\mathcal{PT}$ symmetric phase comes from the breaking of time reversal symmetry, that is $[\mathcal{T},\mathcal{H}]\neq0$.
Single-photon transmission in $\mathcal{PT}$ symmetry broken phase
------------------------------------------------------------------
In this subsection, we will continue to study the dynamics of single-photon transmission in the $\mathcal{PT}$ symmetry broken phase, where $J<\gamma/\sqrt{2}$. On the one hand, we consider that the single photon is initially excited in the passive cavity, then the dynamics of system is obtained as $$\begin{aligned}
\alpha_b(t)&=&-\frac{e^{-i\omega t}}{\delta^2}[J^2+(J^2-\gamma^2)\cosh(\delta t)+\gamma\delta\sinh(\delta t)],\nonumber \\ \\
\beta_b(t)&=&-\frac{iJe^{-i\omega t}}{\delta^2}[\gamma\cosh(\delta t)-\gamma-\delta\sinh(\delta t)],\\
\xi_b(t)&=&-\frac{2J^2e^{-i\omega t}\sinh^2(\frac{\delta t}{2})}{\delta^2}.\end{aligned}$$ where $\delta=\sqrt{\gamma^2-2J^2}$. On the other hand, when the single photon is initially excited in the active cavity, the dynamics of the system is described by $$\begin{aligned}
\alpha'_b(t)&=&-\frac{2J^2e^{-i\omega t}\sinh^2(\frac{\delta t}{2})}{\delta^2},\\
\beta'_b(t)&=&-\frac{iJe^{-i\omega t}}{\delta^2}[\gamma\cosh(\delta t)-\gamma+\delta\sinh(\delta t)],\\
\xi'_b(t)&=&-\frac{e^{-i\omega t}}{\delta^2}[J^2+(J^2-\gamma^2)\cosh(\delta t)-\gamma\delta\sinh(\delta t)].\nonumber \\\end{aligned}$$
![(Color online) The illustration of dynamics of the system when the single photon is initially excited in the passive (a) and active (b) cavities. The parameters are set as $\omega=5\gamma, J=0.5\gamma$, and all parameters are in units of $\gamma=1$. Under these parameters, the system is in the $\mathcal{PT}$ symmetry broken phase.[]{data-label="bb"}](bps "fig:"){width="8cm"} ![(Color online) The illustration of dynamics of the system when the single photon is initially excited in the passive (a) and active (b) cavities. The parameters are set as $\omega=5\gamma, J=0.5\gamma$, and all parameters are in units of $\gamma=1$. Under these parameters, the system is in the $\mathcal{PT}$ symmetry broken phase.[]{data-label="bb"}](bas "fig:"){width="8cm"}
In Fig. \[bb\], we depict the dynamics of the system when it is in the $\mathcal{PT}$ symmetry broken phase. Obviously, it shows a completely different behavior compared with the case in the $\mathcal{PT}$ symmetric phase. As shown in Fig. \[bb\](a), when the single photon is initially excited in the passive cavity, the photon will experience a loss firstly and then the gain in the active cavity will compensate the loss. As a result, the probability for finding the photon in the cavities will increase as the time elapse. As for the central cavity, the incident photon will hop to it, but the obtained photons will jump to the other two cavities due to the coherent coupling until the gained photon from the active cavity jumped back to it. Furthermore, at the time $t=\text{arctanh}[\gamma\delta/(\gamma^2-J^2)]/\delta$, the probability for finding the photon in the passive or the central cavities achieve their smallest values simultaneously. On the other hand, the probability for finding a photon in the active cavity will increase monotonously due to the combinational effect of the photonic hopping from the central cavity and the gain from the surrounding environment.
In Fig. \[bb\](b), we show the results when the single photon is initially excited in the active cavity. In such a situation, the gain effect will take action from the very beginning and the probabilities for finding photons in all of the cavities will undoubtedly increase as the time evolution.
Comparing the results in Figs. \[bb\] (a) and (b), we also observe the unidirectional single-photon transmission in $\mathcal{PT}$ symmetry broken phase. It can be explained from the following two aspects. Firstly, similar to that in $\mathcal{PT}$ symmetric phase, the time reversal symmetry breaking of the Hamiltonian naturally results in the different transmission behaviors for the photons initially excited in the left and right sides of the system. However, the periodical triangle functions which describe the dynamics of the system in the $\mathcal{PT}$ symmetric phase is replaced by the monotonous hyperbola function in the broken phase. Therefore, the fixed phase difference between $\alpha_s(t)$ and $\xi'_s(t)$ does not hold any longer for $\alpha_b(t)$ and $\xi'_b(t)$. Secondly, the unidirectional transmission also comes from the field localization in the $\mathcal{PT}$ symmetry broken phase. As shown in Sec. \[model\], the long lifetime eigen-state $|E_+\rangle$ has a lager distribution weight in the $1$th cavity, that is, the photon is localized in the active cavity when the system is in the $\mathcal{PT}$ symmetry broken phase. As a result, for the single-photon excited in the active cavity, the overlap between the initial state and $|E_+\rangle$ is much larger than that excited in the passive cavity, and leading to a different transmission behavior.
conclusion
==========
In conclusion, we have studied the $\mathcal{PT}$ symmetric phase transition by demonstrating the spontaneous symmetry breaking in an optical trimmer with balanced loss and gain. In the $\mathcal{PT}$ symmetric phase, all of the eigen values are real and the corresponding eigen states show a balanced distribution in the passive and active cavities. In the $\mathcal{PT}$ symmetry broken phase, the imaginary parts of the eigen values appear and the supermode with long lifetime is characterized by a strong field localization in the active cavity. Comparing with the optical molecule/dimmer system, which was broadly studied recently, the critical coupling strength of the phase transition is much smaller in our trimmer structure and is therefore easier to be realized experimentally. As a signature of the $\mathcal{PT}$ symmetric phase transition, we subsequently find the dramatically different single-photon transmission behaviors when the system is in the two phases. Our results show that the regular periodical oscillation is replaced by the non-periodical behavior as the system transfers from the $\mathcal{PT}$ symmetric phase to $\mathcal{PT}$ symmetry broken phase. Moreover, we find a unidirectional single-photon transmission phenomenon in both phases. We hope our study about the $\mathcal{PT}$ symmetry in optical trimmer will be helpful for the designing of photonic device based on coupled-cavity system.
[**Note added.**]{} Recently, we have learned of recent related work about the $\mathcal{PT}$ symmetry trimmer systems, which cares about the single-photon transmission at the EP [@Jin].
This work is supported by the National Natural Science Foundation of China (under Grant Nos. 11404021 and 11504241), and the Fundamental Research Funds for the Central Universities (under Grant Nos. 2412016KJ015 and 2412016KJ004).
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{
"pile_set_name": "ArXiv"
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|
---
abstract: 'We propose a method for zeroth order stochastic convex optimization that attains the suboptimality rate of $\tilde{\mathcal{O}}(n^{7}T^{-1/2})$ after $T$ queries for a convex bounded function $f:{\mathbb R}^n\to{\mathbb R}$. The method is based on a random walk (the *Ball Walk*) on the epigraph of the function. The randomized approach circumvents the problem of gradient estimation, and appears to be less sensitive to noisy function evaluations compared to noiseless zeroth order methods.'
author:
- 'Tengyuan Liang [^1]'
- 'Hariharan Narayanan [^2]'
- 'Alexander Rakhlin [^3]'
bibliography:
- 'Stochastic-Convex-Optimization.bib'
title: 'On Zeroth-Order Stochastic Convex Optimization via Random Walks'
---
Introduction
============
Let $f$ be a convex real-valued function on a closed convex domain $K\subset \mathbb{R}^n$. Within the oracle model of optimization, one sequentially obtains noisy information about this unknown function with the aim of computing an $\epsilon$-minimizer of $f$. In this paper we consider the setting of *stochastic zeroth order optimization*: at step $t$, the oracle reveals a noisy value of the function at a point queried by the algorithm. This model is very basic, and can be viewed through the lens of learning: what is the amount of information one needs to collect in order to identify a near-optimal point, if all that is known about the objective is that it is convex?
The amount of information can be quantified by the number of oracle calls, and it is known that this information-based complexity for minimization of a convex Lipschitz function $f$ to within an error $\epsilon>0$ scales as $\Omega(n^2\epsilon^{-2})$ [@Shamir12]. Yet, to the best of our knowledge, at present time there is no algorithm that comes even close to obtaining the desired $n^2$ dependence on the dimension while simultaneously having the $\epsilon^{-2}$ dependence on accuracy. The seminal work of [@NemYud83] introduces an optimization method for the noiseless zeroth order optimization with the $n^7$ dependence on the dimension, but the authors concede that extending this to the stochastic setting will worsen the power (this dependence is left as an unspecified polynomial in $n$). The noiseless zeroth-order method of [@NemYud83] was extended to the noisy case for the slightly harder problem of regret minimization in [@AgaFosHsuKakRak13siam], where the authors proved an $\tilde{\mathcal{O}}(n^{33}\epsilon^{-2})$ upper bound[^4] on the *regret* of the procedure, a more difficult objective. As a consequence, the same upper bound holds for the problem of optimization via averaging of the trajectory (see [@polyak1992acceleration; @hazan2011beyond; @AgaFosHsuKakRak13siam]).
In the present paper, we describe a method with an $\tilde{\mathcal{O}}(n^{14}\epsilon^{-2})$ oracle complexity (or, equivalently, the $\tilde{\mathcal{O}}(n^{7}T^{-1/2})$ decay of suboptimality after $T$ steps). While not very practical for problems in high enough dimension, the method should be viewed as making progress towards closing the large theoretical gap. Further, the algorithm is based on random walks and is quite different from the classical techniques. These more classical approaches can be roughly divided into two categories: attempting to estimate the gradient using noisy function evaluations, or attempting to find a zero-th order method that is robust to noise. [@NemYud83] discuss the distinction between these two general plans of attack. The first is unlikely to yield the $1/\epsilon^2$ dependence on the accuracy, while the second appears to suffer from an adverse scaling with the dimension under noisy evaluations. The random walk approach can be viewed as yet another possible technique. While the present paper still leaves a large gap to the lower bound, there is hope for improvement using randomized methods such as a random walk. The reason we are optimistic about this approach is because randomized methods appear to be more robust to noise. Ideally, one would hope to use randomness in function evaluation as an asset rather than a disadvantage, thus “riding on the noise”.
![A nearly uniform sample is obtained via a random walk, and its average (red diamond) is computed. The convex set is then cut at the $y$-coordinate of the average point thus reducing the volume.[]{data-label="fig:graphics_epigraph"}](epigraph.pdf){width="1.6in"}
Let us informally describe the method. We start with an $n+1$-dimensional convex body formed by the epigraph of the function and cut off at a value for the maximum of the function over the convex set $K$ (see Figure \[fig:graphics\_epigraph\]). We run several Ball Walks on this body to generate near-uniform samples, in the spirit of the work of [@bertsimas2004solving]. Having obtained the samples, we take an average to approximate the center of mass, cut the convex set, and reshape. We continue in this fashion for $\tilde{\mathcal{O}}(n\log(1/\epsilon))$ iterations. The main technical difficulty is in analyzing the modified Ball Walk: unlike the related work of [@bertsimas2004solving] in the noiseless setting, we have a very restricted access to the convex body. To verify if the current point is inside the body, we adaptively sample the function value to obtain a confidence interval (in the “vertical direction”) for the true value. If the confidence interval does not contain the current point, we proceed as if the inside/outside information were correct. Otherwise, we incur an additional error from not being able to resolve the question. Let us remark that our method is naturally parallelized, shaving off another factor of $n$ in terms of the number of queries per machine.
Let us mention very briefly that recent work has also considered a more restrictive oracle model, whereby one can obtain two function values with the same noise instance [@nesterov2011random; @agarwal2010optimal; @Duchi13]. This problem is markedly more simple, as one may form a good estimate of the gradient in any desired direction. Another simplifying assumption considered in the literature is an additional shape constraint such as smoothness of the objective (e.g. [@jamieson2012query]). In contrast, we only assume convexity and boundedness.
Notation
========
A noisy observation of the function value at a point $x\in K$ is denoted by $O_{1}\circ f (x)$, and the average of $m$ repeated queries is denoted by $O_m \circ f(x)$. We assume that the noise is sub-Gaussian with zero mean. We denote an optimal value $x^*\in{\mathop{\rm arg\min}}_{x \in K} f(x)$. An affine transformation in $\mathbb{R}^{n+1}$ is denoted by $\mathcal{T} : \mathbb{R}^{n+1} \mapsto \mathbb{R}^{n+1}$. The region enclosed by the convex function $f$ and hyperplane $y\leq C_t$ is denoted by $K_t = \{(x,y)\in\mathbb{R}^{n+1}, C_t\geq y\geq f(x)\}$. It is easy to check that this is a convex set in $\mathbb{R}^{n+1}$. Let $\Delta \in \mathbb{R}^{n+1}$ denote the vertical vector linking $(x,f(x))\rightarrow (x,y)$ and $|\Delta|$ denotes its length, which equals to $y-f(x)$. We assume that an initial value $C_0$, an upper bound on the function over the set, is given. However, we do not assume that the function is Lipschitz.
Our goal is to bound the number of the noisy oracle calls given the target accuracy $\epsilon$. Assume that the convex set $K=K_0$ is contained in the axis-aligned cube of width $1$ centered at the origin. At the final epoch $T$, the remaining convex body $K_T$ will contain a cube of width $\epsilon$.
Random Walk on Convex Body
==========================
In this section we will introduce the ball walk algorithm in the *noiseless* oracle setting. Analysis of the Ball Walk algorithm was developed in @kannan1997random, and was later modified in @bertsimas2004solving to solve noiseless convex programs. Our algorithm for the noisy oracle setting builds on the theoretical properties of the noiseless Ball Walk.
The random walk algorithm consists of three main steps: “Cut", “Round" and “Sample". We can assume that we start from $t=1$ and $C_0 = 1/2$, without loss of generality. Assume that $\tilde{\mathcal{O}}(n)$ near uniform distributed samples (a warm start that will be maintained throughout the procedure) are provided. This can be done in time that is independent of $\epsilon$. Let $n_t = \tilde{\mathcal{O}}(n)$ for all $t$.
1. [ Cut]{} the region $K_t$ at epoch $t$, enclosed by $y \geq f(\vec{x})$ and $y \leq C_t$, at the last coordinate of the average computed in the third step of the previous iteration. About $2/3$ of the random samples are still inside $K_t$, due to Lemma \[shrink.vol\] below.
2. [ Round]{} the convex body $K_t$ using an affine transformation $\mathcal{T}_t$ to a near isotropic position and denote the resulting convex body by $\mathcal{T}_t(K_t)$. The affine transformation $\mathcal{T}_t$ is calculated using half of the near-uniform samples left after the “Cut” procedure. ($1/3$ of the samples used in this step, and $1/3$ left untouched.)
3. [ Sample]{} $n_{t} = \tilde{\mathcal{O}}(n)$ nearly independent uniform samples $X^t_1, X^t_2, ...., X^t_{n_t}$ (in the sense of Lemma \[mix.ball\]) using “Ball Walk" in the convex body $\mathcal{T}_t(K_t)$ based on the “warm start" samples left after the “Round” procedure. Since there are about $\frac{1}{3} n_{t-1}$ of these seeds left, we run three independent chains of Ball Walk to ensure that we have $n_t$ samples after mixing. Set the new $C_{t} = \frac{1}{n_t} \sum_{i} (\mathcal{T}_t^{-1}\circ X^t_i )[n+1]$, here $X[n+1] \in \mathbb{R}$ denotes the last coordinate of the vector $X$. Go back to step 1.
The following lemmas \[theta.lemma\]-\[mix.ball\] are useful in proving the theoretical guarantee of the Ball Walk algorithm. The first lemma is taken from @kannan1997random [Corollary 5.2].
\[theta.lemma\] We call a convex body $K$ is in $\theta$-near isotropic position if for any vector $v$, $$(1-\theta) \| v\|_2^2 \leq \mathbb{E} \langle x, v\rangle^2 \leq (1+\theta) \| v\|_2^2$$ where the expectation is taken over uniform distribution inside the convex body $K$. Let $\theta<1/2$. If a convex body $K$ is in $\theta$-near isotropic position and $B$ is a unit ball, then $$(1-2\theta)B \subseteq K \subseteq (1+2\theta) (n+1)B.$$
The second lemma assures the constant factor of volume shrinkage in each epoch. It is proved in @bertsimas2004solving [Lemma 7].
\[shrink.vol\] The volume of the covex body $K_t$ drops by a factor of $\frac{2}{3}$ with high probability in each epoch.
The third lemma is first introduced in @kannan1997random in a weaker version and later improved in @rudelson1999random. The version we are using can be found in @bertsimas2004solving [Corollary 11].
Let $K$ be a convex set, $$O(p n \log n \max\{p, \log n\})$$ random samples are sufficient to find an affine transformation $\mathcal{T}$ to bring $K$ into $1/4$-near-isotropic position, with probability at least $1-\frac{1}{2^{p-1}}$.
When we take $p= O(\log n)$, we can conclude that with overwhelming probability (with probability at least $1-\frac{1}{n^\alpha}$, where $\alpha$ is arbitrary in the sense that it only affects the constant in the big $O$ notation), the transformation $\mathcal{T}$ bring $K$ into near isotropic position with $O(n \log^3 n)$ random samples.
The last lemma about the mixing time of Ball Walk is proved in @kannan1997random [Theorem 2.2].
\[mix.ball\] Given a convex body $K$ satisfying $B \subseteq K \subseteq d B$, a positive integer $N$ and $\epsilon>0$, we can generate a set of $N$ random points $\{v_1, \ldots, v_N\}$ in $K$ that are
- almost uniform in the sense that the distribution of each one is at most $\epsilon$ away from the uniform in total variation distance, and
- almost (pairwise) independent in the sense that for every $1\leq i < j \leq N$ and every two measurable subsets $A$ and $B$ of $K$, $$|P(v_i \in A, v_j \in B) - P(v_i \in A) P(v_j \in B)| \leq \epsilon.$$
The algorithm uses only $\tilde{\mathcal{O}}(n^3d^2+N n^2 d^2)$ calls to the oracle.
Given the above lemmas, we can give a precise upper bound on the number of oracle calls need in the noiseless setting. The proof is given in @bertsimas2004solving. We sketch the main idea here for completeness.
\[noiseless.thm\] Each iteration of the random walk algorithm in the noiseless case uses at most $\tilde{\mathcal{O}}(n^4)$ number of oracle calls. Further, the algorithm can be implemented in at most $\tilde{\mathcal{O}}(n^5)$, with high probability.
According to @bertsimas2004solving [Theorem 12], the number of oracle calls in each iteration is upper bounded by $\tilde{\mathcal{O}}(n^4)$. The volume ratio between the initial convex body and final convex body is $$\frac{{\rm Vol} (K_T)}{{\rm Vol} (K_0)} > \epsilon^n.$$ According to Lemma \[shrink.vol\], $$\left( \frac{2}{3} \right)^T \geq \frac{{\rm Vol} (K_T)}{{\rm Vol} (K_0)} > \epsilon^n$$ Thus $T = O(n \log \frac{1}{\epsilon})$. So the total number of oracle calls is bounded by $\tilde{\mathcal{O}}(n^5)$.
Adaptive Query Algorithm
========================
Based on the Ball Walk analysis in the noiseless setting, the remaining difficulty lies in bounding the misclassification error, i.e. the error of incorrectly classifying the query point as inside or outside the convex body. We use an adaptive query algorithm to address this problem near optimally. The intuitive statistical idea is: keep doubling the number of samples until we get enough “confidence" to tell whether a point is inside or not. The adaptive query algorithm is illustrated in the following with full details.
Suppose we would like to decide whether a point $(x,y)\in \mathbb{R}^{n+1}$ is inside the current epigraph. The $m$-sample noisy oracle returns $O_m\circ f(x)$. Let $Z$ denote a standard normal $\mathcal{N}(0,1)$ (or sub-Gaussian tail random variable; the proof is almost identical), and $C$ a level to be determined later. We make the following adaptive decision:
- $(x,y)$ is [ OUTSIDE]{} - [ $Out$]{} if $y \leq O_m\circ f(x) - \frac{C}{\sqrt{m}}$. The probability of making this decision is $$P({\rm Outside}) = P\left(Z \geq \sqrt{m}(y-f(x))+C\right).$$
- $(x,y)$ is [ INSIDE]{} - [ $In$]{} if $y \geq O_m\circ f(x) + \frac{C}{\sqrt{m}}$. The probability of making this decision is $$P({\rm Inside}) = P\left(Z \leq \sqrt{m}(y-f(x))-C\right).$$
- [POSTPONE]{} decision - [ $Post$]{} decision if $y \in \left[O_m\circ f(x) - \frac{C}{\sqrt{m}},O_m\circ f(x) + \frac{C}{\sqrt{m}}\right]$. The probability of making this decision is $$P({\rm Postpone}) = P\left(\sqrt{m}(y-f(x))-C \leq Z \leq \sqrt{m}(y-f(x))+C\right).$$
The Adaptive Query algorithm is:
- Set a dictionary of sample size $m$ being the set $S = \left\{2^0, 2^1, \ldots, 2^{k},\ldots \right\}$.
- Take $m$ from the dictionary and construct the test sequentially. Stop when we made a decision either [OUTSIDE]{} or [INSIDE]{}. Otherwise continuously increase $m$ from the dictionary.
\[adapt.query\] With probability at least $1 - 2\cdot \left( \log_2 \frac{4C^2}{|\Delta|^2}+1\right) \cdot \exp\left(-\frac{C^2}{2}\right)
$, where $|\Delta| = y - f(x)$, $m = \frac{4C^2}{(y - f(x))^2}$ query numbers are enough to ensure correct decision.
First suppose the query point $(x,y)$ is inside the convex body, then $|\Delta| = y - f(x) >0$. Define the event $E = \left\{ \mbox{Classified as Inside using}~m \leq \frac{4C^2}{(y-f(x))^2}~\mbox{query samples} \right\}$ $$\begin{aligned}
P(E^c) & = P(\text{Out}_{m=2^0}) + P(\text{Post}_{m\in S,m<2^1}, \text{Out}_{m=2^1}) \\
&~~~~~~~~~~ + \ldots+ P(\text{Post}_{m\in S, m<4C^2/|\Delta|^2}, \{\text{In}_{m = 4C^2/|\Delta|^2} \}^c) \\
&\leq P(\text{Out}_{m=2^0}) + P(\text{Out}_{m=2^1})+\ldots+P(\{\text{In}_{m = 4C^2/|\Delta|^2} \}^c)\\
& \leq \sum_{i < \log_2 (4C^2/|\Delta|^2)} P\left(Z \geq \sqrt{m}(y-f(x))+C, m=2^i\right) \\
&~~~~~~~~~~ + P(Z \geq \sqrt{m}(y-f(x))-C, m=4C^2/|\Delta|^2)\\
& \leq \log_2 \frac{4C^2}{|\Delta|^2} \cdot \exp\left(-\frac{C^2}{2}\right) + \exp\left(-\frac{C^2}{2}\right) = \left( \log_2 \frac{4C^2}{|\Delta|^2} +1\right) \cdot \exp\left(-\frac{C^2}{2}\right).\end{aligned}$$ Similarly, suppose the query point $(x,y)$ is outside the convex body, then $|\Delta| = y - f(x) <0$. Define a set $E = \left\{ \mbox{Classified as Outside using}~m \leq \frac{4C^2}{(y-f(x))^2}~\mbox{query samples}\right\}$ $$\begin{aligned}
P(E^c) & = P(\text{In}_{m=2^0}) + P(\text{Post}_{m\in S,m<2^1}, \text{In}_{m=2^1}) \\
&~~~~~~~~~~ +\ldots+ P(\text{Post}_{m\in S, m<4C^2/|\Delta|^2}, \{\text{Out}_{m = 4C^2/|\Delta|^2} \}^c) \\
&\leq P(\text{In}_{m=2^0}) + P(\text{In}_{m=2^1})+\ldots+P(\{\text{Out}_{m = 4C^2/|\Delta|^2} \}^c)\\
& \leq \sum_{i < \log_2 (4C^2/|\Delta|^2)} P\left(Z \leq \sqrt{m}(y-f(x))-C, m=2^i\right) \\
&~~~~~~~~~~ + P(Z \leq \sqrt{m}(y-f(x))+C, m=4C^2/|\Delta|^2)\\
& \leq \log_2 \frac{4C^2}{|\Delta|^2} \cdot \exp\left(-\frac{C^2}{2}\right) + \exp\left(-\frac{C^2}{2}\right)= \left( \log_2 \frac{4C^2}{|\Delta|^2} +1\right) \cdot \exp\left(-\frac{C^2}{2}\right).\end{aligned}$$
As we can see, if $|\Delta|>\frac{1}{n^k}$ (polynomial decay in terms of $n$), as long as $C = O(\sqrt{\log n})$ with a constant big enough (say $C = \sqrt{2 (\ell+1) \log n}$), the error probability is $o(\frac{1}{n^\ell})$. This probability can be arbitrary small with polynomial decay in terms of of $n$.
Stochastic Convex Optimization
==============================
The algorithm to solve stochastic convex optimization problem given noisy oracle in our paper is a combination of random walk in convex body and adaptive hypothesis testing, as illustrated in the following.
- Perform the random walk algorithm as in noiseless case.
- When establishing whether a point is inside or outside, use the adaptive query algorithm.
In order to analyze the expected number of oracle calls used in this algorithm, we will first introduce some lemmas revealing the geometry of convex body and property of the level set function of the given convex function.
![Graphical proof of Lemma \[geo.lm\]. $K$ denotes the convex body, $B$ the inscribed ball, and $H$ the hyperplane. The shaded area is the “Double Cone” inside the convex body $K$.[]{data-label="fig:graphics_cone"}](cone.pdf){width="1.6in"}
\[geo.lm\] Let $\theta<1/2$. For a convex body $K\in \mathbb{R}^n$ in $\theta$-near isotropic position, and a arbitrary hyperplane $H \in \mathbb{R}^{n-1}$, the following inequality holds $${\rm Vol}(K \cap H) \leq \frac{n}{2-4\theta} \cdot {\rm Vol}(K)$$
As shown in @kannan1997random, we can always find a unit ball $B$ inside the convex body $K$ such that $$\label{iso.bound}
(1-2\theta)B \subseteq K \subseteq (1+2\theta) (n+1)B.$$ Consider the intersection $K\cap H$. We know that this intersection is a convex body in $\mathbb{R}^{n-1}$. Find the farthest points $P$ (possibly two) on each side of $K \cap H$ on the surface of the unit ball $B$ to $K\cap H$. Connect $P$ with the boundary of $K\cap H$, we get a “Cone" or “Double Cone" $C$ with volume $$\frac{2-4\theta}{n} \cdot {\rm Vol}(K \cap H) \leq \frac{(1-2\theta) \cdot d(B)}{n} \cdot {\rm Vol}(K \cap H) \leq {\rm Vol}(C) \leq {\rm Vol}(K).$$ Thus proof completed. The graphical illustration of this lemma is in Figure \[fig:graphics\_cone\].
\[dist.levelset\] Fix a convex function $f$ and the associated enclosed convex body $K$. For any point $(x,y) \in K$, $\Delta \in \mathbb{R}^{n+1}$ denotes the vertical vector linking $(x,f(x))\rightarrow (x,y)$. Denote the affine transformation that brings the convex body $K \subset \mathbb{R}^{n+1}$ to $\theta$-near isotropic position ($\theta<1/2$) as $\mathcal{T}$. Consider the uniform distribution on $\mathcal{T}(K)$. Then the distribution of the scalar $|\mathcal{T}(\Delta)|$ has the following properties:
- $|\mathcal{T}(\Delta)| = c \cdot |\Delta|$, with the constant factor depending on $\mathcal{T}$.
- The probability measure $\mathcal{P}(\cdot)$ of $|\mathcal{T}(\Delta)|$ satisfies $d \mathcal{P}(s) \leq \frac{n+1}{2-4\theta} \cdot ds$
Affine transformation keeps the ratio on a line and keeps the parallel property. So property 1 is proved. Property 2 is a direct consequence of Lemma \[geo.lm\], where the hyperplanes are the constructed according to the value $|\mathcal{T}(\Delta)|$.
\[noisy.thm\] There exist an event $E$ with probability at least $P(E) \geq 1 - o(1)$. On the event $E$, the expected number of noisy oracle calls is at most $\tilde{\mathcal{O}}(n^{14} \frac{1}{\epsilon^2})$.
Following @kannan1997random [Theorem 4.1 Remark, Theorem 4.4], samples from the random walk can achieve closeness to the uniform distribution in total variation sense very quickly. More explicitly, if we need $T$ steps to achieve a precision $\epsilon$, then $\epsilon/n^{10}$ can be achieved in almost the same number of steps: we lose only a factor of $O(\log n)$. Thus precision in total variation sense is not a crucial issue in sampling, and we can always assume the samples are drawn from the uniform distribution.
Next, a $1/4$-near isotropic position behaves like isotropic position in our complexity analysis: no additional $n$ factor is involved, and the only difference is in terms of the constant. Therefore, without loss of generality, we may assume the convex body is in the isotropic position after the transformation $\mathcal{T}$.
As we can see in Lemma \[adapt.query\], if the current query point is far away from the boundary along the vertical direction $\Delta$ (that is, $|\Delta|$ is large, which is equivalent to $|\mathcal{T}(\Delta)|$ being large), we can tell whether or not the point is inside with high confidence within $\frac{4C^2}{|\Delta|^2}$ oracle calls. On the one hand, as the query point approaches the boundary, the number of oracle calls goes to infinity. One the other hand, the probability of getting very close to boundary is small. Thus in terms of theoretical analysis, there is a trade-off between whether we want to spend more oracle calls at a given point that is close to boundary or, alternatively, put the probability of the point being close to boundary into bad event scenario and thus save the oracle calls. Hence in analyzing the algorithm, there is a trade-off that determines the best point where we “give up”. We will call this “give up band” as $\delta$-boundary in the following proof, as illustrated in Figure \[fig:graphics\_band\]. We remark that after the $\mathcal{T}$ transformation, the direction in which we obtain noisy information is not vertical, but this does not impact the analysis.
![$\delta$-boundary illustration: the red lines denote the direction of the vector $\mathcal{T}(\Delta)$, which is the direction in which we can do adaptive querying. Because of the affine transformation $\mathcal{T}$, it is not necessarily vertical. []{data-label="fig:graphics_band"}](delta_band.pdf){width="1.6in"}
According to Lemma \[dist.levelset\], we have $|\mathcal{T}(\Delta)| = c \cdot |\Delta|$, where $c$ is a constant factor. Because of , and since an $\epsilon$ cube is inscribed at the final epoch, $c \leq \frac{n}{\epsilon}$. We now define a $\delta$-boundary as the mass $|\mathcal{T}(\Delta)| \leq \delta$. The procedure (for the purposes of analysis only) will give up on the $|\Delta| = y - f(x) = | \mathcal{T}(\Delta) | /c \leq \delta/c$ band and put this mass in our error term. (This giving up on the area near the boundary can be seen as the main source of looseness in the analysis, but we do not know how to avoid it). Then the error probability can be written in two parts, one coming from the probability of giving up in the band, the other coming from the statistical error of adaptive testing procedure. More precisely, $$\begin{aligned}
P({\rm Error}) & \leq P(|\mathcal{T}(\Delta)| \leq \delta) + P(E^c, |\mathcal{T}(\Delta)| \geq \delta)\\
& \leq \int_0^\delta d P(|\mathcal{T}(\Delta)|) + P(E^c, |\Delta| \geq \delta/c)\\
& \leq n \delta + \left( \log_2 \frac{4C^2}{\delta^2 / c^2} +1\right) \cdot \exp\left(-\frac{C^2}{2}\right)\end{aligned}$$ where event $E^c$ is defined in the same way as in Lemma \[adapt.query\].
Since the above error is per one step of the random walk, we need to take $\delta = \tilde{\mathcal{O}}(1/n^6)$ so that $n\delta$ accumulated after $\tilde{\mathcal{O}}(n^5)$ steps is $o(1)$. We have that $\frac{4C^2}{\delta^2 / c^2}$ is at most polynomial in terms of $n$. Hence, for $C = O(\sqrt{\log n})$ with constant big enough, we can ensure $P({\rm Error}) = o(1/n^5)$. This last statement follows from Lemma \[dist.levelset\]; we remark that this bound is sharp because when the convex body is a high dimensional cone, the bound is exact. This is the error for each point we query as in noiseless case. The total number of query in noiseless case is $\tilde{\mathcal{O}}(n^5)$, thus the total error behaves as $o(1)$ by choosing $\delta$ small and $C$ big.
On the complement of the “Error" event, we have each step query complexity is bounded by $m = \frac{4C^2}{|\Delta|^2}$ (Lemma \[adapt.query\]). Thus the expected number of queries is $$\int_\delta^1 \frac{4C^2}{|\Delta|^2} d \mathcal{P}(|\mathcal{T}(\Delta)|).$$ Because of the uniform distribution on the convex body, the expected number of queries $\mathbb{E} N_q$ for each point is bounded by $$\mathbb{E} N_q = \int_\delta^1 \frac{4C^2}{|\Delta|^2} d \mathcal{P}(|\mathcal{T}(\Delta)|) \leq \int_\delta^1 \frac{4C^2}{s^2 / c^2} \cdot \frac{n+1}{2} d s \leq \tilde{\mathcal{O}}\left( n^9 \cdot \frac{1}{\epsilon^2}\right).$$ (by Lemma \[dist.levelset\] , the distribution of $|\mathcal{T}(\Delta)|$ has the relation $d \mathcal{P}(s) \leq \frac{(n+1)}{2} ds$.) We conclude that the total number of oracle queries is $\tilde{\mathcal{O}}(n^5) \cdot \mathbb{E} N_q$ , which is $\tilde{\mathcal{O}}(n^{14} \frac{1}{\epsilon^2})$.
\
[^1]: [email protected]
[^2]: [email protected]
[^3]: [email protected]
[^4]: The $\tilde{O}(\cdot)$ notation disregards polylogarithmic terms in $n$ and $1/\epsilon$.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'In a regular full exponential family, the maximum likelihood estimator (MLE) need not exist in the traditional sense, but the MLE may exist in the Barndorff-Nielsen completion of the family. Existing algorithms for finding the MLE in the Barndorff-Nielsen completion solve many linear programs; they are slow in small problems and too slow for large problems. We provide new, fast, and scalable methodology for finding the MLE in the Barndorff-Nielsen completion based on approximate null eigenvectors of the Fisher information matrix. Convergence of Fisher information follows from cumulant generating function convergence, conditions for which are given.'
bibliography:
- 'conjure.bib'
---
[**Computationally efficient likelihood inference in exponential families when the maximum likelihood estimator does not exist**]{}
Daniel J. Eck
*Department of Biostatistics, Yale School of Public Health.*\
*[email protected]*
Charles J. Geyer
*Department of Statistics, University of Minnesota*\
*[email protected]*
Key Words: Barndorff-Nielsen completion of exponential families; Convergence of moments; Cumulant generating function convergence; Generalized affine functions
\[firstpage\]
Introduction
============
We develop an inferential framework for regular full discrete exponential families when the observed value of the canonical statistic lies on the boundary of its convex support. Then the maximum likelihood estimator (MLE) for the canonical parameter cannot exist [@barndorff-nielsen Theorem 9.13]. But the MLE may exist in a completion of the exponential family. Completions for exponential families have been described (in order of increasing generality) by @barndorff-nielsen [pp. 154–156], @brown [pp. 191–201], @csiszar, and @geyer [unpublished PhD thesis, Chapter 4]. The latter two are equivalent for full exponential families, but the last, most general, has much stronger algebraic properties that help with theory, so we use it. It also is the only completion that is a completion under no regularity conditions whatsoever (other than exponential family). It works for curved exponential families and other non-full exponential families ([@csiszar] works for non-full but closed convex families). Following @geyer-gdor we will call all of these completions the Barndorff-Nielsen completion without fuss about the technical details differentiating them.
@geyer-gdor developed ways to do hypothesis tests and confidence intervals when the MLE in an exponential family does not exist in the conventional sense. The hypothesis test scheme was credited to Fienberg (personal communication — an answer he gave to a question at the end of a talk). The confidence interval scheme generates one-sided non-asymptotic confidence intervals, because the MLE fails to exist in the conventional sense when the canonical statistic is on the boundary of its convex support, and this is an inherently one-sided situation, and conventional asymptotics do not work near the boundary.
To simplify both explanation and computation, @geyer-gdor assumed the regularity conditions that @brown assumed for his completion. These conditions of Brown hold for nearly all applications known to us (applications for which a more general completion are required include aster models [@aster1] and Markov spatial point processes [@points]). We will also need to use Brown’s conditions to guarantee our methods work.
The issue of when the MLE exists in the conventional sense and what to do when it doesn’t is very important because of the wide use of generalized linear models for discrete data and log-linear models for categorical data. In every application of these, existing statistical software gives completely invalid results when the MLE does not exist in the conventional sense, but most such software either does not check for this problem or does very weak checks that have high probability of both false positives and false negatives. Moreover, even if these checks correctly detect nonexistence of the MLE in the conventional sense, conventional software implements no valid procedures for statistical inference when this happens. When this issue is detected most users will go to smaller statistical models for which the MLE seems to exist, even though such models may neither fit the data nor address the questions of scientific interest. @geyer-gdor gives examples with valid inference. Authoritative textbooks, such as @agresti [Section 6.5], discuss the issue but provide no solutions.
Thus a solution to this issue that is efficiently computable would be very important. The algorithms of @geyer [@geyer-gdor] and @albert-anderson are based on doing many linear programs. The algorithm of @geyer-gdor is the most efficient; it is mostly due to Fukuda, who provided the underlying C code for the computational geometry functions of R package `rcdd` [@geyer-rcdd], which this algorithm uses. This algorithm does at most $n$ linear programs, where $n$ is the number of cases of a generalized linear model (GLM) or the number of cells in a contingency table, in order to determine the existence of the MLE in the conventional sense. Each of these linear programs has $p$ variables, where $p$ is the number of parameters of the model, and up to $n$ inequality constraints. Since linear programming can take time exponential in $n$ when pivoting algorithms are used, and since such algorithms are necessary in computational geometry to get correct answers despite inaccuracy of computer arithmetic (see the warnings about the need to use rational arithmetic in the documentation for R package `rcdd` [@geyer-rcdd]), these algorithms can be very slow. Typically, they take several minutes of computer time for toy problems and can take longer than users are willing to wait for real applications. These algorithms do have the virtue that if they use infinite-precision rational arithmetic, then their calculations are exact, as good as a mathematical proof. Previous theoretical discussions of these issues that do not provide algorithms [@barndorff-nielsen; @brown; @csiszar] use the notions of faces of convex sets or tangent cones or normal cones and all of these are much harder to compute than the algorithm of @geyer-gdor. So they provide no direction toward efficient computing.
Because computational geometry is so slow and does not scale to large problems, we abandon it and return to calculations using the inexact computer arithmetic provided by computer hardware. Conventional maximum likelihood computations come close, in a sense, to finding the MLE in the Barndorff-Nielsen completion. They go uphill on the likelihood function until they meet their convergence criteria and stop. At this point, the canonical parameter estimates are still infinitely far away from their analogs for the MLE in the completion, but the corresponding probability distributions are close in total variation norm. Here we show that they are also close in the sense of moment generating function convergence (Theorem \[convex-poly-thm\] below) and consequently moments of all orders are also close. The MLE in the completion is not only a limit of distributions in the original family but also a distribution in the original family conditioned on the affine hull of a face of the effective domain of the log likelihood supremum function [@geyer Theorem 4.3, special cases of which were known to other authors]. To do valid statistical inference when the MLE does not exist in the conventional sense, we need to know this affine hull.
This affine hull is the support of the canonical statistic under the MLE distribution (in the completion). Hence it is a translate of the null space of the Fisher information matrix, which (for an exponential family) is the variance-covariance matrix of the canonical statistic. This affine hull must contain the mean vector of the canonical statistic under the MLE distribution. Hence knowing the mean vector and variance-covariance matrix of the canonical statistic under the MLE distribution allows us to do valid statistical inference, and our conventional maximum likelihood calculation (go uphill until things don’t change much in an iteration) will give us good approximations of them (relative to the inexactness of computer arithmetic).
We will get nearly the correct affine hull if we can guess the correct null space of the Fisher information matrix from its eigenvalues and eigenvectors computed using inexact computer arithmetic. We will not be able to do this when the statistical model has an ill-conditioned model matrix (the model matrix for categorical data analysis being the model matrix when it is recast as a Poisson regression). Ill-conditioning will add spurious nearly zero eigenvalues that arise from the ill-conditioning rather than the concentration of the MLE distribution on the correct affine hull. We will suppose that the model matrix is not ill-conditioned. If a sequence of parameter estimates maximizes the likelihood, then the corresponding sequence of probability density functions (PDFs) has subsequences converging to PDFs of MLE distributions in the Barndorff-Nielsen completion [@geyer Theorem 4.1]. If the MLE distribution is unique, as it always is for a full exponential family [@geyer-gdor Section 3.8], then all of these MLE PDFs will correspond to the same probability distribution. For a curved exponential family, the MLE need not be unique, even when it exists in the conventional sense.
Motivating example {#sec:motivating-example}
==================
Consider the case of complete separation in the logistic regression model as an example of a discrete exponential family with data on the boundary of the convex support of the canonical statistic. Suppose that we have one predictor vector $x$ having values 10, 20, 30, 40, 60, 70, 80, 90, and suppose the components of the response vector $y$ are 0, 0, 0, 0, 1, 1, 1, 1. Then the simple logistic regression model that has linear predictor $\eta = \beta_0 + \beta_1 x$ exhibits failure of the MLE to exist in the conventional sense. This example is the same as that of @agresti [Section 6.5.1].
For an exponential family, the submodel canonical statistic is $M^T y$, where $M$ is the model matrix [@geyer-gdor Section 3.9]. Figure \[fig:boundary\] shows the observed value of the canonical statistic vector and the support (all possible values) of this vector.
![Observed value and support of the submodel canonical statistic vector $M^T y$ for the example of Section \[sec:motivating-example\]. Solid dot is the observed value of this statistic.[]{data-label="fig:boundary"}](coursenotesscript-2){width="3.5in"}
As is obvious from the figure, the observed value of the canonical statistic is on the boundary of the convex support, in which case the MLE does not exist in the conventional sense [@geyer-gdor Theorem 4]. In general, this figure is too computationally intensive and too high-dimensional to draw. So our methods do not use such figures. It is here to develop intuition. In our methodology, this degeneracy follows from the Fisher information matrix at the apparent MLE being nearly the zero matrix.
In this example, like in Example 1 of @geyer-gdor, the MLE in the Barndorff-Nielsen completion corresponds to a completely degenerate distribution. This MLE distribution says no other data than what was observed could be observed. But the sample is not the population and estimates are not parameters. So this degeneracy is not a problem. To illustrate the uncertainty of estimation we follow Figure 2 of @geyer-gdor, which shows confidence intervals (necessarily one-sided) for the saturated model mean value parameters.
![One-sided 95% confidence intervals for saturated model mean value parameters. Bars are the intervals; $\mu(x)$ is the probability of observing response value one when the predictor value is $x$. Solid dots are the observed data.[]{data-label="fig:intervals"}](coursenotesscript-too-1){width="3.5in"}
Our Figure \[fig:intervals\] shows that, as would be expected from so little data, the confidence intervals are very wide. The MLE in the completion says the probability of observing a response equal to one jumps from zero to one somewhere between 40 and 60. The confidence intervals show that we are fairly sure that this probability goes from near zero at $x = 10$ to near one at $x = 90$ but we are very unsure where jumps are if there are any. These intervals were constructed using the theory of @geyer-gdor [Section 3.16]. The actual computations follow some later course notes [@infinity].
Our theory allows for inference in not only the complete separation example but also in any discrete regular full exponential family where the MLE does not exist in the traditional sense. For further motivation, see the examples in Section 2 of [@geyer-gdor]. We redo Example 2.3 of [@geyer-gdor] in our Section 7.2, and we find that our methodology produces the same inferences as theirs in a fraction of the time.
Laplace transforms and standard exponential families {#sec:laptr}
====================================================
Let $\lambda$ be a positive Borel measure on a finite-dimensional vector space $E$. The *log Laplace transform* of $\lambda$ is the function $c : {E^{\textstyle{*}}}\to {\overline{{\mathbb{R}}}}$ defined by $$\label{eq:cumfun}
c(\theta) = \log \int e^{{\langle x, \theta \rangle}} \, \lambda(d x),
\qquad \theta \in {E^{\textstyle{*}}},$$ where ${E^{\textstyle{*}}}$ is the dual space of $E$, where ${\langle {\,\cdot\,}, {\,\cdot\,}\rangle}$ is the canonical bilinear form placing $E$ and ${E^{\textstyle{*}}}$ in duality, and where ${\overline{{\mathbb{R}}}}$ is the extended real number system, which adds the values $- \infty$ and $+ \infty$ to the real numbers with the obvious extensions to the arithmetic and topology [@rock-wets Section 1.E].
If one prefers, one can take $E = {E^{\textstyle{*}}}= {\mathbb{R}}^p$ for some $p$, and define $${\langle x, \theta \rangle} = \sum_{i = 1}^p x_i \theta_i,
\qquad \text{$x \in {\mathbb{R}}^p$ and $\theta \in {\mathbb{R}}^p$},$$ but the coordinate-free view of vector spaces offers more generality and more elegance. Also, as we are about to see, if $E$ is the sample space of a standard exponential family, then a subset of ${E^{\textstyle{*}}}$ is the canonical parameter space, and the distinction between $E$ and ${E^{\textstyle{*}}}$ helps remind us that we should not consider these two spaces to be the same space.
A log Laplace transform is a lower semicontinuous convex function that nowhere takes the value $- \infty$ (the value $+ \infty$ is allowed and occurs where the integral in does not exist) [@geyer Theorem 2.1]. The *effective domain* of an extended-real-valued convex function $c$ on ${E^{\textstyle{*}}}$ is $$\operatorname{dom}c = {\{\, \theta \in {E^{\textstyle{*}}}: c(\theta) < + \infty \,\}}.$$
For every $\theta \in \operatorname{dom}c$, the function $f_\theta : E \to {\mathbb{R}}$ defined by $$\label{eq:densities}
f_\theta(x) = e^{{\langle x, \theta \rangle} - c(\theta)}, \qquad x \in E,$$ is a probability density with respect to $\lambda$. The set $
\mathcal{F} = {\{\, f_\theta : \theta \in \Theta \,\}},
$ where $\Theta$ is any nonempty subset of $\operatorname{dom}c$, is called a *standard exponential family of densities with respect to $\lambda$*. This family is *full* if $\Theta = \operatorname{dom}c$. We also say $\mathcal{F}$ is the standard exponential family *generated by* $\lambda$ having canonical parameter space $\Theta$, and $\lambda$ is the *generating measure* of $\mathcal{F}$.
The log likelihood of this family is have log likelihood $$\label{log-like}
l(\theta) = {\langle x, \theta \rangle} - c(\theta).$$
A general exponential family [@geyer Chapter 1] is a family of probability distributions having a sufficient statistic $X$ taking values in a finite-dimensional vector space $E$ that induces a family of distributions on $E$ that have a standard exponential family of densities with respect to some generating measure. Reduction by sufficiency loses no statistical information, so the theory of standard exponential families tells us everything about general exponential families [@geyer Section 1.2].
In the context of general exponential families $X$ is called the *canonical statistic* and $\theta$ the *canonical parameter* (the terms *natural statistic* and *natural parameter* are also used). The set $\Theta$ is the canonical parameter space of the family, the set $\operatorname{dom}c$ is the canonical parameter space of the full family having the same generating measure. A full exponential family is said to be *regular* if its canonical parameter space $\operatorname{dom}c$ is an open subset of ${E^{\textstyle{*}}}$.
The cumulant generating function (CGF) of the distribution of the canonical statistic for parameter value $\theta$ is the function $k_\theta$ defined by $$\label{eq:cum-gen-fun}
\begin{split}
k_\theta(t)
& =
\log \int e^{{\langle x, t \rangle}} f_\theta(x) \, \lambda(d x)
\\
& =
c(\theta + t) - c(\theta)
\end{split}$$ provided this distribution has a CGF, which it does if and only if $k_\theta$ is finite on a neighborhood of zero, that is, if and only if $\theta \in \operatorname{int}(\operatorname{dom}c)$. Thus every distribution in a full family has a CGF if and only if the family is regular. Derivatives of $k_\theta$ evaluated at zero are the cumulants of the distribution for $\theta$. These are the same as derivatives of $c$ evaluated at $\theta$.
Generalized affine functions
============================
Characterization on affine spaces
---------------------------------
Exponential families defined on affine spaces instead of vector spaces are in many ways more elegant [@geyer Sections 1.4 and 1.5 and Chapter 4]. To start, a family of densities with respect to a positive Borel measure on an affine space is a *standard exponential family* if the log densities are affine functions. Following @geyer [Chapter 4], we complete the exponential family by taking pointwise limits of densities, allowing $+ \infty$ and $- \infty$ as limits.
We call these limits *generalized affine functions*. Real-valued affine functions on an affine space are functions that are are both convex and concave. *Generalized affine functions* on an affine space are extended-real-valued functions that are are both concave and convex [@geyer Chapter 4]. (For a definition of extended-real-valued convex functions see @rock-convex [Chapter 4].)
We thus have two characterizations of generalized affine functions: functions that are both convex and concave and functions that are limits of sequences of affine functions. Further characterizations will be given below.
Let $h_n$ denote a sequence of affine functions that are log densities in a standard exponential family with respect to $\lambda$, that is, $\int e^{h_n} \, d \lambda = 1$ for all $n$. Since $e^{h_n} \to e^h$ pointwise if and only if $h_n \to h$ pointwise, the idea of completing an exponential family naturally leads to the the study of generalized affine functions.
If $h : E \to {\overline{{\mathbb{R}}}}$ is a generalized affine function, we use the notation $$\begin{aligned}
h^{- 1}({\mathbb{R}}) & = {\{\, x \in E : h(x) \in {\mathbb{R}}\,\}}
\\
h^{- 1}(\infty) & = {\{\, x \in E : h(x) = \infty \,\}}
\\
h^{- 1}(-\infty) & = {\{\, x \in E : h(x) = -\infty \,\}}\end{aligned}$$
\[recurse\] An extended-real-valued function $h$ on a finite-dimensional affine space $E$ is generalized affine if and only if one of the following cases holds
- $h^{- 1}(\infty) = E$,
- $h^{- 1}(-\infty) = E$,
- $h^{- 1}({\mathbb{R}}) = E$ and $h$ is an affine function, or
- there is a hyperplane $H$ such that $h(x) = \infty$ for all points on one side of $H$, $h(x) = - \infty$ for all points on the other side of $H$, and $h$ restricted to $H$ is a generalized affine function.
All theorems for which a proof does not follow the theorem statement are proved in either the appendix or the supplementary material.
The intention is that this theorem is applied recursively. If we are in case (d), then the restriction of $h$ to $H$ is another generalized affine function to which the theorem applies. Since a nested sequence of hyperplanes can have length at most the dimension of $E$, the recursion always terminates.
Topology
--------
Let $G(E)$ denote the space of generalized affine functions on a finite-dimensional affine space $E$ with the topology of pointwise convergence.
\[compact-Hausdorff\] $G(E)$ is a compact Hausdorff space.
\[first-countable\] $G(E)$ is a first countable topological space.
\[sequentially-compact\] $G(E)$ is sequentially compact.
Sequentially compact means every sequence has a (pointwise) convergent subsequence. That this follows from the two preceding theorems is well known [@counterexamples p. 22, gives a proof].
The space $G(E)$ is not metrizable, unless $E$ is zero-dimensional [@geyer penultimate paragraph of Section 3.3]. So we cannot use $\delta$-$\varepsilon$ arguments, but we can use arguments involving sequences, using sequential compactness.
Let $\lambda$ be a positive Borel measure on $E$, and let $\mathcal{H}$ be a nonempty subset of $G(E)$ such that $$\label{eq:proper}
\int e^h \, d \lambda = 1, \qquad h \in \mathcal{H}.$$ Then, following @geyer [Chapter 4], we call $\mathcal{H}$ a *standard generalized exponential family* of log densities with respect to $\lambda$. Let $\overline{\mathcal{H}}$ denote the closure of $\mathcal{H}$ in $G(E)$.
\[closure\] Maximum likelihood estimates always exist in the closure $\overline{\mathcal{H}}$.
Suppose $x$ is the observed value of the canonical statistic. Then there exists a sequence $h_n$ in $\mathcal{H}$ such that $$h_n(x) \to \sup_{h \in \mathcal{H}} h(x).$$ This sequence has a convergent subsequence $h_{n_k} \to h$ in $G(E)$. This limit $h$ is in $\overline{\mathcal{H}}$ and maximizes the likelihood.
We claim this is the right way to think about completion of exponential families. For full exponential families or even closed convex exponential families the closure only contains *proper* log probability densities ($h$ that satisfy the equation in ). This is shown by @geyer [Chapter 2] and also by @csiszar.
For curved exponential families and for general non-full exponential families, applying Fatou’s lemma to pointwise convergence in $G(E)$ gives only $$\label{eq:improper}
0 \le \int e^h \, d \lambda \le 1, \qquad h \in \overline{\mathcal{H}}.$$ When the integral in is strictly less than one we say $h$ is an *improper* log probability density. The examples in @geyer [Chapter 4] show that improper probability densities cannot be avoided in curved exponential families.
@geyer [Theorem 4.3] shows that this closure of an exponential family can be thought of as a union of exponential families, so this generalizes the conception of @brown of the closure as an *aggregate exponential family*. Thus our method generalizes all previous methods of completing exponential families.
Admittedly, this characterization of the completion of an exponential family is very different from any other in its ignoring of parameters. Only log densities appear. Unless one wants to call them parameters — and that conflicts with the usual definition of parameters as real-valued — parameters just do not appear.
So in the next section, we bring parameters back.
Characterization on vector spaces
---------------------------------
In this section we take sample space $E$ to be vector space (which, of course, is also an affine space, so the results of the preceding section continue to hold). Recall from Section \[sec:laptr\] above, that ${E^{\textstyle{*}}}$ denotes the dual space of $E$, which contains the canonical parameter space of the exponential family.
\[vec-char\] An extended-real-valued function $h$ on a finite-dimensional vector space $E$ is generalized affine if and only if there exist finite sequences (perhaps of length zero) of vectors $\eta_1$, $\ldots,$ $\eta_j$ in in ${E^{\textstyle{*}}}$ and scalars $\delta_1$, $\ldots,$ $\delta_j$ such that $\eta_1$, $\ldots,$ $\eta_j$ are linearly independent and $h$ has the following form. Define $H_0 = E$ and, inductively, for integers $i$ such that $0 < i \le j$ $$\begin{aligned}
H_i & = {\{\, x \in H_{i - 1} : {\langle x, \eta_i \rangle} = \delta_i \,\}}
\\
C_i^+ & = {\{\, x \in H_{i - 1} : {\langle x, \eta_i \rangle} > \delta_i \,\}}
\\
C_i^- & = {\{\, x \in H_{i - 1} : {\langle x, \eta_i \rangle} < \delta_i \,\}}\end{aligned}$$ all of these sets (if any) being nonempty. Then $h(x) = + \infty$ whenever $x \in C_i^+$ for any $i$, $h(x) = - \infty$ whenever $x \in C_i^-$ for any $i$, and $h$ is either affine or constant on $H_j$, where $+ \infty$ and $-\infty$ are allowed for constant values.
The “if any” refers to the case where the sequences have length zero, in which case the theorem asserts that $h$ that $h$ is affine on $E$ or constant on $E$.
As we saw in the preceding section, we are interested in likelihood maximizing sequences. Here we represent the likelihood maximizing sequence in the coordinates of the linearly independent $\eta$ vectors that characterize the generalized affine function $h$ according to its Theorem \[vec-char\] representation. Let $\theta_n$ be a likelihood maximizing sequence of canonical parameter vectors, that is, $$\label{max-like-seq}
l(\theta_n) \to \sup_{\theta \in \Theta} l(\theta),
\qquad \text{as $n \to \infty$},$$ where the log likelihood $l$ is given by and where $\Theta$ is the canonical parameter space of the family. To make connection with the preceding section, define $$h_\theta(x) = l(\theta) = {\langle x, \theta \rangle} - c(\theta).$$ Then $h_{\theta_n}$ is a sequence of affine functions, which has a subsequence that converges (in $G(E)$) to some generalized affine function $h \in \overline{\mathcal{H}}$, which maximizes the likelihood: $$\label{h-theta}
h(x) = \sup_{\theta \in \Theta} l(\theta).$$ The following lemma gives us a better understanding of the convergence $h_{\theta_n} \to h$.
\[properties-1\] Suppose that a generalized affine function $h$ on a finite dimensional vector space $E$ is finite at at least one point. Represent $h$ as in Theorem \[vec-char\], and extend $\eta_1$, $\ldots,$ $\eta_j$ to be a basis $\eta_1$, $\ldots,$ $\eta_p$ for $E^{\textstyle *}$. Suppose $h_n$ is a sequence of affine functions converging to $h$ in $G(E)$. Then there are sequences of scalars $a_n$ and $b_{i, n}$ such that $$\label{affine-seq}
h_n(y) = a_n + \sum_{i = 1}^j b_{i, n} \left({\langle y, \eta_i \rangle} -
\delta_i\right)
+ \sum_{i = j + 1}^p b_{i, n} {\langle y, \eta_i \rangle}, \qquad y \in E,$$ and, as $n \to \infty$, we have
1. $b_{i, n} \to \infty$, for $1 \le i \le j$,
2. $b_{i, n} / b_{i - 1, n} \to 0$, for $2 \le i \le j$,
3. $b_{i, n}$ converges, for $i > j$, and
4. $a_n$ converges.
In the first sum is empty when $j = 0$ and the second sum is empty when $j = p$. Such empty sums are zero by convention.
The results given in Lemma \[properties-1\] are applicable to generalized affine functions in full generality. The case of interest to us, however, is when $h_n = h_{\theta_n}$ is the likelihood maximizing sequence constructed above.
\[properties-2\] For data $x$ from a regular full exponential family defined on a vector space $E$, suppose $\theta_{n}$ is a likelihood maximizing sequence satisfying with log densities $h_n = h_{\theta_n}$ defined by converging pointwise to a generalized affine function $h$. Characterize $h$ and $h_n$ as in Theorem \[vec-char\] and Lemma \[properties-1\]. Define $\psi_n = \sum_{i=j+1}^p b_{i,n}{\langle x,\eta_i \rangle}$. Then conclusions *(a)* and *(b)* of Lemma \[properties-1\] hold in this setting and $$\psi_n \to {\theta^{\textstyle{*}}}, \quad \text{as} \; n \to \infty,$$ where ${\theta^{\textstyle{*}}}$ is the MLE of the exponential family conditioned to $H_j$.
In case $j = p$ the conclusion $\psi_n \to {\theta^{\textstyle{*}}}$ is the trivial zero converges to zero. The original exponential family conditioned on the event $H_j$ is what @geyer-gdor calls the limiting conditional model (LCM).
The conditions of Lemma \[properties-1\] are satisfied by our assumptions so all conclusions of Lemma \[properties-1\] are satisfied. As a consequence, $\psi_n \to {\theta^{\textstyle{*}}}$ as $n \to \infty$. The fact that ${\theta^{\textstyle{*}}}$ is the MLE of the LCM restricted to $H_j$ follows from our assumption that $\theta_n$ is a likelihood maximizing sequence.
Taken together, Theorem \[vec-char\], Lemma \[properties-1\], and Corollary \[properties-2\] provide a theory of maximum likelihood estimation in the completions of exponential families that is the theory of the preceding section with canonical parameters brought back.
Convergence theorems
====================
Cumulant generating function convergence
----------------------------------------
We now show CGF convergence along likelihood maximizing sequences . This implies convergence in distribution and convergence of moments of all orders.
Theorems \[main\] and \[convex-poly-thm\] in this section say when CGF convergence occurs. Their conditions are somewhat unnatural (especially those of Theorem \[main\]). However, the example in Section 4 of the supplementary material shows not only that some conditions are necessary to obtain CGF convergence (it does not occur for all full discrete exponential families) but also that the conditions of Theorem \[main\] are sharp, being just what is needed to rule out that example.
The CGF of the distribution having log density that is the generalized affine function $h$ is defined by $$\kappa(t) = \log\int e^{\langle y,t \rangle}e^{h(y)} \, \lambda(d y),$$ and similarly $$\kappa_{n}(t) =
\log\int e^{\langle y,t \rangle}e^{h_n(y)} \, \lambda(d y)$$ where we assume $h_n$ are the log densities for a likelihood maximizing sequence such that $h_n \to h$ pointwise. The next theorem characterizes when $\kappa_n \to \kappa$ pointwise.
Let $c_A$ denote the log Laplace transform of the restriction of $\lambda$ to the set $A$, that is, $$c_A(\theta) = \log\int_A e^{{\langle y,\theta \rangle}} \, \lambda(d y),$$ where, as usual, the value of the integral is taken to be $+ \infty$ when the integral does not exist (a convention that will hold for the rest of this section).
\[main\] Let $E$ be a finite-dimensional vector space of dimension $p$. For data $x \in E$ from a regular full exponential family with natural parameter space $\Theta \subseteq {E^{\textstyle{*}}}$ and generating measure $\lambda$, assume that all LCMs are regular exponential families. Suppose that $\theta_{n}$ is a likelihood maximizing sequence satisfying with log densities $h_n$ converging pointwise to a generalized affine function $h$. Characterize $h$ as in Theorem \[vec-char\]. When $j \geq 2$, and for $i = 1,...,j-1$, define $$\label{DiF}
\begin{split}
D_i &= \{y \in C_i^-: {\langle y,\eta_k \rangle} > \delta_k, \;
\text{some} \; k > i \}, \\
F &= E \setminus \cup_{i=1}^{j-1}D_i
= \{y: {\langle y,\eta_i \rangle} \leq \delta_i, \; 1 \leq i \leq j\},
\end{split}$$ and assume that $$\label{main-bound}
\sup_{\theta \in \Theta}\sup_{y \in \cup_{i=1}^{j-1}D_i}
e^{{\langle y,\theta \rangle} - c_{\cup_{i=1}^{j-1}D_i}(\theta)} < \infty
\quad \text{or} \quad
\lambda\left(\cup_{i=1}^{j-1}D_i\right) = 0.$$ Then $\kappa_n(t)$ converges to $\kappa(t)$ pointwise for all $t$ in a neighborhood of 0.
Discrete exponential families automatically satisfy when $$\inf_{y \in \cup_{i=1}^{j-1}D_i}\lambda(\{y\}) > 0.$$ In this setting, $e^{{\langle y,\theta \rangle} - c_{\cup_{i=1}^{j-1}D_i}(\theta)}$ corresponds to the probability mass function for the random variable conditional on the occurrence of $\cup_{i=1}^{j-1}D_i$. Thus, $$\begin{aligned}
&\sup_{\theta \in \Theta}\sup_{y\in\cup_{i=1}^{j-1}D_i}\left(
e^{{\langle y,\theta \rangle} - c_{\cup_{i=1}^{j-1}D_i}(\theta)}\right) \\
&\qquad = \sup_{\theta \in \Theta}\sup_{y\in\cup_{i=1}^{j-1}D_i}\left(
\frac{e^{{\langle y,\theta \rangle}}\lambda(\{y\})}{\lambda(\{y\})
\sum_{x\in \cup_{i=1}^{j-1}D_i} e^{{\langle x,\theta \rangle}}
\lambda(\{x\})}\right) \\
&\qquad \leq \sup_{y\in\cup_{i=1}^{j-1}D_i} \left(1/\lambda(\{y\})\right)
< \infty.\end{aligned}$$ Therefore, Theorem \[main\] is applicable for the non-existence of the maximum likelihood estimator that may arise in logistic and multinomial regression. We show in the next theorem that discrete families with convex polyhedral support $K$ also satisfy under additional regularity conditions that hold in practical applications. When $K$ is convex polyhedron, we can write $
K = \{y : {\langle y, \alpha_i \rangle} \leq a_i, \; \text{for} \; i = 1,...,m \},
$ as in [@rock-wets Theorem 6.46]. When the MLE does not exist, the data $x \in K$ is on the boundary of $K$. Denote the active set of indices corresponding to the boundary $K$ containing $x$ by $
I(x) = \{ i : {\langle x, \alpha_i \rangle} = a_i \}.
$ In preparation for Theorem \[convex-poly-thm\] we define the normal cone $N_K(x)$, the tangent cone $T_K(x)$, and faces of convex sets and then state conditions required on $K$.
The normal cone of a convex set $K$ in the finite dimensional vector space $E$ at a point $x \in K$ is $$N_K(x) = {\{\, \eta \in {E^{\textstyle{*}}}: {\langle y - x, \eta \rangle} \leq 0
\; \text{for all} \; y \in K \,\}}.$$
The tangent cone of a convex set $K$ in the finite dimensional vector space $E$ at a point $x \in K$ is $$T_K(x) = \operatorname{cl}{\{\, s(y - x) : y \in K \; {\mathrel{\rm and}}\; s \geq 0 \,\}}$$ where $\operatorname{cl}$ denotes the set closure operation.
When $K$ is a convex polyhedron, $N_K(x)$ and $T_K(x)$ are both convex polyhedron with formulas given in [@rock-wets Theorem 6.46]. These formulas are $$\begin{aligned}
T_K(x) &= \{y : {\langle y, \alpha_i \rangle} \leq 0 \; \text{for all} \;
i \in I(x)\}, \\
N_K(x) &= \{c_1 \alpha_1 + \cdots + c_m \alpha_m : c_i \geq 0 \;
\text{for} \; i \in I(x), \; c_i = 0 \; \text{for} \; i \notin I(x) \}.\end{aligned}$$
A *face* of a convex set $K$ is a convex subset $F$ of $K$ such that every (closed) line segment in $K$ with a relative interior point in $F$ has both endpoints in $F$. An *exposed face* of $K$ is a face where a certain linear function achieves its maximum over $K$ [@rock-convex p. 162].
The conditions required on $K$ for our theory to hold are from @brown [pp. 193–197]. These conditions are:
- The support of the exponential family is a countable set $X$.
- The exponential family is regular.
- Every $x \in X$ is contained in the relative interior of an exposed face $F$ of the convex support $K$.
- The convex support of the measure $\lambda|F$ equals $F$, where $\lambda$ is the generating measure for the exponential family.
Conditions (i) and (ii) are already assumed in Theorem \[main\]. It is now shown that discrete exponential families satisfy under the above conditions.
\[convex-poly-thm\] Assume the conditions of Theorem \[main\] with the omission of when $j \geq 2$. Let $K$ denote the convex support of the exponential family. Assume that the exponential family satisfies the conditions of Brown. Then holds.
Consequences of CGF convergence
-------------------------------
Theorems \[main\] and \[convex-poly-thm\] both verify CGF convergence along likelihood maximizing sequences on neighborhoods of 0. The next theorems show that CGF convergence on neighborhoods of 0 is enough to imply convergence in distribution and of moments of all orders. Therefore moments of distributions with log densities that are affine functions converge along likelihood maximizing sequences to those of a limiting distributions whose log density is a generalized affine function.
Suppose that $X$ is a random vector in a finite-dimensional vector space $E$ having a moment generating function (MGF) $\varphi_X$, then $$\varphi_X(t) = \varphi_{{\langle X, t \rangle}}(1), \qquad t \in {E^{\textstyle{*}}},$$ regardless of whether the MGF exist or not. It follows that the MGF of ${\langle X, t \rangle}$ for all $t$ determine the MGF of $X$ and vice versa, when these MGF exist. More generally, $$\label{eq:cramer-wold}
\varphi_{{\langle X, t \rangle}}(s) = \varphi_X(s t),
\qquad t \in E^{\textstyle *} {\mathrel{\rm and}}s \in {\mathbb{R}}.$$ This observation applied to characteristic functions rather than MGF is called the Cramér-Wold theorem. In that context it is more trivial because characteristic functions always exist.
If $v_1$, $\ldots,$ $v_d$ is a basis for a vector space $E$, then there exists a unique dual basis $w_1$, $\ldots,$ $w_d$ for $E^{\textstyle *}$ that satisfies $$\label{eq:dual-bases}
{\langle v_i, w_j \rangle} = \begin{cases} 1, & i = j \\ 0, & i \neq j \end{cases}$$ [@halmos Theorem 2 of Section 15].
\[MVmgf-1\] If $X$ is a random vector in $E$ having an MGF, then the random scalar ${\langle X, t \rangle}$ has an MGF for all $t \in {E^{\textstyle{*}}}$. Conversely, if ${\langle X, t \rangle}$ has an MGF for all $t \in {E^{\textstyle{*}}}$, then $X$ has an MGF.
\[MVmgf-2\] Suppose $X_n$, $n = 1$, $2$, $\ldots$ is a sequence of random vectors, and suppose their moment generating functions converge pointwise on a neighborhood $W$ of zero. Then $$\label{eq:weak}
X_n {\stackrel{d}{\longrightarrow}}X,$$ and $X$ has an MGF $\varphi_X$, and $$\varphi_{X_n}(t) \to \varphi_X(t), \qquad t \in E^{\textstyle *}.$$
\[MVmgf-3\] Under the assumptions of Theorem \[MVmgf-2\], suppose $t_1$, $t_2$, $\ldots,$ $t_k$ are vectors defined on ${E^{\textstyle{*}}}$, the dual space of $E$. Then $
\prod_{i = 1}^k {\langle X_n, t_i \rangle}
$ is uniformly integrable so $$\operatorname{E}\left\{ \prod_{i = 1}^k {\langle X_n, t_i \rangle} \right\}
\to
\operatorname{E}\left\{ \prod_{i = 1}^k {\langle X, t_i \rangle} \right\}.$$
The combination of Theorems \[main\]-\[MVmgf-3\] provide a methodology for statistical inference along likelihood maximizing sequences when the MLE is in the Barndorff-Nielsen completion. In particular, we have convergence in distribution and convergence of moments of all orders along likelihood maximizing sequence. The limiting distribution in this context is a generalized exponential family with density $e^h$ where $h$ is a generalized affine function.
Convergence of null spaces of Fisher information
------------------------------------------------
Our method for finding the MLE in the Barndorff-Nielsen completion relies on finding the null space of the Fisher information matrix. We need to show that we have convergence for that. In order to prove this we need an appropriate notion of convergence of vector subspaces.
Painlevé-Kuratowski set convergence [@rock-wets Section 4.A] can be defined as follows (@rock-wets give many equivalent characterizations). If $C_n$ is a sequence of sets in ${\mathbb{R}}^p$ and $C$ is another set in ${\mathbb{R}}^p$, then we say $C_n \to C$ if
- For every $x \in C$ there exists a subsequence $n_k$ of the natural numbers and there exist $x_{n_k} \in C_{n_k}$ such that $x_{n_k} \to x$.
- For every sequence $x_n \to x$ in ${\mathbb{R}}^p$ such that there exists a natural number $N$ such that $x_n \in C_n$ whenever $n \geq N$, we have $x \in C$.
\[implem\] Suppose that $A_n \in {\mathbb{R}}^{p \times p}$ is a sequence of positive semidefinite matrices and $A_n \to A$ componentwise. Fix $\varepsilon > 0$ less than half of the least nonzero eigenvalue of $A$ unless $A$ is the zero matrix in which case $\varepsilon > 0$ may be chosen arbitrarily. Let $V_n$ denote the subspace spanned by the eigenvectors of $A_n$ corresponding to eigenvalues that are less than $\varepsilon$. Let $V$ denote the null space of $A$. Then $V_n \to V$ (Painlevé-Kuratowski).
Calculating the MLE in the completion
=====================================
Assumptions
-----------
So far everything has been for general exponential families except for Theorems \[main\] and \[convex-poly-thm\], the later of which assumes the conditions of @brown, and those conditions hold for GLM and log-linear models for categorical data analysis.
Now, following @geyer-gdor we restrict our attention to discrete GLM. This, in effect, includes log-linear models for contingency tables because we can always assume Poisson sampling, which makes them equivalent to GLM \[[@agresti Section 8.6.7]; [@geyer-gdor Section 3.17]\].
The form of the MLE in the completion
-------------------------------------
### First characterization {#sec:first-characterization}
Suppose we know the *affine support* of the MLE distribution in the completion. This is the smallest affine set that contains the canonical statistic with probability one. Denote the affine support by $A$. An affine set is a translate of a vector subspace. Since the observed value of the canonical statistic is contained in $A$ with probability one, and the canonical statistic for a GLM is $M^T Y$, where $M$ is the model matrix, $Y$ is the response vector, and $y$ its observed value [@geyer-gdor Section 3.9], we have $A = M^T y + V$ for some vector space $V$.
Then the LCM in which the MLE in the completion is found is the OM conditioned on the event $$M^T (Y - y) \in V, \qquad \text{almost surely}$$ [@geyer Theorem 4.3]. Suppose we characterize $V$ as the subspace where a finite set of linear equalities are satisfied $$V = {\{\, w \in {\mathbb{R}}^p : {\langle w, \eta_i \rangle} = 0, \ i = 1, \ldots, j \,\}}.$$ Then the LCM is the OM conditioned on the event $${\langle M^T (Y - y), \eta_i \rangle} = {\langle Y - y, M \eta_i \rangle} = 0,
\qquad i = 1, \ldots, j.$$ From this we see that the vectors $\eta_1$, $\ldots,$ $\eta_j$ span the null space of the Fisher information matrix for the LCM, which our Theorems \[convex-poly-thm\] and \[MVmgf-3\] say is well approximated by the Fisher information matrix for the OM at parameter values that are close to maximizing the likelihood.
The vector subspace spanned by the vectors $\eta_1$, $\ldots,$ $\eta_j$ is called the *constancy space* of the LCM in [@geyer-gdor].
### Second characterization {#sec:second-characterization}
Any vector $\delta$ in the canonical parameter space of an exponential family is called a *direction of recession* (DOR) if the likelihood function is nondecreasing in that direction or, equivalently, if $$\label{eq:dor}
{\langle Y, M \delta \rangle} \le {\langle y, M \delta \rangle}, \qquad \text{almost surely},$$ where $Y$ denotes the response vector considered as a random vector, $y$ denotes its observed value, and $M$ is the model matrix \[[@geyer Section 2.2]; [@geyer-gdor Theorem 3 and the following discussion and Section 3.9]\].
If we can find a DOR, then the MLE in the completion is a distribution in the LCM, which is the family of distributions in the original model (OM) conditioned on the event $$\label{eq:lcm}
{\langle Y, M \delta \rangle} = {\langle y, M \delta \rangle}, \qquad \text{almost surely},$$ or a distribution in the completion of the LCM \[[@geyer Chapter 2]; [@geyer-gdor Section 3]\].
Define $\zeta = M \delta$. In light of the only way can hold is if $\zeta_i \neq 0$ implies $Y_i = y_i$ almost surely.
Thus the distributions in the LCM are the distributions in the OM conditioned on this event. Moreover the MLE in the LCM is easily found using standard software. We simply change the model by removing the components of the response that are fixed in the LCM. Using R function GLM this is done using the optional argument `subset = zeta == 0`, where `zeta` is the R object corresponding to the vector $\zeta$.
If the MLE for the LCM exists in the conventional sense, then we have solved the problem of finding the MLE in the completion of the OM. If not we have to solve the problem of finding the MLE in the completion of the LCM we found, and that is done as we did before. And so forth. The iteration must terminate because each LCM has smaller dimension than the one before. @geyer [Chapter 2] gives details.
### Third characterization {#sec:third-characterization}
@geyer-gdor [Section 3] shows that the recursion in the preceding section can be avoided by use of a *generic direction of recession* (GDOR), which is a DOR in the relative interior of the set of all DOR.
Calculating limiting conditional models
---------------------------------------
### Based on the first characterization {#sec:based-first}
We do not need a DOR because we only use that to determine the affine support of the LCM, and we can estimate that by other methods. Suppose $\eta_1$, $\ldots,$ $\eta_j$ and other notation are as in Section \[sec:first-characterization\] above. The LCM is the OM conditioned on the event $$\label{eq:condition:one}
{\langle Y, M \eta_i \rangle} = {\langle y, M \eta_i \rangle},
\qquad \text{almost surely for $i \in 1, \ldots, j$}.$$ We have no readily available way to fit a GLM subject to .
We know, however, (Section \[sec:second-characterization\] above) that the event fixes some components of the response vector at their observed values and leaves the rest entirely unconstrained. Those components, that are entirely unconstrained are those for which the corresponding component of $M \eta_i$ is zero (or, taking account of the inexactness of computer arithmetic, nearly zero) for all $i = 1$, $\ldots,$ $j$.
### Based on the second characterizations {#sec:based-second}
We can find a DOR by minimizing the function $f$ defined by $$\label{eq:f}
f(a) = \max_{i \in 1, \ldots, m} \sum_{k = 1}^j a_k {\langle v_i, \eta_k \rangle},$$ where $v_1$, $\ldots,$ $v_m$ are vectors that generate the tangent cone for the GLM, which are defined in @geyer-gdor [Sections 3.10 and 3.11] and for which R code to calculating them is given in the technical reports accompanying [@geyer-gdor], and where $\eta_1$, $\ldots,$ $\eta_j$ are as in Section \[sec:third-characterization\] above.
We minimize $f$ over all unit vectors $a$, to avoid unbounded domain (if we minimized over all vectors, the optimal value might be $- \infty$, and no solution would exist) and also to avoid the zero vector being a solution.
If $\bar{a}_1$, $\ldots,$ $\bar{a}_j$ are the components of the solution, the DOR is $$\delta = \sum_{k = 1}^j \bar{a}_k \eta_k,$$ and the LCM is the OM conditioned on the event that every component of the response vector for which the corresponding component of $\zeta = M \delta$ is nonzero is constrained to be equal to its observed value.
We fit the model using the `subset` argument of R function `glm` as explained in Section \[sec:second-characterization\] above.
We can use ideas from Section \[sec:first-characterization\] to tell us whether we need to iterate. We already know the dimension $j$ of the constancy space of the MLE in the completion. If the LCM determined by this GDOR has a constancy space of this dimension, then we have the correct LCM and do not need to iterate. R function `glm` will figure out the dimension of the constancy space (how many coefficients it needs to drop to get an identifiable model) on its own.
### Based on the third characterization {#sec:based-third}
As far as we know, there is no way to calculate a GDOR except by using the very time consuming computational geometry calculations explained by [@geyer-gdor].
Examples
========
Complete separation example
---------------------------
We return to the motivating example of Section 2. Here we see that the Fisher information matrix has only null eigenvectors. Thus the LCM is completely degenerate at the one point set containing only the observed value of the canonical statistic of this exponential family.
We adopt the techniques of Section 3.16.2 of [@geyer-gdor] to make inferences about mean-value parameters (success probability considered as a function of the predictor $x$). This is outlined in Section 2. One-sided confidence intervals are seen in Figure \[fig:intervals\]. As stated in Section 2, the actual computations follow some later course notes [@infinity].
Example in Section 2.3 of [@geyer-gdor]
---------------------------------------
coefficient $\hat{\eta}$ ${\hat{\eta}_{\text{gdor}}}$
---------------- -------------- ------------------------------
intercept -1 -1
$v1$ 1 1
$v2$ 1 1
$v3$ 1 1
$v5$ 1 1
$v1 : v2$ -1 -1
$v1 : v3$ -1 -1
$v1 : v5$ -1 -1
$v2 : v3$ -1 -1
$v2 : v5$ -1 -1
$v3 : v5$ -1 -1
$v1 : v2 : v3$ 1 1
$v1 : v3 : v5$ 1 1
$v2 : v3 : v5$ 1 1
: The estimated null eigenvector of the Fisher information matrix (column 2) and the gdor computed by [@geyer-gdor] (column 3). Only nonzero components are shown.
\[ex1-tab1\]
This example consists of a $2 \times 2 \times \cdots \times 2$ contingency table with seven dimensions hence $2^7 = 128$ cells. These data now have a permanent location [@datasets]. There is one response variable $y$ that gives the cell counts and seven categorical predictors $v_1$, $\ldots$, $v_7$ that specify the cells of the contingency table. We fit a generalized linear regression model where $y$ is taken to be Poisson distributed. We consider a model with all three-way interactions included but no higher-order terms. @geyer-gdor shows the MLE in this example does not exist in the traditional sense, and then computes a generic direction of recession using the repeated linear programming with R package `rcdd` (Section \[sec:based-third\]). In this example there is only single null eigenvector of the Fisher information matrix, which consequently must be a generic direction of recession. Therefore all of our methods of determining the support of the LCM in Sections \[sec:based-first\], \[sec:based-second\], and \[sec:based-third\] must do the same thing.
Table \[ex1-tab1\] displays the comparison between the characterizations in Sections \[sec:based-second\] and \[sec:based-third\]. The vector $\hat{\eta}$ is the estimated null eigenvector of the Fisher information matrix using our implementation. The vector ${\hat{\eta}_{\text{gdor}}}$ is the estimated gdor in [@geyer-gdor]. The $\hat{\eta}$ vector is identical to ${\hat{\eta}_{\text{gdor}}}$ up to six decimal places (the results in Table \[ex1-tab1\] are rounded). Therefore, the inferences resulting from these two distinct approaches is identical up to rounding. The only material difference between our implementation and the linear programming in [@geyer-gdor] is computational time. Our implementation estimates $\eta$ in 0.017 seconds of computer time, while the functions in the `rcdd` package estimates ${\hat{\eta}_{\text{gdor}}}$ in 4.481 seconds of computer time. This is a big difference for a relatively small amount of data. Inference for the MLE in the LCM are included in the supplementary materials.
Big data example
----------------
This example uses the other dataset at [@datasets]. It shows our methods are much faster than the linear programming method of [@geyer-gdor] for recovering directions of recession (Sections \[sec:second-characterization\] and \[sec:third-characterization\]). The characterization in Section \[sec:first-characterization\] is even faster since no direction of recession is computed. This dataset consists of five categorical variables with four levels each and a response variable $y$ that is Poisson distributed. A model with all four-way interaction terms is fit to this data. It may seem that the four way interaction model is too large (1024 data points vs 781 parameters) but $\chi^2$ tests select this model over simpler models, see Table \[bdtest\].
null model alternative model df Deviance Pr($>\chi^2$)
------------ ------------------- ----- ---------- ---------------
m1 m4 765 904.8 0.00034
m2 m4 675 799.2 0.00066
m3 m4 405 534.4 0.00002
: Model comparisons for Example 2. The model m1 is the main-effects only model, m2 is the model with all two way interactions, m3 is the model with all three way interactions, and m4 is the model with all four way interactions.
\[bdtest\]
We estimate that the dimension of the null space of the estimated Fisher information matrix is 23. In the Section \[sec:based-second\] characterization we minimize $f$ over $a \in {\mathbb{R}}^{23}$ in , $\|a\| = 1$ to find a DOR. The resulting vector ${\hat{\eta}_{\text{gdor}}}= \sum_{k=1}^{23}a_k\hat{\eta}_k$ is a GDOR since it satisfies conditions (20a) and (20b) of [@geyer-gdor]. Fitting the model, estimating the dimension of the null space of estimated Fisher information, finding $a$, and estimating the support of the LCM took less than 2 seconds of computer time. In the Section \[sec:based-third\] characterization, the functions in the `rcdd` package perform the same tasks in 334701 seconds (roughly 3.8 days) of computer time. The two different methods estimated different GDORs but they estimate the same support for the LCM. Inferences for the MLE in the LCM are included in the supplementary materials. One-sided 95% confidence intervals for mean-value parameters that correspond to components of the canonical statistic which are on the boundary of their support (MLE equal to 0) are also included in the supplementary materials. We provide a new method for calculating these intervals that has not been previously published, but whose concept is found in @geyer-gdor in the penultimate paragraph of Section 3.16.2.
Let $I$ denote the index set of the components of the response vector on which we condition the OM to get the LCM, and let $Y_I$ and $y_I$ denote these components considered as a random vector and as an observed value, respectively. Let $\theta = M \beta$ denote the saturated model canonical parameter (usually called “linear predictor” in GLM theory) with $\beta$ being the submodel canonical parameter vector. Then endpoints for a $100 (1 - \alpha)\%$ confidence interval for a scalar parameter $g(\beta)$ are $$\label{eq:g-optim}
\min_{\substack{\gamma \in \Gamma_\text{lim} \\
\operatorname{pr}_{\hat{\beta} + \gamma}(Y_I = y_I) \ge \alpha}}
g(\hat{\beta} + \gamma)
\qquad \text{and} \qquad
\max_{\substack{\gamma \in \Gamma_\text{lim} \\
\operatorname{pr}_{\hat{\beta} + \gamma}(Y_I = y_I) \ge \alpha}}
g(\hat{\beta} + \gamma)$$ where $\Gamma_\text{lim}$ is a basis for the null space of Fisher information. At least one of is at the end of the range of this parameter (otherwise we can use conventional two-sided intervals). For Poisson sampling, let $\mu = \exp(\theta)$ denote the mean value parameter (here $\exp$ operates componentwise like the R function of the same name does), then $
\operatorname{pr}_\beta(Y_I = y_I) = \exp\left( - \sum_{i \in I} \mu_i \right).
$ We take the confidence interval problem to be $$\label{eq:poisson-ci-problem}
\begin{split}
\text{maximize} & \quad \mu_k
\\
\text{subject to} & \quad - \sum_{i \in I} \mu_i \ge \log(\alpha)
\end{split}$$ where $\mu$ is taken to be the function of $\gamma$ described in . The optimization in can be done for any $k \in I$. Implementation details are included in Sections 10.7.1 and 10.7.2 in the supplementary materials.
One-sided 95% confidence intervals for mean-valued parameters whose MLE is equal to 0 are displayed in Table \[one-sided-bd\]. The full table is included in the supplementary materials. Some of the intervals in Table \[one-sided-bd\] are relatively wide which represents non-trivial uncertainty about the observed MLE being 0.
X1 X2 X3 X4 X5 lower bound upper bound
---- ---- ---- ---- ---- ------------- -------------
b c c b a 0 0.60
c c c b a 0 2.28
d c c b a 0 1.47
d d c b a 0 2.99
a c d b a 0 0.02
: One-sided 95% confidence intervals for 5 out of 82 mean-valued parameters whose MLE is equal to 0.
\[one-sided-bd\]
Discussion
==========
The chance of observing a canonical statistic on the boundary of its support increases when the dimension of the model increases. Researchers naturally want to include all possibly relevant covariates in an analysis, and this will often result in the MLE not existing in the conventional sense. Our methods provide a computationally inexpensive solution to this problem.
The theory of generalized affine functions and the geometry of exponential families allows GLM software to provide a MLE when the observed value of the canonical statistic is on the boundary of its support. In such settings, the MLE does not exist in the traditional sense and is said to belong to the Barndorff-Nielsen completion of the exponential family [@barndorff-nielsen; @brown; @geyer-gdor; @csiszar] when the supremum of the log likelihood is finite. [@barndorff-nielsen; @brown; @csiszar] all provided a MLE when it exists in the Barndorff-Nielsen completion of the family and [@geyer-gdor] provided estimates of variability under the conditions of [@brown]. We do the same here using the theory of generalized affine functions.
The limiting distribution evaluated along the iterates of such an optimization is a generalized exponential family taking the form of a generalized affine function with structure given by Theorem \[vec-char\]. Cumulant generating functions converge along this sequence of iterates (Theorems \[main\] and \[convex-poly-thm\]) as well as estimates of moments of all orders (Theorem \[MVmgf-3\]) for distributions taking estimated parameter values along this sequence of iterates. We can then use the null eigenvectors of estimated Fisher information to find a DOR and the support of a LCM. Parameter estimation in the LCM is conducted in the traditional manner using GLM software. One-sided confidence intervals for mean-value and canonical parameters that are observed to be on the boundary can also be provided.
The costs of computing a DOR and the support of a LCM are minimal compared to the repeated linear programming in the `rcdd` package, especially when the dimension of the data is large. This is where the desirability of our approach stems from. It is much faster to let optimization software, such as `glm` in R, simply go uphill on the log likelihood of the exponential family until a convergence tolerance is reached. Our examples show what kind of time saving is possible using our methods on small and large datasets.
Technical appendix
==================
First consider the case when $j = 0$, the sequences of $\eta$ vectors and scalars $\delta$ are both of length zero. There are no sets $C^{+}$ and $C^{-}$ in this setting and $h$ is affine on $E$. From Lemma \[properties-1\] we have $\psi_n = \theta_n$. From Corollary \[properties-2\], $\theta_n \to {\theta^{\textstyle{*}}}$ as $n\to\infty$. We observe that $c(\theta_n) \to c({\theta^{\textstyle{*}}})$ from continuity of the cumulant function. The existence of the MLE in this setting implies that there is a neighborhood about 0 denoted by $W$ such that ${\theta^{\textstyle{*}}}+ W \subset \operatorname{int}(\operatorname{dom}c)$. Pick $t \in W$ and observe that $c(\theta_n + t) \to c({\theta^{\textstyle{*}}}+ t)$. Therefore $\kappa_n(t) \to \kappa(t)$ when $j = 0$.
Now consider the case when $j=1$. Define $c_1(\theta) = \log\int_{H_1}e^{{\langle y,\theta \rangle}}\lambda(dy)$ for all $\theta \in \operatorname{int}(\operatorname{dom}\, c_1)$. In this scenario we have $$\begin{aligned}
\kappa_n(t) &= c\left(\psi_n + t + b_{1,n}\eta_1 \right)
- c\left(\psi_n + b_{1,n}\eta_1 \right) \\
&= c\left(\psi_n + t + b_{1,n}\eta_j \right)
- c\left(\psi_n + b_{1,n}\eta_1 \right) \pm b_{1,n}\delta_1 \\
&= \left[c\left(\psi_n + t + b_{1,n}\eta_1 \right)
- b_{1,n}\delta_1\right]
- \left[c\left(\psi_n + b_{1,n}\eta_1 \right)
- b_{1,n}\delta_1 \right].\end{aligned}$$ From [@geyer Theorem 2.2], we know that $$\label{eq:key-too}
\begin{split}
c\left({\theta^{\textstyle{*}}}+ t + s\eta_1 \right) - s\delta_1
&\to c_1\left({\theta^{\textstyle{*}}}+ t\right), \\
c\left({\theta^{\textstyle{*}}}+ s\eta_1 \right) - s\delta_1
&\to c_1\left({\theta^{\textstyle{*}}}\right),
\end{split}$$ as $s \to \infty$ since $\delta_1 \geq {\langle y, \eta_1 \rangle}$ for all $y \in H_1$. The left hand side of is a convex function of $\theta$ and the right hand side is a proper convex function. If $\operatorname{int}(\operatorname{dom}\, c_1)$ is nonempty, which holds whenever $\operatorname{int}(\operatorname{dom}\, c)$ is nonempty, then the convergence in is uniform on compact subsets of $\operatorname{int}(\operatorname{dom}\, c_1)$ [@rock-wets Theorem 7.17]. Also [@rock-wets Theorem 7.14], uniform convergence on compact sets is the same as continuous convergence. Using continuous convergence, we have that both $$\begin{aligned}
c\left(\psi_n + t + b_{1,n}\eta_1 \right)
- b_{1,n}\delta_1 &\to c_1\left({\theta^{\textstyle{*}}}+ t\right), \\
c\left(\psi_n + b_{1,n}\eta_1 \right)
- b_{1,n}\delta_1 &\to c_1\left({\theta^{\textstyle{*}}}\right), \end{aligned}$$ where $b_{1,n} \to \infty$ as $n \to \infty$ by Lemma \[properties-1\]. Thus $$\begin{aligned}
\kappa_n(t) &= c(\theta_n + t) - c(\theta_n)
\to c_1\left({\theta^{\textstyle{*}}}+ t\right) - c_1\left({\theta^{\textstyle{*}}}\right) \\
&= \log\int_{H_1} e^{{\langle y + t,{\theta^{\textstyle{*}}}\rangle} - c({\theta^{\textstyle{*}}})} \lambda(dy)
= \log\int_{H_1} e^{{\langle y,t \rangle} + h(y)} \lambda(dy) \\
&= \log\int e^{{\langle y,t \rangle} + h(y)} \lambda(dy) = \kappa(t).\end{aligned}$$ This concludes the proof when $j=1$.
For the rest of the proof we will assume that $1 < j \leq p$ where dim($E$) = $p$. Represent the sequence $\theta_n$ in coordinate form as $$\label{coordinates}
\theta_n = b_{1,n}\eta_1 + b_{2,n}\eta_2 + \cdots + b_{p,n}\eta_p,$$ with scalars $b_{i,n}$, $i = 1,...,p$. For $0 < j < p$, we know that $\psi_n \to {\theta^{\textstyle{*}}}$ as $n \to \infty$ from Corollary \[properties-2\]. The existence of the MLE in this setting implies that there is a neighborhood about 0, denoted by $W$, such that ${\theta^{\textstyle{*}}}+ W \subset \operatorname{int}(\operatorname{dom}c)$. Pick $t \in W$, fix $\varepsilon > 0$, and construct $\varepsilon$-boxes about ${\theta^{\textstyle{*}}}$ and ${\theta^{\textstyle{*}}}+ t$, denoted by $\mathcal{N}_{0,\varepsilon}({\theta^{\textstyle{*}}})$ and $\mathcal{N}_{t,\varepsilon}({\theta^{\textstyle{*}}})$ respectively, such that both $\mathcal{N}_{0,\varepsilon}({\theta^{\textstyle{*}}}), \mathcal{N}_{t,\varepsilon}({\theta^{\textstyle{*}}})
\subset \operatorname{int}\left(\operatorname{dom}\, c\right)$. Let $V_{t,\varepsilon}$ be the set of vertices of $\mathcal{N}_{t,\varepsilon}({\theta^{\textstyle{*}}})$. For all $y \in E$ define $$\label{bounds}
M_{t,\varepsilon}(y) =
\max_{v \in V_{t,\varepsilon}}\{\langle v, y \rangle\},
\qquad
\widetilde{M}_{t,\varepsilon}(y) =
\min_{v \in V_{t,\varepsilon}}\{\langle v, y \rangle\}.$$ From the conclusions of Lemma \[properties-1\] and Corollary \[properties-2\], we can pick an integer $N$ such that $
{\langle y,\psi_n + t \rangle} \leq M_{t,\varepsilon}(y)
$ and $b_{(i+1),n}/b_{i,n} < 1$ for all $n > N$ and $i = 1,...,j-1$. For all $y \in F$, we have $$\label{add-ass-1}
\begin{split}
{\langle y,\theta_n + t \rangle} - \sum_{i=1}^j b_{i,n}\delta_i
&= {\langle y,\psi_n + t \rangle} + \sum_{i=1}^j
b_{i,n}\left({\langle y,\eta_i \rangle} - \delta_i\right) \\
&\leq M_{t,\varepsilon}(y)
\end{split}$$ for all $n > N$. The integrability of $e^{M_{t,\varepsilon}(y)}$ and $e^{\widetilde{M}_{t,\varepsilon}(y)}$ follows from $$\begin{aligned}
&\int e^{\widetilde{M}_{t,\varepsilon}(y)}\lambda(dy)
\leq \int e^{M_{t,\varepsilon}(y)}\lambda(dy)
=\sum_{v \in V_{t,\varepsilon}} \int\limits_{\{y:\; \langle y,v \rangle
= M_{t,\varepsilon}(y)\}} e^{\langle y,v \rangle} \lambda(dy) \\
&\qquad\leq
\sum_{v \in V_{t,\varepsilon}} \int e^{\langle y,v \rangle} \lambda(dy)
< \infty.\end{aligned}$$ Therefore, $${\langle y,\psi_n + t \rangle}
+ \sum_{i=1}^j b_{i,n}\left({\langle y,\eta_i \rangle} - \delta_i\right)
\to \left\{\begin{array}{cc}
{\langle y,{\theta^{\textstyle{*}}}+ t \rangle}, & y \in H_j, \\
-\infty, & y \in F \setminus H_j.
\end{array}\right.$$ which implies that $$\label{cF}
c_F(\theta_n + t) - c_F(\theta_n) \to c_{H_j}({\theta^{\textstyle{*}}}+ t) - c_{H_j}({\theta^{\textstyle{*}}})$$ by dominated convergence. To complete the proof, we need to verify that $$\label{last-condition}
\begin{split}
&c(\theta_n + t) - c(\theta_n) = c_F(\theta_n + t) - c_F(\theta_n) \\
&\qquad+ c_{\cup_{i=1}^{j-1}D_i}(\theta_n + t)
- c_{\cup_{i=1}^{j-1}D_i}(\theta_n) \\
&\to c_{H_j}({\theta^{\textstyle{*}}}+ t) - c_{H_j}({\theta^{\textstyle{*}}}).
\end{split}$$ We know that holds when $
\lambda(\cup_{i=1}^{j-1}D_i) = 0
$ in because of . Now suppose that $\lambda(\cup_{i=1}^{j-1}D_i) > 0$. We have, $$\label{add-lim}
{\langle y,\psi_n + t \rangle}
+ \sum_{i=1}^j b_{i,n}\left({\langle y,\eta_i \rangle} - \delta_i\right)
\to -\infty, \qquad y \in \cup_{i=1}^{j-1}D_i,$$ and $$\label{add-ass-2}
\begin{split}
&\exp\left(c_{\cup_{i=1}^{j-1}D_i}(\theta_n + t)
- c_{\cup_{i=1}^{j-1}D_i}(\theta_n)\right) \\
&\qquad= \int_{\cup_{i=1}^{j-1}D_i} e^{{\langle y,\theta_n + t \rangle}
- c_{\cup_{i=1}^{j-1}D_i}(\theta_n)} \lambda(dy) \\
&\qquad \leq \int_{\cup_{i=1}^{j-1}D_i} e^{M_{t,\varepsilon}(y)
- \widetilde{M}_{0,\varepsilon}(y) + {\langle y, \theta_n \rangle}
- c_{\cup_{i=1}^{j-1}D_i}(\theta_n)} \lambda(dy) \\
&\qquad \leq \sup_{y\in \cup_{i=1}^{j-1}D_i} \left(e^{{\langle y, \theta_n \rangle}
- c_{\cup_{i=1}^{j-1}D_i}(\theta_n)}\right)
\lambda\left(\cup_{i=1}^{j-1}D_i\right) \\
&\qquad\qquad \times \int_{\cup_{i=1}^{j-1}D_i} e^{M_{t,\varepsilon}(y)
- \widetilde{M}_{0,\varepsilon}(y)}\lambda(dy) \\
&\qquad \leq \sup_{\theta \in \Theta}\sup_{y\in \cup_{i=1}^{j-1}D_i}
\left(e^{{\langle y, \theta \rangle}- c_{\cup_{i=1}^{j-1}D_i}(\theta)}\right)
\lambda\left(\cup_{i=1}^{j-1}D_i\right) \\
&\qquad\qquad \times \int_{\cup_{i=1}^{j-1}D_i} e^{M_{t,\varepsilon}(y)
- \widetilde{M}_{0,\varepsilon}(y)}\lambda(dy) \; < \; \infty
\end{split}$$ for all $n > N$ by the assumption given by . The assumption that the exponential family is discrete and full implies that $\int e^h(y)\lambda(dy) = 1$ [@geyer Theorem 2.7]. This in turn implies that $\lambda(C_i^+) = 0$ for all $i = 1,...,j$ which then implies that $
c(\theta) = c_F(\theta) + c_{\cup_{i=1}^{j-1}D_i}(\theta).
$ Putting , , and together we can conclude that holds as $n\to\infty$ by dominated convergence and $$\label{end}
\begin{split}
&c_{H_j}({\theta^{\textstyle{*}}}+ t) - c_{H_j}({\theta^{\textstyle{*}}}) \\
&\qquad= \log\int_{H_j} e^{{\langle y,{\theta^{\textstyle{*}}}+ t \rangle}} \lambda(dy)
- \log\int_{H_j} e^{{\langle y,{\theta^{\textstyle{*}}}\rangle}} \lambda(dy) \\
&\qquad= \log\int e^{{\langle y,t \rangle} + h(y)}\lambda(dy)
= \kappa(t).
\end{split}$$ for all $t \in W$. This verifies CGF convergence on neighborhoods of 0 which completes the proof.
Represent $h$ as in Theorem \[vec-char\]. Denote the normal cone of the convex polyhedron support $K$ at the data $x$ by $N_K(x)$. We show that a sequence of scalars ${\delta^{\textstyle{*}}}_i$ and a linearly independent set of vectors ${\eta^{\textstyle{*}}}_i \in {E^{\textstyle{*}}}$ can be chosen so that ${\eta^{\textstyle{*}}}_i \in N_K(x)$, and $$\label{newH}
\begin{split}
H_i &= \{y\in H_{i-1}: {\langle y,{\eta^{\textstyle{*}}}_i \rangle} = {\delta^{\textstyle{*}}}_i\}, \\
C_i^+ &= \{y\in H_{i-1}: {\langle y,{\eta^{\textstyle{*}}}_i \rangle} > {\delta^{\textstyle{*}}}_i\}, \\
C_i^- &= \{y\in H_{i-1}: {\langle y,{\eta^{\textstyle{*}}}_i \rangle} < {\delta^{\textstyle{*}}}_i\},
\end{split}$$ for $i = 1,...,j$ where $H_0 = E$ so that holds. We will prove this by induction with the hypothesis $\operatorname{H}(m)$, $m = 1,...,j$, that holds for $i \leq m$ where the vectors ${\eta^{\textstyle{*}}}_i \in N_K(x)$ $i = 1,...,m$. We first verify the basis of the induction. The assumption that the exponential family is discrete and full implies that $\int e^h(y)\lambda(dy) = 1$ [@geyer Theorem 2.7]. This in turn implies that $\lambda(C_k^+) = 0$ for all $k = 1,...,j$. This then implies that $
K \subseteq \{y\in E : {\langle y,\eta_1 \rangle} \leq \delta_1\} = H_1 \cup C_1^-.
$ Thus $\eta_1 \in N_K(x)$ and the base of the induction holds with $\eta_1 = {\eta^{\textstyle{*}}}_1$ and $\delta_1 = {\delta^{\textstyle{*}}}_1$.
We now show that $\operatorname{H}(m+1)$ follows from $\operatorname{H}(m)$ for $m = 1,...,j-1$. We first establish that $K\cap H_m$ is an exposed face of $K$. This is needed to show that holds for $i = 1,...,m+1$. Let $L_K$ be the collection of closed line segments with endpoints in $K$. Arbitrarily choose $l \in L_K$ such that an interior point $y \in l$ is such that $y \in K\cap H_m$. We can write $y = \gamma a + (1-\gamma)b$, $0 < \gamma < 1$, where $a$ and $b$ are the endpoints of $l$. Since $a,b \in K$ by construction, we have that ${\langle a-x,{\eta^{\textstyle{*}}}_m \rangle} \leq 0$ and ${\langle b-x,{\eta^{\textstyle{*}}}_m \rangle} \leq 0$ because ${\eta^{\textstyle{*}}}_m \in N_K(x)$ by $\operatorname{H}(m)$. Now, $$\begin{aligned}
0 &\geq {\langle a-x,{\eta^{\textstyle{*}}}_m \rangle} = {\langle a-y+y-x,{\eta^{\textstyle{*}}}_m \rangle} \\
&= {\langle a-y,{\eta^{\textstyle{*}}}_m \rangle}
= {\langle a-(\gamma a + (1-\gamma)b), {\eta^{\textstyle{*}}}_m \rangle} \\
&= (1-\gamma){\langle a -b, {\eta^{\textstyle{*}}}_m \rangle}\end{aligned}$$ and $$\begin{aligned}
0 &\geq {\langle b-x,{\eta^{\textstyle{*}}}_m \rangle} = {\langle b-y+y-x,{\eta^{\textstyle{*}}}_m \rangle} \\
&= {\langle b-y,{\eta^{\textstyle{*}}}_m \rangle}
= {\langle b-(\gamma a + (1-\gamma)b), {\eta^{\textstyle{*}}}_m \rangle} \\
&= -\gamma{\langle a-b, {\eta^{\textstyle{*}}}_m \rangle}. \end{aligned}$$ Therefore $a,b \in K\cap H_m$ and this verifies that $K\cap H_m$ is a face of $K$ since $l$ was chosen arbitrarily. The function $
y \mapsto {\langle y - x,{\eta^{\textstyle{*}}}_m \rangle} - {\delta^{\textstyle{*}}}_m,
$ defined on $K$, is maximized over $K\cap H_m$. Therefore $K\cap H_m$ is an exposed face of $K$ by definition. The exposed face $
K\cap H_m = K\cap(H_{m+1}\cup C_{m+1}^-)
$ since $\lambda(C_{m+1}^+) = 0$ and the convex support of the measure $\lambda|H_m$ is $H_m$ by assumption. Thus, $\eta_{m+1} \in N_{K\cap H_m}(x)$.
The sets $K$ and $H_m$ are both convex and are therefore regular at every point [@rock-wets Theorem 6.20]. We can write $
N_{K\cap H_m}(x) = N_K(x) + N_{H_m}(x)
$ since $K$ and $H_m$ are convex sets that cannot be separated where + denotes Minkowski addition in this case [@rock-wets Theorem 6.42]. The normal cone $N_{H_m}(x)$ has the form $$\begin{aligned}
N_{H_m}(x) &= \{\eta \in {E^{\textstyle{*}}}: {\langle y-x,\eta \rangle} \leq 0 \;
\text{for all} \; y \in H_m \} \\
&= \{\eta \in {E^{\textstyle{*}}}: {\langle y-x,\eta \rangle} \leq 0 \; \text{for all} \;
y \in E \; \\
&\qquad \text{such that} \; {\langle y-x,\eta_i \rangle} = 0, \;
i = 1,...,m\} \\
&= \left\{\sum_{i=1}^m a_i\eta_i : \; a_i \in {\mathbb{R}}, \; i = 1,...,m \right\}. \end{aligned}$$ Therefore, we can write $$\label{etareform}
\eta_{m+1} = {\eta^{\textstyle{*}}}_{m+1} + \sum_{i=1}^m a_{m,i}{\eta^{\textstyle{*}}}_i$$ where ${\eta^{\textstyle{*}}}_{m+1} \in N_K(x)$ and $a_{m,i} \in {\mathbb{R}}$, $i = 1,...,m$. For $y \in H_{m+1}$, we have that $$\begin{aligned}
{\langle y,{\eta^{\textstyle{*}}}_{m+1} \rangle} &= {\langle y,\eta_{m+1} \rangle} -
\sum_{i=1}^ma_{m,i}{\langle y,\eta_{i} \rangle} \\
&= \delta_{m+1} - \sum_{i=1}^m a_{m,i}\delta_i. \end{aligned}$$ Let ${\delta^{\textstyle{*}}}_{m+1} = \delta_{m+1} - \sum_{i=1}^m a_{m,i}\delta_i$. We can therefore write $$H_{m+1} = \left\{y \in H_m :
{\langle y,{\eta^{\textstyle{*}}}_{m+1} \rangle} = {\delta^{\textstyle{*}}}_{m+1}\right\}$$ and $$\label{showCplus}
\begin{split}
C_{m+1}^+ &= \left\{y \in H_m : {\langle y,\eta_{m+1} \rangle}
> \delta_{m+1}\right\} \\
&= \left\{y \in H_m : {\langle y,{\eta^{\textstyle{*}}}_{m+1} \rangle}
+ \sum_{i=1}^m a_{m,i}\delta_i > \delta_{m+1}\right\} \\
&= \left\{y \in H_m : {\langle y,{\eta^{\textstyle{*}}}_{m+1} \rangle} > \delta_{m+1}
- \sum_{i=1}^m a_{m,i}\delta_i\right\} \\
&= \left\{y \in H_m : {\langle y,{\eta^{\textstyle{*}}}_{m+1} \rangle} > {\delta^{\textstyle{*}}}_{m+1}\right\}.
\end{split}$$ A similar argument to that of verifies that $$C_i^- = \left\{y \in H_m : {\langle y,{\eta^{\textstyle{*}}}_{m+1} \rangle}
< {\delta^{\textstyle{*}}}_{m+1}\right\}.$$ This confirms that holds for $i = 1,...m+1$ and this establishes that $\operatorname{H}(m+1)$ follows from $\operatorname{H}(m)$. Define the sets $D_i$ in with starred quantities replacing the unstarred quantities. Since the vectors ${\eta^{\textstyle{*}}}_1,...,{\eta^{\textstyle{*}}}_j \in N_K(x)$, the sets $K \cap D_i$ are all empty for all $i = 1,...,j-1$. Therefore holds with $
\lambda\left(\cup_{i=1}^{j-1}D_i\right) = 0.
$ This completes the proof.
We first consider the case that $A$ is positive definite and $V = \{0\}$. We can write $A_n = A + (A_n - A)$ where $(A_n - A)$ is a perturbation of $A$ for large $n$. From Weyl’s inequality [@weyl], we have that all eigenvalues of $A_n$ are bounded above zero for large $n$ and $V_n = \{0\}$ as a result. Therefore, $V_n \to V$ as $n\to\infty$ when $A$ is positive definite.
Now consider the case that $A$ is not strictly positive definite. Without loss of generality, let $x \in V$ be a unit vector. For all $0 < \gamma \leq \varepsilon$, let $V_n(\gamma)$ denote the subspace spanned by the eigenvectors of $A_n$ corresponding to eigenvalues that are less than $\gamma$. By construction, $V_n(\gamma) \subseteq V_n$.
From [@rock-wets Example 10.28], if $A$ has $k$ zero eigenvalues, then for sufficiently large $N_1$ there are exactly $k$ eigenvalues of $A_n$ are less than $\varepsilon$ and $p - k$ eigenvalues of $A_n$ greater than $\varepsilon$ for all $n > N_1$. The same is true with respect to $\gamma$ for all $n$ greater than $N_2$. Thus $j_n(\gamma) = j_n(\varepsilon)$ which implies that $V_n(\gamma) = V_n$ for all $n > \max\{N_1,N_2\}$.
We now verify part (i) of Painlevé-Kuratowski set convergence with respect to $V_n(\gamma)$. Let $N_3$ be such that $x^TA_nx < \gamma^2$ for all $n \geq N_3$. Let ${\lambda_{k,n}}$ and ${e_{k,n}}$ be the eigenvalues and eigenvectors of $A_n$, with the eigenvalues listed in decreasing orders. Without loss of generality, we assume that the eigenvectors are orthonormal. Then, $$\begin{aligned}
x &= \sum_{k=1}^p(x^T{e_{k,n}}){e_{k,n}}, \qquad
1 = \|x\|^2 = \sum_{k=1}^p (x^T{e_{k,n}})^2, \\
x^TA_nx &= \sum_{k=1}^p {\lambda_{k,n}}(x^T{e_{k,n}})^2.\end{aligned}$$ There have to be eigenvectors ${e_{k,n}}$ such that $x^T{e_{k,n}}\geq 1/\sqrt{p}$ with corresponding eigenvalues ${\lambda_{k,n}}$ that are very small since ${\lambda_{k,n}}(x^T{e_{k,n}})^2 < \gamma$. But conversely, any eigenvalues ${\lambda_{k,n}}$ such that ${\lambda_{k,n}}\geq \gamma$ must have $${\lambda_{k,n}}(x^T{e_{k,n}})^2 < \gamma^2
\implies (x^T{e_{k,n}})^2 < \gamma^2/{\lambda_{k,n}}\leq \gamma.$$ Define $j_n(\gamma) = |\{{\lambda_{k,n}}:{\lambda_{k,n}}\leq \gamma\}|$ and $
x_n = \sum_{k=p-j_n(\gamma)+1}^p (x^T{e_{k,n}}){e_{k,n}}$ where $x_n \in V_n(\gamma)$ by construction. Now, $$\begin{aligned}
\|x - x_n\| &= \|\sum_{k=1}^p (x^T{e_{k,n}}){e_{k,n}}- \sum_{k=p-j_n(\gamma)+1}^p (x^T{e_{k,n}}){e_{k,n}}\| \\
&= \| \sum_{k=1}^{p-j_n(\gamma)}(x^T{e_{k,n}}){e_{k,n}}\| \\
&\leq \sum_{k=1}^{p-j_n(\gamma)}|x^T{e_{k,n}}| \\
&\leq (p-j_n)\sqrt{\gamma} \\
&\leq p\sqrt{\gamma}\end{aligned}$$ for all $n \geq N_3$. Therefore, for every $x \in V$, there exists a sequence $x_n \in V_n(\gamma) \subseteq V_n$ such that $x_n \to x$ since this argument holds for all $0 < \gamma \leq \varepsilon$. This establishes part (i) of Painlevé-Kuratowski set convergence.
We now show part (ii) of Painlevé-Kuratowski set convergence. Suppose that $x_n \to x \in {\mathbb{R}}^p$ and there exists a natural number $N_4$ such that $x_n \in V_n(\gamma)$ whenever $n \geq N_4$, and we will establish that $x \in V$. From hypothesis, we have that $x_n^TA_nx_n \to x^TAx$. Without loss of generality, we assume that $x$ is a unit vector and that $|x_n^TA_nx_n - x^TAx| \leq \gamma$ for all $n \geq N_5$. From the assumption that $x_n \in V_n(\gamma)$ we have $$\label{deriv}
\begin{split}
x_n^TA_nx_n &= \sum_{k=1}^p {\lambda_{k,n}}(x_n^T{e_{k,n}})^2
= \sum_{k=p-j_n(\gamma)+1}^p {\lambda_{k,n}}(x_n^T{e_{k,n}})^2
\leq \gamma
\end{split}$$ for all $n \geq N_4$. The reverse triangle inequality gives $$||x_n^TA_nx_n| - |x^TAx|| \leq |x_n^TA_nx_n - x^TAx| \leq \gamma$$ and implies $
|x^TAx| \leq 2\gamma
$ for all $n \geq \max\{N_4,N_5\}$. Since this argument holds for all $0 < \gamma < \varepsilon$, we can conclude that $x \in V$. This establishes part (ii) of Painlevé-Kuratowski set convergence with respect to $V_n(\gamma)$. Therefore $V_n \to V$ and this completes the proof.
Supplementary materials {#supplementary-materials .unnumbered}
=======================
The supplement to “Computationally efficient likelihood inference in exponential families when the maximum likelihood estimator does not exist” is available upon request. The proofs of Theorems 1-3, Theorem 5, Lemma 1 and Theorems 8-10 in the main text and all of the code producing our examples can be seen in the supplementary materials.
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{
"pile_set_name": "ArXiv"
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abstract: 'A recently developed dynamical mean-field theory in the iterated perturbation theory approximation was used as a basis for construction of the “first principles” calculation scheme for investigating electronic structure of strongly correlated electron systems. This scheme is based on Local Density Approximation (LDA) in the framework of the Linearized Muffin-Tin-Orbitals (LMTO) method. The classical example of the doped Mott-insulator La$_{1-x}$Sr$_x$TiO$_3$ was studied by the new method and the results showed qualitative improvement in agreement with experimental photoemission spectra.'
author:
- |
V.I. Anisimov, A.I.Poteryaev, M.A.Korotin, A.O.Anokhin\
Institute of Metal Physics, Ekaterinburg,GSP-170,Russia\
\
G.Kotliar\
Serin Physics Laboratory,\
Rutgers University, Piscataway, New Jersey 08854, USA
title: 'First-principles calculations of the electronic structure and spectra of strongly correlated systems:dynamical mean-field theory'
---
Introduction
============
The accurate calculation of the electronic structure of materials starting from first principles is a challenging problem in condensed matter science since unfortunately, except for small molecules, it is impossible to solve many-electron problem without severe approximations.
For materials where the kinetic energy of the electrons is more important than the Coulomb interactions, the most successful first principles method is the Density Functional theory (DFT) within the Local (Spin-) Density Approximation (L(S)DA)[@lda], where the many-body problem is mapped into a non-interacting system with a one-electron exchange-correlation potential approximated by that of the homogeneous electron gas.
It is by now, generally accepted that the spin density functional theory in the local approximation is a reliable starting point for first principle calculations of material properties of weakly correlated solids (For a review see [@RMP]). The situation is very different when we consider more strongly correlated materials, (systems containing f and d electrons). In a very simplified view LDA can be regarded as a Hartree-Fock approximation with orbital-independent (averaged) one-electron potential. This approximation is very crude for strongly correlated systems, where the on-cite Coulomb interaction between d- (or f-) electrons of transition metal (or rare-earth metal) ions (Coulomb parameter $U$) is strong enough to overcome kinetic energy which is of the order of band width $W$. In the result LDA gives qualitatively wrong answer even for such simple systems as Mott insulators with integer number of electrons per cite (so-called “undoped Mott insulators”) . For example insulators CoO and La$%
_2$CuO$_4$ are predicted to be metallic by LDA.
The search for a “first principle” computational scheme of physical properties of strongly correlated electron systems which is as successful as the LDA in weakly correlated systems, is highly desirable given the considerable importance of this class of materials and is a subject of intensive current research. Notable examples of first principle schemes that have been applied to srongly correlated electron systems are the LDA+U method [@ldau] which is akin to orbital-spin-unrestricted Hartree-Fock method using a basis of LDA wave functions, ab initio unrestricted Hartree Fock calculations [@HF] and the use of constrained LDA to derive model parameters of model hamiltonians which are then treated by exact diagonalization of small clusters or other approximations [@hybertsen].
Many interesting effects, such as orbital and charge ordering in transition metal compounds were successfully described by LDA+U method [@review]. However for strongly correlated metals Hartree-Fock approximation is too crude and more sophisticated approaches are needed.
Recently the dynamical mean-field theory was developed [@kotliar] which is based on the mapping of lattice models onto quantum impurity models subject to a self-consistency condition. The resulting impurity model can be solved by various approaches (Quantum Monte Carlo, exact diagonalization) but the most promising for the possible use in “realistic” calculation scheme is Iterated Perturbation Theory (IPT) approximation, which was proved to give results in a good agreement with more rigorous methods.
This paper is the first in a series where we plan to integrate recent develompements of the dynamical mean field approach with state of the art band structure calculation techniques to generate an “ab initio” scheme for the calculation of the electronic structure of correlated solids. For a review of the historical development of the dynamical mean field approach in its various implementations see ref [@kotliar] . In this paper we implement the dynamical mean-field theory in the iterated perturbation theory approximation, and carry out the band structure calculations using a LMTO basis . The calculational scheme is described in section 2 . We present results obtained applying this method to La$_{1-x}$Sr$_x$TiO$_3$ which is a classical example of strongly correlated metal.
The calculation scheme
======================
In order to be able to implement the achievements of Hubbard model theory to LDA one needs the method which could be mapped on tight-binding model. The Linearized Muffin-Tin Orbitals (LMTO) method in orthogonal approximation [@lmto] can be naturally presented as tight-binding calculation scheme (in real space representation):
$$H_{LMTO}=\sum\limits_{ilm,jl^{\prime }m^{\prime },\sigma }(\delta
_{ilm,jl^{\prime }m^{\prime }}\,\epsilon _{il}\,\widehat{n}_{ilm\sigma
}+t_{ilm,jl^{\prime }m^{\prime }}\widehat{\,c}_{ilm\sigma \,}^{\dagger }%
\widehat{c}_{jl^{\prime }m^{\prime }\sigma })$$
($i$ - site index, $lm$ - orbital indexes).
As we have mentioned above, LDA one-electron potential is orbital - independent and Coulomb interaction between d-electrons is taken into account in this potential in an averaged way. In order to generalize this Hamiltonian by including Coulomb correlations, one must add interaction term:
$$H_{int}=\frac 12\sum_{ ilmm^{\prime }\sigma \sigma ^{\prime } \\ %
m\sigma \neq m^{\prime }\sigma ^{\prime } } U_{il}\widehat{\,n}%
_{ilm\sigma }\,\widehat{n}_{ilm^{\prime }\sigma ^{\prime }}$$
We neglected for a while exchange terms and dependence of Coulomb parameter $%
U$ on the particular pair of orbitals $mm^{\prime }$. Through the following we will assume that only for one shell $l_d$ of one type of atoms $i_d$ (for example d-orbitals of the transition metal ions) Coulomb interaction will be taken into account ($U_{il}=U\delta _{il,i_dl_d}$), and in the following iesndex $il$ will be omitted. All other orbitals will be considered as resulting in the itinerant bands and well described by LDA. Such separation of the electronic states into localized and itinerant is close in spirit to the Anderson model.
To avoid double-counting one must in the same time subtract the averaged Coulomb interaction energy term, which we assume is present in LDA. Unfortunately there is no direct connection between Hubbard model and LDA (because LDA is based on the homogeneous electron gas theory and not on the localized atomic type orbitals representation) and it is impossible to express rigorously LDA-energy through d-d Coulomb interaction parameter $U$. However it is known that LDA total energy as a function of the total number of electrons is a good approximation and the value of the Coulomb parameter $%
U$ obtained in LDA calculation agrees well with experimental data and results of the more rigorous calculations [@Ucalc].That leads us to the suggestion that a good approximation for the LDA part of the Coulomb interaction energy will be:
$$E_{Coul}=\frac 12Un_d(n_d-1)$$
($n_d=\sum\limits_{m\sigma }n_{m\sigma }$ total number of d-electrons).
In LDA-Hamiltonian $\epsilon _{d}$ has a meaning of the LDA-one-electron eigenvalue for d-orbitals. It is known that in LDA eigenvalue is the derivative of the total energy over the occupancy of the orbital: $$\epsilon _{d}=\frac d{dn_d}E_{LDA}$$
If we want to introduce new $\epsilon _d^0$ where d-d Coulomb interaction is excluded we must define them as: $$\epsilon _d^0=\frac d{dn_d}(E_{LDA}-E_{Coul})=\epsilon _{d%
}-U(n_d-\frac 12)$$
Then new Hamiltonian will have the form: $$\begin{aligned}
H &=&H^0+H_{int} \nonumber \\
H^0 &=&\sum\limits_{ilm,jl^{\prime }m^{\prime },\sigma }(\delta
_{ilm,jl^{\prime }m^{\prime }}\,\epsilon _{il}^0\,\widehat{n}_{ilm\sigma
}+t_{ilm,jl^{\prime }m^{\prime }}\widehat{\,c}_{ilm\sigma \,}^{\dagger }%
\widehat{c}_{jl^{\prime }m^{\prime }\sigma })\end{aligned}$$
In reciprocal space matrix elements of the operator $H^0$ are: $$H_{qlm,q^{\prime }l^{\prime }m^{\prime }}^0({\bf k})=H_{qlm,q^{\prime
}l^{\prime }m^{\prime }}^{LDA}({\bf k})-\delta _{qlm,q^{\prime }l^{\prime
}m^{\prime }}\delta _{ql,i_dl_d}U(n_d-\frac 12)$$
($q$ is an index of the atom in the elementary unit cell).
In the dynamical mean-field theory the effect of Coulomb correlation is described by self-energy operator in local approximation. The Green function is: $$G_{qlm,q^{\prime }l^{\prime }m^{\prime }}(i\omega )=\frac 1{V_B}\int d{\bf %
k\,[}i\omega +\mu -H_{qlm,q^{\prime }l^{\prime }m^{\prime }}^0({\bf k}%
)-\delta _{qlm,q^{\prime }l^{\prime }m^{\prime }}\delta _{ql,i_dl_d}\Sigma
(i\omega )]^{-1}$$ ($[...]^{-1}$ means inversion of the matrix, integration is over Brillouin zone, $\mu $ is chemical potential, $V_B$ is a volume of Brillouin zone).
In the following we will consider paramagnetic case, orbital and spin degenerate system, so that self-energy $\Sigma (i\omega )$ does not depend on orbital and spin indexes. One can define effective Anderson model Green function through: $$G(i\omega )=G_{i_dl_dm,i_dl_dm}(i\omega )=(i\omega +\mu -\Delta (i\omega
)-\Sigma (i\omega ))^{-1}$$ where $\Delta (i\omega )$ is effective impurity hybridization function. The effective medium “bath” Green function $G^0$ is defined as: $$G^0(i\omega )=(i\omega +\widetilde{\mu }-\Delta (i\omega
))^{-1}=(G^{-1}(i\omega )+\Sigma (i\omega )+\widetilde{\mu }-\mu )^{-1}$$ ($\widetilde{\mu }$ is chemical potential of the effective medium).
The chemical potential of the effective medium $\widetilde{\mu }$ is varied to satisfy Luttinger theorem condition: $$\frac 1\beta \sum\limits_{i\omega _n}e^{i\omega _n0^{+}}G(i\omega _n)\frac
d{d(i\omega _n)}\Sigma (i\omega _n)=0$$
In iterated perturbation theory approximation the [*anzatz*]{} for the self-energy is based on the second order perturbation theory term calculated with “bath” Green function $G^0$: $$\Sigma ^0(i\omega _s)=-(N-1)U^2\frac 1{\beta ^2}\sum\limits_{i\omega
_n}\sum\limits_{ip_m}G^0(i\omega _m+ip_n)G^0(i\omega _m)G^0(i\omega _s-ip_n)
\label{Sigma0}$$
$N$ is a degeneracy of orbitals including spin, $\beta =\frac 1{kT}$, Matsubara frequencies $\omega _s=\frac{(2s+1)\pi }\beta ;p_n=\frac{2n\pi }%
\beta $ $;s,n$ integer numbers.
The term $\Sigma ^0$ is renormalized to insure correct atomic limit: $$\Sigma (i\omega )=Un(N-1)+\frac{A\Sigma ^0(i\omega )}{1-B\Sigma ^0(i\omega )}$$ ($n$ is orbital occupation $n=\frac 1\beta \sum\limits_{i\omega
_n}e^{i\omega _n0^{+}}G(i\omega _n)$), $$\begin{aligned}
B &=&\frac{U[1-(N-1)n]-\mu +\widetilde{\mu }}{U^2(N-1)n_0(1-n_0)} \\
A &=&\frac{n[1-(N-1)n]+(N-2)D[n]}{n_0(1-n_0)} \\
n_0 &=&\frac 1\beta \sum\limits_{i\omega _n}e^{i\omega _n0^{+}}G^0(i\omega
_n)\end{aligned}$$ correlation function $D[n]\equiv <\widehat{n}\,\widehat{n}>_{CPA}$ is calculated using Coherent Potential Approximation (CPA) for the Green function with parameter $\delta \mu $ chosen to preserve orbital occupation $%
n$ : $$\begin{aligned}
G_{CPA}(i\omega ) &=&\frac{[1-n(N-1)]}{i\omega +\mu -\Delta (i\omega
)+\delta \mu }+\frac{n(N-1)}{i\omega +\mu -\Delta (i\omega )-U+\delta \mu }
\\
n &=&\frac 1\beta \sum\limits_{i\omega _n}e^{i\omega _n0^{+}}G_{CPA}(i\omega
_n) \\
D[n] &=&n\,\sum\limits_{i\omega _n}e^{i\omega _n0^{+}}\frac 1{i\omega +\mu
-\Delta (i\omega _n)-U+\delta \mu }\end{aligned}$$
The Matsubara frequency convolution in (\[Sigma0\]) was performed with time variables representation using Fast Fourier Transform algorithm for transition from energy to time variables and back: $$\begin{aligned}
G^0(\tau ) &=&\frac 1\beta \sum\limits_{i\omega _n}e^{-i\omega _n\tau
}G^0(i\omega _n) \\
\Sigma (\tau ) &=&-(N-1)U^2G^0(\tau )G^0(\tau )G^0(-\tau ) \\
\Sigma ^0(i\omega _n) &=&\int\limits_0^\beta d\tau \,e^{i\omega _n\tau
}\,\Sigma (\tau )\end{aligned}$$
The serious problem is to perform integration in [**k**]{}-space over Brillouin zone. For this we used generalized Lambin-Vigneron algorithm [@lambin]. We define new matrix $H({\bf k},z)$ as: $$H({\bf k},z)=H^0({\bf k)+}\Sigma (z)$$ $z$ - complex energy, term $\Sigma (z)$ is added only to diagonal element of $H$-matrix corresponding to d-orbitals. In this matrix notations Green function is : $$G(z)=\frac 1{V_B}\int d{\bf k[}z-H({\bf k},z)]^{-1}$$ After diagonalization, $H({\bf k},z)$ matrix can be expressed through diagonal matrix of its eigenvalues $D({\bf k},z)$ and eigenvectors matrix $U(%
{\bf k},z)$ : $$H({\bf k},z)=U({\bf k},z)D({\bf k},z)U^{-1}({\bf k},z)$$ and Green function: $$G(z)=\frac 1{V_B}\int d{\bf k}U({\bf k},z)[z-D({\bf k},z)]^{-1}U^{-1}({\bf k}%
,z)$$ A particular matrix element of Green function is calculated as: $$G_{ij}(z)=\sum\limits_n\frac 1{V_B}\int d{\bf k}\frac{U_{in}({\bf k}%
,z)U_{nj}^{-1}({\bf k},z)}{z-D_n({\bf k},z)} \label{Gf1}$$
In analytical tetrahedron method the irreducible wedge of the Brillouin zone is divided into a set of tetrahedra and the total integral is calculated as a sum over the tetrahedra. To perform integration over a given tetrahedron with four corners at vectors ${\bf k}_i$ ($i=1,2,3,4$) denominator of the fraction in equation (\[Gf1\]) is interpolated as a linear function in [**k**]{}-space. In the result the integral over one tetrahedron is expressed through the values of numerator and denominator at the corners of the tetrahedron: $$\sum\limits_n\frac 1{V_B}\int\limits_vd{\bf k}\frac{U_{in}({\bf k}%
,z)U_{nj}^{-1}({\bf k},z)}{z-D_n({\bf k},z)}=\sum\limits_n\sum%
\limits_{i=1}^4r_i^nU_{in}({\bf k}_i,z)U_{nj}^{-1}({\bf k}_i,z)\frac v{V_B}
\label{Gftetra}$$ $v$ is tetrahedron volume $$\begin{aligned}
r_i^n &=&\frac{(z-D_n({\bf k}_i,z))^2}{\prod\limits_{k(\neq i)}(D_n({\bf k}%
_k,z)-D_n({\bf k}_i,z))}+ \\
&&\sum\limits_{j(\neq i)}\frac{(z-D_n({\bf k}_j,z))^3}{\prod\limits_{k(\neq
j)}(D_n({\bf k}_k,z)-D_n({\bf k}_j,z))}\frac{\ln [(z-D_n({\bf k}%
_j,z))/(z-D_n({\bf k}_i,z)]}{(D_n({\bf k}_i,z)-D_n({\bf k}_j,z))} \nonumber\end{aligned}$$
The self-energy $\Sigma (i\omega _n)$ and Green function $G(i\omega _n)$ are calculated at the imaginary Matsubara frequencies $i\omega _n=i\pi
(2n+1)/\beta $ . It is enough to calculate expectation values, such as orbital occupancies $n$ , but in order to calculate spectral properties one need to know Green function on the real axis. The real axis equivalent of equations (\[Sigma0\]) is much more complicated and hard to implement numerically than Matsubara frequencies version. It is much more convenient to perform analytical continuation from imaginary energy values to the real ones. For such continuation we have used Padé approximant algorithm [@pade]. If one has a set of the complex energies $z_i$ $(i=1,...,M)$ and the set of values of the analytical function $u_i$ , then the approximant is defined as continued fraction: $$C_M(z)=\frac{a_1}{1+}\frac{a_2(z-z_{2)}}{1+}...\frac{a_M(z-z_{M-1})}1$$ where the coefficients $a_i$ are to be determined so that: $$C_M(z_i)=u_{i,}\ i=1,...,M$$ The coefficients $a_i$ are then given by the recursion: $$\begin{aligned}
a_i &=&g_i(z_i),\ g_1(z_i)=u_i,\;i=1,...,M \\
g_p(z) &=&\frac{g_{p-1}(z_{p-1})-g_{p-1}(z)}{(z-z_{p-1})g_{p-1}(z)},\;p\geq 2\end{aligned}$$ The recursion formula for continued fraction finally yields: $$C_M(z)=A_M(z)/B_M(z)$$ where $$\begin{aligned}
A_{n+1}(z) &=&A_n(z)+(z-z_n)a_{n+1}A_{n-1}(z) \nonumber \\
B_{n+1}(z) &=&B_n(z)+(z-z_n)a_{n+1}B_{n-1}(z)\end{aligned}$$ and $$A_0=0,\;A_1=a_1,\;B_0=B_1=1$$
We have found that the most convenient way is to use analytical continuation not for the Green function $G$ but only for self-energy $\Sigma $, and then to calculate $G$ directly on the real axis through the Brillouin zone integration (\[Gftetra\]).
Results
=======
We have applied the above described calculation scheme to the doped Mott insulator La$_{1-x}$Sr$_x$TiO$_3$. LaTiO$_3$ is a Pauli-paramagnetic metal at room temperature and below T$_N$=125 K antiferromagnetic insulator with a very small gap value ( 0.2 eV) . Doping by a very small value of Sr (few percent) leads to the transition to paramagnetic metal with a large effective mass. As photoemission spectra of this system also show strong deviation from the noninteracting electrons picture, La$_{1-x}$Sr$_x$TiO$_3$ is regarded as an example of strongly correlated metal.
The crystal structure of LaTiO$_3$ is slightly distorted cubic perovskite. The Ti ions have octahedral coordination of oxygen ions and $t_{2g}$-$e_g$ crystal field splitting of d-shell is strong enough to survive in solid. On Fig.1 the total and partial DOS of paramagnetic LaTiO$_3$ are presented as obtained in LDA calculations (LMTO method). On 3 eV above O2p-band there is Ti-3d-band splitted on $t_{2g}$ and $e_g$ subbands which are well separated from each other. Ti$^{4+}$-ions have d$^1$ configuration and $t_{2g}$ band is 1/6 filled.
As only $t_{2g}$ band is partially filled and $e_g$ band is completely empty, it is reasonable to consider Coulomb correlations between $t_{2g}-$electrons only and degeneracy factor $N$ in Eq. (\[Sigma0\]) is equal 6. The value of Coulomb parameter $U$ was calculated by the supercell procedure [@Ucalc] regarding only $t_{2g}-$electrons as localized ones and allowing $e_g-$electrons participate in the screening. This calculation resulted in a value 3 eV. As the localization must lead to the energy gap between electrons with the same spin, the effective Coulomb interaction will be reduced by the value of exchange parameter $J$=1 eV. So we have used effective Coulomb parameter $U_{eff}$=2 eV. The results of the calculation for x=0.06 (doping by Sr was immitated by the decreasing on x the total number of electrons) are presented in the form of the $t_{2g}$-DOS on Fig.2. Its general form is the same as for model calculations: strong quasiparticle peak on the Fermi energy and incoherent subbands below and above it corresponding to the lower and upper Hubbard bands.
The appearance of the incoherent lower Hubbard band in our DOS leads to qualitatively better agreement with photoemission spectra. On Fig.3 the experimental XPS for La$_{1-x}$Sr$_x$TiO$_3$ (x=0.06) [@spectr] is presented with non-interacting (LDA) and interacting (IPT) occupied DOS broadened to imitate experimental resolution. The main correlation effect: simultaneous presence of coherent and incoherent band in XPS is successfully reproduced in IPT calculation. However, as one can see, IPT overestimates the strength of the coherent subband.
Conclusions
===========
In this publication we described how one can interface methods for realistic band structure calculations with the recently developed dynamical mean field technique to obtain a fully “ab initio” method for calculating the electronic spectra of solids.
With respect to earlier calculations, this work introduces several methodological advances: the dynamical mean field equations are incorporated into a realistic electronic structure calculation scheme, with parameters obtained from a first principle calculation and with the realistic orbital degeneracy of the compound.
To check our method we applied to doped titanates for which a large body of model calculation studies using dynamical mean field theory exists. There results are very encouraging considering the experimental uncertainties of the analysis of the photoemission spectra of these compounds.
We have used two relative accurate (but still approximate ) methods for the solution of the band structure aspect and the correlation aspects of this problem: the LMTO in the ASA approximation and the IPT approximation. In principle, one can use other techniques for handling these two aspects of the problem and further application to more complicated materials are necessary to determine the degree of quantitative accuracy of the method.
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Figure captions
===============
Fig. 1. Noninteracting ($U$=0) total and partial density of states (DOS) for LaTiO$_3.$
Fig. 2. Partial ($t_{2g})$ DOS obtained in IPT calculations in comparison with noninteracting DOS.
Fig. 3. Experimental and theoretical photoemission spectra of La$_{1-x}$Sr$_x$TiO$_3$ (x=0.06).
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"pile_set_name": "ArXiv"
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---
abstract: |
The charged fermion masses of the three generations exhibit the two strong hierarchies $m_{3} \gg m_2 \gg m_1$. We assume that also neutrino masses satisfy $m_{\nu 3} > m_{\nu 2} > m_{\nu 1}$ and derive the consequences of the hierarchical spectra on the fermionic mixing patterns. The quark and lepton mixing matrices are built in a general framework with their matrix elements expressed in terms of the four fermion mass ratios $m_u/m_c$, $m_c/m_t$, $m_d/m_s$, and $m_s/m_b$ and $m_e/m_\mu$, $m_\mu/m_\tau$, $m_{\nu 1}/m_{\nu 2}$, and $m_{\nu
2}/m_{\nu 3}$, for the quark and lepton sector, respectively. In this framework, we show that the resulting mixing matrices are consistent with data for both quarks and leptons, despite the large leptonic mixing angles. The minimal assumption we take is the one of hierarchical masses and minimal flavour symmetry breaking that strongly follows from phenomenology. No special structure of the mass matrices has to be assumed that cannot be motivated by this minimal assumption. This analysis allows us to predict the neutrino mass spectrum and set the mass of the lightest neutrino well below $0.01\,{\ensuremath{\mathrm{eV}}}$. The method also gives the $1\,\sigma$ allowed ranges for the leptonic mixing matrix elements. Contrary to the common expectation, leptonic mixing angles are found to be determined solely by the four leptonic mass ratios without any relation to symmetry considerations as commonly used in flavor model building. Still, our formulae can be used to build up a flavor model that predicts the observed hierarchies in the masses—the mixing follows then from the procedure which is developed in this work.
bibliography:
- 'dmhp.bib'
---
TTP14-029
[ **The double mass hierarchy pattern:\
simultaneously understanding\
quark and lepton mixing** ]{}\
Wolfgang Gregor Hollik$^*$ [^1] and Ulises Jesús Saldaña Salazar$^{*,\dag}$ [^2]\
$^*$ *Institut für Theoretische Teilchenphysik, Karlsruhe Institute of Technology*
*Engesserstraße 7, D-76131 Karlsruhe, Germany*
$^\dag$ *Instituto de Física, Universidad Nacional Autónoma de México*
*Apdo. Postal 20-364, 01000, México D.F., México*
**Keywords:** quark and lepton masses and mixing, CP violation\
**PACS:** 12.15.Ff, 12.15.Hh, 14.60.Pq
Introduction
============
The Standard Model of particle physics (SM) describes the interactions among elementary particles at high energies with great success. In spite of this, the setup of the SM lacks an explanation of the origin of fermion masses and mixing. In particular, for the quark sector, one observes six masses, three mixing angles and one phase. It is a simple exercise to relate the quark mixing matrix to the fundamental parameters of the theory, the Yukawa couplings. Generally, however, it is said that mixing angles as well as the masses are *independent* free parameters. Is there really no functional relation between the quark masses and the corresponding mixing matrix elements? There are many models in the literature that want to give an explanation of the mixing matrix elements in terms of the masses [@Gatto:1968ss; @Cabibbo:1968vn; @Tanaka:1969bw; @Oakes:1969vm; @Genz:1973sn; @Pagels:1974qg; @Ebrahim:1977hb; @Fritzsch:1977za; @Weinberg:1977hb; @DeRujula:1977ry; @Wilczek:1977uh; @Mohapatra:1977rj; @Wilczek:1978xi; @Wyler:1978fj; @Fritzsch:1986sn; @Frampton:1991aya; @Rosner:1992qa; @Raby:1995uv; @Ito:1996zt; @Xing:1996hi; @Xue:1996fm; @Barbieri:1996ww; @Falcone:1998us; @Mondragon:1998gy; @Mondragon:1999jt; @Fritzsch:1999ee; @Branco:2010tx; @Canales:2013cga]. Most of them put assumptions on a specific texture in the original mass matrices. We shall show, by contrast, that the pure phenomenological observation of strong hierarchies in the quark masses leads to a functional description of the mixing matrix elements in terms of mass ratios. The consequences in the mixing of this phenomenological observation have already been studied [@Fritzsch:1986sn; @Hall:1993ni; @Xing:1996hi; @Rasin:1997pn; @Rasin:1998je; @Fritzsch:1999rb; @Fritzsch:1999ee; @Xing:2012zv]. Our approach differs from the previous ones in many aspects: i) we take the Singular Value Decomposition of the complex mass matrices as a starting point offering a generic treatment for both quarks and leptons; ii) by means of an approximation theorem we mathematically formulate the steps to build the reparametrization of the mixing matrix in terms of the singular values (fermion masses); iii) we rotate the mass matrices in all three planes of family space whereas before, the 1-3 rotation was neglected; iv) as the two unitary rotations in the 2-3 and 1-3 plane involve an approximation ($m_{f,1} =0$ and $m_{f,2}= 0$, respectively) we consider for the first time a modified method of perturbation theory to add the effect of the terms neglected; v) we do not consider the complex CP phases as free parameters and show that a minimal choice is sufficient to explain CP data; vi) we provide explicit formulae for the mixing angles in terms of only mass ratios.
The applicability of this formulation to the leptonic mixing is not clear *a priori*. First, neutrino masses do not show any strong hierarchy, at best a very mild one. Second, the leptonic mixing matrix exhibits large mixing, while the one in the quark sector is rather close to the unit matrix. This picture seems to suggest two quite different origins for the respective mixing matrices: quark masses strongly dominating the mixing patterns, whereas geometrical factors found from symmetries shaping the leptonic mixing, with only a weak intervention from the lepton masses [@Ishimori:2010au; @King:2013eh].
Fermion masses, on the other hand, are also as puzzling as the mixing matrices: the top quark mass is by far the largest among the charged fermions, there are six orders of magnitude separating the top quark from the electron mass, six orders of magnitude separating the largest neutrino mass from the electron mass (assuming a neutrino mass scale of $0.1\,{\ensuremath{\mathrm{eV}}}$). There are three orders of magnitude between the masses of the up-type quarks, whereas two orders of magnitudes in the down-quark sector. Top and bottom quark are separated by two orders of magnitude—the lightest charged lepton and the heaviest quark by again six orders of magnitude. Within each (charged) fermion species ($f = u,
d, e$), the masses follow a hierarchy $m_{f,3} \gg m_{f,2} \gg m_{f,1}$, $$\begin{aligned}
m_u : m_c : m_t \approx 10^{-6} : 10^{-3} : 1, &\quad\quad
m_d : m_s : m_b \approx 10^{-4} : 10^{-2} : 1, \\
m_e : m_{\mu} : m_{\tau} &\approx 10^{-4} : 10^{-2} : 1,
\end{aligned}$$ while the two squared mass differences measured from neutrino oscillations obey a much weaker hierarchy, $$\begin{aligned}
\Delta m_{21}^2 : \Delta m_{31(32)}^2 \approx 10^{-2} : 1.\end{aligned}$$
Quark masses plus mixing parameters sum up to ten arbitrary physical parameters in the SM. Consideration of neutrino masses, whether Dirac or Majorana, adds at least ten more parameters to the count. Two more complex phases and a possibly arbitrary number of masses for sterile neutrinos appear in the more general cases including Majorana neutrinos [@Schechter:1980gr]. The SM *per se* seems to lack a course of action on how to relate the mixing matrix elements to the corresponding fermion masses.
The first realization of a mixing angle in terms of the masses is commonly assigned to Gatto et al. [@Gatto:1968ss] which is referred to as the Gatto-Sartori-Tonin relation. This relation is an expression of the Cabibbo angle commonly written as $$\label{eq:GSTrel}
\theta_{12}^q \approx \sqrt{\frac{m_d}{m_s}},$$ where originally, the authors of [@Gatto:1968ss] found a similar relation in terms of light meson masses from the demand of weak self-masses being free from quadratic divergences. In a footnote, they break it down to an elementary discussion in a “naive quark model” and state $$\tan^2\theta = \frac{m_n - m_p}{m_p} = \frac{m_n}{m_\lambda},$$ where $m_n$, $m_p$, and $m_\lambda$ are the old notations of down-, up-, and strange-quark masses (moreover, the second equal sign was misleadingly written as a minus sign). The first work referring to [@Gatto:1968ss] as origin of “$\tan\theta = m_n / m_\lambda $” was [@Tanaka:1969bw] (even though with a typo in the abstract). For small angles, $\tan\theta \approx \theta$ and we are at Eq. . Since $\sqrt{m_d / m_s}$ is an astonishingly good approximation for the Cabibbo angle, we will show in the course of this paper how to rearrive at this expression in a formal way of parametrizing mixing matrices in terms of invariants.
The work of [@Gatto:1968ss] was followed by derivations of the same formula focused on the derivation in a more model-building related approach using left-right symmetric scenarios [@Gatto:1968ss; @Cabibbo:1968vn; @Tanaka:1969bw; @Oakes:1969vm; @Genz:1973sn; @Mohapatra:1977rj; @Fritzsch:1979zq]. In the same decade, a model independent approach was initiated where mass matrices with different null matrix elements (“texture zeros”) were considered [@Fritzsch:1977vd; @Ramond:1993kv; @Branco:1999nb; @Roberts:2001zy; @Fritzsch:2002ga; @Gupta:2012dma]; similar relations were then found for other mixing angles. Subsequently, horizontal or family discrete symmetries were used in order to relate the three families in a non-trivial fashion [@Pakvasa:1975ti; @Pakvasa:1977in; @Derman:1978rx; @Wyler:1978fj; @Yamanaka:1981pa; @Yahalom:1983kf; @Wilczek:1977uh; @Wilczek:1978xi]. In their initial stage, though, the experimental uncertainty in the mixing angles and fermion masses was still too large as to build a stable model consistent with the unstable phenomenology. This approach was vigorously resurrected in the last decade when precision measurements for neutrino oscillations started [@King:2013eh; @Altarelli:2014dca; @King:2014nza]. Relations between the neutrino mixing angles and lepton mass hierarchies were found [@Fritzsch:2006sm; @Fritzsch:2009sm] where the values for the three neutrino masses are compatible with what follows from our method, though $\theta_{13}$ was predicted too low (only about $3^\circ$). Nevertheless, up to now, no complex mass matrix with a well-motivated constrained set of parameters has been found to entirely and successfully postdict the Cabibbo-Kobayashi-Maskawa (CKM) quark mixing matrix or to predict the Pontecorvo-Maki-Nakagawa-Sakata (PMNS) matrix in the lepton sector. In this work, we do not focus on a specific model predicting mixing angles, but give explicit relations following from a model independent treatment based on the observation of the two strong hierarchies $m_3 \gg m_2 \gg m_1$ in the charged fermion masses. Moreover, we dare to apply the same fomulae to the neutrino mixing and derive the PMNS angles with astonishingly good agreement.
This paper is organized in the following way: first, we start discussing the generic treatment of mixing matrices following from hierarchical mass matrices in Section \[sec:massmix\], where we focus on the mathematical derivation of relations among fermion mass ratios and mixing angles. This result gets applied to the phenomenological data in Section \[sec:applpheno\]. Finally, we conclude. In the appendices, we review the current status of input data, give a brief statement about the applicability of the method elaborated in this work, comment on the hierarchical structure of the mass matrices as a consequence of hierarchical masses and minimal flavor symmetry breaking, and provide the explicit, approximative formulae that gave the results of Section \[sec:applpheno\].
Mass and mixing matrices {#sec:massmix}
========================
Let us extend the SM by three right-handed neutrinos to have a more symmetric treatment of the problem in the quark and lepton sector. Dirac neutrinos alone still leave the question open why the Yukawa couplings for neutrinos are so much smaller than for the charged fermions. Nonetheless, in the description of fermion mixings in terms of fermion masses this assumption does not play a rôle and later we take an effective neutrino mass matrix without the need to specify whether neutrinos are Dirac or Majorana. The most general, renormalizable and gauge invariant construction of fermion mass matrices follows from the Yukawa Lagrangian $$\begin{aligned}
-{\cal L}_Y = \sum_{f=d,e}
{\cal Y}_f^{ij} \, \overline{\psi}_{fL,i} \, \Phi \, \psi_{fR,j} +
\sum_{f=u,\nu} {\cal Y}_f^{ij} \,
\overline{\psi}_{fL,i} \, (i\sigma_2\Phi^*) \, \psi_{fR,j} + {\text{H.c.}},\end{aligned}$$ where $i,j = 1,2,3$ are family indices and summation over them is implicitly understood. The generic fermion fields are denoted as $\psi_f$, where the left-handed fermions are grouped into ${\ensuremath{\mathrm{SU}}}(2)_L$ doublets and the right-handed ones are the usual singlets. The Higgs doublet is given by $\Phi = (\phi^+, \phi^0)$ whereas its nonvanishing vacuum expectation value $v = \langle \phi^0 \rangle = 174\,{\ensuremath{\mathrm{GeV}}}$. The spontaneous breakdown of electroweak symmetry gives rise to four Dirac mass matrices of the form $$\begin{aligned}
{\cal M}_f = v {\cal Y}_f.\end{aligned}$$ These mass matrices are $3 \times 3$ complex arbitrary matrices; each of them is diagonalized by a biunitary transformation $$\begin{aligned}
\label{eq:SVD}
D_f = L^f {\cal M}_f {R^f}^\dagger,\end{aligned}$$ where $D_f$ is a diagonal matrix with real and positive entries while $L^f$ and $R^f$ are two unitary matrices acting in family space on left- and right-handed fermions of type $f$ respectively. Both transformations, $L^f$ and $R^f$, correspond to the unitary matrices appearing in the *Singular Value Decomposition* of ${\cal
M}_f$. These unitary matrices transform the sets of three left- or three right-handed fermion fields each from the interaction basis to the physical mass basis $$\begin{aligned}
\psi^\prime_{f,L} = L^f \psi_{f,L} \quad\quad {\text{and}} \quad\quad
\psi^\prime_{f,R} = R^f \psi_{f,R}.\end{aligned}$$ The mass eigenstates are therewith $\psi^\prime_f$. In return, the diagonal weak charged current interactions are no longer diagonal, and mix different fermion families. This occurs as a consequence of the mismatch between the two different left unitary matrices acting inside the same fermion sector which results in the observable mixing matrices in the charged current interactions $$\begin{aligned}
V_{\text{CKM}} = L^{u} {L^{d}}^\dagger \quad\quad
{\text{and}} \quad\quad
U_\text{PMNS} = L^e {L^{\nu}}^\dagger.\end{aligned}$$
The double mass hierarchy pattern (DMHP)
----------------------------------------
The singular values of the diagonal matrix $D_f$ in Eq. are to be identified with the measured fermion masses (see \[app:data\]). An interesting and not yet exploited fact is that the observed hierarchies in the masses (singular values) can be used to approximate the original mass matrices by lower-rank matrices as stated in the Schmidt-Mirsky approximation theorem [@Schmidt1907433; @EckartYoung; @Mirsky; @Golub1987317].[^3]
The left and right unitary matrices, $L^f$ and $R^f$ are decomposed into the left and right singular vectors, $l_{f,i}$ and $r_{f,i}$ ($i=1,2,3$), and built up as ${L^f}^\dagger = [l_{f,1}, l_{f,2},
l_{f,3}]$ and ${R^f}^\dagger = [r_{f,1}, r_{f,2}, r_{f,3}]$. Each pair of singular vectors correspond to the singular value $m_{f,i}$. For square matrices when all three singular values can be ordered as $m_{f,3} >m_{f,2} > m_{f,1} \geq 0$, the decomposition is unique up to a shared complex phase for each pair of singular vectors.[^4]
The number of non-zero singular values equals the rank of the mass matrix ${\cal M}_f$. The mass matrix can be written in terms of its singular values with the respective left and right singular vectors as a sum of rank one matrices, $$\begin{aligned}
\label{eq:dmhp}
{\cal M}_f = \left[ \left( l_{f,1}
\frac{m_{f,1}}{m_{f,2}} r_{f,1}^\dagger + l_{f,2}
r_{f,2}^\dagger\right)\frac{m_{f,2}}{m_{f,3}} + l_{f,3}
r_{f,3}^\dagger\right]m_{f,3}.\end{aligned}$$ Any hierarchy among the singular values is of major interest to us as it leads to a lower-rank approximation ${\cal M}_f^{r}$ ($r={\text{rank}}[{\cal M}_f^{r}] <3$). The lower-rank approximation is the closest matrix of the given rank to the original matrix, where “close” has to be specified (see \[app:applic\]). We obtain it by keeping the largest singular values and setting the smaller ones equal to zero. The lower rank matrices are unique if and only if all the kept singular values are larger than those set to zero.
Because of $m_{f,3} \gg m_{f,2} \gg m_{f,1}$, Eq. \[eq:dmhp\] provides a powerful way to appreciate the double hierarchy of its singular values and the emerging relation to its rank by the use of Schmidt-Mirsky’s approximation theorem. As both types of quarks and charged lepton masses satisfy those two hierarchies, we conclude, that their mass matrices can be safely approximated as either matrices of rank one or rank two, depending on how strong their double mass hierarchy pattern (DMHP) is.
As illustrated in Eq. , this expression points also to the fact that the fermion mass ratios $m_{f,1}/m_{f,2}$ and $m_{f,2}/m_{f,3}$ play the dominant rôle in determining the structure of the mass matrix whereas $m_{f,3}$ sets the overall mass scale. Only those two ratios will be necessary in the determination of the mixing parameters, since the overall mass scale can be factored out. For later use, we abbreviate $\hat{m}_{f,1} = m_{f,1}/m_{f,3}$ and $\hat{m}_{f,2}
= m_{f,2}/m_{f,3}$. In the following, the hat ($\,\hat{}\,$) denotes the division by the largest mass $m_{f,3}$.
#### The four mass ratios parametrization
The fact, that only two mass ratios for each fermion species are independent parameters, gives four independent mass ratios in each sector (quarks and leptons). An important remark at this point is, that also four parameters are needed to fully parametrize the mixing. This observation shall be used to build up the mixing matrix. In the standard parametrization, those four values are three angles and one phase—additional phases are to be rotated away by redefinition of the fermion fields. The case of Majorana neutrinos does not allow to rotate away the phases for the neutrinos, so two “Majorana phases” are left. In the following, we will leave aside the issue of Majorana phases and only discuss the Dirac phases. We shall show that it is possible to use the four mass ratios of each fermion sector to entirely parametrize the mixing without introduction of new parameters.
It is interesting to note, that a complete parametrization of the fermion mixing in terms of the fermion mass ratios only works in the two- and three-family case. To completely parametrize the mixing matrix, for $n>1$ families, we need $(n-1)^2$ mixing parameters. On the other hand, $n-1$ mass ratios are independent for each fermion species. Therefore, only when the number of mass ratios in the corresponding fermion sector is equal to or larger than the number of mixing parameters, $2(n-1) \geq
(n-1)^2$, this parametrization will be possible. In general, this only works out for two or three families.
The lower-rank approximations
-----------------------------
Let us investigate the effect of neglecting the first generation masses. From now on we will work with the singular values normalized by the largest one. In the $\hat{m}_{f,1} \to 0$ limit, the application of Schmidt-Mirsky’s approximation theorem to the mass matrices is consistent with the rank-two approximation. As we are neglecting all contributions $\mathcal{O}(\hat m_{f,1})$ we shall take into account all corrections of the same order later on to get a more precise result and reduce the error stemming from this approximation.
The rank two mass matrices are then given by $$\begin{aligned}
\hat{\cal M}_f^{r=2} =
\left[l_{f,2} \hat{m}_{f,2} r_{f,2}^\dagger +
l_{f,3} r_{f,3}^\dagger\right] = \begin{pmatrix}
0 & 0 & 0 \\
0 & \hat m^f_{22} & \hat m^f_{23} \\
0 & \hat m^f_{32} & \hat m^f_{33}
\end{pmatrix}.\end{aligned}$$ In general, all the matrix elements should be different from zero. However, it is crucial to establish a connection between a lower-rank approximation and its origin to the Yukawa interactions. That is, $\hat{m}_{f,1} = 0$ is equivalent to decoupling the first fermion family from the Higgs field, $Y_{1j}^f = 0 = Y_{j1}^f$. Effectively, thus, we are left with a $2\times 2$ mass matrix. In the 1-1 sector, in contrast, a phase freedom corresponding to ${\ensuremath{\mathrm{U}}}(1)$ rotations for the left- and right-handed fields is left, where the second and third generation share one common phase.
Up to now, we have only used the hierarchy $m_{f,2} \gg m_{f,1}$ to decouple the first generation masses. According to the lower-rank approximation theorem, the rank-two approximation differs in every element from the full rank matrix, whereas its norm, for any chosen one, only changes slightly. The DMHP furthermore shows $m_{f,3} \gg
m_{f,2}$ which can be exploited to further approximate the initial mass matrix by a rank-one matrix, $$\begin{aligned}
\label{eq:rank-1}
\hat{\cal M}_f^{r=1} =
l_{f,3} r_{f,3}^\dagger =
\begin{pmatrix}
0 & 0 & 0 \\
0 & 0 & 0 \\
0 & 0 & 1
\end{pmatrix}.\end{aligned}$$ Successively reducing the rank of the mass matrices helps to simplify the parametrization without loosing track of the parameters. It is, however, not necessary to work in the very crude rank one approximation, but sufficient to consider as a starting point the rank two approximation.
Eq. reveals a left-over ${\ensuremath{\mathrm{U}}}(2)$ rotation in the 1-2 plane and one common ${\ensuremath{\mathrm{U}}}(1)$ factor for the third generation. We want to emphasize that the described picture of lower-rank approximations follows what is discussed in the literature as minimally broken flavor symmetry [@Barbieri:1995uv; @Barbieri:1996ww; @Barbieri:1997tu]. In the limit of vanishing Yukawa couplings, the SM exhibits a $[{\ensuremath{\mathrm{U}}}(3)]^5$ global flavor symmetry ($[{\ensuremath{\mathrm{U}}}(3)]^6$ if right-handed neutrinos are considered). Each individual ${\ensuremath{\mathrm{U}}}(3)$ flavor symmetry gets gradually broken $${\ensuremath{\mathrm{U}}}(3) \;\stackrel{M_3}{\longrightarrow}\;
{\ensuremath{\mathrm{U}}}(2)\;\stackrel{M_2}{\longrightarrow}\;
{\ensuremath{\mathrm{U}}}(1)\;\stackrel{M_1}{\longrightarrow}\; \text{nothing},$$ with $M_3 > M_2 > M_1$ which simultaneously occurs in the up- and down sector and trivial ${\ensuremath{\mathrm{U}}}(1)$s are left out for readability. After the first symmetry breaking step at $M_3$, one global phase freedom is left for the third generation that is combined to a global ${\ensuremath{\mathrm{U}}}(1)$ for the second and third after the following symmetry breaking. There is one residual ${\ensuremath{\mathrm{U}}}(1)$ symmetry left for all fermions in each sector at the end which is either baryon or lepton number. It is not only safe to work with $M_3 \gg M_2$—where we are at the ${\ensuremath{\mathrm{U}}}(2)$ flavor symmetries of [@Barbieri:1995uv; @Barbieri:1996ww; @Barbieri:1997tu], but even $M_2 \gg M_1$ which allows to work with the rank-two approximation at a sufficiently low scale and perform the final symmetry breaking step at say the electroweak scale.
${\ensuremath{\mathrm{U}}}(2)$ symmetric Yukawa couplings give a well-motivated and frequently used setup to study flavor physics in supersymmetric [@Crivellin:2008mq; @Crivellin:2011sj] and unified [@Barbieri:1996ww] theories and are still a viable tool to discuss recent results in flavor physics [@Barbieri:2012uh; @Buras:2012sd]. Application to lepton flavor physics was also considered [@Carone:1997qg; @Tanimoto:1997zw; @Hall:1998cu; @Barbieri:1999pe], recently also in the context of $[{\ensuremath{\mathrm{U}}}(3)]^5$ breaking [@Blankenburg:2012nx]. The implication of ${\ensuremath{\mathrm{U}}}(2)$ flavor symmetries which can be used in a weaker symmetry assignment [@Aranda:1999kc], is the arrangement of the first two families into one doublet whereas the third family transforms as a singlet under the flavor symmetry. This assignment can be achieved with the minimal discrete symmetry $S_3$ [@Hall:1995es; @Kubo:2003iw; @Morisi:2006pf; @Feruglio:2007hi; @Teshima:2011wg] that was applied to neutrinos [@Jora:2009gz] as well as quarks [@Canales:2013cga].
The important point in the discussion of fermion mixings in terms of fermion masses via lower-rank approximations is, that we implicitly assume the maximal $[{\ensuremath{\mathrm{U}}}(3)]^6$ flavor symmetry broken with each symmetry breaking step *occurring simultaneously* for each subgroup $[{\ensuremath{\mathrm{U}}}(3)]^6 = {\ensuremath{\mathrm{U}}}(3)_Q \times {\ensuremath{\mathrm{U}}}(3)_u \times {\ensuremath{\mathrm{U}}}(3)_d \times {\ensuremath{\mathrm{U}}}(3)_L
\times {\ensuremath{\mathrm{U}}}(3)_e \times {\ensuremath{\mathrm{U}}}(3)_\nu$.
#### Order of independent rotations
To parametrize the three-fold mixing, we follow the commonly used three successive rotations depending on one angle and one phase each. The order of these transformations needs to follow the consecutive breakdown of the initial ${\ensuremath{\mathrm{U}}}(3)$ symmetry as implied by the hierarchy in the masses. Therefore, $$\begin{aligned}
\label{eq:Euler-order}
L^f = L^f_{12}(\theta_{12}^f,\delta^f_{12})
L^f_{13}(\theta_{13}^f,\delta^f_{13})
L^f_{23}(\theta_{23}^f,\delta^f_{23}),\end{aligned}$$ where each individual rotation is parametrized by one angle $\theta^f_{ij}$ and one phase $\delta^f_{ij}$.[^5]
Note that this set of rotations diagonalize the mass matrices for each fermion type. The resulting mixing matrices are the product of all the individual rotations $$V_{\text{CKM}} = L^u {L^d}^\dag = L^u_{12} L^u_{13} L^u_{23}
{L^d_{23}}^\dag {L^d_{13}}^\dag {L^d_{12}}^\dag$$ and $$U_{\text{PMNS}} = L^e {L^\nu}^\dag = L^e_{12} L^e_{13} L^e_{23}
{L^\nu_{23}}^\dag {L^\nu_{13}}^\dag {L^\nu_{12}}^\dag.$$ By convention, up- and down-type rotations are exchanged for leptons.
The effective $2 \times 2$ mass matrix
--------------------------------------
It is instructive to first study the two-family limit in the rank-two approximation following from $\hat m_{f,1} \ll 1$. The second hierarchy $m_{f,2} \ll m_{f,3}$ implies a $2\times 2$ mass matrix of the form $$\label{eq:2by2mass}
\hat{\bf m}^f = \begin{pmatrix}
\hat m^f_{ss} & \hat m^f_{sl} \\ \hat m^f_{ls} & \hat m^f_{ll}
\end{pmatrix},$$ with hierarchical elements $|\hat m^f_{ll}|^2 \gg |\hat m^f_{sl}|^2,
|\hat m^f_{ls}|^2 \gg |\hat m^f_{ss}|^2$ and where we are now generically treating two fermion families whose singular values obey the hierarchy, $\sigma_l \gg \sigma_s$. In general, the matrix elements are complex numbers. The labelling $s$ and $l$ refers to the corresponding smaller and larger singular value, respectively. It can be shown that the order of magnitude of $\hat m^f_{ss}$ is about $\mathcal{O}(|\hat{m}^f_{sl}|^2)$ (see \[app:m11\]). In the following, we work with the approximation $\hat m^f_{ss} = 0$.
Unlike most considerations, we take the outcome of the DMHP and minimal flavor symmetry breaking to set the magnitudes of the off-diagonals equal—the phases are not constrained, such that $$|\hat{m}^f_{sl}| = |\hat{m}^f_{ls}|
\,\qquad\text{not}\qquad
\hat{m}^f_{sl} = (\hat{m}^f_{ls})^*,$$ as implied by the requirement of an Hermitian mass matrix. We only need normal mass matrices.[^6] In both cases (normal and Hermitian), the left and right Hermitian products are diagonalized by the same unitary transformation. For a normal mass matrix, however, the phases can be arranged in a way that the off-diagonal magnitudes do not have to be the same. We only constrain the matrix of absolute values to be symmetric, whereas the phases can be arbitrary: $$\label{eq:approxmassmat}
\hat{\bf m}^f = \begin{pmatrix}
0 & |\hat m^f_{sl}| e^{i\delta_{sl}^f} \\
|\hat m^f_{sl}| e^{i\delta_{ls}^f} & \hat m^f_{ll}
\end{pmatrix}.$$
As a self-consistency check, it is important to verify that the required hierarchy in all the mass matrix elements of the full-rank scenario actually is respected when expressing the matrix elements in terms of the masses (singular values).
#### Reparametrization in terms of the singular values
Due to our lack of knowledge of right-handed flavor mixing, the relevant object that determines our phenomenology is the Hermitian product ${\mathbf{n}}^f = {\mathbf{m}}^f\,({\mathbf{m}}^f)^\dag$, which exhibits two invariants: ${\operatorname{tr}}{\mathbf{n}}^f = \sigma^{f2}_s +
\sigma^{f2}_l$ and $\det{\mathbf{n}}^f = \sigma^{f2}_s
\sigma^{f2}_l$. The small and large singular value are denoted by $\sigma^f_s$ and $\sigma^f_l$, respectively. Through means of the two invariants, we find $$\begin{aligned}
\label{eq:twoInvariants}
|{\hat m}^f_{sl}| =
\sqrt{\hat{\sigma}_{sl}^f}, \quad\quad {\text{and}} \quad\quad |\hat
m^f_{ll}| = 1 - \hat{\sigma}_{sl}^f,\end{aligned}$$ where we have expressed for a generic treatment the normalized ratio of the small singular value over the large one as $\hat{\sigma}^f_{sl}
\equiv \sigma^f_s / \sigma^f_l$.
This reparametrization nicely shows the result of the Schmidt-Mirsky approximation theorem: on the one hand, $|\hat m^f_{ll}|^2 \gg |\hat
m^f_{sl}|^2$, while on the other hand, $|\hat m^f_{ll}| = 1$ is the only non-vanishing matrix element in the limit $\hat{\sigma}^f_s \rightarrow
0$.
The left unitary transformation corresponding to the diagonalization of this matrix is given by $$\begin{aligned}
\label{eq:2diag}
L_{sl}^f(\hat{\sigma}^f_{sl},\delta^f_{sl}) = \frac{1}{\sqrt{1+\hat{\sigma}^f_{sl}}}
\begin{pmatrix}
1 & e^{-i\delta^f_{sl}} \sqrt{\hat{\sigma}^f_{sl}} \\
- e^{i\delta^f_{sl}} \sqrt{\hat{\sigma}^f_{sl}} & 1
\end{pmatrix}.\end{aligned}$$ This result has been already discussed previously by many authors [@Fritzsch:1977za; @Weinberg:1977hb; @Raby:1995uv; @Fritzsch:1999ee]. The mixing angle can be obtained from $\tan\theta_{sl}^f = \sqrt{\hat\sigma^f_{sl}}$.[^7] Note that this relation indeed is the Gatto-Sartori-Tonin result, see Eq..
#### The two-family mixing matrix
Eq. diagonalizes the mass matrix of one fermion type. In the weak charged current, an $a$-type fermion ($a = u,e$) meets a $b$-type fermion ($b=d,\nu$), so we need two such diagonalizations to describe fermion mixing in the charged current interactions. Anyway, two unitary $2\times 2$ rotations do not commute, and the new mixing parameters are not just the sum or difference of the former ones: $\theta_{sl} \neq \theta^a_{sl} \pm \theta^b_{sl}$ and $\delta \neq
\delta^a_{sl} \pm \delta^b_{sl}$. Explicitly, $$\label{eq:twofold}
V_{sl} = L_{sl}^a {L_{sl}^b}^\dag = {\operatorname{diag}}(1, e^{-i\delta_{sl}^a})
\begin{pmatrix}
\sqrt{1 - \lambda^2} e^{-i\delta_0} & \lambda e^{-i\delta} \\
- \lambda e^{i\delta} & \sqrt{1 - \lambda^2} e^{i\delta_0}
\end{pmatrix} {\operatorname{diag}}(1, e^{i \delta_{sl}^a}),$$ where we factored out the phase $\delta_{sl}^a$. This choice is completely arbitrary, the same is true for $\delta_{sl}^b$. The relevant phases *inside* the matrix only depend on the *difference*. The mixing can then be obtained in the following way $$\begin{aligned}
\label{eq:theta}
\lambda =\sin\theta_{sl} &= \sqrt{\frac{\hat{\sigma}^a_{sl} +
\hat{\sigma}^b_{sl} - 2 \sqrt{ \hat{\sigma}^a_{sl}
\hat{\sigma}^b_{sl}} \cos(\delta^a_{sl} -
\delta^b_{sl})}{(1+\hat{\sigma}^a_{sl})(1+\hat{\sigma}^b_{sl})}}
, \\
\tan\delta &= \frac{\hat{\sigma}^b_{sl} \sin(\delta^a_{sl} -
\delta^b_{sl})}{\hat{\sigma}^a_{sl} - \hat{\sigma}^b_{sl}
\cos(\delta^a_{sl} -
\delta^b_{sl})}, \\
\tan\delta_0 &= \frac{\hat{\sigma}^a_{sl} \hat{\sigma}^b_{sl}
\sin(\delta^a_{sl} - \delta^b_{sl})}{1 +\hat{\sigma}^a_{sl}
\hat{\sigma}^b_{sl} \cos(\delta^a_{sl} - \delta^b_{sl})}.\end{aligned}$$ The functional dependence on the two initial complex phases is found to be only their difference. From the hierarchies $\hat\sigma^x_{sl} =
\sigma^x_s / \sigma^x_l \ll 1$ (for $x=a,b$) follow the new phases to be approximately given by $\tan\delta \approx - \tan(\delta_{sl}^a -
\delta_{sl}^b)$ and $\tan\delta_0 \approx 0$. For the full-rank scenario, however, this simple conclusion cannot be drawn—it actually holds for the “initial” 2-3 rotation, but not anymore when subsequent rotations are added.
#### Comment on the complex phases
In general, the complex phases of the initial mass matrix elements are not constrained to a particular value. The employed matrix invariants only restrict the moduli of the matrix elements, the phases are unconstrained. There is nevertheless an ambiguity in those phases that is not necessary to set up a full parametrization of fermion mixing in the SM. The standard parametrization uses three successive rotations with $\theta_{ij} \in [0,\frac{\pi}{2}]$ and one complex phase $\delta_{CP} \in [0,2\pi)$. These four parameters are sufficient to describe both mixing and CP violation in each fermion sector (unless we want to include a description of Majorana phases for neutrinos). In contrast, we have four mass ratios—and the freedom to put either real or purely imaginary matrix elements. This last choice can be achieved by restricting all phases to be either maximal CP violating ($\pi/2$ or $3\pi/2$) or CP conserving ($0$ or $\pi$). Interestingly, at the end, there is no freedom in phase choices at all and we find that only the 1-2 phase is allowed to be maximally CP violating, which indeed follows from a symmetry argument.
The full-rank picture
---------------------
Working in the lower-rank approximations, we are neglecting the first generation mass ($\hat{m}_{f,1} = 0$) in the 2-3 rotation and the second generation mass ($\hat{m}_{f,2} = 0$) while performing the 1-3 rotation. The last transformation that appears in Eq. acting in the 1-2 plane needs no approximation. It affects only the upper left $2\times 2$ submatrix and is an exact diagonalization. In all cases, the mass matrices are of the form , where the elements are properly distributed over the $3\times 3$ matrix elements. All residual matrix elements are zero. The same holds for the arising rotation matrices that are $3\times 3$ generalizations of Eq. .
Working in the leading order approximations shows a subtle inconsistency: neglecting $\mathcal{O}(\hat m_{f,2})$ terms in the 1-3 rotation means actually ignoring a large effect, because $\mathcal{O}(\hat m_{f,1}) = \mathcal{O}(\hat m_{f,2}^2)$. Moreover, to include $\mathcal{O}(\hat m_{f,1})$ contributions in the 1-3 rotation following the initial rotation in the 2-3 plane, we first have to consider contributions of the same order that were missing in the initial rotation. Therefore, we briefly discuss how to consistently include corrections of missing pieces to improve the result.
#### Inclusion of corrections
We include the corrections as correcting (small) rotations. This procedure is crucial in view of the symmetry breaking chain from an enhanced flavor symmetry, as $[{\ensuremath{\mathrm{U}}}(3)]^3$ (corresponding to a rank-zero mass matrix), down to the least symmetry left over. Since each breaking step is done by a small parameter, we do not disturb much by adding perturbations. Moreover, by repeatedly applying rotations, this guarantees from the very beginning normalized eigenvectors, and furthermore, an inclusion of formally higher order terms in perturbation theory. This can be seen from the following example of two real rotations, where $\hat\epsilon \lesssim
\hat{\sigma}^f_{sl}$:[^8] $$\begin{aligned}
{L^{f}_{sl}}^{(p=1)} = {L^{f}_{sl}}^{(1)}(\pm\hat
\epsilon){L^{f}_{sl}}^{(0)} (\hat{\sigma}^f_{sl}) =
\begin{pmatrix} \cos{\theta^f_{sl}}^{(p=1)} &
\sin{\theta^f_{sl}}^{(p=1)} \\ -\sin{\theta^f_{sl}}^{(p=1)}&
\cos{\theta^f_{sl}}^{(p=1)}
\end{pmatrix},\end{aligned}$$ and the new angle is given by $$\begin{aligned}
\sin{\theta^f_{sl}}^{(p=1)} =
\frac{\sqrt{\hat{\sigma}_{sl}^{f}} \pm \sqrt{\hat
\epsilon}}{\sqrt{(1+\hat{\sigma}_{sl}^{f})(1+\hat\epsilon)}}.\end{aligned}$$ For real rotations, the requirement $\hat\epsilon \lesssim
\hat{\sigma}^f_{sl}$ is irrelevant, because ${\ensuremath{\mathrm{O}}}(2)$ rotations commute. Therefore, there is also no need to specify any order in the addition of correcting rotations in each $i$-$j$ plane.
Inverting this procedure shows that it is equivalent to add the perturbation term $$\begin{aligned}
\label{eq:perturb}
-\sqrt{\hat\epsilon} \left[ 1+(\hat{\sigma}^f_{sl})^2 -2\hat{\sigma}^f_{sl}
+ \sqrt{\hat {\epsilon}\hat{\sigma}^f_{sl} }
(\hat{\sigma}^f_{sl}-1)\right](1-\hat \epsilon)\end{aligned}$$ to the off diagonal matrix elements $s$-$l$ and $l$-$s$.
Continuing this, an arbitrary number of correcting rotations could be added in each $2\times 2$ rotation: $$\begin{aligned}
\label{eq:theta-pert}
\sin{\theta^f_{sl}}^{(p=n)} =
\frac{\sum_{j=0}^{n}(-1)^{\delta_j}\sqrt{\hat{a}_j} +
\mathcal{O}\left(\left[\sqrt{\hat a_i \hat a_j \hat a_k}\right]_{i
\neq j \neq k}\right)}
{\sqrt{(1+\hat{a}_{0})(1+\hat{a}_1)(1+\hat{a}_2) \cdots (1+\hat{a}_n)}},\end{aligned}$$ where we have denoted $\hat{a}_0 \equiv \hat{\sigma}_{sl}^f$ and $\hat
a_{i>0}$ for the parameters of the following rotations. Each $(-1)^{\delta_i}$ is the orientation of the $i$-th rotation, which is either clockwise or counterclockwise (plus or minus). We neglect in Eq. all trilinear and higher products of $\hat
a_i$, where no $\hat a_i^2$ and no even products appear. Let us emphasize here, nevertheless, that these correcting rotations do not follow the traditional procedure of perturbation theory where we could naively think that the following new correcting rotation is a power of the previous one. Inclusion of new correcting rotations requires a careful treatment. We have found to be sufficient to include two correcting rotations to the mixing matrix parametrization which are the contributions $\mathcal{O}(\hat m_{f,1})$, $\mathcal{O}(\hat
m_{f,2}^2)$, and $\mathcal{O}( \hat m_{f,1} \cdot \hat m_{f,2} )$ which are of the same order as the neglected terms in each case.
#### First rotation: The 2-3 sector
Starting from the rank-two approximation, we loose track of all $\sqrt{\hat{m}_{f,1}}$ contributions in the mass matrix. However, all correcting rotations have to be consistent with the initial approximation ($\hat{m}_{f,1} \to 0$) and, moreover, all “higher order” contributions ($\sim\hat{m}_{f,2}^2$, $\sim\hat{m}_{f,1}^2$) are already covered as can be seen from . We therefore conclude, that all reasonable rotations in the 2-3 plane can be expressed as $$\begin{aligned}
{L^f_{23}}^{(p=2)} ={L^{f}_{23}}^{(2)}(\hat{m}_{f,1}\cdot\hat{m}_{f,2})
{L^{f}_{23}}^{(1)}(\hat{m}_{f,1}){L^{f}_{23}}^{(0)} (\hat{m}_{f,2}).\end{aligned}$$ Additionally, in principle, there is a freedom in the choice of the complex phase, which can be boiled down to the two different sign choices.
#### Second rotation: The 1-3 sector
What follows is the same procedure in the 1-3 sector after the 2-3 rotations have been done. In this case, the $p=2$ leading correcting rotations are $$\begin{aligned}
{L^f_{13}}^{(p=2)} =
{L^{f}_{13}}^{(2)}(\hat{m}_{f,1}\cdot\hat{m}_{f,2}) {L^{f}_{13}}^{(1)}(\hat{m}_{f,2}^2)
{L^{f}_{13}}^{(0)} (\hat{m}_{f,1}).\end{aligned}$$
#### Last rotation: The 1-2 sector
No approximation is left anymore, therefore the exact rotation is expressed as $$\begin{aligned}
{L^f_{12}} = L^f_{12} (\frac{\hat{m}_{f,1}}{\hat{m}_{f,2}}, \delta^f_{12}),\end{aligned}$$ where we now explicitly put the phase $\delta^f_{12}$. This occurrence is very clear from the rank evolution: in the rank-one approximation, there is the freedom of a ${\ensuremath{\mathrm{U}}}(2)$ rotation left in the 1-2 block. The initial 2-3 and 1-3 rotations can always be taken real, the only possible phase then sits in the 1-2 rotation.
The necessity of correcting rotations is very apparent from the flavor symmetry breaking chain: First, in the rank-two approximation we have $$\begin{pmatrix} 0 & 0 & 0 \\ 0 & X & X \\ 0 & X & X \end{pmatrix}
\stackrel{L_{23}^{(0)}}{\longrightarrow}
\begin{pmatrix} 0 & 0 & 0 \\ 0 & X & 0 \\ 0 & 0 & X \end{pmatrix}.$$ After performing the symmetry breaking step to the full-rank matrix, we get contributions in all matrix elements not larger than $\mathcal{O}(\sqrt{\hat{m}_{f,1}})$—also in off-diagonal components that were already rotated away: $$\begin{pmatrix} * & * & * \\ * & X & * \\ * & * & X \end{pmatrix}.$$ So, we indeed have to consider higher order corrections to the initial rotation. The correcting rotations also do not spoil the required hierarchy. After the successive 2-3 and 1-3 rotations there is a contribution shuffled into the 1-1 entry which is $\sim s_{13}^2 m_{33}
\sim \mathcal{O}(\hat{m}_{f,1})$ and therefore of higher order compared to $\mathcal{O}(\hat{m}_1/\hat{m}_2)$, the original 1-1 element.
Applying the DMHP to phenomenology {#sec:applpheno}
==================================
By building up the mixing matrices following the procedure of the previous section, there appears the impression of an arbitrariness in the choice of complex phases. This arbitrariness can be attenuated taking into account some well motivated considerations. First, complex phases appear pairwise in the up- and down-type fermion sectors. We therefore have the freedom to keep track of them in only one sector and set all phases in the other one equal to zero. The charged current mixing matrix is therefore constructed in the following way: $$\begin{aligned}
V_\text{CKM} &= L^u {L^d}^\dag, \nonumber\\
L^u &= L^u_{12}\left(\frac{m_u}{m_c}\right)
L^u_{13}\left(\frac{m_um_c}{m_t^2}\right)
L^u_{13}\left(\frac{m_c^2}{m_t^2}\right)
L^u_{13}\left(\frac{m_u}{m_t}\right)
L^u_{23}\left(\frac{m_um_c}{m_t^2}\right)
L^u_{23}\left(\frac{m_u}{m_t}\right)
L^u_{23}\left(\frac{m_c}{m_t}\right), \\
{L^d}^\dag &= {L^d_{23}}^\dag\left(\frac{m_s}{m_b}, \delta^{(0)}_{23}\right)
{L^d_{23}}^\dag\left(\frac{m_d}{m_b}, \delta^{(1)}_{23}\right)
{L^d_{23}}^\dag\left(\frac{m_dm_s}{m_b^2}, \delta^{(2)}_{23}\right)
\;\times \nonumber \\ &\quad
{L^d_{13}}^\dag\left(\frac{m_d}{m_b}, \delta^{(0)}_{13}\right)
{L^d_{13}}^\dag\left(\frac{m_s^2}{m_b^2}, \delta^{(1)}_{13}\right)
{L^d_{13}}^\dag\left(\frac{m_dm_s}{m_b^2}, \delta^{(2)}_{13}\right)
{L^d_{12}}^\dag\left(\frac{m_d}{m_s}, \delta_{12}\right),
\label{eq:down} \\
U_\text{PMNS} &= L^e {L^\nu}^\dag, \nonumber \\
L^e &= L^e_{12}\left(\frac{m_e}{m_\mu}\right)
L^e_{13}\left(\frac{m_\mu^2}{m_\tau^2}\right)
L^e_{13}\left(\frac{m_em_\mu}{m_\tau^2}\right)
L^e_{13}\left(\frac{m_e}{m_\tau}\right)
L^e_{23}\left(\frac{m_em_\mu}{m_\tau^2}\right)
L^e_{23}\left(\frac{m_e}{m_\tau}\right)
L^e_{23}\left(\frac{m_\mu}{m_\tau}\right), \\
{L^\nu}^\dag &= {L^\nu_{23}}^\dag\left(\frac{m_{\nu2}}{m_{\nu3}}, \delta^{(0)}_{23}\right)
{L^\nu_{23}}^\dag\left(\frac{m_{\nu1}}{m_{\nu3}}, \delta^{(1)}_{23}\right)
{L^\nu_{23}}^\dag\left(\frac{m_{\nu1}m_{\nu2}}{m_{\nu3}^2}, \delta^{(2)}_{23}\right)
\;\times \nonumber \\ &\quad
{L^\nu_{13}}^\dag\left(\frac{m_{\nu1}}{m_{\nu3}}, \delta^{(0)}_{13}\right)
{L^\nu_{13}}^\dag\left(\frac{m_{\nu2}^2}{m_{\nu3}^2}, \delta^{(1)}_{13}\right)
{L^\nu_{13}}^\dag\left(\frac{m_{\nu1}m_{\nu2}}{m_{\nu3}^2}, \delta^{(2)}_{13}\right)
{L^\nu_{12}}^\dag\left(\frac{m_{\nu1}}{m_{\nu2}}, \delta_{12}\right).
\label{eq:neutrino}\end{aligned}$$
The method itself is not quite arbitrary at all. For the CKM mixing it gives well-separated regions that have to be entered with a specific choice for the phases (see Fig. \[fig:CKMphases\]). Since both quark masses as well as CKM mixing matrix entries are rather well measured, this observations allows us to set the phases. We find only one distinct choice. Moreover, we make a *minimal* choice: on the one hand, we allow CP phases to be either maximally CP violating or CP conserving. On the other hand, we find, that the only maximally CP violating phase has to be in the 1-2 rotation of the down-type quarks or neutrinos, respectively. This can be seen from Fig. \[fig:cabibbo\_jarlskog\] where the three bands correspond to a phase $\delta_{12} = 0,
\frac{\pi}{2} \text{ and } \pi$.
The previously derived subsequent rotations only depend on four mass ratios in each fermion sector and have to be faced with phenomenological data. As input values we are using the quark and lepton masses only (see \[app:data\]) and then give a prediction for the neutrino masses to be in agreement with observations of neutrino mixing in this setup.
![Distribution of allowed values in the $V_{ub}$-$V_{cb}$ plane. The small red points show allowed regions where the masses were varied in their $1\,\sigma$ regimes, the blue crosses show the values coming from the central values of the masses. Right: zoom into the phenomenological viable region. There are only three distinct phase choices leading to both small values for $V_{ub}$ and $V_{cb}$.[]{data-label="fig:CKMphases"}](CKMphases){width="\textwidth"}
![Distribution of allowed values in the $V_{ub}$-$V_{cb}$ plane. The small red points show allowed regions where the masses were varied in their $1\,\sigma$ regimes, the blue crosses show the values coming from the central values of the masses. Right: zoom into the phenomenological viable region. There are only three distinct phase choices leading to both small values for $V_{ub}$ and $V_{cb}$.[]{data-label="fig:CKMphases"}](CKMphases_zoom){width="\textwidth"}
![The left plot shows the regions for the Cabibbo angle (more exactly $V_{us}$—clearly the solution $\delta_{12} =
\frac{\pi}{2}$ is favoured (which corresponds to the stripe in the center). On the right side, the rephasing invariant $J_q$ is shown against $V_{ub}$. Color code as in Fig. \[fig:CKMphases\]: red dots are points with masses varied in the $1\,\sigma$ regimes, blue crosses are the central values.[]{data-label="fig:cabibbo_jarlskog"}](CKM_cabibbo){width="\textwidth"}
![The left plot shows the regions for the Cabibbo angle (more exactly $V_{us}$—clearly the solution $\delta_{12} =
\frac{\pi}{2}$ is favoured (which corresponds to the stripe in the center). On the right side, the rephasing invariant $J_q$ is shown against $V_{ub}$. Color code as in Fig. \[fig:CKMphases\]: red dots are points with masses varied in the $1\,\sigma$ regimes, blue crosses are the central values.[]{data-label="fig:cabibbo_jarlskog"}](CKM_jarlskog){width="\textwidth"}
Minimal or maximal CP violation
-------------------------------
The nature of the complex phases and its impact in the mixing matrix elements needs further investigation. Giving a solution to this problem is, however, outside the scope of this work. We shall use our observation to distribute the CP violating phase properly and leave the origin of CP violation for later work.
A final comment can be done, though, that guarantees the uniqueness of the parametrization. In Fig. \[fig:CKMphases\], we show the maximally allowed ranges for the mixing matrix elements $V_{ub}$ and $V_{cb}$. The amount of data points was constructed choosing the quark masses from their $1\,\sigma$ regimes and randomly taking every phase in the final paramerization from the set $\lbrace 0, \frac{\pi}{2},
\pi\rbrace$. It is sufficient to constrain oneself to this set which gives the minimal and maximal allowed amount of CP violation [@Masina:2006ad]—and connected to that minimal and maximal mixing. The latter can be seen from Eq. for the two-generation sub-case: the phase difference $\delta^a_{sl} -
\delta^b_{sl}$ controls the magnitude of the mixing angle between minimal ($\delta^a_{sl} - \delta^b_{sl} = 0$) and maximal ($\pi$) mixing.
The fact, that only *one* combination of phases survives, is astonishing: note that all possible combinations in Eq. are generically $3^7 = 2187$ choosing from $\lbrace 0, \frac{\pi}{2},
\pi\rbrace$. Still, after taking $\delta_{12} = \frac{\pi}{2}$ and constraining the remnant phases to be either zero or $\pi$, 64 combinations are left. It is therefore not *a priori* clear that the mass ratios alone give the right mixing. The functional dependence on the mass ratios, however, is unique once the phases are set. We therefore use this description to determine the position of the maximal CP phase, where in contrast the other phases give relative minus signs. The maximal CP violating phase in the neutrino 1-2 mixing is somehwat different to what was found in connection with maximal atmospheric mixing [@Masina:2005hf].
$\delta_{12}$ $\delta_{13}^{(0)}$ $\delta_{13}^{(1)}$ $\delta_{13}^{(2)}$ $\delta_{23}^{(0)}$ $\delta_{23}^{(1)}$ $\delta_{23}^{(2)}$
------ ----------------- --------------------- --------------------- --------------------- --------------------- --------------------- ---------------------
CKM $\frac{\pi}{2}$ 0 $\pi$ $\pi$ $0$ $\pi$ $\pi$
PMNS $\frac{\pi}{2}$ 0 $\pi$ $\pi$ $\pi$ $\pi$ $0$
: The choice of phases in Eqs. and leading to the mixing matrices shown in and .[]{data-label="tab:phases"}
Projected values of $V_{\text{CKM}}^{\text{th}}$ and $J_q$
----------------------------------------------------------
Consideration of all the aforementioned prescriptions gives the following numbers for the magnitude of the mixing matrix elements (see \[app:Formulae\] for the explicit formulae of the mixing angles and the Jarlskog invariant), $$\begin{aligned}
\label{eq:postCKM}
|V_{\text{CKM}}^{\text{th}}| =
\begin{pmatrix}
0.974^{+0.004}_{-0.003} & 0.225^{+0.016}_{-0.011} & 0.0031^{+0.0018}_{-0.0015} \\
0.225^{+0.016}_{-0.011} & 0.974^{+0.004}_{-0.003} & 0.039^{+0.005}_{-0.004} \\
0.0087^{+0.0010}_{-0.0008} & 0.038^{+0.004}_{-0.004} & 0.9992^{+0.0002}_{-0.0001}
\end{pmatrix}\end{aligned}$$ and the following amount of CP violation as measured by the Jarlskog invariant, $$\begin{aligned}
J_q = {\ensuremath{\operatorname{Im}}}(V_{us}V_{cb}V_{ub}^*V_{cs}^*) = ( 2.6^{+1.3}_{-1.0} ) \times 10^{-5},\end{aligned}$$ where all quantities here are seen to be in quite good agreement within the errors compared to the global fit result given by the PDG 2014 [@Agashe:2014kda] (see \[app:data\] for present knowledge on masses and mixings). Note that generically, the amount of CP violation is much larger (Fig. \[fig:cabibbo\_jarlskog\]) and a small value of $V_{ub}$ is connected to a small $J_q$, as expected.
Lepton sector {#sec:lept}
-------------
Quark masses show a very strong hierarchy. Charged lepton masses also do. Neutrinos, though, do not do. Is it really viable to apply the DMHP also to lepton mixing? Leptonic mixing angles are large, this observation may hint to a different mechanism. However, mass ratios for neutrinos are also large. The parametrization of fermion mixing in terms of mass ratios allows to also cope with large mixings by large mass ratios. Nevertheless, we have to include a solid examination of the errors in this approximation and see whether the same procedure as for quarks is viable also for leptons.
Are neutrino masses hierarchical? Neither the quasidegenerate solution nor the strong hierarchy are excluded yet. A hierarchical mass spectrum in any case predicts a very light lightest neutrino (it still can be exactly massless—in this case we would only have a rank two mass matrix), where degenerate masses are likely to be tested in the near future.
The power of the mixing parametrization in terms of mass ratios lies in its invertibility: the formulae give us a unique description of the missing mass ratio once the mixing angle is measured. The pattern of neutrino masses brings us into the comfortable situation of nearly disentangling the 1-2 from the 2-3 mixing, because $\Delta m_{21}^2 /
\Delta m_{31}^2 \ll 1$. Additionally, the 1-2 mixing angle has the smallest error in the global fit.
#### Predicted neutrino masses
We do not focus on a specific model behind the theory of neutrino masses. It is sufficient to consider an effective neutrino mass matrix irrespective of the UV completion behind. To embed our description into a theory of neutrino flavor, it definitely matters if neutrinos are Dirac or Majorana. The size of the masses, however, allows to neglect RG running in any case. Therefore, we also ignore the nature of the neutrino mass operator. Since we take the magnitudes of the Dirac masses symmetric for quarks, the only difference would be the off-diagonal phase. Having this similarity in mind, the 1-2 approximation for neutrinos follows directly from Eq. and the determining equation for the missing mass ratio from Eq. with obvious relabelings: $$\label{eq:Ue2}
|U_{e2}| \approx \sqrt{\frac{\hat{m}_{e\mu} +
\hat{m}_{\nu 12} - 2 \sqrt{ \hat{m}_{e\mu} \hat{m}_{\nu 12}} \cos(\delta^e_{12} -
\delta^\nu_{12})}{(1+\hat{m}_{e\mu})(1+\hat{m}_{\nu 12})}},$$ where the mass ratios are $\hat m_{e\mu} = m_e / m_\mu$ and $\hat
m_{\nu12} = m_{\nu1} / m_{\nu2}$. The three individual neutrino masses[^9] are obtained via the mass squared differences: $$\label{eq:neutrino-mas-def}
\begin{aligned}
m_{\nu2} &= \sqrt{ \Delta m_{21}^2 / ( 1 - \hat{m}^2_{\nu12} ) }, \\
m_{\nu1} &= \sqrt{ m_{\nu2}^2 - \Delta m_{21}^2}, \\
m_{\nu3} &= \sqrt{\Delta m_{31}^2 - \Delta m_{21}^2 + m_{\nu2}^2}.
\end{aligned}$$ In Eq. , there appears the phase difference $\delta^e_{12} - \delta^\nu_{12}$. Although a twofold rotation shows no CP violation, this phase has to be considered because it appears last in the order of successive rotations. Moreover, we observed a maximal CP phase in the quark 1-2 sector. Albeit there is no connection between quark and lepton mixing at this stage, we shall keep the assignment $\delta^e_{12} - \delta^\nu_{12} = \frac{\pi}{2}$ and get $$\label{eq:numassratio}
\hat{m}_{\nu1} = \frac{|U_{e2}|^2 ( 1 + \hat m_e ) - \hat m_e}{1 -
|U_{e2}|^2 ( 1 + \hat m_e)} =
0.41 \ldots 0.45$$ using $\hat m_e = 0.00474$ and $|U_{e2}| = \sin\theta_{12} = 0.54
\ldots 0.56$. The masses are calculated as $$\begin{aligned}
m_{\nu1} &= ( 0.0041 \pm 0.0015 )\,{\ensuremath{\mathrm{eV}}}, \\
m_{\nu2} &= ( 0.0096 \pm 0.0005 )\,{\ensuremath{\mathrm{eV}}}, \\
m_{\nu3} &= ( 0.050 \pm 0.001 )\,{\ensuremath{\mathrm{eV}}}.\end{aligned}$$ The errors were propagated from the $\Delta m^2$ and added linearly to be more conservative. Within $3\,\sigma$, the lightest neutrino can be massless. This prediction, however, will significantly improve with the improved errors on $\Delta m_{21}^2$.
The minimally and maximally allowed neutrino masses (corresponding to $\delta^e_{12} - \delta^\nu_{12} = 0, \pi$) are very close:
min (in ${\ensuremath{\mathrm{eV}}}$) max (in ${\ensuremath{\mathrm{eV}}}$)
--------------------------------------- ---------------------------------------
$m_{\nu1} = 0.0029 \pm 0.0017$ $m_{\nu1} = 0.0062 \pm 0.0017$
$m_{\nu2} = 0.0091 \pm 0.0003$ $m_{\nu2} = 0.011 \pm 0.001$
$m_{\nu3} = 0.050 \pm 0.001$ $m_{\nu3} = 0.050 \pm 0.001$
In any case, the lightest neutrino is much lighter than $0.01\,{\ensuremath{\mathrm{eV}}}$.
{width="\textwidth"}
{width="\textwidth"}
#### $U_{\text{PMNS}}^{\text{th}}$ as implied by the four leptonic mass ratios
Albeit the hierarchy is not as strong as for quarks and charged neutrinos, we dare to use the same description and show that indeed large mass ratios in the four mass ratio parametrization also lead to large mixing angles. The applicability of the whole method depends on hierarchical masses. In \[app:applic\] we give a simple criterion parameter to check whether the lower-rank approximations are good approximations. Indeed, the deviation from unity is only a few percent. Therefore, we safely use the previous described procedure.
With the predicted neutrino masses (which only know about $|U_{e2}|$) and the knowledge of the charged fermion mass ratios, the leptonic mixing matrix exhibits the following numerical values $$\begin{aligned}
\label{eq:prePMNS}
|U_{\text{PMNS}}^{\text{th}}| =
\begin{pmatrix}
0.83^{+0.04}_{-0.05} & 0.54^{+0.06}_{-0.09} & 0.14 \pm 0.03 \\
0.38^{+0.04}_{-0.06} & 0.57^{+0.03}_{-0.04} & 0.73 \pm 0.02 \\
0.41^{+0.04}_{-0.06} & 0.61^{+0.03}_{-0.04} & 0.67 \pm 0.02
\end{pmatrix},\end{aligned}$$ whereas the implied amount of CP violation is displayed as $$\begin{aligned}
J_\ell = {\ensuremath{\operatorname{Im}}}(U_{e2}U_{\mu 3}U_{e 3}^*U_{\mu 2}^*) = 0.031^{+0.006}_{-0.007} .\end{aligned}$$
We remark an astonishingly good agreement with the measured values (see \[app:data\]) and observe a close-to-maximal CP violation in the lepton sector! ($\delta_\text{CP} = 70^\circ$ from the central values: $J_\ell = J^\text{max}_\ell \sin\delta_\text{CP}$, the error on $J^\text{max}_\ell$ is nevertheless compatible with maximal CP violation, $\delta_\text{CP} = 90^\circ$.)
About precision
---------------
The goal of the presented work is not to be a precision analysis of quark and lepton mixing. The projected values of the mixing matrices are rather a rough-and-ready estimate compatible though very well with experimental data. We wanted to show that the knowledge of fermion masses is sufficient to describe their mixing accepting a hierarchical nature.
The errors that are presented in Eqs. and follow from the uncertainties in the masses. Better precision in the determination of quark masses leads to better discrimination in future whether the described procedure is valid. The estimates are not too bad, nevertheless, we ignored radiative corrections to the mixing matrices and constrain ourselves on a tree-level discussion. One-loop corrections to the masses or Yukawa couplings would be suppressed by factors $Y_{ij} Y_{jk} Y_{kl} /
(16\pi^2)$ and are therefore in the range of the errors for the masses. Renormalization group running of the parameters is also negligible: quark mixing angles do basically not run. The running of fermion mixing parameters depends on a factor $(m_{i} + m_{j} ) /
(m_{i} - m_{j})$ which is small for the hierarchical spectra. Especially neutrino masses and mixings run only slightly in the scenario which is under consideration in this work.
Conclusions
===========
We investigated the long-standing question of understanding the functional description of the mixing matrices in terms of the fermion masses. The pure phenomenological observation of strong hierarchies among the charged fermion masses $m_{f,3} \gg m_{f,2} \gg m_{f,1}$ guides the way to a parametrization of fermion mixings in terms of mass ratios without further assumptions. By solely exploiting the mathematical properties of the mass matrices, namely their Singular Value Decomposition, and making use of the double mass hierarchy pattern (DMHP), we have shown that four mass ratios in each fermion sector and a maximal CP violating phase in the 1-2 rotation are sufficient to reproduce the numerical quantities of the fermionic mixing matrices. Hierarchical masses guarantee a unique decomposition into singular vectors up to a complex phase shared by the respective pair of singular vectors of a singular value. This uniqueness theorem dissolves the common ambiguities found in the literature originated in the freedom of weak bases. Schmidt-Mirsky’s approximation theorem has been used to approximate the hierarchical mass matrices by lower-rank matrices that are the closest one to the given full rank matrix. The connection of each lower rank approximation to the nature of the Yukawa interactions, $m_{f,i} = 0 \rightarrow Y^f_{ij} = 0 = Y_{ji}^f$, helps to simplify the reparametrization of the mass matrix without loosing track of the parameters. This connection is established via the minimal breaking of maximal flavor symmetry $\left[{\ensuremath{\mathrm{U}}}(3)\right]^3 \to \left[{\ensuremath{\mathrm{U}}}(2)\right]^3
\to \left[{\ensuremath{\mathrm{U}}}(1)\right]^3 \rightarrow {\ensuremath{\mathrm{U}}}(1)_F$ in each fermion sector, where the remnant ${\ensuremath{\mathrm{U}}}(1)_F$ symmetry is either baryon or lepton number. The approximation, however, neglects sizeable terms in the mass matrices that have been consistently added by use of correcting rotations. The arbitrariness of complex phases is reduced by requiring them to be either maximally CP violating ($\pi/2$) or CP conserving ($\pi$). This assumption is motivated by the fact that the four mass ratios should be enough to serve as mixing parameters in the unitary $3\times 3$ mixing matrix.
We found a remarkably good agreement of the projected magnitudes of both the CKM and PMNS matrix elements and reproduce the Jarlskog invariant of the quark sector quite well. The strength of this description in terms of mass ratios lies in its invertibility. In the leptonic sector, we have calculated the neutrino mass spectrum following from the inversion of the formulae in the 1-2 mixing sector and the measured mass squared differences. The lightest neutrino has a mass well below $0.01\,{\ensuremath{\mathrm{eV}}}$, while the largest neutrino mass lies around $0.05\,{\ensuremath{\mathrm{eV}}}$. We therefore conclude that, if also in the neutrino sector the mixing is determined by the mass ratios without any further contribution, the electron neutrino mass escapes its nearby measurement from tritium decay. Moreover, we give a prediction for the leptonic CP phase close to maximal, $\delta^\nu_\text{CP} \approx 90^\circ$.
Hence, contrary to the common expectation, leptonic mixing angles are found to be determined solely by the four leptonic mass ratios: $m_e/m_\mu$, $m_\mu/m_\tau$, $m_{\nu 1}/m_{\nu 2}$, and $m_{\nu
2}/m_{\nu 3}$ without any relation to the geometrical factors observed in most flavor models. Notwithstanding, we see a great power of the described method in the application to flavor model building: once a model gives hierarchical masses, the mixing follows from this hierarchy. In contrast, our approach gives viable patterns and textures for mass matrices in terms of the singular values (fermion masses). We explicitly leave the question of a model behind open. Likewise, the origin of CP violation stays unexplained, though our observation about the distribution of CP phases gives an important starting point.
Acknowledgements {#acknowledgements .unnumbered}
================
UJSS would like to acknowledge useful and detailed discussions about this idea in its initial stage with A. Mondragón and E. Jiménez. The authors want to thank the following people for their critical assessment on this work during its different stages: M. Spinrath, U. Nierste, C. Wiegand, M. Zoller, J. Hoff, M. Hoeschele, and K. Melnikov. Also, the authors are indebted to U. Nierste for a careful reading of the manuscript and his detailed comments on it. UJSS wants to acknowledge financial support from the Karlsruhe House of Young Scientists (KHYS) for his stay in Karlsruhe. WGH acknowledges support by the DFG-funded research training group GRK 1694 “Elementarteilchenphysik bei höchster Energie und höchster Präzision”. For last, UJSS is grateful to the Institut für Theoretische Teilchenphysik (TTP) for its warm hospitality during the realization and completion of this work.
State of the art in the fermion masses and mixing matrices {#app:data}
==========================================================
In this section, we collect the current knowledge about fermion mixing data and specify the input values we use in the following for the masses.
For all numerical evaluations made in this work, we stick to the updated values of the quark mixing matrix [@Agashe:2014kda], $$\label{eq:CKMPDG}
|V_{\text{CKM}}| =
\begin{pmatrix}
0.97427 \pm 0.00014 & 0.22536 \pm 0.00061 & 0.00355\pm 0.00015 \\
0.22522 \pm 0.00061 & 0.97343 \pm 0.00015 & 0.0414 \pm 0.0012 \\
0.00886^{+0.00033}_{-0.00032} & 0.0405^{+0.0011}_{-0.0012} &
0.99914\pm 0.00005 \end{pmatrix},$$ with the Jarlskog invariant equal to $ J_q =
(3.06^{+0.21}_{-0.20})\times 10^{-5}$. In the standard parametrization by the Particle Data Group (PDG), the central values give the following mixing angles, $$\theta_{12}^q \approx 13.3^\circ, \qquad
\theta_{13}^q \approx 0.2^\circ, \qquad
\theta_{23}^q \approx 2.4^\circ.$$
The most recent update on the $3\sigma$ allowed ranges of the elements of the PMNS mixing matrix are given by [@Gonzalez-Garcia:2014bfa], $$|U_{PMNS}| =
\begin{pmatrix}
0.801\rightarrow 0.845 & 0.514 \rightarrow 0.580 & 0.137 \rightarrow 0.158 \\
0.225 \rightarrow 0.517 & 0.441 \rightarrow 0.699 & 0.614 \rightarrow 0.793 \\
0.246 \rightarrow 0.529 & 0.464 \rightarrow 0.713 & 0.590
\rightarrow 0.776
\end{pmatrix}.$$ Where the best fit points of the mixing angles are $$\theta_{12}^\ell = 33.48^\circ, \qquad
\theta_{13}^\ell = 8.50^\circ, \qquad
\theta_{23}^\ell = 42.3^\circ.$$ The maximal value of the leptonic Jarlskog invariant is given by $J_\ell^\text{max} = 0.033 \pm 0.010$ and different from zero at more than $3\,\sigma$—still, the proper $J_\ell$ has first to be multiplied by $\sin\delta_\mathrm{CP}$ and is supposed to be smaller.
The study of the mixing matrices in terms of the masses is done at the scale of the $Z$ boson mass. The input values for the numerical calculations are obtained using the experimental values of the quark masses as given by the PDG Review 2014 [@Agashe:2014kda] and running them to the scale of the $Z$ boson determining the electroweak scale. We include highest precision running in QCD by the virtue of the RunDec package [@Chetyrkin:2000yt]. For completeness, we show the input values and their uncertainties as well as the resulting outputs in Table \[Table:Quark-masses\].
--------------------------------------------------------------------------------------------------------
**input** **output**
-------------------------------------------------------------------- -----------------------------------
$m_u(2\,{\ensuremath{\mathrm{GeV}}}) = 0.0023^{+0.0007}_{-0.0005}$ $m_u(M_Z) =
0.0013^{+0.0004}_{-0.0003}$
$m_d(2\,{\ensuremath{\mathrm{GeV}}}) = 0.0048^{+0.0005}_{-0.0003}$ $m_d(M_Z) =
0.0028^{+0.0003}_{-0.0002}$
$m_s(2\,{\ensuremath{\mathrm{GeV}}}) = 0.095 \pm 0.005$ $m_s(M_Z) =
0.055 \pm 0.003$
$m_c(m_c) = 1.275 \pm 0.025$ $m_c(M_Z) =
0.622 \pm 0.012$
$m_b(m_b) = 4.18 \pm 0.03$ $m_b(M_Z) =
2.85 \pm 0.02$
$m_t(\mathrm{OS}) = 173.07 \pm 1.24$ $m_t(M_Z) =
172.16^{+1.47}_{-1.46}$
--------------------------------------------------------------------------------------------------------
: The quark masses are run to the $Z$ boson mass scale by virtue of the RunDec package [@Chetyrkin:2000yt]. The mass inputs correspond to the experimental measured values while the outputs, evaluated at the $Z$ pole, include the resummation of higher order corrections from QCD by the RG running. RunDec takes properly into account the decoupling of heavy quarks below their scale. All masses are given in ${\ensuremath{\mathrm{GeV}}}$.[]{data-label="Table:Quark-masses"}
The reported measured on-shell values in MeV for the charged lepton masses are, $$\begin{aligned}
m_e=0.510998928, \quad\quad m_\mu=105.6583715, \quad\quad m_\tau=1776.82 \pm 0.16,\end{aligned}$$ where we have neglected the tiny experimental errors in the first two generation masses. The recent changes of this values affect only the few last digits. Therefore, we safely trust the results of [@Xing:2007fb] for their values at the $Z$ scale (in ${\ensuremath{\mathrm{MeV}}}$): $$\begin{aligned}
\label{eq:leptmass}
m_e(M_Z)=0.486570161, \quad\quad
m_\mu(M_Z)=102.7181359, \quad\quad
m_\tau(M_Z)=1746.24^{+0.20}_{-0.19}.\end{aligned}$$
The nine mass ratios are of essential use in the evaluation of the analytic formulae to describe fermion mixing. We show our input values determined from Table \[Table:Quark-masses\] and Eq. in Table \[Table:Q-massratios\].
$f$ $m_{f,1}/m_{f,2}$ $m_{f,1}/m_{f,3}$ $m_{f,2}/m_{f,3}$
----- ------------------------------ -------------------------------------- ---------------------
$u$ $0.0021^{+0.0007}_{-0.0005}$ $(7.6^{+2.4}_{-1.8} )\times 10^{-6}$ $0.0036 \pm 0.0001$
$d$ $0.051^{+0.009}_{-0.006}$ $(9.8^{+1.1}_{-0.7})\times 10^{-4}$ $0.019\pm 0.0012$
$e$ 0.00474 0.000279 0.0588
: Charged fermions mass ratios at the $M_Z$ scale.[]{data-label="Table:Q-massratios"}
In the case of neutrinos, only two squared mass differences have been measured whose values are taken from [@Gonzalez-Garcia:2014bfa], $$\label{eq:Deltam2}
\begin{aligned}
{\text{NO:}} \quad \Delta m_{31}^2 = +2.457 \pm 0.002 \times 10^{-3}\,{\ensuremath{\mathrm{eV}}}^2,
\quad\quad&
{\text{IO:}} \quad \Delta m_{32}^2 = -2.448 \pm 0.047 \times 10^{-3}\,{\ensuremath{\mathrm{eV}}}^2,\\
\Delta m_{21}^2 = 7.50^{+0.19}_{-0.17} &\times 10^{-5}\,{\ensuremath{\mathrm{eV}}}^2,
\end{aligned}$$ where NO and IO stand for normal and inverted ordering, respectively.
Still, the most recent direct bound on the neutrino mass scale stems from tritium beta decay experiments: $m(\nu_e) \lesssim 2\,{\ensuremath{\mathrm{eV}}}$ at $95\%$ C.L. [@Aseev:2011dq]. The KATRIN experiment is going to improve this bound by one order of magnitude [@Osipowicz:2001sq].
Applicability of the method {#app:applic}
===========================
The Schmidt-Mirsky theorem relates the validity of the lower rank approximation to a measure of being close to the full rank matrix. This measure has to be a scalar parameter and can be any norm. In the original formulation, the Frobenius norm was used, which is also the most natural choice since it is the square root over the sum of squared singular values and directly related to one of the invariants of the mass matrix $$\begin{aligned}
\parallel {\cal M}_f \parallel_{{\bf{F}}} \,\, = \sqrt{\sum_{i=1,2,3} m_{f,i}^2}.\end{aligned}$$ The use of this norm serves as a way to define a criterion which allows us to distinguish when the hierarchy is strong enough as to safely make an approximation. In this regard, we define the parameter $x_f^r$ as, $$\begin{aligned}
\label{eq:crit-parametr} x_f^r \equiv
\frac{\sqrt{(r-1)m_{f,2}^2 + m_{f,3}^2}}{\parallel {\cal
M}_f \parallel_{{\bf{F}}}} = \sqrt{\frac{(r-1)m_{f,2}^2 +
m_{f,3}^2}{m_{f,1}^2+m_{f,2}^2+m_{f,3}^2}},\end{aligned}$$ where $r={\operatorname{rank}}[{\cal M}^r_f] \in \lbrace 1,
2\rbrace$. The approximation becomes better the closer $x^r_f$ is to one and is exact in the $x_f^r \rightarrow 1$ limit. Eq. is actually the ratio of the lower rank approximated mass matrix norm with the original norm. Hence, $x_f^r$ is a measure of the applicability of the method. Table \[Table:xrf\] shows the different values obtained of $x_f^r$ for the several charged fermion masses. The values in the rank one approximation, $r=1$, for all practical purposes equal to one, though for both charged and neutral leptons deviate in the per mill and percent regime, respectively. From here we can already understand why the quark mixing matrix is so close to the unit matrix which is the trivial mixing matrix in the rank one approximation. In a similar manner, the very mild hierarchy for neutrinos leads to a stronger deviation from the rank one approximation and therefore larger mixing angles.
$x^{r}_f$ $u$ $d$ $e$ $\nu$
----------- ---------- ---------- ---------- ----------
$r=1$ 0.999993 0.999816 0.998274 0.978894
$r=2$ 0.999999 0.999999 0.999999 0.996773
: Values of the criterion parameter $x_f^r \equiv
\sqrt{[(r-1)m_{f,2}^2 + m_{f,3}^2
]/(m_{f,1}^2+m_{f,2}^2+m_{f,3}^2)}$, for the different cases of the fermion masses, where $x_f^r$ provides a measure of the applicability of the method. The fact that all cases here are sufficiently close to one guarantees the safe use of the lowest rank approximations. Even for neutrinos, $x^2_\nu$ is close to one, where we exploit the prediciton for neutrino masses from Sec. \[sec:lept\].[]{data-label="Table:xrf"}
Hierarchical mass matrices {#app:m11}
==========================
We show how to derive the hierarchical structure of the mass matrices by the use of the lower-rank approximation theorem and the principle of minimal flavor violation. Let us consider the two-flavor case and the mass matrix $$\begin{aligned}
\bf m = \begin{pmatrix}
m_{ss} & m_{sl} \\ m_{ls} & m_{ll}
\end{pmatrix},\end{aligned}$$ with the two singular values $\sigma_s$ and $\sigma_l$ respecting the hierarchy $\sigma_s \ll \sigma_l$.
We decompose the mass matrix in terms of the Singular Value decomposition $$\begin{aligned}
\label{eq:svd}
\mathbf{L}\,\mathbf{m}\,\mathbf{R}^\dagger = {\operatorname{diag}}(\sigma_s, \sigma_l),\end{aligned}$$ where the left and right unitary matrices diagonalize the Hermitian products $$\begin{aligned}
\label{eq:LRdiag}
\mathbf{L}\, \mathbf{m}\,\mathbf{m}^\dagger\, \mathbf{L}^\dagger =
\begin{pmatrix}
\sigma^2_s & 0 \\ 0 & \sigma^2_l
\end{pmatrix}
= \mathbf{R}\, \mathbf{m}^\dagger\, \mathbf{m}\, \mathbf{R}^\dagger.\end{aligned}$$ Each Hermitian product can be expressed as a sum of rank one matrices with the components of $\mathbf{L}$ and $\mathbf{R}$, $$\begin{aligned}
\label{eq:left}
\mathbf{m}\,\mathbf{m}^\dagger = \sigma_s^2
\begin{pmatrix}
|L_{11}|^2 & L_{11}L_{21}^* \\ L^*_{11}L_{21} & |L_{21}|^2
\end{pmatrix} +
\sigma_l^2
\begin{pmatrix}
|L_{12}|^2 & L_{12}L_{22}^* \\ L^*_{12}L_{22} & |L_{22}|^2
\end{pmatrix}\end{aligned}$$ and $$\begin{aligned}
\mathbf{m}^\dagger\, \mathbf{m} = \sigma_s^2
\begin{pmatrix}
|R_{11}|^2 & R_{11}R_{21}^* \\ R^*_{11}R_{21} & |R_{21}|^2
\end{pmatrix} +
\sigma_l^2
\begin{pmatrix}
|R_{12}|^2 & R_{12}R_{22}^* \\ R^*_{12}R_{22} & |R_{22}|^2
\end{pmatrix}.\end{aligned}$$ Due to our lack of knowledge of right-handed flavor mixing, the relevant object that determines our phenomenology is the left Hermitian product, $\mathbf{m}\, \mathbf{m}^\dagger$.
#### Applying Schmidt-Mirsky’s approximation theorem
Consider the rank-one approximation in Eq. by $\hat\sigma = \sigma_s / \sigma_l = 0$ normalized with respect to the larger singular value $$\begin{aligned}
\hat{\mathbf{m}}^{r=1} (\hat{\mathbf{m}}^{r=1})^\dagger =
\begin{pmatrix}
|L_{12}|^2 & L_{12}L_{22}^* \\ L^*_{12}L_{22} & |L_{22}|^2
\end{pmatrix}.\end{aligned}$$ The components of the left unitary matrix depend on $\hat\sigma$. In the limit $\hat\sigma \to 0$, there is trivial mixing and the rank one left Hermitian product is $$\begin{aligned}
\hat{\mathbf{m}}^{r=1}(\hat{\mathbf{m}}^{r=1})^\dagger =
\begin{pmatrix}
0 & 0 \\ 0 & 1
\end{pmatrix}.\end{aligned}$$
A small breaking of the $[{\ensuremath{\mathrm{U}}}(1)]^2$ symmetry for the massless fermions implies only a small deviation from the trivial mixing: $$\begin{aligned}
|\mathbf{L}| \sim \begin{pmatrix}
1 & \theta \\ \theta & 1
\end{pmatrix}.\end{aligned}$$ The mixing angle is related to the parameter of symmetry breaking $\hat\sigma$ and it is an easy exercise to derive $\theta \sim
\sqrt{\hat\sigma}$ from Eq. .
We then get an estimate on the magnitudes of each element in Eq. $$\begin{aligned}
\label{apC-mmdagger}
|\hat{\mathbf{m}}\,
\hat{\mathbf{m}}^\dagger| \sim
\begin{pmatrix}
\mathcal{O}(\theta^2) & \mathcal{O}(\theta) \\
\mathcal{O}(\theta) & 1 + \mathcal{O}(\theta^2)
\end{pmatrix}.\end{aligned}$$
The explicit form of the mass matrix $\mathbf{m}$ stays unknown as long as we have no information about $\mathbf{R}$. However, the minimal breaking of the maximal flavor symmetry applies to both chiralities simultaneously and the argument from above is the same for the right Hermitian product. We therefore know that $\mathbf{L}$ and $\mathbf{R}$ have the same moduli and get the hierarchical structure of $\mathbf{m}$: $$\begin{aligned}
\hat{\mathbf{m}} =
\begin{pmatrix}
m_{ss} & m_{sl} \\
m_{ls} & m_{ll}
\end{pmatrix} \sim \begin{pmatrix}
{\cal O}(\theta^2) & {\cal O}(\theta) \\
{\cal O}(\theta) & 1 + \mathcal{O}(\theta^2)
\end{pmatrix},\end{aligned}$$ with $|m_{sl}| = |m_{ls}|$ as a natural consequence of hierarchical masses and minimal flavor symmetry breaking. The hierarchical structure for the mass matrix and its Hermitian product is the same. Hence, due to the strong hierarchy in the masses we can neglect the role of $|m_{ss}|^2 \sim \theta^4$ in working with the leading order contributions in $\theta$ and assume $m_{ss} = 0$ as done in Eq. . This gives corrections to the Gatto-Satori-Tonin relation, $\tan\theta = \sqrt{\sigma_s / \sigma_l} =
\sqrt{\hat\sigma}$, which are $\mathcal{O}(\theta^3) =
\mathcal{O}(\hat\sigma\sqrt{\hat\sigma})$ and therefore neglected.
Explicit approximate formulae for the mixing angles and the Jarlskog invariant {#app:Formulae}
==============================================================================
The explicit formulae for the distinct mixing matrix elements in terms of the mass ratios is rather lengthy. We opt then, to show only the three mixing angles, used in the standard parametrization, with the corresponding Jarlskog invariant. This allows to express the mixing angles in terms of three moduli of the mixing matrix $$\begin{aligned}
\sin\theta_{23}^{f=q,\ell} =
\frac{\left|V^{f=q,\ell}_{23}\right|}{\sqrt{1-\left|V_{13}^{f=q,\ell}\right|^2}},
\quad
\sin\theta_{12}^{f=q,\ell} =
\frac{\left|V^{f=q,\ell}_{12}\right|}{\sqrt{1-\left|V_{13}^{f=q,\ell}\right|^2}},
\quad
\sin\theta_{13}^{f=q,\ell} = \left|V^{f=q,\ell}_{13}\right|\;.\end{aligned}$$ In the four mass ratios parametrization it is more natural to give not the formulae of the mixing angles in terms of the masses but rather of the aforementioned moduli $$\begin{aligned}
|V_{12}^{f=q,\ell}| &\approx \sqrt{\frac{\hat{m}^a_{12}+\hat{m}^b_{12}}{(1+\hat{m}^a_{12})(1+\hat{m}^b_{12})}}
\label{eq:theta12}\;, \\
|V^{f=q,\ell}_{23}| &\approx \mp
\frac{\sqrt{\hat{m}_{13}^a}+\sqrt{\hat{m}_{13}^b}+\sqrt{\hat{m}_{23}^a}\mp
\sqrt{\hat{m}_{23}^b}+\sqrt{\hat{m}_{13}^a\hat{m}_{23}^a}\pm
\sqrt{\hat{m}_{13}^b\hat{m}_{23}^b}}
{\sqrt{(1+\hat{m}_{13}^a)(1+\hat{m}_{13}^b)(1+\hat{m}_{23}^a)(1+\hat{m}_{23}^b)(1+\hat{m}_{13}^a\hat{m}_{23}^a)
(1+\hat{m}_{13}^b\hat{m}_{23}^b)}}
\label{eq:theta23}\;, \\
|V_{13}^{f=q,\ell}| &\approx
\mp|V_{23}^{f=q,\ell}|\sqrt{\frac{\hat{m}_{12}^a}{1+\hat{m}_{12}^a}} \;+
\nonumber \\ &\;
\frac{\sqrt{\hat{m}^a_{13}}-\sqrt{\hat{m}^b_{13}}+\sqrt{\hat{m}^a_{13}\hat{m}^a_{23}}+
\sqrt{\hat{m}^b_{13}\hat{m}^b_{23}}+ \hat{m}^a_{23}+\hat{m}^b_{23}}
{\sqrt{(1+\hat{m}_{13}^a)(1+\hat{m}_{13}^a\hat{m}_{23}^a)\left(1+(\hat{m}_{23}^a)^2\right)(1+\hat{m}_{12}^a)(1+\hat{m}^b_{13})(1+\hat{m}^b_{13}\hat{m}^b_{23})\left(1+(\hat{m}^b_{23})^2\right)}}\;,
\label{eq:theta13}\end{aligned}$$ where we have denoted $\hat{m}^{a(b)}_{ij} = m^{a(b)}_i / m^{a(b)}_j$, the upper and lower signs in Eq. \[eq:theta23\] correspond to $q$ and $\ell$, respectively. The two fermion species of each sector are $a=
u,e$ and $b = d,\nu$.
The Jarlskog invariant is given by, $$\begin{aligned}
J_{f=q,\ell} \approx \cos\theta_{12}^b \sin\theta_{12}^b
\sin\theta_{23}^{f=q,\ell}
\left(\sin\theta_{12}^a\sin\theta_{23}^{f=q,\ell}
+ \sin\theta_{13}^a-\sin\theta_{13}^b\right),\end{aligned}$$ where $$\begin{aligned}
\sin\theta_{12}^{a(b)} =
\sqrt{\frac{\hat{m}^{a(b)}_{12}}{1+\hat{m}^{a(b)}_{12}}}
\quad {\text{and}} \quad
\sin\theta_{13}^{a(b)} \approx \frac{\pm\sqrt{\hat{m}^{a(b)}_{13}}+ \sqrt{\hat{m}^{a(b)}_{13}\hat{m}^{a(b)}_{23}} + \hat{m}^{a(b)}_{23}}
{\sqrt{\left(1+\hat{m}^{a(b)}_{13}\right)\left(1+\hat{m}^{a(b)}_{13}\hat{m}^{a(b)}_{23}\right)\left(1+(\hat{m}^{a(b)}_{23})^2\right)}}.\end{aligned}$$ The approximate relations here given differ from the complete one in $\sim 1$% order.
[^1]: E-mail: `[email protected]`
[^2]: E-mail: `[email protected]`
[^3]: It is often wrongly called the Eckart-Young-Mirsky or simply Eckart-Young theorem, see [@Stewart1993551] for an early history on the Singular Value Decomposition.
[^4]: In the case of degeneracy among some of the singular values, there is no longer a unique Singular Value Decomposition for ${\cal M}_f$. This matters in the discussion of degenerate neutrino masses.
[^5]: Later, when reparametrizing the individual rotations in terms of the masses we will see that some of these six mixing parameters are unphysical while the rest can be expressed solely by two mass ratios.
[^6]: A matrix is normal if the left and right Hermitian products are the same: $\mathbf{m}\,\mathbf{m}^\dag =
\mathbf{m}^\dag\,\mathbf{m}$.
[^7]: Another solution can be found, that behaves wrongly in the limit $\hat{\sigma}^f_{sl} \to 0$ and gives maximal mixing $\tan\theta^f_{sl} \to \infty$ instead of zero mixing.
[^8]: The two signs reflect the freedom of choice for a clockwise or counterclockwise correcting rotation.
[^9]: We are implicitly assuming normal ordering. Inverted ordering is excluded by construction because it is not hierarchical in the minimal flavor symmetry breaking chain.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We propose that the flat decay phase in the first 10$^2$–10$^4$ seconds of the X–ray light curve of Gamma Ray Bursts can be interpreted as prolonged activity of the central engine, producing shells of decreasing bulk Lorentz factors $\Gamma$. The internal dissipation of these late shells produces a continuous and smooth emission, usually dominant in X–rays and sometimes in the optical. When $\Gamma$ of the late shells is larger than $1/\theta_j$, where $\theta_j$ is the jet opening angle, we see only a portion of the emitting surface. Eventually, $\Gamma$ becomes smaller than $1/\theta_j$, and the entire emitting surface is visible. When $\Gamma=1/\theta_j$ there is a break in the light curve, and the plateau ends. During the plateau phase, we see the sum of the “late–prompt” emission (due to late internal dissipation), and the “real afterglow” emission (due to external shocks). A variety of different optical and X–ray light curves is possible, explaining why the X–ray and the optical light curves often do not track each other, and why they often do not have simultaneous breaks.'
author:
- Gabriele Ghisellini
title: 'The shallow phase of X–ray afterglows'
---
[ address=[INAF – Osservatorio Astronomico di Brera, Merate, Italy]{} ]{}
Introduction
============
The so called “Steep–Flat–Steep" behavior [@gt05; @nousek05] of the early (up to $\sim$a day) X–ray afterglow was unpredicted before we could observe it with [*Swift*]{}. It has been interpreted in several ways (for reviews, see e.g. [@zhang07]) none of which seems conclusive. The spectral slope does not change across the temporal break from the shallow to the normal decay phase, ruling out a changing spectral break as a viable explanation. An hydrodynamical or geometrical nature of the break is instead preferred. Furthermore, the X–ray and optical lightcurves often do not track one another (e.g. [@pana06; @pana07]) suggesting a possible different origin.
To solve these difficulties Uhm & Beloborodov [@uhm07] and Genet, Daigne & Mochkovitch [@genet07] suggested that the X–ray plateau emission is not due to the forward, but to the reverse shock running into ejecta of relatively small (and decreasing) Lorentz factors. This however requires an appropriate $\Gamma$–distribution of the ejecta, and also the suppression of the X–ray flux produced by the forward shock.
We ([@gg07]) instead suggested that the plateau phase of the X–ray emission (and sometimes even of the optical) is due to a prolonged activity of the central engine (see also [@lp07]), responsible for a “late–prompt” phase: after the early “standard" prompt the central engine continues to produce for a long time (i.e. days) shells of progressively lower power and bulk Lorentz factor. The dissipation process during this and the early phases occur at similar radii (namely close to the transparency radius). The reason for the shallow decay phase, and for the break ending it, is that the $\Gamma$–factors of the late shells are motonically decreasing, allowing to see an increasing portion of the emitting surface, until all of it is visible. Then the break occurs when $\Gamma=1/\theta_j$.
![ Cartoon of the proposed model, and schematic illustration of the different components contributing to the X–ray and optical light curves, as labelled. Scales are arbitrary. The early prompt phase is erratic, with shells of varying $\Gamma$ and power. Then the central engine produces shells of progressively less power and bulk Lorentz factors, producing a smoother light curve. Since the average $\Gamma$–factor is decreasing, the observer sees an increasing portion of the emitting area, until all of it becomes visible when $\Gamma \sim 1/\theta_j$. When this occurs there is a break in the light curve, associated with the ending of the shallow phase. The case illustrated here is only one (likely the most common) possible case, when the X–ray flux is dominated by late prompt emission (solid line, the dotted line corresponds to an extrapolation at very late times), while the optical flux is dominated by the real afterglow (dashed). Adapted from [@gg07]. ](ghisellini_f1.eps){height="0.5\textheight"}
The shallow X–ray afterglow phase
=================================
The time ending the shallow phase
---------------------------------
Willingale et al. [@willi07] have proposed to described the X–ray afterglow light curve with a rising exponential connecting to a power law function. The end of the shallow phase is the junction between the exponential and the power law, and it is called $T_a$. They showed that interpreting $T_a$ as a jet break time one obtains, for the [*Swift*]{} bursts in their sample, a good correlation between the peak energy of the prompt spectrum, $E_{\rm peak}$, and the collimation corrected energetics $E_\gamma$, with a small scatter and a slope identical to the so called Ghirlanda relation [@ggl04] (which identifies as a jet break time the break in the optical light curve, occurring usually much later), challenging the physical nature of the Ghirlanda relation. Nava et al. [@nava07] have then investigated this issue with a larger sample, finding that the correlation found by [@willi07] does not have the same slope of the Ghirlanda one, and it is not as tight. More importantly, they demonstrated that $T_a$ does not play any role in the construction of the correlation found by [@willi07], which is instead (entirely) a by–product of the the $E_{\rm peak}$–$E_{\rm iso}$ correlation (the so called “Amati" relation, [@ama02]). In fact there is no (anti)–correlation between $T_a$ and $E_{\rm iso}$ (“a la Frail", [@frail01]) for GRBs of the same $E_{\rm peak}$ (see [@nava07] for more details and figures).
Prolonged central engine activity
---------------------------------
The time $T_a$ is not a jet break time, still it may be produced by a mechanism very similar to the process responsible for the jet break visible during the deceleration of the fireball. Consider the accretion onto the newly formed black hole, and suppose that it occurs in two phases. The first is short, intense, erratic, corresponding to the early prompt phase of GRBs. The second is longer, smoother, with a rate decreasing in time, corresponding to the late prompt emission. The first accretion mode might correspond to the accretion of the equatorial core material which failed to form the black hole in the first place. It can form a very dense accreting torus, which can sustain a strong magnetic field, which in turn efficiently extracts the rotational energy of the black hole. After this phase, some fall–back material may also be accreted, with a density smaller than in the early phases. The magnetic field that this matter can sustain is weaker than before, with a corresponding smaller power extracted from the black hole spin. This may well correspond to production of shells of smaller $\Gamma$–factors. These shells can dissipate part of their energy with the same mechanism of the early ones. Occasionally, in this late prompt phase, the central engine may produce a faster than average shell, originating the late flares often observed in the Swift/XRT light curves.
In the scenario we have proposed, there is a simple relation between the function describing how $\Gamma$ decreases in time and the observed decay slopes before and after $T_a$. Assume that the plateau phase is described by $L(t)\propto t^{-\alpha_2}$, followed by a steeper decay $L(t)\propto t^{-\alpha_3}$. Then, by geometry alone, one can derive that ([@gg07]): $$\Gamma \, \propto t^{-(\alpha_3-\alpha_2)/2}$$ We can also estimate how the barion loading of the late shells changes in time. Assume $L(t) \propto \eta \Gamma\dot M c^2$, and consider for semplicity $t>T_a$, when all the jet is visible. Then, for constant $\eta$ we have: $$\dot M \, \propto \, t^{-(\alpha_2+\alpha_3)/2}$$ If we insert the average values of $\alpha_3$ and $\alpha_2$ ($\sim 1.25\pm 0.25$ and $\sim 0.6\pm 0.3$, respectively, see [@pana06]) we approximately have $\dot M\propto t^{-1}$ and $\Gamma\propto t^{-1/3}$. This means that the total energy (i.e. integrated over time, $E =\int \Gamma \dot M c^2 dt$, beginning from the start of the plateau phase) involved in the late phase is smaller than the energy spent during the early prompt.
Observational tests
-------------------
If we allow for [*two*]{} origins for the emission during and after the X–ray plateau phase (one due to the late prompt and the other due to the conventional forward shock), we can account for a variety of cases: both the optical and the X–rays are late prompt emission or forward shock emission; or X–rays and optical are “decoupled”, one due to late prompt and the other to the forward shock. One obvious way to check these possibilities is through the simultaneous spectral energy distribution (SED), which can confirm or not if the X–ray and the IR–optical fluxes belong to the same component. If the emission in the two bands have a different origin they should not “interfere" with one another, requiring that the X–ray spectrum breaks at low energies, and the optical at high ($\sim$UV) energies. The unknown extinction due to the host galaxy material may be a complication, but infrared data can help. The SED so obtained may clearly show if the IR–optical and X–ray emission belong (or not) to two different components.
Since in our scenario the late central activity is not energetically demanding, another test concerns the total kinetic energy of the fireball after its radiative phase, using the radio data, as done e.g. for GRB 970508 [@frail00]. Should the derived energetics be smaller than what required by e.g. the refreshed shock scenario, one could exclude this possibility, and instead favor our scenario.
In cases in which the late prompt emission ends, the underlying forward shock emission can be revealed. In the light curve, this should appear as a steep–flat transition at late times (not to be confused with the usual steep–flat–steep X–ray decay). This can also be confirmed by the corresponding SEDs.
I gratefully thank all my collaborators: A. Celotti, C. Firmani, G. Ghirlanda, M. Nardini, L. Nava and F. Tavecchio.
[1]{} L. Amati, F. Frontera, M. Tavani, et al., 2002, [*A&A*]{}, [**390**]{}, 81 D.A. Frail, E. Waxman & S.R. Kulkarni, 2000, [*ApJ*]{}, [**537**]{}, 191 D.A. Frail, S.R. Kulkarni, Sari, R. et al., 2001, [*ApJ*]{}, [**562**]{}, L55 F. Genet, F. Daigne, & R. Mochkovitch, 2007, [*MNRAS*]{}, [**381**]{}, 732 G. Ghirlanda, G. Ghisellini & D, Lazzati, 2004, [*ApJ*]{}, [**616**]{}, 331 G. Ghisellini, G. Ghirlanda, L. Nava & C. Firmani, 2007, [*ApJ*]{}, [**658**]{}, L75 D. Lazzati & R. Perna, 2007, [*MNRAS*]{}, [**375**]{}, L46 L. Nava, G. Ghisellini, G. Ghirlanda et al., 2007, [*MNRAS*]{}, [**377**]{}, 1464 J.A. Nousek, C. Kouveliotou, D. Grupe, et al., 2005, [*ApJ*]{}, [**642**]{}, 389 A. Panaitescu, P. Meszaros, D. Burrows, N. Nousek et al. 2006, [*MNRAS*]{}, [**369**]{}, 2059 A. Panaitescu, 2007, [*Il Nuovo Cimento*]{}, in press (astro–ph/0607396) G. Tagliaferri, M. Goad, G. Chincarini, et al., 2005, [*Nature*]{}, [**436**]{}, 985 L.Z. Uhm & A.M. Beloborodov, 2007, [*ApJ*]{}, [**665**]{}, L93 R. Willingale, P.T. O’Brien, J.P. Osborne, et al., 2007, [*ApJ*]{}, [**662**]{}, 1093 B. Zhang, 2007, [*Advances in Space Research*]{}, [**40**]{}, Issue 8, p. 1186 (astro–ph/0611774)
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{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'This paper is an extended version of an expository talk given at the workshop “Topology of stratified spaces” at MSRI Berkeley in September 2008. It gives an introduction and overview about recent developments on the interaction of the theories of characteristic classes and mixed Hodge theory for singular spaces in the complex algebraic context. It uses M. Saito’s deep theory of mixed Hodge modules as a“black box", thinking about them as “constructible or perverse sheaves of Hodge structures", having the same functorial calculus of Grothendieck functors. For the “constant Hodge sheaf", one gets the “motivic characteristic classes" of Brasselet-Schürmann-Yokura, whereas the classes of the “intersection homology Hodge sheaf" were studied by Cappell-Maxim-Shaneson. The classes associated to “good" variation of mixed Hodge structures where studied in connection with understanding the monodromy action by Cappell-Libgober-Maxim-Shaneson and the author. There are two versions of these characteristic classes. The K-theoretical classes capture information about the graded pieces of the filtered de Rham complex of the filtered D-module underlying a mixed Hodge module. Application of a suitable Todd class transformation then gives classes in homology. These classes are functorial for proper pushdown and exterior products, together with some other properties one would expect for a “good" theory of characteristic classes for singular spaces. For “good" variation of mixed Hodge structures they have an explicit classical description in terms of “logarithmic de Rham complexes". On a point space they correspond to a specialization of the Hodge polynomial of a mixed Hodge structure, which one gets by forgetting the weight filtration. Finally also some relations to other subjects of the conference, like index theorems, signature, L-classes, elliptic genera and motivic characteristic classes for singular spaces, will be indicated.'
address: 'Dr. J. Schürmann : Mathematische Institut, Universität Münster, Einsteinstr. 62, 48149 Münster, Germany.'
author:
- Jörg Schürmann
title: Characteristic classes of mixed Hodge modules
---
Introduction
============
This paper gives an introduction and overview about recent developments on the interaction of the theories of characteristic classes and mixed Hodge theory for singular spaces in the complex algebraic context. The reader is not assumed to have any background on one of these subjects, and the paper can also be used as a bridge for communication between researchers on one of these subjects.\
General references for the theory of characteristic classes of singular spaces is the survey [@SY] as well as the paper [@Y] in these proceedings. As references for mixed Hodge theory one can use [@PS; @Voi], as well as the nice paper [@P] for explaining the motivic viewpoint to mixed Hodge theory. Finally as an introduction to M. Saito’s deep theory of mixed Hodge modules one can use [@PS]\[chap. 14\], [@Sa2] as well as the introduction [@Sab]. The theory of mixed Hodge modules is used here more or less as a“black box", thinking about them as “constructible or perverse sheaves of Hodge structures", having the same functorial calculus of Grothendieck functors. The underlying theory of constructible and perverse sheaves can be found in [@BBD; @KS; @Sc].\
For the “constant Hodge sheaf" ${\mathbb{Q}}^H_Z$ one gets the “motivic characteristic classes" of Brasselet-Schürmann-Yokura [@BSY] as explained in [@Y] in these proceedings. The classes of the “intersection homology Hodge sheaf" $IC_Z^H$ were studied by Cappell-Maxim-Shaneson in [@CMS0; @CMS]. Also, the classes associated to “good" variation of mixed Hodge structures where studied via Atiyah-Meyer type formulae by Cappell-Libgober-Maxim-Shaneson in [@CLMS; @CLMS2]. For a summary compare also with [@MSc].\
There are two versions of these characteristic classes, the [*motivic Chern class transformation*]{} $MHC_y$ and the [*motivic Hirzebruch class transformation*]{} $MHT_{y*}$. The $K$-theoretical classes $MHC_y$ capture information about the graded pieces of the filtered de Rham complex of the filtered D-module underlying a mixed Hodge module. Application of a suitable twisting $td_{(1+y)}$ of the Todd class transformation $td_*$ of Baum-Fulton-MacPherson [@BFM; @Fu] then gives the classes $MHT_{y*}= td_{(1+y)}\circ MHC_y$ in homology. It is the [*motivic Hirzebruch class transformation*]{} $MHT_{y*}$, which unifies
1. the (rationalized) [*Chern class transformation*]{} $c_*$ of MacPherson [@M],
2. the [*Todd class transformation*]{} $td_*$ of Baum-Fulton-MacPherson [@BFM], and
3. the [*$L$-class transformation*]{} $L_*$ of Cappell-Shaneson [@CS]
for $y=-1,0$ and $1$ respectively (compare with [@BSY; @SY] and also with [@Y] in these proceedins). But in this paper we focus on the $K$-theoretical classes $MHC_y$, because these imply then also the corresponding results for $MHT_{y*}$ just by application of the (twisted) Todd class transformation. So the [*motivic Chern class transformation*]{} $MHC_y$ studied here is really the basic one!\
Here we explain the functorial calculus of these classes, stating first in a very precise form the key results used from Saito’s theory of mixed Hodge modules, and explaing then how to get from this the basic results about the motivic Chern class transformation $MHC_y$. Moreover these results are illustrated by many interesting examples. For the convenience of the reader, the most general results are only stated in the end of the paper. In fact, while most of the paper is a detailed survey of the K-theoretical version of the theory as developed in [@BSY; @CLMS; @CLMS2; @MSc], it is this last section which contains new results on the important functorial properties of these characteristic classes. The first two section do not use mixed Hodge modules and are formulated in the (now) classical language of (variation of) mixed Hodge structures. Here is the plan of the paper:
Section 2
: gives an introduction to pure and mixed Hodge structures and the corresponding Hodge genera like $E$-polynomial and $\chi_y$-genus. These are suitable generating functions of Hodge numbers with $\chi_y$ using only the Hodge filtration $F$, whereas the $E$-polynomial also uses the weight filtration. We also carefully explain, why only the $\chi_y$-genus can be further generalized to characteristic classes, i.e. why one has to forget the weight filtration for applications to characteristic classes.
Section 3
: motivates and explains the notion of a variation (or family) of pure and mixed Hodge structures over a smooth (or maybe singular) base. Basic examples come from the cohomology of the fibers of a family of complex algebraic varieties. We also introduce the notion of a “good” variation of mixed Hodge structures on a complex algebraic manifold $M$, to shorten the notion for a graded polarizable variation of mixed Hodge structures on $M$, which is [*admissible*]{} in the sense of Steenbrink-Zucker [@SZ] and Kashiwara [@Ka], with [*quasi-unipotent monodromy*]{} at infinity, i.e. with respect to a compactification $\bar{M}$ of $M$ by a compact complex algebraic manifold $\bar{M}$, with complement $D:=\bar{M}\backslash M$ a normal crossing divisor with smooth irreducible components. Later on these will give the basic example of so called “smooth” mixed Hodge modules. And for these good variations we introduce a simple [*cohomological*]{} characterstic class transformtion $MHC^y$, which behaves nicely with respect to smooth pullback, duality and (exterior) products. As a first approximation to more general mixed Hodge modules and their characteristic classes, we also study in detail functorial properties of the canonical Deligne extension across a normal crossing divisor $D$ at infinity (as above), leading to [*cohomological*]{} characteristic classes $MHC^y(j_*(\cdot))$ defined in terms of “logarithmic de Rham complexes". These classes of good variations have been studied in detail in [@CLMS; @CLMS2; @MSc], and most results described here are new functorial reformulations of the results from these sources.
Section 4
: starts with an introduction to Saito’s functorial theory of algebraic mixed Hodge modules, explaining its power in many examples, e.g. how to get a pure Hodge structure on the global Intersection cohomology $IH^*(Z)$ of a compact complex algebraic variety $Z$. From this we deduce the basic calculus of Grothendieck groups $K_0(MHM(\cdot ))$ of mixed Hodge modules needed for our motivic Chern class transformation $MHC_y$. We also explain the relation to the motivic view point coming from relative Grothendieck groups of complex algebraic varieties.
Section 5.1
: is devoted to the definition of our motivic characteristic [*homology class*]{} transformations $MHC_y$ and $MHT_{y*}$ for mixed Hodge modules. By Saito’s theory they commute with push down for proper morphisms, and on a compact space one gets back the corresponding $\chi_y$-genus by pushing down to a point, i.e. by taking the degree of these characteristic homology classes.
Sections 5.2-5.3
: finally explain other important functoriality properties, like
1. Multiplicativity for exterior products.
2. The behaviour under smooth pullback given by a Verdier Riemann-Roch formula.
3. A “going up and down” formula for proper smooth morphisms.
4. Multiplicativity between $MHC^y$ and $MHC_y$ for a suitable (co)homological pairing in the context of a morphism with smooth target. As special cases one gets from this interesting Atiyah and Atiyah-Meyer type formulae (as studied in [@CLMS; @CLMS2; @MSc]).
5. The relation between $MHC_y$ and duality, i.e. the Grothendieck duality transformation for coherent sheaves ond the Verdier duality for mixed Hodge modules.
6. The identification of $MHT_{-1*}$ with the (rationalized) Chern class transformation $c_*\otimes{\mathbb{Q}}$ of MacPherson for the underlying constructible sheaf complex or function.
Note that such a functorial calculus is expected for any good theory of functorial characteristic classes of singular spaces (compare [@BSY; @SY]):
1. For MacPherson’s Chern class transformation $c_*$ compare with [@BSY; @Ke; @M; @SY].
2. For Baum-Fulton-MacPherson’s Todd class transformation $td_*$ compare with [@BFM; @BFM2; @BSY; @Fu; @FM; @SY].
3. For Cappel-Shaneson’s $L$-class transformation $L_*$ compare with [@BCS; @Ba; @Ba2; @BSY; @CS; @SY; @Si; @Woo].
Note that the counterpart of mixed Hodge modules in these theories are constructible functions and sheaves (for $c_*$), coherent sheaves (for $td_*$) and selfdual perverse or constructible sheaf complexes (for $L_*$). The cohomological counterpart of the smooth mixed Hodge modules (i.e. good variation of mixed Hodge structures) are locally constant functions and sheaves (for $c^*$), locally free coherent sheaves or vector bundles (for the Chern character $ch^*$) and selfdual local systems (for $L^*$ and the $KO$-classes of Meyer [@Mey]).\
In this paper we concentrate mainly on pointing out the relation and analogy to the $L$-class story related to important signature invariants, because these are the subject of many other talks from the conference given in more topological terms. Finally also some relations to other themes of the conference, like index theorems, $L^2$-cohomology, elliptic genera and motivic characteristic classes for singular spaces, will be indicated.
Hodge structures and genera
===========================
Pure Hodge structures
---------------------
Let $M$ be a compact [*Kähler manifold*]{} (e.g. a complex projective manifold) of complex dimension $m$. By classical Hodge theory one gets the decomposition (for $0\leq n \leq 2m$) $$\label{h-dec}
H^n(M,{\mathbb{C}})=\oplus_{p+q=n}\; H^{p,q}(M)$$ of the complex cohomology of $M$ into the spaces $H^{p,q}(M)$ of harmonic forms of type $(p,q)$. This decomposition doesn’t depend on the choice of a Kähler form (or metric) on $M$, and for a complex algebraic manifold $M$ it is of algebraic nature. Here it is more natural to work with the [*Hodge filtration*]{} $$\label{h-fil}
F^i(M):=\oplus_{p\geq i}\;H^{p,q}(M)$$ so that $ H^{p,q}(M)=F^p(M)\cap \overline{F^q(M)}$, with $\overline{F^q(M)}$ the complex conjugate of $F^q(M)$ with respect to the real structure $H^n(M,{\mathbb{C}})=H^n(M,{\mathbb{R}})\otimes {\mathbb{C}}$. If $$\begin{CD} \Omega_M^{\bullet}=[{{\mathcal O}}_M @> d >> \cdots @> d >> \Omega^m_M]
\end{CD}$$ denotes the usual holomorphic de Rham complex (with ${{\mathcal O}}_M$ in degree zero), then one gets $$H^*(M,{\mathbb{C}})=H^*(M,\Omega_M^{\bullet})$$ by the holomorphic Poincaré-lemma, and the Hodge filtration is induced from the “stupid" decreasing filtration $$\label{dr-fil} \begin{CD}
F^p\Omega_M^{\bullet}=[0@>>> \cdots 0 @>>> \Omega^p_M @> d >> \cdots @> d >> \Omega^m_M] \:.
\end{CD}$$ More precisely, the corresponding [*Hodge to de Rham spectral-sequence*]{} degenerates at $E_1$, with $$\label{Hpq}
E_1^{p,q}=H^{q}(M,\Omega^p_M)\simeq H^{p,q}(M)\:.$$ More generally, the same results are true for a compact complex manifold $M$, which is only [*bimeromorphic to a Kähler manifold*]{} (compare e.g. [@PS]\[cor.2.30\]). This is especially true for a compact complex algebraic manifold $M$. Moreover in this case one can calculate by Serre’s GAGA-theorem $H^*(M,\Omega_M^{\bullet})$ also with the algebraic (filtered) de Rham complex in the Zariski topology.\
Abstracting these properties, one can say the $H^n(M,{\mathbb{Q}})$ gets an induced [*pure Hodge structure of weight $n$*]{} in the following sense:
Let $V$ be a finite dimesional rational vector space. A (rational) Hodge structure of weight $n$ on $V$ is a decomposition $$V_{{\mathbb{C}}}:=V\otimes_{{\mathbb{Q}}}{\mathbb{C}}= \oplus_{p+q=n}\; V^{p,q}, \quad \text{with $V^{q,p}=\overline{V^{p,q}}$}
\quad \text{(Hodge decomposition).}$$ In terms of the (decreasing) [*Hodge filtration*]{} $F^iV_{{\mathbb{C}}}:=\oplus_{p\geq i}\;V^{p,q}$, this is equivalent to the condition $$F^pV\cap \overline{F^qV}= \{0\} \quad \text{whenever $p+q=n+1$} \quad \text{($n$-opposed filtration).}$$ Then $V^{p,q}=F^p\cap \overline{F^q}$, with $h^{p,q}(V):=dim(V^{p,q})$ the corresponding [*Hodge number*]{}.
If $V,V'$ are rational vector spaces with Hodge structures of weight $n$ and $m$, then $V\otimes V'$ gets an induced Hodge structure of weight $n+m$, with Hodge filtration $$\label{F-tensor}
F^k(V\otimes V')_{{\mathbb{C}}}:= \oplus_{i+j=k}\; F^iV_{{\mathbb{C}}}\otimes F^jV'_{{\mathbb{C}}} \:.$$ Similarly the dual vector space $V^{\vee}$ gets an induced Hodge structure of weight $-n$, with $$\label{F-dual}
F^k( V^{\vee}_{{\mathbb{C}}}):=(F^{-k}V_{{\mathbb{C}}})^{\vee} \:.$$ A basic example is the [*Tate Hodge structure*]{} of weight $-2n\in {\mathbb{Z}}$ given by the one dimensional rational vector space $${\mathbb{Q}}(n):=(2\pi i)^n\cdot {\mathbb{Q}}\subset {\mathbb{C}},\quad \text{with ${\mathbb{Q}}(n)_{{\mathbb{C}}}=({\mathbb{Q}}(n)_{{\mathbb{C}}})^{-n,-n}$.}$$ Then integration defines an isomorphism $H^2(P^1({\mathbb{C}}),{\mathbb{Q}})\simeq {\mathbb{Q}}(-1)$, with ${\mathbb{Q}}(-n)={\mathbb{Q}}(-1)^{\otimes n}$, ${\mathbb{Q}}(1)={\mathbb{Q}}(-1)^{\vee}$ and ${\mathbb{Q}}(n)={\mathbb{Q}}(1)^{\otimes n}$ for $n>0$.
\[pol\] A [*polarization*]{} of a rational Hodge structure $V$ of weight $n$ is a rational $(-1)^n$-symmetric bilinear form $S$ on $V$ such that $$S(F^p,F^{n-p+1})=0 \quad \text{for all $p$ and $i^{p-q}S(u,\bar{u})> 0$ for all
$0\neq u\in V^{p,q}$.}$$ So for $n$ [*even*]{} one gets in particular $$\label{positive}
(-1)^{p-n/2}S(u,\bar{u})> 0 \quad \text{for all $q$ and
$0\neq u\in V^{p,q}$.}$$ $V$ is called [*polarizable*]{}, if such a polarization exists.
For example the cohomology $H^n(M,{\mathbb{Q}})$ of a projective manifold is polarizable by the choice of a suitable Kähler form! Also note that a polarization of a rational Hodge structure $V$ of weight $n$ induces an isomorphism of Hodge structures (of weight $n$): $$V \simeq V^{\vee}(-n):=V^{\vee}\otimes_{{\mathbb{Q}}}{\mathbb{Q}}(-n)\:.$$ So if we choose the isomorphism of rational vector spaces ${\mathbb{Q}}(-n)=(2\pi i)^{-n}\cdot {\mathbb{Q}}\simeq {\mathbb{Q}}$, then a polarisation induces a $(-1)^n$-symmetric duality isomorphism $V\simeq V^{\vee}$.
Mixed Hodge structures
----------------------
The cohomology (with compact support) $H^n_{(c)}(X,{\mathbb{Q}})$ of a singular or non-compact complex algebraic variety can’t have a pure Hodge structure in general, but by Deligne [@De1; @De3] it carries a canonical functorial (graded polarizable) [*mixed Hodge structure*]{} in the following sense:
A finite dimensional rational vector space $V$ has a mixed Hodge structure, if there is a (finite) increasing [*weight filtration*]{} $W=W_{\bullet}$ on $V$ (by rational subvector spaces), and a (finite) decreasing Hodge filtration $F=F^{\bullet}$ on $V_{{\mathbb{C}}}$, such that $F$ induces a Hodge structure of weight $n$ on $Gr^W_nV:=W_nV/W_{n-1}V$ for all $n$. Such a mixed Hodge structure is called [*(graded) polarizable*]{} if each graded piece $Gr^W_nV$ is polarizable.
A morphism of mixed Hodge structures is just a homomorphism of rational vector spaces compatible with both filtrations. Such a morphism is then [*strictly*]{} compatible with both filtrations so that the category $mHs^{(p)}$ of (graded polarizable) mixed Hodge structures is an abelian category, with $Gr^W_*, Gr_F^*$ and $Gr_F^*Gr^W_*$ preserving short exact sequences. $mHs^{(p)}$ is also endowed with a tensor product $\otimes$ and a duality $(\cdot)^{\vee}$, where the corresponding Hodge and weight filtrations are defined as in (\[F-tensor\]) and (\[F-dual\]). So for a complex algebraic variety $X$ one can consider its cohomology class $$[H^*_{(c)}(X)]:=\sum_{i}\; (-1)^i�\cdot [H^i_{(c)}(X,{\mathbb{Q}})] \in K_0(mHs^{(p)})$$ in the Grothendieck group $K_0(mHs^{(p)})$ of (graded polarizable) mixed Hodge structures. The functoriality of Deligne’s mixed Hodge structure means in particular, that for a closed complex algebraic subvariety $Y\subset X$, with open complement $U=X\backslash Y$, the corresponding long exact cohomology sequence $$\label{long-H}
\cdots H^i_{c}(U,{\mathbb{Q}})\to H^i_{c}(X,{\mathbb{Q}}) \to H^i_{c}(Y,{\mathbb{Q}})\to \cdots$$ is an exact sequence of mixed Hodge structures. Similarly the Künneth isomorphism $$\label{K-H}
H^*_{c}(X,{\mathbb{Q}})\otimes H^*_{c}(Z,{\mathbb{Q}}) \simeq H^*_{c}(X\times Z,{\mathbb{Q}})$$ for complex algebraic varieties $X,Z$ is an isomorphism of mixed Hodge structures. Let us denote by $K_0(var/pt)$ the Grothendieck group of complex algebraic varieties, i.e. the free abelian group of isomorphism classes $[X]$ of such varieties divided out by the [*additivity relation*]{} $$[X]= [Y]+ [X\backslash Y]$$ for $Y\subset X$ a closed complex subvariety. This is then a commutative ring with addition resp. multiplication induced by the disjoint union resp. the product of varieties. So by (\[long-H\]) and (\[K-H\]) we get an induced ring homomorphism $$\label{H-ring}
\chi_{Hdg}: K_0(var/pt)\to K_0(mHs^{(p)});\: [X]\mapsto [H^*_{c}(X)] \:.$$
Hodge genera
------------
The [*E-polynomial*]{} $$\label{E}
E(V):=\sum_{p,q}\; h^{p,q}(V)\cdot u^pv^q \in {\mathbb{Z}}[u^{\pm1},v^{\pm 1}]$$ of a rational mixed Hodge structure $V$ with [*Hodge numbers*]{} $$h^{p,q}(V):=dim_{{\mathbb{C}}} Gr_F^pGr^W_{p+q}(V_{{\mathbb{C}}}) \:,$$ induces a [*ring*]{} homomorphism $$E: K_0(mHs^{(p)})\to {\mathbb{Z}}[u^{\pm1},v^{\pm 1}]\:,\quad \text{with $E({\mathbb{Q}}(-1))=uv$.}$$ Note that $E(V)(u,v)$ is [*symmetric*]{} in $u$ and $v$, since $h(V)=\sum_n\;h(W_nV)$ and $V^{q,p}=\overline{V^{p,q}}$ for a pure Hodge structure. With respect to [*duality*]{} one has in addition the relation $$\label{E-dual}
E(V^{\vee})(u,v) = E(V)(u^{-1},v^{-1}) \:.$$
Later on we will be mainly interested in the following specialized ring homomorphism $$\chi_y:=E(-y,1): K_0(mHs^{(p)})\to {\mathbb{Z}}[y^{\pm1}]\:,\quad \text{with $\chi_y({\mathbb{Q}}(-1))=-y$,}$$ defined by $$\label{chi-y-H}
\chi_y(V):= \sum_{p}\; dim_{{\mathbb{C}}}(Gr_F^p(V_{{\mathbb{C}}}))\cdot (-y)^p \:.$$ So here one uses only the Hodge and forgets the weight filtration of a mixed Hodge structure. With respect to [*duality*]{} one has then the relation $$\label{chi-y-dual}
\chi_y(V^{\vee}) = \chi_{1/y}(V) \:.$$
Note that $\chi_{-1}(V)=dim(V)$ and for a pure polarized Hodge structure $V$ of weight $n$ one has by $\chi_1(V)=(-1)^n\chi_1(V^{\vee})=(-1)^n\chi_1(V)$ and (\[positive\]): $$\chi_1(V)=\begin{cases}
0 &\text{for $n$ odd,}\\
sign(V)&\text{for $n$ even,}
\end{cases}$$ where $sign$ denotes the [*signature*]{} of the induced symmetric bilinear form $(-1)^{n/2}\cdot S$ on $V$. A similar but deeper result is the famous [*Hodge index theorem*]{} (compare e.g. [@Voi]\[thm.6.3.3\])): $$\chi_1([H^*(M)])=sign(H^m(M,{\mathbb{Q}}))$$ for $M$ a compact Kähler manifold of complex even dimension $m=2n$. Here the right side denotes the signature of the symmetric intersection pairing $$\begin{CD}
H^m(M,{\mathbb{Q}})\times H^m(M,{\mathbb{Q}})@> \cup >> H^{2m}(M,{\mathbb{Q}})\simeq {\mathbb{Q}}\:.
\end{CD}$$
The advantage of $\chi_y$ compared to $E$ (and the use of $-y$ in the definition) comes from the following
Question
: Let $E(X):=E([H^*(X)])$ for $X$ a complex algebraic variety. For $M$ a compact complex algebraic manifold one gets by (\[Hpq\]): $$E(M)=\sum_{p,q\geq 0}\; (-1)^{p+q}\cdot dim_{{\mathbb{C}}}H^q(M,\Omega_M^p)\cdot u^pv^q \:.$$ Is there a [*(normalized multiplicative) characteristic class*]{} $$cl^*: Iso({\mathbb{C}}-VB(M))\to H^*(M)[u^{\pm1},v^{\pm 1}]$$ of complex vector bundles such that the E-polynomial is a [*characteristic number*]{} in the sense that $$\label{genus}
E(M)=\sharp(M):=deg(cl^*(TM)\cap[M])\in H^*(pt)[u^{\pm1},v^{\pm 1}]$$ for any compact complex algebraic manifold $M$ with fundamental class $[M]$?
So the cohomology class $cl^*(V)\in H^*(M)[u^{\pm1},v^{\pm 1}]$ should only depend on the isomorphism class of the complex vector bundle $V$ over $M$ and commute with pullback. Multiplicativity says $$cl^*(V)=cl^*(V')\cup cl^*(V'') \in H^*(M)[u^{\pm1},v^{\pm 1}]$$ for any short exact sequence $0\to V'\to V\to V''\to 0$ of complex vector bundles on $M$. Finally $cl^*$ is normalized if $cl^*(trivial)=1\in H^*(M)$ for any trivial vector bundle. Then the answer to this question is [*NO*]{} because there are unramified coverings $p: M'\to M$ of elliptic curves $M,M'$ of (any) degree $d>0$. Then $p^*TM\simeq TM'$ and $p_*([M'])=d\cdot [M]$ so that the projection formula would give for the topological characteristic numbers the relation $$\sharp(M')= d\cdot \sharp(M)\:.$$ But one has $$E(M)=(1-u)(1-v)=E(M')\neq 0$$ so that the equality $E(M)=\sharp(M)$ is not possible! Here wo don’t need to ask $cl^*$ to be multiplicative or normalized. But if we use the invariant $\chi_y(X):=\chi_y([H^*(X)])$, then $\chi_y(M)=0$ for an elliptic curve, and $\chi_y(M)$ is a characteristic number in the sense above by the famous [*generalized Hirzebruch Riemann Roch theorem*]{} ([@H]):
\[gHRR\] There is a unique normalized multiplicative characteristic class $$T_y^*: Iso({\mathbb{C}}-VB(M))\to H^*(M,{\mathbb{Q}})[y]$$ such that $$\chi_y(M)=deg (T_y^*(TM)\cap [M]) = \langle T_y^*(TM), [M]\rangle
\in {\mathbb{Z}}[y]\subset {\mathbb{Q}}[y]$$ for any compact complex algebraic manifold $M$. Here $\langle \cdot,\cdot\rangle$ is the Kronecker pairing between cohomology and homology.
The [*Hirzebruch class*]{} $T^*_y$ and $\chi_y$-genus unify the following (total) characteristic classes and numbers: $$T^*_y=
\begin{cases}
c^* \:\text{,the Chern class}\\
td^*\:\text{,the Todd class}\\
L^* \:\text{,the L class}\\
\end{cases} \quad \text{and $\chi_y=$}
\begin{cases}
\chi \:\text{,the Euler characteristic}\\
\chi_a \:\text{,the arithmetic genus}\\
sign \:\text{,the signature}
\end{cases}
\:\text{for $y=$}
\begin{cases}
-1\\ 0\\1\:. \end{cases}$$
In fact $(gHRR)$ is just a cohomological version of the following $K$-theoretical calculation. Let $M$ be a compact complex algebraic manifold, so that $$\label{chiy-M} \begin{split}
\chi_y(M)&=\sum_{p,q\geq 0}\; (-1)^{p+q}\cdot dim_{{\mathbb{C}}}H^q(M,\Omega_M^p)\cdot (-y)^p \\
&=\sum_{p\geq 0}\; \chi(H^*(M,\Omega_M^p))\cdot y^p \:.
\end{split}$$ Let us denote by $K^0_{an}(Y)$ (or $G_0^{an}(Y)$) the Grothendieck group of the exact (or abelian) category of holomorphic vector bundles (or coherent ${{\mathcal O}}_Y$-module sheaves) on the complex variety $Y$, i.e. the free abelian group of isomorphism classes $V$ of such vector bundles (or sheaves), divided out by the relation $$[V]=[V']+[V''] \quad \text{for any short exact sequence $0\to V'\to V\to V''\to 0$.}$$ Then $G_0^{an}(Y)$ (or $K^0_{an}(Y)$) is of (co)homological nature, with $$f_*: G_0^{an}(X)\to G_0^{an}(Y);\: [{{\mathcal F}}]\mapsto \sum_{i\geq 0}\;(-1)^i\cdot [R^if_*{{\mathcal F}}]$$ the functorial pushdown for a proper holomorphic map $f: X\to Y$. In particular for $X$ compact, the constant map $k: X\to pt$ is proper, with $$\chi(H^*(X,{{\mathcal F}}))=k_*([{{\mathcal F}}])\in G^{an}_0(pt)\simeq K^0_{an}(pt)\simeq {\mathbb{Z}}\:.$$ Moreover, the tensor product $\otimes_{{{\mathcal O}}_Y}$ induces a natural pairing $$\cap=\otimes: K^0_{an}(Y)\times G_0^{an}(Y) \to G_0^{an}(Y)\:,$$ where we identify a holomorphic vector bundle $V$ with its locally free coherent sheaf of sections ${{\mathcal V}}$. So for $X$ compact we can define a [*Kronecker pairing*]{} $$K^0_{an}(X)\times G_0^{an}(X) \to G_0^{an}(pt)\simeq {\mathbb{Z}};\:
\langle [{{\mathcal V}}],[{{\mathcal F}}] \rangle:=k_*([{{\mathcal V}}\otimes_{{{\mathcal O}}_X} {{\mathcal F}}])\:.$$ The [*total $\lambda$-class*]{} of the dual vector bundle $$\lambda_y(V^{\vee}):=\sum_{i\geq 0}\; \Lambda^i(V^{\vee})\cdot y^i$$ defines a multiplicative characteristic class $$\lambda_y((\cdot)^{\vee}): K^0_{an}(Y)\to K^0_{an}(Y)[y] \:.$$ And for a compact complex algebraic manifold $M$ one gets the equality $$\label{gHHR-K} \begin{split}
\chi_y(M)&= \sum_{i\geq 0}\; k_*[\Omega^i_M]\cdot y^i\\
&=\langle \lambda_y(T^*M), [{{\mathcal O}}_M] \rangle \in G_0^{an}(pt)[y]\simeq {\mathbb{Z}}[y] \:.
\end{split}$$
Characteristic classes of variations of mixed Hodge structures
==============================================================
This section explains the definition of [*cohomological*]{} characteristic classes associated to good variations of mixed Hodge structures on complex algebraic and analytic manifolds. These were previously considered in [@CLMS; @CLMS2; @MSc] in connection with Atiyah-Meyer type formulae of Hodge-theoretic nature. Here we also consider important functorial properties of these classes.
Variation of Hodge structures
-----------------------------
Let $f: X\to Y$ be a [*proper smooth*]{} morphism of complex algebraic varieties or a [*projective smooth*]{} morphism of complex analytic varieties. Then the higher direct image sheaf $L=L^n:=R^nf_*{\mathbb{Q}}_X$ is a [*locally constant sheaf*]{} on $Y$ with finite dimensional stalks $$L_y=(R^nf_*{\mathbb{Q}}_X)_y=H^n(\{f=y\},{\mathbb{Q}})$$ for $y\in Y$. Let ${{\mathcal L}}:=L\otimes_{{\mathbb{Q}}_Y} {{\mathcal O}}_Y\simeq R^nf_*(\Omega^{\bullet}_{X/Y})$ denote the corresponding holomorphic vector bundle (or locally free sheaf), with $\Omega^{\bullet}_{X/Y}$ the [*relative holomorphic de Rham complex*]{}. Then the stupid filtration of $\Omega^{\bullet}_{X/Y}$ determines a decreasing filtration $F$ of ${{\mathcal L}}$ by holomorphic subbundles $F^p{{\mathcal L}}$, with $$\label{F-relDR}
Gr_F^p((R^{p+q}f_*{\mathbb{Q}}_X)\otimes_{{\mathbb{Q}}_Y} {{\mathcal O}}_Y)\simeq R^qf_*(\Omega^{p}_{X/Y})\:,$$ inducing for all $y\in Y$ the Hodge filtration $F$ on the cohomology $$H^n(\{f=y\},{\mathbb{Q}})\otimes {\mathbb{C}}\simeq {{\mathcal L}}|_y$$ of the compact and smooth algebraic fiber $\{f=y\}$ (compare [@PS]\[chap.10\]). If $Y$ (and therefore also $X$ is smooth), then ${{\mathcal L}}$ gets an induced [*integrable Gauss-Manin connection*]{} $$\nabla: {{\mathcal L}}\to {{\mathcal L}}\otimes_{{{\mathcal O}}_Y}\Omega_Y^1
,\quad \text{with $L\simeq kern(\nabla)$ and $\nabla\circ \nabla=0$,}$$ satisfying the [*Griffith’s transversality*]{} condition $$\label{Griff}
\nabla(F^p{{\mathcal L}})\subset F^{p-1}{{\mathcal L}}\otimes_{{{\mathcal O}}_Y}\Omega_Y^1 \quad \text{for all $p$.}$$ This motivates the following
A [*holomorphic family*]{} $(L,F)$ of Hodge structures of weight $n$ on the reduced complex space $Y$ is a local system $L$ with rational coefficients and finite dimensional stalks on $Y$, and a decreasing filtration $F$ of ${{\mathcal L}}=L\otimes_{{\mathbb{Q}}_Y}{{\mathcal O}}_Y$ by holomorphic subbbundles $F^p{{\mathcal L}}$ such that $F$ determines by $L_y\otimes_{\mathbb{Q}}{\mathbb{C}}\simeq {{\mathcal L}}|_y$ a pure Hodge structure of weight $n$ on each stalk $L_y$ ($y\in Y$).
If $Y$ is a smooth complex manifold, then such a holomorphic family $(L,F)$ is called a [*variation*]{} of Hodge structures of weight $n$, if one has in addition for the induced connection $\nabla: {{\mathcal L}}\to {{\mathcal L}}\otimes_{{{\mathcal O}}_Y}\Omega_Y^1$ the Griffith’s transversality (\[Griff\]).
Finally a [*polarization*]{} of $(L,F)$ is a pairing of local systems $S: L\otimes_{{\mathbb{Q}}_Y} L
\to {\mathbb{Q}}_y$, that induces a polarization of Hodge structures on each stalk $L_y$ ($y\in Y$).
For example in the geometric case above, one can get such a polarization on $L=R^nf_*{\mathbb{Q}}_X$ for $f: X\to Y$ a [*projective smooth*]{} morphism of complex algebraic (or analytic) varieties. The existence of a polarization is needed for example for the following important result of Schmid ([@Sch]\[thm.7.22\]):
\[rigid\] Let $Y$ be a connected complex manifold Zarisky open in a compact complex analytic manifold $\bar{Y}$, with $(L,F)$ a polarizable variation of pure Hodge structures on $Y$. Then $H^0(Y,L)$ gets an induced Hodge structure such that the evaluation map $H^0(Y,L)\to L_y$ is an isomorphism of Hodge structures for all $y\in Y$. In particular the variation $(L,F)$ is constant, if the underlying local system $L$ is constant.
Variation of mixed Hodge structures
-----------------------------------
If one considers a morphism $f: X\to Y$ of complex algebraic varieties with $Y$ smooth, which is a topological fibration with possible singular or non-compact fiber, then the locally constant direct image sheaves $L=L^n:=R^nf_*{\mathbb{Q}}_X$ ($n\geq 0$) are [*variations of mixed Hodge structures*]{} in the sense of the following definitions.
Let $Y$ be a reduced complex analytic space. A [*holomorphic family of mixed Hodge structures*]{} on $Y$ consists of the following data:
1. a local system $L$ of rational vector spaces on $Y$ with finite dimensional stalks,
2. a finite decreasing [*Hodge filtration*]{} $F$ of ${{\mathcal L}}=L\otimes_{{\mathbb{Q}}_Y} {{\mathcal O}}_Y$ by holomorphic subbundles $F^p{{\mathcal L}}$,
3. a finite increasing [*weight filtration*]{} $W$ of $L$ by local subsystems $W_nL$,
such that the induced filtrations on ${{\mathcal L}}_y\simeq L_y\otimes_{{\mathbb{Q}}}{\mathbb{C}}$ and $L_y$ define a mixed Hodge structure on all stalks $L_y$ ($y\in Y$).
If $Y$ is a smooth complex manifold, then such a holomorphic family $(L,F,W)$ is called a [*variation of mixed Hodge structures*]{}, if one has in addition for the induced connection $\nabla: {{\mathcal L}}\to {{\mathcal L}}\otimes_{{{\mathcal O}}_Y}\Omega_Y^1$ the Griffith’s transversality (\[Griff\]).
Finally $(L,F,W)$ is called [*graded polarizable*]{}, if the induced family (or variation) of pure Hodge structures $Gr^W_nL$ (with the induced Hodge filtration $F$) is polarizable for all $n$.
With the obvious notion of morphisms, the categories $FmHs^{(p)}(Y)$ (or $VmHs^{(p)}(Y)$) of (graded polarizable) families (or variations) of mixed Hodge structures on $Y$ become abelian categories with a tensor product $\otimes$ and duality $(\cdot)^{\vee}$. Again any such morphism is strictly compatible with the Hodge and weight filtrations. Moreover, one has for a holomorphic map $f: X\to Y$ (of complex manifolds) a functorial pullback $$f^*: FmHs^{(p)}(Y)\to FmHs^{(p)}(X) \quad \text{(or $f^*: VmHs^{(p)}(Y)\to VmHs^{(p)}(X)$),}$$ comuting with tensor product $\otimes$ and duality $(\cdot)^{\vee}$. On a point space $pt$ one gets just back the category $$FmHs^{(p)}(pt)=VmHs^{(p)}(pt) =mHs^{(p)}$$ of (graded polarizable) mixed Hodge structures. Using the pullback under the constant map $k: Y\to pt$, we get the constant family (or variation) of Tate Hodge structures ${\mathbb{Q}}_Y(n):=k^*{\mathbb{Q}}(n)$ on $Y$.
Cohomological characteristic classes
------------------------------------
The Grothendieck group $K^0_{an}(Y)$ of holomorphic vector bundles on the complex variety $Y$ is a commutative ring with multiplication induced by $\otimes$ and has a duality involution induced by $(\cdot)^{\vee}$. For a holomorphic map $f: X\to Y$ one has a functorial pullback $f^*$ of rings with involutions. Similarly for $K^0_{an}(Y)[y^{\pm 1}]$, if we extend the duality involution by $$([V]\cdot y^k)^{\vee}:=[V^{\vee}]\cdot (1/y)^k \:.$$
For a family (or variation) of mixed Hodge structures $(L,F,W)$ on $Y$ let us introduce the characteristic class $$\label{MHC-coh}
MHC^y((L,F,W)):=\sum_p\; [Gr^p_F({{\mathcal L}})]\cdot (-y)^p \in K^0_{an}(Y)[y^{\pm 1}]\:.$$ Since morphisms of families (or variations) of mixed Hodge structures are strictly compatible with the Hodge filtrations, we get an induced group homomorphism of Grothendieck groups: $$MHC^y: K_0(FmHs^{(p)}(Y))\to K^0_{an}(Y)[y^{\pm 1}] \quad
\text{or $MHC^y: K_0(VmHs^{(p)}(Y))\to K^0_{an}(Y)[y^{\pm 1}]$.}$$
Note that $MHC^{-1}((L,F,W))=[{{\mathcal L}}]\in K^0_{an}(Y)$ is just the class of the associated holomorphic vector bundle. And for $Y=pt$ a point, we get back the $\chi_y$-genus: $$\chi_y=MHC^y: K_0(mHs^{(p)})= K_0(FmHs^{(p)}(pt))\to K^0_{an}(pt)[y^{\pm 1}]
={\mathbb{Z}}[y^{\pm 1}]\:.$$
The transformation $$MHC^y: K_0(FmHs^{(p)}(Y))\to K^0_{an}(Y)[y^{\pm 1}] \quad
\text{or $MHC^y: K_0(VmHs^{(p)}(Y))\to K^0_{an}(Y)[y^{\pm 1}]$}$$ is contravariant functorial. It is a transformation of commutative rings with unit, i.e. it commutes with products and respects the units: $MHC^y([{\mathbb{Q}}_Y(0)])=[{{\mathcal O}}_Y]$. Similarly it respects the duality involutions: $$MHC^y([(L,F,W)^{\vee}])=\sum_p\; [(Gr^{-p}_F({{\mathcal L}}))^{\vee}]\cdot (-y)^p=
\left(MHC^y([(L,F,W)])\right)^{\vee}\:.$$
\[ex:smooth\] Let $f: X\to Y$ be a [*proper smooth*]{} morphism of complex algebraic varieties or a [*projective smooth*]{} morphism of complex analytic varieties, so that the higher direct image sheaf $L^n:=R^nf_*{\mathbb{Q}}_X$ ($n\geq 0$) with the induced Hodge filtration as in (\[F-relDR\]) defines a holomorphic family of pure Hodge structures on $Y$. If $m$ is the complex dimension of the fibers, then $L_n=0$ for $n>2m$ so that one can define $$[Rf_*{\mathbb{Q}}_X]:=\sum_{n=0}^{2m}\; (-1)^n\cdot [(R^nf_*{\mathbb{Q}}_X,F)] \in K_0(FmHs(Y))\:.$$ Then one gets by (\[F-relDR\]): $$\begin{split} MHC^y([Rf_*{\mathbb{Q}}_X])
&= \sum_{p,q\geq 0}\;(-1)^{p+q}\cdot [R^qf_*\Omega^p_{X/Y}]\cdot (-y)^p\\
&= \sum_{p\geq 0}\; f_*[\Omega^p_{X/Y}]\cdot y^p\\
&=: f_*\left(\lambda_y(T^*_{X/Y})\right) \in K^0_{an}(Y)[y] \:.
\end{split}$$ Assume moreover that
1. $Y$ is a connected complex manifold Zarisky open in a compact complex analytic manifold $\bar{Y}$,
2. All direct images sheaves $L^n:=R^nf_*{\mathbb{Q}}_X$ ($n\geq 0$) are [*constant*]{}.
Then one gets by the [*rigidity*]{} theorem \[rigid\] (for $z\in Y$): $$f_*\left(\lambda_y(T^*_{X/Y})\right)=\chi_y(\{f=z\})\cdot [{{\mathcal O}}_Y]\in K^0_{an}(Y)[y]\:.$$
Let $f: X\to Y$ be a [*smooth*]{} morphism of compact complex algebraic manifolds, with $Y$ connected. Let $T^*_{X/Y}$ be the relative holomorphic cotangent bundle of the fibers, fitting into the short exact sequence $$0\to f^*T^*Y \to T^*X \to T^*_{X/Y}\to 0 \:.$$ Assume all direct images sheaves $L^n:=R^nf_*{\mathbb{Q}}_X$ ($n\geq 0$) are [*constant*]{}, i.e. $\pi_1(Y)$ acts trivially on the cohomology $H^*(\{f=z\})$ of the fiber. Then one gets the multiplicativity of the $\chi_y$-genus (with $k: Y\to pt$ the constant map): $$\begin{split}\label{multipl-chiy}
\chi_y(X)&= (k\circ f)_* [\lambda_y(T^*X)]\\
&= k_*f_*\left( [\lambda_y(T^*_{X/Y})]\otimes f^*[\lambda_y(T^*Y)]\right)\\
&=k_*\left( \chi_y(\{f=z\})\cdot [\lambda_y(T^*Y)]\right)\\
&=\chi_y(\{f=z\})\cdot \chi_y(Y) \:.
\end{split}$$
The multiplicativity relation (\[multipl-chiy\]) specializes for $y=1$ to the classical multiplicativity formula $$sign(X)= sign(\{f=z\})\cdot sign(Y)$$ of Chern-Hirzebruch-Serre [@CHS] for the signature of an oriented fibration of smooth coherently oriented compact manifolds, if $\pi_1(Y)$ acts trivially on the cohomology $H^*(\{f=z\})$ of the fiber. So it is a Hodge theoretic counterpart of this. Moreover, the corresponding Euler characteristic formula for $y=-1$ $$\chi(X)= \chi(\{f=z\})\cdot \chi(Y)$$ is even true [*without*]{} $\pi_1(Y)$ acting trivially on the cohomology $H^*(\{f=z\})$ of the fiber!
The Chern-Hirzebruch-Serre signature formula was motivational for many subsequent works which studied monodromy contributions to invariants (genera and characteristic classes), e.g. see [@At; @BCS; @CMS; @CMS0; @CMS; @CLMS; @CLMS2; @CS; @MSc; @Mey].
Instead of working with holomorphic vector bundles, we can of course also use only the underlying topological complex vector bundles, which gives the forgetful transformation $$For: K^0_{an}(Y)\to K^0_{top}(Y)\:.$$ Here the target can also be viewed as the even part of ${\mathbb{Z}}_2$-graded topological complex K-cohomology. Of course, $For$ is contravariant functorial and commutes with product $\otimes$ and duality $(\cdot)^{\vee}$. This duality induces a ${\mathbb{Z}}_2$-grading on $K^0_{top}(Y)[1/2]$ by splitting into the (anti-)invariant part, and similarly for $K^0_{an}(Y)[1/2]$. Then the (anti-)invariant part of $K^0_{top}(Y)[1/2]$ can be identified with the even part of ${\mathbb{Z}}_4$-graded topological real K-theory $KO^0_{top}(Y)[1/2]$ (and $KO^2_{top}(Y)[1/2]$).\
Assume now that $(L,F)$ is a holomorphic family of pure Hodge structures of weight $n$ on the complex variety $Y$, with a polarization $S: L\otimes_{{\mathbb{Q}}_Y} L\to {\mathbb{Q}}_Y$. This induces an isomorphism of families of pure Hodge structures (of weight $n$): $$L \simeq L^{\vee}(-n):=L^{\vee}\otimes {\mathbb{Q}}_Y(-n)\:.$$ So if we choose the isomorphism of rational local systems ${\mathbb{Q}}_Y(-n)=(2\pi i)^n\cdot {\mathbb{Q}}_Y\simeq {\mathbb{Q}}_Y$, then the polarisation induces a $(-1)^n$-symmetric duality isomorphism $L\simeq L^{\vee}$ of the underlying local systems. And for such an (anti)symmetric selfdual local system $L$ Meyer [@Mey] has introduced a $KO$-characteristic class $$[L]_{KO}\in KO^0_{top}(Y)[1/2]\oplus KO^2_{top}(Y)[1/2])=K^0_{top}(Y)[1/2]$$ so that for $Y$ a compact oriented manifold of even real dimension $2m$ the following [*twisted signature formula*]{} is true: $$\label{tw-sig}
sign(H^m(Y,L))= \langle ch^*(\Psi^2([L]_{KO})), L^*(TM)\cap [M]\rangle \:.$$ Here $H^m(Y,L)$ gets an induced (anti)symmetric duality, with $sign(H^m(Y,L)):=0$ in case of an antisymmetric pairing. Moreover $ch^*$ is the Chern character, $\Psi^2$ the second Adams operation and $L^*$ is the Hirzebruch-Thom $L$-class.\
We now explain that $[L]_{KO}$ agrees up to some universal signs with $For(MHC^1((L,F))$. The underlying topological complex vector bundle of ${{\mathcal L}}$ has a natural real structure so that as a topological complex vector bundle one gets an orthogonal decomposition $${{\mathcal L}}=\oplus_{p+q=n}\; {{\mathcal H}}^{p,q} \quad \text{with ${{\mathcal H}}^{p,q}=F^p{{\mathcal L}}\cap\overline{F^q{{\mathcal L}}}
=\overline{{{\mathcal H}}^{q,p}}$,}$$ with $$\label{MHC1}
For(MHC^1((L,F))= \sum_{p\;even,q}\; [{{\mathcal H}}^{p,q}]- \sum_{p\;odd,q}\; [{{\mathcal H}}^{p,q}]\:.$$ If $n$ is even, then both sums of the right hand side in (\[MHC1\]) are invariant under conjugation. And $(-1)^{-n/2}\cdot S$ is by (\[positive\]) positive resp. negative definite on the corresponding real vector bundle $(\oplus_{p\;even,q}\; {{\mathcal H}}^{p,q})_{{\mathbb{R}}}$ resp. $(\oplus_{p\;odd,q}\; {{\mathcal H}}^{p,q})_{{\mathbb{R}}}$. So if we choose the pairing $(-1)^{n/2}\cdot S$ for the isomorphism $L\simeq L^{\vee}$, then this agrees with the splitting introduced by Meyer [@Mey] in the definition of his $KO$-characteristic class $[L]_{KO}$ associated to this [*symmetric*]{} duality isomorphism of $L$: $$For(MHC^1((L,F))= [L]_{KO} \in KO^0_{top}(Y)[1/2]\:.$$ Similarly, if $n$ is odd, both sums of the right hand side in (\[MHC1\]) are exchanged under conjugation. If we choose the pairing $(-1)^{(n+1)/2}\cdot S$ for the isomorphism $L\simeq L^{\vee}$, then this agrees by definition \[pol\] with the splitting introduced by Meyer [@Mey] in the definition of his $KO$-characteristic class $[L]_{KO}$ associated to this [*antisymmetric*]{} duality isomorphism of $L$: $$For(MHC^1((L,F))= [L]_{KO} \in KO^2_{top}(Y)[1/2]\:.$$
Let $(L,F)$ be a holomorphic family of pure Hodge structures of weight $n$ on the complex variety $Y$, with a polarization $S$ chosen. Then the class $[L]_{KO}$ introduced in [@Mey] for the duality isomorphism coming from the pairing $(-1)^{n(n+1)/2}\cdot S$ is equal to $$For(MHC^1((L,F))= [L]_{KO}\in KO^0_{top}(Y)[1/2]\oplus KO^2_{top}(Y)[1/2]
= K^0_{top}(Y)[1/2]\:.$$ It is therefore independent of the choice of the polarisation $S$. Moreover, this identification is functorial under pullback and compatible with products (as defined in [@Mey]\[p.26\] for (anti)symmetic selfdual local systems).
There are Hodge theoretic counterparts of the twisted signature formula (\[tw-sig\]). Here we formulate a corresponding K-theoretical result. Let $(L,F,W)$ be a variation of mixed Hodge structures on the $m$-dimensional complex manifold $M$. Then $$H^n(M,L)\simeq H^n(M,DR({{\mathcal L}}))$$ gets an induced (decreasing) $F$ filtration coming from the filtration of the holomorphic de Rham complex of the vector bundle ${{\mathcal L}}$ with its integrable connection $\nabla$: $$\begin{CD} DR({{\mathcal L}})=[{{\mathcal L}}@> \nabla >> \cdots @> \nabla >> {{\mathcal L}}\otimes_{{{\mathcal O}}_M} \Omega^m_M]
\end{CD}$$ (with ${{\mathcal L}}$ in degree zero), defined by $$\label{dr-fil-ext} \begin{CD}
F^p DR({{\mathcal L}}) =[F^p{{\mathcal L}}@> \nabla >> \cdots @> \nabla >> F^{p-m}{{\mathcal L}}\otimes_{{{\mathcal O}}_M} \Omega^m_M] \:.
\end{CD}$$ Note that here we are using the Griffith’s transversality (\[Griff\])!\
The following result is due to Deligne and Zucker ([@Zu]\[thm.2.9, lem.2.11\]) in the case of a compact Kähler manifold, whereas the case of a compact complex algebraic manifold follows from Saito’s general results as explained in the next section.
Assume $M$ is a compact Kähler manifold or a compact complex algebraic manifold, with $(L,F,W)$ a graded polarizable variation of mixed (or pure) Hodge structures on $M$. Then $H^n(M,L)\simeq H^n(M,DR({{\mathcal L}}))$ gets an induced mixed (or pure) Hodge structure with $F$ the Hodge filtration. Moreover, the corresponding [*Hodge to de Rham spectral-sequence*]{} degenerates at $E_1$ so that $$Gr_F^p(H^n(M,L)) \simeq H^{n}(M,Gr_F^pDR({{\mathcal L}})) \quad \text{for all $n,p$.}$$
Therefore one gets as a corollary (compare [@CLMS; @CLMS2; @MSc]): $$\begin{split}
\chi_y(H^*(M,L))
&= \sum_{n,p}\; (-1)^n\cdot dim_{{\mathbb{C}}}\left(H^{n}(M,Gr_F^pDR({{\mathcal L}}))\right) \cdot (-y)^p\\
&= \sum_p\; \chi\left(H^{*}(M,Gr_F^pDR({{\mathcal L}}))\right) \cdot (-y)^p\\
&= \sum_{p,i}\; (-1)^i\cdot \chi\left(H^{*}(M,Gr_F^{p-i}({{\mathcal L}})\otimes_{{{\mathcal O}}_M}\Omega^i_M)
\right)\cdot (-y)^p \\
&=k_*\left( MHC^y(L)\otimes \lambda_y(T^*M)\right) \\
&=:\langle MHC^y(L), \lambda_y(T^*M)\cap [{{\mathcal O}}_M] \rangle \in {\mathbb{Z}}[y^{\pm 1}]\:.
\end{split}$$
Good variation of mixed Hodge structures
----------------------------------------
For later use let us introduce the following
\[good\] Let $M$ be a complex algebraic manifold. A graded polarizable variation of mixed Hodge structures $(L,F,W)$ on $M$ is called good, if it is [*admissible*]{} in the sense of Steenbrink-Zucker [@SZ] and Kashiwara [@Ka], with [*quasi-unipotent monodromy*]{} at infinity, i.e. with respect to a compactification $\bar{M}$ of $M$ by a compact complex algebraic manifold $\bar{M}$, with complement $D:=\bar{M}\backslash M$ a normal crossing divisor with smooth irreducible components.
\[good-var\] Two important examples for such a good variation of mixed Hodge structures are the following:
1. A polarizable variation of [*pure*]{} Hodge structures is always admissible by a deep theorem of Schmid [@Sch]\[thm.6.16\]. So it is good precisely when it has quasi-unipotent monodromy at infinity.
2. Consider a morphism $f: X\to Y$ of complex algebraic varieties with $Y$ smooth, which is a topological fibration with possible singular or non-compact fiber. Then the locally constant direct image sheaves $R^nf_*{\mathbb{Q}}_X$ and $R^nf_!{\mathbb{Q}}_X$ ($n\geq 0$) are good variations of mixed Hodge structures (compare with remark \[geometric\]).
This class of good variations on $M$ is again an abelian category $VmHs^g(M)$ stable under tensor product $\otimes$, duality $(\cdot)^{\vee}$ and pullback $f^*$ for $f$ an algebraic morphism of complex algebraic manifolds. Moreover, in this case all vector bundles $F^p{{\mathcal L}}$ of the Hodge filtration carry the structure of a unique underlying complex algebraic vector bundle (in the Zariski topology), so that the characteristic class transformation $MHC^y$ can be seen as a natural contravariant transformation of rings with involution $$MHC^y: K_0(VmHs^g(M)) \to K^0_{alg}(M)[y^{\pm 1}] \:.$$
In fact, consider a (partial) compactification $\bar{M}$ of $M$ as above, with $D:=\bar{M}\backslash M$ a normal crossing divisor with smooth irreducible components and $j: M\to \bar{M}$ the open inclusion. Then the holomorphic vector bundle ${{\mathcal L}}$ with integrable connection $\nabla$ corresponding to $L$ has a unique [*canonical Deligne extension*]{} $(\overline{{{\mathcal L}}},\overline{\nabla})$ to a holomorphic vector bundle $\overline{{{\mathcal L}}}$ on $\bar{M}$, with [*meromorphic*]{} integrable connection $$\label{D-ext}
\overline{\nabla}: \overline{{{\mathcal L}}}\to
\overline{{{\mathcal L}}}\otimes_{{{\mathcal O}}_{\bar{M}}} \Omega^1_{\bar{M}}(log(D))$$ having [*logarithmic poles*]{} along $D$. Here the [*residues*]{} of $\overline{\nabla}$ along $D$ have real eigenvalues, since $L$ has [*quasi-unipotent monodromy*]{} along $D$. And the canonical extension is characterized by the property, that all these eigenvalues are in the half-open intervall $[0,1[\;$. Moreover, also the Hodge filtration $F$ of ${{\mathcal L}}$ extends uniquely to a filtration $\bar{F}$ of $\overline{{{\mathcal L}}}$ by holomorphic subvector bundles $$F^p\overline{{{\mathcal L}}}:=j_*(F^p{{\mathcal L}}) \cap \overline{{{\mathcal L}}} \subset j_*{{\mathcal L}}\:,$$ since $L$ is [*admissible*]{} along $D$. Finally the Griffith’s transversality extends to $$\label{Griff-alg}
\overline{\nabla}(F^p\overline{{{\mathcal L}}})\subset F^{p-1}\overline{{{\mathcal L}}}\otimes_{{{\mathcal O}}_{\bar{M}}} \Omega^1_{\bar{M}}(log(D)) \quad\text{for all $p$.}$$ For more details see [@De]\[prop.5.4\] and [@PS]\[sec.11.1, sec.14.4\].\
If we choose $\bar{M}$ as a compact algebraic manifold, then we can apply Serre’s GAGA theorem to conclude that $\overline{{{\mathcal L}}}$ and all $F^p\overline{{{\mathcal L}}}$ are [*algebraic*]{} vector bundles, with $\overline{\nabla}$ an [*algebraic*]{} meromorphic connection.
\[rem-del\] The canonical Deligne extension $\overline{{{\mathcal L}}}$ (as above) with its Hodge filtration $F$ has the following compabilities (compare [@De]\[part II\]):
smooth pullback
: Let $f:\bar{M'} \to \bar{M}$ be a smooth morphism so that $D':=f^{-1}(D)$ is also a normal crossing divisor with smooth irreducible components on $\bar{M'}$ with complement $M'$. Then one has $$f^*\left(\overline{{{\mathcal L}}}\right) \simeq \overline{f^*{{\mathcal L}}} \quad \text{and}
f^*\left(F^{p}\overline{{{\mathcal L}}}\right) \simeq F^p\overline{f^*{{\mathcal L}}}\:\:\text{for all $p$.}$$
exterior product
: Let $L$ and $L'$ be two good variations on $M$ and $M'$. Then their canonical Deligne extensions satisfy $$\overline{{{\mathcal L}}\boxtimes_{{{\mathcal O}}_{M\times M'}} {{\mathcal L}}'}
\simeq \overline{{{\mathcal L}}}\boxtimes_{{{\mathcal O}}_{\bar{M}\times \bar{M}'}} \overline{{{\mathcal L}}'}\:,$$ since the residues of the corresponding meromorphic connections are compatible. Then one has for all $p$: $$F^p\left(\overline{{{\mathcal L}}\boxtimes_{{{\mathcal O}}_{M\times M'}} {{\mathcal L}}'}\right)\simeq \oplus_{i+k=p}\; \left(F^i\overline{{{\mathcal L}}}\right)\boxtimes_{{{\mathcal O}}_{\bar{M}\times \bar{M}'}} \left(F^k\overline{{{\mathcal L}}'}\right) \:.$$
tensor product
: In general the canonical Deligne extensions of two good variations $L$ and $L'$ on $M$ are [*not*]{} compatible with tensor products, because of the choice of different residues for the corresponding meromorphic connections. This problem doesn’t appear if one of these variations, lets say $L'$, is already defined on $\bar{M}$. Let $L$ resp. $L'$ be a good variation on $M$ resp. $\bar{M}$. Then their canonical Deligne extensions satisfy $$\overline{{{\mathcal L}}\otimes_{{{\mathcal O}}_M} ({{\mathcal L}}'|M)}
\simeq \overline{{{\mathcal L}}}\otimes_{{{\mathcal O}}_{\bar{M}}} {{\mathcal L}}'\:,$$ and one has for all $p$: $$F^p\left(\overline{{{\mathcal L}}\otimes_{{{\mathcal O}}_M} ({{\mathcal L}}'|M)}\right)\simeq \oplus_{i+k=p}\; \left(F^i\overline{{{\mathcal L}}}\right)\otimes_{{{\mathcal O}}_{\bar{M}}} \left(F^k{{\mathcal L}}'\right) \:.$$
Let $\bar{M}$ be a partial compactification of $M$ as before, i.e. we don’t assume that $\bar{M}$ is compact, with $m:=dim_{{\mathbb{C}}}(M)$. Then the [*logarithmic de Rham complex*]{} $$\begin{CD} DR_{log}\left(\overline{{{\mathcal L}}}\right)
:=[\overline{{{\mathcal L}}} @> \overline{\nabla} >> \cdots @> \overline{\nabla} >> \overline{{{\mathcal L}}}\otimes_{{{\mathcal O}}_{\bar{M}}} \Omega^m_{\bar{M}}(log(D))]
\end{CD}$$ (with $\overline{{{\mathcal L}}}$ in degree zero) is quasi-isomorphic to $Rj_*L$, so that $$H^*(M,L) \simeq H^*\left(\bar{M},DR_{log}\left(\overline{{{\mathcal L}}}\right)\right) \:.$$ So these cohomology groups get an induced (decreasing) $F$-filtration coming from the filtration $$\label{drlog-fil} \begin{CD}
F^p DR_{log}\left(\overline{{{\mathcal L}}}\right) =
[F^p\overline{{{\mathcal L}}} @> \overline{\nabla} >> \cdots @> \overline{\nabla} >> F^{p-m}\overline{{{\mathcal L}}}\otimes_{{{\mathcal O}}_{\bar{M}}} \Omega^m_{\bar{M}}(log(D))] \:.
\end{CD}$$
For $\bar{M}$ a compact algebraic manifold, this is again the Hodge filtration of an induced mixed Hodge structure on $H^*(M,L)$ (compare with corollary \[MHM3\]).
\[j\*VMHS\] Assume $\bar{M}$ is a smooth algebraic compactification of the algebraic manifold $M$ with the complement $D$ a normal crossing divisor with smooth irreducible components. Let $(L,F,W)$ be a good variation of mixed Hodge structures on $M$. Then $H^n(M,L)\simeq H^*\left(\bar{M},DR_{log}\left(\overline{{{\mathcal L}}}\right)\right)$ gets an induced mixed Hodge structure with $F$ the Hodge filtration. Moreover, the corresponding [*Hodge to de Rham spectral-sequence*]{} degenerates at $E_1$ so that $$Gr_F^p(H^n(M,L)) \simeq H^{n}\left(M,Gr_F^pDR_{log}\left(\overline{{{\mathcal L}}}\right)\right) \quad \text{for all $n,p$.}$$
Therefore one gets as a corollary (compare [@CLMS; @CLMS2; @MSc]): $$\label{pairing1}\begin{split}
\chi_y(H^*(M,L))
&= \sum_{n,p}\; (-1)^n\cdot dim_{{\mathbb{C}}}\left(H^{n}\left(M,Gr_F^pDR_{log}\left(\overline{{{\mathcal L}}}\right)\right)\right) \cdot (-y)^p\\
&= \sum_p\; \chi\left(H^{*}\left(M,Gr_F^pDR_{log}\left(\overline{{{\mathcal L}}}\right)\right)\right) \cdot (-y)^p\\
&= \sum_{p,i}\; (-1)^i\cdot \chi\left(H^{*}\left(M,Gr_F^{p-i}\left(\overline{{{\mathcal L}}}\right)\otimes_{{{\mathcal O}}_{\bar{M}}}\Omega^i_{\bar{M}}(log(D)) \right)
\right)\cdot (-y)^p \\
&=:\langle MHC^y(Rj_*L), \lambda_y\left(\Omega^1_{\bar{M}}(log(D)) \right)\cap [{{\mathcal O}}_{\bar{M}}] \rangle \in {\mathbb{Z}}[y^{\pm 1}]\:.
\end{split}$$
Here we use the notion $$\label{MHCj*-coh}
MHC^y(Rj_*L):=\sum_p\; [Gr^p_F\left(\overline{{{\mathcal L}}}\right)]\cdot (-y)^p \in K^0_{alg}(\bar{M})[y^{\pm 1}]\:.$$
Remark \[rem-del\] then implies the
\[Cor-j\*\] Let $\bar{M}$ be a smooth algebraic partial compactifiction of the algebraic manifold $M$ with the complement $D$ a normal crossing divisor with smooth irreducible components. Then $MHC^y(Rj_*(\cdot))$ induces a transformation $$MHC^y(j_*(\cdot)): K_0(VmHs^{g}(M))\to K^0_{alg}(\bar{M})[y^{\pm 1}] \:.$$
1. This is contravariant functorial for a smooth morphism $f: \bar{M}'\to \bar{M}$ of such partial compactifications, i.e. $$f^*\left(MHC^y(j_*(\cdot))\right) \simeq MHC^y\left(j'_*(f^*(\cdot))\right) \:.$$
2. It commutes with exterior products for two good variations $L,L'$: $$MHC^y\left((j\times j')_*[(L\boxtimes_{{\mathbb{Q}}_{M\times M'}} L']\right)=
MHC^y(j_*[L]) \boxtimes MHC^y(j'_*[(L']) \:.$$
3. Let $L$ resp. $L'$ be a good variation on $M$ resp. $\bar{M}$. Then $MHC^y(j_*[\cdot])$ is multiplicative in the sense that $$MHC^y\left(j_*[(L\otimes_{{\mathbb{Q}}_M} (L'|M)]\right)=
MHC^y(j_*[L]) \otimes MHC^y([L']) \:.$$
Calculus of mixed Hodge modules
===============================
Mixed Hodge modules.
--------------------
Before discussing extensions of the characteristic cohomology classes $MHC^y$ to the singular setting, we need to briefly recall some aspects of Saito’s theory [@Sa0; @Sa1; @Sa2; @Sa4; @Sa5] of algebraic mixed Hodge modules, which play the role of singular extensions of good variations of mixed Hodge structures.\
To each complex algebraic variety $Z$, Saito associated a category $MHM(Z)$ of [*algebraic mixed Hodge modules*]{} on $Z$ (cf. [@Sa0; @Sa1]). If $Z$ is smooth, an object of this category consists of an algebraic (regular) holonomic $D$-module $({{\mathcal M}},F)$ with a good filtration $F$ together with a perverse sheaf $K$ of rational vector spaces, both endowed a finite increasing filtration $W$ such that $$\alpha: DR({{\mathcal M}})^{an}\simeq K\otimes_{{\mathbb{Q}}_Z} {\mathbb{C}}_Z \quad \text{is compatible with $W$}$$ under the Riemann-Hilbert correspondence coming from the (shifted) analytic de Rham complex (with $\alpha$ a chosen isomorphism). Here we use left $D$-modules, and the sheaf ${{\mathcal D}}_Z$ of algebraic differential operators on $Z$ has the increasing filtration $F$ with $F_i{{\mathcal D}}_Z$ given by the differential operators of degree $\leq i$ ($i\in {\mathbb{Z}}$). Then a [*good*]{} filtration $F$ of the algebraic holonomic $D$-module ${{\mathcal M}}$ is given by a bounded from below, increasing and exhaustive filtration $F_p{{\mathcal M}}$ by [*coherent*]{} algebraic ${{\mathcal O}}_Z$-modules such that $$\label{D-filt}
F_i{{\mathcal D}}_Z\left( F_p{{\mathcal M}}\right) \subset F_{p+i}{{\mathcal M}}\; \quad \text{for all $i,p$, and this is an equality for $i$ big enough.}$$
In general, for a singular variety $Z$ one works with suitable local embeddings into manifolds and corresponding filtered $D$-modules supported on $Z$. In addition, these objects are required to satisfy a long list of complicated properties (not needed here). The [*forgetful*]{} functor $rat$ is defined as $$rat: MHM(Z)\to Perv({\mathbb{Q}}_Z);\: \left(({{\mathcal M}},F),K,W\right)\mapsto K\:.$$
\[MHM1\] $MHM(Z)$ is an abelian category with $rat: MHM(Z)\to Perv({\mathbb{Q}}_Z)$ exact and [*faithful*]{}. It extends to a functor $$rat: D^bMHM(Z) \to D^b_c({\mathbb{Q}}_Z)$$ to the derived category of complexes of ${\mathbb{Q}}$-sheaves with algebraically constructible cohomology. There are functors $$f_*, \;f_! ,\; f^*,\; f^!,\; \otimes,\; \boxtimes,\; {{\mathcal D}}\quad \text{on $D^bMHM(Z)$} \:,$$ which are “lifts" via $rat$ of the similar (derived) functors defined on $D^b_c({\mathbb{Q}}_Z)$, with $(f^*,f_*)$ and $(f_!,f^!)$ also pairs of [*adjoint*]{} functors. One has a natural map $f_!\to f_*$, which is an isomorphism for $f$ proper. Here ${{\mathcal D}}$ is a duality involution ${{\mathcal D}}^2\simeq id$ “lifting" the Verdier duality functor, with $${{\mathcal D}}\circ f^* \simeq f^!\circ {{\mathcal D}}\quad \text{and} \quad
{{\mathcal D}}\circ f_* \simeq f_!\circ {{\mathcal D}}\:.$$
Compare with [@Sa1]\[thm.0.1 and sec.4\] for more details (as well as with [@Sa4] for a more general formal abstraction). The usual truncation $\tau_{\leq}$ on $D^bMHM(Z)$ corresponds to the [*perverse truncation*]{} ${^p\tau}_{\leq}$ on $D^b_c(Z)$ so that $$rat\circ H= \;^p{{\mathcal H}}\circ rat\:,$$ where $H$ stands for the cohomological functor in $D^bMHM(Z)$ and $\;^p{{\mathcal H}}$ denotes the perverse cohomology (always with respect to the self-dual middle perversity).
\[smooth\] Let $M$ be a complex algebraic manifold of pure complex dimension $m$, with $(L,F,W)$ a good variation of mixed Hodge structures on $M$. Then ${{\mathcal L}}$ with its integrable connection $\nabla$ is a holonomic (left) $D$-module with $\alpha: DR({{\mathcal L}})^{an}\simeq L[m]$, where this time we use the shifted de Rham complex $$\begin{CD}
DR({{\mathcal L}}):=[{{\mathcal L}}@> \nabla >> \cdots @> \nabla >> {{\mathcal L}}\otimes_{{{\mathcal O}}_M} \Omega^m_M]
\end{CD}$$ with ${{\mathcal L}}$ in degree $-m$, so that $DR({{\mathcal L}})^{an}\simeq L[m]$ is a perverse sheaf on $M$. The filtration $F$ induces by Griffith’s transversality (\[Griff\]) a good filtration $F_p({{\mathcal L}}):=F^{-p}{{\mathcal L}}$ as a filtered $D$-module. As explained before, this comes from an underlying algebraic filtered $D$-module. Finally $\alpha$ is compatible with the induced filtration $W$ defined by $$W^i(L[m]):=W^{i-m}L[m] \quad \text{ and} \quad
W^i({{\mathcal L}}):=(W^{i-m}L)\otimes_{{\mathbb{Q}}_M} {{\mathcal O}}_M \:.$$ And this defines a mixed Hodge module ${{\mathcal M}}$ on $M$, with $rat({{\mathcal M}})[-m]$ a local system on $M$.
A mixed Hodge module ${{\mathcal M}}$ on the pure $m$-dimensional complex algebraic manifold $M$ is called [*smooth*]{}, if $rat({{\mathcal M}})[-m]$ is a local system on $M$. Then this example corresponds to [@Sa1]\[thm.0.2\], whereas the next theorem corresponds to [@Sa1]\[thm.3.27 and rem. on p.313\]:
\[MHM2\] Let $M$ be a pure $m$-dimensional complex algebraic manifold. Associating to a good variation of mixed Hodge structures ${\mathbb{V}}=(L,F,W)$ on $M$ the mixed Hodge module ${{\mathcal M}}:={\mathbb{V}}_H$ as in example (\[smooth\]) defines an equivalence of categories $$MHM(M)_{sm}\simeq VmHs^g(M)$$ between the categories of smooth mixed Hodge modules $MHM(M)_{sm}$ and good variation of mixed Hodge structures on $M$. This commutes with exterior product $\boxtimes$ and pullback $$f^*: VmHs^g(M)\to VmHs^g(M') \quad \text{resp.}\quad f^*[m'-m]: MHM(M)\to MHM(M')$$ for an algebraic morphism of smooth algebraic manifolds $M,M'$ of dimension $m,m'$. For $M=pt$ a point, one gets in particular an equivalence $$MHM(pt)\simeq mHs^p \:.$$
\[geometric\] These two theorems explain why a geometic variations of mixed Hodge structures as in Example \[good-var\](2) is good.
By the last identification of the theorem, there exists a unique Tate object ${\mathbb{Q}}^H(n)
\in MHM(pt)$ such that $rat({\mathbb{Q}}^H(n))={\mathbb{Q}}(n)$ and ${\mathbb{Q}}^H(n)$ is of type $(-n,-n)$: $$MHM(pt)\ni {\mathbb{Q}}^H(n) \simeq {\mathbb{Q}}(n)\in mHs^p \:.$$ For a complex variety $Z$ with constant map $k: Z\to pt$, define $${\mathbb{Q}}_Z^H(n):=k_Z^*{\mathbb{Q}}^H(n) \in D^bMHM(Z), \quad \text{with $rat({\mathbb{Q}}_Z^H(n))={\mathbb{Q}}_Z(n)$.}$$ So tensoring with $ {\mathbb{Q}}_Z^H(n)$ defines the Tate twist $\cdot (n)$ of mixed Hodge modules. To simplify the notations, let ${\mathbb{Q}}_Z^H:={\mathbb{Q}}_Z^H(0)$. If $Z$ is *smooth* of complex dimension $n$ then ${\mathbb{Q}}_Z[n]$ is perverse on $Z$, and ${\mathbb{Q}}_Z^H[n]\in MHM(Z)$ is a single mixed Hodge module, explicitly described by $${\mathbb{Q}}_Z^H[n]=(({{\mathcal O}}_Z, F), {\mathbb{Q}}_Z[n], W), \quad \text{with $gr^F_i=0=gr^W_{i+n}$ for all $ i \neq 0$.}$$
It follows from the definition that every ${{\mathcal M}}\in MHM(Z)$ has a finite increasing [*weight filtration*]{} $W$ so that the functor $M \to Gr^W_kM$ is exact. We say that ${{\mathcal M}}\in D^bMHM(Z)$ has [*weights $\leq n$ (resp. $\geq n$)*]{} if $Gr_j^WH^iM=0$ for all $j>n+i$ (resp. $j<n+i$). ${{\mathcal M}}$ is called [*pure of weight $n$*]{}, if it has weights both $\leq n$ and $\geq n$. For the following results compare with [@Sa1]\[prop.2.26 and (4.5.2)\]:
If $f$ is a map of algebraic varieties, then $f_!$ and $f^*$ preserve weight $\leq n$, and $f_*$ and $f^!$ preserve weight $\geq n$. If $f$ is smooth of pure complex fiber dimension $m$, then $f^!\simeq f^*[2m](m)$ so that $f^*,f^!$ preserve pure objects for $f$ smooth. Moreover, if ${{\mathcal M}}\in D^bMHM(X)$ is pure and $f:X \to
Y$ is proper, then $f_*{{\mathcal M}}\in D^bMHM(Y)$ is pure of the same weight as ${{\mathcal M}}$.
Similarly the duality functor ${{\mathcal D}}$ exchanges “weight $\leq n$” and “weight $\geq -n$”, in particular it preserves pure objects. Finally let $j: U\to Z$ be the inclusion of a Zariski open subset. Then the [*intermediate extension*]{} functor $$j_{!*}: MHM(U) \to MHM(Z):\; {{\mathcal M}}\mapsto
Im\left( H^0(j_!{{\mathcal M}}) \to H^0(j_*({{\mathcal M}})\right)$$ preserves weight $\leq n$ and $\geq n$, in particular it preserves pure objects (of weight $n$).
We say that ${{\mathcal M}}\in D^bMHM(Z)$ is supported on $S\subset Z$ if and only if $rat({{\mathcal M}})$ is supported on $S$. There are the abelian subcategories $MH(Z,k)^p \subset MHM(Z)$ of pure Hodge modules of weight $k$, which in the algebraic context are assumed to be polarizable (and extendable at infinity).
For each $k \in {\mathbb{Z}}$, the abelian category $MH(Z,k)^p$ is semi-simple, in the sense that every pure Hodge module on $Z$ can be uniquely written as a finite direct sum of pure Hodge modules with strict support in irreducible closed subvarieties of $Z$. Let $MH_S(Z,k)^p$ denote the subcategory of [*pure Hodge modules of weight $k$ with strict support in $S$*]{}. Then every ${{\mathcal M}}\in MH_S(Z,k)^p$ is generically a good variation of Hodge structures ${\mathbb{V}}_U$ of weight $k-d$ ($d:=dim\;S$) on a Zariski dense smooth open subset $U \subset S$ (i.e. ${\mathbb{V}}_U$ is polarizable with quasi-unipotent monodromy at infinity). This follows from theorem \[MHM2\] and the fact, that a perverse sheaf is generically a shifted local system on a smooth dense Zariski open subset $U\subset S$. Conversely, every such good variation of Hodge structures ${\mathbb{V}}$ on such an $U$ corresponds by theorem \[MHM2\] to a pure Hodge module ${\mathbb{V}}_H$ on $U$, which can be extended in an unique way to a pure Hodge module $j_{!*}{\mathbb{V}}_H$ on $S$ with strict support. Under this correspondence, for $M \in MH_S(Z,k)^p$ we have that $$rat({{\mathcal M}})=IC_S({\mathbb{V}})$$ is the [*twisted intersection cohomology complex*]{} for ${\mathbb{V}}$ the corresponding variation of Hodge structures. Similarly $$\label{duality-IC}
{{\mathcal D}}(j_{!*}{\mathbb{V}}_H) \simeq j_{!*}({\mathbb{V}}^{\vee}_H) (d)\:.$$
Moreover, a [*polarization*]{} of ${{\mathcal M}}\in MH_S(Z,k)^p$ corresponds to an isomorphism of Hodge modules (compare [@PS]\[def.14.35, rem.14.36\]) $$\label{pol-HM}
S: {{\mathcal M}}\simeq {{\mathcal D}}({{\mathcal M}})(-k)\:,$$ whose restriction to $U$ gives a polarization of ${\mathbb{V}}$. In particular it induces a self-duality isomorphism $$S: rat({{\mathcal M}}) \simeq {{\mathcal D}}(rat({{\mathcal M}}))(-k) \simeq {{\mathcal D}}(rat({{\mathcal M}}))$$ of the underlying twisted intersection cohomology complex, if an isomorphism ${\mathbb{Q}}_U(-k)\simeq {\mathbb{Q}}_U$ is chosen.\
So if $U$ is smooth of pure complex dimension $n$, then ${\mathbb{Q}}_U^H[n]$ is a pure Hodge module of weight $n$. If moreover $j: U \hookrightarrow
Z$ is a Zariski-open dense subset in $Z$, then the [*intermediate extension* ]{} $j_{!*}$ for mixed Hodge modules (cf. also with [@BBD]) preserves the weights. This shows that if $Z$ is a complex algebraic variety of pure dimension $n$ and $j: U \hookrightarrow Z$ is the inclusion of a smooth Zariski-open dense subset then the intersection cohomology module $IC_Z^H:=j_{!*}({\mathbb{Q}}_U^H[n])$ is pure of weight $n$, with underlying perverse sheaf $rat(IC_Z^H)=IC_Z$.\
Note that the stability of a pure object ${{\mathcal M}}\in MHM(X)$ under a proper morphism $f: X\to Y$ implies the famous [*decomposition theorem*]{} of [@BBD] in the context of pure Hodge modules ([@Sa1]\[(4.5.4) on p.324\]): $$\label{decomp}
f_*{{\mathcal M}}\simeq \oplus_i\; H^if_*{{\mathcal M}}[-i]\:, \quad \text{with $H^if_*{{\mathcal M}}$ semi-simple for all $i$.}$$
Assume $Y$ is pure-dimesional, with $f: X\to Y$ a [*resolution of singularities*]{}, i.e. $X$ is smooth with $f$ a proper morphism, which generically is an isomorphism on some Zariski dense open subset $U$. Then ${\mathbb{Q}}^H_X$ is pure, since $X$ is smooth, and $IC^H_Y$ has to be the direct summand of $H^0f_*{\mathbb{Q}}^H_X$ which corresponds to ${\mathbb{Q}}^H_U$.
Assume $Y$ is pure-dimesional, with $f: X\to Y$ a [*resolution of singularities*]{}. Then $IC^H_Y$ is a direct summand of $f_*{\mathbb{Q}}^H_X \in D^bMHM(Y)$.
Finally we get the following results about the existence of a mixed Hodge structure on the cohomology (with compact support) $H^i_{(c)}(Z,M)$ for ${{\mathcal M}}\in D^bMHM(Z)$.
\[MHM3\] Let $Z$ be a complex algebraic variety with constant map $k: Z\to pt$. Then the cohomology (with compact support) $H^i_{(c)}(Z,{{\mathcal M}})$ of ${{\mathcal M}}\in D^bMHM(Z)$ gets an induced graded polarizable mixed Hodge structure: $$H^i_{(c)}(Z,{{\mathcal M}})=H^i(k_{*(!)}M)\in MHM(pt)\simeq mHs^p\:.$$ In particular:
1. The rational cohomology (with compact support) $H^i_{(c)}(Z,{\mathbb{Q}})$ of $Z$ gets an induced graded polarizable mixed Hodge structure by: $$H^i(Z,{\mathbb{Q}})= rat(H^i(k_*k^*{\mathbb{Q}}^H)) \quad \text{and} \quad
H^i_{c}(Z,{\mathbb{Q}})= rat(H^i(k_!k^*{\mathbb{Q}}^H)) \:.$$
2. Let ${\mathbb{V}}_U$ be a good variation of mixed Hodge structures on a smooth pure $n$-dimensional complex variety $U$, which is Zariski open and dense in a variety $Z$, with $j: U\to Z$ the open inclusion. Then the global twisted Intersection cohomology (with compact support) $$IH_{(c)}^i(Z,{\mathbb{V}}):=H^i_{(c)}(Z, IC_Z({\mathbb{V}})[-n])$$ gets a mixed Hodge structure by $$IH_{(c)}^i(Z,{\mathbb{V}})=H^i(k_{*(!)}IC_Z({\mathbb{V}})[-n]) =H^i(k_{*(!)}j_{!*}({\mathbb{V}})[-n]) \:.$$ If $Z$ is compact, with ${\mathbb{V}}$ a polarizable variation of pure Hodge structures of weight $w$, then also $IH^i(Z,{\mathbb{V}})$ has a (polarizable) pure Hodge structure of weight $w+i$.
3. Let ${\mathbb{V}}$ be a good variation of mixed Hodge structures on a smooth (pure dimensional) complex manifold $M$, which is Zariski open and dense in complex algebraic manifold $\bar{M}$, with complement $D$ a normal crossing divisor with smooth irreducible components. Then $H^i(M,{\mathbb{V}})$ gets a mixed Hodge structure by $$H^i(M,{\mathbb{V}})\simeq H^i(\bar{M},j_*{\mathbb{V}})\simeq H^i(k_*j_*{\mathbb{V}})\:,$$ with $j: U\to Z$ the open inclusion.
Let us point out some important properties of these mixed Hodge structures:
1. By a deep theorem of Saito ([@Sa5]\[thm.0.2,cor.4.3\]), the mixed Hodge structure on $H^i_{(c)}(Z,{\mathbb{Q}})$ defined as above coincides with the classical mixed Hodge structure constructed by Deligne ([@De1; @De3]).
2. Assume we are in the context of (3) above with $Z=\bar{M}$ projective and ${\mathbb{V}}$ a good variation of pure Hodge structures on $U=M$. Then the pure Hodge structure of (2) on the global Intersection cohomology $IH^i(Z,{\mathbb{V}})$ agrees with that of [@CKS; @KK] defined in terms of $L^2$-cohomology with respect to a Kähler metric with Poincaré singularities along $D$ (compare [@Sa1]\[rem.3.15\]). The case of a $1$-dimensional complex algebraic curve $Z=\bar{M}$ due to Zucker [@Zu]\[thm.7.12\] is used in the work of Saito [@Sa0]\[(5.3.8.2)\] in the proof of the stability of pure Hodge modules under projective morphisms [@Sa0]\[thm.5.3.1\] (compare also with the detailed discussion of this $1$-dimensional case in [@Sab]).
3. Assume we are in the context of (3) above with $\bar{M}$ compact. Then the mixed Hodge structure on $H^i(M,{\mathbb{V}})$ is the one of theorem \[j\*VMHS\], whose Hodge filtration $F$ comes from the filtered logarithmic de Rham complex (compare [@Sa1]\[sec.3.10, prop.3.11\]).
Grothendieck groups of algebraic mixed Hodge modules. {#Grot}
-----------------------------------------------------
In this section, we describe the functorial calculus of Grothendieck groups of algebraic mixed Hodge modules. Let $Z$ be a complex algebraic variety. By associating to (the class of) a complex the alternating sum of (the classes of) its cohomology objects, we obtain the following identification (e.g. compare \[[@KS], p. 77\], \[[@Sc], Lemma 3.3.1\]) $$K_0(D^bMHM(Z))=K_0(MHM(Z)).$$ In particular, if $Z$ is a point, then $$K_0(D^bMHM(pt))=K_0(mHs^p),$$ and the latter is a commutative ring with respect to the tensor product, with unit $[{\mathbb{Q}}^H]$. Then we have for any complex ${{\mathcal M}}^{\bullet} \in D^bMHM(Z)$ the identification $$\label{i1}
[{{\mathcal M}}^{\bullet}]=\sum_{i \in {\mathbb{Z}}} (-1)^i [H^i({{\mathcal M}}^{\bullet})] \in
K_0(D^bMHM(Z)) \cong K_0(MHM(Z)).$$ In particular, if for any ${{\mathcal M}}\in MHM(Z)$ and $k \in {\mathbb{Z}}$ we regard ${{\mathcal M}}[-k]$ as a complex concentrated in degree $k$, then $$\label{i2}
\left[ {{\mathcal M}}[-k] \right]= (-1)^k [{{\mathcal M}}] \in K_0(MHM(Z)).$$ All functors $f_*$, $f_!$, $f^*$, $f^!$, $\otimes$, $\boxtimes$, ${{\mathcal D}}$ induce corresponding functors on $K_0(MHM(\cdot))$. Moreover, $K_0(MHM(Z))$ becomes a $K_0(MHM(pt))$-module, with the multiplication induced by the exact exterior product with a point space: $$\boxtimes : MHM(Z) \times MHM(pt) \to MHM(Z \times \{pt\}) \simeq
MHM(Z).$$ Also note that $${{\mathcal M}}\otimes {\mathbb{Q}}^H_Z \simeq {{\mathcal M}}\boxtimes
{\mathbb{Q}}^H_{pt} \simeq {{\mathcal M}}$$ for all ${{\mathcal M}}\in MHM(Z)$. Therefore, $K_0(MHM(Z))$ is a unitary $K_0(MHM(pt))$-module. The functors $f_*$, $f_!$, $f^*$, $f^!$ commute with exterior products (and $f^*$ also commutes with the tensor product $\otimes$), so that the induced maps at the level of Grothendieck groups $K_0(MHM(\cdot))$ are $K_0(MHM(pt))$-linear. Similarly ${{\mathcal D}}$ defines an involution on $K_0(MHM(\cdot))$. Moreover, by the functor $$rat:K_0(MHM(Z)) \to K_0(D^b_c({\mathbb{Q}}_Z)) \simeq K_0(Perv({\mathbb{Q}}_Z)),$$ all these transformations lift the corresponding transformations from the (topological) level of Grothendieck groups of constructible (or perverse) sheaves.
The Grothendieck group $K_0(MHM(Z))$ has two different type of generators:
1. It is generated by the classes of pure Hodge modules $[IC_S({\mathbb{V}})]$ with strict support in an irreducible complex algebraic subset $S\subset Z$, with ${\mathbb{V}}$ a good variation of (pure) Hodge structures on a dense Zariski open smooth subset $U$ of $S$. These generators behave well under duality.
2. It is generated by the classes $f_*[j_*{\mathbb{V}}]$, with $f: \bar{M}\to Z$ a proper morphisms from the smooth complex algebraic manifold $\bar{M}$, $j: M\to \bar{M}$ the inclusion of a Zariski open and dense subset $M$, with complement $D$ a normal crossing divisor with smooth irreducible components, and ${\mathbb{V}}$ a good variation of mixed (or if one wants also pure) Hodge structures on $M$. These generators will be used in the next section about characteristic classes of mixed Hodge modules.
Here (1) follows from the fact, that a mixed Hodge module has a finite weight filtration, whose graded pieces are pure Hodge modules, i.e. are finite direct sums of pure Hodge modules $IC_S({\mathbb{V}})$ with strict support $S$ as above. (2) follows by induction from resolution of singularities and from the existence of a standard distinguished triangle associated to a closed inclusion.\
Let $i: Y\to Z$ be a closed inclusion of complex algebraic varieties with open complement $j: U=Z\backslash Y\to Z$. Then one has by Saito’s work [@Sa1]\[(4.4.1)\] the following functorial distinguished triangle in $D^bMHM(Z)$: $$\label{triangle}
\begin{CD}
j_!j^* @> ad_j >> id @> ad_i >> i_*i^* @> [1]>> \:.
\end{CD}$$ Here the maps $ad$ are the adjunction maps, with $i_*=i_!$ since $i$ is proper. If $f: Z \to X$ is a complex algebraic morphism, then we can apply $f_!$ to get another distinguished triangle $$\label{triangle2}
\begin{CD}
f_!j_!j^*{\mathbb{Q}}^H_Z @> ad_j >> f_!{\mathbb{Q}}^H_Z @> ad_i >> f_!i_!i^*{\mathbb{Q}}^H_Z @> [1]>> \:.
\end{CD}$$ On the level of Grothendieck groups, we get the important [*additivity relation*]{} $$f_![{\mathbb{Q}}^H_Z] = (f\circ j)_![{\mathbb{Q}}^H_U] + (f\circ i)_![{\mathbb{Q}}^H_Y] \in K_0(D^bMHM(X))=K_0(MHM(X))\:.$$
\[chiHdg\] One has a natural group homomorphism $$\chi_{Hdg}: K_0(var/X) \to K_0(MHM(X)); [f:Z\to X]\mapsto [f_!{\mathbb{Q}}^H_Z]\:,$$ which commutes with pushdown $f_!$, exterior product $\boxtimes$ and pullback $g^*$. For $X=pt$ this corresponds to the ring homomorphism (\[H-ring\]) under the identification of $MHM(pt)\simeq mHs^p$.
Here $K_0(var/X)$ is the motivic [*relative Grothendieck group*]{} of complex algebraic varieties over $X$, i.e. the free abelian group generated by isomorphism classes $[f]=[f: Z\to X]$ of morphisms $f$ to $X$, divided out be the [*additivity relation*]{} $$[f]=[f\circ i] + [f\circ j]$$ for a closed inclusion $i: Y\to Z$ with open complement $j: U=Z\backslash Y\to Z$. The pushdown $f_!$, exterior product $\boxtimes$ and pullback $g^*$ for these relative Grothendieck groups are defined by composition, exterior product and pullback of arrows. The fact that $\chi_{Hdg}$ commutes with exterior product $\boxtimes$ (or pullback $g^*$) follows then from the corresponding Künneth (or base change) theorem for the functor $$f_!: D^bMHM(Z)\to D^bMHM(X)$$ (contained in Saito’s work [@Sa4] and [@Sa1]\[(4.4.3)\]).\
Let ${\mathbb{L}}:=[{\mathbb{A}}^1_{{\mathbb{C}}}]\in K_0(var/pt)$ be the class of the affine line so that $$\chi_{Hdg}({\mathbb{L}})=[H^2(P^1({\mathbb{C}}),{\mathbb{Q}})]=[ {\mathbb{Q}}(-1)]\in K_0(MHM(pt))=K_0(mHs^p)$$ is the Lefschetz class $[ {\mathbb{Q}}(-1)]$. This is invertible in $K_0(MHM(pt))=K_0(mHs^p)$ so that the transformation $\chi_{Hdg}$ of corollary \[chiHdg\] factorizes over the localization $$M_0(var/X):=K_0(var/X)[{\mathbb{L}}^{-1}] \:.$$ Altogether we get the following diagram of natural transformations commuting with $f_!$, $\boxtimes$ and $g^*$: $$\label{motfunct} \begin{CD}
F(X) @< can << M_0(var/X) @<<< K_0(var/X) \\
@A \chi_{stalk} AA @VV \chi_{Hdg}V \\
K_0(D^b_c(X)) @<< rat < K_0(MHM(X)) \:.
\end{CD}$$
Here $F(X)$ is the group of algebraically constructible functions on $X$ generated by $1_Z$ for $Z\subset X$ a closed complex algebraic subset, with $ \chi_{stalk}$ given by the Euler characteristic of the stalk complexes (compare [@Sc]\[sec.2.3\]). The pushdown $f_!$ for algebraically constructible functions is defined for a morphism $f: Y\to X$ by $$f_!(1_Z)(x):=\chi\left(H^*_c(Z\cap \{f=x\},{\mathbb{Q}})\right) \quad \text{for $x\in X$,}$$ so that the horizontal arrow $can$ is given by $$can: [f: Y\to X]\mapsto f_!(1_Y) \:, \quad \text{with $can({\mathbb{L}})=1_{pt}$.}$$
The advantage of $M_0(var/X)$ compared to $K_0(var/X)$ is the fact, that it has an induced [*duality*]{} involution ${{\mathcal D}}: M_0(var/X)\to M_0(var/X)$ characterized uniquely by (compare [@Bi]): $${{\mathcal D}}\left([f: M\to X]\right)={\mathbb{L}}^{-m}\cdot [f: M\to X]$$ for $f: M\to X$ a proper morphism with $M$ smooth and pure $m$-dimensional. This “motivic duality" ${{\mathcal D}}$ commutes with pushdown $f_!$ for proper $f$, so that $\chi_{Hdg}$ also commutes with duality by $$\label{motduality}\begin{split}
\chi_{Hdg}\left({{\mathcal D}}[id_M]\right) &= \chi_{Hdg}\left({\mathbb{L}}^{-m}\cdot [Id_M]\right)
=[{\mathbb{Q}}^H_M(m)]\\
&= [{\mathbb{Q}}^H_M[2m](m)]= [{{\mathcal D}}( {\mathbb{Q}}^H_M)]= {{\mathcal D}}\left(\chi_{Hdg}\left([Id_M]\right)\right)
\end{split}$$ for $M$ smooth and pure $m$-dimensional. In fact by resolution of singularities and “additivity", $K_0(var/X)$ is generated by such classes $f_![id_M]=[f: M\to X]$.\
Then all the transformations of (\[motfunct\]) [*commute with duality*]{}, were $K_0(D^b_c(X))$ gets this involution from Verdier duality, and ${{\mathcal D}}=id$ for algebraically constructible functions by $\chi\left([ {\mathbb{Q}}(-1)]\right)=1_{pt}$ (compare also with [@Sc]\[sec.6.0.6\]). Similarly they commute with $f_*$ and $g^!$ defined by the relations (compare [@Bi]): $${{\mathcal D}}\circ g^* = g^!\circ {{\mathcal D}}\quad \text{and} \quad
{{\mathcal D}}\circ f_* = f_!\circ {{\mathcal D}}\:.$$ For example for an open inclusion $j: M\to \bar{M}$ one gets $$\label{j-dual}
\chi_{Hdg}\left(j_*[id_M]\right)=j_*[{\mathbb{Q}}^H_M] \:.$$
Characteristic classes of mixed Hodge modules
=============================================
Homological characteristic classes
----------------------------------
In this section we explain the theory of $K$-theoretical characteristic homology classes of mixed Hodge modules based on the following result of Saito (compare with [@Sa0]\[sec.2.3\] and [@Sa5]\[sec.1\] for the first part, and with [@Sa1]\[sec.3.10, prop.3.11\]) for the part (2)):
\[grDR\] Let $Z$ be a complex algebraic variety. Then there is a functor of triangulated categories $$Gr^F_pDR: D^bMHM(Z) \to D^b_{coh}(Z)$$ commuting with proper push-down, with $Gr^F_pDR({{\mathcal M}})=0$ for almost all $p$ and ${{\mathcal M}}$ fixed, where $D^b_{coh}(Z)$ is the bounded derived category of sheaves of algebraic ${{\mathcal O}}_Z$-modules with coherent cohomology sheaves. If $M$ is a (pure $m$-dimensional) complex algebraic manifold, then one has in addition:
1. Let ${{\mathcal M}}\in MHM(M)$ be a single mixed Hodge module. Then $Gr^F_pDR({{\mathcal M}})$ is the corresponding complex associated to the de Rham complex of the underlying algebraic left $D$-module ${{\mathcal M}}$ with its integrable connection $\nabla$: $$\begin{CD} DR({{\mathcal M}})=[{{\mathcal M}}@> \nabla >> \cdots @> \nabla >> {{\mathcal M}}\otimes_{{{\mathcal O}}_M} \Omega^m_M]
\end{CD}$$ with ${{\mathcal M}}$ in degree $-m$, filtered by $$\begin{CD}
F_p DR({{\mathcal M}}) =[F_p{{\mathcal M}}@> \nabla >> \cdots @> \nabla >> F_{p+m}{{\mathcal M}}\otimes_{{{\mathcal O}}_M} \Omega^m_M] \:.
\end{CD}$$
2. Let $\bar{M}$ be a smooth partial compactification of the complex algebraic manifold $M$ with complement $D$ a normal crossing divisor with smooth irreducible components, with $j: M\to \bar{M}$ the open inclusion. Let ${\mathbb{V}}=(L,F,W)$ be a good" variation of mixed Hodge structures on $M$. Then the filtered de Rham complex $$(DR(j_*{\mathbb{V}}),F) \quad \text{of}\quad j_*{\mathbb{V}}\in MHM(\bar{M})[-m]\subset D^bMHM(\bar{M})$$ is filtered quasi-isomorphic to the logarithmic de Rham complex $DR_{log}({{\mathcal L}})$ with the increasing filtration $F_{-p}:=F^{p}$ ($p\in {\mathbb{Z}}$) associated to the decreasing $F$-filtration (\[drlog-fil\]). In particular $Gr^F_{-p}DR(j_*{\mathbb{V}})$ ($p\in {\mathbb{Z}}$) is quasi-isomorphic to $$\begin{CD}
Gr_F^{p}DR_{log}\left(\overline{{{\mathcal L}}}\right) =
[Gr_F^{p}\overline{{{\mathcal L}}} @> Gr\;\overline{\nabla} >> \cdots @> Gr\;\overline{\nabla} >> Gr_F^{p-m}\overline{{{\mathcal L}}}\otimes_{{{\mathcal O}}_{\bar{M}}} \Omega^m_{\bar{M}}(log(D))] \:.
\end{CD}$$
Here the filtration $F_p DR({{\mathcal M}})$ of the de Rham complex is well defined, since the action of the integrable connection $\nabla$ is given in local coordinates $(z_1,\dots, z_m)$ by $$\nabla(\cdot) = \sum_{i=1}^m\: \frac{\partial}{\partial z_i}(\cdot) \otimes dz_i \:,
\quad \text{with
$ \frac{\partial}{\partial z_i}\in F_1{{\mathcal D}}_M$,}$$ so that $\nabla(F_p{{\mathcal M}})\subset
F_{p+1}{{\mathcal M}}$ for all $p$ by (\[D-filt\]). For later use, let us point that the maps $Gr\;\nabla$ and $Gr\;\overline{\nabla}$ in the complexes $$Gr^F_{p}DR({{\mathcal M}}) \quad \text{and} \quad
Gr_F^{p}DR_{log}\left(\overline{{{\mathcal L}}}\right)$$ are ${{\mathcal O}}$-linear!
\[vanishing\] Let $M$ be a pure $m$-dimensional complex algebraic manifold. Then $$Gr^F_{-p}DR({\mathbb{Q}}^H_M)\simeq \Omega^p_M[-p] \in D^b_{coh}(M)$$ for $0\leq p \leq m$, and $Gr^F_{-p}DR({\mathbb{Q}}^H_M)\simeq 0$ otherwise. Assume in addition that $f:M\to Y$ is a resolution of singularities of the pure dimensional complex algebraic variety $Y$. Then $IC^H_Y$ is a direct summand of $f_*{\mathbb{Q}}^H_M \in D^bMHM(Y)$ so that by functoriality $gr^F_{-p}DR(IC^H_Y)$ is a direct summand of $Rf_*\Omega^p_M[-p] \in D^b_{coh}(Y)$. In particular $$Gr^F_{-p}DR(IC^H_Y)\simeq 0 \quad \text{ for $p<0$ or $p>m$.}$$
The transformations $Gr^F_pDR$ ($p\in {\mathbb{Z}}$) induce functors on the level of Grothendieck groups. Therefore, if $G_0(Z) \simeq K_0(D^b_{coh}(Z))$ denotes the Grothendieck group of coherent [*algebraic*]{} ${{\mathcal O}}_Z$-sheaves on $Z$, we get group homomorphisms $$Gr^F_pDR: K_0(MHM(Z))=K_0(D^bMHM(Z))\to K_0(D^b_{coh}(Z))\simeq G_0(Z)\:.$$
The [*motivic Hodge Chern class transformation*]{} $$MHC_y: K_0(MHM(Z)) \to G_0(Z) \otimes {\mathbb{Z}}[y^{\pm 1}]$$ is defined by $$\label{MHC}
[{{\mathcal M}}] \mapsto \sum_{i,p}\; (-1)^{i} [{{\mathcal H}}^i ( Gr^F_{-p} DR({{\mathcal M}}) )] \cdot
(-y)^p\:.$$
So this characteristic class captures information from the graded pieces of the filtered de Rham complex of the filtered $D$-module underlying a mixed Hodge module ${{\mathcal M}}\in MHM(Z)$, instead of the graded pieces of the filtered $D$-module itself (as more often studied). Let $p'=min\{p|\;F_p{{\mathcal M}}\neq 0\}$. Using theorem \[grDR\](1) for a local embedding $Z\hookrightarrow M$ of $Z$ into a complex algebraic manifold $M$ of dimension $m$, one gets $$Gr^F_{p} DR({{\mathcal M}})=0 \quad \text{for $p<p'-m$, and} \quad
Gr^F_{p'-m} DR({{\mathcal M}}) \simeq \left(F_{p'}{{\mathcal M}}\right) \otimes_{{{\mathcal O}}_M} \omega_M$$ is a coherent ${{\mathcal O}}_Z$-sheaf independent of the local embedding. Here we are using left $D$-modules (related to variation of Hodge structures), whereas for this question the corresponding filtered right $D$-module (as used in [@Sa3]) $${{\mathcal M}}^r:= {{\mathcal M}}\otimes_{{{\mathcal O}}_M} \omega_M \quad \text{with} \quad
F_p{{\mathcal M}}^r:=\left(F_{p+m}{{\mathcal M}}\right) \otimes_{{{\mathcal O}}_M} \omega_M$$ would better work. Then the coefficient of the “top-dimensional" power of $y$ in $MHC_y\left([{{\mathcal M}}]\right)$: $$MHC_y\left([{{\mathcal M}}]\right)= [F_{p'}{{\mathcal M}}\otimes_{{{\mathcal O}}_M} \omega_M ]\otimes (-y)^{m-p'}
+ \sum_{i<m-p'} (\cdots) \cdot y^i \in G_0(Z)[y^{\pm 1}]$$ is given by the class $ [F_{p'}{{\mathcal M}}\otimes_{{{\mathcal O}}_M} \omega_M ] \in G_0(Z)$ of this coherent ${{\mathcal O}}_Z$-sheaf (up to a sign). Using resolution of singularities, one gets for example for an $m$-dimensional complex algebraic variety $Z$, that $$MHC_y([{\mathbb{Q}}^H_Z]) = [\pi_*\omega_M] \cdot y^m + \sum_{i<m} (\cdots) \cdot y^i \in G_0(Z)[y^{\pm 1}]\:,$$ with $\pi: M\to Z$ any resolution of singularities of $Z$ (compare [@Sa5]\[cor.0.3\]). More generally, for an irreducible complex variety $Z$ and ${{\mathcal M}}=IC^H_Z({{\mathcal L}})$ a pure Hodge module with strict support $Z$, the corresponding coherent ${{\mathcal O}}_Z$-sheaf $$S_Z({{\mathcal L}}):=F_{p'}IC^H_Z({{\mathcal L}}) \otimes_{{{\mathcal O}}_M} \omega_M$$ only depends on $Z$ and the good variation of Hodge structures ${{\mathcal L}}$ on a Zariski open smooth subset of $Z$, and it behaves much like a dualizing sheaf. Its formal properties are studied in Saito’s proof given in [@Sa3] of a conjecture of Kollar. So the “top-dimensional" power of $y$ in $MHC_y\left([IC^H_Z({{\mathcal L}})]\right)$ exactly picks out (up to a sign) the class $[S_Z({{\mathcal L}})]\in G_0(Z)$ of this interesting coherent sheaf $S_Z({{\mathcal L}})$ on $Z$.\
Let $td_{(1+y)}$ be the [*twisted Todd transformation*]{} $$\label{td} \begin{split}
td_{(1+y)}:\: &G_0(Z) \otimes {\mathbb{Z}}[y^{\pm 1}] \to H_*(Z) \otimes
{\mathbb{Q}}[y^{\pm 1}, (1+y)^{-1}]\;;\\
&[{{\mathcal F}}] \mapsto \sum_{k \geq 0}\; td_k([{{\mathcal F}}]) \cdot (1+y)^{-k}\:,
\end{split}$$ where $H_*(\cdot)$ stands either for Chow homology groups $CH_*(\cdot)$ or for Borel-Moore homology groups $H_{2*}^{BM}(\cdot)$ (in even degrees), and $td_k$ is the degree $k$ component in $H_{k}(Z)$ of the [*Todd class transformation*]{} $td_*: G_0(Z) \to H_{*}(Z) \otimes {\mathbb{Q}}$ of Baum-Fulton-MacPherson [@BFM], which is linearly extended over ${\mathbb{Z}}[y^{\pm 1}]$ (compare also with [@Fu]\[chap.18\] and [@FM]\[Part II\]).
The (un)normalized [*motivic Hirzebruch class transformations*]{} $MHT_{y*}$ (and $MH\tilde{T}_{y*}$) are defined by the composition $$\label{MHT}
MHT_{y*} :=td_{(1+y)} \circ MHC_y: K_0(MHM(Z)) \to H_{*}(Z)
\otimes {\mathbb{Q}}[y^{\pm 1},(1+y)^{-1}]$$ and $$\label{MHT2}
MH{T}_{y*} :=td_* \circ MHC_y: K_0(MHM(Z)) \to H_{*}(Z)
\otimes {\mathbb{Q}}[y^{\pm 1}] \:.$$
By precomposing with the transformation $\chi_{Hdg}$ from corollary \[chiHdg\] one gets similar transformations $$mC_y:=MHC_y\circ \chi_{Hdg},\; T_{y*}:=MHT_{y*}\circ \chi_{Hdg} \quad
\text{and} \quad \tilde{T}_{y*}:=MH\tilde{T}_{y*}\circ \chi_{Hdg}$$ defined on the relative Grothendieck group of complex algebraic varieties $K_0(Var/\cdot)$ as studied in [@BSY]. Then it is the (normalized) motivic Hirzebruch class transformation $T_{y*}$, which “unifies" in a functorial way
1. the (rationalized) Chern class transformation $c_*$ of MacPherson [@M],
2. the Todd class transformation $td_*$ of Baum-Fulton-MacPherson [@BFM], and
3. the $L$-class transformation $L_*$ of Cappell-Shaneson [@CS]
for $y=-1,0$ and $1$ respectively (compare with [@BSY; @SY] and also with [@Y] in these proceedings).
In this paper we work most the time only with the more important $K$-theoretical transformation $MHC_y$. The corresponding results for $MHT_{y*}$ follow from this by the known properties of the Todd class transformation $td_*$ (compare [@BFM; @Fu; @FM]).
\[pt\]Let ${\mathbb{V}}=(V,F,W) \in MHM(pt)=mHs^p$ be a (graded polarizable) mixed Hodge structure. Then: $$\label{point}
MHC_y([{\mathbb{V}}])= \sum_p\:
\text{dim}_{{\mathbb{C}}} (Gr^p_F V_{{\mathbb{C}}}) \cdot (-y)^p = \chi_y([{\mathbb{V}}]) \in {\mathbb{Z}}[y^{\pm 1}]
=G_0(pt)\otimes {\mathbb{Z}}[y^{\pm 1}]\:.$$ So over a point the transformation $MHC_y$ coincides with the $\chi_y$-genus ring homomorphism $\chi_y:K_0(mHs^p) \to
{\mathbb{Z}}[y^{\pm 1}]$ (and similarly for $MH\tilde{T}_{y*}$ and $MHT_{y*}$).
The [*motivic Chern resp. Hirzebruch class*]{} $C_y(Z)$ resp. $T_{y*}(Z)$ of a complex algebraic variety $Z$ is defined by $$C_y(Z):=MHC_y([{\mathbb{Q}}_Z^H]) \quad \text{and} \quad
T_{y*}(Z):=MHT_{y*}([{\mathbb{Q}}_Z^H]) \:.$$ Similarly, if $U$ is a pure $n$-dimensional complex algebraic manifold, and ${{\mathcal L}}$ is a local system on $U$ underlying a good variation of mixed Hodge structures, we define [*twisted motivic Chern resp. Hirzebruch characteristic classes*]{} by (compare [@CLMS; @CLMS2; @MSc]) $$\label{tHc}
C_y(U; {{\mathcal L}}):=MHC_y([{{\mathcal L}}^H]) \quad \text{and} \quad
T_{y*}(U; {{\mathcal L}}):=MHT_{y*}([{{\mathcal L}}^H]) \:,$$ where ${{\mathcal L}}^H[n]$ is the smooth mixed Hodge module on $U$ with underlying perverse sheaf ${{\mathcal L}}[n]$. Assume in addition, that $U$ is dense and Zariski open in the complex algebraic variety $Z$. Let $IC^H_Z, IC_Z^H({{\mathcal L}}) \in MHM(Z)$ be the (twisted) intersection homology (mixed) Hodge module on $Z$, whose underlying perverse sheaf is $IC_Z$ resp. $IC_Z({{\mathcal L}})$. Then we define [*Intersection characteristic classes*]{} by (compare [@BSY; @CMS; @CLMS2; @MSc]): $$IC_y(Z):=MHC_y\left(\left[IC^H_Z[-n]\right]\right) \quad \text{and} \quad
IT_{y*}(Z):=MHT_{y*}\left(\left[ IC^H_Z[-n]\right]\right)$$ and similarly, $$IC_y(Z; {{\mathcal L}}):=MHC_y\left(\left[IC_Z^H({{\mathcal L}})[-n] \right]\right) \:\: \text{and} \:\:
IT_{y*}(Z;{{\mathcal L}}):=MHT_{y*}\left(\left[ IC^H_Z({{\mathcal L}})[-n]\right]\right) \:.$$
By definition and theorem \[grDR\], the transformations $MHC_y$ and $MHT_{y*}$ [*commute with proper push-forward*]{}. The following [*normalization*]{} property holds (cf. [@BSY]): If $M$ is smooth, then $$C_y(Z)=\lambda_y(T^*M)\cap [{{\mathcal O}}_M] \quad \text{and} \quad
T_{y*}(Z)=T_y^*(TM) \cap [M]\:,$$ where $T_y^*(TM)$ is the cohomology Hirzebruch class of $M$ as in theorem \[gHRR\].
Let $Z$ be a compact (possibly singular) complex algebraic variety, with $k:Z \to pt$ the proper constant map to a point. Then for ${{\mathcal M}}\in D^bMHM(Z)$ the pushdown $$k_*(MHC_y({{\mathcal M}}))=MHC_y(k_*{{\mathcal M}})= \chi_y\left([H^*(Z,{{\mathcal M}})]\right)$$ is the Hodge genus $$\label{defchi} \chi_y([H^*(Z,{{\mathcal M}})])=\sum_{i,p}\;
(-1)^i dim_{{\mathbb{C}}} (Gr^p_F H^i(Z,{{\mathcal M}})) \cdot (-y)^p \:.$$ In particular:
1. If $Z$ is smooth, then $$k_*C_y(Z)=\chi_y(Z):=\chi_y\left([H^*(Z,{\mathbb{Q}})]\right)$$ and $$k_*C_y(Z;{{\mathcal L}})=\chi_y(Z;{{\mathcal L}}):=\chi_y\left([H^*(Z,{{\mathcal L}})]\right)\:.$$
2. If $Z$ is pure dimensional, then $$k_*IC_y(Z)=I\chi_y(Z):=\chi_y\left([IH^*(Z,{\mathbb{Q}})]\right)$$ and $$k_*IC_y(Z;{{\mathcal L}})=I\chi_y(Z;{{\mathcal L}}):=\chi_y\left([IH^*(Z,{{\mathcal L}})]\right)\:.$$
Note that for $Z$ compact $$I\chi_{-1}(Z)=\chi([IH^*(Z;{\mathbb{Q}})]$$ is the [*intersection (co)homology Euler characteristic*]{} of $Z$, whereas for $Z$ projective, $$I\chi_1(Z)=sign\left(IH^*(Z,{\mathbb{Q}})\right)$$ is the [*intersection (co)homology signature*]{} of $Z$ due to Goresky-MacPherson [@GM2]. In fact this follows as in the smooth context from Saito’s (relative version of the) Hodge index theorem for intersection cohomology ([@Sa0]\[thm.5.3.2\]). Finally $\chi_0(Z)$ and $I\chi_0(Z)$ are two possible extensions to singular varieties of the [*arithmetic genus*]{}. Here it makes sense to take $y=0$, since one has by Example \[vanishing\]: $$k_*IC_y(Z)=I\chi_y(Z) \in {\mathbb{Z}}[y]\:.$$
It is conjectured that for a pure $n$-dimensional compact variety $Z$: $${IT_1}_*(Z)\stackrel{?}{=} L_*(Z) \in H_{2*}(Z,{\mathbb{Q}})$$ is the Goresky-MacPherson homology $L$-class [@GM2] of the Witt space $Z$ ([@BSY], Remark 5.4). Similarly one should expect for a pure-dimensional compact variety $Z$, that $$\alpha(IC_1(Z)) \stackrel{?}{=} \triangle(Z) \in KO^{top}_0(Z)[1/2]\oplus KO^{top}_2(Z)[1/2]\simeq K^{top}_0(Z)[1/2]\:,$$ where $\alpha: G_0(Z)\to K_0^{top}(Z)$ is the K-theoretical Riemann-Roch transformation of Baum-Fulton-MacPherson [@BFM2], and $\triangle(Z)$ is the [*Sullivan class*]{} of the Witt space $Z$ (compare with [@Ba2] in these proceedings). These conjectured equalities are true for a smooth $Z$, or more generally for a pure $n$-dimensional compact complex algebraic variety $Z$ with a [*small resolution*]{} of singularities $f: M\to Z$, in which case one has $f_*({\mathbb{Q}}_M^H)=IC^H_Z[-n]$ so that $$IT_{1*}(Z) = f_*T_{1*}(M) = f_*L_*(M)= L_*(Z) \:,$$ and $$\alpha\left(IC_1(Z)\right) = f_*\left(\alpha(C_1(M))\right) = f_*\triangle(M)= \triangle(Z)\:.$$ Here the functoriality $f_*L_*(M)= L_*(Z)$ and $f_*\triangle(M)= \triangle(Z)$ for a small resolution follows e.g. from the work [@Woo], which allows one to think of the characteristic classes $L_*$ and $\triangle$ as covariant functors for suitable Witt groups of selfdual constructible sheaf complexes.\
In particular, the classes $f_*C_1(M)$ and $f_*T_{1*}(M)$ do not depend on the choice of a small resolution. In fact the same functoriality argument applies to (compare [@CMS; @MSc]) $$IC_y(Z)=f_*C_y(M) \in G_0(Z)\otimes{\mathbb{Z}}[y] \quad \text{and} \quad IT_{y*}(Z)=f_*T_{y*}(M)\in
H_{2*}(Z)\otimes {\mathbb{Q}}[y,(1+y)^{-1}] \:.$$ Note that in general a complex variety $Z$ doesn’t have a small resolution, and even if it exists, it is in general not unique. This type of independence question were discussed by Totaro [@To], pointing out the relation to the famous [*elliptic genus and classes*]{} (compare also with [@Lib; @Wae] in these proceedings). Note that we get such a result for the K-theoretical class $$IC_y(Z)=f_*C_y(M) \in G_0(Z)\otimes{\mathbb{Z}}[y] \:!$$
Calculus of characteristic classes
----------------------------------
So far we only discussed the functoriality of $MHC_y$ with respect to proper push down, and the corresponding relation to Hodge genera for compact $Z$ coming from the push down for the proper constant map $k: Z\to pt$. Now we explain some other important functoriality properties. Their proof is based on the following (e.g. see [@MSc]\[(4.6)\]):
\[MHMopen\] Let $\bar{M}$ be a smooth partial compactification of the complex algebraic manifold $M$ with complement $D$ a normal crossing divisor with smooth irreducible components, with $j: M\to \bar{M}$ the open inclusion. Let ${\mathbb{V}}=(L,F,W)$ be a good variation of mixed Hodge structures on $M$. Then the filtered de Rham complex $$(DR(j_*{\mathbb{V}}),F) \quad \text{of}\quad j_*{\mathbb{V}}\in MHM(\bar{M})[-m]\subset D^bMHM(\bar{M})$$ is by theorem \[grDR\](2) filtered quasi-isomorphic to the logarithmic de Rham complex $DR_{log}({{\mathcal L}})$ with the increasing filtration $F_{-p}:=F^{p}$ ($p\in {\mathbb{Z}}$) associated to the decreasing $F$-filtration (\[drlog-fil\]). Then $$\begin{split}\label{MHCj*-fomula}
MHC_y(j_*{\mathbb{V}}) &= \sum_{i,p}\; (-1)^{i} [{{\mathcal H}}^i ( Gr_F^{p} DR_{log}({{\mathcal L}}) )] \cdot (-y)^p\\
&=\sum_{p}\; [Gr_F^{p} DR_{log}({{\mathcal L}}) ] \cdot (-y)^p\\
&\stackrel{(\ast)}{=} \sum_{i,p}\; (-1)^{i} [ Gr_F^{p-i}({{\mathcal L}}) \otimes_{{{\mathcal O}}_{\bar{M}}} \Omega_{\bar{M}}^i(log(D))] \cdot (-y)^p\\
&=MHC^y(Rj_*L) \cap \left(\lambda_y\left( \Omega_{\bar{M}}^1(log(D))\right)\cap [{{\mathcal O}}_{\bar{M}}] \right)\:.
\end{split}$$ In particular for $j=id: M\to M$ we get the following [*Atiyah-Meyer type formula*]{} (compare [@CLMS; @CLMS2; @MSc]): $$\label{MHCy-fomula}
MHC_y({\mathbb{V}}) = MHC^y(L) \cap \left(\lambda_y(T^*M)\cap [{{\mathcal O}}_M] \right)\:.$$
The formula (\[MHCj\*-fomula\]) is a class version of the formula (\[pairing1\]) of theorem \[j\*VMHS\], which one gets back from (\[MHCj\*-fomula\]) by pushing down to a point for the proper constant map $k: \bar{M}\to pt$ on the compactification $\bar{M}$ of $M$.
Also note that in the equality ($\ast$) above we use the fact that the complex $Gr_F^{p} DR_{log}({{\mathcal L}})$ has coherent (locally free) objects, with ${{\mathcal O}}_{\bar{M}}$-linear maps between them.
The formula (\[MHCj\*-fomula\]) describes a [*splitting*]{} of the characteristic class $MHC_y(j_*{\mathbb{V}})$ into
1. a cohomological term $MHC^y(Rj_*L)$ capturing the information of the good variation of mixed Hodge structures $L$, and
2. the homological term $$\lambda_y\left( \Omega_{\bar{M}}^1(log(D))\right)\cap [{{\mathcal O}}_{\bar{M}}] = MHC_y(j_*{\mathbb{Q}}^H_M)$$ capturing the information of the underlying space or embedding $j: M\to \bar{M}$.
The term $MHC^y(Rj_*L)$ has by corollary \[Cor-j\*\] good functorial behavior with respect to exterior and suitable tensor products, as well as for smooth pullbacks. For the exterior products one gets similarly (compare [@De]\[prop.3.2\]): $$\Omega_{\bar{M}\times \bar{M}'}^1(log(D\times M'\cup M\times D')) \simeq \left(\Omega_{\bar{M}}^1(log(D))\right) \boxtimes
\left( \Omega_{\bar{M'}}^1(log(D'))\right)$$ so that $$\lambda_y\left( \Omega_{\bar{M}\times \bar{M}'}^1(log(D\times M'\cup M\times D'))\right)\cap
[{{\mathcal O}}_{\bar{M}\times \bar{M}'}] =$$ $$\left(\lambda_y\left( \Omega_{\bar{M}}^1(log(D))\right)\cap [{{\mathcal O}}_{\bar{M}}] \right)\boxtimes
\left(\lambda_y\left( \Omega_{\bar{M'}}^1(log(D'))\right)\cap [{{\mathcal O}}_{\bar{M'}}] \right)$$ for the product of two partial compactifications as in example \[MHMopen\]. But the Grothendieck group $K_0(MHM(Z))$ of mixed Hodge modules on the complex variety $Z$ is generated by classes of the form $f_*(j_*[{\mathbb{V}}])$, with $f: \bar{M}\to Z$ proper and $M,\bar{M},{\mathbb{V}}$ as before. Finally one also has the multiplicativity $$(f\times f')_*= f_*\boxtimes f'_*$$ for the push down for proper maps $f: \bar{M}\to Z$ and $f': \bar{M}'\to Z'$ on the level of Grothendieck groups $K_0(MHM(\cdot))$ as well as for $G_0(\cdot)\otimes{\mathbb{Z}}[y^{\pm 1}]$. Then one gets (as in [@BSY]\[Proof of Cor. 2.1(3)\]) from corollary \[Cor-j\*\] and the example \[MHMopen\] the following
The motivic Chern class transformation $MHC_y$ commutes with exterior products: $$MHC_y([M\boxtimes M'])=
MHC_y([M]\boxtimes [M'])= MHC_y([M]) \boxtimes MHC_y([M'])$$ for $M\in D^bMHM(Z)$ and $M'\in D^bMHM(Z')$.
Next we explain the behaviour of $MHC_y$ for smooth pullbacks. Consider a cartesian diagram of morphisms of complex algebraic varieties $$\begin{CD}
\bar{M}' @> g' >> \bar{M}\\
@V f' VV @VV f V\\
Z' @>> g > Z \:,
\end{CD}$$ with $g$ smooth, $f$ proper and $M,\bar{M},{\mathbb{V}}$ as before. Then also $g'$ is smooth and $f'$ is proper, and one has the [*base change isomorphism*]{} $$g^*f_*=f'_*g'^*$$ on the level of Grothendieck groups $K_0(MHM(\cdot))$ as well as for $G_0(\cdot)\otimes{\mathbb{Z}}[y^{\pm 1}]$. Finally for the induced partial compactification $\bar{M}'$ of $M':=g'^{-1}(M)$, with complement $D'$ the induced normal crossing divisor with smooth irreducible components, one has a short exact sequence of vector bundles on $\bar{M}'$: $$0\to g'^*\left(\Omega_{\bar{M}}^1(log(D))\right)\to \Omega_{\bar{M}'}^1(log(D'))
\to T_{g'}^* \to 0\:,$$ with $T_{g'}^*$ the relative cotangent bundle along the fibers of the smooth morphism $g'$. And by base change one has $T_{g'}^*=f'^*(T_g^*)$. So for the corresponding lambda classes we get $$\begin{split}
\lambda_y\left( \Omega_{\bar{M}'}^1(log(D'))\right) &=
\left(g'^*\lambda_y\left( \Omega_{\bar{M}}^1(log(D))\right)\right)\otimes \lambda_y(T_{g'}^*)\\
&=\left(g'^*\lambda_y\left( \Omega_{\bar{M}}^1(log(D))\right)\right)\otimes
f'^*\lambda_y(T_g^*) \:.
\end{split}$$
Finally (compare also with [@BSY]\[Proof of Cor. 2.1(4)\]), by using the [*projection formula*]{} $$\lambda_y(T_g^*) \otimes f'_*(\cdot)= f'_*\left( f'^*\lambda_y(T_g^*)\otimes (\cdot)\right)\; : \:,\:\: G_0(\bar{M}')\otimes{\mathbb{Z}}[y^{\pm 1}]\to G_0(Z')\otimes{\mathbb{Z}}[y^{\pm 1}]$$ one gets from corollary \[Cor-j\*\] and the example \[MHMopen\] the following
\[VRR\] For a smooth morphism $g: Z'\to Z$ of complex algebraic varieties one has for the motivic Chern class transformation the following [*Verdier Riemann-Roch formula*]{}: $$\lambda_y(T_g^*) \cap
g^*MHC_y([M])= MHC_y(g^*[M])= MHC_y([g^*M])$$ for $M\in D^bMHM(Z)$. In particular $$g^*MHC_y([M])= MHC_y(g^*[M])= MHC_y([g^*M])$$ for $g$ an étale morphism (i.e. a smooth morphism with zero dimensional fibers), or in more topological terms, for $g$ an unramified covering. The most important special case is that of an open embedding.
If moreover $g$ is also proper, then one gets from corollary \[VRR\] and the projection formula the following result:
\[updown\] Let $g: Z'\to Z$ be a smooth and proper morphism of complex algebraic varieties. Then one has for the motivic Chern class transformation the following [*going up und down formula*]{}: $$\begin{split}
MHC_y(g_*g^*[M])&=g_*MHC_y(g^*[M])\\
&=g_*\left( \lambda_y(T_g^*) \cap g^*MHC_y([M])\right)\\
&=\left(g_*\lambda_y(T_g^*)\right) \cap MHC_y([M])
\end{split}$$ for $M\in D^bMHM(Z)$, with $$g_*\left(\lambda_y(T_g^*)\right):= \sum_{p,q\geq 0}\;(-1)^q\cdot [R^qg_*(\Omega^p_{Z'/Z})]\cdot y^p
\in K^0_{alg}(Z)[y]$$ the algebraic cohomology class being given (as in example \[ex:smooth\]) by $$MHC^y([Rg_*{\mathbb{Q}}_{Z'}])= \sum_{p,q\geq 0}\;(-1)^q\cdot [R^qg_*(\Omega^p_{Z'/Z})]\cdot y^p\:.$$ Note that all higher direct image sheaves $R^qg_*(\Omega^p_{Z'/Z})$ are locally free in this case, since $g$ is a smooth and proper morphism of complex algebraic varieties (compare with [@De2]). In particular $$g_*C_y(Z')= \left(g_*\lambda_y(T_g^*)\right) \cap C_y(Z) \:,$$ and $$g_*IC_y(Z')= \left(g_*\lambda_y(T_g^*)\right) \cap IC_y(Z)$$ for $Z$ and $Z'$ pure dimensional. If moreover $Z,Z'$ are compact, with $k: Z\to pt$ the constant proper map, then $$\chi_y(g^*[{{\mathcal M}}])=k_*g_*MHC_y(g^*[M])=\langle g_*\lambda_y(T_g^*), MHC_y([M])\rangle \:.$$ In particular $$\chi_y(Z')= \langle g_*\lambda_y(T_g^*), C_y(Z)\rangle \quad \text{and} \quad
I\chi_y(Z')= \langle g_*\lambda_y(T_g^*), IC_y(Z)\rangle \:.$$
The result of this corollary can also be seen form a different view point, by making the “going up and down" calculation already on the level of Grothendieck groups of mixed Hodge modules, where this time one only needs the assumption that $f: Z'\to Z$ is proper (to get the projection formula): $$f_*f^*[{{\mathcal M}}]=[f_*f^*{{\mathcal M}}]=[f_*({\mathbb{Q}}^H_{Z'}\otimes f^*{{\mathcal M}})] = [f_* {\mathbb{Q}}^H_{Z'}]\otimes [{{\mathcal M}}] \in K_0(MHM(Z))$$ for ${{\mathcal M}}\in D^bMHM(Z)$. The problem for a singular $Z$ is then, that we do not have a precise relation between $$[f_* {\mathbb{Q}}^H_{Z'}] \in K_0(MHM(Z)) \quad \text{and} \quad [Rf_* {\mathbb{Q}}_{Z'}]\in K_0(FmHs^p(Z))\:.$$
What is [*missing*]{} up to now is the right notion of a good variation (or family) of mixed Hodge structures on a [*singular*]{} complex algebraic variety $Z$! This class should at least contain:
1. The higher direct image local systems $R^i f_* {\mathbb{Q}}_{Z'}$ ($i\in {\mathbb{Z}}$) for a smooth and proper morphism $f: Z'\to Z$ of complex algebraic varieties.
2. The pullback $g^*{{\mathcal L}}$ of a good variation of mixed Hodge structures ${{\mathcal L}}$ on a smooth complex algebraic manifold $M$ under an algebraic morphism $g: Z\to M$.
At the moment we have to assume that $Z$ is smooth (and pure dimensional) so that one can use theorem \[MHM2\].
Nevertheless, in case (2) above we can already prove the following interesting result (compare with [@MSc]\[sec.4.1\] for a similar result for $MHT_{y*}$ in the case when $f$ is a closed embedding):
\[pairing-class\] Let $f: Z\to N$ be a morphism of complex algebraic varieties, with $N$ smooth and pure $n$-dimensional. Then one has a natural [*pairing*]{} $$f^*(\cdot) \cap (\cdot): K_0(VmHs^g(N)) \times K_0(MHM(Z)) \to K_0(MHM(Z))\:,$$ $$([{{\mathcal L}}],[{{\mathcal M}}])\mapsto [f^*\left({{\mathcal L}}^H\right)\otimes {{\mathcal M}}] \:.$$ Here ${{\mathcal L}}^H[m]$ is the smooth mixed Hodge module on $N$ with underlying perverse sheaf ${{\mathcal L}}[m]$. One also has a similar pairing on (co)homological level: $$f^*(\cdot) \cap (\cdot): K^0_{alg}(N)\otimes {\mathbb{Z}}[y^{\pm 1}] \times G_0(Z)\otimes {\mathbb{Z}}[y^{\pm 1}] \to G_0(Z)\otimes {\mathbb{Z}}[y^{\pm 1}] \:,$$ $$([{{\mathcal V}}]\cdot y^i,[{{\mathcal F}}]\cdot y^j)\mapsto [f^*({{\mathcal V}})\otimes {{\mathcal F}}]\cdot y^{i+j} \:.$$ And the motivic Chern class transformations $MHC^y$ and $MHC_y$ commute with these natural pairings: $$\begin{split}
MHC_y\left([f^*\left({{\mathcal L}}^H\right)\otimes {{\mathcal M}}]\right) &= MHC^y\left([f^*{{\mathcal L}}]\right) \cap MHC_y([{{\mathcal M}}])\\
&= f^*\left(MHC^y\left([{{\mathcal L}}]\right)\right) \cap MHC_y([{{\mathcal M}}])
\end{split}$$ for ${{\mathcal L}}\in VmHs^g(N)$ and ${{\mathcal M}}\in D^bMHM(Z)$.
For the proof we can once more assume ${{\mathcal M}}=g_*j_*{\mathbb{V}}$ for $g: \bar{M}\to Z$ proper, with $\bar{M}$ a pure-dimensional smooth complex algebraic manifold, $j: M\to \bar{M}$ a Zariski open inclusion with complement $D$ a normal crossing divisor with smooth irreducible components, and finally ${\mathbb{V}}$ a good variation of mixed Hodge structures on $M$. Using the projection formula, it is then enough to prove $$MHC_y\left([g^*f^*\left({{\mathcal L}}^H\right)\otimes j_*{\mathbb{V}}]\right) = MHC^y\left([g^*f^*{{\mathcal L}}]\right) \cap MHC_y([j_*{\mathbb{V}}]) \:.$$ But $g^*f^*{{\mathcal L}}$ is a good variation of mixed Hodge structures on $\bar{M}$, so that by example \[MHMopen\] and corollary \[Cor-j\*\](3) both sides are equal to $$\left(MHC^y(g^*f^*{{\mathcal L}}) \otimes MHC^y(j_*{\mathbb{V}})\right) \cap \left(\lambda_y\left( \Omega_{\bar{M}}^1(log(D))\right)\cap [{{\mathcal O}}_{\bar{M}}] \right)\:.$$
As an application of the very special case $f=id: Z\to N$ the identity of a complex algebraic manifold $Z$, with $$MHC_y([{\mathbb{Q}}^H_Z])= \lambda_y(T^*Z) \cap [{{\mathcal O}}_Z] \:,$$ one gets the Atiyah-Meyer type formula (\[MHCy-fomula\]) as well as the following result (cf. [@CLMS; @CLMS2; @MSc]):
Let $g: Z' \to Z$ be a proper morphism of complex algebraic varieties, with $Z$ smooth and connected. Assume that for a given ${{\mathcal M}}\in D^bMHM(Z')$ all direct image sheaves $$R^ig_*rat({{\mathcal M}})\quad \text{($i\in {\mathbb{Z}})$ are locally constant}\:,$$ e.g. $g$ is a locally trivial fibration and ${{\mathcal M}}={\mathbb{Q}}^H_{Z'}$ or ${{\mathcal M}}=IC^H_{Z'}$ (for $Z'$ pure dimensional), so that they all underlie a good variation of mixed Hodge structures. Then one can define $$[Rg_*rat({{\mathcal M}})]:=\sum_{i\in {\mathbb{Z}}}\; (-1)^i\cdot [R^ig_*rat({{\mathcal M}})] \in K_0(VmHs^g(Z))\:,$$ with $$\begin{split}
g_*MHC_y([{{\mathcal M}}])&= MHC_y(g_*[M])\\
&= MHC^y([Rg_*rat({{\mathcal M}})])\otimes
\left(\lambda_y(T^*Z) \cap [{{\mathcal O}}_Z]\right) \:.
\end{split}$$
As a final application we mention the
\[AM-IC\] Let $f: Z\to N$ be a morphism of complex algebraic varieties, with $N$ smooth and pure $n$-dimensional (e.g. a closed embedding). Assume also $Z$ is pure $m$-dimensional. Then one has for a good variation of mixed Hodge structures ${{\mathcal L}}$ on $N$ the equality $$IC^H_Z(f^*{{\mathcal L}})[-m] \simeq f^*{{\mathcal L}}^H \otimes IC^H_Z[-m] \in MHM(Z)[-m]\subset D^bMHM(Z)\:,$$ so that $$IC_y(Z;f^*{{\mathcal L}}) = MHC^y(f^*{{\mathcal L}}) \cap IC_y(Z) = f^*\left(MHC^y({{\mathcal L}})\right) \cap IC_y(Z) \:.$$ If in addition $Z$ is also compact, then one gets by pushing down to a point: $$I\chi_y(Z;f^*{{\mathcal L}})= \langle MHC^y(f^*{{\mathcal L}}), IC_y(Z)\rangle \:.$$
This example should be seen as a Hodge theoretical version of the corresponding result of Banagl-Cappell-Shaneson [@BCS] for the $L$-classes $L_*(IC_Z(L))$ of a selfdual [*Poincaré local system*]{} $L$ on all of $Z$. The special case of example \[AM-IC\] for $f$ a closed inclusion was already explained in [@MSc]\[sec.4.1\].
Finally note that all the results of this section can easily be applied to the (un)normalized [*motivic Hirzebruch class transformation*]{} $MHT_{y*}$ (and $MH\tilde{T}_{y*}$), because the [*Todd class transformation*]{} $td_*: G_0(\cdot) \to
H_*(\cdot)$ of Baum-Fulton-MacPherson [@BFM] has the following properties (compare also with [@Fu]\[chapter 18\] and [@FM]\[Part II\]):
Functoriality
: The Todd class transformation $td_*$ commutes with pushdown $f_*$ for a proper morphism $f: Z\to X$: $$td_*\left(f_*\left([{{\mathcal F}}]\right)\right)=
f_*\left(td_*\left([{{\mathcal F}}]\right)\right) \quad \text{for $[{{\mathcal F}}]\in G_0(Z)$.}$$
Multiplicativity for exterior products
: The Todd class transformation $td_*$ commutes with exterior products: $$td_*\left([{{\mathcal F}}\boxtimes {{\mathcal F}}']\right)=
td_*\left([{{\mathcal F}}]\right) \boxtimes td_*\left([{{\mathcal F}}']\right) \quad \text{for
$[{{\mathcal F}}]\in G_0(Z)$ and $[{{\mathcal F}}']\in G_0(Z')$.}$$
VRR for smooth pullbacks
: For a smooth morphism $g: Z'\to Z$ of complex algebraic varieties one has for the Todd class transformation $td_*$ the following Verdier Riemann-Roch formula: $$td^*(T_g) \cap
g^*td_*([{{\mathcal F}}])= td_*(g^*[{{\mathcal F}}])= td_*([g^*{{\mathcal F}}]) \quad \text{for $[{{\mathcal F}}]\in G_0(Z)$.}$$
Multiplicativity
: Let $ch^*: K^0_{alg}(\cdot)\to H^*(\cdot)\otimes {\mathbb{Q}}$ be the cohomological [*Chern character*]{} to the cohomolgy $H^*(\cdot)$ given by the operational Chow ring $CH^*(\cdot)$ or the usual cohomology $H^{2*}(\cdot,{\mathbb{Z}})$ in even degrees. Then one has the multiplicativity relation $$td_*([{{\mathcal V}}\otimes {{\mathcal F}}])= ch^*([{{\mathcal V}}]) \cap td_*([{{\mathcal F}}])$$ for $[{{\mathcal V}}]\in K^0_{alg}(Z)$ and $[{{\mathcal F}}]\in G_0(Z)$, with $Z$ a (possible singular) complex algebraic variety.
Characteristic classes and duality
----------------------------------
In this final section we explain the characteristic class version of the duality formula (\[chi-y-dual\]) for the $\chi_y$-genus. We also show that the specialization of $MHT_{y*}$ for $y=-1$ exists and is equal to the rationalized MacPherson Chern class $c_*$ of the underlying constructible sheaf complex. The starting point is the following result [@Sa0]\[sec.2.4.4\]:
Let $M$ be a pure $m$-dimensional complex algebraic manifold. Then one has for ${{\mathcal M}}\in D^bMHM(M)$ the [*duality*]{} result (for $j\in {\mathbb{Z}}$) $$\label{duality-classes1}
Gr_j^F(DR({{\mathcal D}}{{\mathcal M}})) \simeq {{\mathcal D}}\left(Gr_{-j}^FDR({{\mathcal M}})\right) \in D^b_{coh}(M) \:.$$ Here ${{\mathcal D}}$ on the left side is the duality of mixed Hodge modules, wheres ${{\mathcal D}}$ on the right hand side is the [*Grothendieck duality*]{} $${{\mathcal D}}=Rhom(\cdot, \omega_M[m]):\: D^b_{coh}(M) \to D^b_{coh}(M)\:,$$ with $\omega_M=\Omega^m_M$ the canonical sheaf of $M$.
A priori this is a duality for the corresponding analytic (cohomology) sheaves. Since ${{\mathcal M}}$ and $DR({{\mathcal M}})$ can be extended to smooth complex algebraic compactification $\bar{M}$, one can apply Serre’s GAGA to get the same result also for the underlying algebraic (cohomology) sheaves.
Let $Z$ be a complex algebraic variety with [*dualizing complex*]{} $\omega_Z^{\bullet}\in D^b_{coh}(Z)$, so that the [*Grothendieck duality transformation*]{} ${{\mathcal D}}=Rhom(\cdot ,\omega_Z^{\bullet})$ induces a duality involution $${{\mathcal D}}: G_0(Z)\to G_0(Z) \:.$$ Extend this to $G_0(Z)\otimes{\mathbb{Z}}[y^{\pm 1}]$ by $y\mapsto 1/y$. Then the motivic Hodge Chern class transformation $MHC_y$ commutes with duality ${{\mathcal D}}$: $$\label{duality-classes2}
MHC_y({{\mathcal D}}(\cdot))= {{\mathcal D}}(MHC_y(\cdot)): K_0(MHM(Z))\to G_0(Z)\otimes{\mathbb{Z}}[y^{\pm 1}] \:.$$
Note that for $Z=pt$ a point this reduces to the duality formula (\[chi-y-dual\]) for the $\chi_y$-genus. For dualizing complexes and (relative) Grothendieck duality we refer to [@Ha; @Con; @LH] as well as [@FM]\[Part I,sec. 7\]). Note that for $M$ smooth of pure dimension $m$, one has $$\omega_M[m] \simeq \omega_M^{\bullet} \in D^b_{coh}(M) \:.$$ Moreover, for a proper morphism $f: X\to Z$ of complex algebraic varieties one has the relative Grothendieck duality isomorphism $$Rf_*\left(Rhom({{\mathcal F}},\omega_X^{\bullet})\right) \simeq Rhom(Rf_*{{\mathcal F}},\omega_Z^{\bullet}) \quad \text{for ${{\mathcal F}}\in D^b_{coh}(X)$,}$$ so that the duality involution $${{\mathcal D}}: G_0(Z)\otimes{\mathbb{Z}}[y^{\pm 1}]\to G_0(Z)\otimes{\mathbb{Z}}[y^{\pm 1}]$$ commutes with proper push down. Since $K_0(MHM(Z))$ is generated by classes $f_*[{{\mathcal M}}]$, with $f: M\to Z$ proper morphism from a pure dimensional complex algebraic manifold $M$ (and ${{\mathcal M}}\in MHM(M)$), it is enough to prove (\[duality-classes2\]) in the case $Z=M$ a pure dimensional complex algebraic manifold, in which case it directly follows from Saito’s result (\[duality-classes1\]).\
For a systematic study of the behaviour of the Grothendieck duality transformation ${{\mathcal D}}: G_0(Z)\to G_0(Z)$ with respect to exterior products and smooth pullback, we refer e.g. to [@FL] and [@FM]\[Part I,sec. 7\], where a corresponding “bivariant” result is stated. Here we only point out that the dualities $(\cdot)^{\vee}$ and ${{\mathcal D}}$ commute with the [*pairings*]{} of corollary \[pairing-class\]: $$\begin{split}
f^*\left((\cdot)^{\vee}\right) &\cap ({{\mathcal D}}(\cdot))= {{\mathcal D}}\left( f^*(\cdot) \cap (\cdot) \right)
:\\ K^0_{alg}(N)\otimes {\mathbb{Z}}[y^{\pm 1}] &\times G_0(Z)\otimes {\mathbb{Z}}[y^{\pm 1}] \to G_0(Z)\otimes {\mathbb{Z}}[y^{\pm 1}] \:,
\end{split}$$ and similarly $$\begin{split}
f^*\left((\cdot)^{\vee}\right) &\cap ({{\mathcal D}}(\cdot))= {{\mathcal D}}\left( f^*(\cdot) \cap (\cdot) \right)
:\\ K_0(VmHs^g(N)) &\times K_0(MHM(Z)) \to K_0(MHM(Z))\:.
\end{split}$$
Here the last equality needs only be checked for classes $[IC_S({{\mathcal L}})]$, with $S\subset Z$ irreducible of dimension $d$ and ${{\mathcal L}}$ a good variation of pure Hodge structures on a Zariski dense open smooth subset $U$ of $S$, and ${\mathbb{V}}$ a good variation of pure Hodge structures on $N$. But then the claim follows from $$f^*({\mathbb{V}})\otimes IC_S({{\mathcal L}})\simeq IC_S(f^*({\mathbb{V}})|U\otimes {{\mathcal L}})$$ and (\[duality-IC\]) in the form $$\begin{aligned}
{\mathcal D}\left(IC_S(f^*(\mathbb{V})|U\otimes {\mathcal L})\right)
&\simeq IC_S\left( (f^*(\mathbb{V})|U\otimes {\mathcal L})^{\vee} \right) (d)\\
& \simeq IC_S\left( f^*(\mathbb{V}^{\vee})|U\otimes {\mathcal L}^{\vee} \right) (d) \:.\end{aligned}$$
Also the Todd class transformation $td_*: G_0(\cdot)\to H_*(\cdot)\otimes {\mathbb{Q}}$ commutes with duality (compare with [@Fu]\[ex.18.3.19\] and [@FM]\[Part I, cor.7.2.3\]), if the duality involution ${{\mathcal D}}: H_*(\cdot)\otimes {\mathbb{Q}}\to H_*(\cdot)\otimes {\mathbb{Q}}$ in homology is defined as ${{\mathcal D}}:= (-1)^i\cdot id$ on $H_i(\cdot)\otimes {\mathbb{Q}}$. So also the unnormalized Hirzebruch class transformation $MH\tilde{T}_{y*}$ commutes with duality, if this duality in homology is extended to $H_*(\cdot)\otimes {\mathbb{Q}}[y^{\pm 1}]$ by “$\;y\mapsto 1/y$".
As a final result of this paper, we have the
\[MHT=C\] Let $Z$ be a complex algebraic variety, with $[{{\mathcal M}}]\in K_0(MHM(Z))$. Then $$MHT_{y*}([{{\mathcal M}}])\in H_*(Z)\otimes{\mathbb{Q}}[y^{\pm 1}] \subset
H_*(Z)\otimes{\mathbb{Q}}[y^{\pm 1},(1+y)^{-1}]\:,$$ so that the specialization $MHT_{-1*}([{{\mathcal M}}])\in H_*(Z)\otimes{\mathbb{Q}}$ for $y=-1$ is defined. Then $$MHT_{-1*}([{{\mathcal M}}]) = c_*([rat({{\mathcal M}})]) =: c_*(\chi_{stalk}([rat({{\mathcal M}})]))
\in H_*(Z)\otimes{\mathbb{Q}}$$ is the rationalized MacPherson Chern class of the underlying constructible sheaf complex $rat({{\mathcal M}})$ (or the constructible function $\chi_{stalk}([rat({{\mathcal M}})])$). In particular $$MHT_{-1*}({{\mathcal D}}[{{\mathcal M}}]) = MHT_{-1*}([{{\mathcal D}}{{\mathcal M}}]) = MHT_{-1*}([{{\mathcal M}}]) \:.$$
Here $\chi_{stalk}$ is the transformation form the diagram (\[motfunct\]). Similarly, all the transformations from this diagram (\[motfunct\]), like $\chi_{stalk}$ and $rat$, commute with duality ${{\mathcal D}}$. This implies already the last claim, since ${{\mathcal D}}=id$ for algebraically constructible functions (compare [@Sc]\[sec.6.0.6\]). So we only need to prove the first part of the proposition. Since $MHT_{-1*}$ and $c_*$ both commute with proper push down, we can assume $[{{\mathcal M}}]=[j_*{\mathbb{V}}]$, with $Z=\bar{M}$ a smooth pure dimensional complex algebraic manifold, $j: M\to \bar{M}$ a Zariski open inclusion with complement $D$ a normal crossing divisor with smooth irreducible components, and ${\mathbb{V}}$ a good variation of mixed Hodge structures on $M$. So $$MH\tilde{T}_{y*}([j_*{\mathbb{V}}])= ch^*\left( MHC^y(Rj_*L)\right) \cap MH\tilde{T}_{y*}([j_*{\mathbb{Q}}_M])
\in H_*(\bar{M})\otimes{\mathbb{Q}}[y^{\pm 1}]$$ by (\[MHCj\*-fomula\]) and the [*multiplicativity*]{} of the Todd class transformation $td_*$. Introduce the [*twisted Chern character*]{} $$ch^{(1+y)}: K^0_{alg}(\cdot)\otimes {\mathbb{Q}}[y^{\pm 1}] \to H^*(\cdot)\otimes{\mathbb{Q}}[y^{\pm 1}]\;:
\:\:[{{\mathcal V}}]\cdot y^j\mapsto \sum_{i\geq 0}\; ch^i([{{\mathcal V}}])\cdot (1+y)^i \cdot y^j\:,$$ with $ch^i([{{\mathcal V}}])\in H^i(\cdot)\otimes{\mathbb{Q}}$ the $i$-th componenent of $ch^*$. Then one easily gets $$MHT_{y*}([j_*{\mathbb{V}}])= ch^{(1+y)}\left( MHC^y(Rj_*L)\right) \cap MHT_{y*}([j_*{\mathbb{Q}}_M])
\in H_*(\bar{M})\otimes{\mathbb{Q}}[y^{\pm 1},(1+y)^{-1}] \:.$$
But $[j_*{\mathbb{Q}}_M]=\chi_{Hdg}(j_*[id_M])$ is by (\[j-dual\]) in the image of $$\chi_{Hdg}: M_0(Var/\bar{M})= K_0(Var/\bar{M})[{\mathbb{L}}^{-1}] \to K_0(MHM(\bar{M})) \:.$$ So for $MHT_{y*}([j_*{\mathbb{Q}}_M])$ we can apply the following special case of proposition \[MHT=C\]:
\[T=c\] The transformation $$T_{y*}=MHT_{y*}\circ \chi_{Hdg}: M_0(Var/Z) \to H_*(Z)\otimes{\mathbb{Q}}[y^{\pm 1},(1+y)^{-1}]$$ takes values in $H_*(Z)\otimes{\mathbb{Q}}[y^{\pm 1}] \subset H_*(Z)\otimes{\mathbb{Q}}[y^{\pm 1},(1+y)^{-1}]$, with $$T_{-1*}=T_{-1*}\circ {{\mathcal D}}= c_*\circ can: M_0(Var/Z) \to H_*(Z)\otimes{\mathbb{Q}}\:.$$
Assuming this lemma, we get from the following commutative diagram, that the specialization $MHT_{-1*}([j_*{\mathbb{V}}])$ for $y=-1$ exists: $$\begin{CD}
H^*(\cdot)\otimes{\mathbb{Q}}[y^{\pm 1}] \times H_*(\cdot)\otimes{\mathbb{Q}}[y^{\pm 1},(1+y)^{-1}]
@> \cap >> H_*(\cdot)\otimes{\mathbb{Q}}[y^{\pm 1},(1+y)^{-1}]\\
@A incl. AA @AA incl. A\\
H^*(\cdot)\otimes{\mathbb{Q}}[y^{\pm 1}] \times H_*(\cdot)\otimes{\mathbb{Q}}[y^{\pm 1}]
@> \cap >> H_*(\cdot)\otimes{\mathbb{Q}}[y^{\pm 1}]\\
@V y=-1 VV @VV y=-1 V\\
H^*(\cdot)\otimes{\mathbb{Q}}\times H_*(\cdot)\otimes{\mathbb{Q}}@> \cap >> H_*(\cdot)\otimes{\mathbb{Q}}\:.
\end{CD}$$
Moreover $ch^{(1+y)}\left( MHC^y(Rj_*L)\right)$ specializes for $y=-1$ just to $$rk(L)=ch^0([\;\overline{{{\mathcal L}}}\;]) \in H^0(\bar{M})\otimes {\mathbb{Q}}\:,$$ with $rk(L)$ the rank of the local system $L$ on $M$. So we get $$MHT_{-1*}([j_*{\mathbb{Q}}_M])=rk(L)\cdot c_*(j_*1_M)= c_*(rk(L)\cdot j_*1_M) \in H_*(\bar{M})\otimes{\mathbb{Q}}\:,$$ with $rk(L)\cdot j_*1_M =\chi_{stalk}(rat([j_*{\mathbb{V}}]))$.\
It remains to prove the lemma \[T=c\]. But all transformations $T_{y*}, {{\mathcal D}}, c_*$ and $can$ commute with pushdown for proper maps. Moreover, by resolution of singularities and additivity, $M_0(Var/Z)$ is generated by classes $[f: N\to Z]\cdot {\mathbb{L}}^k$ ($k\in {\mathbb{Z}}$), with $N$ smooth pure $n$-dimensional and $f$ proper. So it is enough to prove that $T_{y*}([id_N]\cdot {\mathbb{L}}^k)\in H_*(N)\otimes{\mathbb{Q}}[y^{\pm 1}]$, with $$T_{y*}([id_N]\cdot {\mathbb{L}}^k)= T_{y*}\left({{\mathcal D}}([id_N]\cdot {\mathbb{L}}^k)\right) = c_*\left(can([id_N]\cdot {\mathbb{L}}^k)\right) \:.$$
But by the [*normalization condition*]{} for our characteristic class transformations one has (compare [@BSY]): $$T_{y*}([id_N])= T_y^*(TN)\cap [N] \in H_*(N)\otimes{\mathbb{Q}}[y] \:,$$ with $T_{-1*}([id_N])= c^*(TN)\cap [N]= c_*(1_N)$. Similarly $$T_{y*}([{\mathbb{L}}])=\chi_y([{\mathbb{Q}}(-1)])=-y \quad \text{ and} \quad can([{\mathbb{L}}])=1_{pt}\:,$$ so that $$T_{y*}([id_N]\cdot {\mathbb{L}}^k)\in H_*(N)\otimes{\mathbb{Q}}[y^{\pm 1}]$$ by the multiplicativity of $MHT_{y*}$ for exterior products (with a point space). Moreover $$T_{-1*}([id_N]\cdot {\mathbb{L}}^k)= c_*(1_N)=c_*\left( can([id_N]\cdot {\mathbb{L}}^k)\right) \:.$$
Finally ${{\mathcal D}}([id_N]\cdot {\mathbb{L}}^k)= [id_N]\cdot {\mathbb{L}}^{k-n}$ by definition of ${{\mathcal D}}$, so that $$T_{-1*}([id_N]\cdot {\mathbb{L}}^k)= T_{-1*}\left({{\mathcal D}}([id_N]\cdot {\mathbb{L}}^k)\right) \:.$$
Acknowledgements {#acknowledgements .unnumbered}
================
This paper is an extended version of an expository talk given at the workshop “Topology of stratified spaces” at MSRI Berkeley in September 2008. Here I would like to thank the organizers (G. Friedman, E. Hunsicker, A. Libgober and L. Maxim) for the invitation to this workshop. I also would like to thank S. Cappell, L. Maxim and S. Yokura for some discussions on the subject of this paper.
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|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'By analyzing the data sets of 17.3, 6.5 and 1.0 pb$^{-1}$ taken, respectively, at $\sqrt s= 3.773$, 3.650 and 3.6648 GeV with the BES-II detector at the BEPC collider, we measure the observed cross sections for $e^+e^-\to \pi^+\pi^-\pi^0\pi^0$, $K^+K^-\pi^0\pi^0$, $2(\pi^+\pi^-\pi^0)$, $K^+K^-\pi^+\pi^-\pi^0\pi^0$ and $3(\pi^+\pi^-)\pi^0\pi^0$ at the three energy points. Based on these cross sections we set the upper limits on the observed cross sections and the branching fractions for $\psi(3770)$ decay into these final states at 90% C.L..'
author:
- |
M. Ablikim$^{1}$, J. Z. Bai$^{1}$, Y. Bai$^{1}$, Y. Ban$^{11}$, X. Cai$^{1}$, H. F. Chen$^{15}$, H. S. Chen$^{1}$, H. X. Chen$^{1}$, J. C. Chen$^{1}$, Jin Chen$^{1}$, X. D. Chen$^{5}$, Y. B. Chen$^{1}$, Y. P. Chu$^{1}$, Y. S. Dai$^{17}$, Z. Y. Deng$^{1}$, S. X. Du$^{1}$, J. Fang$^{1}$, C. D. Fu$^{14}$, C. S. Gao$^{1}$, Y. N. Gao$^{14}$, S. D. Gu$^{1}$, Y. T. Gu$^{4}$, Y. N. Guo$^{1}$, K. L. He$^{1}$, M. He$^{12}$, Y. K. Heng$^{1}$, J. Hou$^{10}$, H. M. Hu$^{1}$, T. Hu$^{1}$, G. S. Huang$^{1}$$^{a}$, X. T. Huang$^{12}$, Y. P. Huang$^{1}$, X. B. Ji$^{1}$, X. S. Jiang$^{1}$, J. B. Jiao$^{12}$, D. P. Jin$^{1}$, S. Jin$^{1}$, Y. F. Lai$^{1}$, H. B. Li$^{1}$, J. Li$^{1}$, L. Li$^{1}$, R. Y. Li$^{1}$, W. D. Li$^{1}$, W. G. Li$^{1}$, X. L. Li$^{1}$, X. N. Li$^{1}$, X. Q. Li$^{10}$, Y. F. Liang$^{13}$, H. B. Liao$^{1}$$^{b}$, B. J. Liu$^{1}$, C. X. Liu$^{1}$, Fang Liu$^{1}$, Feng Liu$^{6}$, H. H. Liu$^{1}$$^{c}$, H. M. Liu$^{1}$, J. B. Liu$^{1}$$^{d}$, J. P. Liu$^{16}$, H. B. Liu$^{4}$, J. Liu$^{1}$, R. G. Liu$^{1}$, S. Liu$^{8}$, Z. A. Liu$^{1}$, F. Lu$^{1}$, G. R. Lu$^{5}$, J. G. Lu$^{1}$, C. L. Luo$^{9}$, F. C. Ma$^{8}$, H. L. Ma$^{1}$, L. L. Ma$^{1}$$^{e}$, Q. M. Ma$^{1}$, M. Q. A. Malik$^{1}$, Z. P. Mao$^{1}$, X. H. Mo$^{1}$, J. Nie$^{1}$, R. G. Ping$^{1}$, N. D. Qi$^{1}$, H. Qin$^{1}$, J. F. Qiu$^{1}$, G. Rong$^{1}$, X. D. Ruan$^{4}$, L. Y. Shan$^{1}$, L. Shang$^{1}$, D. L. Shen$^{1}$, X. Y. Shen$^{1}$, H. Y. Sheng$^{1}$, H. S. Sun$^{1}$, S. S. Sun$^{1}$, Y. Z. Sun$^{1}$, Z. J. Sun$^{1}$, X. Tang$^{1}$, J. P. Tian$^{14}$, G. L. Tong$^{1}$, X. Wan$^{1}$, L. Wang$^{1}$, L. L. Wang$^{1}$, L. S. Wang$^{1}$, P. Wang$^{1}$, P. L. Wang$^{1}$, W. F. Wang$^{1}$$^{f}$, Y. F. Wang$^{1}$, Z. Wang$^{1}$, Z. Y. Wang$^{1}$, C. L. Wei$^{1}$, D. H. Wei$^{3}$, Y. Weng$^{1}$, N. Wu$^{1}$, X. M. Xia$^{1}$, X. X. Xie$^{1}$, G. F. Xu$^{1}$, X. P. Xu$^{6}$, Y. Xu$^{10}$, M. L. Yan$^{15}$, H. X. Yang$^{1}$, M. Yang$^{1}$, Y. X. Yang$^{3}$, M. H. Ye$^{2}$, Y. X. Ye$^{15}$, C. X. Yu$^{10}$, G. W. Yu$^{1}$, C. Z. Yuan$^{1}$, Y. Yuan$^{1}$, S. L. Zang$^{1}$$^{g}$, Y. Zeng$^{7}$, B. X. Zhang$^{1}$, B. Y. Zhang$^{1}$, C. C. Zhang$^{1}$, D. H. Zhang$^{1}$, H. Q. Zhang$^{1}$, H. Y. Zhang$^{1}$, J. W. Zhang$^{1}$, J. Y. Zhang$^{1}$, X. Y. Zhang$^{12}$, Y. Y. Zhang$^{13}$, Z. X. Zhang$^{11}$, Z. P. Zhang$^{15}$, D. X. Zhao$^{1}$, J. W. Zhao$^{1}$, M. G. Zhao$^{1}$, P. P. Zhao$^{1}$, B. Zheng$^{1}$, H. Q. Zheng$^{11}$, J. P. Zheng$^{1}$, Z. P. Zheng$^{1}$, B. Zhong$^{9}$ L. Zhou$^{1}$, K. J. Zhu$^{1}$, Q. M. Zhu$^{1}$, X. W. Zhu$^{1}$, Y. C. Zhu$^{1}$, Y. S. Zhu$^{1}$, Z. A. Zhu$^{1}$, Z. L. Zhu$^{3}$, B. A. Zhuang$^{1}$, B. S. Zou$^{1}$\
(BES Collaboration)\
title: ' Measurements of the observed cross sections for $e^+e^-\rightarrow$ exclusive light hadrons containing $\pi^0\pi^0$ at $\sqrt s= 3.773$, 3.650 and 3.6648 GeV'
---
-0.2cm -0.2cm
Introduction
============
In the past thirty years, it is expected that almost all of the $\psi(3770)$ decay into $D\bar D$ meson pairs. However, the earlier published data [@pdg04] show that the $\psi(3770)$ production cross section exceeds the $D\bar D$ production cross section by about 38% [@hepex_0506051]. Recently, CLEO Collaboration measured the cross section for $e^+e^-\to\psi(3770)\to$ non-$D\bar
D$ to be $(-0.01\pm0.08^{+0.41}_{-0.30})$ nb [@prl96_092002]. However, by analyzing different data samples with different methods, BES Collaboration extracted the branching fraction for $\psi(3770)\to$ non$-D\bar D$ decay to be $(15\pm5)\%$ [@plb641_145; @prl97_121801; @plb659_74; @prd76_000000; @pdg07], which means that there may exist substantial non-$D\bar D$ final states in the $\psi(3770)$ decays, or there are some new structure or physics effects which may partially be responsible for the large ${\rm
non-}D\bar{D}$ branching fraction of the $\psi(3770)$ decays measured by the BES collaboration [@prl101_102004; @plb668_263].
BES Collaboration found the first ${\rm non-}D\bar{D}$ decay mode of $\psi(3770)$, i.e. $\psi(3770)\rightarrow J/\psi\pi^+\pi^-$ for the first time, and extracted the branching fraction for $\psi(3770)\to
J/\psi\pi^+\pi^-$ to be $(0.34\pm0.14\pm0.09)\%$ [@hepnp28_325; @plb605_63]. Latter, CLEO Collaboration confirmed the BES observation [@prl96_082004] and observed more $\psi(3770)$ exclusive non-$D\bar D$ decays, $\psi(3770)\to
J/\psi\pi^0\pi^0$, $J/\psi\pi^0$, $J/\psi\eta$ [@prl96_082004], $\gamma\chi_{cJ}(J=0,1,2)$ [@prl96_182002; @prd74_031106] and $\phi\eta$ [@prd74_012005], etc. However, the sum of these measured branching fractions for $\psi(3770)\to$ exclusive non-$D\bar D$ decays is not more than 2%. BES and CLEO Collaborations have also made many efforts to search for $\psi(3770)$ exclusive charmless decays by comparing the cross sections for $e^+e^-\to$ exclusive light hadrons measured at the peak of $\psi(3770)$ and at some continuum energy points [@prd70_077101; @prd72_072007; @plb650_111; @plb656_30; @epjc52_805] [@prd74_012005; @prl96_032003; @prd73_012002].
In this Letter, we report another effort to search for $\psi(3770)$ exclusive charmless decays, which are performed by studying some processes containing $\pi^0\pi^0$ mesons in the final states with the same method as the one used in Refs. [@plb650_111; @plb656_30; @epjc52_805]. We measure the observed cross sections for $e^+e^-\to \pi^+\pi^-\pi^0\pi^0$, $K^+K^-\pi^0\pi^0$, $2(\pi^+\pi^-\pi^0)$, $K^+K^-\pi^+\pi^-\pi^0\pi^0$ and $3(\pi^+\pi^-)\pi^0\pi^0$ at $\sqrt
s = 3.773$, 3.650 and 3.6648 GeV. Then, we set the upper limits on the observed cross sections and the branching fractions for $\psi(3770)$ decay into these final states at 90% C.L.. The data used in the analysis were taken at the center-of-mass energies $\sqrt{s} = 3.773$, 3.650 and 3.6648 GeV with the BES-II detector at the BEPC collider, corresponding to the integrated luminosities of 17.3, 6.5 and 1.0 pb$^{-1}$, respectively. In the Letter, we call, respectively, the three data sets as the $\psi(3770)$ resonance data, the continuum data 1 and the continuum data 2.
BES-II detector
===============
The BES-II is a conventional cylindrical magnetic detector [@nima344_319; @nima458_627] operated at the Beijing Electron-Positron Collider (BEPC). A 12-layer vertex chamber (VC) surrounding the beryllium beam pipe provides input to the event trigger, as well as coordinate information. A forty-layer main drift chamber (MDC) located just outside the VC yields precise measurements of charged particle trajectories with a solid angle coverage of $85\%$ of 4$\pi$; it also provides ionization energy loss ($dE/dx$) measurements which are used for particle identification. Momentum resolution of $1.7\%\sqrt{1+p^2}$ ($p$ in GeV/$c$) and $dE/dx$ resolution of $8.5\%$ for Bhabha scattering electrons are obtained for the data taken at $\sqrt s= 3.773$ GeV. An array of 48 scintillation counters surrounding the MDC measures the time of flight (TOF) of charged particles with a resolution of about 180 ps for electrons. Outside the TOF, a 12 radiation length, lead-gas barrel shower counter (BSC), operating in limited streamer mode, measures the energies of electrons and photons over $80\%$ of the total solid angle with an energy resolution of $\sigma_E/E=0.22/\sqrt{E}$ ($E$ in GeV) and spatial resolutions of $\sigma_{\phi}=7.9$ mrad and $\sigma_z=2.3$ cm for electrons. A solenoidal magnet outside the BSC provides a 0.4 T magnetic field in the central tracking region of the detector. Three double-layer muon counters instrument the magnet flux return and serve to identify muons with momentum greater than 0.5 GeV/c. They cover $68\%$ of the total solid angle.
Event selection {#evtsel}
===============
The exclusive light hadron final states of $\pi^+\pi^-\pi^0\pi^0$, $K^+K^-\pi^0\pi^0$, $2(\pi^+\pi^-\pi^0)$, $K^+K^-\pi^+\pi^-\pi^0\pi^0$ and $3(\pi^+\pi^-)\pi^0\pi^0$ are studied by examining the different photon combinations from $\pi^+\pi^-\gamma_1\gamma_2\gamma_3\gamma_4$, $K^+K^-\gamma_1\gamma_2\gamma_3\gamma_4$, $2(\pi^+\pi^-)\gamma_1\gamma_2\gamma_3\gamma_4$, $K^+K^-\pi^+\pi^-\gamma_1\gamma_2\gamma_3\gamma_4$ and $3(\pi^+\pi^-)\gamma_1\gamma_2\gamma_3\gamma_4$, respectively. The $\pi^0$ mesons are reconstructed through the decay $\pi^0\to\gamma\gamma$.
For each candidate event, it is required that there are at least two charged tracks to be well reconstructed in the MDC with good helix fits and at least four neutral tracks to be well reconstructed in the BSC. The charged track is required to have the polar angle satisfying $|\cos\theta|<0.85$, and originate from the interaction region $V_{xy}<2.0$ cm and $|V_{z}|<20.0$ cm. Here, $V_{xy}$ and $|V_{z}|$ are the closest approaches of the charged track in the $xy$ plane and in the $z$ direction.
The charged particles are identified by using the $dE/dx$ and TOF measurements, with which the combined confidence levels $CL_{\pi}$ and $CL_{K}$ for pion and kaon hypotheses are calculated. The candidate charged tracks satisfying $CL_{\pi}>$0.001 and $CL_{K}>CL_{\pi}$ are identified as pion and kaon, respectively.
The good photons are selected with the BSC measurements. They are required to satisfy the following criteria: the energy deposited in the BSC is greater than 50 MeV, the electromagnetic shower starts in the first 5 layers, the angle between the photon and the nearest charged track is greater than 22$^\circ$ [@plb597_39] and the opening angle between the cluster development direction and the photon emission direction is less than 37$^\circ$ [@plb597_39].
For each selected candidate event, there may be several different charged or neutral track combinations satisfying the above selection criteria for exclusive light hadron final states. Each combination is subjected to an energy-momentum conservative kinematic fit. Candidates with a fit probability larger than 1$\%$ are accepted. If more than one combination satisfies the selection criteria in an event, only the combination with the largest fit probability is retained.
At center-of-mass energy of 3.773 GeV, the events of $\psi(2S)\rightarrow(\gamma)J/\psi\pi^0\pi^0$, with $J/\psi\to
e^+e^-$ or $J/\psi\to\mu^+\mu^-$ may be misidentified as the $\pi^+\pi^-\pi^0\pi^0$ final state due to misidentifying $e^+e^-$ or $\mu^+\mu^-$ pair as $\pi^+\pi^-$ pair. We suppress these events by requiring the invariant mass of $\pi^+\pi^-$ combination from each selected $\pi^+\pi^-\pi^0\pi^0$ candidate event to be less than 3.0 GeV/c$^2$.
The $K^+K^-\pi^+\pi^-\pi^0\pi^0$ final state suffers more contaminations from $D\bar D$ decays than the other four modes. To eliminate these contaminations, we exclude the events from $D\bar D$ decays by rejecting those in which the $D$ and $\bar D$ mesons can be reconstructed in the decay modes of $D^0\to K^-\pi^+\pi^0$ and $\bar{D^0}\to K^+\pi^-\pi^0$ [@npb727_395]. The residual contaminations from the other $D$ meson decays are further removed based on Monte Carlo simulation.
Data Analysis {#sec:data_analysis}
=============
For each $\gamma_1\gamma_2\gamma_3\gamma_4$ combination satisfying the above selection criteria, there are three possible photon combinations (\[$\pi^0_{\gamma_1\gamma_2}\pi^0_{\gamma_3\gamma_4}$\], \[$\pi^0_{\gamma_1\gamma_3}\pi^0_{\gamma_2\gamma_4}$\],\[$\pi^0_{\gamma_1\gamma_4}\pi^0_{\gamma_2\gamma_3}$\]) which may be reconstructed as $\pi^0\pi^0$ meson pairs. We calculate the invariant masses for the $\gamma_i\gamma_j$ and $\gamma_{i^\prime}\gamma_{j^{\prime}}$ combinations, $M_{\gamma_i\gamma_j}$ and $M_{\gamma_{i^{\prime}}\gamma_{j^{\prime}}}$($i$, $j$, $i^{\prime}$, $j^{\prime}$ = 1, or 2, or 3, or 4; $i < j$, $i^{\prime} <
j^{\prime}$, $i < i^{\prime}$ and $j < j^{\prime}$) with the fitted momentum vectors from the kinematic fit. Figure \[fig:dot\] shows the scatter plot for $M_{\gamma_{i^{\prime}}\gamma_{j^{\prime}}}$ versus $M_{\gamma_i\gamma_j}$ of the candidates for the $\pi^+\pi^-\pi^0\pi^0$, $K^+K^-\pi^0\pi^0$, $2(\pi^+\pi^-\pi^0)$, $K^+K^-\pi^+\pi^-\pi^0\pi^0$ and $3(\pi^+\pi^-)\pi^0\pi^0$ final states selected from the data including the three possible combinations.
In Fig. \[fig:dot\], the cluster around the $\pi^0\pi^0$ signal region indicates the production of the process containing $\pi^0\pi^0$ mesons in the final state. In the following analysis, the mass window of $\pm3\sigma_{M_{\gamma\gamma}}$($\pm$60 MeV/$c^2$) around the $\pi^0$ nominal mass is taken as the $\pi^0$ signal region, where $\sigma_{M_{\gamma\gamma}}$ is the $\pi^0$ mass resolution determined by Monte Carlo simulation. In each figure, projecting the events with $M_{\gamma_{i^{\prime}}\gamma_{j^{\prime}}}$ in the $\pi^0$ signal region onto $M_{\gamma_{i}\gamma_{j}}$ axis, we obtain the distribution of the $\gamma_i\gamma_j$ invariant masses for each process as shown in Fig. \[fig:inv\]. Fitting each of the $\gamma_i\gamma_j$ invariant mass spectra in Fig. \[fig:inv\] with a Gaussian function for the $\pi^0$ signal and a polynomial for the background yields the number $N_{\rm obs}^{\pi^0_{\gamma_i\gamma_j}}$ of the events for each process observed from each of the data sets. However, in each event, there may be more than one combination entering the $\gamma_i\gamma_j$ invariant mass spectra. The number $N^{\rm rc}$ of the repeated counting events in the observed number $N_{\rm obs}^{\pi^0_{\gamma_i\gamma_j}}$ of $\pi^0$s from the fit is subtracted from the number of $N_{\rm obs}^{\pi^0_{\gamma_i\gamma_j}}$. The number of $N^{\rm rc}$ is accounted via the number of $N_{\rm obs}^{\pi^0_{\gamma_i\gamma_j}}$, the number of the repeated counting events and the number of all the events in the $\pi^0$ signal region, as well as the repeated counting rate of the combinatorial background events which is estimated with the events in sideband region.
![The scatter plots of $M_{\gamma_{i^{\prime}}\gamma_{j^{\prime}}}$ versus $M_{\gamma_i\gamma_j}$ of the candidates for $e^+e^-\to$ (a) $\pi^+\pi^-\pi^0\pi^0$, (b) $K^+K^-\pi^0\pi^0$, (c) $2(\pi^+\pi^-\pi^0)$, (d) $K^+K^-\pi^+\pi^-\pi^0\pi^0$ and (e) $3(\pi^+\pi^-)\pi^0\pi^0$ selected from the $\psi(3770)$ resonance data (left), the continuum data 1 (middle) and the continuum data 2 (right).[]{data-label="fig:dot"}](fig1.eps "fig:"){width="8cm"} (-145,-15) [**$M_{\gamma_{i}\gamma_{j}}$(GeV/$c^2$)**]{} (-245,70) (-180,205) [**(a)**]{} (-100,205) [**(${\rm a^{\prime}}$)**]{} (-23,205) [**(${\rm a^{\prime\prime}}$)**]{} (-180,162) [**(b)**]{} (-100,162) [**(${\rm b^{\prime}}$)**]{} (-23,162) [**(${\rm b^{\prime\prime}}$)**]{} (-180,120) [**(c)**]{} (-100,120) [**(${\rm c^{\prime}}$)**]{} (-23,120) [**(${\rm c^{\prime\prime}}$)**]{} (-180,77) [**(d)**]{} (-100,77) [**(${\rm d^{\prime}}$)**]{} (-23,77) [**(${\rm d^{\prime\prime}}$)**]{} (-180,37) [**(e)**]{} (-100,37) [**(${\rm e^{\prime}}$)**]{} (-23,37) [**(${\rm e^{\prime\prime}}$)**]{}
![ The distributions of the invariant masses for the $\gamma_i\gamma_j$ combinations from the candidates for $e^+e^-\to$ (a) $\pi^+\pi^-\pi^0\pi^0$, (b) $K^+K^-\pi^0\pi^0$, (c) $2(\pi^+\pi^-\pi^0)$, (d) $K^+K^-\pi^+\pi^-\pi^0\pi^0$ and (e) $3(\pi^+\pi^-)\pi^0\pi^0$ from the $\psi(3770)$ resonance data (left), the continuum data 1 (middle) and the continuum data 2 (right), where the pairs of arrows show the $\pi^0$ signal region.[]{data-label="fig:inv"}](fig2.eps "fig:"){width="8cm"} (-145,-15)[**$M_{\gamma_{i}\gamma_{j}}$(GeV/$c^2$)**]{} (-245,50) (-176,200) [**(a)**]{} (-102,200) [**(${\rm
a^{\prime}}$)**]{} (-28,200) [**(${\rm
a^{\prime\prime}}$)**]{} (-176,159) [**(b)**]{} (-102,159) [**(${\rm b^{\prime}}$)**]{} (-28,159) [**(${\rm b^{\prime\prime}}$)**]{} (-176,119) [**(c)**]{} (-102,119) [**(${\rm
c^{\prime}}$)**]{} (-28,119) [**(${\rm
c^{\prime\prime}}$)**]{} (-176,77) [**(d)**]{} (-102,77) [**(${\rm d^{\prime}}$)**]{} (-28,77) [**(${\rm d^{\prime\prime}}$)**]{} (-176,38) [**(e)**]{} (-102,38) [**(${\rm
e^{\prime}}$)**]{} (-28,38) [**(${\rm
e^{\prime\prime}}$)**]{}
Using the number of events observed in a sideband region which is the mass window of $\pm60$ MeV/$c^2$ around 0.335 GeV$/c^2$ in the $\gamma_i^{\prime}\gamma_j^{\prime}$ invariant mass spectra, we can estimate the contributions of the combinatorial $\gamma_{i^\prime}\gamma_{j^\prime}$ background in the $\pi^0_{\gamma_{i^\prime}\gamma_{j^\prime}}$ signal region. We obtain the number of these contributions, $N^{\pi^0_{\gamma_i\gamma_j}}_{\rm sid}$, by normalizing the fitted number of $\pi^0$s observed in the sideband region. Here, the normalization factor is the ratio of the number of the background events in the signal region over the number of the background events in the sideband region.
However, there are only a few events in Fig. \[fig:inv\](${\rm
b^{\prime\prime}}$), (${\rm d^{\prime\prime}}$) and (${\rm
e^{\prime\prime}}$). Counting the events with $M_{\gamma\gamma}$ within the $\pi^0$ signal region, we obtain 1, 2 and 3 candidate events for $e^+e^-\to K^+K^-\pi^0\pi^0$, $K^+K^-\pi^+\pi^-\pi^0\pi^0$ and $3(\pi^+\pi^-)\pi^0\pi^0$ from the continuum data 2, respectively. In this case, we set the upper limit $N^{\rm up}$ on the number of the signal events at 90% confidence level (C.L.). Here, we use the Feldman-Cousins method [@prd57_3873] and ignore the background.
Other background subtraction
============================
In Section \[sec:data\_analysis\], we have considered the combinatorial $\gamma\gamma$ background in the $\pi^0\pi^0$ reconstruction. However, there are still some other kinds of backgrounds from $J/\psi$ and $\psi(3686)$ decays due to ISR returns, from the other final states due to the misidentification between charged pion and kaon, from the decays of $\psi(3770) \to
J/\psi\pp$, $J/\psi\pi^0\pi^0$, $J/\psi\pi^0$, $\gamma \chi_{cJ}
\hspace{0.1cm}(J=0,1,2)$, and from $D\bar D$ decays. In order to directly compare the cross sections for $e^+e^-\to f$ ($f$ represents exclusive light hadron final state) measured at the peak of $\psi(3770)$ and at some continuum energy points, we need to remove these background events. The method of background subtraction has been described in detail in Ref. [@plb650_111]. Monte Carlo study shows that the contaminations from the decays of $\psi(3770)
\to J/\psi\pp$, $J/\psi\pi^0\pi^0$, $J/\psi\pi^0$ and $\psi(3770)
\to\gamma \chi_{cJ}$ can be neglected [@plb650_111] and the contaminations from the final states with an extra photon can be ignored.
For the process of $K^+K^-\pi^+\pi^-\pi^0\pi^0$, even though we have removed the main contaminations from $D\bar D$ decays in the previous event selection (see section \[evtsel\]), there are still some residual contaminations from $D\bar D$ decays, which can satisfy the selection criteria for the $K^+K^-\pi^+\pi^-\pi^0\pi^0$ final state. The number of these residual contaminations are further removed based on Monte Carlo simulation.
Since we don’t know the details about the resonance and the continuum amplitudes, we neglect the possible interference between them in the following analysis. In this case, by subtracting $N^{\rm
rc}$, $N^{\pi^0_{\gamma_i\gamma_j}}_{\rm sid}$ and the number $N^{\rm b}$ of these contaminations from the number $N^{\pi^0_{\gamma_i\gamma_j}}_{\rm obs}$ of the candidate events, we obtain the net numbers $N^{\rm net}$ of the signal events for each process. The numbers of $N^{\pi^0_{\gamma_i\gamma_j}}_{\rm obs}$, $N^{\rm rc}$, $N^{\pi^0_{\gamma_i\gamma_j}}_{\rm sid}$, $N^{\rm b}$ and $N^{\rm net}$ (or $N^{\rm up}$) are summarized in Tabs. \[tab:crs3773\], \[tab:crs3650\] and \[tab:crs36648\].
Results
=======
Monte Carlo efficiency
----------------------
We estimate the detection efficiency $\epsilon$ for $e^+e^-\to f$ by using a phase space generator including initial state radiation and vacuum polarization corrections [@yf41_377] with $1/s$ cross section energy dependence. Final state radiation [@cpc79_291] decreases the detection efficiency not more than 0.5%. The Monte Carlo events are generated based on the Monte Carlo simulation for the BES-II detector [@nima552_344]. By analyzing these Monte Carlo events, we obtain the detection efficiency for each process at each center-of-mass energy. They are summarized in the fifth columns of Tabs. \[tab:crs3773\], \[tab:crs3650\] and \[tab:crs36648\]. The detection efficiencies listed in the tables do not include the branching fraction for $\pi^0\to\gamma\gamma$.
Observed cross sections
-----------------------
The observed cross section for $e^+e^- \to f$ is obtained by dividing the number $N^{\rm net}$ of the signal events by the integrated luminosity $\mathcal{L}$ of the data set, the detection efficiency $\epsilon$ and the square of the branching fraction ${\mathcal B}^2(\pi^0\to\gamma\gamma)$ for $\pi^0\to\gamma\gamma$, $$\sigma_{e^+e^-\to f} = \frac{N^{\rm net}} {\mathcal{L}
\times \epsilon
\times {\mathcal B}^2(\pi^0\to\gamma\gamma)
}.
\label{eq:crs}$$ Inserting the numbers of $N^{\rm net}$, $\mathcal{L}$, $\epsilon$ and ${\mathcal B}(\pi^0\to\gamma\gamma)$ in Eq. (\[eq:crs\]), we obtain $\sigma_{e^+e^-\to f}$ for each process at $\sqrt s= 3.773$, 3.650 and 3.6648 GeV, respectively. They are summarized in the last columns of Tabs. \[tab:crs3773\], \[tab:crs3650\] and \[tab:crs36648\], where the first error is statistical and the second systematic. The systematic error in the cross section measurement arises mainly from the uncertainties in integrated luminosity of the data set ($\sim2.1\%$ [@plb641_145; @prl97_121801]), photon selection ($\sim2.0\%$, per photon), tracking efficiency ($\sim2.0\%$ per track), particle identification ($\sim0.5\%$ per pion or kaon), kinematic fit ($\sim1.5\%$), Monte Carlo statistics ($\sim(3.3\sim5.7)\%$), branching fraction quoted from PDG ($\sim0.03\%$ for $B(\pi^0\to
\gamma\gamma)$), background subtraction ($\sim(0.0\sim6.1)\%$), fitting to the mass spectra ($\sim(2.4\sim9.5)\%$), and Monte Carlo modeling ($\sim6.0\%$). Adding these uncertainties in quadrature yields the total systematic error $\Delta_{\rm sys}$ for each process at each center-of-mass energy.
Upper limits on the observed cross sections
-------------------------------------------
The upper limits on the observed cross sections for $e^+e^-\to
K^+K^-\pi^0\pi^0$, $K^+K^-\pi^+\pi^-\pi^0\pi^0$ and $3(\pi^+\pi^-)\pi^0\pi^0$ at $\sqrt s=3.6648$ GeV are set by $$\begin{aligned}
\sigma^{\rm up}_{e^+e^-\to f}=
\frac{N^{\rm up}} {\mathcal{L} \times
\epsilon
\times (1-\Delta_{\rm sys})
\times {\mathcal B}^2(\pi^0\to\gamma\gamma)
},
\label{eq:crsup}\end{aligned}$$ where $N^{\rm up}$ is the upper limit on the number of the signal events setting based on Feldman-Cousins method [@prd57_3873], and $\Delta_{\rm sys}$ is the systematic error in the cross section measurement. Inserting the numbers of $N^{\rm up}$, $\mathcal{L}$, $\epsilon$, $\Delta_{\rm sys}$ and ${\mathcal
B}(\pi^0\to\gamma\gamma)$ in Eq. (\[eq:crsup\]), we obtain $\sigma^{\rm up}_{e^+e^-\to f}$ for these processes at $\sqrt s=
3.6648$ GeV, which are also summarized in the last column of Tab. \[tab:crs36648\].
Upper limits on the observed cross sections and the branching fractions for $\psi(3770)\to f$
---------------------------------------------------------------------------------------------
In the $\psi(3770)$ resonance region [^1], if we ignore the possible interference effects between the continuum and resonance amplitudes and the difference of the vacuum polarization corrections at $\sqrt s= 3.773$ and 3.650 GeV, the contributions of the continuum production for $e^+e^- \to f$ at $\sqrt s= 3.773$ GeV can be expected by these measured at $\sqrt s=
3.650$ GeV. In this case, the observed cross section for $\psi(3770)\to f$ at $\sqrt s= 3.773$ GeV can be written as $$\sigma_{\psi(3770)\to f}=
\sigma^{\rm 3.773\hspace{0.05cm}GeV}_{e^+e^-\to f} -
f_{\rm co}\times \sigma^{\rm 3.650\hspace{0.05cm}GeV}_{e^+e^- \to f},
\label{eq:obscrs}$$ where $\sigma^{\rm 3.773\hspace{0.05cm}GeV}_{e^+e^- \to f}$ and $\sigma^{\rm 3.650 \hspace{0.05cm}GeV}_{e^+e^- \to f}$ are the observed cross sections for $e^+e^- \to f$ measured at $\sqrt s=
3.773$ and 3.650 GeV, respectively; $f_{\rm co}$ is the factor accounting for the 1/s dependence of the cross section. Neglecting the difference in the corrections for ISR and vacuum polarization effects at the two energy points, we have $f_{\rm co} =
(3.650/3.773)^2$. With the $\sigma^{\rm
3.773\hspace{0.05cm}GeV}_{e^+e^- \to f}$ and $\sigma^{\rm 3.650
\hspace{0.05cm}GeV}_{e^+e^- \to f}$ listed in Tabs. \[tab:crs3773\] and \[tab:crs3650\], we determine $\sigma_{\psi(3770)\to f}$ at $\sqrt s= 3.773$ GeV for each process. They are summarized in the second column of Tab. \[tab:up\_psipp\], where the first error is the statistical, the second is the independent systematic arising from the uncertainties in the Monte Carlo statistics, in the fit to the mass spectrum and in the background subtraction, and the third is the common systematic error arising from the other uncertainties as discussed in the subsection B.
Assuming that the cross section for $\psi(3770) \to f$ follows a Gaussian distribution, we set the upper limit on the observed cross section for $\psi(3770)\to f$ at $\sqrt s= 3.773$ GeV, $\sigma^{\rm
up}_{\psi(3770) \to f}$, which are summarized in the third column of Tab. \[tab:up\_psipp\].
The upper limit on the branching fraction ${\mathcal B}^{\rm
up}_{\psi(3770)\to f}$ for $\psi(3770)\to f$ is set by dividing its upper limit on the observed cross section $\sigma^{\rm
up}_{\psi(3770) \to f}$ by the observed cross section $\sigma^{\rm
obs}_{\psi(3770)}=(7.15\pm0.27\pm0.27)$ nb [@plb650_111; @prl97_121801; @plb652_238] for the $\psi(3770)$ production at $\sqrt s= 3.773$ GeV and a factor $(1-\Delta
\sigma^{\rm obs}_{\psi(3770)})$, where $\Delta \sigma^{\rm
obs}_{\psi(3770)}$ is the relative error of the $\sigma^{\rm obs}_{\psi(3770)}$. The results on ${\mathcal B}^{\rm up}_{\psi(3770)\to f}$ are summarized in the last column of Tab. \[tab:up\_psipp\].
**Summary**
===========
In summary, by analyzing the data sets of 17.3, 6.5 and 1.0 pb$^{-1}$ taken, respectively, at $\sqrt s= 3.773$, 3.650 and 3.6648 GeV with the BES-II detector at the BEPC collider, we measured the observed cross sections for $e^+e^-\to \pi^+\pi^-\pi^0\pi^0$, $K^+K^-\pi^0\pi^0$, $2(\pi^+\pi^-\pi^0)$, $K^+K^-\pi^+\pi^-\pi^0\pi^0$ and $3(\pi^+\pi^-)\pi^0\pi^0$ at the three energy points. Based on the measured cross sections for these processes at $\sqrt s = 3.773$ and 3.650 GeV, as well as the observed cross section for the $\psi(3770)$ production at $\sqrt s=
3.773$ GeV, we also set the upper limits on the observed cross sections and the branching fractions for $\psi(3770)$ decay into these final states at $90\%$ C.L.. These measurements provide useful experimental information for better understanding the possible excess of the cross section for the $\psi(3770)$ production relative to the cross section for the $D\bar D$ production and the mechanism of the continuum light hadron production.
Acknowledgments
===============
The BES collaboration thanks the staff of BEPC for their hard efforts. This work is supported in part by the National Natural Science Foundation of China under contracts Nos. 10491300, 10225524, 10225525, 10425523, the Chinese Academy of Sciences under contract No. KJ 95T-03, the 100 Talents Program of CAS under Contract Nos. U-11, U-24, U-25, the Knowledge Innovation Project of CAS under Contract Nos. U-602, U-34 (IHEP), the National Natural Science Foundation of China under Contract No. 10225522 (Tsinghua University).
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$e^+e^-\to f$ $N^{\pi^0_{\gamma_i\gamma_j}}_{\rm obs}$ $N^{\rm rc}$ $N^{\pi^0_{\gamma_i\gamma_j}}_{\rm sid}$ $N^{\rm b}$ $N^{\rm net}$ $\epsilon(\%)$ $\Delta_{\rm sys}(\%)$ $\sigma^{\rm obs}$ \[pb\]
------------------------------ ------------------------------------------ --------------- ------------------------------------------ ------------- ---------------- ---------------- ------------------------ ---------------------------
$\pi^+\pi^-\pi^0\pi^0$ $259.7\pm21.8$ $5.2\pm7.2$ $29.8\pm10.4$ $7.1\pm2.2$ $217.6\pm25.3$ $6.00\pm0.21$ 13.0 $214.8\pm25.0\pm27.9$
$K^+K^-\pi^0\pi^0$ $19.8\pm5.6$ $0.0\pm0.0$ $4.0\pm3.1$ $2.0\pm0.5$ $13.8\pm6.4$ $3.06\pm0.15$ 15.1 $26.7\pm12.4\pm4.0$
$2(\pi^+\pi^-\pi^0)$ $374.6\pm29.0$ $31.5\pm10.6$ $29.2\pm15.6$ $8.5\pm1.4$ $305.4\pm34.6$ $1.72\pm0.07$ 14.1 $1051.5\pm119.2\pm148.3$
$K^+K^-\pi^+\pi^-\pi^0\pi^0$ $38.2\pm9.5$ $7.1\pm4.3$ $0.7\pm4.3$ $5.9\pm1.5$ $24.5\pm11.4$ $0.78\pm0.03$ 16.8 $186.0\pm86.4\pm31.2$
$3(\pi^+\pi^-)\pi^0\pi^0$ $81.2\pm14.2$ $17.0\pm5.1$ $2.4\pm5.2$ $2.6\pm0.7$ $59.2\pm16.0$ $0.36\pm0.02$ 18.5 $973.8\pm262.8\pm180.2$
\[tab:crs3773\]
$e^+e^-\to f$ $N^{\pi^0_{\gamma_i\gamma_j}}_{\rm obs}$ $N^{\rm rc}$ $N^{\pi^0_{\gamma_i\gamma_j}}_{\rm sid}$ $N^{\rm b}$ $N^{\rm net}$ $\epsilon(\%)$ $\Delta_{\rm sys}(\%)$ $\sigma^{\rm obs}$ \[pb\]
------------------------------ ------------------------------------------ -------------- ------------------------------------------ ------------- ---------------- ---------------- ------------------------ ---------------------------
$\pi^+\pi^-\pi^0\pi^0$ $132.6\pm15.4$ $8.0\pm2.8$ $6.6\pm7.1$ $2.3\pm0.8$ $115.7\pm17.2$ $6.24\pm0.23$ 13.5 $292.2\pm43.4\pm39.4$
$K^+K^-\pi^0\pi^0$ $12.0\pm3.9$ $0.0\pm0.0$ $0.8\pm1.4$ $1.0\pm0.3$ $10.2\pm4.2$ $2.96\pm0.14$ 14.8 $54.3\pm22.1\pm8.0$
$2(\pi^+\pi^-\pi^0)$ $147.6\pm16.2$ $18.3\pm8.6$ $8.7\pm7.7$ $0.3\pm0.2$ $120.3\pm19.9$ $1.76\pm0.07$ 15.3 $1077.3\pm178.1\pm164.8$
$K^+K^-\pi^+\pi^-\pi^0\pi^0$ $25.7\pm5.8$ $7.0\pm2.6$ $0.7\pm1.5$ $0.2\pm0.1$ $17.8\pm6.5$ $0.74\pm0.03$ 15.1 $379.1\pm139.1\pm57.2$
$3(\pi^+\pi^-)\pi^0\pi^0$ $25.7\pm6.4$ $7.0\pm2.6$ $0.0\pm0.0$ $0.0\pm0.0$ $18.7\pm6.9$ $0.35\pm0.02$ 19.1 $842.1\pm311.1\pm160.8$
\[tab:crs3650\]
$e^+e^-\to f$ $N^{\pi^0_{\gamma_i\gamma_j}}_{\rm obs}$ $N^{\rm rc}$ $N^{\pi^0_{\gamma_i\gamma_j}}_{\rm sid}$ $N^{\rm b}$ $N^{\rm net}$ (or $N^{\rm up}$ ) $\epsilon(\%)$ $\Delta_{\rm sys}(\%)$ $\sigma^{\rm obs}$ (or $\sigma^{\rm up}$) \[pb\]
------------------------------ ------------------------------------------ -------------- ------------------------------------------ ------------- ---------------------------------- ---------------- ------------------------ --------------------------------------------------
$\pi^+\pi^-\pi^0\pi^0$ $24.9\pm5.3$ $1.0\pm1.0$ $0.0\pm0.0$ $0.4\pm0.1$ $23.5\pm5.4$ $6.06\pm0.20$ 12.8 $397.3\pm91.3\pm50.9$
$K^+K^-\pi^0\pi^0$ 1 - - - $<4.36$ $3.14\pm0.14$ 12.0 $<161.7$
$2(\pi^+\pi^-\pi^0)$ $16.9\pm4.8$ $2.0\pm1.4$ $0.0\pm0.0$ $0.1\pm0.1$ $14.8\pm5.0$ $1.73\pm0.06$ 13.9 $876.4\pm296.1\pm121.8$
$K^+K^-\pi^+\pi^-\pi^0\pi^0$ 2 - - - $<5.91$ $0.77\pm0.03$ 15.1 $<926.2$
$3(\pi^+\pi^-)\pi^0\pi^0$ 3 - - - $<7.42$ $0.35\pm0.02$ 17.2 $<2623.1$
\[tab:crs36648\]
[lccl]{} Decay Mode &$\sigma_{\psi(3770)\to f}$ &$\sigma^{\rm
up}_{\psi(3770)\to f}$ &$B^{\rm up}_{\psi(3770)\to f}$\
&\[pb\]&\[pb\]&\[$\times$10$ ^{-3}$\]\
$\pi^+\pi^-\pi^0\pi^0$ &$-58.6\pm47.7\pm25.7\pm6.5$ &$<61.1 $&$<8.9$\
$K^+K^-\pi^0\pi^0$ &$-24.1\pm24.1\pm5.8\pm2.7$ &$<28.9 $&$<4.2 $\
$2(\pi^+\pi^-\pi^0)$ &$43.3\pm204.9\pm95.4\pm5.7$ &$<399.5$&$<58.5$\
$K^+K^-\pi^+\pi^-\pi^0\pi^0$&$-168.8\pm156.2\pm32.9\pm22.3$&$<182.1$&$<26.7$\
$3(\pi^+\pi^-)\pi^0\pi^0$ &$185.7\pm392.2\pm122.0\pm29.9$&$<801.6$&$<117.4$\
\[tab:up\_psipp\]
[^1]: Assuming that there is only one $\psi(3770)$ in the energy region from 3.70 to 3.87 GeV.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: |
Barium stars and technetium-poor, extrinsic S stars are binary systems with a white dwarf companion, and with orbital elements similar to those of symbiotic systems. One may thus wonder whether these various families of binary systems involving red giant stars are somehow related. This question is actually twofold:\
(i) Do barium and binary S stars exhibit some symbiotic activity?\
(ii) Do symbiotic systems exhibit overabundances of s-process elements like barium and S stars?\
This paper reviews the current situation regarding these two questions.
author:
- Alain Jorissen
title: 'The link between symbiotic stars and chemically-peculiar red giants'
---
\#1 1.25in .125in .25in
The zoo of red giant stars {#Sect:zoo}
==========================
Symbiotic stars (SyS), barium stars and technetium-poor S stars are three families involving giant stars where binarity plays a key role. Thanks to the progress of UV astronomy, the binary nature of SyS now goes undisputed as their UV spectra bear the signature of a hot component, generally a white dwarf (WD), whereas the optical spectrum is dominated by the red giant (Mikołajewska, this conference).
Barium stars, first identified by Bidelman & Keenan (1951), are G and K giants where carbon and elements heavier than Fe, like Ba and Sr, have surface abundances in excess of the solar value (e.g. Wallerstein at al. 1997 and references therein), i.e., \[X/Fe\] $ = \log \left[\left(N({\rm X})/N({\rm Fe})\right) / \left(N_\odot({\rm X})/N_\odot({\rm Fe})\right)\right] > 0$, where $N(\rm X)$ stands for the abundance of element X. Heavy elements like Sr and Ba are synthesized by the so-called [ *s-process*]{} of nucleosynthesis, a sequence of neutron captures starting on abundant seed nuclei like iron-group elements (Burbidge et al. 1957; Wallerstein et al. 1997). The operation of the s-process is commonly associated with thermal pulses (TPs) occurring on the asymptotic giant branch (AGB; e.g. Goriely & Mowlavi 2000). AGB stars have a complex internal structure, consisting of a carbon-oxygen core, helium- and hydrogen-burning shells and a deep convective envelope (Olofsson & Habing 2003). TPs are a recurrent thermal instability affecting the He-burning shell. Right after a TP, the inner boundary of the convective envelope may penetrate the intershell region where the s-process operated. As a result of this mixing process (the so-called ‘third dredge-up’ – 3DUP), s-process-enriched material is brought to the stellar surface. Barium stars are, however, too warm and of too low a luminosity to be thermally-pulsing AGB (TP-AGB) stars (Scalo 1976). With the discovery that the barium stars are all single-line spectroscopic binaries (McClure et al. 1980; McClure 1983; Jorissen et al. 1998), their chemical peculiarities have been ascribed to mass transfer across the binary system. The unseen companion is almost certainly a WD (McClure & Woodsworth 1990). In a former state of the binary system, that WD was a TP-AGB star where the s-process and the 3DUP were operating. If mass transfer processes (like wind accretion or Roche lobe overflow – RLOF) were able at that stage to dump s-process-rich and C-rich material from the AGB star onto its companion, that companion would be turned into a barium star (see Jorissen 2003 for a review). This is the so-called [*binary paradigm*]{} to account for the [*barium syndrome*]{}.
The stars of spectral type S (first identified by Merrill in 1922) exhibit chemical peculiarities very similar to those of barium stars. However, the S star family hosts two kinds of stars, the so-called [*extrinsic*]{} and [*intrinsic*]{} S stars, having very different evolutionary status (Van Eck & Jorissen 2000). They are best distinguished by the presence or absence of Tc lines, a heavy element with no stable isotopes (Van Eck & Jorissen 1999). Intrinsic S stars exhibit Tc lines, and are AGB stars where the s-process is operating, as indicated above. There is thus no need to invoke mass transfer across a binary system to account for their chemical peculiarities. On the contrary, extrinsic S stars have no Tc lines, and are all binary stars (Jorissen et al. 1998). They are the cool analogs of barium stars. For both barium and extrinsic S stars (collectively referred to as peculiar red giants – PRG – in the following), the companion mass inferred from the observed mass-function distribution is consistent with that companion being a WD (McClure & Woodsworth 1990; Jorissen et al. 1998; North et al. 2000).
Since both PRG and SyS are binary systems consisting of a red giant and a WD, the relation between these families ought to be elucidated; more precisely: (i) Do PRG exhibit symbiotic activity? (ii) Do SyS exhibit the barium syndrome?
Do PRG exhibit symbiotic activity?
==================================
Physical conditions required to trigger symbiotic activity {#Sect:formulae}
----------------------------------------------------------
Before reviewing the symbiotic activity observed in the various classes of PRG (Sect. \[Sect:PRGsymbio\]), it is useful to formally identify the physical conditions required to trigger symbiotic activity. The key to this activity lies in the luminosity of a hot ($T >
50\;000$ K) companion star, estimated to be at least 10 (Mürset et al. 1991; Yungelson et al. 1995), which emits UV radiation that ionizes the wind from the cool star, giving rise to the rich emission-line spectrum (Nussbaumer & Vogel 1987). Different physical processes, all related to the accretion of mass by the companion, may be at the origin of this high luminosity (Yungelson et al. 1995; Iben & Tutukov 1996): (i) steady hydrogen burning at the surface of a WD (ii) thermonuclear flashes at the surface of a WD \[associated with symbiotic novae\] (iii) release of gravitational energy associated with accretion, partially converted into radiative energy. Regimes (i) and (ii) correspond to accretion rates respectively above and below some critical value $\dot{M}^{\rm acc}_{\rm crit}$ (Eq. b of Table \[Tab:formulae\], about $5\times 10^{-8}$ M$_\odot$ y$^{-1}$ for a 0.6 WD). Regime (iii) applies to main-sequence or neutron-star accretors, and to WD accretors in between H-burning flashes (since the accretion luminosity is smaller than the H-burning luminosity at any given WD mass $M_h$; see Table \[Tab:formulae\] and Fig. 5 of Iben & Tutukov 1996). Formulae relating the luminosity $L_h$ of the compact companion to its mass $M_h$ for these three regimes are provided in Table \[Tab:formulae\]. A convenient analytical formula for the accretion rate $\Macc$ is available in the framework of the Bondi-Hoyle regime of wind accretion (Bondi & Hoyle 1944), although detailed hydrodynamical simulations (Theuns, Boffin & Jorissen 1996; Mastrodemos & Morris 1998; Folini & Walder 2000) have shown that the Bondi-Hoyle accretion rates (Eq. c in Table \[Tab:formulae\] with $\eta \sim 1$) are generally an order of magnitude too large when the wind velocity is of the same order as the orbital velocity, as it is the case for SyS. The parameter $\eta$ appearing in Eq. (c) of Table \[Tab:formulae\] must therefore be taken of the order of 0.1 to reflect the results of numerical simulations when $v_{\rm wind} /
v_{\rm orb} \lesssim 1$.
To summarize, Table \[Tab:formulae\] provides relationships that may in principle be used to propagate the condition $L_h > 10$ defining a SyS into constraints on the various physical parameters involved (the hot companion mass $M_h$, the cool star mass $M_c$, its luminosity $L_c$ and temperature $T_c$, the metallicity $Z$ and the orbital period $P$), namely: $$\begin{aligned}
10\; {\rm L}_\odot < L_h & = & f\;[M_h, \Macc]\\
& = & f\;[M_h, \Macc(\dot{M}_c^{\rm
wind},v_{\rm wind},M_c,P)]\\
& = & f\left[M_h, \Macc\left(\dot{M}_c^{\rm wind} [ M_c, T_c, L_c
( M_c, T_c, Z )], v_{\rm wind}(L_c,Z),M_c,P)\right)\right]\\
& = & f\;[M_h, M_c, T_c, Z, P]. \end{aligned}$$
The metallicity $Z$ enters the discussion through the mass-loss properties of the cool star, especially its wind velocity (Eq. e of Table \[Tab:formulae\]; Van Loon 2000). In the various empirical parametrizations reviewed by Zijlstra (1995), the wind mass loss rate $\dot{M}_c^{\rm wind}$ depends explicitely upon $M_c$, $L_c$ and the stellar radius $R_c$, which transforms into a function of $M_c, T_c$ and $Z$ using the Stefan-Boltzmann formula to eliminate $R_c$ and the evolutionary track to express $L_c$ as a function of $M_c, T_c$ and $Z$. All the empirical formulae reviewed by Zijlstra (1995) predict that [*the mass loss rate increases with increasing $L_c$ and $R_c$*]{}. This will turn out to be a very important property to understand the occurrence of – or lack of – symbiotic activity in the various families of PRG.
The above discussion assumes that the system is detached, which sets yet another constraint, namely $P > P_{\rm RLOF}(R_c,M_c,M_h)$ (Eq. d of Table \[Tab:formulae\]) where $P_{\rm RLOF}(R_c,M_c,M_h)$ is the orbital period of the (semi-detached) system with a cool star of radius $R_c$ filling its Roche lobe.
\
[**The operation of a given regime is dictated by the ratio $\dot{M}_h^{\rm acc}/\dot{M}_{\rm crit}^{\rm acc}$:**]{}\
where\
$\dot{M}_{\rm crit}^{\rm acc}= 10^{-9.31 + 4.12 M_h - 1.42
M_h^2}$\
$\dot{M}_h^{\rm acc} = -\dot{M}_c^{\rm wind}
\;
\eta \; \mu^2 \; \frac{k^4} {\left[ 1 + k^2 + \left(\frac{c}{v_{\rm
wind}}\right)^2\right]^{3/2}}$
\
$\mu = M_h/(M_h + M_c)$\
$k = v_{\rm orb} / v_{\rm wind} = 30 \; \left(\frac{M_h + M_c}{P}\right)^{1/3} \; / \; v_{\rm wind}$\
$P > P_{\rm RLOF} [R_c, M_c, M_h ] =
\frac{3\times 10^{-4}\;R_c^{3/2}}{(M_c + M_h)^{1/2} (0.38 + 0.2 \log
(M_c/M_h))^{3/2}}$\
$
\begin{array}{llll}
\hspace*{15mm} \eta \sim & 1 & {\rm if}\; k^{-1} = v_{\rm wind} / v_{\rm orb} >> 1 & \hspace{\fill} {\rm (Bondi-Hoyle\; regime)}\\
\hspace*{15mm} \eta \sim & 0.1 & {\rm if}\; k^{-1} = v_{\rm wind} / v_{\rm orb} \le 1
&\\
\end{array}
$\
$c =$ [sound velocity]{}\
$v_{\rm wind} \propto L_c^{1/4} Z^{1/2}$\
$\dot{M}_c^{\rm wind} = \dot{M}_c^{\rm wind} [ M_c, T_c, L_c
( M_c, T_c, Z ) ]$\
References:
Symbiotic activity among PRG {#Sect:PRGsymbio}
----------------------------
### Ba stars.
Symbiotic activity among barium stars is basically inexistent, except for the barium supergiant 56 Peg and for HD 46407, a barium star with one of the shortest orbital periods (456.6 d). 56 Peg is an X-ray source with a hot WD (Schindler et al. 1982; Dominy & Lambert 1983; Schwope et al. 2000). HD 46407 exhibits dust obscuration episodes (Jorissen 1994, 1997) reminiscent of those observed in symbiotic Miras (Munari & Whitelock 1989), although to a much lesser extent. This quasi-absence of symbiotic activity among barium stars is not surprising given their rather low mass-loss rates $\dot{M}_c^{\rm wind}$ (Drake, Simon, & Linsky 1987), consistent with their luminosities of RGB (rather than AGB) stars (Scalo 1976). Equation (c) of Table \[Tab:formulae\] then indicates that the accretion rate by the companion will be low as well ($< 10^{-10}$ M$_\odot$ y$^{-1}$; Jorissen 1997). If anything, [*this situation leads to H-flashes on the WD companion (regime ii)*]{}, although no such events have yet been reported for barium stars.
### S stars.
Van Eck & Jorissen (2002; their Table 2) have collected all S stars where signatures of symbiotic activity have been reported. All of these – with the exception of the Henize S stars (see below) – result from serendipitous discoveries, and thus rely on different diagnostics of symbiotic activity which are not equally sensitive to $\dot{M}_h^{\rm acc}$. For instance, emission is not observed in the long-period system HD 49368 (=V613 Mon; $P \sim 3000$ d) despite a strong UV excess (Ake 1996, priv. comm.). A similar situation is encountered for the SyS EG And (see in particular the discussion in Sect. 4.4 of Munari 1994).
Therefore, to find any systematics (like correlation with orbital period) requires a more systematic approach. The 66 binary S stars from the Henize sample (Van Eck & Jorissen 2000) offers such an opportunity. These stars were searched for emission, resulting in the discovery of two new SyS (Hen4-18 and Hen4-121, following the SIMBAD terminology) and of two marginal cases (Hen4-134 and Hen4-137; Van Eck & Jorissen 2002). Their profiles, displayed in Fig. \[Fig:HaP\], are typical of SyS, since they closely resemble those labelled ‘S-3’ by Van Winckel, Duerbeck & Schwarz (1993; see Lee 2000 for a discussion of the formation mechanism of the emission line in SyS). Figure \[Fig:HaP\] reveals that symbiotic S stars with emission are found in the narrow period range 600 – 800 d (as noted above, S stars with longer periods may exhibit other signatures of symbiotic activity, though).
The absence of emission among the shortest-period systems is an interesting result, which confirms that the orbital period is not the primary parameter controlling symbiotic activity, as inferred from Table \[Tab:formulae\]. The key parameter is rather the accretion rate $\dot{M}_h^{\rm acc}$, which is a combination of several parameters, including $P$ and $\dot{M}_c^{\rm wind}$ (Eq. c of Table \[Tab:formulae\]). The absence of symbiotic activity among short-period S stars is likely due to their low mass-loss rates $\dot{M}_c^{\rm wind}$, which are in turn a consequence of the fact that S stars with short orbital periods cannot be located very far up the giant branch. They must thus have smaller radii, luminosities, and hence mass loss rates, than S stars with longer orbital periods. The orbital period indeed imposes a maximum admissible radius $R_c$ corresponding to the critical Roche radius (Eq. d of Table \[Tab:formulae\]). Put differently, this condition means that to any given spectral type corresponds a minimum admissible orbital period for unevolved (i.e., pre-mass-transfer) [*detached*]{} systems. Such a correlation is clearly apparent on Fig. \[Fig:P\], which shows that the minimum orbital period among K giants is 40 d, increasing to 200 d for M giants, and even to 400 d for giants later than M0 III. A similar correlation has been found by Mürset & Schmid (1999) among SyS (see also Harries & Howarth 2000), despite the fact that SyS are not really unevolved systems.
Finally, it must be noted that the period range of red SyS (excluding symbiotic novae and symbiotic Miras) matches fairly well that of symbiotic S stars with emission and, moreover, corresponds to the short-period tail of the M III binaries. The same holds true for the period distribution of yellow SyS, which corresponds to the short-period tail of barium stars ([*but not*]{} to the short-period tail of K III binaries!). The reason for this is clear: barium stars and yellow SyS are two families which involve WD companions, unlike K III binaries which may also involve main sequence companions. Therefore, barium and yellow SyS share the same evolutionary history, namely one component has gone through the AGB. The fact that these systems once contained a very extended star sets a lower limit on their orbital period, as discussed above in relation with Eq. d (for a more detailed discussion of these aspects, see Jorissen 2003). But this restriction does not apply to the sample of K III binaries, hence they may contain systems with much shorter orbital periods than barium stars and yellow SyS.
[lllrrcllll]{}\
\
Name & Sp. Typ. & \[Fe/H\] & & & \[Ba/Fe\] & $V \sin \; i$ & nebula & Ref.\
& & & & &&\
\
\
\
V417 Cen & G8-K2 & $\sim 0.0$ & & $-1$ & 0.5 & 70 & y & (5,11)\
HDE 330036& G5 & $\sim 0.0$ & $-14$ & $+4$ & 0.6 & 100 & PN & (5,14)\
AS 201 & G5 & $\sim 0.0$ & & $+7$ & 0.4 & 25 & y & (5,12)\
V471/V741 Per & G5 & ? &$-$12 & $-$9 & $> 0$ & &PN & (2)\
St H$_\alpha$ 190 & G5 & 0.0 & $\sim 10$ & $-$35 & $\sim 0.5$ & 100 & bip. outf.& (10,13)\
Wray 157 & G5 & ?\
Hen 1591 & $<$ K4 & ?\
\
\
\
UKS Ce-1 & C4,5Jch & ? & +20 & +20 & $>0$ &&& (6)\
S 32 & C1,1CH & ? & +325 & $-$30 & $>0$ & &&(6,14)\
Hen 2-467 & K0 & -1.1 & $-$109 & $-$12 & +0.8 &&n & (4,16)\
BD-21:3873 & K2 & -1.1 & +204 & +37 & +0.5 & & n &(3,15,16)\
& & -1.3 & & & +0.3 & &&(9)\
AG Dra & K2 & -1.3 & $-$148 & +41 & +0.5 & & n &(8,16)\
CD -43:14304 & K7 &? & +27 & $-41$ & ? & &&(7)\
\
\
& & -1.0 & & & $< 0.2$ &&& (1)\
\
[References: (1) Edvardsson et al., 1993, A&A, 275, 101 (2) Grauer & Bond, 1981, PASP, 93, 630 (3) Pereira et al., 1997, AJ, 114, 2128 (4) Pereira et al., 1998, AJ, 116, 1977 (5) Pereira et al., 2003, this conference (6) Schmid, 1994, A&A, 284, 156 (7) Schmid et al., 1998, A&A, 329, 986 (8) Smith et al., 1996, A&A, 315, 179 (9) Smith et al., 1997, A&A, 324, 97 (10) Smith et al., 2001, ApJ, 556, L55 (11) Van Winckel et al., 1994, A&A, 285, 241 (12) Schwarz, 1991, A&A, 243, 469 (13) Munari et al., 2001, A&A, 369, L1 (14) Schmid & Nussbaumer, 1993, A&A, 268, 159 (15) Munari & Patat, 1993, A&A, 277, 195 (16) Corradi et al., 1999, A&A, 343, 841]{}
Do SyS exhibit the barium syndrome?
===================================
Physical conditions required to trigger the s-process {#Sect:s-process}
-----------------------------------------------------
Here again, we start with a formal discussion of the conditions required to trigger the operation of the s-process in AGB stars, before considering the question whether SyS exhibit the barium syndrome. These conditions are:\
$$\fbox{$
\begin{array}{lll}
M_h & > & 0.5\; {\rm M}_\odot\\
Z & < & Z_\odot
\end{array}
$}$$
The first condition on the core mass of the AGB star (which is identical to the mass $M_h$ of the WD in the present SyS) expresses the fact that the AGB star must have gone through the TP phase (Sect. \[Sect:zoo\]). Wagenhuber & Groenewegen (1998, their Fig. 7) provide the AGB core mass at the first TP, for AGB stars of various metallicities and initial masses, from which the first condition is derived.
The second condition, expressing that the efficiency of the s-process is higher in low-metallicity AGB stars, was first suggested by Clayton (1988). This efficiency may be expressed by a single quantity, $n_c$, the number of neutrons captured per (iron) seed nuclei. For example, $n_c \simeq 138 - 56 = 82$ is required for $^{138}$Ba to be synthesized from $^{56}$Fe. Assuming that there are no strong neutron poisons, all neutrons will be captured by Fe and its daughter nuclei, so that $n_c = N({\rm neutron\; supply})/N({\rm
Fe})$. Neutrons are supplied by a ‘neutron source’, namely $^{13}$C($\alpha$,n)$^{16}$O as it is currently believed for AGB stars (see e.g., Wallerstein et al. 1997), with $^{13}$C resulting from the so-called proton-mixing scenario (e.g., Goriely & Mowlavi 2000). In this scenario, protons from the convective envelope are mixed in layers enriched in $^{12}$C by the former TP, resulting in the synthesis of $^{13}$C through the chain $^{12}$C(p,$\gamma)^{13}$N($\beta)^{13}$C. In the framework of the proton-mixing scenario, $^{13}$C may be considered as ‘primary’ (in the sense of galactic chemical evolution), since it is synthesized from primary species, namely hydrogen from the envelope and $^{12}$C resulting from the $3\alpha$ reaction. Assuming that there is no leak in the neutron production by the $^{13}$C source, all available $^{13}$C nuclei will yield neutrons, so that $n_c = N({\rm neutron\; supply})/N({\rm Fe}) = N(^{13}{\rm C})/N({\rm
Fe}) \propto 1/Z$.
This expectation seems to be borne out by empirical evidence; see the discussion in Jorissen & Boffin (1992) and Jorissen et al. (1998; their Sect. 8) showing that binarity is not a sufficient condition to form a barium star, but that a subsolar metallicity seems to be required as well.
At this point, it should be remarked, however, that metallicity is also likely to have an impact on the wind accretion rate: at high $Z$, the wind velocity is larger (Eq. e of Table \[Tab:formulae\]; also Zuckerman & Dyck 1989) and therefore the accretion rate is smaller, since $\dot{M}_h^{\rm acc}
\propto v_{\rm wind}^{-4}$ approximately. Therefore, barium stars may not form at high metallicities, not only because the s-process in the former companion AGB star is less efficient, but also because accretion of its wind is less efficient.
Because of this sensitivity upon metallicity, s-type yellow SyS, d’-type yellow SyS and red SyS have to be considered separately in the following, since they belong to different galactic populations.
s-Type yellow SyS are PRG
-------------------------
All known yellow SyS are listed in Table \[Tab:yellow\], which reveals that all the stars studied so far exhibit the barium syndrome. Yellow SyS with a [*stellar*]{} infrared continuum (s-type, as opposed to the dusty d’-type; see below) are clearly halo objects, as revealed by their low metallicities and high space velocities (CD $-43:14304$ may be an exception; however, it is of spectral type K7, and should perhaps not be included in the family of yellow SyS). The presence of the barium syndrome among a family of binary stars belonging to the halo fully supports the discussion of Sect. \[Sect:s-process\] about the conditions required for s-processing. It should be added at this point that s-type yellow SyS, with their metallicities lower than classical barium stars, may be expected to be, on average, more luminous than the latter (see Fig. 11 of Smith et al. 1996 comparing the luminosity function of Pop.I and Pop.II K giants). This is a direct consequence of the fact that evolutionary tracks shift towards the blue in the Hertzsprung-Russell (HR) diagram as metallicity decreases, as shown in Fig. \[Fig:mdBa\]b. Fig. \[Fig:mdBa\]a confirms that the yellow SyS AG Dra and BD $-21:3873$ are indeed more luminous than classical barium stars. This difference in the average luminosity – and hence mass-loss rate – of the two populations thus explains why yellow SyS, despite hosting a K giant, exhibit symbiotic activity whereas barium stars do not. The larger mass-loss rates for the cool components of s-type yellow SyS – as compared to Ba stars – may be inferred from the comparison of their IRAS \[12\] $-$ \[25\] color indices, which reflect the amount of dust present in the system: (\[12\] $-$ \[25\])$_{\rm Ba} < $ 0.1, as compared to 0.45 for AG Dra (Smith et al. 1996). Mürset et al. (1991) and Drake et al. (1987) provide direct measurements (or upper limits) for the mass loss rates of AG Dra and of Ba stars, respectively, which confirm the above conclusion.
[*Metal-deficient barium stars*]{} (with metallicities in the range $-1.1$ to $-1.8$ comparable to that of yellow SyS) were identified by Luck & Bond (1991) and Mennessier et al. (1997), and occupy the same region of the HR diagram as yellow SyS (Fig. \[Fig:mdBa\]b). The question thus arises why metal-deficient barium stars are not SyS. Different answers must be seeked, depending upon their absolute visual magnitudes $M_{\rm V}$. The most luminous systems, with $M_{\rm V} < -2$, are likely located on the TP-AGB[^1], so that their Ba syndrome may be explained by internal nucleosynthesis. They ought thus not be binaries, and therefore cannot be SyS! HD 104340 (open circle in Fig. \[Fig:mdBa\]b), a metal-deficient Ba star studied by Junqueira & Pereira (2001), provides a good illustration of this situation, since it lies above the TP-AGB threshold and 15 unpublished CORAVEL radial-velocity measurements spanning 7 y do not reveal any clear orbital motion.
The less luminous and warmest among metal-deficient Ba stars, clumping around $M_{\rm V} \sim +1$ in the HR diagram, are also sometimes classified as CH stars (crosses in Fig. \[Fig:mdBa\]b). They are not losing mass at a large enough rate to trigger any symbiotic activity, as revealed by their small \[12\] $-$ \[25\] color indices ($< 0.3$; Smith et al. 1996).
Finally, at intermediate luminosities ($-2 \lesssim M_{\rm V} \lesssim
+1$), metal-deficient Ba stars are not luminous enough to be TP-AGB (hence they should be binaries), but yet their mass loss rates must be large enough to trigger symbiotic activity, since the yellow SyS belong to the same luminosity range. It would thus be of great interest to check (i) the binary nature of those metal-deficient Ba stars[^2] with intermediate luminosities, and (ii) their suspected symbiotic activity.
d’-type yellow SyS: young post-PN systems?
------------------------------------------
Yellow SyS of type d’ (Allen 1982; Schmid & Nussbaumer 1993) differ from their s-type counterparts in several respects (Table \[Tab:yellow\]): they host a complex circumstellar environment (including cool dust, bipolar outflows, extended optical nebulae or emission-line spectra closely resembling those of planetary nebulae), the cool components have early spectral types (F to early K), they are often fast rotators (with the possible exception of M 1-2 =V471 Per; Grauer & Bond 1981) and, finally, they belong to the galactic disk unlike s-type yellow SyS which belong to the halo.
All these arguments suggest that the hot component in d’-type SyS just evolved from the AGB to the WD stage. The rather cool dust (Schmid & Nussbaumer 1993) is a relic from the mass lost by the AGB star. The optical nebulae observed in d’-type SyS are most likely genuine planetary nebulae rather than the nebulae associated with the ionized wind of the cool component (Corradi et al. 1999). This is especially clear for AS 201 which actually hosts [*two*]{} nebulae (Schwarz 1991): a large fossil planetary nebula detected by direct imaging, and a small nebula formed in the wind of the current cool component. Finally, the rapid rotation of the cool component has likely been caused by spin accretion from the former AGB wind like in WIRRING systems (Jeffries & Stevens 1996; see also Jorissen 2003). The fact that the cool star has not yet been slowed down by magnetic braking is another indication that the mass transfer occurred fairly recently (Theuns et al. 1996). Finally, one may wonder whether the much earlier spectral types encountered among yellow d’-type SyS as compared to s-type SyS bear some relationship to their fast rotation (departure from thermal equilibrium?).
No extrinsic PRG among red SyS! Why?
------------------------------------
A small number of galactic red SyS (UV Aur, SS 38, AS 210, HD 59643 = NQ Gem and V335 Vul) contain a cool carbon star as cool component, corresponding to a frequency of 5/176 = 0.03 in the catalogue of Belczyński et al. (2000). This small frequency contrasts with that prevailing in the Magellanic Clouds, where 6 out of 11 SyS contain cool carbon stars (Mürset, Schild & Vogel 1996). The frequency of carbon-rich SyS actually reflects the number ratio of C to M stars in the parent galaxy, and this number ratio in turn reflects the metallicity of the population (Richer 1989). Therefore, these carbon SyS are likely [*intrinsic*]{} carbon stars (Mürset et al. 1996), i.e., TP-AGB stars where the carbon observed in the atmosphere results from the 3DUP (see Sect. \[Sect:zoo\]).
The question then arises why there are no [*extrinsic*]{} C or S stars among SyS (Mürset & Schmid 1999), namely cool components polluted by carbon-rich matter from the former TP-AGB companion. Or in other words, why do red SyS not comply with the [*binary paradigm*]{} (Sect. \[Sect:zoo\])? There are at least three possible explanations for the fact that red SyS contain M rather than S giants:\
$\bullet$ the hot companion is a main sequence star rather than a WD;\
$\bullet$ the former AGB star did not go through the TP-AGB, i.e., $M_h <
0.5$ (see Sect. \[Sect:s-process\]);\
$\bullet$ the former AGB star did go through the TP-AGB, but its high metallicity hindered the efficiency of the s-process and of the mass transfer (see Sect. \[Sect:s-process\]).\
The first question is difficult to address on observational grounds. Let us just mention here that the eccentricity – period diagram may, in some cases, be used to distinguish systems with WD companions from systems with main-sequence companions. As discussed by Jorissen et al. (1998, their Sect. 6 and Fig. 4), binary systems with $e < 0.1$ and $P > 300$ d most likely host a WD companion, whereas systems with $ e > 0.23\; ( \log P({\rm d}) - 1)$ are likely to host main sequence companions. Since most SyS have nearly circular orbits (Mikołajewska and Hinkle et al., this conference), they are likely to host WD companions indeed.
The second possibility ($M_h < 0.5$ ) applies to a number of red SyS with companion masses fairly accurately determined (see also Mikołajewska, this conference), like AX Per (0.4 ), EG And ($0.4\pm0.1$ ), SY Mus ($0.43\pm0.05$ ), RW Hya ($0.48\pm0.06$ ; Mürset et al. 2000 and references therein). There are, however, several other red SyS which do not fulfill this condition, either marginally (BX Mon: $0.55\pm0.26$ ) or more significantly (FG Ser: $0.60\pm0.15$ ; AR Pav: $0.75\pm0.15$ ; M" urset et al. 2000; Schild et al. 2001). For comparison, the mass of the WD companion in barium systems peaks at 0.67 ($\pm0.09$) (North et al. 2000), in agreement with the requirement that the AGB progenitor went through the TP-AGB phase.
Therefore, the third possibility (high $Z$) must be invoked to account for the lack of barium syndrome in systems like FG Ser or AR Pav for example, which have $M_h > 0.5$ . Do red SyS indeed belong to a high-metallicity population? There are contradictory arguments in that respect. The distribution of carbon abundances in the cool components of SyS derived by Schmidt and Mikołajewska (this conference) is representative of red giants having slightly subsolar metallicities (\[Fe/H\] $\sim -0.3$ to $-0.5$). On the contrary, Whitelock & Munari (1992) showed that the $JHK$ colors of red SyS resemble more the colors of bulge-like M giants than those of normal M giants in the solar neighborhood. They argue that this color difference may be related to the higher metallicity of bulge-like giants, and, hence, of red SyS. A subsequent kinematical analysis (Munari 1994) confirmed that red SyS belong to the bulge/thick-disk population. A direct high-resolution spectroscopic determination of the metallicities of red SyS is needed to definitely settle that question. It must be hoped that such a study does indeed confirm the expectation of high metallicities for red SyS, otherwise answers to the lack of barium syndrome different from those discussed here would have to be found.
To conclude, a synopsis of the different families of PRG and SyS stars, and their relationship in terms of presence or absence of symbiotic activity and barium syndrome, is presented in Fig. \[Fig:synopsis\].
This paper greatly benefited from discussions with H.M. Schmid and C. Pereira about the nature of d’-SyS. A.J. is Research Associate of the [*Fonds National de la Recherche Scientifique*]{} (Belgium).
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[^1]: According to Lattanzio (1991), the first TP in a 1 AGB star of metallicity \[Fe/H\]$=-1.8$ occurs at $M_{\rm bol} = -3$, corresponding to $M_{\rm V} \sim -2$
[^2]: Primary targets with independent confirmation of their halo nature are HD 5424 (binary with $P = 1881$ d; Jorissen et al. 1998), HD 55496, HD 104340, HD 148897 and HD 206983. Secondary targets, with their halo classification relying only on the Mennessier et al. (1997) analysis, are HD 15589, HD 43389 (binary with $P = 1689$ d; Jorissen et al. 1998), HD 123396, HD 139409, HD 187762 and CD$-$27:2233
|
{
"pile_set_name": "ArXiv"
}
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---
author:
- Zhiqiang Yu
- |
Zhiqiang Yu\
[^1]\
[School of Mathematical Sciences, East China Normal University, Shanghai 200241, China]{}
title: 'Pre-modular fusion categories of small global dimensions'
---
\[section\] \[theo\][Proposition]{} \[theo\][Lemma]{} \[theo\][Corollary]{} \[theo\][Conjecture]{} \[theo\][Definition]{} \[theo\][Example]{} \[theo\][Question]{} \[theo\][Remark]{}
We first prove an analogue of Lagrange theorem of global dimensions of fusion categories, then we consider classification of pre-modular fusion categories of some small integer global dimensions.
[**Keywords:**]{} Global dimension; pre-modular fusion category; spherical fusion category
Mathematics Subject Classification 2010: 18D10 $\cdot$ 16T05
Introduction
============
Throughout this paper, let ${\mathbb{C}}$ be the field of complex numbers, ${\mathbb{C}}^{*}:={\mathbb{C}}\backslash {\{0}\}$, $\mathbb{Q}$ and $\overline{\mathbb{Q}}$ denote the field of rationals and its algebraic closure, respectively. For $r\in \mathbb{N}$, let ${\mathbb{Z}}_r:={\mathbb{Z}}/r$. Categories are assumed to be semisimple ${\mathbb{C}}$-linear finite abelian categories. For any finite abelian category ${\mathcal{C}}$, let ${\mathcal{O}}({\mathcal{C}})$ be the set of isomorphism classes of simple objects of ${\mathcal{C}}$. The cardinal of ${\mathcal{O}}({\mathcal{C}})$ is called rank of ${\mathcal{C}}$, and will denoted by $rank({\mathcal{C}})$.
A fusion category ${\mathcal{C}}$ is spherical, if ${\mathcal{C}}$ admits a pivotal structure $j$, which is a natural isomorphism from identity tensor functor $id_{\mathcal{C}}$ to double dual tensor functor $(-)^{**}$, and the pivotal structure satisfies $dim(X)=dim(X^*)$ for any object $X$ of ${\mathcal{C}}$, where $dim(X)$ is the quantum dimension of $X$ defined by $j$, see section \[preliminaries\] for definition.
We know that all fusion categories of prime FP-dimensions are pointed [@ENO1 Corollary 8.30]. However, this is not the case for global dimensions. In [@O3 Example 5.1.2], Ostrik classified all spherical fusion categories of integer global dimension less than or equal to $5$. Specifically, these fusion categories are pointed, or tensor equivalent to an Ising category ${\mathcal{I}}$ or equivalent to a Deligne tensor product $YL\boxtimes \overline{YL}$, where $YL$ is a Yang-Lee fusion category and $ \overline{YL}$ is a Galois conjugate of $YL$. It’s easy to see that $YL\boxtimes \overline{YL}$ is not pointed and $dim(YL\boxtimes \overline{YL})=5$. Therefore, this is a non-trivial task to classify spherical fusion categories of given small global dimensions. Meanwhile, for a given global dimension, it follows from [@O3 Theorem 1.1.1] that there are finite many tensor equivalence classes of spherical fusion categories, then it is doable to classify spherical fusion categories of small global dimensions.
When classifying spherical fusion categories by global dimensions, one of the main difficulties is to restrict the rank of fusion category for a given dimension, and there have no general methods, see [@O3 Lemma 4.2.2]. Recall that a spherical fusion category ${\mathcal{C}}$ is a pre-modular fusion category if ${\mathcal{C}}$ is braided. There are some classification results of (super-) modular fusion categories of small ranks, see [@BGNPRW; @BNRW; @O1; @O2; @RSW]. So, in this paper, we turn our attentions to pre-modular fusion categories of small integer dimensions.
For fusion categories over field ${\mathbb{C}}$, it is well-known that Lagrange theorem for FP-dimension is true. That is, for any fusion category ${\mathcal{C}}$ and fusion subcategory ${\mathcal{D}}\subseteq{\mathcal{C}}$, the ratio $\frac{FPdim({\mathcal{C}})}{FPdim({\mathcal{D}})}$ is an algebraic integer [@ENO1 Proposition 8.15]. In Theorem \[Lagrange\] we prove an analogue of Lagrange theorem of global dimension. However, $\frac{FPdim({\mathcal{C}})}{FPdim({\mathcal{D}})}\neq\frac{dim({\mathcal{C}})}{dim({\mathcal{D}})}$ in general, see Remark \[nonequratios\]. Then we can use Theorem \[Lagrange\] to restrict global dimensions of pre-modular fusion subcategories of ${\mathcal{C}}$. Together with techniques developed in [@O2; @O3], we can apply some well-known classifications results on pre-modular fusion categories.
The organization of this paper is as follows. In section \[preliminaries\], we recall some basic notions and notations of fusion categories, such as global dimensions, formal codegrees and $d$-numbers introduced in [@O1]. In subsection \[subsection3.1\], we give a proof of Lagrange theorem of global dimension of fusion categories in Theorem \[Lagrange\]. In subsection \[subsection3.2\], we first prove that pre-modular fusion categories of dimension $7$ are pointed in Theorem \[moddimen7\], then in Proposition \[propdimen8\] we show that pre-modular fusion categories of global dimension $8$ are always weakly integral if they are not simple. In subsection \[subsection3.3\], we show that spherical fusion categories of dimension $6$ are weakly integral (Theorem \[spherical6\]), and we give a partial result on classification of pre-modular of dimension $10$ in Proposition \[propdimen10\].
Preliminaries
=============
Spherical fusion category
-------------------------
Let ${\mathcal{C}}$ be a fusion category over ${\mathbb{C}}$. Homomorphism FPdim(-) of ${\mathcal{C}}$ is the unique homomorphism from Grothendick ring $Gr({\mathcal{C}})$ to ${\mathbb{C}}$ such that $FPdim(X)\geq1$ is an algebraic integer for all objects $X\in{\mathcal{O}}({\mathcal{C}})$ [@ENO1 Theorem 8.6], and $FPdim(X)$ is called the Frobenius-Perron dimension of object $X$. The Frobenius-Perron dimension $FPdim({\mathcal{C}})$ of fusion category ${\mathcal{C}}$ is defined by $$\begin{aligned}
FPdim({\mathcal{C}}):=\sum_{X\in{\mathcal{O}}({\mathcal{C}})}FPdim(X)^2.\end{aligned}$$
Fusion category ${\mathcal{C}}$ is weakly integral, if $FPdim({\mathcal{C}})\in{\mathbb{Z}}$; ${\mathcal{C}}$ is integral, if $FPdim(X)\in{\mathbb{Z}}$ for all $X\in{\mathcal{O}}({\mathcal{C}})$. A fusion category ${\mathcal{C}}$ is pointed if and only if all simple objects of ${\mathcal{C}}$ have FP-dimension $1$, so ${\mathcal{C}}\cong Vec_G^\omega$ in this case, where $Vec_G^\omega$ is the category of $G$-graded finite-dimension vector space over ${\mathbb{C}}$, $\omega\in Z^3(G,{\mathbb{C}}^*)$ is a $3$-cocycle. In the following, we use ${\mathcal{C}}_{pt}$ and ${\mathcal{C}}_{int}$ to denote the maximal pointed fusion subcategory and the maximal integral fusion subcategory of ${\mathcal{C}}$, respectively.
Assume $X\in{\mathcal{C}}$ is an object, let $(X^*, ev_X,coev_X)$ be a left dual object of $X$. That is, there exist morphisms $coev_X:I\to X\otimes X^*$ and $ev_X:X^*\otimes X\to I$ satisfying the following equations $$\begin{aligned}
id_X\otimes ev_X \circ coev_X\otimes id_X=id_X,\quad ev_X \otimes id_{X^*} \circ id_{X^*}\otimes coev_X=id_{X^*}.\end{aligned}$$ Here, we suppress the associator and unit constraints of ${\mathcal{C}}$.
In a fusion category ${\mathcal{C}}$, since $X\cong X^{**}$ and $Hom_{\mathcal{C}}(X,X)\cong {\mathbb{C}}$ for all $X\in{\mathcal{O}}({\mathcal{C}})$, up to scalar, there is a unique isomorphism $\alpha_X:X\to X^{**}$. Then for any morphism $f:X\to X$, we define trace of $f$ as the following scalar $$\begin{aligned}
tr(\alpha_X\circ f)=ev_{X^*}\circ (\alpha_X\circ f)\otimes id_{X^*}\circ coev_X: I\to I.\end{aligned}$$ Following [@ENO1; @Mu1], then we define square norm of simple object $X$ as $$\begin{aligned}
|X|^2:=tr(\alpha_X) tr((\alpha_X^*)^{-1}),\end{aligned}$$ and global dimension (or, categorical dimension) of fusion category ${\mathcal{C}}$ as $$\begin{aligned}
dim({\mathcal{C}}):=\sum_{X\in{\mathcal{O}}({\mathcal{C}})}|X|^2.\end{aligned}$$
Fusion category ${\mathcal{C}}$ is said to be pseudo-unitary if $dim({\mathcal{C}})=FPdim({\mathcal{C}})$. It is easy to see that global dimension $dim({\mathcal{C}})$ is independent of the choice of isomorphisms ${\{\alpha_X|X\in{\mathcal{O}}({\mathcal{C}})}\}$. Moreover, $dim({\mathcal{C}})\geq1$ is a positive algebraic integer [@ENO1 Theorem 2.3] for an arbitrary fusion category ${\mathcal{C}}$. It follows from [@ENO1 Proposition 8.22] that the ratio $\frac{dim({\mathcal{C}})}{FPdim({\mathcal{C}})}\leq1$ is an algebraic integer. In addition, $dim({\mathcal{C}})>\frac{4}{3}$ if ${\mathcal{C}}$ is a spherical fusion category [@O3 Theorem 1.1.2]. Definitely, for non-trivial pseudo-unitary fusion categories ${\mathcal{C}}$, $dim({\mathcal{C}})\leq 2$. For more properties of global dimension, we refer the readers to references [@EGNO; @ENO1; @Mu1; @O3].
Let ${\mathcal{C}}$ be a pivotal fusion category with pivotal structure $j$, which is a natural isomorphism from $id_{\mathcal{C}}$ to the double dual tensor functor $(-)^{**}$. Then we define $dim_j(X):=tr(j_X)$, the quantum (or categorical) dimension of $X$ determined by $j$. A direct computation shows that $j_{X^*}=((j_X)^*)^{-1}$ [@EGNO Exercise 4.7.9], so $\overline{dim_j(X)}=dim_j(X^*)$ by [@ENO1 Proposition 2.9]. Hence, $$\begin{aligned}
dim({\mathcal{C}})=\sum_{X\in{\mathcal{O}}({\mathcal{C}})}dim_j(X)dim_j(X^*)=\sum_{X\in{\mathcal{O}}({\mathcal{C}})}dim_j(X)\overline{dim_j(X)}.\end{aligned}$$
It is well-known that $dim_j(-)$ induces a homomorphism from $Gr({\mathcal{C}})$ to ${\mathbb{C}}$ [@EGNO Proposition 4.7.12]. It was conjectured in [@ENO1 Conjecture 2.8] that every fusion category admits a pivotal structure. Pivotal fusion category ${\mathcal{C}}$ is spherical, if $dim_j(X)=dim_j(X^*)$ for any object $X$ of ${\mathcal{C}}$. Thus, for spherical fusion category ${\mathcal{C}}$, $$\begin{aligned}
dim({\mathcal{C}})=\sum_{X\in{\mathcal{O}}({\mathcal{C}})}dim_j(X)^2.\end{aligned}$$ We fix a spherical structure $j$ of ${\mathcal{C}}$, and we use $dim(X)$ instead of $dim_j(X)$ to denote the dimension of $X$ below.
Notice that for an arbitrary spherical fusion category ${\mathcal{C}}$, we can consider twist ${\mathcal{C}}^\sigma$ of ${\mathcal{C}}$, where $\sigma$ belongs to Galois group $Gal(\overline{\mathbb{Q}}/\mathbb{Q})$. Specifically, ${\mathcal{C}}^\sigma$ is a fusion category with same monoidal functor $\otimes$ as ${\mathcal{C}}$, but associator of ${\mathcal{C}}^\sigma$ is obtained by composing the one of ${\mathcal{C}}$ with automorphism $\sigma$. Moreover, $dim({\mathcal{C}}^\sigma)=\sigma(dim({\mathcal{C}}))$.
In the last, we say a fusion category ${\mathcal{C}}$ is simple, if ${\mathcal{C}}$ does not contain any fusion subcategory other than ${\mathcal{C}}$ and $Vec$. For example, pointed fusion category $Vec^\omega_{{\mathbb{Z}}_p}$ is simple for any prime $p$, where $\omega\in Z^3({\mathbb{Z}}_p,{\mathbb{C}}^*)$ is a $3$-cocycle; and fusion category $Rep(G)$ is simple if and only if $G$ is a finite simple group, where $Rep(G)$ is the category of finite-dimensional representations of $G$ over ${\mathbb{C}}$.
Pre-modular fusion category {#subsection2.2}
---------------------------
Fusion category ${\mathcal{C}}$ is a braided fusion category, if for any $X,Y,Z\in{\mathcal{C}}$, there exists a natural isomorphism $c_{X,Y}:X\otimes Y\to Y\otimes X$, and braiding $c$ satisfies $c_{X,I}=c_{I,X}=id_X$, $c_{X\otimes Y,Z}=c_{X,Z}\otimes id_Y \circ id_X\otimes c_{Y,Z}$, $c_{Z,X\otimes Y}=id_X\otimes c_{Z,Y}\circ c_{Z,X}\otimes id_Y$, here we suppress the associativity isomorphism of ${\mathcal{C}}$.
Let ${\mathcal{D}}$ be a fusion subcategory of braided fusion category $({\mathcal{C}},c)$, the centralizer ${\mathcal{D}}'$ [@Mu] of ${\mathcal{D}}$ is the fusion subcategory generated by all simple objects $X$ of ${\mathcal{C}}$ such that $c_{Y,X} c_{X,Y}=id_{X\otimes Y}$, $\forall Y\in{\mathcal{D}}$. In particular, we call ${\mathcal{C}}'$ the Müger center of ${\mathcal{C}}$.
Braided fusion category ${\mathcal{C}}$ is a pre-modular (or ribbon) category, if ${\mathcal{C}}$ is spherical. For pre-modular fusion category ${\mathcal{C}}$, we can define $S$-matrix $S=(s_{X,Y})_{X,Y\in{\mathcal{O}}({\mathcal{C}})}$ and $T$-matrix $T=(T_{X,Y})_{X,Y\in{\mathcal{O}}({\mathcal{C}})}$ of ${\mathcal{C}}$ [@BK]. Specifically, $s_{X,Y}=tr(c_{Y,X}c_{X,Y})$ and $T_{X,Y}=\delta_{X,Y^*}\theta_X$ for all $X,Y\in{\mathcal{O}}({\mathcal{C}})$, where $\theta$ is the ribbon structure of ${\mathcal{C}}$.
Consequently, pre-modular fusion category ${\mathcal{C}}$ is modular if and only if its $S$-matrix is non-degenerate [@BK; @DrGNO2; @EGNO; @Mu], equivalently Müger center ${\mathcal{C}}'=Vec$, where $Vec$ is the category of finite-dimensional vectors spaces over ${\mathbb{C}}$. We use ${\mathcal{C}}(G,\eta)$ is the pointed modular fusion category determined by metric group $(G,\eta)$ below, see [@DrGNO2 Appendix A]. Meanwhile, a pre-modular fusion category ${\mathcal{C}}$ is super-modular, if ${\mathcal{C}}'\cong sVec$, where $sVec$ is the category of finite-dimensional super-vectors spaces over ${\mathbb{C}}$.
Then it follows from [@Mu Lemma 5.4] or [@DrGNO2 Lemma 3.28] that rank of a super-modular fusion category ${\mathcal{C}}$ must be even. In fact, given a super-modular fusion category ${\mathcal{C}}$, let ${\mathcal{C}}'=\langle\chi\rangle=sVec$. Based on the partition of set ${\mathcal{O}}({\mathcal{C}})=\Pi_0\cup \Pi_1$ of [@BGNPRW], we can define the so-called naive fusion rule: for arbitrary simple objects $X, Y, Z\in\Pi_0$, $$\begin{aligned}
\widehat{N}_{X,Y}^Z:=dim_{\mathbb{C}}(Hom(X\otimes Y,Z))+dim_{\mathbb{C}}(Hom(X\otimes Y,\chi\otimes Z))\end{aligned}$$ Indeed, for any $W\in\Pi_1$ there is a unique $V\in\Pi_0$ such that $W=\chi\otimes V$.
Then there is a non-degenerate symmetric matrix $\widehat{S}$ such that the $S$-matrix of ${\mathcal{C}}$ is $\left(
\begin{array}{cc}
\widehat{S} & \widehat{S} \\
\widehat{S} & \widehat{S} \\
\end{array}
\right)
$, and $\widehat{S}$ has orthogonal rows. Moreover, as $\Pi_0$ is close under taking dual objects, the naive fusion rule and $\Pi_0$ generate a unital fusion ring $R$ satisfying that all homomorphisms from $R$ to ${\mathbb{C}}$ have form $\phi(X)=\frac{\widehat{s}_{X,Y}}{\widehat{s}_{I,Y}}$ for some $Y\in\Pi_0$, $\forall X\in\Pi_0$, see [@BGNPRW Proposition 2.7] for details and [@Yu] for some applications. In particular, we can define homomorphism FPdim(-) of fusion ring $R$ as follows: $$\begin{aligned}
\underline{FPdim}(X):= \frac{\widehat{s}_{X,Y}}{\widehat{s}_{I,Y}}, ~\forall X\in\Pi_0, ~\text{ for some $Y\in \Pi_0$}.
\end{aligned}$$ For FP-dimensions of fusion rings, see [@EGNO $\S 3.3$]. Notice that $\underline{FPdim}(X)$ is not FP-dimension of simple object $X$ in ${\mathcal{C}}$, as it is determined by the naive fusion rule $\widehat{N}$.
Formal codegrees and $d$-numbers
--------------------------------
Given a fusion category ${\mathcal{C}}$, let $Irr(Gr({\mathcal{C}}))$ be the set of isomorphism classes of irreducible representations of $Gr({\mathcal{C}})$ over ${\mathbb{C}}$. For irreducible representations $\chi,\chi'\in Irr(Gr({\mathcal{C}}))$, let $Tr_\chi$ be ordinary trace function on representation $\chi$. Then up to scalar, by [@Lusztig; @O1] there exists a unique central element $\alpha_\chi:=\sum_{X\in{\mathcal{O}}({\mathcal{C}})}Tr_\chi(X)X^*\in Gr({\mathcal{C}})\otimes_{\mathbb{Z}}{\mathbb{C}}$ such that $\chi'(\alpha_\chi)=0$ if $\chi\ncong\chi'$, and $f_\chi=\chi(\alpha_\chi)$ is a positive algebraic integer. In fact, it is well-defined for all fusion rings [@Lusztig].
We call these algebraic integers $f_\chi$ $(\forall \chi\in Irr(Gr({\mathcal{C}})))$ formal codegrees of ${\mathcal{C}}$ [@O1]. It is proved in [@O2 Corollary 2.14] that $\frac{dim({\mathcal{C}})}{f_\chi}$ are also algebraic integers. And formal codegrees of fusion category ${\mathcal{C}}$ satisfy the following equation [@O2 Proposition 2.10] $$\begin{aligned}
\label{classequation}
\sum_{\chi\in Irr(Gr({\mathcal{C}}))}\frac{\chi(1)}{f_\chi}=1.\end{aligned}$$ Obviously, we have $f_\chi\geq1$. Moreover, if spherical fusion category ${\mathcal{C}}\ncong Vec$, then $f_\chi\geq\sqrt{\frac{2rank({\mathcal{C}})}{rank({\mathcal{C}})+1}}\geq\sqrt{\frac{4}{3}}$ for all $\chi\in Irr(Gr({\mathcal{C}}))$ by [@O3 Theorem 4.2.1].
Note that if there exists a homomorphisms $\phi$ from $Gr({\mathcal{C}})$ to ${\mathbb{C}}$, definitely it is an irreducible representation of $Gr({\mathcal{C}})$. Then the corresponding formal codegrees $f_\phi$ of ${\mathcal{C}}$ is given by $$\begin{aligned}
\sum_{X\in{\mathcal{O}}({\mathcal{C}})}\phi(X) \phi(X^*)=\sum_{X\in{\mathcal{O}}({\mathcal{C}})}\phi(XX^*)=\sum_{X\in{\mathcal{O}}({\mathcal{C}})}|\phi(X)|^2.
\end{aligned}$$ In particular, for arbitrary fusion category ${\mathcal{C}}$, $FPdim(-)$ is an irreducible representation of Grothendieck ring $Gr({\mathcal{C}})$, so $FPdim({\mathcal{C}})$ and its Galois conjugates are formal codegrees of ${\mathcal{C}}$. Similarly, if ${\mathcal{C}}$ is pivotal then $dim({\mathcal{C}})$ is also a formal codegree.
For pre-modular fusion category ${\mathcal{C}}$, let $X,Y\in{\mathcal{O}}({\mathcal{C}})$, then map $h_X(Y):=\frac{s_{X,Y}}{s_{I,X}}$ defines a homomorphism from Grothendieck ring $Gr({\mathcal{C}})$ to ${\mathbb{C}}$ [@EGNO Proposition 8.13.11]. In particular, if ${\mathcal{C}}$ is modular, then Verlinde formula [@BK; @EGNO] says that set ${\{h_X|\forall X\in{\mathcal{O}}({\mathcal{C}})}\}$ is a complete set of homomorphisms from Grothendieck ring $Gr({\mathcal{C}})$ to ${\mathbb{C}}$. Therefore, if ${\mathcal{C}}$ is modular, then formal codegrees are equal to $\frac{dim({\mathcal{C}})}{dim(X)^2}$ for objects $X\in{\mathcal{O}}({\mathcal{C}})$. Indeed, for $s_{I,X}=dim(X)$, Verlinde formula implies that $$\begin{aligned}
\sum_{Y\in{\mathcal{O}}({\mathcal{C}})}h_X(Y)h_X(Y^*)=\sum_{Y\in{\mathcal{O}}({\mathcal{C}})}\frac{s_{X,Y}}{dim(X)}\frac{s_{X,Y^*}}{dim(X)}
=\frac{\sum_{Y\in{\mathcal{O}}({\mathcal{C}})}s_{X,Y}s_{X,Y^*}}{dim(X)^2}=\frac{dim({\mathcal{C}})}{dim(X)^2}.\end{aligned}$$ Therefore, if $dim(X)^2\in{\mathbb{Z}}$ for all $X\in{\mathcal{O}}({\mathcal{C}})$, then ${\mathcal{C}}$ is weakly integral, as the rational number $FPdim({\mathcal{C}})=\frac{dim({\mathcal{C}})}{dim(Y)^2}$ (for certain $Y\in{\mathcal{O}}({\mathcal{C}})$) is an algebraic integer.
Recall that an algebraic integer $\alpha$ is a $d$-number [@O1 Definition 1.1], if in the algebraic integer ring, the ideal generated by $\alpha$ is invariant under action of Galois group $Gal(\overline{\mathbb{Q}}/\mathbb{Q})$. Equivalently, there exists a polynomial $f(x)=x^n+a_1x^{n-1}+\cdots a_{n-1}x+a_n$ with integer coefficients such that $f(\alpha)=0$ and $(a_n)^i|(a_i)^n$, for all $1\leq i\leq n$. See [@O1 Lemma 2.7] for more equivalent conditions about $d$-number.
For example, $dim(X)$ of simple objects $X$ of pre-modular fusion category ${\mathcal{C}}$ are $d$-numbers [@O1 Theorem 1.8]. Moreover, it was proved in [@O1 Theorem 1.2] that formal codegrees of fusion categories are $d$-numbers. However, formal codegrees of fusion rings are not $d$-numbers in general, see [@O1 Example 1.6]. Thus, we can use this property to detect whether a fusion ring is categorifiable, this is called $d$-number test.
In addition, following [@EGNO Definition 7.21.13], we say that an algebraic integer $\alpha$ is totally positive, if $\alpha$ is still positive under any embedding of algebraic integers into field ${\mathbb{C}}$. For example, $\sqrt{5}$ is not a totally positive integer. For any fusion category ${\mathcal{C}}$, let $X\in{\mathcal{O}}({\mathcal{C}})$, $FPdim(X)^2$ is totally positive, thus $FPdim({\mathcal{C}})$ is totally positive. Indeed, all formal codegrees of ${\mathcal{C}}$ are totally positive by [@O2 Remark 2.12].
We also use the following theorem [@ENO1 Corollary 8.53], which is called cyclotomic test in [@O2 Proposition 2.1].
\[cyclotomic\]Given a fusion category ${\mathcal{C}}$, let $\chi $ be an irreducible representation of Grothendieck ring $Gr({\mathcal{C}})$. Then $\chi$ is defined over $\mathbb{Q}(\xi)$ for some root of unity $\xi$.
Therefore, $FPdim(X),FPdim({\mathcal{C}})\in{\mathbb{Z}}[\xi]$, $\forall X\in{\mathcal{O}}({\mathcal{C}})$. That is, the Galois groups of minimal polynomials defining FP-dimension of simple objects and ${\mathcal{C}}$ have to be abelian. For this purpose, we use program GAP to compute Galois groups of minimal polynomials of $FPdim({\mathcal{C}})$, also we use GAP to do $d$-number test.
Main result {#mainresult}
===========
In this section, we first prove the Lagrange theorem for global dimension of fusion categories. Then we consider classifications of pre-modular fusion categories of global dimensions $6,7,8$ and $10$, respectively.
Lagrange theorem of global dimension {#subsection3.1}
------------------------------------
Given a fusion category ${\mathcal{C}}$, an algebra $A\in{\mathcal{C}}$ is connected, if $dim_{\mathbb{C}}(Hom_{\mathcal{C}}(I,A))=1$. In a braided fusion category ${\mathcal{C}}$, recall that a commutative algebra $A$ is said to be an étale algebra, if ${\mathcal{C}}_A$ is semisimple [@DMNO Proposition 2.7], where ${\mathcal{C}}_A$ is the category of right $A$-modules in ${\mathcal{C}}$. For a connected étale algebra $A$ in ${\mathcal{C}}$, the subcategory $ {\mathcal{C}}^0_A\subseteq{\mathcal{C}}_A$ of dyslectic (or local) modules of $A$ in $ {\mathcal{C}}$ is a braided fusion category. See [@DMNO; @KiO] for details about étale algebras and their dyslectic modules.
Now we are ready to give a proof of the Lagrange theorem for global dimension of fusion categories. For pseudo-unitary fusion categories, this is [@ENO1 Proposition 8.15].
\[Lagrange\]Let ${\mathcal{C}}$ be a fusion category, and let ${\mathcal{D}}$ be a fusion subcategory of ${\mathcal{C}}$. Then $\frac{dim({\mathcal{C}})}{dim({\mathcal{D}})}$ is an algebraic integer.
We can assume that ${\mathcal{C}}$ is a spherical fusion category. In fact, if not, it follows from [@ENO1 Remark 3.1] and [@ENO1 Proposition 5.14] that pivotalization $\widetilde{{\mathcal{C}}}$ of ${\mathcal{C}}$ is a spherical fusion category. In addition, we have equality $dim(\widetilde{{\mathcal{C}}})=2dim({\mathcal{C}})$ [@EGNO Remark 7.21.11]. Consequently, $\frac{dim({\mathcal{C}})}{dim({\mathcal{D}})}=\frac{dim(\widetilde{{\mathcal{C}}})}{dim(\widetilde{{\mathcal{D}}})}$.
For any fusion subcategory ${\mathcal{D}}\subseteq{\mathcal{C}}$, up to isomorphism, [@DMNO Theorem 4.10] says that there exists a unique connected étale subalgebra $A\subseteq{\mathcal{I}}(I)$ corresponding to ${\mathcal{D}}$, which satisfies $\frac{FPdim({\mathcal{C}})}{FPdim({\mathcal{D}})}=FPdim(A)$, where ${\mathcal{I}}$ is the right adjoint functor of forgetful tensor functor $F:{\mathcal{Z}}({\mathcal{C}})\to{\mathcal{C}}$. Since ${\mathcal{C}}$ is a spherical fusion category, Drinfeld center ${\mathcal{Z}}({\mathcal{C}})$ is a modular fusion category by [@Mu2 Theorem 6.4]. Note that $A=\oplus_{X\in{\mathcal{O}}({\mathcal{Z}}({\mathcal{C}}))}[A:X]X$, where $[A:X]:=dim_{\mathbb{C}}(Hom_{{\mathcal{Z}}({\mathcal{C}})}(A,X))$, hence $$\begin{aligned}
dim(A)=\sum_{X\in{\mathcal{O}}({\mathcal{Z}}({\mathcal{C}}))}[A:X]dim(X),\end{aligned}$$ If $[A:X]$ is non-zero, then $dim(X)=\frac{dim({\mathcal{C}})}{f_\chi}$ by [@O2 Theorem 2.13] for some formal codegree $f_\chi$ of ${\mathcal{C}}$. In particular, since all formal codegrees of ${\mathcal{C}}$ are positive algebraic integers, we obtain $dim(A)>0$. Thus by [@DMNO Remark 3.4], étale algebra $A$ is a rigid ${\mathcal{C}}$-algebra in sense of [@KiO Definition 1.11].
Notice that [@DMNO Theorem 4.10] shows that there exists a braided equivalence of non-degenerate fusion categories ${\mathcal{Z}}({\mathcal{D}})\cong{\mathcal{Z}}({\mathcal{C}})^0_A$. Meanwhile, $\theta_{{\mathcal{I}}(I)}=id_{{\mathcal{I}}(I)}$ by [@NS Theorem 4.1] or [@O3 Theorem 2.7], where $\theta$ is the ribbon structure of ${\mathcal{Z}}({\mathcal{C}})$. Then $\theta_A=id_A$ as $A\subseteq{\mathcal{I}}(I)$, and [@KiO Theorem 4.5] says that ${\mathcal{Z}}({\mathcal{D}})\cong{\mathcal{Z}}({\mathcal{C}})^0_A$ as modular fusion categories. Moreover, $dim({\mathcal{Z}}({\mathcal{D}}))=dim({\mathcal{D}})^2$ by [@ENO1 Theorem 2.15] or [@Mu2 Theorem 1.2]. Then we deduce from [@KiO Theorem 4.5] again that $$dim({\mathcal{D}})^2=dim({\mathcal{Z}}({\mathcal{D}}))=dim({\mathcal{Z}}({\mathcal{C}})^0_A)=\frac{dim({\mathcal{Z}}({\mathcal{C}}))}{dim(A)^2}=\frac{dim({\mathcal{C}})^2}{dim(A)^2},$$ so the ratio $\frac{dim({\mathcal{C}})}{dim({\mathcal{D}})}=dim(A)$ is an algebraic integer, as desired.
\[nonequratios\]Let ${\mathcal{C}}$ be a fusion category. For fusion subcategory ${\mathcal{D}}\subseteq{\mathcal{C}}$, Theorem \[Lagrange\] also says that $\frac{dim({\mathcal{C}})}{dim({\mathcal{D}})}=\frac{FPdim({\mathcal{C}})}{FPdim({\mathcal{D}})}$ if and only if $FPdim(A)=dim(A)$. This equality fails in general, however. For example, let ${\mathcal{D}}=YL$ be the Yang-Lee fusion category of global dimension $\frac{5+\sqrt{5}}{2}$, its conjugate $\overline{YL}$ has global dimension $\frac{5-\sqrt{5}}{2}$. Assume ${\mathcal{C}}=YL\boxtimes \overline{YL}$, so $dim({\mathcal{C}})=5$ and $FPdim({\mathcal{C}})=(\frac{5+\sqrt{5}}{2})^2=\frac{15+5\sqrt{5}}{2}$ since $FPdim(YL)=\frac{5+\sqrt{5}}{2}$. These two ratios are not equal obviously.
Given two tensor categories ${\mathcal{C}}$ and ${\mathcal{D}}$, recall that a tensor functor $F:{\mathcal{C}}\to{\mathcal{D}}$ is said to be surjective, if every simple object of ${\mathcal{D}}$ is a subobject of $F(X)$ for some object $X\in{\mathcal{C}}$; $F$ is injective if $F$ is bijective on sets of morphisms [@EGNO Definition 1.8.3].
Same as [@ENO1 Corollary 8.11], we have the following corollary:
\[quotient\]Let ${\mathcal{C}}$ and ${\mathcal{D}}$ be fusion categories, assume $F:{\mathcal{C}}\to{\mathcal{D}}$ is a surjective tensor functor. Then $\frac{dim({\mathcal{C}})}{dim({\mathcal{D}})}$ is an algebraic integer.
Since $F$ is a surjective tensor functor, we can regard ${\mathcal{D}}$ as an indecomposable left ${\mathcal{C}}$-module category via tensor functor $F$. Let ${\mathcal{C}}^*_{\mathcal{D}}$ and ${\mathcal{D}}_{\mathcal{D}}^*$ be the tensor categories of left module functors of ${\mathcal{C}}$ and ${\mathcal{D}}$ with respect to module category ${\mathcal{D}}$, respectively. This is the so-called exact pair $(F,{\mathcal{D}})$, see [@EGNO $\S 7.17$] and [@ENO1] for details.
Therefore, we deduce from [@EGNO Corollary 7.12.13] that both ${\mathcal{C}}^*_{\mathcal{D}}$ and ${\mathcal{D}}_{\mathcal{D}}^*$ are fusion categories. Then we obtain an injective tensor functor $F^*:{\mathcal{D}}^*_{\mathcal{D}}\to{\mathcal{C}}^*_{\mathcal{D}}$ by [@ENO1 Proposition 5.3]. It follows from [@EGNO Proposition 9.3.9] that we have the following equation $$\begin{aligned}
\frac{dim({\mathcal{C}})}{dim({\mathcal{D}})}= \frac{dim({\mathcal{C}}^*_{\mathcal{D}})}{dim({\mathcal{D}}_{\mathcal{D}}^*)},\end{aligned}$$ which is an algebraic integer by Theorem \[Lagrange\].
In Corollary \[quotient\], it is easy to see that $\frac{dim({\mathcal{C}})}{dim({\mathcal{D}})} \neq\frac{FPdim({\mathcal{C}})}{FPdim({\mathcal{D}})}$ in general. For example, let ${\mathcal{D}}$ be a fusion category such that $dim({\mathcal{D}})\neq FPdim({\mathcal{D}})$, and ${\mathcal{C}}={\mathcal{Z}}({\mathcal{D}})$. Let $F:{\mathcal{Z}}({\mathcal{D}})\to{\mathcal{D}}$ be the forgetful tensor functor, so $F$ is surjective. Then [@ENO1 Theorem 2.15] and [@ENO1 Proposition 8.12] say that $\frac{dim({\mathcal{C}})}{dim({\mathcal{D}})}=dim({\mathcal{D}})\neq FPdim({\mathcal{D}})=\frac{FPdim({\mathcal{C}})}{FPdim({\mathcal{D}})}$.
Pre-modular fusion categories of dimensions $7$ {#subsection3.2}
-----------------------------------------------
Spherical fusion categories ${\mathcal{C}}$ of integer dimension with $dim({\mathcal{C}})\leq 5$ were classified completely in [@O3 Example 5.1.2]. Explicitly, ${\mathcal{C}}$ is pointed, or ${\mathcal{C}}$ is equivalent to an Ising category ${\mathcal{I}}$, or ${\mathcal{C}}$ is equivalent to the Deligne tensor product $YL\boxtimes \overline{YL}$.
We first classify pre-modular fusion categories of dimension $6$.
\[premoddim6\]Let ${\mathcal{C}}$ be a pre-modular fusion category of global dimension $6$, then ${\mathcal{C}}$ is an integral fusion category.
We consider Müger center ${\mathcal{C}}'$ of ${\mathcal{C}}$. Obviously, ${\mathcal{C}}$ is integral if ${\mathcal{C}}$ is symmetric. If ${\mathcal{C}}'$ is a proper subcategory of ${\mathcal{C}}$, then $dim({\mathcal{C}}')=1,2,3$ by Theorem \[Lagrange\]. If ${\mathcal{C}}'=Rep(G)$ is Tannakian, then ${\mathcal{C}}\cong{\mathcal{D}}^G$ is integral by [@O3 Example 5.1.2]. If ${\mathcal{C}}'\cong sVec$, then $rank({\mathcal{C}})=4,6$. [@O3 Remark 4.2.3] shows that $rank({\mathcal{C}})=6$ if and only if ${\mathcal{C}}$ is pointed. If $rank({\mathcal{C}})=4$, then for any simple object $X\in{\mathcal{C}}$, $dim(X)^2\in{\{1,2}\}$. Meanwhile, [@Yu Corollary 3.4] shows that $\frac{6}{2dim(X)^2}$ is an algebraic integer, this is impossible. If ${\mathcal{C}}$ is modular, when $rank({\mathcal{C}})<6$, results of [@BNRW; @RSW] imply that there is no such modular fusion category. Therefore, ${\mathcal{C}}$ is an integral fusion category.
For any modular fusion category ${\mathcal{A}}$, let ${\mathcal{E}}$ be an arbitrary symmetric fusion subcategory of ${\mathcal{A}}$, then ${\mathcal{E}}\subseteq{\mathcal{E}}'$ by definition. [@DrGNO2 Theorem 3.10] and Theorem \[Lagrange\] show that $\frac{dim({\mathcal{A}})}{dim({\mathcal{E}})^2}$ is an algebraic integer. Assume $p$ is a prime. Let ${\mathcal{C}}$ be a pre-modular fusion category of global dimension $p$. Since ${\mathcal{C}}'$ is a symmetric fusion subcategory of ${\mathcal{C}}$, and $dim({\mathcal{C}}')=FPdim({\mathcal{C}}')$ is an integer, Theorem \[Lagrange\] shows that rational number $\frac{p}{dim({\mathcal{C}}')}$ is an algebraic integer. Then ${\mathcal{C}}$ is symmetric if $p=dim({\mathcal{C}}')$, otherwise ${\mathcal{C}}$ is modular. If ${\mathcal{C}}$ is symmetric, obviously ${\mathcal{C}}\cong Rep({\mathbb{Z}}_p)$ or ${\mathcal{C}}\cong sVec$, so ${\mathcal{C}}$ is a pointed fusion category.
We assume that pre-modular fusion category ${\mathcal{C}}$ is not pointed below. In particular, ${\mathcal{C}}$ is modular and ${\mathcal{C}}_{int}={\mathcal{C}}_{pt}=Vec$ by Theorem \[Lagrange\]. Moreover, when prime $p>5$, it follows from [@BNRW; @RSW] that dimensions of modular fusion categories of rank less than $6$ do not equal to $p$, then [@O3 Lemma 4.2.2] shows that $6\leq rank({\mathcal{C}})\leq p-1$. Let ${\mathcal{D}}\subseteq{\mathcal{C}}$ be an arbitrary fusion subcategory, note that ${\mathcal{D}}\cap{\mathcal{D}}'$ is symmetric, where ${\mathcal{D}}'$ is the centralizer of ${\mathcal{D}}$, so ${\mathcal{D}}\cap{\mathcal{D}}'=Vec$. Therefore, ${\mathcal{D}}$ is also a modular fusion category. Consequently, it follows from [@DrGNO2 Theorem 3.13] that ${\mathcal{C}}\cong{\mathcal{A}}_1\boxtimes\cdots\boxtimes{\mathcal{A}}_s$, where ${\mathcal{A}}_i$ are simple modular fusion subcategories of ${\mathcal{C}}$, $1\leq i\leq s<rank({\mathcal{C}})\leq p$.
Note that $rank({\mathcal{C}})=\prod_{i=1}^s rank({\mathcal{A}}_i)$. Hence, in order to classify modular fusion categories of prime global dimensions, first we have to consider classification of simple non-pointed modular fusion categories of small ranks. When $dim({\mathcal{C}})=7,11,13$, classification results of [@BNRW; @RSW] say that modular fusion category ${\mathcal{C}}$ must be simple.
To classify modular fusion categories of global dimension $7$, we need to prove the following two lemmas.
\[lemmdimen7(1)\]If there exists a fusion category ${\mathcal{C}}$ of rank $6$ and $FPdim({\mathcal{C}})=\frac{21+7\sqrt{5}}{2}$, moreover ${\mathcal{C}}_{int}=Vec$, and $FPdim(X)\in \mathbb{Q}(\sqrt{5})$ for all $ X\in{\mathcal{O}}({\mathcal{C}})$. Then $$\begin{aligned}
FPdim({\mathcal{C}})=\sum_{X\in{\mathcal{O}}({\mathcal{C}})}FPdim(X)^2=1+4\times(\frac{1+ \sqrt{5}}{2})^2+(\frac{3+\sqrt{5}}{2})^2
\end{aligned}$$ is the unique decomposition into sum of squares of FP-dimensions of simple objects.
Let $FPdim({\mathcal{C}})=1+\sum^5_{i=1}\frac{a_i+b_i\sqrt{5}}{2}$, where $a_1,b_i$ are positive rational numbers, and $FPdim(X_i)^2=\frac{a_i+b_i\sqrt{5}}{2}$, $1\leq\forall i\leq5$. Indeed, by assumption if $b_i\neq0$. If $b_i<0$, since $\frac{a_i+b_i\sqrt{5}}{2}$ is square of FP-dimension of simple objects, while $\frac{a_i-b_i\sqrt{5}}{2}>\frac{a_i+b_i\sqrt{5}}{2}$, this contradicts to property of FP-dimension of simple object [@EGNO Proposition 3.3.4]. If $a_i\leq0$, then by definition $\frac{a_i+b_i\sqrt{5}}{2}$ can not totally positive algebraic integers.
Next, we show $a_i$ and $b_i$ are integers. Let $\alpha_i=\frac{a_i+b_i\sqrt{5}}{2}$, then $\alpha_i$ are roots of equations $4x^2-4a_ix+a_i^2-5b_i^2=0$. Hence, $a_i\in{\mathbb{Z}}$. Since $\frac{a_i+b_i\sqrt{5}}{2}$ is a totally positive algebraic integer, $\frac{a_i-b_i\sqrt{5}}{2}$ is also positive. Let $4m=a_i^2-5b_i^2$, where $m$ is a positive integer, then $b_i=\sqrt{\frac{a_i^2-4m}{5}} $ is a rational number, which means $b_i$ is an integer.
Assume $b_i\leq b_j$ whenever $i\leq j$. Therefore, $(b_1,b_2,b_3,b_4,b_5)=(1,1,1,2,2)$ or $(1,1,1,1,3)$. In the first case, if $X_5\in{\mathcal{O}}({\mathcal{C}})$, and $$\begin{aligned}
FPdim(X_5)^2=\frac{a_5+2\sqrt{5}}{2}=(\frac{\alpha_5+\beta_5\sqrt{5}}{2})^2
=\frac{\alpha_5^2+5\beta_5^2}{4}+\frac{\alpha_5\beta_5\sqrt{5}}{2},\end{aligned}$$ then $\left\{
\begin{array}{ll}
\alpha_5=2 \\
\beta_5=1
\end{array}
\right.$ or $\left\{
\begin{array}{ll}
\alpha_5=1 \\
\beta_5=2
\end{array}
\right.$. This is impossible, as both $\frac{2+\sqrt{5}}{2}$ and $\frac{1+2\sqrt{5}}{2}$ are not an algebraic integers. In the second case, similarly, we have $\left\{
\begin{array}{ll}
\alpha_5=1 \\
\beta_5=3
\end{array}
\right.$ or $\left\{
\begin{array}{ll}
\alpha_5=3 \\
\beta_5=1
\end{array}
\right.$. If $FPdim(X_5)=\frac{1+3\sqrt{5}}{2}$, then $FPdim(X_5)^2=\frac{23+3\sqrt{5}}{2}$, this is impossible. Then we obtain that $FPdim(X_5)=\frac{3+\sqrt{5}}{2}$ and $FPdim(X_j)=\frac{1+\sqrt{5}}{2}$, for $1\leq j\leq4$.
\[lemmdimen7(2)\]Let ${\mathcal{C}}$ be a fusion category of rank $6$ and $FPdim({\mathcal{C}})=\frac{21+7\sqrt{5}}{2}$, moreover ${\mathcal{C}}_{int}=Vec$, and $FPdim(X)\in \mathbb{Q}(\sqrt{5})$ for all $ X\in{\mathcal{O}}({\mathcal{C}})$. Then exists exactly two self-dual simple objects in ${\mathcal{O}}({\mathcal{C}})$.
Assume ${\mathcal{O}}({\mathcal{C}})={\{I,V,W, X,Y,Z}\}$. By Lemma \[lemmdimen7(1)\], let $FPdim(Z)=\frac{3+\sqrt{5}}{2}$, FP-dimensions of $V,W,X,Y$ are all equal to $\frac{1+\sqrt{5}}{2}$. Obviously, $Z$ is self-dual.
If $V,W,X,Y$ are self-dual simple objects. Let $A\ncong X$ be an arbitrary simple object of FP-dimension $\frac{1+\sqrt{5}}{2}$, then by computing FP-dimension of $X\otimes A$, $X\otimes A=Z$. Meanwhile, $X\otimes X=I\oplus B$, where $B$ is another simple object. We claim $B \ncong X$. Indeed, if $B\cong X$, then ${\mathcal{C}}$ contains a Yang-Lee fusion category $YL=\langle X\rangle$. However, the ratio $\frac{FPdim({\mathcal{C}})}{FPdim(YL)}$ is not an algebraic integer, this contradicts to [@ENO1 Proposition 8.15]. Again, $B\otimes X\cong Z$. While, $(B\otimes X)\otimes X=Z\otimes X$, and $$\begin{aligned}
(B\otimes X)\otimes X\cong B\otimes (X\otimes X)\cong B\otimes (I\oplus B)\supseteq I,\end{aligned}$$ which implies that $X$ is a dual object of $Z$, this is impossible.
If $X,Y$ are self-dual simple objects, and $V^*=W$. Then $X\otimes X=I\oplus A$ and $Y\otimes Y=I\oplus B$, where $A,B$ are self-dual simple objects. Since ${\mathcal{C}}$ does not contain a Yang-Lee fusion category, we must have $A\cong Y$ and $B\cong X$. In addition, we have $Y\otimes X\cong Z$. However, $$\begin{aligned}
Z\otimes X\cong (Y\otimes X)\otimes X\cong Y\otimes (X\otimes X)\cong Y\otimes (I\oplus Y)\supseteq I,\end{aligned}$$ which again implies that $X$ is a dual object of $Z$, this is impossible.
\[moddimen7\]Let ${\mathcal{C}}$ be a modular fusion category of global dimension $7$. Then is pointed. That is, ${\mathcal{C}}\cong{\mathcal{C}}({\mathbb{Z}}_7,\eta)$.
On the contrary, assume that ${\mathcal{C}}$ is not pointed, then previously argument says that ${\mathcal{C}}$ is a simple modular fusion category of rank $6$. In particular, homomorphisms $dim(-)$ and $FPdim(-)$ are not in the same orbits under the action of Galois group $Gal(\overline{\mathbb{Q}}/\mathbb{Q})$, and each orbit of $dim(-)$ and $FPdim(-)$ has at least two homomorphisms. Hence, homomorphisms from $Gr({\mathcal{C}})$ to ${\mathbb{C}}$ are divided into two or three or four orbits.
(1).
: There are two orbits of homomorphisms from $Gr({\mathcal{C}})$ to ${\mathbb{C}}$.
(i).
: If orbit of $dim(-)$ contains $4$ homomorphisms. Let $f$ be a formal codegree of ${\mathcal{C}}$, which is a Galois conjugate of $FPdim({\mathcal{C}})$. Then [@O2 Proposition 2.10] (or equation \[classequation\]) means that $$\begin{aligned}
\frac{4}{7}+\frac{1}{f}+\frac{1}{FPdim({\mathcal{C}})}=1\end{aligned}$$ While all formal codegrees of ${\mathcal{C}}$ are bigger than $1$ and $FPdim({\mathcal{C}})>7$, moreover $FPdim({\mathcal{C}})\cdot f|7^2$ by [@O2 Corollary 2.14]. Therefore, $FPdim({\mathcal{C}})$ and $f$ are roots of equation $x^2-21 x+49=0$, so $FPdim({\mathcal{C}})=\frac{21+7\sqrt{5}}{2}$ and $f=\frac{21-7\sqrt{5}}{2}$. Since ${\mathcal{C}}$ is modular, all formal codegrees are like $\sum_{X\in{\mathcal{O}}({\mathcal{C}})}h_Y(X) h_Y(X^*)$ for simple objects $Y$, then $$dim(X)^2\in{\{1,\frac{3+\sqrt{5}}{2},\frac{3-\sqrt{5}}{2}}\}, ~\forall X\in{\mathcal{O}}({\mathcal{C}}).$$ Note that there are four simple objects $X$ of ${\mathcal{C}}$ which satisfy $dim(X)^2=1$. Moreover, simple object $X$ is self-dual if $dim(X)^2=\frac{3\pm\sqrt{5}}{2}$. Since ${\mathcal{C}}$ is spherical, $dim(A^*)=dim(A)$ for any simple object $A$ of ${\mathcal{C}}$. Then we see that there are at least four simple objects are self-dual, by Lemma \[lemmdimen7(2)\] such fusion category ${\mathcal{C}}$ does not exist.
(ii).
: If the orbit of $dim(-)$ has $3$ homomorphisms. Let $f_1\leq f_2\leq f_3=FPdim({\mathcal{C}})$ be distinct Galois conjugates of $FPdim({\mathcal{C}})$. By equation \[classequation\], we have $$\begin{aligned}
\frac{3}{7}+\frac{1}{f_1}+\frac{1}{f_2}+\frac{1}{f_3}=1,\end{aligned}$$ so, $f_1>\frac{7}{4}$, $f_2>\frac{7}{2}$ and $f_3>7$. Again, $f_1f_2f_3|7^3$, so $f_1f_2f_3=49$ or $f_1f_2f_3=343$. If $f_1f_2f_3=49$, then $d$-number test [@O1 Lemma 2.7] shows that $f_1, f_2,f_3$ are roots of equation $x^3-14x^2+28x-49=0$, which fails to satisfy cyclotomic test in Theorem \[cyclotomic\]; and similarly, if $f_1f_2f_3=343$, then $f_1, f_2,f_3$ are roots of equation $x^3-\alpha x^2+196x-343=0$, where $\alpha=f_1+f_2+f_3$. Obviously, $d$-number test [@O1 Lemma 2.7] shows that $7|\alpha$, then $\alpha\geq14$. Meanwhile, $$\begin{aligned}
\alpha<(f_1-\frac{3}{4})f_2+(f_3-6)f_1+(f_2-\frac{5}{2})f_3< 196-\frac{3}{2}\alpha,\end{aligned}$$ then $\alpha\leq78$. However, for these integers, a direct computation shows that there is no solution for cyclotomic test in Theorem \[cyclotomic\] if $\alpha\neq 28$. While when $\alpha=28$, roots of equation $x^3-28 x^2+196x-343=0$ are equal to $7,\frac{21\pm7\sqrt{5}}{2}$. By assumption the orbit of $FPdim(-)$ also have three homomorphisms, equation $x^3-28 x^2+196x-343=0$ must not have integer root, this is a contradiction.
(iii).
: If the orbit of $dim(-)$ contains $2$ homomorphisms. Then ${\mathcal{C}}$ contain a unique self-dual non-trivial simple object $X$ such that $dim(X)^2=1$. Let $f_1\leq f_2\leq f_3\leq f_4=FPdim({\mathcal{C}})$ be conjugated formal codegrees of ${\mathcal{C}}$. Assume they are roots of equation $x^4-\beta_1x^3+\beta_2x^2-\beta_3x+\beta_4=0$ $(*)$, where $\beta_i$ are positive integers, $1\leq i\leq4$. Hence equation \[classequation\] shows that $$\begin{aligned}
\frac{2}{7}+\frac{1}{f_1}+\frac{1}{f_2}+\frac{1}{f_3}+\frac{1}{f_4}=1.\end{aligned}$$ Then $f_1>\frac{7}{5}$, $f_2>\frac{14}{5}$, $f_3>\frac{21}{5}$ and $f_4>7$. Hence $\beta_1\geq21$ and $\beta_3>10\beta_1$. Note that $\beta_2\geq 6\sqrt{\beta_4}$, and $\beta_2= 6\sqrt{\beta_4}$ if and only if $f_1=f_2=f_3=f_4$, so $\beta_2> 6\sqrt{\beta_4}$ and $\beta_1>4\sqrt[4]{\beta_4}$. We know that $\beta_4|7^4$, then $\beta_4=343,2401$. Meanwhile, let $\beta_2=7^yn$, where $n$ is a positive integer such that $(7,n)=1$, $d$-number test [@O1 Lemma 2.7] shows $\beta_4^2|\beta_2^4$, so $y\geq2$. Also, $$\begin{aligned}
&\beta_2=\sum_{1\leq i<j\leq4}f_if_j
\\&<f_1(f_2+f_3)+(f_2-\frac{9}{5})f_1f_4+(f_1-\frac{2}{5})f_2f_3+
(f_3-\frac{16}{5})f_2f_4+(f_1-\frac{2}{5})f_3f_4
\\&<\beta_3-f_4(\frac{16}{5}f_2+\frac{2}{5}f_3+\frac{1}{5}f_1).\end{aligned}$$ When $\beta_4=343$, then $\beta_1=21$, $\beta_3=245$, and $98\leq\beta_2\leq\beta_3-77=168$. So $\beta_2=98, 147$ for $49|\beta_2$. In these two cases, they do not satisfy cyclotomic test in Theorem \[cyclotomic\]. When $\beta_4=2401$, then $\beta_3=1715$, $35\leq\beta_1\leq168$. Since $1715=\beta_3>21f_4$, $\beta_2\leq\beta_3-890=825$. While, except when $\beta_2=392$ and $\beta_1=35$, either all roots of equation $(*)$ are not real or they don’t satisfy cyclotomic test in Theorem \[cyclotomic\]. When $\beta_2=392$ and $\beta_1=35$, roots of equation $x^4-35x^3+392x^2-1715x+2401=0$ are $7,7,\frac{21\pm7\sqrt{5}}{2}$, this is a contradiction by argument of subcase $(\mathbf{ii})$.
(2).
: There are three orbits of homomorphisms under Galois group’s action. If each orbits has two homomorphisms. Let $f_1\leq f_2$ and $f_3\leq f_4$ be conjugated formal codegrees of ${\mathcal{C}}$. Hence, by equation \[classequation\] $$\begin{aligned}
\frac{2}{7}+\frac{1}{f_1}+\frac{1}{f_2}+\frac{1}{f_3}+\frac{1}{f_4}=1.\end{aligned}$$ Assume $f_2=FPdim({\mathcal{C}})$, then $f_1f_2=49$, in addition, $\frac{1}{f_1}+\frac{1}{f_2}<\frac{5}{7}$ and $f_1+f_2\geq 2\sqrt{f_1f_2}=14$. While $FPdim({\mathcal{C}})>7$, so $f_1+f_2=21, 28$. If $f_1+f_2=28$, then $\frac{1}{f_3}+\frac{1}{f_4}=\frac{1}{7}$, however, there is no real solutions in equation $x^2-7x+49=0$. Therefore, $f_1+f_2=21$ and $f_3=f_4=7$, but subcase $(\mathbf{i})$ shows that modular fusion category of dimension $7$ has to be pointed, this is impossible. If there exists a homomorphism $\phi$ from $Gr({\mathcal{C}})$ to ${\mathbb{C}}$ fixed by group $Gal(\overline{\mathbb{Q}}/\mathbb{Q})$, then formal codegree $\sum_{X\in{\mathcal{O}}({\mathcal{C}})}|\phi(X)|^2=7$ by [@O3 Theorem 1.1.2] since it is an integer and divides $7$. Hence, arguments of subcases $(\mathbf{ii})$ and $(\mathbf{iii})$ say that there is a contradiction.
(3).
: If there are four orbits, then two homomorphisms from $Gr({\mathcal{C}})$ to ${\mathbb{C}}$ are fixed by group $Gal(\overline{\mathbb{Q}}/\mathbb{Q})$. Thus, both of the corresponding formal codegrees are $7$ by [@O3 Theorem 1.1.2]. Combined with global dimension of ${\mathcal{C}}$, so there are four formal codegrees $7,7,7,7$. Again, this situation can be reduced to subcase $(\mathbf{i})$.
In summary, modular fusion categories of global dimension $7$ are pointed.
Hence, Theorem \[moddimen7\] implies that pre-modular fusion categories of global dimension $7$ are braided equivalent to $Rep({\mathbb{Z}}_7)$ or ${\mathcal{C}}({\mathbb{Z}}_7,\eta)$.
Let $p$ be a prime, is pre-modular fusion category of dimension $p$ pointed or braided equivalent to Deligne tensor product $YL\boxtimes \overline{YL}$?
Weakly integral pre-modular fusion categories of dimension $8$ were classified in [@Yu Remark 4.7], for example.
\[propdimen8\]Let ${\mathcal{C}}$ be a pre-modular fusion category of global dimension $8$. If ${\mathcal{C}}$ is not simple, then ${\mathcal{C}}$ is weakly integral.
If ${\mathcal{C}}$ contains a non-trivial Tannakian subcategory $Rep(G)$, then ${\mathcal{C}}\cong{\mathcal{D}}^G$ with $dim({\mathcal{D}})=\frac{8}{|G|}$ by [@DrGNO2 Proposition 4.26]. Then ${\mathcal{D}}$ is weakly integral by [@O3 Example 5.1.2], so is ${\mathcal{C}}$ by [@DrGNO2 Corollary 4.27]. We always assume ${\mathcal{C}}$ does not contain non-trivial Tannakian subcategory below. Then it suffices to show that ${\mathcal{C}}$ is weakly integral when ${\mathcal{C}}'=sVec$ or ${\mathcal{C}}'=Vec$.
If ${\mathcal{C}}$ is a super-modular fusion category, then [@DrGNO2 Lemma 3.28] implies that $rank({\mathcal{C}})$ is even, and $rank({\mathcal{C}})\leq8$ by [@O3 Remark 4.2.3]. We deduce from [@BGNPRW] and [@O3 Remark 4.2.3] that ${\mathcal{C}}\cong sVec\boxtimes{\mathcal{A}}$, where ${\mathcal{A}}$ is a modular fusion category of global dimension $4$. Consequently, ${\mathcal{C}}$ is a weakly integral fusion category.
If ${\mathcal{C}}$ is a modular fusion category. Let ${\mathcal{A}}$ be an arbitrary proper fusion subcategory of ${\mathcal{C}}$. Then $dim({\mathcal{A}}')dim({\mathcal{A}})=dim({\mathcal{C}})=8$ by [@DrGNO2 Theorem 3.10], where ${\mathcal{A}}'$ be the centralizer of ${\mathcal{A}}$ in ${\mathcal{C}}$. Since symmetric fusion category ${\mathcal{A}}\cap{\mathcal{A}}'\subseteq{\mathcal{A}}$ and ${\mathcal{A}}\cap{\mathcal{A}}'\subseteq{\mathcal{A}}'$, it follows from Theorem \[Lagrange\] that $dim({\mathcal{A}}\cap{\mathcal{A}}')^2|8$. So $dim({\mathcal{A}}\cap{\mathcal{A}}')=1$ or $2$.
Since ${\mathcal{C}}$ does not contain non-trivial Tannakian subcategory, ${\mathcal{A}}$ and ${\mathcal{A}}'$ are both modular or both super-modular. If ${\mathcal{A}}\cap{\mathcal{A}}'=Vec$, then ${\mathcal{C}}\cong{\mathcal{A}}\boxtimes{\mathcal{A}}'$ by [@DrGNO2 Theorem 3.13], and $rank({\mathcal{A}})=2,3,4$. Hence ${\mathcal{A}}$ and ${\mathcal{A}}'$ are modular fusion categories of global dimension $2$ or $4$ by [@RSW], so ${\mathcal{C}}$ is weakly integral. If ${\mathcal{A}}\cap{\mathcal{A}}'=sVec$, then ${\mathcal{C}}$ contains a super-modular fusion category ${\mathcal{B}}$ of dimension $4$ by [@DrGNO2 Theorem 3.10]. By [@O3 Example 5.1.2] and [@DrGNO2 Corollary A.19] ${\mathcal{B}}\cong sVec\boxtimes{\mathcal{C}}({\mathbb{Z}}_2,\eta)$. Then ${\mathcal{C}}\cong{\mathcal{C}}({\mathbb{Z}}_2,\eta)\boxtimes{\mathcal{C}}({\mathbb{Z}}_2,\eta)'$ by [@DrGNO2 Theorem 3.13], so centralizer ${\mathcal{C}}({\mathbb{Z}}_2,\eta)'$ in ${\mathcal{C}}$ is either an Ising category or pointed by [@O3 Example 5.1.2], again ${\mathcal{C}}$ is weakly integral.
Let ${\mathcal{B}}$ be a pre-modular fusion category of global dimension $9$, using same methods as Proposition \[propdimen8\], it can be proved that ${\mathcal{B}}$ is pointed if ${\mathcal{B}}$ is not simple.
Let ${\mathcal{A}}:=YL\boxtimes \overline{YL}$, then modular fusion category ${\mathcal{A}}\boxtimes{\mathcal{A}}$ has global dimension $25$ and $rank({\mathcal{A}}\boxtimes{\mathcal{A}})=16$. Moreover, there exist modular fusion categories ${\mathcal{C}}$ of rank $3$ [@O2], whose global dimensions are roots of equation $x^3-14x^2+49x-49=0$. Assume ${\mathcal{C}}_i$ ($1\leq i\leq3$) are three twists of ${\mathcal{C}}$ such that the corresponding global dimensions are exactly conjugated roots of previous equation. Let ${\mathcal{D}}:={\mathcal{C}}_1\boxtimes{\mathcal{C}}_2\boxtimes{\mathcal{C}}_3$, then ${\mathcal{D}}$ has global dimension $49$ with $rank({\mathcal{D}})=27$. Thus, it is much more complicated to classify pre-modular fusion categories of global dimension $p^2$ when prime $p$ is larger than $3$.
Spherical fusion categories of other dimensions {#subsection3.3}
-----------------------------------------------
In this subsection, we show that spherical fusion category ${\mathcal{C}}$ is weakly integral when $dim({\mathcal{C}})=6$. Obviously, there exists pre-modular fusion category ${\mathcal{C}}$ which are not weakly integral when $dim({\mathcal{C}})=10$, ${\mathcal{C}}\cong YL\boxtimes \overline{YL}\boxtimes sVec$, for example. In the last, we classify pre-modular fusion categories of dimension $10$ which are not simple.
Now, we classify spherical fusion categories of dimension $6$, which generalizes result of Proposition \[premoddim6\]. The proof of the following theorem is same as Theorem \[moddimen7\].
\[spherical6\]Spherical fusion categories of dimension $6$ are weakly integral.
Let ${\mathcal{C}}$ be spherical fusion category of dimension $6$. It follows from [@O3 Remark 4.2.3] that ${\mathcal{C}}$ is pointed if and only if ${\mathcal{C}}$ has rank $6$. Moreover, if $rank({\mathcal{C}})=3$, this is a direct result of [@O2 Theorem 1.1]. Below we assume that ${\mathcal{C}}$ is not weakly integral, in particular, homomorphisms $dim(-)$ and $FPdim(-)$ are not same. Hence, $Gr({\mathcal{C}})$ is commutative and $rank({\mathcal{C}})=4,5$. Note that homomorphisms from $Gr({\mathcal{C}})$ to ${\mathbb{C}}$ are divided into two or three orbits under the action of Galois group $Gal(\overline{\mathbb{Q}}/\mathbb{Q})$. Moreover, $dim({\mathcal{C}}_{int})=FPdim({\mathcal{C}}_{int})$ equals to $1,2,3$ by Theorem \[Lagrange\], so ${\mathcal{C}}_{int}={\mathcal{C}}_{pt}$. Here, we only give a proof when $rank({\mathcal{C}})=4$, the other is same.
We know that orbits of $dim(-)$ and $FPdim(-)$ contain $2$ homomorphisms, respectively. Let $f_2$ be a formal codegree of ${\mathcal{C}}$, which is a Galois conjugate of $f_1=FPdim({\mathcal{C}})$. Then equation \[classequation\] says that $$\begin{aligned}
\frac{1}{3}+\frac{1}{f_1}+\frac{1}{f_2}=1,\end{aligned}$$ in addition, $f_1f_2|36$ by [@ENO1 Proposition 8.22] and $f_1+f_2>6$. Hence, $f_1f_2=12,18,36$. If $f_1f_2=12$, then $f_1=6$ and $f_2=2$, this is impossible.
If $f_1f_2=18$, then $f_1=6+3\sqrt{2}$ and $f_1=6-3\sqrt{2}$. If $FPdim({\mathcal{C}}_{int})=3$, then there is a unique simple object $X$ such that $FPdim(X)^2=3+3\sqrt{2}$. Let $G({\mathcal{C}})={\mathbb{Z}}_3=\langle g\rangle$, then $g\otimes X=X$. However, $X\otimes X=I\oplus g\oplus g^2 \oplus n X$, so $nFPdim(X)=3\sqrt{2}$, this is impossible. If $FPdim({\mathcal{C}}_{int})=2$, then as in Lemma \[lemmdimen7(1)\] there exist positive integers $\alpha_i,\beta_i$ such that $$\begin{aligned}
6+3\sqrt{2}=2+\frac{a_1+b_1\sqrt{2}}{2}+\frac{a_2+b_2\sqrt{2}}{2}
=2+(\frac{\alpha_1+\beta_1\sqrt{2}}{2})^2+(\frac{\alpha_2+\beta_2\sqrt{2}}{2})^2.\end{aligned}$$ So, $(b_1,b_2)=(5,1),(4,2),(3,3)$. These cases can be excluded through a direct computation. The remaining cases can be proved similarly, we omit the proof.
Hence, spherical fusion categories ${\mathcal{C}}$ of dimension $6$ are weakly integral. Then the classification of spherical fusion categories ${\mathcal{C}}$ follows from [@EGO Theorem 1.1].
Next, we classify pre-modular fusion categories of dimension $10$. We begin with classification of super-modular fusion categories of dimension $10$.
In [@BPRZ], some techniques for classifying super-modular ${\mathcal{C}}$ are established, these are similar to that of [@BNRW]. Recall that Galois group $Gal({\mathcal{C}}):=Gal(\mathbb{Q}(\widehat{S})/\mathbb{Q})$ is an abelian group, where $\mathbb{Q}(\widehat{S})$ is the extension field of $\mathbb{Q}$ by coefficients of $\widehat{S}$, see subsection \[subsection2.2\] for definition of matrix $\widehat{S}$. Moreover, for any $\sigma\in Gal({\mathcal{C}})$, let $\widehat{\sigma}$ be the induced morphism of $\sigma$ on set $\Pi_0$, then we have $$\begin{aligned}
\sigma(\frac{\widehat{s}_{X,Y}}{\widehat{s}_{I,Y}})
=\frac{\widehat{s}_{X,\widehat{\sigma}(Y)}}{\widehat{s}_{I,\widehat{\sigma}(Y)}}, ~\forall X,Y\in\Pi_0.\end{aligned}$$
\[lemmdimen10\]Let ${\mathcal{C}}$ be a super-modular fusion category of global dimension $10$. Then ${\mathcal{C}}\cong sVec\boxtimes{\mathcal{C}}({\mathbb{Z}}_5,\eta)$ or ${\mathcal{C}}\cong (YL\boxtimes \overline{YL})\boxtimes sVec$.
It follows from [@BGNPRW] that super-modular fusion categories ${\mathcal{C}}$ of dimension $10$ have rank $8$ or $10$. If $rank({\mathcal{C}})=10$, then ${\mathcal{C}}$ is pointed by [@O3 Remark 4.2.3], and ${\mathcal{C}}\cong sVec\boxtimes{\mathcal{C}}({\mathbb{Z}}_5,\eta)$ by [@DrGNO2 Corollary A.19]. If $rank({\mathcal{C}})=8$, we show that ${\mathcal{C}}$ contains a proper fusion subcategory ${\mathcal{A}}$, which is not equivalent to ${\mathcal{C}}'$.
On the contrary, assume all fusion subcategories of ${\mathcal{C}}$ are equivalent to $Vec$, ${\mathcal{C}}'\cong sVec$ or ${\mathcal{C}}$. By [@BPRZ Lemma 4.1] we find that super-modular fusion category ${\mathcal{C}}$ of rank $8$ is pointed if ${\mathcal{C}}$ is not self-dual. So, when $dim({\mathcal{C}})=10$ and $rank({\mathcal{C}})=8$, super-modular fusion category ${\mathcal{C}}$ must be self-dual. Hence, coefficients of $\widehat{S}$ belong to the field of real numbers. Let $\Pi_0={\{X_0,X_1,X_2,X_3}\}$, $X_0=I$. By [@BPRZ Theorem 3.1], it suffices to show that such super-modular fusion category does not exist when $Gal({\mathcal{C}})={\mathbb{Z}}_4=\langle(0123)\rangle$, ${\mathbb{Z}}_3=\langle(012)\rangle$ or ${\mathbb{Z}}_2=\langle(01)(23)\rangle$.
If $Gal({\mathcal{C}})={\mathbb{Z}}_4=\langle(0123)\rangle$, then there exists $\sigma\in Gal({\mathcal{C}})$ such that $$\begin{aligned}
\underline{FPdim}(X_i)= \frac{\widehat{s}_{X_i,Y}}{\widehat{s}_{I,Y}}
=\frac{\widehat{s}_{X_i,\widehat{\sigma}(I)}}{\widehat{s}_{I,\widehat{\sigma}(I)}}
=\sigma(\frac{\widehat{s}_{X_i,I}}{\widehat{s}_{I,I}})=\sigma(dim(X_i)).\end{aligned}$$ Hence, $\sum\limits_{\substack{i=0}}^3\underline{FPdim}(X_i)^2=5$. Since $\underline{FPdim}(X_i)\geq1$, $\underline{FPdim}(X_i)<2$ for all $i$. By [@EGNO Corollary 3.3.16], $\underline{FPdim}(X_i)=2cos(\frac{\pi}{n_i})$, where $n_i\geq3$ are positive integers, this is impossible by a direct computation.
If $Gal({\mathcal{C}})={\mathbb{Z}}_3=\langle(012)\rangle$. For fusion ring $R$, the orbit of $dim(-)$ under Galois group $Gal(\overline{\mathbb{Q}}/\mathbb{Q})$ has at least three homomorphisms. In fact, given an arbitrary $\sigma\in Gal({\mathcal{C}})$, [@BPRZ Equation (2)] shows that $$\begin{aligned}
\sigma(\frac{\widehat{s}_{X,X_i}}{\widehat{s}_{I,X_i}})
=\frac{\widehat{s}_{X,\widehat{\sigma}(X_i)}}{\widehat{s}_{I,\widehat{\sigma}(X_i)}}
=\frac{\widehat{s}_{X, X_{\sigma(i)}}}{\widehat{s}_{I,X_{\sigma(i)}}}
, ~\forall X,X_i\in\Pi_0.\end{aligned}$$ and $dim(X)=\frac{\widehat{s}_{X,X_0}}{\widehat{s}_{I,X_0}}$. Therefore, the orbit of $\underline{FPdim}(-)$ belongs to orbit of $dim(-)$ or it is fixed by $Gal(\overline{\mathbb{Q}}/\mathbb{Q})$ . In the first case, above calculation shows that it is impossible. In the second case, it means that $\underline{FPdim}(X_i)$ takes values in the ring of integers and $\underline{FPdim}(X_i)=\frac{\widehat{s}_{X_i,X_3}}{\widehat{s}_{I,X_3}}
=\frac{\widehat{s}_{X_i,X_3}}{dim(X_3)}$, then property of $\widehat{S}$ [@BGNPRW Proposition 2.7] shows that $$\begin{aligned}
\sum_{i=0}^3\widehat{s}_{X_i,X_3}\overline{\widehat{s}_{X_i,X_3}}
=\sum_{i=0}^3\widehat{s}_{X_i,X_3} \widehat{s}_{X_i,X_3}
=dim(X_3)^2\sum_{i=0}^3\underline{FPdim}(X_i)^2=5.\end{aligned}$$ Since $dim(X_3)^2=\frac{5}{\sum_{i=0}^3\underline{FPdim}(X_i)^2}$ is a rational number, which is also an algebraic integer, it follows that $dim(X_3)^2\in{\mathbb{Z}}$. So, the integer sum $\sum_{i=0}^3\underline{FPdim}(X_i)^2$ is equal to $1$ or $5$, this is a contradiction for $\underline{FPdim}(X_i)$ are positive integers, $0\leq i\leq3$.
If $Gal({\mathcal{C}})={\mathbb{Z}}_2=\langle(01)(23)\rangle$. We see that the orbit of $dim(-)$ under Galois group $Gal(\overline{\mathbb{Q}}/\mathbb{Q})$ has at least two homomorphisms. It follows from [@BPRZ Theorem 3.9] that $\mathbb{Q}(\widehat{S})=\mathbb{Q}(dim(X_1))$ is a non-trivial ${\mathbb{Z}}_2$-extension of $\mathbb{Q}$. Let $\sigma$ be the non-trivial element of Galois group $Gal({\mathcal{C}})$, then by [@BGNPRW Proposition 2.7] and [@BPRZ Equation (2)], $$\begin{aligned}
\sigma(5)=5=\sum_{i=0}^3\sigma(\frac{\widehat{s}_{X_i,X_0}}{\widehat{s}_{X_0,X_0}})^2
=\sum_{i=0}^3(\frac{\widehat{s}_{X_i,X_1}}{\widehat{s}_{X_0,X_1}})^2
=\frac{\sum_{i=0}^3(\widehat{s}_{X_i,X_1})^2}{dim(X_1)^2} =\frac{5}{dim(X_1)^2},\end{aligned}$$ which shows that $dim(X_1)=\pm1$, this is a contradiction.
Let ${\mathcal{A}}$ be an arbitrary proper fusion subcategory of ${\mathcal{C}}$ such that ${\mathcal{A}}\ncong {\mathcal{C}}'$, then by [@DrGNO2 Theorem 3.10], we have equation $$\begin{aligned}
dim({\mathcal{A}})dim(A')=dim({\mathcal{C}})dim({\mathcal{C}}'\cap{\mathcal{A}}).\end{aligned}$$ Because ${\mathcal{A}}\cap{\mathcal{A}}'$ is symmetric, $dim({\mathcal{A}}\cap{\mathcal{A}}')$ is an integer, and Theorem \[Lagrange\] imply that $\frac{10dim({\mathcal{A}}\cap{\mathcal{C}}')}{dim({\mathcal{A}}\cap{\mathcal{A}}')^2}$ is a positive integer. If ${\mathcal{C}}'\subseteq{\mathcal{A}}$, then ${\mathcal{A}}\cap{\mathcal{A}}'=sVec$. That is, ${\mathcal{A}}$ is a super-modular fusion category of rank less than $8$. Since $\frac{dim({\mathcal{C}})}{dim({\mathcal{A}})}$ is an algebraic integer and ${\mathcal{C}}$ is not pointed, it follows from [@BGNPRW section 3] that ${\mathcal{A}}\cong sVec\boxtimes {\mathcal{B}}$, where ${\mathcal{B}}$ is a Yang-Lee fusion category. If ${\mathcal{C}}'\cap{\mathcal{A}}=Vec$, then ${\mathcal{A}}\cap{\mathcal{A}}'=Vec$, so ${\mathcal{A}}$ is modular and ${\mathcal{A}}'$ is super-modular. By [@DrGNO2 Theorem 3.13], ${\mathcal{C}}\cong{\mathcal{A}}\boxtimes{\mathcal{A}}'$, and $rank({\mathcal{A}})=2$ or $4$. Then [@RSW] or [@BGNPRW] show that ${\mathcal{C}}\cong (YL\boxtimes \overline{YL})\boxtimes sVec$ as $dim({\mathcal{C}})=10$.
The proof of Lemma \[lemmdimen10\] also shows that there does not exit super-modular fusion categories ${\mathcal{C}}$ of dimension $14$ with $rank({\mathcal{C}})=8$.
The following lemma is a direct result of Theorem \[Lagrange\].
\[premodular2p\]Let ${\mathcal{C}}$ be a pre-modular fusion category of global dimension $2p$, where $p$ is an odd prime. If ${\mathcal{A}}\subseteq{\mathcal{C}}$ is a non-trivial symmetric fusion subcategory, then ${\mathcal{A}}\subseteq{\mathcal{C}}'$.
If ${\mathcal{C}}$ is symmetric, this is trivial. We assume ${\mathcal{C}}$ is not symmetric below, indeed we claim that ${\mathcal{A}}={\mathcal{C}}'$. By [@DrGNO2 Theorem 3.10], we have equation $$\begin{aligned}
dim({\mathcal{A}})dim(A')=dim({\mathcal{C}})dim({\mathcal{C}}'\cap{\mathcal{A}}).\end{aligned}$$ If ${\mathcal{C}}'\cap{\mathcal{A}}=Vec$, while ${\mathcal{A}}\subseteq{\mathcal{A}}'$, where ${\mathcal{A}}'$ is the centralizer of ${\mathcal{A}}$ in ${\mathcal{C}}$, then Theorem \[Lagrange\] implies that $dim({\mathcal{A}})|dim({\mathcal{A}}')$, so $dim({\mathcal{A}})^2|2p$, this is a contradiction. So, ${\mathcal{A}}\subseteq{\mathcal{C}}'$. Since $dim({\mathcal{A}})\mid dim({\mathcal{C}}')\mid dim({\mathcal{C}})=2p$ and ${\mathcal{C}}$ is not symmetric, ${\mathcal{A}}={\mathcal{C}}'$ as claimed.
\[propdimen10\]Let ${\mathcal{C}}$ be a pre-modular fusion category of global dimension $10$. If ${\mathcal{C}}$ is not simple, then ${\mathcal{C}}$ is pointed, or ${\mathcal{C}}\cong Rep(D_{10})$, or ${\mathcal{C}}\cong (YL\boxtimes \overline{YL})^{{\mathbb{Z}}_2}$, or ${\mathcal{C}}\cong (YL\boxtimes \overline{YL})\boxtimes{\mathcal{D}}$, where ${\mathcal{D}}$ is a pointed pre-modular fusion category of dimension $2$, $D_{10}$ is the dihedral group of order $10$.
Let ${\mathcal{A}}$ be a non-trivial fusion subcategory of ${\mathcal{C}}$ with $dim({\mathcal{A}})\in {\mathbb{Z}}$. It follows from Theorem \[Lagrange\] that $dim({\mathcal{A}})=2$ or $5$, then ${\mathcal{A}}$ is symmetric or ${\mathcal{A}}$ is modular. If ${\mathcal{A}}$ is modular, then ${\mathcal{C}}\cong{\mathcal{A}}\boxtimes{\mathcal{D}}$ by [@DrGNO2 Theorem 3.13], where ${\mathcal{D}}$ is the centralizer of ${\mathcal{A}}$ in ${\mathcal{C}}$. Therefore, [@O3 Example 5.1.2] shows that ${\mathcal{A}}\cong{\mathcal{C}}({\mathbb{Z}}_2,\eta)$ or ${\mathcal{A}}\cong YL\boxtimes \overline{YL}$. Moreover, note that the structure of ${\mathcal{D}}$ can also be deduced from [@O3 Example 5.1.2].
If ${\mathcal{A}}$ is symmetric, then ${\mathcal{A}}={\mathcal{C}}'$ by Lemma \[premodular2p\]. When $dim({\mathcal{A}})=5$, then ${\mathcal{A}}$ is a Tannakian fusion category. Hence ${\mathcal{C}}\cong {\mathcal{D}}^{{\mathbb{Z}}_5}\cong Rep({\mathbb{Z}}_5)\boxtimes{\mathcal{C}}({\mathbb{Z}}_2,\eta)$, where ${\mathcal{D}}={\mathcal{C}}_{{\mathbb{Z}}_5}$ is a modular fusion category of dimension $2$ by [@DrGNO2 Proposition 4.30]. When $dim({\mathcal{A}})=2$, then ${\mathcal{A}}$ is braided equivalent to $Rep({\mathbb{Z}}_2)$ or $sVec$. In the first case, ${\mathcal{C}}\cong{\mathcal{B}}^{{\mathbb{Z}}_2}$, where ${\mathcal{B}}$ is a modular fusion category of dimension $5$. Hence [@O3 Example 5.1.2] says that ${\mathcal{C}}\cong (YL\boxtimes \overline{YL})^{{\mathbb{Z}}_2}$, or ${\mathcal{C}}\cong (YL\boxtimes \overline{YL})\boxtimes Rep({\mathbb{Z}}_2)$, or ${\mathcal{C}}$ is pointed, or ${\mathcal{C}}\cong Rep(D_{10})$. If ${\mathcal{A}}\cong sVec$, then this is exactly classification of Lemma \[lemmdimen10\].
Now let ${\mathcal{A}}$ be a non-trivial fusion subcategory such that $dim({\mathcal{A}})\notin{\mathbb{Z}}$. Then it suffices to classify ${\mathcal{C}}$ when ${\mathcal{C}}'=Vec$. Notice that ${\mathcal{A}}\cap{\mathcal{A}}'$ is a symmetric fusion subcategory of integer global dimension. If ${\mathcal{A}}\cap{\mathcal{A}}'$ is non-trivial, we are done. If ${\mathcal{A}}\cap{\mathcal{A}}'=Vec$, then ${\mathcal{A}}$ and ${\mathcal{A}}'$ are modular and ${\mathcal{C}}\cong{\mathcal{A}}\boxtimes{\mathcal{A}}'$ by [@DrGNO2 Theorem 3.13]. Note that $rank({\mathcal{A}})rank({\mathcal{A}}')=rank({\mathcal{C}})$ and $rank({\mathcal{C}})\leq10$ by [@O3 Lemma 4.2.2], then classification follows from [@BNRW; @RSW].
Acknowledgements {#acknowledgements .unnumbered}
================
The author is grateful to Prof. Ostrik for insightful conversations on étale algebras and totally positive algebraic integers, particularly for providing reference [@KiO]. Part of this paper was written during a visit of the author at University of Oregon supported by China Scholarship Council (grant No. 201806140143), he appreciates the Department of Mathematics for their warm hospitality.
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[^1]: Email:[email protected]
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abstract: 'Energy levels, radiative rates and lifetimes are calculated among the lowest 98 levels of the $n \le$ 4 configurations of Be-like Al X. The [grasp]{} (General-purpose Relativistic Atomic Structure Package) is adopted and data are provided for all E1, E2, M1 and M2 transitions. Similar data are also obtained with the Flexible Atomic Code ([fac]{}) to assess the accuracy of the calculations. Based on comparisons between calculations with the two codes as well as with available measurements, our listed energy levels are assessed to be accurate to better than 0.3%. However, the accuracy for radiative rates and lifetimes is estimated to be about 20%. Collision strengths are also calculated for which the Dirac Atomic R-matrix Code ([darc]{}) is used. A wide energy range (up to 380 Ryd) is considered and resonances resolved in a fine energy mesh in the thresholds region. The collision strengths are subsequently averaged over a Maxwellian velocity distribution to determine effective collision strengths up to a temperature of 1.6$\times$10$^7$ K. Our results are compared with the previous (limited) atomic data and significant differences (up to a factor of 4) are noted for several transitions, particularly those which are not allowed in $jj$ coupling.'
author:
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Kanti M. Aggarwal$^{1}$[^1] and Francis P. Keenan$^{1}$\
$^{1}$Astrophysics Research Centre, School of Mathematics and Physics, Queen’s University Belfast, Belfast BT7 1NN, Northern Ireland, UK
date: 'Accepted 2013 November 21. Received 2013 November 14; in original form 2013 August 1'
title: 'Energy levels, radiative rates and electron impact excitation rates for transitions in Al X[^2]'
---
\[firstpage\]
atomic data – atomic processes
Introduction
============
Emission lines of Al ions are widely detected in a variety of plasmas, including solar and lasing plasmas – see for example, [@el1] and [-@gu1]. Their observation provides diagnostics for which atomic data are required for a range of parameters, such as energy levels, radiative rates (A- values), and excitation rates or equivalently the effective collision strengths ($\Upsilon$), which are obtained from the electron impact collision strengths ($\Omega$). Emission lines of many Al ions are included in the CHIANTI database at [http://www.chiantidatabase.org/]{}, and for Al X are listed in the 35-60,600 ${\rm \AA}$ wavelength range in the [*Atomic Line List*]{} (v2.04) of Peter van Hoof at ${\tt {\verb+http://www.pa.uky.edu/~peter/atomic/+}}$. Of particular interest are two transitions, namely 2s$^2$ $^1$S$_0$–2s2p $^3$P$^o_1$ and 2s2p $^3$P$^o_1$–2p$^2$ $^1$D$_2$, at wavelengths of $\sim$ 637 and 670 $\rm \AA$, respectively. These have been measured in solar spectra from the Solar Ultraviolet Measurement of Emitted Radiation (SUMER) instrument on board the Solar and Heliospheric Observatory (SOHO). The ratio of their intensities is temperature sensitive (Landi et al. 2001) and hence provides an excellent diagnostic. Since these lines are close in wavelength, they are readily detected by a single spectrograph, thus providing an additional advantage of accuracy.
Unfortunately, existing atomic data for Al X are very limited, particularly for collision strengths ($\Omega$) and effective collision strengths ($\Upsilon$). [@zs92] calculated $\Omega$ for 45 transitions among the lowest 10 levels of Be-like ions with 8 $\le$ Z $\le$ 92, but did not report results for Al X. The only data available for $\Upsilon$ are those of [@fpk1], who provided analytical expressions to derive interpolated values of $\Upsilon$ based on $R$-matrix calculations for Be-like ions between C III and Si XI. Their results are only for transitions among the lowest 10 levels of the $n$=2 configurations, and are incorporated in the CHIANTI database. No calculation has to our knowledge been performed with the $R$-matrix code which explicitly includes the contribution of resonances to the determination of $\Upsilon$. This may be highly significant, particularly for the forbidden transitions, as noted earlier for another Be-like ion, i.e. Ti XIX [@tixix].
To analyse solar observations, [@el1] adopted the atomic data for Al X in CHIANTI. Since these data are limited to transitions within the $n$=2 configurations, they also calculated $\Upsilon$ for a larger model, i.e. 98 levels of the $n \le$ 4 configurations. For this they adopted the Hebrew University Lawrence Livermore Atomic Code (HULLAC) of [-@hullac], based on the well-known and widely-used [*distorted-wave*]{} (DW) method. Since resonance contributions are not normally included in DW calculations, the results for $\Upsilon$ may be significantly underestimated. Therefore, they included their contribution using the isolated resonance approximation. As a consequence, temperatures deduced from solar observations using two different sets of atomic data are significantly different, i.e. 10$^{6.47}$ and 10$^{5.75}$ K from CHIANTI and HULLAC, respectively.
The isolated resonance approximation takes into account the resonance contribution to a large extent, but $\Upsilon$ can still be greatly underestimated, as demonstrated by [@mo34; @fe25] for transitions in Mo XXXIV and Fe XXV, respectively. Furthermore, [@el1] did not list any atomic data and these are not available on any website. Therefore, in this work we report atomic data for energy levels, A- values, $\Omega$ and $\Upsilon$ for all transitions among the lowest 98 levels of the $n \le$ 4 configurations of Al X. To determine the atomic structure (i.e. calculate energy levels and A- values) we employ the fully relativistic [grasp]{} (General-purpose Relativistic Atomic Structure Package) code. Our version was originally developed by [@grasp0] and is referred to as GRASP0, but has been significantly revised by one of its authors (Dr. P. H. Norrington), and is available at the website [http://web.am.qub.ac.uk/DARC/]{}. It is a fully relativistic code, based on the $jj$ coupling scheme, and includes higher-order relativistic corrections arising from the Breit (magnetic) interaction and quantum electrodynamics (QED) effects (vacuum polarisation and Lamb shift). Furthermore, as in our earlier work, we have used the option of [*extended average level*]{} (EAL). Under this option a weighted (proportional to 2$j$+1) trace of the Hamiltonian matrix is minimised. However, the results obtained for energy levels as well as radiative rates are comparable to other options, such as [*average level*]{} (AL), as noted by [-@kr35; -@xe54] for Kr and Xe ions.
For the scattering calculations, we have adopted the [*Dirac Atomic $R$-matrix Code*]{} ([darc]{}) of P. H. Norrington and I. P. Grant ([http://web.am.qub.ac.uk/DARC/]{}). This is a relativistic version of the standard $R$-matrix code and is based on the $jj$ coupling scheme. For this reason, the accuracy of calculated data (for $\Omega$ and subsequently $\Upsilon$) should be higher, particularly for transitions among the [*fine-structure*]{} levels of a state, because resonances through the energies of degenerating levels are also taken into account.
Energy levels
=============
The lowest 98 levels of Al X belong to the 17 configurations: (1s$^2$) 2$\ell$2$\ell'$, 2$\ell$3$\ell'$ and 2$\ell$4$\ell'$. Our level energies calculated from [grasp]{}, [*without*]{} and [*with*]{} the inclusion of Breit and QED effects, are listed in Table 1. [@mz] have compiled and critically evaluated experimentally-measured energy levels of Al X, which are available at the NIST (National Institute of Standards and Technology) website [http://www.nist.gov/pml/data/asd.cfm]{}, and included in Table 1 for comparison. However, NIST energies are not available for many levels, particularly of the 2$\ell$4$\ell'$ configurations, and for some of the levels their results are the same (i.e. non degenerate) – see for example: the 2s4f $^3$F$^o_{2,3,4}$ levels. Also included in the table, for comparison purpose, are energies obtained from the [*Flexible Atomic Code*]{} ([fac]{}) code of [@fac]. These results listed under FAC1 include the same CI (configuration interaction) as in [grasp]{}.
The inclusion of the Breit and QED effects (GRASP2) does not significantly alter the energies obtained with their exclusion (GRASP1), as both sets of results agree within 0.01 Ryd, and their orderings are also the same. Similarly, there is no discrepancy in the ordering of levels between GRASP and NIST, and the energy differences for common levels are generally within 0.01 Ryd. However, particularly for two levels, namely 2p3s $^1$P$^o_1$ and 2p3d $^1$F$^o_3$, the NIST energies are higher by up to 0.06 Ryd. For some of the levels, including 2p3s $^1$P$^o_1$, the NIST energies are not highly accurate as they have placed a question mark on these.
Our FAC1 energies show the same ordering as those of GRASP and NIST, and agree with our GRASP2 calculations within 0.05 Ryd. The inclusion of the 2$\ell$5$\ell'$ configurations, labelled FAC2 calculations in Table 1, makes no appreciable difference. Small discrepancies in the [grasp]{} and [fac]{} energies, also noted for several other ions, are primarily due to the different ways that the calculations of central potential for radial orbitals and recoupling schemes of angular parts have been performed – see the detailed discussion in the [fac]{} manual (). However, for the levels in common our GRASP2 energies are slightly closer to those of NIST than those from FAC. Therefore, based on the comparisons shown, our GRASP2 energy levels listed in Table 1 are assessed to be accurate to better than 0.3%.
Radiative rates
===============
Since the A- values of [@zs92] are limited to E1 transitions among the lowest 10 levels of Al X, we here provide a complete set of data for all transitions among the 98 levels. Furthermore, A- values are calculated for four types of transitions, namely electric dipole (E1), electric quadrupole (E2), magnetic dipole (M1), and magnetic quadrupole (M2). Generally, E1 transitions are dominant, but occasionally other types of transitions are also prominent and are therefore required for a complete plasma model. The absorption oscillator strength ($f_{ij}$) and radiative rate A$_{ji}$ (in s$^{-1}$) for all types of transition $i \to j$ are related by the following expression [@rhg]:
$$f_{ij} = \frac{mc}{8{\pi}^2{e^2}}{\lambda^2_{ji}} \frac{{\omega}_j}{{\omega}_i} A_{ji}
= 1.49 \times 10^{-16} \lambda^2_{ji} (\omega_j/\omega_i) A_{ji}$$
where $m$ and $e$ are the electron mass and charge, respectively, $c$ the velocity of light, $\lambda_{ji}$ the transition energy/wavelength in $\rm \AA$, and $\omega_i$ and $\omega_j$ the statistical weights of the lower ($i$) and upper ($j$) levels, respectively. However, the relationships between oscillator strength f$_{ij}$ (dimensionless) and the line strength S (in atomic unit, 1 a.u. = 6.460$\times$10$^{-36}$ cm$^2$ esu$^2$) with the A- values are different for different types of transitions – see Eqs. (2–5) of [@tixix].
In Table 2 we list transition energies/wavelengths ($\lambda$, in $\rm \AA$), radiative rates (A$_{ji}$, in s$^{-1}$), oscillator strengths (f$_{ij}$, dimensionless), and line strengths (S, in a.u.) for all 1468 electric dipole (E1) transitions among the 98 levels of Al X. For conciseness, results are only listed in the length form, which are considered to be more accurate. However, below we discuss the velocity/length form ratio, as this provides some assessment of the accuracy of the results. Also note that the [*indices*]{} used to represent the lower and upper levels of a transition are defined in Table 1. There are 1754 electric quadrupole (E2), 1424 magnetic dipole (M1), and 1792 magnetic quadrupole (M2) transitions among the same 98 levels. However, for these only the A-values are listed in Table 2, and the corresponding results for f- values can be easily obtained through Eq. (1).
As noted in section 1, f-values for Al X are available in the literature [@zs92] for only a limited number of transitions. Therefore, as with energy levels we have also performed a calculation with the [fac]{} code of [@fac]. Our calculated f- values from both [grasp]{} and [fac]{}, as well as their ratio, are listed in Table 3 for some representative transitions among the lowest 20 levels of Al X. For these transitions the agreement between the two sets of f- values is better than 20%, particularly for the strong ones with f $\ge$ 0.01. However, there are 4 exceptions, namely 21–71 (2p3s $^3$P$^o_0$ – 2p4p $^3$P$_1$), 31–85 (2p3p $^3$P$_1$ – 2p4d $^3$P$^o_2$), 46–76 (2p3d $^1$P$^o_1$ – 2p4p $^1$D$_2$), and 46–82 (2p3d $^1$P$^o_1$ – 2p4f $^3$F$_2$), for which the discrepancies are larger, but still less than 50%. These discrepancies are partly due to the corresponding differences in the energy levels of the two calculations.
A general criterion to assess the accuracy of f- or A- values is to compare the ratio of their length and velocity forms. This should ideally be close to unity but often is not, because the two formulations are not exactly the same. Therefore, we also include in Table 3 the ratio of the velocity and length forms. For a majority (89%) of the strong E1 transitions (f $\ge$ 0.01) the ratio is within 20% of unity, but discrepancies for some are higher, although mostly within a factor of two. However, for a few ($\sim$ 8%) weak(er) transitions (f $\le$ 10$^{-3}$) the two forms of the f- value differ by up to several orders of magnitude – see for example: 34–37, 56–58, 67-75, and 71–73. For all of these transitions, $\Delta$E$_{ij}$ is very low and a small variation in this has a large effect on the f- value. A few other transitions with significant $\Delta$E$_{ij}$ for which the two forms disagree by over 20% are: 3–12 (f $\sim$ 10$^{-6}$), 6–15 (f $\sim$ 10$^{-3}$), and 10–15 (f $\sim$ 10$^{-5}$), as shown in Table 3. Finally, as noted for the energy levels in section 2, inclusion of additional CI with the $n$ = 5 configurations does not make any appreciable effect on the f- or A- values. To be specific, discrepancies in A- values for all strong E1 transitions are less than $\sim$ 20% with those listed in Tables 2 and 3. Therefore, based on this and other comparisons already discussed, we are confident that for almost all strong E1 transitions listed in Table 2, our f- values (and other related parameters) are accurate to better than 20%. However, for the weaker E1 and other types of transitions (i.e. E2, M1 and M2) the accuracy may be comparatively lower. Finally, we note that these conclusions are similar to those arrived earlier for transitions of another Be-like ion, i.e. Ti XIX [@tixix].
Lifetimes
=========
The lifetime $\tau$ for a level $j$ is defined as follows (Woodgate 1970):
$${\tau}_j = \frac{1}{{\sum_{i}^{}} A_{ji}}.$$
Since this is a measurable quantity, it facilitates an assessment of the accuracy of the A- values. Therefore, in Table 1 we have also listed our calculated lifetimes. Generally, A- values for E1 transitions dominate, but for higher accuracy we have also included the contributions from E2, M1 and M2. Their inclusion is particularly useful for those levels which do not connect via E1 transitions. [@et1] have measured the lifetime of the 2s2p $^1$P$^o_1$ level to be 175$\pm$15 ps, which compares very well with our result of 160 ps and the 173 ps theoretical value of Andersson et al. (2009). They have also measured lifetimes corresponding to the 2s2p $^1$P$^o_1$–2p$^2$ $^1$D$_2$ (5–9) and 2s2p $^1$P$^o_1$–2p$^2$ $^1$S$_0$ (5–10) transitions to be 1080$\pm$80 and 112$\pm$12 ps, respectively, which compare favourably with our results of 1072 and 105 ps, respectively.
Collision strengths
===================
The collision strength for electron impact excitation ($\Omega$) is related to the better-known parameter collision cross section ($\sigma_{ij}$, $\pi{a_0}^2$) by the following equation [@bt]:
$$\Omega_{ij}(E) = {k^2_i}\omega_i\sigma_{ij}(E)$$
where ${k^2_i}$ is the incident energy of the electron and $\omega_i$ is the statistical weight of the initial state. Since $\Omega$ is a symmetric and dimensionless quantity, results for it are preferred over those of $\sigma_{ij}$.
As in our earlier work, such as on Ti XIX [@tixix], for calculating $\Omega$ we have adopted the [*Dirac Atomic $R$-matrix Code*]{} ([darc]{}) of P. H. Norrington and I. P. Grant, available at the website [http://web.am.qub.ac.uk/DARC/]{}. This code includes the relativistic effects, which are very important for high Z ions, but are equally significant for taking into account degeneracy among the fine-structure levels of a state of an ion with lower Z, such as Al X. The [darc]{} code is based on the $jj$ coupling scheme and uses the Dirac-Coulomb Hamiltonian in an $R$-matrix approach. The $R$-matrix radius adopted for Al X is 6.4 au, and 55 continuum orbitals have been included for each channel angular momentum in the expansion of the wavefunction. This large expansion is computationally more demanding but allows us to compute $\Omega$ up to an energy of 380 Ryd, $\sim$355 Ryd [*above*]{} the highest threshold considered in this work. Furthermore, this large energy range is sufficient to calculate values of effective collision strength $\Upsilon$ (see section 6) up to T$_e$ = 1.8 $\times$10$^{7}$ K, more than an order of magnitude higher than the temperature of maximum abundance in ionisation equilibrium for Al X, i.e. 1.3 $\times$10$^{6}$ K [@pb]. The maximum number of channels for a partial wave is 428 and the corresponding size of the Hamiltonian matrix is 23,579. To achieve convergence of $\Omega$ for a majority of transitions and at all energies, we have included all partial waves with angular momentum $J \le$ 40.5. Additionally, to account for higher neglected partial waves, we have included a top-up, based on the Coulomb-Bethe [@ab] and geometric series approximations for allowed and forbidden transitions, respectively. These contributions enhance the accuracy of our calculated values of $\Omega$s, particularly at the higher end of the energy range.
In Table 4 we list our values of $\Omega$ for resonance transitions of Al X at energies [*above*]{} thresholds. The indices used to represent the levels of a transition have already been defined in Table 1. Unfortunately, no similar data are available for comparison purposes as already noted in section 1. One way to assess the accuracy of these results is to compare with the similar calculations from [fac]{}, as undertaken in our work on Ti XIX [@tixix]. However, such a comparison is not very useful, particularly for the forbidden transitions, because often there are anomalies in the FAC calculations, as shown in Fig. 6 of [@mgxi; @caxix]. Nevertheless, our listed results for $\Omega$ should be helpful for future comparisons.
Effective collision strengths
=============================
As well as energy levels and radiative rates, excitation and de-excitation rates are required for plasma modelling, which are determined from the collision strengths ($\Omega$). However, $\Omega$ does not vary smoothly with increasing energy, particularly at energies in between the thresholds. The threshold energy region is generally dominated by numerous closed-channel (Feshbach) resonances, especially for (semi) forbidden transitions. Therefore, values of $\Omega$ should be calculated in a fine energy mesh to accurately account for their contribution. Additionally, in many plasmas electrons have a wide distribution of velocities, and therefore it is more appropriate to average values of $\Omega$ over a suitable distribution. For astrophysical applications the most appropriate and commonly used distribution is [*Maxwellian*]{}, although any other distribution may also be applied if suitable for a particular plasma. Such an averaged value, known as [*effective*]{} collision strength ($\Upsilon$) [@bt] is:
$$\Upsilon(T_e) = \int_{0}^{\infty} {\Omega}(E) \, {\rm exp}(-E_j/kT_e) \,d(E_j/{kT_e}),$$
where $k$ is Boltzmann constant, T$_e$ electron temperature in K, and E$_j$ the electron energy with respect to the final (excited) state. Once the value of $\Upsilon$ is known the corresponding results for the excitation q(i,j) and de-excitation q(j,i) rates can be easily obtained from the following equations:
$$q(i,j) = \frac{8.63 \times 10^{-6}}{{\omega_i}{T_e^{1/2}}} \Upsilon \, {\rm exp}(-E_{ij}/{kT_e}) \hspace*{1.0 cm}{\rm cm^3s^{-1}}$$
and $$q(j,i) = \frac{8.63 \times 10^{-6}}{{\omega_j}{T_e^{1/2}}} \Upsilon \hspace*{1.0 cm}{\rm cm^3 s^{-1}},$$ where $\omega_i$ and $\omega_j$ are the statistical weights of the initial ($i$) and final ($j$) states, respectively, and E$_{ij}$ is the transition energy. The contribution of resonances often enhances the values of $\Upsilon$ over those of the background collision strengths ($\Omega_B$), particularly for the (semi) forbidden transitions. This enhancement can be dominant (by up to an order of magnitude or even more), but depends on the type of a transition as well as the temperature. Generally, the enhancement in $\Upsilon$ is greater at lower temperatures. Similarly, values of $\Omega$ should be calculated over a wide energy range (above thresholds) to obtain convergence of the integral in Eq. (4), as demonstrated in Fig. 7 of [@ni11]. If calculations of $\Omega$ are performed only for a limited range of energy, it is still necessary to include the contribution of $\Omega$ at high energies. For this the high energy limits recommended by [@bt] for a range of transitions may be adopted. However, in our work there is no such need because calculations for $\Omega$ have already been performed up to sufficiently high energies, as clarified in section 5.
To delineate resonances, we have performed our calculations of $\Omega$ at over $\sim$ 13,000 energies in the thresholds region. Close to thresholds ($\sim$0.1 Ryd above a threshold) the energy mesh is 0.001 Ryd, and away from thresholds is 0.002 Ryd. This fine resolution accounts for the majority of resonances, and their density and importance can be appreciated from transitions of Ti XIX shown in Figs. 7–12 of [@tixix]. For transitions of Al X we observe similar dense resonances, as shown in Figs. 1 and 2 for two important lines, namely 2s$^2$ $^1$S$_0$–2s2p $^3$P$^o_1$ (1–3) and 2s2p $^3$P$^o_1$–2p$^2$ $^1$D$_2$ (3–9). Both of these are allowed transitions in $jj$ coupling and yet resonances are not only dense but also highly significant in magnitude. Furthermore, resonances are spread over a range of 20 Ryd (equivalently over 3$\times$10$^6$ K), and hence make an appreciable contribution to $\Upsilon$ over the entire range of temperatures of interest for transitions of Al X.
Our calculated values of $\Upsilon$ are listed in Table 5 over a wide temperature range up to 10$^{7.2}$ K, suitable for applications to a wide range of laboratory and astrophysical plasmas. Corresponding data at any desired temperature can either be easily interpolated, because $\Upsilon$ is a slowly varying function of T$_e$, or may be requested from the first author. As noted in section 1, [@fpk1] have reported values of $\Upsilon$ for transitions among the lowest 10 levels of Al X. In Table 6 we compare results for $\Upsilon$ at three temperatures of 10$^{5.9}$, 10$^{6.1}$ and 10$^{6.3}$ K, which are most relevant for Al X [@pb]. The interpolated values of $\Upsilon$ listed by Keenan et al. are based on the $R$-matrix calculations for C III, O V, Ne VII and Si XI, which were performed in $LS$ coupling. Since their work is not based on direct calculations for Al X, differences with their results are not unexpected. For transitions which are allowed in $LS$ coupling, such as 1–5 and 5–9/10, there is no discrepancy between our results and those of Keenan et al. This is because such allowed transitions do not normally have a significant contribution from resonances. However, there are differences between the two sets of $\Upsilon$ of up a factor of two for several (but not all) transitions which are allowed in $LS$ coupling but not in $jj$, such as 2/3/4–6/7/8, i.e. 2s2p $^3$P$^o_{0,1,2}$ – 2p$^2$ $^3$P$_{0,1,2}$. For other transitions (particularly forbidden) the $\Upsilon$ values of Keenan et al. are [*underestimated*]{} by up to a factor of 4.
Conclusions
===========
Energies and lifetimes for the lowest 98 levels of Al X belonging to the $n \le$ 4 configurations have been reported, for which the [grasp]{} code has been adopted. Also listed are results for radiative rates for four types of transitions (E1, E2, M1 and M2). Based on a variety of comparisons with available measurements, as well as with analogous calculations with the [fac]{} code, our results for radiative rates, oscillator strengths and line strengths are judged to be accurate to better than 20% for a majority of strong transitions. Similarly, energy levels are assessed to be accurate to $\sim$0.3%. Measurements of lifetimes are available for only three levels for which there are no discrepancies with theory.
Results have also been reported for collision strengths over a wide range of energy, but only for resonance transitions. However, corresponding results for effective collision strengths are listed for [*all*]{} transitions among the 98 levels of Al X and over a wide range of temperature, suitable for applications in a variety of plasmas. For calculations of $\Upsilon$, resonances in the thresholds energy region for many transitions are noted to be as dominant as for Ti XIX [@tixix]. Their inclusion in the determination of $\Upsilon$ has significantly enhanced the results. Since no prior calculations with comparable accuracy and complexity are available, it is not straightforward to assess the uncertainty of our values of $\Upsilon$. However, comparisons between our results and those interpolated by [@fpk1] have been made for transitions among the lowest 10 levels, and the latter are found to be underestimated by up to a factor of four for several transitions, mostly forbidden.
Since a large range of partial waves has been considered to achieve convergence of $\Omega$ at all energies and contribution of higher neglected partial waves has been included, our results for $\Omega$ should be accurate to better than 20%. This assessment of accuracy is mainly based on comparisons of similar data for Be-like ion of Ti. Similarly, to calculate values of $\Upsilon$ up to T$_e$ = 10$^{7.2}$ K, we have included a wide energy range for $\Omega$ and have also resolved resonances in a fine energy mesh to account for their contribution. Hence, we see no apparent limitations in our data. Moreover, as for Ti XIX [@tixix], we estimate the accuracy of our results for $\Upsilon$ to be better than 20% for most transitions. However, there is scope for improvement, especially for transitions involving levels of the $n$ = 4 configurations. This can perhaps be achieved by the inclusion of levels of the $n$ = 5 configurations in the collisional calculations. At present we believe the reported results for radiative and excitation rates for transitions in Al X are the most exhaustive and accurate available to date. The complete set of atomic data should be highly useful for modelling astrophysical and fusion plasmas.
Acknowledgments {#acknowledgments .unnumbered}
===============
KMA is thankful to AWE Aldermaston for financial support.
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Index Configuration Level NIST GRASP1 GRASP2 FAC1 FAC2 $\tau$ (s)
------- --------------- ------------- --------- ---------- ---------- ---------- ---------- ------------
1 2s$^2$ $^1$S$ _0$ 0.00000 0.00000 0.00000 0.00000 0.00000 ........
2 2s2p $^3$P$^o_0$ 1.41381 1.41694 1.41897 1.42722 1.42661 ........
3 2s2p $^3$P$^o_1$ 1.42885 1.43399 1.43387 1.44188 1.44128 5.963-06
4 2s2p $^3$P$^o_2$ 1.46194 1.46944 1.46662 1.47416 1.47357 1.246-00
5 2s2p $^1$P$^o_1$ 2.73827 2.81662 2.81633 2.81532 2.81053 1.600-10
6 2p$^2$ $^3$P$ _0$ 3.68675 3.71751 3.71907 3.73711 3.73633 2.154-10
7 2p$^2$ $^3$P$ _1$ 3.70446 3.73616 3.73652 3.75430 3.75353 2.129-10
8 2p$^2$ $^3$P$ _2$ 3.73337 3.76904 3.76518 3.78247 3.78171 2.096-10
9 2p$^2$ $^1$D$ _2$ 4.09826 4.17709 4.17504 4.19138 4.18716 1.050-09
10 2p$^2$ $^1$S$ _0$ 5.04644 5.17695 5.17938 5.18781 5.18508 1.052-10
11 2s3s $^3$S$ _1$ 16.9109 16.90263 16.89542 16.89461 16.89432 5.201-12
12 2s3s $^1$S$ _0$ 17.1721 17.16367 17.15693 17.17011 17.16978 1.435-11
13 2s3p $^1$P$^o_1$ 17.5314 17.53546 17.52796 17.53879 17.53775 2.139-12
14 2s3p $^3$P$^o_0$ 17.55638 17.55008 17.56230 17.56240 4.322-10
15 2s3p $^3$P$^o_1$ 17.56265 17.55555 17.56749 17.56755 4.143-11
16 2s3p $^3$P$^o_2$ 17.57084 17.56323 17.57501 17.57512 4.037-10
17 2s3d $^3$D$ _1$ 17.9142 17.91422 17.90599 17.92123 17.91911 1.076-12
18 2s3d $^3$D$ _2$ 17.9162 17.91616 17.90756 17.92272 17.92059 1.079-12
19 2s3d $^3$D$ _3$ 17.9182 17.91907 17.91022 17.92529 17.92316 1.084-12
20 2s3d $^1$D$ _2$ 18.1555 18.18722 18.17882 18.19304 18.18938 1.608-12
21 2p3s $^3$P$^o_0$ 18.69695 18.69110 18.71734 18.71745 6.603-12
22 2p3s $^3$P$^o_1$ 18.71325 18.70615 18.73218 18.73226 6.538-12
23 2p3s $^3$P$^o_2$ 18.7460 18.75178 18.74171 18.76709 18.76720 6.422-12
24 2p3s $^1$P$^o_1$ 19.0625 19.01237 19.00401 19.03602 19.03186 5.332-12
25 2p3p $^1$P$ _1$ 19.0894 19.09350 19.08552 19.11202 19.11216 3.777-12
26 2p3p $^3$D$ _1$ 19.1578 19.16750 19.16008 19.18591 19.18569 7.090-12
27 2p3p $^3$D$ _2$ 19.1721 19.18332 19.17515 19.20029 19.20005 7.590-12
28 2p3p $^3$D$ _3$ 19.0240 19.21937 19.20844 19.23319 19.23295 7.516-12
29 2p3p $^3$S$ _1$ 19.3160 19.33016 19.32141 19.34755 19.34700 4.454-12
30 2p3p $^3$P$ _0$ 19.39104 19.38473 19.43306 19.43188 4.346-12
31 2p3p $^3$P$ _1$ 19.3980 19.40812 19.40025 19.44680 19.44568 4.348-12
32 2p3p $^3$P$ _2$ 19.4137 19.42677 19.41709 19.46398 19.46286 4.347-12
33 2p3d $^3$F$^o_2$ 19.47683 19.46968 19.50477 19.50257 1.062-11
34 2p3d $^3$F$^o_3$ 19.50513 19.49594 19.53497 19.53229 1.144-10
35 2p3d $^1$D$^o_2$ 19.5155 19.52021 19.50998 19.54890 19.54802 3.121-12
36 2p3d $^3$F$^o_4$ 19.53388 19.52225 19.55770 19.55499 2.471-09
37 2p3p $^1$D$ _2$ 19.5778 19.61598 19.60698 19.66078 19.65660 2.935-12
38 2p3d $^3$D$^o_1$ 19.6893 19.69761 19.68861 19.72802 19.72823 8.663-13
39 2p3d $^3$D$^o_2$ 19.7012 19.70542 19.69591 19.73859 19.73878 8.831-13
40 2p3d $^3$D$^o_3$ 19.7138 19.72105 19.70995 19.75056 19.75077 8.622-13
41 2p3d $^3$P$^o_2$ 19.7762 19.79012 19.77978 19.81691 19.81661 1.553-12
42 2p3d $^3$P$^o_1$ 19.7898 19.80156 19.79099 19.82829 19.82796 1.579-12
43 2p3d $^3$P$^o_0$ 19.80750 19.79743 19.83558 19.83523 1.606-12
44 2p3p $^1$S$ _0$ 19.94986 19.94280 20.00059 19.98775 5.527-12
45 2p3d $^1$F$^o_3$ 19.9828 20.03858 20.02755 20.06982 20.06305 7.032-13
46 2p3d $^1$P$^o_1$ 20.08601 20.07597 20.11448 20.11208 1.174-12
47 2s4s $^3$S$ _1$ 22.54933 22.54097 22.54646 22.54537 1.037-11
48 2s4s $^1$S$ _0$ 22.65071 22.64271 22.65172 22.64843 1.106-11
Index Configuration Level NIST GRASP1 GRASP2 FAC1 FAC2 $\tau$ (s)
------- --------------- ------------- --------- ---------- ---------- ---------- ---------- ------------
49 2s4p $^3$P$^o_0$ 22.79976 22.79178 22.80264 22.80264 3.219-11
50 2s4p $^3$P$^o_1$ 22.80151 22.79331 22.80412 22.80410 3.045-11
51 2s4p $^3$P$^o_2$ 22.80575 22.79723 22.80786 22.80787 3.258-11
52 2s4p $^1$P$^o_1$ 22.83208 22.82350 22.83676 22.83426 3.598-12
53 2s4d $^3$D$ _1$ 22.94439 22.93591 22.94425 22.94326 2.738-12
54 2s4d $^3$D$ _2$ 22.94507 22.93645 22.94479 22.94380 2.743-12
55 2s4d $^3$D$ _3$ 22.94610 22.93741 22.94574 22.94474 2.750-12
56 2s4d $^1$D$ _2$ 23.0328 23.03637 23.02777 23.03406 23.03223 3.107-12
57 2s4f $^3$F$^o_2$ 23.0420 23.03790 23.02937 23.03614 23.03421 6.766-12
58 2s4f $^3$F$^o_3$ 23.0420 23.03825 23.02961 23.03639 23.03446 6.766-12
59 2s4f $^3$F$^o_4$ 23.0420 23.03872 23.03003 23.03679 23.03485 6.767-12
60 2s4f $^1$F$^o_3$ 23.06257 23.05392 23.06192 23.05970 6.749-12
61 2p4s $^3$P$^o_0$ 24.21374 24.20663 24.23547 24.23546 1.115-11
62 2p4s $^3$P$^o_1$ 24.22541 24.21768 24.24657 24.24586 1.071-11
63 2p4s $^3$P$^o_2$ 24.26942 24.25816 24.28604 24.28604 1.066-11
64 2p4s $^1$P$^o_1$ 24.33218 24.32131 24.35227 24.34046 7.459-12
65 2p4p $^1$P$ _1$ 24.39368 24.38599 24.41716 24.41709 5.881-12
66 2p4p $^3$D$ _1$ 24.42626 24.41706 24.44825 24.44790 6.445-12
67 2p4p $^3$D$ _2$ 24.42918 24.42038 24.45195 24.45182 7.901-12
68 2p4p $^3$D$ _3$ 24.46580 24.45399 24.48457 24.48453 7.977-12
69 2p4p $^3$S$ _1$ 24.48873 24.47906 24.51326 24.50573 6.182-12
70 2p4p $^3$P$ _0$ 24.49044 24.48248 24.52186 24.52000 7.285-12
71 2p4p $^3$P$ _1$ 24.52093 24.50993 24.54626 24.54261 6.635-12
72 2p4p $^3$P$ _2$ 24.52528 24.51423 24.55234 24.55105 7.233-12
73 2p4d $^3$F$^o_2$ 24.53974 24.53225 24.56341 24.56166 9.250-12
74 2p4d $^3$F$^o_3$ 24.56480 24.55592 24.58731 24.58512 1.027-11
75 2p4d $^1$D$^o_2$ 24.57451 24.56504 24.59586 24.59471 5.064-12
76 2p4p $^1$D$ _2$ 24.5755 24.59319 24.58233 24.62476 24.61978 5.473-12
77 2p4d $^3$F$^o_4$ 24.59736 24.58565 24.61617 24.61389 1.423-11
78 2p4d $^3$D$^o_1$ 24.61927 24.61082 24.64026 24.63980 2.096-12
79 2p4d $^3$D$^o_2$ 24.63018 24.62033 24.65013 24.64908 2.414-12
80 2p4f $^1$F$ _3$ 24.63738 24.62916 24.65589 24.65263 6.803-12
81 2p4f $^3$F$ _3$ 24.64055 24.63240 24.65934 24.65823 6.938-12
82 2p4f $^3$F$ _2$ 24.64195 24.63346 24.66072 24.66063 6.880-12
83 2p4f $^3$F$ _4$ 24.64430 24.63607 24.66310 24.65907 7.045-12
84 2p4d $^3$D$^o_3$ 24.64876 24.63750 24.66663 24.66666 2.100-12
85 2p4d $^3$P$^o_2$ 24.67077 24.65945 24.68876 24.68505 2.914-12
86 2p4d $^3$P$^o_1$ 24.67783 24.66650 24.69593 24.69122 3.141-12
87 2p4d $^3$P$^o_0$ 24.68175 24.67065 24.70015 24.69477 3.344-12
88 2p4f $^3$G$ _3$ 24.68824 24.67701 24.70310 24.69764 6.913-12
89 2p4f $^3$G$ _4$ 24.69295 24.68171 24.70791 24.70163 7.024-12
90 2p4f $^3$G$ _5$ 24.70949 24.69767 24.72356 24.71287 6.980-12
91 2p4f $^3$D$ _3$ 24.71783 24.70694 24.73342 24.75159 6.831-12
92 2p4f $^3$D$ _2$ 24.72363 24.71272 24.73956 24.73334 6.818-12
93 2p4f $^1$G$ _4$ 24.72378 24.71209 24.73907 24.72144 7.701-12
94 2p4f $^3$D$ _1$ 24.73681 24.72532 24.75171 24.72754 6.820-12
95 2p4p $^1$S$ _0$ 24.74104 24.73058 24.78352 24.73931 1.064-11
96 2p4f $^1$D$ _2$ 24.74808 24.73671 24.76430 24.76358 6.805-12
97 2p4d $^1$F$^o_3$ 24.77573 24.76474 24.79181 24.77893 1.472-12
98 2p4d $^1$P$^o_1$ 24.78736 24.77677 24.80363 24.79443 2.300-12
[NIST: [http://www.nist.gov/pml/data/asd.cfm]{}\
GRASP1: Coulomb energies\
GRASP2: QED corrected energies\
FAC1: Energies from the FAC for 98 level calculations\
FAC2: Energies from the FAC for 166 level calculations\
]{}
$i$ $j$ f (GRASP) f (FAC) Vel./Len. f(GRASP)/f(FAC)
----- ----- ----------- ----------- ----------- ----------------- --
1 3 3.046$-$5 3.039$-$5 0.70 1.00
1 5 2.942$-$1 2.942$-$1 0.97 1.00
1 13 5.324$-$1 5.445$-$1 0.97 0.98
1 15 2.478$-$2 2.183$-$2 0.97 1.14
2 7 1.124$-$1 1.127$-$1 0.90 1.00
2 11 3.321$-$2 3.394$-$2 0.94 0.98
2 17 7.100$-$1 7.095$-$1 0.98 1.00
3 6 3.688$-$2 3.698$-$2 0.90 1.00
3 7 2.790$-$2 2.797$-$2 0.90 1.00
3 8 4.719$-$2 4.731$-$2 0.91 1.00
3 9 2.755$-$5 2.640$-$5 0.61 1.04
3 10 4.082$-$6 4.097$-$6 1.70 1.00
3 11 3.334$-$2 3.404$-$2 0.94 0.98
3 12 1.076$-$6 1.377$-$6 0.46 0.78
3 17 1.775$-$1 1.773$-$1 0.98 1.00
3 18 5.318$-$1 5.315$-$1 0.98 1.00
4 7 2.743$-$2 2.751$-$2 0.90 1.00
4 8 8.326$-$2 8.348$-$2 0.90 1.00
4 9 3.187$-$4 3.139$-$4 0.93 1.02
4 11 3.356$-$2 3.423$-$2 0.94 0.98
4 17 7.105$-$3 7.107$-$3 0.98 1.00
4 18 1.064$-$1 1.063$-$1 0.98 1.00
4 19 5.947$-$1 5.945$-$1 0.98 1.00
5 8 1.638$-$4 1.632$-$4 1.50 1.00
5 9 1.048$-$1 1.061$-$1 1.30 0.99
5 10 7.063$-$2 7.054$-$2 0.58 1.00
5 12 1.406$-$2 1.456$-$2 0.79 0.97
5 20 5.462$-$1 5.424$-$1 1.00 1.01
6 15 1.296$-$3 1.365$-$3 1.70 0.95
7 14 4.245$-$4 4.460$-$4 1.70 0.95
7 15 2.922$-$4 3.098$-$4 1.70 0.94
7 16 6.034$-$4 6.250$-$4 1.60 0.97
8 13 1.141$-$4 1.106$-$4 0.96 1.03
8 15 2.565$-$4 2.766$-$4 1.80 0.93
8 16 1.004$-$3 1.048$-$3 1.70 0.96
9 13 1.181$-$2 1.201$-$2 0.62 0.98
9 15 6.397$-$4 5.647$-$4 0.66 1.13
9 16 1.847$-$6 2.138$-$6 1.90 0.86
10 13 2.320$-$3 1.955$-$3 0.26 1.19
10 15 8.195$-$5 5.692$-$5 0.15 1.44
11 14 3.051$-$2 3.080$-$2 1.10 0.99
11 15 8.808$-$2 8.943$-$2 1.10 0.98
11 16 1.560$-$1 1.573$-$1 1.10 0.99
13 20 1.154$-$1 1.155$-$1 0.79 1.00
----- ----- ----------- ----------- ----------- ----------- ----------- ----------- ----------- --
$i$ $j$ 50 100 150 200 250 300 350
1 2 1.708$-$3 6.687$-$4 3.528$-$4 2.180$-$4 1.481$-$4 1.070$-$4 8.115$-$5
1 3 5.675$-$3 2.584$-$3 1.630$-$3 1.195$-$3 9.725$-$4 8.336$-$4 7.602$-$4
1 4 8.495$-$3 3.323$-$3 1.752$-$3 1.082$-$3 7.350$-$4 5.314$-$4 4.027$-$4
1 5 1.958$+$0 2.275$+$0 2.361$+$0 2.286$+$0 2.251$+$0 2.206$+$0 2.279$+$0
1 6 4.860$-$5 1.852$-$5 1.131$-$5 8.698$-$6 7.501$-$6 6.860$-$6 6.481$-$6
1 7 1.206$-$4 3.403$-$5 1.399$-$5 7.060$-$6 4.056$-$6 2.540$-$6 1.700$-$6
1 8 2.439$-$4 1.062$-$4 7.504$-$5 6.460$-$5 6.021$-$5 5.810$-$5 5.694$-$5
1 9 1.217$-$2 1.273$-$2 1.300$-$2 1.315$-$2 1.324$-$2 1.331$-$2 1.334$-$2
1 10 4.128$-$3 3.532$-$3 3.216$-$3 3.020$-$3 2.888$-$3 2.793$-$3 2.722$-$3
1 11 1.295$-$3 4.340$-$4 2.162$-$4 1.297$-$4 8.662$-$5 6.202$-$5 4.661$-$5
1 12 6.828$-$2 7.410$-$2 7.622$-$2 7.737$-$2 7.814$-$2 7.872$-$2 7.918$-$2
1 13 7.765$-$2 1.371$-$1 1.779$-$1 2.084$-$1 2.328$-$1 2.532$-$1 2.712$-$1
1 14 3.425$-$4 9.731$-$5 4.461$-$5 2.538$-$5 1.633$-$5 1.135$-$5 8.339$-$6
1 15 5.179$-$3 7.740$-$3 9.822$-$3 1.144$-$2 1.276$-$2 1.386$-$2 1.484$-$2
1 16 1.705$-$3 4.843$-$4 2.220$-$4 1.263$-$4 8.125$-$5 5.649$-$5 4.150$-$5
1 17 1.806$-$3 5.195$-$4 2.390$-$4 1.360$-$4 8.740$-$5 6.076$-$5 4.461$-$5
1 18 3.014$-$3 8.719$-$4 4.051$-$4 2.338$-$4 1.530$-$4 1.088$-$4 8.194$-$5
1 19 4.212$-$3 1.211$-$3 5.574$-$4 3.172$-$4 2.038$-$4 1.417$-$4 1.040$-$4
1 20 1.222$-$1 1.606$-$1 1.763$-$1 1.846$-$1 1.894$-$1 1.926$-$1 1.946$-$1
1 21 7.219$-$6 2.127$-$6 9.620$-$7 5.386$-$7 3.412$-$7 2.344$-$7 1.704$-$7
1 22 5.483$-$5 6.518$-$5 7.987$-$5 9.248$-$5 1.030$-$4 1.121$-$4 1.206$-$4
1 23 3.537$-$5 1.043$-$5 4.721$-$6 2.644$-$6 1.675$-$6 1.151$-$6 8.370$-$7
1 24 2.573$-$3 4.370$-$3 5.682$-$3 6.696$-$3 7.515$-$3 8.208$-$3 8.847$-$3
1 25 9.645$-$5 6.424$-$5 4.745$-$5 3.707$-$5 3.011$-$5 2.515$-$5 2.146$-$5
1 26 7.385$-$5 2.769$-$5 1.486$-$5 9.490$-$6 6.706$-$6 5.058$-$6 3.993$-$6
1 27 1.200$-$4 4.268$-$5 2.257$-$5 1.472$-$5 1.093$-$5 8.830$-$6 7.552$-$6
1 28 1.582$-$4 5.301$-$5 2.551$-$5 1.476$-$5 9.546$-$6 6.648$-$6 4.885$-$6
1 29 5.365$-$5 1.732$-$5 8.380$-$6 4.941$-$6 3.270$-$6 2.334$-$6 1.754$-$6
1 30 9.022$-$6 3.664$-$6 2.649$-$6 2.333$-$6 2.205$-$6 2.145$-$6 2.113$-$6
1 31 2.166$-$5 5.233$-$6 2.040$-$6 1.014$-$6 5.849$-$7 3.716$-$7 2.533$-$7
1 32 4.262$-$5 1.928$-$5 1.521$-$5 1.402$-$5 1.356$-$5 1.334$-$5 1.323$-$5
1 33 9.028$-$5 4.142$-$5 3.065$-$5 2.549$-$5 2.216$-$5 1.973$-$5 1.784$-$5
1 34 1.035$-$4 2.745$-$5 1.348$-$5 8.746$-$6 6.617$-$6 5.488$-$6 4.823$-$6
1 35 1.284$-$4 9.889$-$5 8.740$-$5 7.823$-$5 7.063$-$5 6.430$-$5 5.898$-$5
1 36 1.281$-$4 3.151$-$5 1.366$-$5 7.577$-$6 4.813$-$6 3.330$-$6 2.443$-$6
1 37 1.036$-$3 1.273$-$3 1.348$-$3 1.377$-$3 1.389$-$3 1.394$-$3 1.396$-$3
1 38 5.883$-$5 6.940$-$5 8.107$-$5 9.052$-$5 9.820$-$5 1.047$-$4 1.108$-$4
1 39 2.497$-$5 5.629$-$6 2.388$-$6 1.365$-$6 9.207$-$7 6.864$-$7 5.460$-$7
1 40 2.571$-$5 6.975$-$6 4.314$-$6 3.632$-$6 3.403$-$6 3.318$-$6 3.288$-$6
1 41 1.322$-$4 3.616$-$5 1.639$-$5 9.316$-$6 6.021$-$6 4.225$-$6 3.142$-$6
1 42 9.320$-$5 3.969$-$5 3.102$-$5 2.927$-$5 2.931$-$5 2.995$-$5 3.090$-$5
1 43 2.743$-$5 7.464$-$6 3.351$-$6 1.881$-$6 1.199$-$6 8.287$-$7 6.066$-$7
1 44 5.905$-$4 5.783$-$4 5.762$-$4 5.761$-$4 5.766$-$4 5.774$-$4 5.783$-$4
1 45 1.108$-$3 1.209$-$3 1.252$-$3 1.280$-$3 1.300$-$3 1.316$-$3 1.330$-$3
1 46 6.973$-$3 1.000$-$2 1.198$-$2 1.347$-$2 1.466$-$2 1.566$-$2 1.658$-$2
1 47 4.959$-$4 1.521$-$4 7.309$-$5 4.301$-$5 2.842$-$5 2.010$-$5 1.508$-$5
1 48 1.337$-$2 1.490$-$2 1.547$-$2 1.578$-$2 1.598$-$2 1.613$-$2 1.625$-$2
1 49 1.600$-$4 4.049$-$5 1.768$-$5 9.791$-$6 6.201$-$6 4.252$-$6 3.104$-$6
1 50 6.261$-$4 3.797$-$4 3.866$-$4 4.202$-$4 4.553$-$4 4.886$-$4 5.208$-$4
----- ----- ----------- ----------- ----------- ----------- ----------- ----------- ----------- --
----- ----- ----------- ----------- ----------- ----------- ----------- ----------- ----------- --
$i$ $j$ 50 100 150 200 250 300 350
1 51 7.968$-$4 2.016$-$4 8.798$-$5 4.874$-$5 3.086$-$5 2.116$-$5 1.545$-$5
1 52 1.720$-$2 2.989$-$2 3.864$-$2 4.533$-$2 5.071$-$2 5.529$-$2 5.950$-$2
1 53 7.155$-$4 1.973$-$4 8.952$-$5 5.061$-$5 3.239$-$5 2.245$-$5 1.645$-$5
1 54 1.193$-$3 3.305$-$4 1.511$-$4 8.643$-$5 5.615$-$5 3.963$-$5 2.967$-$5
1 55 1.667$-$3 4.597$-$4 2.086$-$4 1.179$-$4 7.546$-$5 5.229$-$5 3.833$-$5
1 56 2.259$-$2 3.019$-$2 3.328$-$2 3.491$-$2 3.590$-$2 3.657$-$2 3.702$-$2
1 57 3.191$-$4 6.620$-$5 2.659$-$5 1.410$-$5 8.701$-$6 5.896$-$6 4.257$-$6
1 58 4.472$-$4 9.363$-$5 3.828$-$5 2.085$-$5 1.330$-$5 9.387$-$6 7.101$-$6
1 59 5.734$-$4 1.189$-$4 4.774$-$5 2.533$-$5 1.562$-$5 1.059$-$5 7.643$-$6
1 60 7.039$-$3 8.601$-$3 9.011$-$3 9.177$-$3 9.265$-$3 9.323$-$3 9.370$-$3
1 61 3.291$-$6 9.160$-$7 4.092$-$7 2.283$-$7 1.447$-$7 9.946$-$8 7.254$-$8
1 62 2.009$-$5 1.559$-$5 1.624$-$5 1.743$-$5 1.860$-$5 1.965$-$5 2.064$-$5
1 63 1.612$-$5 4.526$-$6 2.029$-$6 1.134$-$6 7.198$-$7 4.955$-$7 3.618$-$7
1 64 1.437$-$4 1.640$-$4 1.868$-$4 2.062$-$4 2.225$-$4 2.363$-$4 2.487$-$4
1 65 2.305$-$5 1.064$-$5 7.057$-$6 5.311$-$6 4.259$-$6 3.551$-$6 3.039$-$6
1 66 2.114$-$5 8.895$-$6 5.593$-$6 4.080$-$6 3.209$-$6 2.640$-$6 2.238$-$6
1 67 3.157$-$5 1.138$-$5 6.698$-$6 4.922$-$6 4.062$-$6 3.573$-$6 3.271$-$6
1 68 4.064$-$5 1.227$-$5 5.681$-$6 3.215$-$6 2.051$-$6 1.414$-$6 1.032$-$6
1 69 2.131$-$5 6.721$-$6 3.437$-$6 2.170$-$6 1.538$-$6 1.171$-$6 9.333$-$7
1 70 6.768$-$6 4.381$-$6 4.084$-$6 4.052$-$6 4.074$-$6 4.108$-$6 4.141$-$6
1 71 1.644$-$5 4.418$-$6 1.942$-$6 1.072$-$6 6.746$-$7 4.634$-$7 3.363$-$7
1 72 1.916$-$5 6.966$-$6 4.749$-$6 4.005$-$6 3.657$-$6 3.454$-$6 3.324$-$6
1 73 3.402$-$5 1.220$-$5 7.824$-$6 6.030$-$6 5.020$-$6 4.350$-$6 3.863$-$6
1 74 4.131$-$5 1.177$-$5 6.655$-$6 4.987$-$6 4.262$-$6 3.892$-$6 3.683$-$6
1 75 4.041$-$5 1.988$-$5 1.492$-$5 1.245$-$5 1.084$-$5 9.658$-$6 8.739$-$6
1 76 1.185$-$4 1.234$-$4 1.223$-$4 1.196$-$4 1.167$-$4 1.141$-$4 1.117$-$4
1 77 5.048$-$5 1.179$-$5 5.015$-$6 2.752$-$6 1.737$-$6 1.196$-$6 8.749$-$7
1 78 6.758$-$5 8.629$-$5 1.020$-$4 1.144$-$4 1.247$-$4 1.335$-$4 1.412$-$4
1 79 2.511$-$5 7.313$-$6 4.018$-$6 2.817$-$6 2.213$-$6 1.846$-$6 1.598$-$6
1 80 1.226$-$5 7.371$-$6 5.516$-$6 4.402$-$6 3.653$-$6 3.117$-$6 2.717$-$6
1 81 3.206$-$6 6.064$-$7 2.584$-$7 1.492$-$7 1.003$-$7 7.370$-$8 5.740$-$8
1 82 1.084$-$5 1.346$-$5 1.516$-$5 1.622$-$5 1.691$-$5 1.738$-$5 1.773$-$5
1 83 9.779$-$6 7.992$-$6 8.248$-$6 8.531$-$6 8.752$-$6 8.923$-$6 9.058$-$6
1 84 1.521$-$5 3.767$-$6 2.166$-$6 1.740$-$6 1.588$-$6 1.525$-$6 1.498$-$6
1 85 4.825$-$5 1.223$-$5 5.396$-$6 3.040$-$6 1.964$-$6 1.385$-$6 1.038$-$6
1 86 3.925$-$5 1.623$-$5 1.289$-$5 1.236$-$5 1.253$-$5 1.291$-$5 1.336$-$5
1 87 1.247$-$5 3.170$-$6 1.382$-$6 7.625$-$7 4.802$-$7 3.291$-$7 2.394$-$7
1 88 7.156$-$6 3.137$-$6 2.168$-$6 1.672$-$6 1.362$-$6 1.148$-$6 9.925$-$7
1 89 8.816$-$6 5.361$-$6 5.167$-$6 5.218$-$6 5.295$-$6 5.367$-$6 5.429$-$6
1 90 9.487$-$6 1.901$-$6 8.236$-$7 4.637$-$7 2.980$-$7 2.076$-$7 1.528$-$7
1 91 8.631$-$6 2.003$-$6 9.824$-$7 6.338$-$7 4.655$-$7 3.675$-$7 3.037$-$7
1 92 2.831$-$5 3.422$-$5 3.858$-$5 4.132$-$5 4.309$-$5 4.429$-$5 4.516$-$5
1 93 4.208$-$5 4.565$-$5 4.894$-$5 5.115$-$5 5.270$-$5 5.384$-$5 5.472$-$5
1 94 3.797$-$6 6.927$-$7 2.597$-$7 1.317$-$7 7.876$-$8 5.216$-$8 3.701$-$8
1 95 2.085$-$4 2.169$-$4 2.255$-$4 2.315$-$4 2.359$-$4 2.394$-$4 2.421$-$4
1 96 9.353$-$5 1.296$-$4 1.483$-$4 1.594$-$4 1.665$-$4 1.712$-$4 1.746$-$4
1 97 1.887$-$4 1.828$-$4 1.831$-$4 1.849$-$4 1.868$-$4 1.887$-$4 1.905$-$4
1 98 1.837$-$3 2.641$-$3 3.166$-$3 3.567$-$3 3.894$-$3 4.171$-$3 4.414$-$3
----- ----- ----------- ----------- ----------- ----------- ----------- ----------- ----------- --
--- ---- ----------- ----------- ----------- ----------- ----------- ----------- -- -- --
I J 5.90 6.10 6.30 5.90 6.10 6.30
1 2 7.824$-$3 6.631$-$3 5.524$-$3 1.161$-$2 1.076$-$2 9.407$-$3
1 3 2.347$-$2 1.989$-$2 1.657$-$2 3.528$-$2 3.275$-$2 2.870$-$2
1 4 3.912$-$2 3.316$-$2 2.762$-$2 5.881$-$2 5.444$-$2 4.753$-$2
1 5 1.296$-$0 1.368$-$0 1.459$-$0 1.280$-$0 1.346$-$0 1.433$-$0
1 6 2.038$-$4 1.807$-$4 1.552$-$4 6.151$-$4 6.846$-$4 6.534$-$4
1 7 6.113$-$4 5.420$-$4 4.657$-$4 1.744$-$3 1.930$-$3 1.834$-$3
1 8 1.019$-$3 9.033$-$4 7.761$-$4 2.827$-$3 3.075$-$3 2.902$-$3
1 9 1.281$-$2 1.249$-$2 1.220$-$2 1.711$-$2 1.736$-$2 1.687$-$2
1 10 4.581$-$3 4.495$-$3 4.383$-$3 7.199$-$3 7.312$-$3 7.000$-$3
2 3 1.467$-$1 1.142$-$1 8.695$-$2 1.504$-$1 1.258$-$1 1.023$-$1
2 4 1.316$-$1 1.009$-$1 7.795$-$2 1.139$-$1 9.819$-$2 8.375$-$2
2 5 1.749$-$2 1.472$-$2 1.231$-$2 3.116$-$2 2.919$-$2 2.533$-$2
2 6 4.106$-$3 3.683$-$3 3.211$-$3 5.641$-$3 5.463$-$3 4.910$-$3
2 7 6.600$-$1 6.922$-$1 7.328$-$1 6.409$-$1 6.727$-$1 7.153$-$1
2 8 5.527$-$3 4.973$-$3 4.356$-$3 1.148$-$2 1.163$-$2 1.061$-$2
2 9 9.489$-$3 8.527$-$3 7.446$-$3 1.695$-$2 1.700$-$2 1.534$-$2
2 10 1.139$-$3 1.034$-$3 9.112$-$4 2.704$-$3 2.638$-$3 2.307$-$3
3 4 4.821$-$1 3.721$-$1 2.844$-$1 5.072$-$1 4.196$-$1 3.430$-$1
3 5 5.247$-$2 4.417$-$2 3.693$-$2 9.302$-$2 8.719$-$2 7.566$-$2
3 6 6.561$-$1 6.892$-$1 7.307$-$1 6.438$-$1 6.755$-$1 7.177$-$1
3 7 5.150$-$1 5.408$-$1 5.734$-$1 5.057$-$1 5.292$-$1 5.588$-$1
3 8 8.488$-$1 8.973$-$1 9.582$-$1 8.240$-$1 8.643$-$1 9.153$-$1
3 9 2.847$-$2 2.558$-$2 2.234$-$2 5.341$-$2 5.340$-$2 4.812$-$2
3 10 3.417$-$3 3.102$-$3 2.734$-$3 8.667$-$3 8.480$-$3 7.434$-$3
4 5 8.744$-$2 7.361$-$2 6.156$-$2 1.553$-$1 1.452$-$1 1.258$-$1
4 6 5.508$-$3 4.959$-$3 4.347$-$3 9.998$-$3 1.008$-$2 9.185$-$3
4 7 8.499$-$1 8.984$-$1 9.590$-$1 8.273$-$1 8.675$-$1 9.186$-$1
4 8 2.517$-$0 2.642$-$0 2.799$-$0 2.437$-$0 2.556$-$0 2.712$-$0
4 9 4.744$-$2 4.263$-$2 3.723$-$2 1.039$-$1 1.037$-$1 9.444$-$2
4 10 5.694$-$3 5.170$-$3 4.556$-$3 1.587$-$2 1.547$-$2 1.351$-$2
5 6 7.206$-$3 6.169$-$3 5.188$-$3 1.264$-$2 1.193$-$2 1.057$-$2
5 7 2.162$-$2 1.851$-$2 1.556$-$2 3.682$-$2 3.439$-$2 3.002$-$2
5 8 3.603$-$2 3.084$-$2 2.594$-$2 7.226$-$2 6.812$-$2 6.101$-$2
5 9 3.629$-$0 3.816$-$0 4.049$-$0 3.517$-$0 3.686$-$0 3.913$-$0
5 10 1.161$-$0 1.225$-$0 1.305$-$0 1.143$-$0 1.204$-$0 1.284$-$0
6 7 7.308$-$2 6.478$-$2 5.610$-$2 1.016$-$1 9.890$-$2 8.872$-$2
6 8 4.909$-$2 4.674$-$2 4.474$-$2 6.447$-$2 6.499$-$2 6.194$-$2
6 9 3.784$-$2 3.390$-$2 2.950$-$2 4.965$-$2 4.779$-$2 4.244$-$2
6 10 4.239$-$3 3.724$-$3 3.158$-$3 1.008$-$2 9.699$-$3 8.374$-$3
7 8 2.009$-$1 1.865$-$1 1.731$-$1 2.703$-$1 2.677$-$1 2.480$-$1
7 9 1.135$-$1 1.017$-$1 8.850$-$2 1.553$-$1 1.498$-$1 1.332$-$1
7 10 1.272$-$2 1.117$-$2 9.473$-$3 3.065$-$2 2.926$-$2 2.514$-$2
8 9 1.892$-$1 1.695$-$1 1.475$-$1 2.696$-$1 2.592$-$1 2.306$-$1
8 10 2.119$-$2 1.862$-$2 1.579$-$2 5.313$-$2 5.059$-$2 4.336$-$2
9 10 9.438$-$2 9.645$-$2 9.929$-$2 1.141$-$1 1.172$-$1 1.181$-$1
--- ---- ----------- ----------- ----------- ----------- ----------- ----------- -- -- --
[RM: Earlier interpolated results of [@fpk1]\
DARC: Present results from the DARC code\
]{}
[^1]: E-mail: [email protected](KMA); [email protected] (FPK)
[^2]: Tables 2 and 5 are available only in the electronic version.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We have observed the eclipsing, post-common envelope white dwarf-brown dwarf binary, SDSS141126.20+200911.1, in the near-IR with the HAWK-I imager, and present here the first direct detection of the dark side of an irradiated brown dwarf in the $H$ band, and a tentative detection in the $K_s$ band. Our analysis of the lightcurves and indicates that the brown dwarf is likely to have an effective temperature of 1300 K, which is not consistent with the effective temperature of 800 K suggested by its mass and radius. As the brown dwarf is already absorbing almost all the white dwarf emission in the $K_s$ band we suggest that this inconsistency may be due to the UV-irradiation from the white dwarf inducing an artificial brightening in the $K_s$ band, similar to that seen for the similar system WD0137-349B, suggesting this brightening may be characteristic of these UV-irradiated binaries.'
author:
- |
S. L. Casewell$^{1}$ [^1], S. P. Littlefair$^{2}$, S. G. Parsons$^{2}$, T.R. Marsh$^{3}$, J. J. Fortney$^{4}$, and M. S. Marley$^{5}$\
$^{1}$Department of Physics and Astronomy, University of Leicester, University Road, Leicester LE1 7RH, UK\
$^{2}$Department of Physics and Astronomy, University of Sheffield, Sheffield, S3 7RH, UK\
$^{3}$Department of Physics, University of Warwick, Gibbet Hill Road, Coventry CV4 7AL, UK\
$^{4}$Department of Astronomy and Astrophysics, University of California, Santa Cruz, CA 95064, USA\
$^{5}$NASA Ames Research Center, MS-245-3, Moffett Field, CA 94035, USA
bibliography:
- 'wd0137\_bib.bib'
date: 'Accepted XXX. Received YYY; in original form ZZZ'
title: 'The direct detection of the irradiated brown dwarf in the white dwarf - brown dwarf binary SDSSJ141126.20+200911.1'
---
\[firstpage\]
brown dwarfs, binaries:eclipsing, white dwarfs,
Introduction
============
Despite recent results reporting the discovery of brown dwarf companions to main sequence stars (e.g. @anderson11 [@siverd12; @bayliss2017; @triaud18]), there are still only thirteen known to date, and they are very rare compared to planetary or stellar companions to main sequence stars [@grether06; @metchev04]. As a result, there are very few systems known to have evolved from these binaries with @steele11 predicting only 0.5% of white dwarfs having brown dwarf companions.
To date only nine post-common envelope systems have been confirmed: GD1400 (WD+L6, P=9.98hrs; @farihi04 [@dobbie05; @burleigh11]), WD0137-349 (WD+L6-L8, P=116min; @maxted06 [@burleigh06]), WD0837+185 (WD+T8, P=4.2hrs; @casewell12), NLTT5306 (WD+L4-L7, P=101.88min; @steele13), SDSS J155720.77+091624.6 (WD+L3-L5, P=2.27hrs; @farihi17), SDSS J1205-0242 (WD+L0, P=71.2min; @parsons17 [@rappaport17]), SDSS J1231+0041 (WD+M/L, P=72.5min; @parsons17), EPIC212235321 (WD+L5, P=68 min; @casewell18) and SDSS J141126.20+200911.1, hereafter SDSS1411+2009 (WD+T5, P=121.73min; @beuermann13 [@littlefair14]). All of these systems have survived a phase of common-envelope evolution, resulting in the close binary system. They are all detached, and likely tidally-locked, resulting in a brown dwarf that is irradiated on one hemisphere, similar to the situation in most hot Jupiter exoplanets. Eventually, these white dwarf-brown dwarf binaries will become cataclysmic variables, such as SDSS1433+1011 in which the substellar donor was recently detected [@hernandez16].
Irradiated brown dwarfs are expected to have very similar atmospheres to irradiated exoplanets, and have been described as the “fourth corner” of the parameter space containing irradiated exoplanets, solar system planets and isolated brown dwarfs [@showman]. For instance, Kelt-9b [@gaudi17] is a 2.88 M$_{\rm Jup}$ planet orbiting a $\sim$10 000 K star. This planet is expected to receive $\sim$700 times more UV irradiation than a planet orbiting the next hottest exoplanet host star (WASP-33). However, the primary star in the Kelt-9 system is still $\sim$3000 K cooler than SDSS1411+1011A and $\sim$6500 K cooler than WD0137-349A. The brown dwarf companion in this latter system has been shown to have an atmosphere that is significantly affected by UV irradiation [@longstaff17; @casewell15]. In fact, Kelt-9b has been shown to have a day-nightside temperature difference of $\sim$500 K, the same as the irradiated brown dwarf WD0137-349B, indicating poor heat redistribution is present in both systems, despite their differences in internal temperature. Studying irradiated brown dwarfs can therefore provide a useful proxy for exoplanet systems, especially to explore the effects of UV irradiation and any resultant photochemistry, as in general hot Jupiter host stars replicating the same conditions must be very large, making them challenging systems to observe. One of the most recently discovered of the post-common envelope systems, and the first eclipsing system to be discovered, SDSS1411+2009, was discovered as part of the Catalina Sky Survey by @drake10. The substellar nature of the companion to the white dwarf was confirmed by @beuermann13. While its period is very similar to that of the well-studied WD0137-349, the white dwarf is cooler with T$_{\rm eff}$=13000$\pm$300 K and log g=7.86$\pm$0.07, giving a mass of 0.53$\pm$0.03 M$_{\odot}$ [@littlefair14]. The brown dwarf mass is calculated to be 50$\pm$2 M$_{\rm Jup}$, and has an estimated spectral type of T5, derived from the secondary’s mass. The $z'$ band eclipse and $K_s$ excess presented in @littlefair14 were used to estimate the dayside spectral type to be between L7 and T1, suggesting significant irradiation.
Observations and data reduction
===============================
We observed SDSSJ1411+2009 with the infrared imager HAWK-I [@kissler08] on the VLT as part of programme 94.C-0032. The data were obtained on the nights of the 2015-04-04, 2015-04-05, and the 2015-03-13 for $J$, $H$ and $K_s$ respectively. The seeing was 1“ in the $J$, and $H$ bands and between 1.5” and 2.5" in the $K_s$ band. We used the fast photometry mode, allowing us to window the detector and reduce the deadtime between frames to a few microseconds, and used exposure times of 5 s in each of the $J$, $H$, and $K_s$ bands. We observed using chip 4, and orientated the 128 pixel window to 120 degrees to also observe a standard star, 2MASS14112391+2008132 which was used to calibrate the photometry. The data were dark-subtracted, flat fielded and extracted using aperture photometry within the ULTRACAM pipeline [@dhillon07].
Results
=======
We used <span style="font-variant:small-caps;">lroche</span>, part of the <span style="font-variant:small-caps;">lcurve</span> software to model the lightcurves (see @copperwheat10 for a description). We sample the posterior probability distributions for model parameters using affine-invariant Markov-chain Monte-Carlo (MCMC) [@foreman-mackey2013]. We used the system parameters given in @beuermann13 and @littlefair14 to set priors on the mass ratio, orbital period, angle of inclination, white dwarf temperature and stellar radii. The covariance matrix from @littlefair14 was used to create multi-variate normal priors for the stellar radii and the inclination. Independent Gaussian priors were used for all other parameters. Since the lightcurves show evidence for red noise, presumably arising from instrumental systematics, we do not use the chi-squared statistic to estimate the likelihood. Instead we model the residuals from the <span style="font-variant:small-caps;">lroche</span> model using a Gaussian process with a Mat[é]{}rn-3/2 kernel and use the likelihood of the residuals [see @mcallister17 for an example of this approach]. Multiple, independent MCMC chains are run from different starting points, and we use the Gelman-Rubin diagnostic, applied to the independent runs, to test for convergence. We also tested that the results were insensitive to the kernel function adopted for the Gaussian process.
We adopt the limb darkening coefficients in @gianninas13 for a 13000 K, log g =8.00 white dwarf for the $y$ band, as there are none available for the near-IR, although as this is within the Raleigh-Jeans tail of the white dwarf spectrum, these coefficients are not expected to deviate much from these values. Additionally, given the S/N of our data, any deviation will have a negligible effect on our fit.
The [lroche]{} model is used to measure the level of the reflection effect caused as the heated side of the brown dwarf moves into view. The brightness temperature of an element on the companion is modeled as: $$T_{c,j}^4 = \left[ T_c \left( \frac{g_j}{g_{{\rm pole}}}\right)^{\beta} \right]^4 + \alpha G_j T_{\rm wd}^4,$$ where $\alpha$ is the fraction of the incident flux which is absorbed (i.e $\alpha = 1 - A$), where $A$ is the albedo. $g_j$ is the surface gravity of the element, $g_{\rm pole}$ is the surface gravity at the pole, $\beta$ is the gravity darkening exponent, for which we adopted a value of 0.45. $G_j$ is a geometric factor which accounts for the fraction of the WD flux absorbed by the companion, taking the full Roche geometry into account. $T_c$ and $T_{\rm wd}$ are the black-body brightness temperatures of the companion and white dwarf respectively. Because our observations are within the Raleigh-Jeans tail of the white dwarf spectrum, the surface brightness of a white dwarf differs from that of the same-temperature black-body by less than 5%. The lightcurve of an irradiated binary in a single band constrains the [*ratio*]{} of brightness temperatures of the two components. Therefore, since a black-body is a reasonable description for the white dwarf, we can say that using $T_c$ in the Planck function gives an accurate prediction of the surface flux of the brown dwarf; these surface fluxes can be compared directly with surface fluxes predicted by irradiated models. The posterior probability distributions for these models are shown in Figures \[cornerh\] and \[cornerk\].
Our model of the system in the $H$ band predicts a nightside temperature of the brown dwarf of $1540^{+90}_{-70}$ K and the fraction of flux from the white dwarf absorbed by the brown dwarf as 0.50$\pm$0.06. The equivalent model for the $K_s$ band predicts 1000$\pm 500$ K and 0.80$\pm$0.15. As the $J$ band eclipse was not observed, we were unable to fit a model to these data, and instead fitted a sine curve to the data to measure the reflection effect as was done in @casewell15 for WD0137-349.
We detect the primary eclipse of the white dwarf in both the $H$ and $K_s$ data (Figures \[zoom1\] and \[zoom2\]). We do not detect the secondary eclipse in any of our data. Our model predicts that the secondary eclipse depth is 0.8 per cent in the $H$ band and 3 per cent in the $K_s$ band, which is smaller than our photometric errors ($\sim$0.05 mags in $H$, and 0.2 mags in $K_s$), and as the secondary eclipse is predicted to last $\sim$ 4 minutes including ingress and egress, we cannot bin our data up to a high enough precision.
The primary eclipse is total, not unexpected, as brown dwarf radii are typically comparable to that of Jupiter, while white dwarfs have radii similar to that of the Earth, hence all the flux we detect is from the nightside of the brown dwarf at this point. This flux is significantly non-zero in the $H$-band, making this the first direct detection of the dark side of an irradiated brown dwarf. The flux in the $K_s$ band is consistent with zero, which is reflected in our large uncertainties on the $K_s$ brightness temperature. Although our model has calculated an average nightside temperature in the $K_s$ band, we have chosen to give the nightside an upper limit of 1500 K to reflect the zero flux.
In addition to the detection of the night-side of the brown dwarf, we are also able to calculate the magnitude of the day-side of the brown dwarf due to the reflection effect in the system, causing sinusoidal variations as the tidally locked brown dwarf orbits the white dwarf. The semi-amplitude of this variability is 0.0019$\pm$0.0003 mJy in the $H$ band, and 0.0039$\pm$0.0006 mJy in the $K_s$ band. This variability is slightly larger than that detected for the WD0137-349AB system [@casewell15] which has a similar period, but a hotter, and less massive white dwarf (T$_{\rm eff}=$16500 K, M=0.4M$_{\odot}$ @maxted06), but the errors are large on these measurements.
We also used the <span style="font-variant:small-caps;">molly</span> software package to search for any emission lines from the brown dwarf in the 28 UVB and VIS XSHOOTER spectra used to measure the radial velocity in @littlefair14. We did not detect H$\alpha$ emission, as is seen for WD0137-349B [@maxted06], or any other emission lines as were detected by @longstaff17 for the same system. As SDSS1411+2009 is 3 magnitudes fainter in the optical than the WD0137-349 system, we phase binned the data and combined the spectra in phase, but still did not detect any emission features from the brown dwarf. The data from the NIR arm of XSHOOTER are of not good enough quality to be used in any analysis.
Discussion
==========
We calculated brightness temperatures for the dayside of the brown dwarf for the $J$ band using a model white dwarf spectrum and the method detailed in @casewell15. For the $H$ and $K_s$ bands where we have models of the system from <span style="font-variant:small-caps;">lcurve</span> we generated a temperature map of the surface of the brown dwarf as was done in @hernandez16. The average dayside and nightside temperatures are reported in Table \[brightness\], although, from the surface map of the brown dwarf we were also able to model the maximum and minimum temperatures present across the surface. These temperatures had a maximum of 1940$\pm$70 K in the $H$ and 2000$\pm$150 K in the $K_s$ bands, and a minimum of 1530$\pm$90 K and 950$\pm$500 K in the $H$ and $K_s$ bands respectively.
We have generated irradiated brown dwarf models using the atmospheric structure model of @marley99, @marley02 and @fortney05 using the log g from @littlefair14 and intrinsic effective temperatures (the temperature the brown dwarf would have in the absence of the white dwarf) ranging from 500 K to 1500 K (Figure \[brightnessfig\]). The white dwarf irradiation was modelled using a 13 000 K black body at the appropriate separation. We have chosen to use surface flux densities in displaying these data, as this removes any uncertainties associated with the radius of the brown dwarf and the distance to the system. While the dayside $H$ and $K_s$ band fluxes are consistent with an irradiated brown dwarf of 1300 K, it is clear that the dayside $K_s$ flux also encompasses temperatures much hotter than 1500 K (the hottest model plotted). This is consistent with our findings in @casewell15, where the $K_s$ band was much brighter than the models predicted.
It can be seen that SDSS1411-2009B has an average difference in day-night side temperatures of 93$\pm$12 K in the $H$ band, and a 360$\pm$80 K day-night difference in the $K_s$ band. As these measurements are derived from the <span style="font-variant:small-caps;">lcurve</span> model, they take into account the errors on the radii and the correlated errors relating to the distance to the binary. The distance from Gaia DR2 is 177$\pm$5 pc [@gaia], compared to 190$\pm$8 in @littlefair14. These distances agree to within 1.5 $\sigma$. Surface flux densities for the brown dwarf were derived from the brightness temperatures of each element using the Planck curve. To compare the nightside fluxes with models, we used non-irradiated cloud free brown dwarf models, again using the atmospheric structure of @marley99, @marley02 and @fortney05. These models, and the nightside fluxes can be seen in Figure \[nightside\]. Both the $H$ and $K_s$ bands are consistent with T$_{\rm eff}$= 1300 K. This raises an interesting conundrum, as the estimated T$_{\rm eff}$ of the brown dwarf using the radius from the lightcurves and the mass from the radial velocity solution combined with evolutionary models of @baraffe03 is $\sim$ 800 K [@littlefair14].
In comparison with a similar system, both SDSS1411-2009B and WD0137-349B have similar brightness temperatures (within the errors) in the $H$ band on both the day and night sides, although the nightside of WD0137-349B is an upper limit, and the dayside temperature is not well constrained at 1585$\pm$329 K compared to 1730$\pm$70 K for SDSSJ1411-2009B. The nightside of SDSS1411-2009B in the $H$ band appears to be hotter in than that of WD0137-349B, despite SDSSJ1411-2009B being of a later spectral type, but the errors on the upper limit mean we can not state this conclusively.
In the $K_s$ band, the dayside of WD0137-349B (2015 K)is hotter than the dayside of SDSSJ1411-2009B (1620 K), as would be expected for a brown dwarf orbiting a hotter white dwarf in a shorter orbit. The nightside brightness temperatures of both objects have large errors, however, as with the $H$ nightside measurements, they may be similar temperatures.
---------- ------------------------ ---------------- --------------------- --------------------
Waveband [Magnitude (WD+BD) ]{} Magnitude (WD)
Dayside Dayside Nightside
$J$ 17.96$\pm$0.04 18.02 1715$^{+95}_{-131}$ -
$H$ 17.80$\pm$0.04 18.18 1730$\pm$70 1530$^{+90}_{-70}$
$K_s$ 18.11$\pm$0.10 18.27 1620$\pm$160 1500
---------- ------------------------ ---------------- --------------------- --------------------
Despite the white dwarf in SDSSJ1411+2009 being $\sim$ 3500 K cooler than the white dwarf in WD0137-349B, there is not a large difference in the SED of the irradiated brown dwarfs in these systems. WD0137-349B emits much more strongly in the ultraviolet (by a factor of $\sim$10) than SDSS1411-2009A does, although the peak of the white dwarf SED is approximately at the same wavelength in both cases. This is likely to be the explanation for the lack of emission lines seen in the atmosphere of SDSSJ1411-2009B. The lack of UV irradiation means SDSSJ1411+2009 is unlikely to have a chromosphere, similar to that suggested for WD0137-349B by @longstaff17. This is also suggested by the lack of H$\alpha$ emission lines in the optical spectra. However, despite this lack of emission lines, the same brightening is seen in the $K_s$ for both WD0137-349B and SDSSJ1411-2009B.
Our nightside brightness temperatures for SDSSJ1411-2009B indicate that in the absence of any heat transport, the T$_{\rm eff}$ of the brown dwarf is 1300 K. Our <span style="font-variant:small-caps;">lcurve</span> modelling of these lightcurves gives an absorb parameter (the fraction of flux from the white dwarf absorbed by the brown dwarf) of 0.50$\pm$0.06 in the $H$ band and 0.80$\pm$0.15 in the $K_s$ band. These parameters mean that if only absorption and reprocessing within the brown dwarf atmosphere is important, SDSS1411J-2009B must be absorbing 50 per cent of the $H$ band flux and 80 per cent of the $K_s$ band flux, in order to produce the dayside brightness temperatures.
However, the brown dwarf effective temperature as estimated from the mass and radius is 800 K [@littlefair14]. If this is the true effective temperature of the brown dwarf, were it an isolated object, then the absorb parameters must be even higher in order to produce enough heat transport to heat the nightside to 1300 K. The absorb parameter for the $K_s$ band is already close to 100 per cent though, which would indicate there is poor energy circulation around the brown dwarf, supported by the 200 K day-nightside difference in the $H$ band.
An additional factor that would affect estimates of temperature and energy circulation, may be fluorescence or emission within the brown dwarf atmosphere. We suggested this is present in WD0137-349B [@casewell15], again causing brightening in the $K_s$ and 4.5 micron bands. If this emission is present, it will increase the dayside flux, particularly in the $K_s$ band, meaning that the absorb parameter is artificially high. In particular it would mean that the brown dwarf needs to absorb a smaller fraction of flux in order to heat the nightside. This scenario is also potentially consistent with a lower T$_{\rm eff}$ of the brown dwarf. Emission from the dayside has artificially increased the flux, leading to an overestimate of the effective temperature.
Observations of Kelt-1b, a T2 dwarf orbiting a main sequence star [@siverd12], seem to support the hypothesis of UV-induced brightening in the $K_s$ band. Kelt-1b, orbiting a 6500 K F5V star lacks the intense UV irradiation of the white dwarf irradiated systems, and does not show this brightening. Indeed eclipse measurements suggest that this object fits very well with a field dwarf template [@croll15; @beatty17].
The only way we can, however, confirm this hypothesis of UV-induced emission is by obtaining spectrophotometry of SDSSJ1411-2009B with JWST. This would allow us to determine if at $K_s$ and 4.5 microns the brown dwarf looks like an isolated field object on the dayside, or whether UV-induced emission lines are present.
Conclusions
===========
We have observed the close, post-common envelope binary SDSS1411+2009 with HAWK-I in the $JHK_s$ bands, and have directly detected the brown dwarf in the $H$ and $K_s$ bands as it eclipses its white dwarf companion. We have determined the brightness temperatures for the day and night-sides of the brown dwarf and measure a temperature difference of only $\sim$200 K, compared to $\sim$500 K for WD0137-349B, a system with a similar period, but a hotter white dwarf primary. From comparing the surface fluxes to models of irradiated and non-irradiated brown dwarfs, we also determine that in general, the models indicate the brown dwarf is consistent with T$_{\rm eff}$=1300 K, but that the mass and radius suggest an effective temperature that is much lower. As the brown dwarf is already absorbing almost all the emission from the white dwarf in the $K_s$ band, this discrepancy suggests that an additional mechanism is making the $K_s$ band brighter. This mechanisms may similar to that suggested in WD0137-349B, hinting this may be a common trait in these systems, and may be due to photochemistry.
Acknowledgements
================
We thank Detlev Koester for providing the white dwarf models. This work is based on observations made with ESO Telescopes at the La Silla Paranal Observatory. This work also makes use of the white dwarf models from Pierre Bergeron:$\sim$bergeron/CoolingModels. S.L. Casewell acknowledges support from the University of Leicester College of Science and Engineering. SPL is supported by STFC grant ST/M001350/1, and TRM is supported by STFC grant ST/L000733. SGP acknowledges the support of the Leverhulme Trust.
Posterior probability distributions
===================================
\[lastpage\]
[^1]: E-mail: [email protected]
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We report the presence of large anisotropies in the sky distributions of powerful extended quasars as well as some other sub-classes of radio galaxies in the 3CRR survey, the most reliable and most intensively studied complete sample of strong steep-spectrum radio sources. The anisotropies lie about a plane passing through the equinoxes and the north celestial pole. Out of a total of 48 quasars in the sample, 33 of them lie in one half of the observed sky and the remaining 15 in the other half. The probability that in a random distribution of 3CRR quasars in the sky, statistical fluctuations could give rise to an asymmetry in observed numbers up to this level is only $\sim 1\%$. Also only about 1/4th of Fanaroff-Riley 1 (FR1) type of radio galaxies lie in the first half of the observed sky and the remainder in the second half. If we include all the observed asymmetries in the sky distributions of quasars and radio galaxies in the 3CRR sample, the probability of their occurrence by a chance combination reduces to $\sim 2 \times 10^{-5}$. Two pertinent but disturbing questions that could be raised here are – firstly why should there be such large anisotropies present in the sky distribution of some of the strongest and most distant discrete sources, implying inhomogeneities in the universe at very large scales (covering a fraction of the universe)? Secondly why should such anisotropies lie about a great circle decided purely by the orientation of earth’s rotation axis and/or the axis of its revolution around the sun? Are these alignments a mere coincidence or do they imply that these axes have a preferential placement in the larger scheme of things, implying an apparent breakdown of the Copernican principle or its more generalization, cosmological principle, upon which the standard cosmological model is based upon?'
author:
- 'Ashok K. Singal'
title: A large anisotropy in the sky distribution of 3CRR quasars and other radio galaxies
---
Introduction
============
Copernican principle states that earth does not have any eminent or privileged position in the universe and therefore an observer’s choice of origin and/or orientation of his/her coordinate system should have no bearing on the appearance of the distant universe. Its natural generalization is the cosmological principle which states that the universe on a sufficiently large scale should appear homogeneous and isotropic, with no preferred directions, to all observers. However to us on earth the universe does show heterogeneous structures up to the scale of superclusters of galaxies and even somewhat beyond, but the conventional wisdom is that it would all appear homogeneous and isotropic when observed on still larger scales, perhaps beyond a couple of hundreds of megaparsecs. Radio galaxies and quasars, the most distant discrete objects (at distances of gigaparsecs and farther) seen in the universe, should trace the distribution of matter in the universe at that large scale and should therefore appear isotropically distributed from any vantage point in the universe, including that on earth.
On the other hand Cosmic Microwave Background Radiation (CMBR) observations from the WMAP satellite have in recent years been reported to show some unexpected anisotropies, which surprisingly seem to be aligned with the ecliptic (Tegmark et al. 2003; de Oliveira-Costa et al. 2004; Ralston & Jain 2004; Schwarz et al. 2004; Land & Magueijo 2005). The alignment of the four normals to the quadrupole and octopole planes in the CMBR with the cosmological dipole and the equinoxes (Copi et al. 2010) could undermine our ideas about the standard cosmological model with very damaging implications. The latest data from the Planck satellite have confirmed the presence of these anisotropies (Ade et al. 2014). Also using a large sample of radio sources from the NRAO VLA Sky Survey (NVSS, Condon et al. 1998), which covers whole sky north of declination $-40^{\circ}$ and contains 1.8 million sources with a flux-density limit $S>3$ mJy at 1.4 GHz, Singal (2011) showed in this faint radio source distribution the presence of a dipole anisotropy which is about 4 times larger than the CMBR dipole (Lineweaver et al. 1996; Hinshaw et al. 2009), presumably of a kinetic origin due to the solar motion with respect to the otherwise isotropic CMBR. These unexpected findings have recently been corroborated by two independent groups (Rubart & Schwarz 2013; Tiwari et al. 2015; also see Singal 2014a for a clarification on some misgivings in the literature about the formulation used in these analyses). The fact that the direction of the two independently derived dipoles, viz. from NVSS and CMBR, coincide implies that there is certainly some peculiarity along this direction in sky which incidentally lies close to the line joining the equinoxes. But the large difference in the inferred motion (as much as a factor of $\sim 4$) cannot be easily explained. A genuine discrepancy in the dipoles inferred with respect to two different cosmic reference frames would imply a relative motion between these frames, not in accordance with our present ideas of cosmology.
A large–scale bulk flow has also been inferred from peculiar velocities of clusters of galaxies (Kashlinsky et al. 2011), though the genuineness of these results has been severely criticized in the literature (Keisler 2009; Osborne et al. 2011). There are reports of the presence of other large–scale alignments in radio and optical polarizations data (Jain and & Ralston 1999; Hutsemekers et al. 2005). It seems the universe might not be all isotropic and homogeneous, as assumed in the cosmological principle. Here we report even larger anisotropies which are seen in the sky distributions of powerful extended quasars and some other sub-classes of radio galaxies.
The Sample
==========
One of the earliest and best studied source of radio galaxies and quasars is the third Cambridge twice revised (3CRR) catalogue (Laing et al. 1983), which is radio complete in the sense that all radio sources brighter than the sensitivity limit ($S_{178}=10.9$ Jy) of the survey are included (and certainly with no spurious entries as each and every source in the sample has been studied in detail). It covers the sky north of declination, $\delta=10^\circ$, except for a zone of avoidance, a band of $\pm 10^\circ$ about the galactic plane ($b=0^\circ$). Also it has a 100% optical identification content with detailed optical spectra to classify radio sources into radio galaxies and quasars. The catalogue with the latest updates is downloadable from http://astroherzberg.org/people/chris-willott/research/3crr/.
The steep spectrum radio sources (radio spectral index $\alpha > 0.5$ with $S\propto \nu^{-\alpha}$) in the 3CRR catalogue are divided broadly into two classes, radio galaxies and quasars, the former further sub-divided into two types, Fanaroff-Riley 1 and 2 (FR1 and FR2), based on their radio morphologies (Fanaroff & Riley 1974) with the quasars almost always resembling the radio morphology of FR2 types. When compared to FR1s, the FR2 types are almost always found amongst the more powerful radio galaxies, overlapping the radio luminosities of quasars. However FR2 radio galaxies in general show narrow emission lines in their optical spectra, while quasars always show broad emission lines. Included among quasars is a small number of what are termed as broad line radio galaxies (BLRGs) or weak quasars (WQ), the latter with broad emission lines seen in polarized optical emission, or/and compact optical nuclei detected in infrared or X-rays. FR2 radio galaxies are further sub-divided by their optical spectra into low excitation galaxies (LEGs) and high excitation galaxies (HEGs). One object (3C386) shows an overlap of LEG and WQ properties (see Grimes et al. 2004), which we have therefore dropped from our sample. Also excluded are a small number of compact steep spectrum sources (CSSS, with angular size $\stackrel{<}{_{\sim}}2$ arcsec) which seem to be a different class (Kapahi et al. 1995). Then we have 23 FR1s, 17 LEGs, 65 HEGs and 48 quasars, making a total of 153 radio sources in our sample.
The conventional wisdom (Laing et al. 1994; Grimes et al. 2004) is that steep spectrum ($\alpha > 0.5$) HEGs and quasars belong to the same parent population, and that it is the orientation of the source in the sky that decides whether it will appear as an HEG or a quasar, the latter when the major radio-axis happens to be within a certain critical angle ($\xi_{c}$) around the observer’s line of sight. HEGs and quasars, in all other respects, are considered to be intrinsically the same. In this orientation-based unified scheme (OUS), because of the smaller inclinations of the radio axes of the quasars with respect to the observer’s line of sight, the observed radio sizes of the quasars will be foreshortened due to the geometry and should appear systematically smaller than those of the HEGs. It is a popular notion that $\xi_{\rm c} \sim 45^{\circ}$ and that in the 3CRR catalogue the observed sizes of quasars are accordingly about a factor of two smaller as compared to those of radio galaxies (Barthel 1989; Urry & Padovani 1995; Peterson 1997).
Results
=======
Recently it was shown that the relative size distributions of quasar and HEGs do not always show the projection effects, predicted by the OUS, when we compare the sources within different redshift bins (Singal 2014b). But what about when the size distributions are compared for different directions in the sky? To test this we divided the sample, starting from the first source in it, into two equal right ascension (RA) regions, region I from RA 0 to 12 and region II from 12 to 24 hours. While in region II the two size distributions differed by a factor of two or so, with quasar sizes being statistically smaller as one would expect due to the foreshortening in the OUS, in region I the sizes appeared statistically indistinguishable, contrary to the predictions of the OUS. This unexpected result prompted us to check their numbers as well in these two sky regions, since to be consistent with the OUS predictions, not only their relative sizes but their relative numbers should also differ by a factor of about two for a ‘canonical’ value of $\xi_{\rm c} \sim 45^{\circ}$. And again we found that while in region II the number of quasars was indeed about half that of HEGs, but in region I there were as many quasars as the HEGs, contrary to what expected according to the OUS. Figure 1 shows normalized cumulative plots of the linear size distributions of HEGs and quasars in the two regions. In region II we do notice the quasar sizes (as well as numbers) to be smaller than those of the HEGs by a factor of about two. However in region I, the differences, if any, in radio sizes or numbers are hardly seen and a Kolmogorov-Smirnov test shows that the two distributions are statistically almost indistinguishable, thereby punching a hole in the unification scheme. Not only does this seem to be a very strong evidence against the OUS (after all the OUS could not hold good in just one half of the sky), but it seems that there could be much more at stake here than just the validity of the OUS.
In the OUS, the ratios in the sizes and numbers of HEGs and quasars could change with redshift depending upon details of the model used as, for example, in the receding-torus-type scheme (Lawrence 1991; Hill et al. 1996) where the critical angle ($\xi_{c}$) may be evolving with redshift or luminosity. But in any case the ratio should not vary with the direction in sky. Therefore while any variations in numbers or sizes with redshift one could try to put down to some sort of cosmological evolution of their properties, irrespective of whether or not unified scheme holds good, but the same type of escape route cannot be available for a variation (over and above what might be due to statistical fluctuations) in the sky distribution. Further, any effects of zone of avoidance ($\pm 10^\circ$ around the galactic plane, $b=0$) should proportionally be the same for both HEGs and quasars, without affecting their number ratios.
It should be noted that even within the unification scheme, HEGs and quasars observationally are not identical and each class has distinct properties and they are identifiable or distinguishable as separate type of objects observationally and each of them should have their own isotropic distribution and there should be no sky-position dependent effect between the two. In fact with or without the unified schemes, from the isotropy expected from the cosmological principle, the number of any type of distant extragalactic objects should not vary with direction in sky, apart from the statistical fluctuations. A close investigation showed that while the HEGs, which are the largest number of the 3CRR constituents, are quite uniformly distributed over the observed sky, the quasars are quite unevenly distributed. While about two thirds (33 out of a total of 48) quasars in the sample lie in region I, the remainder one third (15 out of 48) appear in region II. In a priori chosen division of the sky in two adjacent and contiguous regions, for a random distribution of the sources one expects to get a binomial distribution. The probability of such a deviation in a binomial distribution to occur at $(33-15)/\sqrt {48} \sim 2.6 \sigma$ level due to statistical fluctuations is only $\sim 0.01$ (Bevington & Robinson 2003).
Could the anisotropy in the distribution of quasars have any local Supercluster or some other local origin? Such a thing, if any, should show up as a difference in the redshift distributions in the two regions. Figure 2 shows the redshift distributions of HEGs and quasars. Apart from the total number of quasars being less in region II, there does not appear to be any gross changes with redshift in the distribution of quasars and as well of HEGs in the two regions. This almost rules out the possibility that the quasar anisotropy has any local origin. Even the weak quasars (WQs), which like the other quasars are also proportionally less in region II, have redshift distributions which are very similar in the two regions, so any anomaly in quasar distribution is certainly not due to the presence of a differential number of WQs.
Sky region N(HEG) N(Q) N(LEG) N(FR1)
------------ -------- ------ -------- -------- -- --
I + II 65 48 17 23
I 32 33 12 6
II 33 15 5 17
: Counts of radio sources in two regions of the sky.
Figure 3 shows a normalized cumulative plot of HEG and quasar distributions in RA in sky. The sky distributions of HEGs and quasars appear very different, with slightly more than two thirds of all quasars lying in region I, while HEGs are distributed quite evenly over the sky. Also plotted in the figure are distributions of LEGs and FR1s. These too show very uneven distributions, with about 70% of LEGs lying in regions I and only about 1/4th of FR1s in that region. Overall the percentage of FR1s varies substantially between the two regions, while in region 1 there are only 7% of the total sources as FR1s, in region II the percentage is as much as 24%. Table 1 gives the number counts of different type of sources in the two regions of the sky. The probabilities of such a deviation to occur in a binomial distribution at $11/\sqrt {23} \sim 2.3 \sigma$ level due to statistical fluctuations is $\sim 0.02$ for FR1s. Further, as asymmetries of quasars and FR1s would have independent binomial probabilities, if these were due to random statistical fluctuations, then their combined probability of occurrence due to being simply a statistical fluctuation is only about $0.01 \times 0.02 \sim 2 \times 10^{-4}$, i.e., a $4 \sigma$ result.
Similarly LEGs also show an asymmetric distribution though at somewhat lower level (Table 1); while in region I there are 12 LEGs, in region II there are only 5 LEGs, implying a $7/\sqrt {17} \sim 1.7 \sigma$ deviation. If we include their probability of occurrence at $\sim 0.09$ as well, then the total combined probability becomes $\sim 2 \times 10^{-5}$. It should be noted that LEGs and FR1s, which may have overlap in some of their properties, are otherwise different type of objects classified by their distinct radio properties, e.g., their different radio morphologies. Moreover LEGs are of higher radio luminosities than the FR1s and are seen at relatively much higher redshifts. Therefore their uneven distributions are not a result of a mix-up in their classifications. Quasars of course stand apart, being the most energetic and most distant of these objects.
To ensure that there is nothing amiss in our probability calculations, we also did Monte Carlo simulations by throwing quasars, LEGS and FR1s randomly in sky and counting the number of times we get a distribution like that in Table 1. A total of 1000 simulations were done, every time starting with a different seed for a random number generator, while in each simulation 100,000 different random throws of 48 quasars, 17 LEGs and 23 FR1s were made. Thus in total a 100 million independent trials were made and out of these 1856 cases were found to have deviations equal to those in Table 1, implying a probability consistent with our calculations of $\sim 2 \times 10^{-5}$.
Figure 4 shows a Lambert azimuthal equal-area projection, mapping the Northern hemisphere onto a circular disc centered on NCP and accurately representing areas in all regions of the hemisphere. All points on a circle at a declination $\delta$ in sky are represented by a circle of radius $\propto \sqrt{1-\sin \delta}$ on the disc. The figure shows plot of HEGs from the 3CRR catalogue; the distribution seems to be fairly uniform on the sky.
Figure 5 shows plot of quasars from the 3CRR catalogue on the sky. To a first order, this division of sky in regions I and II happens to yield almost the maximum asymmetry visible in the quasar distribution, and it amounts to passing a great circle between the equinoxes (intersection points of the equatorial plane and the ecliptic) and the north celestial pole (NCP). First thing we want to be sure is that the observed number of quasars in region I being double or so of that in region II is not due to any instrumental/observational selection effects in these two regions in the 3CRR catalogue. This is guaranteed by the fact that virtually no difference is seen between the numbers of HEGs from these two regions in the same catalogue (Figure 4), which could not have happened if there were any such selection effects. It confirms that the quasar anomaly is not due to any observational selection effects, as any selection effects would not treat HEGs and quasars differentially, which were first radio selected and only later categorized as HEGs or quasars from their optical/infrared properties. The same argument can also be applied for the absence of any influence of our Galaxy on various distributions, as the Galaxy could not have affected distribution of different type of objects differently. Even otherwise the quasar asymmetry in Figure 5 seems to have no correlation with the galactic plane.
Comparing the regions between RA 06 to 12 hours and 12 to 18 hours in top half of Figure 5, we notice that there are 22 quasars between RA 06 to 12 hours while there are only 10 between 12 to 18 hours, giving a ratio of 2.2 in quasar numbers between these two regions. Actually with about 10% of the region from 06 to 12 hours overlapping the zone of avoidance, one would rather expect a proportionally smaller number of quasars in that region as compared to that in RA 12 to 18 hours, contrary to what actually seen. The total number of sources in the bottom half of the figure is less as compared to that in the top half, mainly because of a large fraction of area in the bottom half overlapping with the zone of avoidance. But even there as well a ratio of 2.2 is found between the region 0 to 6 hour (11 quasars) and that between 18 to 24 hour (5 quasars). From this it is clear that the asymmetry in quasar distribution is not due to a local excess (i.e., any local clustering) in neighborhood of some point in sky and that this excess in RA range 0 to 12 hour as compared to 12 to 24 hour is fairly widely distributed. Also there seems to be no effect of the Galactic latitude on the quasar distribution outside the zone of avoidance. The Super-galactic plane ($B=0$) too does not seem to have any relation with the distribution of quasars on sky. This of course is expected as quasars are at much higher redshifts as compared to that of the local Virgo supercluster. Therefore being two to three orders of magnitude more distant than the Virgo Supercluster, quasars can in no way be physically related to it or some other local objects.
Figure 6 shows the distribution of FR1 types of radio galaxies in the sky. It is clear that FR1 radio galaxies also have a highly asymmetric number distribution between the two regions, though in opposite sense to that of quasars. The distribution is particularly asymmetric about the line joining the Autumn equinox (RA=12 hour) to the NCP. While there are 13 FR1s between RA=12 to 18 hour, there is only 1 FR1 radio galaxy between RA range from 6 to 12 hour, and that too lies close to the boundary at 12 hour. The area covered by galactic plane in the region RA 06 to 12 hours is only $\sim 10\%$, so that does not resolve the asymmetry. Nor is this order of magnitude difference explained even if we exclude a couple of FR1s (M84; M87 or Virgo A) which lie close to the Super-galactic plane ($B=0$). If we drop the two FR1s close to the Super-galactic plane and adjust for the $10\%$ galactic plane coverage, then we have approximately 10 versus 1 FR1s in the two regions which may imply a $9/\sqrt {11} \sim 2.7 \sigma$ fluctuation, with a $\sim 0.007$ chance probability.
Discussion and conclusions
==========================
It is interesting that while relatively low redshift (up to $z\sim 0.2$) FR1s have excess between 12 to 18 hr RA, high redshifted quasars (up to $z\sim 2$) have an excess in the RA range from 6 to 12 hour, in direction where FR1s are almost non-existent. This shows not only an anisotropic universe but also a direct evidence of the presence of large scale inhomogeneities. It should be noted that the scale spanned by FR1s in the universe (up to $\sim$ a gigaparsec) is almost an order of magnitude larger than the scale at which inhomogeneities (Super-clusters, Great-Wall, Voids etc.) have till now been seen through optical observations. And of course quasars further cover a scale an order of magnitude larger than FR1s. This in fact is the largest scale in which discrete objects have been seen in the universe and any anisotropy or inhomogeneity on that scale is certainly a cause of worry as it will negate the cosmological principle.
These results are robust. There is little likelihood that these anomalies could be due to some missing or even spurious sources in the 3CRR catalogue, a radio complete sample of sources, in the sense that all source above the sensitivity limit of the catalogue have been detected and listed.
It is to be noted that a large scale dipole anisotropy in radio source distribution at much fainter levels was seen earlier, and was interpreted due to motion of the solar system with respect to an average universe. The derived direction of motion matched with that inferred from the CMBR, though the magnitude was found to be about a factor of four larger (Singal 2011) than for CMBR (Lineweaver et al. 1996; Hinshaw et al. 2009). These apparently anomalous results have recently been vindicated by the findings of two independent groups (Rubart & Schwarz 2013; Tiwari et al. 2014).
However the anisotropies pointed out here in the 3CRR sample could not be caused by a motion of the solar system as it could not give rise to different anisotropies for different kind of objects. We have seen that while powerful HEGs numbers are evenly distributed, quasars and LEGs have more numbers in region I, but the less luminous FR1’s are found to be more in region II. It is as if different regions of the sky were more amenable to one kind of source types than the other. Nor could these be attributed to some effect of our Galaxy or some effect of local Supercluster. Any such things would have affected all type of different objects in roughly the same way, but as we have seen the HEGs, LEGs, quasars and FR1s have very different asymmetries in their distributions.
There is certainly something intriguing. Is there a breakdown of the Copernican principle as things seen in two regions of sky, divided purely by a coordinate system based on earth’s orientation in space, show very large anisotropies in extragalactic source distributions? Why should the equinox points should have any bearing on the large scale distribution of matter in the universe? The only way to still retain the cosmological principle will be to doubt the reliability of the 3CRR survey, which will come very much of a surprise to almost all radio astronomers who take the 3CRR sample to be a true representation of strong radio source population. It should be noted that in the last three decades, since the 3CRR sample was formed (Laing et al. 1983), there have been a few, if any, changes due to addition of missing sources or deletions of spurious sources, and it is unlikely that the problem would get resolved that way. Many more deeper surveys covering all sky are certainly required in order to resolve this enigma, but even if deeper or more complete southern surveys show the absence of these anisotropies in the sky distribution of quasars and/or other radio galaxies, it will still remain to be explained why these anomalies are present in the strong 3CRR sample in the Northern hemisphere. After all many important studies like the number counts, luminosity function and/or cosmological evolution of other properties of radio population have been made using the 3CRR source distributions as an important ingredient, where an implicit assumption was an isotropic distribution of radio sources in the 3CRR sample (or at least presence of no such large anomalies), whether for quasars or for other radio objects. Even if in future it does turn out that one could explain away these anomalies due to some ill-understood subtle local effect, it might still require at least a rethinking on some of these earlier results. The OUS at least seems to be ousted as it cannot be valid only in one half of the sky as implied by the number and size ratios. A further confirmation of the asymmetries will of course be much vicissitudinous for all astronomers and cosmologists as well, since cosmological principle is the basis on which almost all modern cosmological theories depend upon as a starting point.
For the fore-mentioned apparent alignment in the CMBR in one particular direction through space, it has to be kept in mind that all such observations are obscured by the disc of the Milky Way galaxy, and one has to be extra careful while interpreting the data. Even there have been speculations whether solar system dust could give rise to sizable level of microwave emission or absorption, leading to a correlation with the ecliptic (Dikarev et al. 2009). But no such effect will be expected in the number distributions of discrete sources. The normals to the four quadrupole and octopole planes are aligned with the direction of the equinoxes and so does the dipole direction representing the overall motion of the solar system in the universe (Schwarz et al. 2004; Copi et al. 2010). Also our plane dividing the two regions of asymmetry passes through the same two equinox points. But it is not clear whether the asymmetries seen by us are related to that in the CMBR, as it is not presently possible to see if the anomalous distribution of radio sources is really related to ecliptic coordinates as the region covered by the 3CRR, unlike equatorial coordinates, is not divided equally in two ecliptic hemispheres. Perhaps an all-sky complete catalogue in future will help resolve this issue. But irrespective of that there is no denying that from the large anisotropies present in the radio sky, independently seen both in the discrete source distributions and in the diffuse CMBR, the Copernican principle seems to be in jeopardy.
Acknowledgements {#acknowledgements .unnumbered}
================
I thank Robert Antonucci for his comments, especially about the Copernican principle.
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{
"pile_set_name": "ArXiv"
}
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---
abstract: 'Participation in [[ ]{}]{} energy demand response programs requires an active role by the [[ ]{}]{}. [[ ]{}]{} contribute flexibility in how they use their appliances as the means to adjust energy consumption, and [[ ]{}]{}. Understanding the collective potential of appliance-level flexibility for [[ ]{}]{} is challenging and complex. [[ ]{}]{}. The findings of this study can be used to design more cost-effective and granular [[ ]{}]{} demand response programs in participatory and decentralized Smart Grids.'
address:
- 'Chair of Computational Social Science, ETH Zurich, Switzerland'
- 'School of Computing, University of Leeds, Leeds, UK'
author:
- Farzam Fanitabasi
- Evangelos Pournaras
bibliography:
- 'reference.bib'
title: 'Appliance-level Flexible Scheduling for Socio-technical Smart Grid Optimization'
---
Appliance Scheduling ,Flexibility ,Demand Response ,Smart Grid ,Distributed Optimization
Introduction {#S:introduction}
============
The European 2030 climate and energy framework has set three key targets for the year 2030: At least 40% reduction in greenhouse gas emissions, 27% share of renewable energy, and 27% improvement in energy efficiency from the 1990 levels [@EU2030]. [[ ]{}Meanwhile, the global electricity consumption in residential sector, which accounts for $30-40\%$ of the total energy usage, is ever increasing]{} [@alberini2011response; @torriti2014review]. [[ ]{}]{} [@li2019climate]. [[ ]{}]{} [@agnetis2013load; @kohlhepp2019large; @yahia2020multi]. [[ ]{}]{} [@qdr2006benefits; @palensky2011demand; @adika2014autonomous].\
[[ ]{}]{} [@mckenna2018simulating; @mammoli2019behavior]. [[ ]{}]{} [@yilmaz2019sensitive]. [[ ]{}]{} [@baldauf2015smart; @yao2016real]. [[ ]{}]{} [@yilmaz2019sensitive], rather they vary based on the appliance type, time-of-use, individual characteristics and behavior [@gyamfi2013residential], conventions [@powells2014peak], monetary incentives [@verbong2013smart], and social practices involving the appliances [@torriti2017understanding]. [[ ]{}]{} [@yahia2020multi; @wang2015robust]. [[ ]{}]{} [@jordehi2019optimisation; @yahia2020multi].\
[[ ]{}]{} To this end, this paper proposes a novel appliance-level energy scheduling framework that relies on consumers’ self-determined flexibility and comfort requirements, to regulate energy demand. [[ ]{}]{} [[ ]{}]{} whilst for individual consumers, the objective is to maximize comfort by using their appliances at the desired time [@chavali2014distributed]. Preventing demand peaks can be achieved by leveraging consumer flexibility in appliance usage, and uniformly distributing the demand across the day [[ ]{}]{} [@palensky2011demand; @spiliotis2016demand; @hassan2013impact]. In this setting, consumer flexibility is considered to be the contribution of alternative appliance usage schedules. For instance, multiple schedules as a result of shifting an appliance usage earlier or later in time from the intended usage time [@spiliotis2016demand; @zhai2019appliance; @d2015demand; @mohsenian2010autonomous]. This flexibility creates a degree of freedom for coordination within the framework to optimize the selection between these alternatives in a way that reduces the peak-load [@powells2014peak; @mckenna2018simulating; @gyamfi2013residential; @verbong2013smart].\
[[ ]{}]{} The two objectives of maximizing comfort and reducing demand peaks can be opposing, as certain appliance usages might be delayed (or advanced), thus lowering consumers’ comfort [@pilgerstorfer2017self; @pournaras2014decentralized]. Moreover, [[ ]{}]{}, such coordination requires selecting a subset of discrete schedules based on a quadratic cost function (minimizing demand variance), which is an NP-hard combinatorial optimization problem [@pilgerstorfer2017self; @de2018complexity]. This calls for approximation mechanisms to find a near-optimal and computationally feasible solution [@molzahn2017survey; @petersen2013taxonomy]. This paper addresses these challenges by introducing a decentralized network of autonomous scheduling agents, each representing a [[ ]{}]{} These agents cooperatively coordinate to select a subset of consumers’ schedules to reduce demand peaks. To optimize agents’ selections, this paper applies the I-EPOS (*Iterative Economic Planning and Optimized Selections*) [@pilgerstorfer2017self] system, to perform fully decentralized, privacy-preserving, and multi-objective combinatorial optimization. Depending on the appliance and its automation level, these coordinated schedules can be executed via smart appliances [@fanti2019cooperative], [[ ]{}]{} [@sattarpour2018multi], [[ ]{}]{} [@haider2016review].\
In summary, the contributions of this paper are the following: (i) A novel appliance-level scheduling framework based on consumers’ self-determined flexibility and comfort requirements, [[ ]{}]{} (ii) A data-driven analysis of appliance-level socio-technical factors, such as cooperation level, and unfairness that influence consumers’ flexibility and [[ ]{}]{} (iii) [[ ]{}]{} a quantitative comparison to related work which reveals that in comparison to improving appliance efficiency, flexible coordinated scheduling can further [[ ]{}]{}. (iv) A new dataset on flexible scheduling of appliances by residential consumers. The rest of this paper is outlined as follows: Sections \[S:relatedWork\] and \[S:overview\] summarize related work, and provide an overview of the framework operations, respectively. Section \[S:applianceScheduling\] introduces the flexible scheduling model, and Section \[S:IEPOS\] illustrates the distributed combinatorial optimization system. In Section \[S:expMethod\] the experimental methodology of the paper along with the dataset, survey, and mobile application are illustrated. Section \[S:evaluation\] shows the experimental evaluation. Finally, Section \[S:conclusion\] concludes this paper and outlines future work.
{width="90.00000%"}
Related Work {#S:relatedWork}
============
Demand response programs for Smart Grids have been subject to extensive research [@jordehi2019optimisation; @haider2016review]. Several studies attempt to model markets and pricing schemes to coordinate consumers’ [[ ]{}]{} Examples include game-theoretic approaches [@mohsenian2010autonomous; @rahimi2010overview; @tushar2015three], heuristic evolutionary algorithms [@logenthiran2012demand], and agent-based techniques [@jordan2018better; @pournaras2017self]. [[ ]{}]{} residential appliance scheduling via demand response programs as a [[ ]{}]{} approach for improving Smart Grid efficiency and utilization [@kohlhepp2019large; @haider2016review; @d2015demand; @adika2014autonomous]. Often, such programs adopt [[ ]{}]{} [@torriti2014review; @kohlhepp2019large]. To perform load-shifting, the consumers’ flexibility in appliance usage is calculated, often using the following two approaches: (i) Estimation of flexibility based on extrapolated consumption data [@d2015demand; @agnetis2013load; @spiliotis2016demand; @zhai2019appliance; @hassan2013impact; @ji2017demand; @drysdale2015flexible; @kwon2014assessment; @dyson2014using; @joe2012optimized; @macdougall2016applying]. (ii) Simulating the operating times of appliances, as well as usage habits [@murray2016understanding; @adika2014autonomous; @d2015demand; @yin2016quantifying; @olivieri2014evaluation; @ali2014demand; @bartusch2014further; @saele2011demand; @nistor2015capability; @halvorsen2001flexibility; @petersen2013taxonomy].\
[[ ]{}]{} [@molzahn2017survey; @muhsen2019domestic; @yahia2018optimal]. [[ ]{}]{} [@molzahn2017survey; @de2018complexity; @soares2014multi], [[ ]{}]{} [@molzahn2017survey; @setlhaolo2014optimal; @shakouri2017multi; @setlhaolo2015optimal; @setlhaolo2016combined]. [[ ]{}]{} [@yahia2018optimal] [[ ]{}]{} [@setlhaolo2014optimal] [[ ]{}]{} [@setlhaolo2016combined], [[ ]{}]{} [@shakouri2017multi]. [[ ]{}]{} [@molzahn2017survey; @yahia2018optimal]. [[ ]{}]{} [@jordehi2019optimisation; @yahia2020multi]. To address this, the framework introduced in this paper leverages consumers’ self-determined flexibility, comfort requirements, [[ ]{}]{} to study the collective potential of appliance load scheduling for reducing peak demands.\
[[ ]{}]{} [@de2018complexity]. Distributed optimization, and genetic algorithms are utilized to cope with this complexity, and to approximate a near-optimal solution between consumers’ demand and the available energy supply of the utility company [@molzahn2017survey]. [[ ]{}]{} [@soares2014multi], [[ ]{}]{} [@muralitharan2016multiobjective], [[ ]{}]{} [@muhsen2019domestic]. However, these studies do not address the distributed and privacy sensitive nature of appliance usage data. To this end, this paper utilizes and expands I-EPOS to provide a distributed, privacy-preserving coordination and optimization scheme for consumers’ schedules [@pilgerstorfer2017self].\
Lastly, recent research also highlight the socio-technical aspects of Smart Grids and emphasized the need to design demand response programs in a more bottom-up, and consumer-centric manner [@mckenna2018simulating; @mammoli2019behavior]. [[ ]{}]{} socio-technical aspect such as age, income, household size, and working hours, as well as increasing attention to privacy, self-determination, and autonomy influence the [[ ]{}]{} of demand response programs [@yilmaz2019sensitive]. Additionally, consumer flexibility does not only depend on the appliances and monetary incentives [@verbong2013smart], but on individual characteristics [[ ]{}]{} [@gyamfi2013residential], conventions [@powells2014peak], and social practices involving the appliances [@torriti2017understanding]. However, flexibility estimations based on extrapolated or simulated data often fail to capture such socio-technical factors, and do not account for the scenarios where consumers’ behavior deviates from the norm (e.g., having guest, or going on holidays). Thus, this paper utilizes a personalized scheduling agent for each consumer. Using this scheduling agent, consumers directly determine their data and privacy preference, schedules, flexibility, usage preference, scheduling constraints, and [[ ]{}]{}, on a daily basis[^1].
**Notation** **Meaning**
---------------------------------------------------- --------------------------------------------------------
$U$ Set of all consumers
$u \in U$ Consumer $u$
$H_u$ Set of all appliances of consumer $u$
$h_i \in H_u$ Appliance $i$, belonging to consumer $u$
$\langle s,d,f \rangle$ Representation of an appliance schedule
$\langle v_1,...,v_{n} \rangle$ Representation of an appliance plan
$P_u$ Set of all plans of consumer $u$
$p_{u,i} \in P_u$ Plan $i$ of consumer $u$
[[ ]{}]{} [[ ]{}]{}
$f_L \colon p_{u,i} \rightarrow \mathbb{R}$ [[ ]{}]{}
$f_G \colon \mathbb{R}^{n} \rightarrow \mathbb{R}$ [[ ]{}]{}
$a_1^t$ Aggregate plans of agents at root at iteration $t$
$a_u^{t}$ Aggregate plans of agents beneath $u$ at iteration $t$
$p_{u,o} \in S_u$ Original plan of agent $u$ with no flexibility
$p_{u,s}^t \in S_u$ Selected plan for agent $u$ at iteration $t$
$p_{u,s}^F\in S_u$ Final selected plan for agent $u$ by I-EPOS
$\lambda := 0\leq \lambda \leq1$ [[ ]{}]{}
$D^t$ Average discomfort of agents at iteration $t$
$\Psi^t$ Unfairness across agents at iteration $t$
: Mathematical notations used in this paper
\[t:Maths\]
Overview & Framework Operations {#S:overview}
===============================
Figure \[fig:Scenario\] [[ ]{}]{} (i) [[ ]{}]{} Via this scheduling agent, consumers indicate their appliances’ schedules [[ ]{}]{}, and usage constraints, such as which appliances should not be used in parallel[^2]. Consumers also indicate the discomfort level they are willing to tolerate as an indicator of flexibility, such as to what extent [[ ]{}]{} can shift in time from the desired operational time. This discomfort level determines how willing they are to contribute to the collective goal, i.e. [[ ]{}]{} (ii) The consumers submit their appliance schedules and flexibility [[ ]{}]{} to the scheduling agent[^3]. By leveraging flexibility, multiple alternative energy consumption patterns are generated using the scheduling agent (Section \[S:applianceScheduling\]), each called a *plan*[^4]. [[ ]{}]{} (iii) [[ ]{}]{} [@sattarpour2018multi; @pournaras2017self], [[ ]{}]{} [@pilgerstorfer2017self; @de2018complexity]. This calls for approximation mechanisms to find a near-optimal solution at low computational cost [@molzahn2017survey; @petersen2013taxonomy]. [[ ]{}]{} a decentralized network of autonomous scheduling agents, each representing [[ ]{}]{} [[ ]{}]{} coordinate to select a subset of consumers’ schedules to reduce demand peaks[^5].. To optimize agents’ selections, this paper utilizes and extends the I-EPOS (*Iterative Economic Planning and Optimized Selections*) [@pilgerstorfer2017self] system, to perform fully decentralized, privacy-preserving, and cooperative combinatorial optimization. (iv) Finally, the coordinated plans are submitted back to the scheduling agent. Note that this paper studies a range of personal appliances, ranging from highly automated (washing machine), to low automation (TV). Depending on the appliance and its automation level[^6], the recommended usage plans can be executed automatically via the home energy management system (HEMS) (in case of washing machines), or [[ ]{}]{} directly to the consumer as recommendations (in case of TV). The above framework can be operationalized in different scales depending on the available computational resources. Ranging from cooperative micro grids (e.g., smart building) where individual appliances form atomic units, to hierarchical scenarios where the lowest levels are households that aggregate to form neighborhoods, and districts. In the next sections the above scenario is illustrated in more detail.
Flexible Appliance Scheduling {#S:applianceScheduling}
=============================
The utilized mathematical notations are presented in Table \[t:Maths\]. A consumer $u \in U$ schedules the usage of appliance $h\in H_u$. A schedule is defined as: $\langle s,d,f \rangle$, where $s$ is the preferred starting time, $d$ is the duration in minutes, and $f$ is the appliance flexibility in minutes. This flexibility means that the consumer is willing to shift its original starting time $s$, either earlier or later, by $f$ minutes. A plan $i$ of consumer $u$ ($p_{u,i}~\in P_u$) is defined as a sequence of real values $\langle v_1,...,v_{n} \rangle$ of size $n=1440$ (i.e., 24\*60, number of minutes in a day). Each $v_j$ represents the energy consumption of the appliance on the $j^{th}$ minute of the day. For each schedule, using the flexibility provided by the consumer, the scheduling agent generates all possible plans ($P_u$) based on Algorithm \[A:planGeneration\]. A schedule with flexibility $f$ results in multiple plans, where the earliest plan starts at $s-f$, and the latest plan at $s+f$.\
Associated with each plan is its *discomfort*, [[ ]{}]{} starting time of the appliance usage. Intuitively, the closer a plan is to the preferred starting time, the lower the discomfort it imposes on the consumer, which in turn increases its [[ ]{}]{} [[ ]{}]{}: $f_L(p_{u,i})$. The plan derived by $f=0$ is represented as $p_{u,o}$ and its discomfort is 0 (i.e., $f_L(p_{u,o})=0$). An example of such a process is shown in Table \[SampleAS\].
/\* number of plans based on flexibility $k \leftarrow 2*f~/~g + 1$ /\* list of all plans including the plan with $f=0$ Initialise array $P$ of size $k$ /\* calculate the plan length $n \leftarrow \text{24h}/g$ (based on the scheduling horizon) $e \leftarrow$ energy consumption of $h$ per $g$
**Appliance** **Plan Start Time** **Discomfort**
--------------- --------------------- ----------------
Kettle 17:58 2
Kettle 17:59 1
Kettle 18:00 0
Kettle 18:01 1
Kettle 18:02 2
: Consumer $u$ schedules the kettle usage at 6pm, for 10’ and has 2’ flexibility: $\langle \text{18:00},10',2' \rangle$. This means that the preferred starting time is 18:00 but the consumer is flexible for the start time to be between 17:58 to 18:02. Hence, 5 different plans are generated, the first one starting from 17:58 and ending at 18:08 and the last one starting from 18:02 and ending at 18:12. Assume that $f_L(p_{u,i})$ calculates the discomfort as the absolute difference of the plan start time from the preferred starting time. The 5 [[ ]{}]{} are listed below:
\[SampleAS\]
Constraints & Preferences {#S:consPrefPlanSpace}
-------------------------
[[ ]{}]{} [[ ]{}]{} does not wish to shower while the oven is on, or use the washing machine after 9 pm. Moreover, consumers’ preferences for different plans are measured by [[ ]{}]{}, with higher preference given to plans with lower discomfort. If the consumer schedules multiple appliances, the scheduling agent combines the plans from appliances. This process is performed as follows: assume consumer $u$ schedules $h_i$ with $f=p$, and $h_j$ with $f=q$. Each schedule generates $2f+1$ plans[^7], hence, $4pq+2p+2q+1$ combined plans. [[ ]{}]{} This process is performed by the scheduling agent.
Consumers’ Cooperation Level
----------------------------
The collective goal of the proposed framework [[ ]{}]{} is measured by the global cost function: $f_G(C)$, where $C = \{p_{u,s}~|~\forall u \in U)\}$ is the set of all selected plans by the consumer, and $p_{u,s} \in P_u$ is the selected plan that consumer $u$ intends to execute. This can be achieved by minimizing the variance of consumers’ total energy demand across the day [@sattarpour2018multi; @pournaras2017self]. To do so, the framework leverages consumers’ flexibility to shift their appliance usage in time. Such an approach requires collective action and cooperation by the consumers, and can reduce consumers’ comfort, due to the shift in appliance usage time. Thus, the consumers can determine their *[[ ]{}]{}* in two phases: (i) plan generation (scheduling), and (ii) plan selection.
### Plan Generation (Scheduling) Phase {#S:PGP}
For a given schedule $\langle s,d,f \rangle$, if consumer $u$ determines a high flexibility value, then the scheduling agent creates a high number of possible plans $|P_u| >> 1$. A high number of possible plans increases the computational complexity and the storage capacity requirements of the agent[^8]. Assume that consumer $u$ needs to submit $k < |P_u|$ plans to the demand response program, and the plans in $P_u$ are sorted based on their [[ ]{}]{} [[ ]{}]{} can utilize various plan sampling mechanisms on $P_u$ in order to indicate [[ ]{}]{}. A simplified version of the social value orientation theory [@mcclintock1989social] is used to study the range of non-cooperative [[ ]{}]{} A non-cooperative consumer samples $k$ plans with the lowest discomfort, while the fully cooperative consumer samples $k$ plans with [[ ]{}]{} Intuitively, due to the popularity of certain actions, the cooperative consumers provide the demand response program with more diverse plans. For instance, using the oven is a very common between 6-7pm [@torriti2017understanding]. Thus, a cooperative consumer that [[ ]{}]{} before 6pm or after 7pm, [[ ]{}]{} to reducing demand peaks.
### Plan Selection Phase
[[ ]{}]{} \[S:IEPOS\]) [[ ]{}]{} $$\label{eq:autonomousPresPlanning}
\arg\min_{s=1}^{k} \Big( (1-\lambda)*f_G(C) + \lambda*f_L(p_{u,s}) \Big)$$ in which $\lambda := 0 \leq \lambda \leq 1$ is the weight assigned to the [[ ]{}]{} and is determined by each consumer, indicating the cooperation level in the plan selection phase, and $k$ is the number of plans for consumer $u$ A higher $\lambda$ value indicates [[ ]{}]{} Various incentivisation schemes (e.g., monetary rewards) can be used to motivate consumers to set lower $\lambda$ values.
Coordinated Plan Selection {#S:IEPOS}
==========================
[[ ]{}]{} [@jordehi2019optimisation; @haider2016review; @pilgerstorfer2017self]. [[ ]{}]{} consumers’ scheduling agents employ the I-EPOS system [@pilgerstorfer2017self] as a [[ ]{}]{}, fully decentralized, self-organizing, and privacy-preserving combinatorial optimization mechanism[^9] . [[ ]{}]{} I-EPOS coordinates and selects a subset of consumers’ plans aiming to minimize the variance of the total energy demand across the day [[ ]{}]{} [@pilgerstorfer2017self]. [[ ]{}]{} I-EPOS agents self-organize in a tree-topology [@pournaras2010self] as a way of structuring their interactions, [[ ]{}]{} I-EPOS performs consecutive learning iterations. Each iteration has two phases: the bottom-up (leaves to root) phase and top-down (root to leaves) phase. During the bottom-up phase of each iteration, agent $u$ selects the plan $p_{u,s}$ which satisfies the following [[ ]{}]{} $$\begin{aligned}
\label{eq:IEPOSExpanded}
p_{u,s}^t = \arg\min_{s=1}^{k}~\Bigg((1-\lambda)\Big({\sigma^2}\big(a_1^{t-1} - a_u^{t-1} + a_u^t \\
-~p_{u,s}^{t-1} + p_{u,s}^t\big)\Big) + \lambda\Big(f_L(p_{u,s}^t)\Big)\Bigg)
\end{aligned}$$ where $\sigma^2$ is the variance function, and $t$ is an iteration of I-EPOS. $a_1^{t-1} = \sum_{u=1}^{|U|} p_{u,s}^{t-1}$ is the *aggregate plan* at iteration $t-1$ of the selected plans of all agents, summed up at the root. $a_u^{t-1}$ and $a_u^t$ shows the aggregate plan calculated by summing up the selected plans of agents in the branch below agent $u$, at iterations $t-1$ and $t$, respectively. $p_{u,s}^{t-1}$ and $p_{u,s}^t$ are the selected plans of agent $u$ at iteration $t-1$ and $t$, respectively. Finally, consumers’ [[ ]{}]{} is included in the objective via the $\lambda$ parameter.\
[[ ]{}]{} In the top-down phase, agents are notified about $a_1^{t-1}$. After the final iteration $F$ is completed, $p_{u,s}^F$ is presented to the consumer by the scheduling agent of consumer $u$. Moreover, at each iteration $t$, the [[ ]{}]{} $D^t$ across all agents, is calculated as: $$\label{AVGDiscomfort}
D^t = \frac{1}{|U|} \sum_{u}^U (f_L(p_{u,s}^t)).$$ [[ ]{}]{} at each iteration $t$, the deviation of discomfort values across all agents is referred to as unfairness $\Psi^t$, calculated as: $$\label{eq:unfairness}
\Psi^t = \sqrt{ \frac{1}{|U|} \sum_{u}^U (f_L(p_{u,s}^t))^2 - (\frac{1}{|U|} \sum_{u}^U f_L(p_{u,s}^t))^2 }.$$ Further elaboration on I-EPOS is out of the scope of this paper and the interested reader is referred to previous work [@pilgerstorfer2017self].
Complexity, Optimality, and Privacy
-----------------------------------
The computational and communication complexity of the above algorithm are $O(kt\log |U|)$ and $O(t\log |U|)$, respectively. Where $k$ is the maximum number of plans per agent, $t$ is the number of iterations, and $|U|$ is the number of agents/consumers [@pilgerstorfer2017self]. This allows for efficiency and scalability to scenarios with higher number of agents. [[ ]{}]{} [@nikolic2019structural] [[ ]{}has revealed that compared to the state-of-art, I-EPOS has a superior performance profile. Regarding optimality, in a solution spaces of size $2^{20}$ I-EPOS empirically finds solutions in the top 33% of all solutions in the first learning iteration, and top 3.35% in the last learning iteration]{} [@pilgerstorfer2017self]. Lastly, the preservation of privacy is based on the fact that each agent only shares the aggregated plan to the parent agent in the tree topology, and does not disclose preferences, $p_{u,s}$, $P_u$, or $\lambda$.
Experimental Methodology {#S:expMethod}
========================
[[ ]{}]{}
Data Collection {#S:dataset}
---------------
![Participant information in the collected dataset. Continent refers to where participants currently reside in. EU stands for Europe, AS for Asia, and AM for the Americas. House size represents the number of bedrooms. House type shows the distribution of participants living in apartments (AP), semi-detached housed (SD) or other types of houses. House year-built shows the distribution of the year the participants’ house was built. These features are specifically chosen to provide a comparison baseline with the REFIT dataset[@murray2017electrical], and to assist in determining consumption profiles[]{data-label="Demography"}](plots/surveyHist.png){width="45.00000%"}
[[ ]{}]{} Moreover, large-scale social studies [[ ]{}]{} at the appliance level are usually costly, over-regulated, and require complex interventions by power utilities. To overcome such limitations that have significantly restricted the scope of earlier research [@zhai2019appliance; @d2015demand; @pournaras2014decentralized], two datasets are combined to make this research feasible: (i) A new real-world dataset[^10] [[ ]{}]{} in Section \[S:overview\]. This dataset contains appliance usage schedules and flexibility from 51 participants[^11] across four days, from 4 to 8 February 2018, [[ ]{}]{} This dataset contains appliance usage schedules and flexibility from 51 participants across four days, from 4 to 8 February 2018, [[ ]{}]{} To the best of the authors’ knowledge, this dataset is the first pilot study which addresses consumers’ self-determined flexibility, as well as [[ ]{}]{} Table \[planDist\] illustrates the distribution of the schedules and plans across different household appliances. This dataset is further analyzed in Section \[S:evaluation\]. (ii) The state-of-the-art REFIT dataset [@murray2017electrical], [[ ]{}]{}
**Appliance** **\# of Schedules** **\# of Plans** **Percentage (%)**
----------------- --------------------- ----------------- --------------------
Computer 62 6237 14.76%
Dish Washer 40 4320 9.52%
Hob 44 6516 10.47%
Kettle 80 5896 19.04%
Oven 97 10026 23.09%
Tumble Dryer 15 2184 3.57%
Washing Machine 82 11570 19.52%
Total 420 46749 100%
: Distribution of schedules and plans among appliances
\[planDist\]
### Mobile Application {#S:plannerApp}
An Android mobile application[^12] (Figure \[plannerAppScreens\]) was developed and distributed among the participants, [[ ]{}]{} [[ ]{}]{} is in charge of receiving the schedules, generating the plans, enforcing constraints and preferences, interacting with I-EPOS, [[ ]{}]{}
### Participants Survey {#survey}
[[ ]{}]{} The participants were invited to answer questions about demographic, household, and energy usage preferences, [[ ]{}]{} The detailed questionnaire and responses are presented in Appendix C.
### Appliance Energy Consumption {#S:consProf}
Figure \[Demography\] shows some general information about the participants and their living situation. [[ ]{}]{} household information, the scheduling agent matches each consumer [[ ]{}]{} to the closest household in the REFIT dataset [@murray2017electrical], [[ ]{}]{} To estimate the consumption profile of a given consumer, the scheduling agent utilizes a linear scoring model of four household features: number of occupants, the built year, the house type, and the number of bedrooms. These features have been empirically assigned weights of 0.533, 0.267, 0.133, and 0.067, respectively. The assignment of these weights is based on the importance of each feature on household energy usage [@baker2008improving; @kim2018characteristics] [^13]. Appendix A illustrates the household matching process and the distribution of profiles among consumers in more detail.
![Developed Android mobile application as the consumers’ scheduling agent: (a) Sample of survey questions, (b) The schedule submission phase. (c) Schedules for different appliances. (d) Selected plan after I-EPOS execution.[]{data-label="plannerAppScreens"}](pictures/appInterface1.pdf){width="\linewidth"}
Plan Sampling Mechanism {#S:sampTech}
-----------------------
This paper utilizes five different plan sampling mechanisms, each sampling 10 plans from $P_u$ for all consumers. Each sampling mechanism indicates a particular [[ ]{}]{} at the plan generation (scheduling) phase (Section \[S:PGP\]).
1. **Top Ranked**: Non-cooperative [[ ]{}]{} consumer; samples the top 10 plans from the plan space ($P$) with the lowest [[ ]{}]{}.
2. **Top Poisson**: Semi-non-cooperative *(semi-selfish)* consumer; samples 10 plans from the plan space, skewed towards lower [[ ]{}]{}. Modeled via a Poisson distribution[^14] on $P_u$ ordered by increasing [[ ]{}]{}.
3. **Uniform**: Balanced *(fair)* consumer; samples 10 plans uniformly distributed across the plan space.
4. **Bottom Poisson**: Semi-cooperative *(semi-altruistic)* consumer; samples 10 plans skewed towards the higher [[ ]{}]{} from the plan space. Modelled via a Poisson distribution^\[note1\]^ on $P$ ordered by decreasing [[ ]{}]{}.
5. **Bottom Ranked**: [[ ]{}]{}cooperative *(altruistic)* consumer; samples the top 10 plans from the plan space with the highest [[ ]{}]{}.
\
\
\
Experimental Design {#S:Methodology}
-------------------
Table \[Setting\] illustrates the I-EPOS parameters used for the experiments. [[ ]{}]{} In each simulation of I-EPOS, the agents are randomly assigned to a position in the tree topology. The topology is a balanced binary tree. The consumers schedules were collected during 4 days. The first 3 days with 51 and the last day with 50 consumers[^15]. The presented results are the average across the four days. The $\lambda$ parameter [[ ]{}]{} is determined based on the survey results of each consumer, specifically the question P7 in Appendix C: “*I would like to accept discomfort to make more efficient energy usage."* The plan sampling mechanisms (Section \[S:sampTech\]) are used as a system-wide setting for all scheduling agents. The scheduling agent samples and provides 10 plans to I-EPOS ($k=10$)[^16]. The plan dimensions (24\*60) are the number of minutes in a day, where the value on each dimension shows the total energy usage on that minute by the consumer. The [[ ]{}]{} The global cost function is MIN-VAR, which minimizes the variance of consumers’ total energy demand, hence reducing demand peaks. Below is the list of experiments and their methodology.
**Parameters** **Value**
--------------------------- ----------------------------------
Number of Executions 10
Number of Iterations 50
Number of Agents 50-51
Number of Plans per Agent 10
$\lambda$ [[ ]{}]{} Consumer Determined
Plan Dimensions 24\*60 (1440)
Network Topology Balance Binary Tree
Local Cost Function Discomfort (shift in start time)
Global Cost Function MIN-VAR (minimizing variance)
: I-EPOS parameters used in experiments.
\[Setting\]
### Flexible Appliance Scheduling & Reducing Demand Peaks
These experiments study the effect of consumers’ flexibility [[ ]{}]{} in scheduling their appliances [[ ]{}]{} Each agent sets its own $\lambda$ value [[ ]{}]{} provided by the corresponding consumer (Section \[S:Methodology\]). In addition, three other fixed system-wide values of $\lambda =0.0$, $0.5$, and $1.0$ are evaluated. The experiments are repeated across the five different plan sampling mechanisms.
### Impact of Individual Appliances on Reducing Demand Peaks
[[ ]{}]{} study the impact of individual appliances on the [[ ]{}]{} To calculate this impact, 7 experiments are performed, each time excluding one of the appliances from the demand response program [[ ]{}]{}
### Increased Efficiency vs. Flexible Coordinated Scheduling {#S:M:IEOFS}
Earlier research has studied the impact of more efficient use of appliances on the Smart Grid; specifically the increased energy efficiency of kettles if consumers avoid overfilling[@murray2016understanding]. [[ ]{}]{}, the following methodology is used: Scenario (a) “Efficient Kettle": the [[ ]{}]{} [@murray2016understanding] are applied to the consumers of the collected dataset, during peak hours (6:30-8:30 and 19:30-21:30). Scenario (b) “Flexible Kettle": the I-EPOS experiments are performed by setting the flexibility of all appliances to 0, except the kettle. [[ ]{}]{}, in both scenarios, the two systems can only use the kettle to [[ ]{}]{}
### Varying Adoption of Recommended Plans
Consumer participation is necessary for demand response programs to achieve their targets[@hassan2016framework]. Given consumer autonomy over the [[ ]{}]{}, their participation and adoption level greatly affect how well the [[ ]{}]{} [@hassan2016framework]. This effect is analyzed by utilizing the following methodology: The consumers are sorted in descending order, based on their $\lambda$ value. The reduced adoption is calculated by changing the $\lambda$ value of the top $n$-percent of the consumers with $\lambda \neq 1$ to 1. The consumers with $\lambda=1$ purely [[ ]{}]{} Utilizing the above methodology, this experiment studies the necessary level of participation and adoption [[ ]{}]{}
Results and Findings {#S:evaluation}
====================
This section illustrates the results and findings [[ ]{}]{} \[S:expMethod\].
Dataset Analysis {#S:datasetAnal}
----------------
Figures \[fig:Flex96\]a and \[fig:Flex96\]b [[ ]{}]{} the inclusion of flexibility makes the likelihood of appliance usage more spread-out across the day. For instance, the [[ ]{}]{} (Figure \[fig:Flex96\]a). However, this probability is more distributed across 6-8pm when flexibility is included. Figure \[fig:applianceKDE\] shows the density distribution[^17] [[ ]{}]{}, while the computer has the longest average duration (300’) out of all appliances, its flexibility is relatively low (47’). On the other hand, the oven has a low duration on average (52’), however, it has a relatively high flexibility (60’). [[ ]{}]{} the oven can contribute more to distributing the energy demand across the day than the computer.
**Appliance** **Morning** **Mid Day** **Evening** **Average**
----------------- ------------- ------------- ------------- -------------
Computer 0.125 0.2 0.175 0.16
Dish Washer 0.46 1.64 1 1.03
Hob 4 1.66 1 2.22
Kettle 1.61 0.95 1.81 1.45
Oven 2.14 0.85 1 1.33
Tumble Dryer 0.66 0.7 1 0.78
Washing Machine 0.25 0.62 0.68 0.51
: Relative flexibility of appliances throughout the day (Median): The morning hours are between 00:00 - 08:59, mid day between 09:00 - 16:59, and evening between 17:00 - 23:59. This splitting is made based on the common demand patters throughout the day [@richardson2010domestic]
\[T:flexDay\]
Table \[T:flexDay\] illustrates the changes in the relative flexibility of the appliances throughout the day. Additionally, Table \[T:scheduleDay\] in Appendix B illustrates the distribution of appliance schedules throughout the day. The values of $\lambda$, [[ ]{}]{} are based on their answer to question P7 (Appendix C). The participants are assigned one of the [[ ]{}5 discrete values between 0 to 1, with the resulting distribution: $\lambda=$1 (5.88%), 0.75 (35.3%), 0.5 (27.5%), 0.25 (21.6%), 0 (27.5%).]{} [[ ]{}]{}
\
\
\
{width="93.00000%"}s
{width="93.00000%"}
{width="93.00000%"}
\[T:applianceRank\]
![Comparison of the [[ ]{}]{} between the efficient kettle and the flexible kettle. Energy consumption of the efficient kettle is based on estimated reduce in energy consumption if the consumers do not overfill[@murray2016understanding]. The flexible kettle is the scenario where only the kettle is flexible and the schedule flexibility of all other appliances is set to 0. The lower-bound is the scenario where appliances are flexible, and upper-bound is where non of the appliance are flexible.[]{data-label="fig:consOverTime-Kettle"}](plots/consumptionOverTime-Kettle.png){width="45.00000%"}
{width="93.00000%"}
Experimental Results {#S:IEPOSRes}
--------------------
This section illustrates the results of the experiments based on the methodology described in Section \[S:Methodology\].
### Flexible Appliance Scheduling & Reducing Demand Peaks {#S:globalCost}
Figure \[fig:IEPOSRes\]a illustrates the [[ ]{}]{} at the final iteration of I-EPOS, for different sampling mechanisms and across different values of $\lambda$. [[ ]{}]{} The general trend across all sampling mechanisms is that with the increase of $\lambda$, the [[ ]{}]{} (Equation \[eq:IEPOSExpanded\]). The Top Poisson is the most sensitive [[ ]{}case, with 44.48% increase in variance from $\lambda=0$ to $\lambda=1$.]{} More drastic is the impact of different sampling mechanisms on [[ ]{}]{}. [[ ]{}]{} changing the plan sampling mechanism from Top Ranked to Top Poisson, [[ ]{}]{} by 54.35% for $\lambda=0$, 54.29% for $\lambda=0.5$, 52.44% for consumer specified, and 20.80% for $\lambda=1$. These differences are due to the entropy and diversity of consumers’ sampled plans [@pournaras2017self; @pournaras2014decentralized]. [[ ]{}]{}, in Top Ranked the agents always sample 10 plans with the lowest [[ ]{}]{} Hence, I-EPOS cannot perform effective optimization in such a non-diverse plan space. Overall, the best performing sampling mechanism is the Bottom Poisson that reduces the [[ ]{}demand variance by 51.71% compared]{} to Top Ranked.
### **Discomfort & Unfairness** {#S:avgLocalcost .unnumbered}
Figure \[fig:IEPOSRes\]b illustrates the average [[ ]{}]{} Within the same plan sampling mechanism, [[ ]{}]{} This is because the higher $\lambda$ values correspond to [[ ]{}]{} cooperative consumers, [[ ]{}]{} [[ ]{}]{} the best performing sampling mechanism regarding reducing demand peaks (Bottom Poisson), results in one of the highest average [[ ]{}]{}. [[ ]{}]{} \[fig:LCKDE\], [[ ]{}]{} For instance, in Top Ranked, the discomfort values are highly concentrated around 0, while in the Bottom Ranked the values are concentrated around 1[^18]. [[ ]{}]{} by changing the plan sampling mechanism from Top Ranked to Top Poisson, Uniform, Bottom Poisson, and Bottom Ranked, the average [[ ]{}]{} rises by a factor of 5.9, 0.5, 1.05, and 0.42, respectively. Figure \[fig:IEPOSRes\]c shows the unfairness calculated by Equation \[eq:unfairness\]. With the exception of the Bottom Ranked sampling mechanism, the unfairness decreases with the increase of $\lambda$. Among the plan sampling mechanism, the average unfairness in Top Ranked, Top Poisson, Uniform, Bottom Poisson, and Bottom Ranked are 0.175, 0.304, 0.216, 0.137, and 0.133, respectively.
### **Peak-time Load-shifting** {#S:totalEnergyDemand .unnumbered}
Figure \[fig:consOverTime\] shows consumers’ aggregated energy demand across different sampling mechanisms. [[ ]{}]{} Additionally, Figure \[fig:consOverTime\] [[ ]{}]{}
### Impact of Individual Appliances on Reducing Demand Peaks {#S:applianceImpact}
Figure \[fig:applianceExc\] illustrates the impact of various appliances on [[ ]{}]{} [[ ]{}]{} across all plan sampling mechanisms, the oven has the highest impact. After the over, the kettle and dish washer are the next appliances with the highest impact. This impact is attributed to a multitude of factors, such as appliance scheduling flexibility, [[ ]{}]{}, relative flexibility (Table \[T:flexDay\]), energy consumption, and the number of plans in the dataset. Table \[T:applianceRank\] ranks the appliances based on these factors, [[ ]{}]{} \[T:applianceExc\] [[ ]{}]{}
### Increased Efficiency vs Flexible Coordinated Scheduling {#S:efficiencyVSflexibility}
Figure \[fig:consOverTime-Kettle\] illustrates the results of scenarios (a) “Efficient Kettle" and (b) “Flexible Kettle", defined in Section \[S:M:IEOFS\]. [[ ]{}]{} the upper-bound is the scenario where none of the appliances are flexible, and the lower-bound is where all appliances are flexible. In scenario (a), the total energy demand decreases by 4.73%, and on average the [[ ]{}]{} reduces by 2.04% compared to the upper-bound, [[ ]{}]{}. In scenario (b), the total energy demand remains the same. However, on average the [[ ]{}]{} reduces by 29.8% compared to the upper-bound. [[ ]{}]{} critical observation here is the role of plan sampling mechanism. In the schemes with lower consumer flexibility, i.e., Top Ranked, Top Poisson (referred to as “Efficiency Superior"), the increased efficiency approach performs better. However, in schemes where consumers are relatively more flexible, i.e., the Uniform, Bottom Poisson, and Bottom Ranked (referred to as “Flexibility Superior"), optimized flexible scheduling performs better. These results show that in scenarios with high overall flexibility and cooperation, flexible coordinated scheduling of appliances can further contribute to the effective [[ ]{}]{}.
### Variable Adoption of Recommended Plans {#S:varParticipation}
Figure \[fig:varParticipation\] illustrates the increase in [[ ]{}]{}, due to the reduced adoption of recommended plans by the consumers. Top Poisson is the most sensitive sampling mechanism, as with 30% reduced participation, the [[ ]{}]{} already increases by 61.83%. The Bottom Ranked is the most resilient sampling mechanism where the [[ ]{}]{} can be reduced by 16.98% even if 40% of consumers do not adopt the recommended plan.
Summary of the Findings {#sec:findings .unnumbered}
=======================
The key findings of this paper are summarized as follows:
- Consumers’ flexibility in appliance scheduling depends on various socio-technical factors, such as the appliance type, usage habits, and the time of the day.
- [[ ]{}]{}
- Among the 7 appliances involved in the experiments, the oven has the most significant role in [[ ]{}]{}
- [[ ]{}]{}
- Decrease in consumer participation and adoption [[ ]{}]{} However, the degree of such effect varies among the plan sampling mechanisms, with Bottom Ranked being the most resilient.
Conclusion and Future Work {#S:conclusion}
==========================
[[ ]{}]{} Further research can address the inclusion of the markets and the role of dynamic energy pricing, such as incentive mechanisms (e.g., discounts). The proposed framework can be evaluated in pilot tests with energy utility companies to analyze the long-term improvement and efficiency. Moreover, the framework and methodology of this paper can be expanded to study additional appliances, such as PV panels, electric heating, and cooling devices, where power consumption can be adjusted as well. Lastly, additional societal and behavioral factors, such as consumers’ environmental awareness, and carbon emissions in appliance usage, can be included in the proposed framework to provide a more comprehensive study.
![Distribution of assigned houses across all consumers. The profiles are based on Table \[T:REFITProfiles\].[]{data-label="fig:ProfDist"}](plots/ProfDist.png){width="0.6\linewidth"}
**House** **Occupancy** **Year Built** **Appliances** **Type** **Size**
----------- --------------- ---------------- ---------------- --------------- ----------
1 2 1975-1980 35 Detached 4 Beds
2 4 - 15 Semi-Detached 4 Beds
3 2 1988 27 Detached 3 Beds
4 2 1850-1899 33 Detached 3 Beds
5 4 1878 44 Mid-Terrace 4 Beds
6 2 2005 49 Detached 4 Beds
7 4 1965-1974 25 Detached 4 Beds
8 2 1966 35 Detached 3 Beds
9 2 1919-1944 24 Detached 2 Beds
10 4 1919-1944 31 Detached 3 Beds
11 1 1945-1963 25 Detached 3 Beds
12 3 1991-1995 26 Detached 3 Beds
13 4 Post 2002 28 Detached 4 Beds
14 1 1965-1974 19 Semi-Detached 3 Beds
15 6 1981-1990 48 Detached 5 Beds
16 3 Mid 60s 22 Detached 3 Beds
17 2 1965-1974 34 Detached 3 Beds
18 4 1945-1964 26 Semi-Detached 3 Beds
19 2 1965-1975 39 Detached 3 Beds
20 4 1981-1990 23 Detached 3 Beds
: Households in the REFIT dataset.[]{data-label="T:REFITProfiles"}
**House / Appliance** **TD** **WM** **Computer** **DW** **Kettle**
----------------------- -------- -------- -------------- -------- ------------
House 1 472 513 29 1379 -
House 2 - 327 - 770 2257
House 3 1373 492 16 1150 1550
House 4 - 56 52 - 1703
House 5 766 66 66 182 2352
House 6 - 369 66 778 2192
House 7 2075 442 - 613 1913
House 8 - 273 19 - 2340
House 9 - 507 - 700 2359
House 10 - 349 - 364 -
House 11 - 91 10 753 1841
House 12 - - - - 2482
House 13 152 203 39 1250 1542
House 14 1476 495 20 532 2521
House 15 - 300 27 1239 -
House 16 1594 373 20 - 1689
House 17 - 377 26 1021 -
House 18 - 161 - - 2448
House 19 1097 293 75 434 2350
House 20 1240 434 - 350 1276
: Appliance energy consumption (kWh) for households in the REFIT dataset. TD: Tumble Dryer, WD: Tasing machine, DW: Dish Washer.[]{data-label="T:REFITHouses"}
**Appliance** **Morning (%)** **Mid Day (%)** **Evening (%)**
----------------- ----------------- ----------------- -----------------
Computer 16 (25.80%) 26 (41.93%) 20 (32.25%)
Dish Washer 2 (5%) 3 (7.5%) 35 (87.5%)
Hob 2 (4.54%) 18 (40.9%) 24 (54.54%)
Kettle 37 (46.25%) 18 (22.5%) 25 (31.25%)
Oven 27 (27.83%) 19 (19.58%) 51 (52.57%)
Tumble Dryer 2 (13.33%) 10 (66.66%) 3 (20%)
Washing Machine 12 (14.63%) 21 (25.60%) 49 (59.75%)
: Distribution of appliance schedules throughout the day. The morning hours are between 00:00 - 08:59, mid day between 09:00 - 16:59, and evening between 17:00 - 23:59. This splitting is made based on the common demand patters throughout the day[@richardson2010domestic][]{data-label="T:scheduleDay"}
\
\
\
\
\
\
**[[ ]{}]{} (E8)** **Lower-Bound** **Computer** **Dish Washer** **Kettle** **Hob** **Oven** **Tumble Dryer** **Washing Machine** **Upper-Bound**
-------------------- ----------------- -------------- ----------------- ------------ --------- ---------- ------------------ --------------------- -----------------
Top Ranked 6.1971 6.1976 6.2165 6.2223 6.2206 6.4095 6.2000 6.2169 6.4346
Top Poisson 2.8289 2.8289 3.006 2.9551 3.4916 4.0458 2.8759 2.9771 5.0961
Uniform 2.9202 2.9231 3.1401 3.0200 3.5315 4.2458 2.9424 3.0255 5.1120
Bottom Poisson 2.5694 2.5737 2.7656 2.6455 2.6049 3.4636 2.5707 2.6234 4.3049
Bottom Ranked 2.7426 2.7446 2.9027 2.8246 2.7845 3.7413 2.7479 2.8780 3.9483
\[T:applianceExc\]
A: Profile Assignment to consumers {#A:profileAssignment}
==================================
The estimation of consumers’ appliance energy consumption is based on the households from REFIT, illustrated in Table \[T:REFITProfiles\]. Based on consumers’ household information, each consumer is assigned to one of the REFIT households. The matching is performed using a linear scoring function: $$\label{eq:profileMatching}
\begin{aligned}
&Score = 0.533*Occupancy~+~0.267*Size \\
&+~0.133*Type~+~0.067*\text{\textit{Year-built}}
\end{aligned}$$ The weights are based on the importance each feature on determining the household. The linear combination is used for the sake of simplicity and interpretability. Though, more complex formulations are possible. For each consumer, the matching scores are calculated for all houses. The house with the highest matching score is assigned to the consumer. The energy consumption of consumers’ appliances are then calculated based on the assigned house. This consumption data is illustrated in Table \[T:REFITHouses\]. If the assigned house does not include one of the consumers’ appliances, the next best-matched house is used to calculate the appliance energy consumption. The energy consumption of the oven and hob are based on average consumption of some models available in the market. Figure \[fig:ProfDist\] illustrates the consumers’ assigned house distribution in the collected dataset.
B: Expanded Experiments and Evaluations {#A:expandedExp}
=======================================
Table \[T:scheduleDay\] shows the distribution of appliance schedules throughout the day. The observed usage patterns, such as high usage of Hob around lunch and dinner times, are in accordance with previous research on appliance usage[@torriti2017understanding]. Figure \[fig:IEPOSRes5\] and Figure \[fig:IEPOSRes100\] illustrate the results of I-EPOS experiment with 5 and 100 plans per agent, respectively. The key findings of the results illustrated in the paper, e.g., increase of [[ ]{}]{} with with increase of $\lambda$, are observed here as well. The higher number of plans results in lower [[ ]{}]{}, but higher average discomfort and unfairness. Table \[T:applianceExc\] shows more detailed results of the impact of individual appliances on demand variance.
C: Detailed Survey {#A:surveyRes}
==================
[lXXl]{} **Category** & **Question** & **Answer (%)** & **ID**\
& 1. What year were you born? & 1962 (2.17%) - 1972 (2.17%) - 1976 (2.17%) - 1978 (2.17%) - 1980 (4.35%) - 1981 (4.35%) - 1983 (4.35%) - 1984 (4.35%) - 1985 (4.35%) - 1986 (8.70%) - 1987 (8.70%) - 1988 (6.52%) - 1989 (6.52%) - 1990 (15.22%) - 1991 (13.04%) - 1994 (4.35%) - 1995 (4.35%) - 1996 (2.17%) & D.1\
& 2. In which country were you born? & Brazil (6.52%) - Colombia (2.17%) - France (4.35%) - Germany (10.87%) - Greece (4.35%) - India (2.17%) - Indonesia (10.87%) - Iran (28.26%) - Italy (4.35%) - The Netherlands (13.04%) - Romania (2.17%) - Russia (2.17%) - Singapore (2.17%) - Switzerland (2.17%) - Turkey (4.35%) & D.2\
& 3. What is your gender? & Male (78.26%) - Female (21.74%) & D.3\
& 4. In which country have you lived the longest? & Brazil (6.52%) - Colombia (2.17%) - France (4.35%) - Germany (8.70%) - Greece (4.35%) - India (2.17%) - Indonesia (10.87%) - Iran (28.26%) - Italy (4.35%) - The Netherlands (15.22%) - Romania (2.17%) - Russia (2.17%) - Switzerland (4.35%) - Turkey (4.35%) & D.4\
& 5. What is the highest level of education you have completed? & Short cycle tertiary (2.17%) - Bachelor or equivalent (13.04%) - Master or equivalent (73.91%) - Doctoral or equivalent (10.87%) & D.5\
& 6. Which of the following best describes your employment status? & Full time (63.04%) - Part-time (4.35%) - Self-employed (2.17%) - Student (23.91%) - Unemployed (6.52%) & D.6\
& 1. What type of house do you live in? & Apartment/Flat (63.04%) - Detached (63.04%) - Mid-terrace (63.04%) - Semi-detached (63.04%) - Other (63.04%) & H.1\
&2. What is the size of your house? & 1 Bed (28.26%) - 2 Beds (32.61%) - 3 Beds (30.43%) - 4 Beds (4.35%) - 5 Beds (2.17%) - 6+ Beds (2.17%) & H.2\
&3. Approximately, when was your house built? & Pre 1900s (8.70%) - 1920-1929 (6.52%) - 1930-1939 (4.35%) - 1950-1959 (2.17%) - 1960-1969 (17.39%) - 1970-1979 (6.52%) - 1980-1989 (15.22%) - 1990-1999 (10.87%) - 2000-2009 (15.22%) - 2010+ (13.04%) & H.3\
&4. What type of house do you live in? & Apartment/Flat (63.04%) - Detached (63.04%) - Mid-terrace (63.04%) - Semi-detached (63.04%) - Other (63.04%) & H.4\
&5. How many people live in the house? & 1 (26.09%) - 2 (28.26%) - 3 (30.43%) - 4 (10.87%) - 6+ (4.35%) & H.5\
&6. Which appliance do you have at home? & Washing machine (80.39%) - Tumble dryer (19.60%) - Computer-laptop (86.27%) - Computer-desktop (33.33%) - Oven (70.50%) - Hob (23.52%) - Electric shower (7.84%) - Dish washer (39.21%) - Electric heater (29.41%) - Air conditioner (13.72%) - Kettle (54.90%) - Microwave (62.74%) - Freezer (68.62%) - Fridge (80.39%) & H.6\
&7. To which devices you would allow automated control for a more efficient energy usage? & Washing machine (82.35%) - Tumble dryer (27.46%) - Computer-laptop (25.49%) - Computer-desktop (11.76%) - Oven (31.37%) - Hob (5.88%) - Electric shower (9.80%) - Dish washer (47.05%) - Electric heater (19.60%) - Air conditioner (17.64%) - Kettle (15.68%) - Microwave (23.52%) - Freezer (47.05%) - Fridge (56.86%) & H.7\
& 1. I am concerned about the amount of my residential energy consumption: & 0 (0%) - 1 (13.46%) - 2 (17.31%) - 3 (46.15%) - 4 (23.08%) & P.1\
&2. I would like to consume lower energy at home: & 0 (0%) - 1 (2.24%) - 2 (13.43%) - 3 (51.49%) - 4 (32.84%) & P.2\
&3. I would like to consume energy at home more efficiently: & 0 (0%) - 1 (1.45%) - 2 (13.04%) - 3 (47.83%) - 4 (37.68%) & P.3\
&4. I would like to make more efficient usage for the following reason: & Reduce my energy bill (66.66%) - Contribute to the grid reliability, e.g. prevent a blackout (17.64%) - Protect the environment (82.35%) - Others do, so I do (9.80%) - Others do not, so I do (1.96%) & P.4\
&5. I would like to use the following means to make a more efficient residential usage of energy: & Lowering my consumption of appliances (66.66%) - Shifting the consumption of appliances at different times (e.g. during off-peak night times) (62.74%) & P.5\
&6. Which of the following would you, as a result of the automated control of residential appliances for a more efficient energy usage, find create discomfort: & Feeling cold in cold winters or feeling warm in warm summers (56.86%) - Extra cost for special equipment and appliances (31.37%) - Changing my overall lifestyle at home (35.29%) - Doing my daily residential activities at different and maybe undesirable times (45.09%) & P.6\
&7. I would like to accept discomfort to make more efficient energy usage: & 0 (9.8%) - 1 (21.6%) - 2 (27.5%) - 3 (35.3%) - 4 (5.8%) & P7\[q:lambdaQuestion\]\
&8. I would like to sacrifice energy efficiency to experience a low discomfort: & 0 (6.52%) - 1 (15.22%) - 2 (43.48%) - 3 (34.78%) - 4 (0%) & P.8\
&9. I would like to be more energy efficient if I know that others are more energy efficient as well: & 0 (8.70%) - 1 (21.74%) - 2 (28.26%) - 3 (36.96%) - 4 (4.35%) & P.9\
&10. I can accept a discomfort caused by energy efficiency if others can accept it as well: & 0 (10.87%) - 1 (21.74%) - 2 (30.43%) - 3 (30.43%) - 4 (6.52%) & P.10\
&11. I would not like to be energy efficient if others are not energy efficient as well: & 0 (34.78%) - 1 (34.78%) - 2 (26.09%) - 3 (4.35%) - 4 (0%) & P.11\
&12. I would not like to experience higher discomfort by energy efficiency if others do not experience higher as well: & 0 (26.09%) - 1 (41.30%) - 2 (26.09%) - 3 (6.52%) - 4 (0%) & P.12\
&13. I would like to allow technology to schedule a more efficient energy usage of my appliances: & 0 (2.17%) - 1 (23.91%) - 2 (54.35%) - 3 (19.57%) - 4 (0%) & P.13\
&14. I am willing to scheduler the use of appliances to make more efficient energy usage: & 0 (0%) - 1 (6.52%) - 2 (23.91%) - 3 (56.52%) - 4 (13.04%) & P.14\
&15. Scheduling of appliance to make more efficient energy usage best works for me: & 30? ahead (25.49%) - 1 hour ahead (23.52%) - 3 hours ahead (29.41%) - 6 hours ahead (15.68%) - 12 hours ahead (19.60%) - 24 hours ahead (9.80%) - 1 week ahead (7.8%) & P.15\
&16. For which discomfort level would you like to overtake control back over an appliance schedule for an efficient energy usage: & 0 (0%) - 1 (26.09%) - 2 (39.13%) - 3 (32.61%) - 4 (2.17%) & P.16\
[^1]: While this approach does require a higher level of engagement from consumers, recent research has shown that values such as increased control and autonomy can improve the acceptance and adoption of such programs[@verbong2013smart; @ellabban2016smart; @murtagh2014qualitative].
[^2]: Scheduling is a well-established approach in literature[@fanti2019cooperative; @chavali2014distributed; @fathi2013adaptive; @rokni2018optimum; @wang2018green; @lahon2019contribution; @de2005multi; @maliah2017collaborative; @georgeff1988communication], and has been utilized in several real-world application domains[@pilgerstorfer2017self; @pournaras2017self; @barbati2012applications].
[^3]: Previous studies have shown that daily energy usage follows a semi-repetitive manner[@halvorsen2001flexibility]. Hence, the consumers can choose to schedule once for each day, and only modify them if required. This greatly reduces the scheduling effort on part of the consumers.
[^4]: Note that a schedule specifies the intended appliance usage. and is defined independent of the appliance energy consumption. But the plans specify the exact energy consumption of the appliance during its use.
[^5]: Crucially, the selected plans should not violate the consumers’ comfort expectations, or the physical constraints of the Smart Grid, for instance, the power generator limits [@spiliotis2016demand].
[^6]: Such as allowing for direct control, and interruptible/deferrable operation.
[^7]: $f$ number of plans with starting time before $s$, another $f$ with starting time after $f$, plus the plan with $f=0$.
[^8]: For example, in a scenario with 100 agents, each with 10 plans, the solution space size is $10^{100}$.
[^9]: Accessible online at <https://github.com/epournaras/EPOS> (last accessed: April 2020)
[^10]: Available online: [@Fanitabasi2019Figshare] (last accessed: April 2020)
[^11]: The participants were recruited through a cross-university campaign.
[^12]: Accessible online at <https://github.com/epournaras/EPOS-Smart-Grid-Scheduler> (last accessed: April 2020)
[^13]: However, as this paper mostly studies the consumers’ scheduling and socio-technical factors affecting it, these weights do not affect the findings. The linear combination is used for the sake of simplicity and interpretability.
[^14]: \[note1\]With a $\lambda$ parameter of 2 for the Poisson distribution.
[^15]: On the last day one of the consumers did not schedule.
[^16]: In Appendix B more experiments with 5 and 100 plans are shown to illustrate the effect of $k$ on the demand response program.
[^17]: Calculated using the Gaussian kernel density estimator with the nrd0[@scott2015multivariate] algorithm for determining the bandwidth.
[^18]: This observation supports the insight regarding differences in plan entropy and diversity across plan sampling mechanisms (Section \[S:globalCost\]).
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'A conjecture connected with quantum physics led N. Katz to discover some amazing mixed character sum identities over a field of $q$ elements, where $q$ is a power of a prime $p >3$. His proof required deep algebro-geometric techniques, and he expressed interest in finding a more straightforward direct proof. The first author recently gave such a proof of his identities when $q \equiv 1 \pmod 4$, and this paper provides such a proof for the remaining case $q \equiv 3 \pmod 4$. Our proofs are valid for all characteristics $p>2$. Along the way we prove some elegant new character sum identities.'
author:
- |
\
\
Ron Evans\
Department of Mathematics\
University of California at San Diego\
La Jolla, CA 92093-0112\
[email protected]\
\
and\
\
John Greene\
Department of Mathematics and Statistics\
University of Minnesota–Duluth\
Duluth, MN 55812\
[email protected]
date: December 2016
title: SOME MIXED CHARACTER SUM IDENTITIES OF KATZ II
---
2010 *Mathematics Subject Classification*. 11T24, 33C05.
*Key words and phrases*. Hypergeometric $\2F1$ character sums over finite fields, Gauss and Jacobi sums, norm-restricted Gauss and Jacobi sums, Eisenstein sums, Hasse–Davenport theorems, quantum physics.
Introduction
============
Let $\F$ be a field of $q$ elements, where $q$ is a power of an odd prime $p$. Throughout this paper, $A$, $B$, $C$, $D$, $\chi$, $\lambda$, $\nu$, $\mu$, $\e$, $\phi$ denote complex multiplicative characters on $\f$, extended to map 0 to 0. Here $\e$ and $\phi$ always denote the trivial and quadratic characters, respectively. Define $\delta(A)$ to be 1 or 0 according as $A$ is trivial or not, and let $\delta(j,k)$ denote the Kronecker delta for $j,k \in \F$.
Much of this paper deals with the extension field $\FF$ of $\F$. Let $M_4$ denote a fixed quartic character on $\FF$ and let $M_8$ denote a fixed octic character on $\FF$ such that $M_8^2 = M_4$.
Define the additive character $\psi$ on $\F$ by $$\psi(y) :=
\exp \Bigg( \frac{2 \pi i}{p} \Big( y^p + y^{p^2} + \dots + y^q \Big) \Bigg),
\quad y \in \F.$$ The corresponding additive character on $\FF$ will be denoted by $\psi_2$.
Recall the definitions of the Gauss and Jacobi sums over $\F$: $$G(A) = \sum_{y \in \F} A(y) \psi(y), \quad
J(A, B) = \sum_{y \in \F} A(y) B(1-y).$$ These sums have the familiar properties $$G(\e) = -1, \quad J(\e,\e) = q-2,$$ and for nontrivial $A$, $$G(A) G(\oA) = A(-1) q, \quad J(A, \oA) = -A(-1),
\quad J(\e, A)=-1.$$ Gauss and Jacobi sums are related by [@BEW p. 59] $$J(A,B) = \frac{G(A) G(B)}{G(AB)}, \quad \text{if } AB \neq \e$$ and $$J(A,\oC) = \frac{A(-1)G(A) G(\oA C)}{G(C)}=A(-1)J(A,\oA C),
\quad \text{if } C \neq \e.$$ The Hasse–Davenport product relation [@BEW p. 351] yields $$\label{eq:1.1}
A(4) G(A) G(A \phi) = G(A^2) G(\phi).$$
As in [@TAMS p. 82], define the hypergeometric $\2F1$ function over $\F$ by $$\label{eq:1.2}
\2F1 \l( \bar A,B \\ C \ear \ x \r)
=\frac{\e (x)}{q}
\sum_{y \in \F} B(y) \oB C(y-1)
\oA(1-xy), \quad x \in \F.$$
For $j, k \in \F$ and $a \in \f$, Katz [@Katz p. 224] defined the mixed exponential sums $$\label{eq:1.3}
\begin{split}
P(j,k): &= \delta(j,k) + \phi(-1)\delta(j,-k) + \\
& \frac{1}{G(\phi)}\sum_{x \in \f}
\phi(a/x - x)\psi(x(j+k)^2 + (a/x)(j-k)^2).
\end{split}$$ Note that $$\label{eq:1.4}
P(j,k)=P(k,j), \quad
P(-j,k) = \phi(-1)P(j,k).$$ Katz proved an equidistribution conjecture of Wootters [@Katz p. 226], [@ASSW] connected with quantum physics by constructing explicit character sums $V(j)$ [@Katz pp. 226–229]) for which the identities $$\label{eq:1.5}
P(j,k) = V(j)V(k)$$ hold for all $j,k \in \F$. (The $q$-dimensional vector $(V(j))_{j \in \F} $ is a minimum uncertainty state, as described by Sussman and Wootters [@SW].) Katz’s proof [@Katz Theorem 10.2] of the identities required the characteristic $p$ to exceed 3, in order to guarantee that various sheaves of ranks 2, 3, and 4 have geometric and arithmetic monodromy groups which are SL(2), SO(3), and SO(4), respectively.
As Katz indicated in [@Katz p. 223], his proof of is quite complex, invoking the theory of Kloosterman sheaves and their rigidity properties, as well as results of Deligne [@Del] and Beilinson, Bernstein, Deligne [@BBD]. Katz [@Katz p. 223] wrote, “It would be interesting to find direct proofs of these identities."
The goal of this paper is to respond to Katz’s challenge by giving a direct proof of ( a “character sum proof" not involving algebraic geometry). This has the benefit of making the demonstration of his useful identities accessible to a wider audience of mathematicians and physicists. Since a direct proof for $q \equiv 1 \pmod 4$ has been given in [@Evans], we will assume from here on that $q \equiv 3 \pmod 4$.
A big advantage of our proof is that it works for all odd characteristics $p$, including $p=3$. As a bonus, we obtain some elegant new double character sum evaluations in –.
Our method of proof is to show (in Section 6) that the double Mellin transforms of both sides of are equal. The Mellin transforms of the left and right sides of are given in Theorems 3.1 and 5.1, respectively. A key feature of our proof is a formula (Theorem 4.1) relating a norm-restricted Jacobi sum over $\FF$ to a hypergeometric $\2F1$ character sum over $\F$. Theorem 4.1 will be applied to prove Theorem 5.3, an identity for a weighted sum of hypergeometric $\2F1$ character sums. Theorem 5.3 is crucial for our proof of in Section 6.
Hypergeometric character sums over finite fields have had a variety of applications in number theory. For some recent examples, see [@BarKal], [@BarSai], [@ElOno], [@FLRST], [@LinTu], [@McPap], [@Salerno].
Since $q \equiv 3 \pmod 4$, we have $\phi(-1)=-1$, and every element $z \in \FF$ has the form $$z= x + iy, \quad x,y \in \F,$$ where $i$ is a fixed primitive fourth root of unity in $\FF$. Write $\oz = x - iy$ and note that $\oz = z^q$. The restriction of $M_8$ to $\F$ equals $\e$ or $\phi$ according as $q$ is congruent to 7 or 3 mod 8. In particular, $$\label{eq:1.6}
M_8(-1) = \phi(2).$$ For a character $C$ on $\F$, we let $CN$ denote the character on $\FF$ obtained by composing $C$ with the norm map $N$ on $\FF$ defined by $$Nz = z\oz \in \F.$$ Given a character $B$ on $\F$, $BCN$ is to be interpreted as the character $(BC)N$, i.e., $BN CN$.
For the same $a$ as in , define $$\tau = -\sqrt{qM_8(-a)},$$ where the choice of square root is fixed. Katz defined the sums $V(j)$ to be the following norm-restricted Gauss sums: $$\label{eq:1.7}
V(j): = \tau^{-1}\phi(j)
\sum_{\substack{z \in \FF \\ Nz=a}} M_8(z)\psi_2(j^2 z), \quad j \in \F.$$ Note that $$\label{eq:1.8}
V(-j) = -V(j).$$
Mellin transform of the sums $V(j)$
===================================
This section begins with some results related to Gauss sums over $\FF$ that will be used in this paper. We use the notation $G_2$ and $J_2$ for Gauss and Jacobi sums over $\FF$, in order to distinguish them from the Gauss and Jacobi sums $G$ and $J$ over $\F$. For any character $\beta$ on $\FF$, we have $$\label{eq:2.1}
G_2(\beta) = G_2(\beta^q);$$ for example, for a character $C$ on $\F$, $G_2(CN M_8)$ equals $G_2(CN \ome)$ or $G_2(CN M_8^3)$ according as $q$ is congruent to 7 or 3 mod 8. The Hasse-Davenport theorem on lifted Gauss sums [@BEW Theorem 11.5.2] gives $$\label{eq:2.2}
G_2(CN) = - G(C)^2.$$ From [@Hiro (4.10)], $$\label{eq:2.3}
G_2(CN M_4) = G_2(CN \omf) = -\oC^2\phi(2)G(C^2\phi)G(\phi).$$ For any character $\beta$ on $\FF$, define $$\label{eq:2.4}
E(\beta): = \sum_{y \in \F} \beta(1+iy).$$ It is easily seen that $$\label{eq:2.5}
E(\beta) = \beta(2) E_2(\beta),$$ where $E_2(\beta)$ is the Eisenstein sum $$\label{eq:2.6}
E_2(\beta): = \sum_{\substack{z \in \FF \\ z+z^q =1}} \beta(z).$$ Let $\beta^*$ denote the restriction of $\beta$ to $\F$. Applying [@BEW Theorem 12.1.1] with $q$ in place of $p$, we can express $E_2(\beta)$ in terms of Gauss sums when $\beta$ is nontrivial, as follows: $$\label{eq:2.7}
E_2(\beta) = \begin{cases}
G_2(\beta)/G(\beta^*) &\mbox{if } \beta^* \ne \e \\
-G_2(\beta)/q & \mbox{if } \beta^* = \e . \end{cases}$$
For any character $\chi$ on $\F$, define the Mellin transform $$\label{eq:2.8}
S(\chi): = \sum_{j \in \f} \chi(j)V(j).$$ In the case that $\chi$ is odd, we may write $\chi = \phi \lambda^2$ for some character $\lambda$ on $\F$. In that case, we may assume without loss of generality that $\lambda$ is even, otherwise replace $\lambda$ by $\phi \lambda$. In summary, when $\chi$ is odd, $$\label{eq:2.9}
\chi = \phi \lambda^2 = \phi \nu^4, \quad \lambda = \nu^2$$ for some character $\nu$ on $\F$.
The next theorem gives an evaluation of $S(\chi)$ in terms of Gauss sums.
If $\chi$ is even, then $S(\chi)=0$. If $\chi$ is odd (so that holds), then $$\label{eq:2.10}
S(\chi)=
\onu(a)\tau^{-1}G_2(\nu N M_8)+\phi\onu(a)\tau^{-1}G_2(\nu N M_8^5).$$
If $\chi$ is even, then $S(\chi)$ vanishes by (1.8) and (2.8). Now assume that $\chi$ is odd, so that $\chi = \phi \nu^4$. Then $$S(\chi)=\frac{\tau^{-1}}{q-1}\sum_{z \in \FF} \sum_{j \in \f}
M_8(z)\psi_2(z j^2) \nu^4(j) \sum_{C} C(N(z)/a).$$ Replace $z$ by $z/j^2$ to get $$S(\chi)=\frac{\tau^{-1}}{q-1}\sum_{C} \oC(a)\sum_{z \in \FF}
M_8(z)\psi_2(z)CN(z)\sum_{j \in \f} \nu^4 \oC^4(j).$$ The sum on $j$ on the right equals $q-1$ when $C \in \{\nu, \phi \nu\}$ and it equals $0$ otherwise. Since $\phi N = M_8^4$, the result now follows from the definition of $G_2$.
Double Mellin transform of $V(j)V(k)$
=====================================
For characters $\chi_1, \chi_2$, define the double Mellin transform $$\label{eq:3.1}
S=S(\chi_1, \chi_2): = \sum_{j, k \in \f} \chi_1(j)\chi_2(k)V(j)V(k).$$ As in , when $\chi_1$ and $\chi_2$ are both odd, $$\label{eq:3.2}
\chi_i = \phi \lambda_i^2 = \phi \nu_i^4, \quad \lambda_i = \nu_i^2,
\quad i=1,2,$$ for some characters $\nu_1$, $\nu_2$ on $\F$. In this case, write $$\label{eq:3.3}
\mu = \nu_1 \nu_2.$$
The following theorem evaluates $S$ in terms of Gauss and Jacobi sums.
If $\chi_1$ or $\chi_2$ is even, then $S=0$. If $\chi_1$ and $\chi_2$ are both odd (so that and hold), then $$\label{eq:3.4}
S=\sum_{i=0}^{1} \phi^i \omu(a)\frac{q }{G_2(\phi^i \omu N)}
\{J_2(\nu_1 N M_8, \phi^i \omu N) + J_2(\nu_1 N M_8^5, \phi^i \omu N)\}.$$
By , $S=S(\chi_1)S(\chi_2)$. By Theorem 2.1, $S=0$ when $\chi_1$ or $\chi_2$ is even. Thus assume that $\chi_1$ and $\chi_2$ are both odd. Then Theorem 2.1 yields $$\label{eq:3.5}
\begin{split}
S=&\sum_{i=0}^{1} \phi^i \omu(a)\frac{M_8(-a) }{q} \ \times \\
&\times \ \{G_2(\nu_1 N M_8)G_2( \phi^i \mu \ono N M_8) +
G_2(\nu_1 N M_8^5)G_2( \phi^i \mu \ono N M_8^5)\}.
\end{split}$$ A straightforward computation with the aid of shows that is equivalent to . The computation is facilitated by noting that $M_8(-a)$ equals $1$ or $-\phi(a)$ according as $q$ is congruent to 7 or 3 mod 8, so that the bracketed expression for $i=0$ in is to be compared to that for $i=1$ in when $q$ is congruent to 3 mod 8.
Identity for a norm-restricted Jacobi sum in terms of a $\2F1$
==============================================================
Let $D$ be a character on $\F$. Define the norm-restricted Jacobi sums $$\label{eq:4.1}
R(D,j):=\sum_{\substack{z \in \FF \\ N(z)=j^4}}
M_8(z)\oD N(1-z), \quad j \in \f.$$
The next theorem provides a formula expressing $R(D,j)$ in terms of a $\2F1$ hypergeometric character sum.
For $j = \pm 1$, $$\label{eq:4.2}
R(D,j) = -\oD(4)J(\phi D^2, \phi).$$ For all other $j \in \f$, $$\label{eq:4.3}
R(D,j)=-\phi(j) q \oD^4(j-1)
\2F1 \l( \bar D,D^2\phi \\ D\phi \ear \
-\left(\frac{j+1}{j-1}\right)^2 \r).$$
Replace $z$ in by $-zj^2$. By , we obtain $$R(D,j)=\phi(2)\sum_{\substack{z \in \FF \\ N(z)=1}}
M_8(z)\oD N(1+zj^2).$$ Each $z$ in the sum must be a square, since $N(z)$ is a square in $\F$. Thus $$R(D,j)=\frac{\phi(2)}{2}\sum_{\substack{z \in \FF \\ N(z)=1}}
M_4(z)\oD N(1+z^2j^2).$$ Writing $z=x+iy$, we have $$R(D,j)=\frac{\phi(2)}{2}\sum_{x^2+y^2=1}
M_4(x+iy)\oD ((1-j^2)^2 + 4 j^2 x^2),$$ where it is understood that the sum is over all $x,y \in \F$ for which $x^2 + y^2 =1$. Thus, since $M_4(\pm i)=M_8(-1)=\phi(2)$, $$\label{eq:4.4}
R(D,j) = \oD^2(1-j^2) + Q(D,j),$$ where $$Q(D,j)=\frac{\phi(2)}{2}\sum_{\substack{x^2+y^2=1 \\ x \ne 0}}
M_4(x+iy)\oD ((1-j^2)^2 + 4 j^2 x^2).$$ Replacing $y$ by $yx$, we have $$Q(D,j)=\frac{\phi(2)}{2}\sum_{\substack{1+y^2=x^{-2} \\ x \ne 0}}
M_4(1+iy)\oD ((1-j^2)^2 + 4 j^2 /(1+ y^2)).$$ Since $\omf = \phi N M_4$, this yields $$\label{eq:4.5}
Q(D,j)=\frac{\phi(2)}{2}\sum_{y \in \F}
\{M_4(1+iy)+\omf(1+iy)\} \oD ((1-j^2)^2 + 4 j^2 /(1+ y^2)).$$
First consider the case where $j = \pm 1$. By and , $$R(D,j)=Q(D,j) = \frac{\oD(4) \phi(2)}{2}\sum_{y \in \F}
\{DN M_4(1+iy)+DN \omf(1+iy)\}.$$ By and , $$\label{eq:4.6}
R(D,j)=\frac{\phi(2)}{2}\{E_2(DN M_4) + E_2(DN \omf)\}.$$ The restriction of $DN M_4$ to $\F$ is $D^2$. Thus by , $$R(D,j)=\frac{\phi(2)}{2 G(D^2)}\{G_2(DN M_4) + G_2(DN \omf)\},$$ if $D^2$ is nontrivial, and $$R(D,j)=\frac{\phi(2)}{-2 q}\{G_2(DN M_4) + G_2(DN \omf)\},$$ if $D^2$ is trivial. By , $$G_2(DN M_4) = G_2(DN \omf) = -\oD(4) \phi(2)G(D^2 \phi)G(\phi).$$ Consequently, $$\label{eq:4.7}
R(D,j) = -\oD(4)J(\phi D^2, \phi)$$ for every $D$, which completes the proof when $j = \pm 1$. Thus assume for the remainder of this proof that $j^2 \ne 1$.
By , $Q(D,j)$ equals $$\frac{\oD^2(1-j^2)}{2\phi(2)}\sum_{y \in \F}
\{M_4(1+iy)+\omf(1+iy)\}
\oD \left(1 + \frac{4 j^2}{(1+ y^2)(1-j^2)^2}\right).$$ By the “binomial theorem" [@TAMS (2.10)], the rightmost factor above equals $$\frac{q}{q-1}\sum_{\chi}
\begin{pmatrix} D \chi \\ \chi \end{pmatrix}
\chi\left(\frac{-4 j^2}{(1+ y^2)(1-j^2)^2}\right),$$ where the “binomial coefficient" over $\F$ is defined by [@TAMS p. 80] $$\begin{pmatrix} A \\ B \end{pmatrix}
= \frac{B(-1)}{q} J(A, \overline{B}).$$ Replacing $\chi$ with $\oc$ and observing that [@TAMS p. 80] $$\begin{pmatrix} D \oc \\ \oc \end{pmatrix}=
D(-1)\begin{pmatrix} \chi \\ \oD \chi \end{pmatrix},$$ we see that $$Q(D,j)=\frac{\oD^2(1-j^2)D(-1)\phi(2)q}{2(q-1)}\sum_{\chi}
\begin{pmatrix} \chi \\ \oD \chi \end{pmatrix}
\chi\left(\frac{-(1-j^2)^2}{4j^2}\right)\kappa(\chi),$$ where $$\kappa(\chi):=\sum_{y \in \F}
\{\chi N M_4(1+iy) + \chi N \omf(1+iy)\}.$$ By and , $$\kappa(\chi)=\chi(4)\{E_2(\chi N M_4) + E_2(\chi N \omf)\}.$$ Comparing and , we see that $$\kappa(\chi)= -2\phi(2)J(\phi \chi^2, \phi)=
2q\phi(2)\oc(4)
\begin{pmatrix} \phi \chi^2 \\ \chi \end{pmatrix},$$ where the last equality follows from the Hasse-Davenport relation . Consequently, $$Q(D,j)=\frac{\oD^2(1-j^2)D(-1)q^2}{q-1}\sum_{\chi}
\begin{pmatrix} \chi \\ \oD \chi \end{pmatrix}
\begin{pmatrix} \phi \chi^2 \\ \chi \end{pmatrix}
\chi\left(\frac{-(1-j^2)^2}{16j^2}\right).$$ Replace $\chi$ by $D \chi$ to get $$Q(D,j)=\frac{\oD(16j^2)q^2}{q-1}\sum_{\chi}
\begin{pmatrix} D\chi \\ \chi \end{pmatrix}
\begin{pmatrix} D^2 \phi \chi^2 \\ D \chi \end{pmatrix}
\chi\left(\frac{-(1-j^2)^2}{16j^2}\right).$$ By [@TAMS (2.15)] with $A=D \chi$, $B=\chi$, and $C=D^2 \phi \chi^2$, $$\begin{split}
&\frac{q^2}{q-1}
\begin{pmatrix} D\chi \\ \chi \end{pmatrix}
\begin{pmatrix} D^2 \phi \chi^2 \\ D \chi \end{pmatrix} \\
&=\frac{q^2}{q-1}
\begin{pmatrix} D^2 \phi \chi^2 \\ \chi \end{pmatrix}
\begin{pmatrix} D^2 \phi \chi \\ D \phi \chi \end{pmatrix}
-\chi(-1)\delta(D \chi) + D(-1) \delta(D^2 \phi \chi),
\end{split}$$ since by [@TAMS (2.6)], $$\begin{pmatrix} D^2 \phi \chi \\ D \end{pmatrix} =
\begin{pmatrix} D^2 \phi \chi \\ D \phi \chi \end{pmatrix}.$$ Thus $$\label{eq:4.8}
\begin{split}
Q(D,j)=&\frac{\oD(16j^2)q^2}{q-1}\sum_{\chi}
\begin{pmatrix} D^2 \phi \chi^2 \\ \chi \end{pmatrix}
\begin{pmatrix} D^2 \phi \chi \\ D \phi \chi \end{pmatrix}
\chi\left(\frac{-(1-j^2)^2}{16j^2}\right) \\
&-\oD^2(1-j^2) -D(-16j^2)\oD^4(1-j^2).
\end{split}$$ By [@TAMS Theorem 4.16] with $A=D$, $B=\phi D^2$, and $x = -(j+1)^2/(j-1)^2$, we have $$\begin{split}
&-\phi(j)D^4(j-1)\oD(16j^2)\frac{q}{q-1}\sum_{\chi}
\begin{pmatrix} D^2 \phi \chi^2 \\ \chi \end{pmatrix}
\begin{pmatrix} D^2 \phi \chi \\ D \phi \chi \end{pmatrix}
\chi\left(\frac{-(1-j^2)^2}{16j^2}\right) = \\
&\2F1 \l( \bar D,D^2\phi \\ D\phi \ear \
-\left(\frac{j+1}{j-1}\right)^2 \r) -\phi(j)D(-16j^2)\oD^4(j+1)/q.
\end{split}$$ Multiply by $-\phi(j)q\oD^4(j-1)$ to get $$\begin{split}
&\oD(16j^2)\frac{q^2}{q-1}\sum_{\chi}
\begin{pmatrix} D^2 \phi \chi^2 \\ \chi \end{pmatrix}
\begin{pmatrix} D^2 \phi \chi \\ D \phi \chi \end{pmatrix}
\chi\left(\frac{-(1-j^2)^2}{16j^2}\right) = \\
&-\phi(j)q\oD^4(j-1)\2F1 \l( \bar D,D^2\phi \\ D\phi \ear \
-\left(\frac{j+1}{j-1}\right)^2 \r)
+D(-16j^2)\oD^4(1 -j^2).
\end{split}$$ Thus by , $$\label{eq:4.9}
\begin{split}
& Q(D,j) + \oD^2(1-j^2) + D(-16j^2) \oD^4(1-j^2) = \\
&-\phi(j)q\oD^4(j-1)\2F1 \l( \bar D,D^2\phi \\ D\phi \ear \
-\left(\frac{j+1}{j-1}\right)^2 \r)
+D(-16j^2)\oD^4(1-j^2).
\end{split}$$ Combining and , we arrive at the desired result .
Double Mellin Transform of $P(j,k)$
===================================
For characters $\chi_1, \chi_2$, define the double Mellin transform $$\label{eq:5.1}
T=T(\chi_1, \chi_2): = \sum_{j, k \in \f} \chi_1(j)\chi_2(k)P(j,k).$$ Note that $T(\chi_1, \chi_2)$ is symmetric in $\chi_1$, $\chi_2$.
The following theorem evaluates $T$.
If $\chi_1$ or $\chi_2$ is even, then $T=0$. If $\chi_1$ and $\chi_2$ are both odd (so that and hold), then $$\label{eq:5.2}
T=\sum_{i=0}^1 \omu \phi^i(a)\frac{G(\phi \mu^2)}{G(\phi)}
\{\sum_{j \in \f} \co(j) h(\mu \phi^i,j) +2(q-1)\delta(\mu \phi^i)\},$$ where for a character $D$ on $\F$ and $j \in \f$, we define $$\label{eq:5.3}
h(D,j):=\sum_{x \in \f} D(x)\phi(1-x) \phi \oD^2(x(j+1)^2 +(j-1)^2).$$
By , $P(j, k)=-P(j,-k)$, so $T=0$ if $\chi_1$ or $\chi_2$ is even. Thus assume that and hold. Replacing $j$ by $jk$ in , we obtain $$\begin{split}
&T=2(q-1)\delta(\mu^4) + \\
&\frac{1}{G(\phi)}\sum_{x,j,k \in \f}
\co(j)\mu^4(k)\phi(a/x - x)\psi(k^2(x(j+1)^2+a(j-1)^2/x)).
\end{split}$$ Since $\delta(\mu^4)=\delta(\mu^2)$, this becomes $$\begin{split}
&T=2(q-1)\delta(\mu^2) + \\
&\frac{1}{G(\phi)}\sum_{x,j,k \in \f}
\co(j)\mu^2(k)\phi(a/x - x)\psi(k(x(j+1)^2+a(j-1)^2/x))(1+\phi(k)).
\end{split}$$ There is no contribution from the $1$ in the rightmost factor $(1+\phi(k))$; to see this, replace $k$ and $x$ by their negatives. Therefore, $$\begin{split}
&T=2(q-1)\delta(\mu^2) + \\
&\frac{G(\phi \mu^2)}{G(\phi)}\sum_{x,j \in \f}
\co(j)\phi(a/x - x)\phi \omu^2(x(j+1)^2+a(j-1)^2/x).
\end{split}$$ It follows that $$\begin{split}
&T=2(q-1)\delta(\mu^2)\frac{G(\phi \mu^2)}{G(\phi)} + \\
&\frac{G(\phi \mu^2)}{G(\phi)}\sum_{x,j \in \f}
\co(j)\phi(a - x)\phi \omu^2(x(j+1)^2+a(j-1)^2)\mu(x)(1+\phi(x)).
\end{split}$$ After replacing $x$ by $ax$ and employing , the desired result readily follows.
We proceed to analyze $h(D,j)$.
We have $$\label{eq:5.4}
h(D,j) = -\phi(j) \oD(16) J(D, \phi), \quad \mbox{ if } j = \pm 1,$$ and for $j \ne \pm 1$ and nontrivial $D$, we have $$\label{eq:5.5}
h(D,j) = \frac{G(\phi)G(D)^2}{G(\phi D^2)}\oD^4(j-1)
\2F1 \l( \bar D,D^2\phi \\ D\phi \ear \
-\left(\frac{j+1}{j-1}\right)^2 \r).$$ Finally, if $j \ne \pm 1$ and $D$ is trivial, then $h(D,j)=0$.
The evaluation in follows directly from the definition of $h(D,j)$ in . The evaluation in is the same as that in [@Evans (5.21)], the proof of which is valid for $q$ congruent to either 1 or 3 mod 4. Finally, let $j \ne \pm 1$. Then since $$h(\e,j) = -1 + \sum_{x \in \F} \phi(1-x) \phi(x(j+1)^2 + (j-1)^2),$$ replacement of $x$ by $1-x(2j^2+2)/(j+1)^2$ shows that $h(\e,j)= -1 + 1 = 0$.
For a character $D$ on $\F$, define $$\label{eq:5.6}
W(D): = \sum_{j \in \f} \phi \no^4(j)h(D,j).$$ Then $W(\e)=2$, and for nontrivial $D$, $$\label{eq:5.7}
W(D)=\frac{-G(\phi)G(D)^2}{qG(\phi D^2)}
\{J_2(\no N M_8, \oD N)+ J_2(\no N M_8^5, \oD N)\}.$$
It follows directly from Lemma 5.2 that $W(\e)=2$. Let $D$ be nontrivial. By Lemma 5.2, $$\label{eq:5.8}
\begin{split}
&W(D)=-2\oD(16)J(D, \phi) \ - \\
&\frac{G(\phi)G(D)^2}{qG(\phi D^2)}
\sum_{\substack{j \in \f \\ j \ne \pm 1}}
\no^4(j)\left(-q\phi(j)\oD^4(j-1)
\2F1 \l( \bar D,D^2\phi \\ D\phi \ear \
-\left(\frac{j+1}{j-1}\right)^2 \r)\right).
\end{split}$$ Thus by –, $$\begin{split}
&W(D) = -2\oD(16)J(D, \phi) -\frac{G(\phi)G(D)^2}{qG(\phi D^2)}
\sum_{\substack{j \in \f \\ j \ne \pm 1}}
\no^4(j)R(D,j) = \\
&-2\oD(16)J(D, \phi) -
\frac{G(\phi)G(D)^2}{qG(\phi D^2)}\{2\oD(4)J(\phi D^2, \phi) \ +
\sum_{j \in \f } \no^4(j)R(D,j)\}.
\end{split}$$ This simplifies to $$\label{eq:5.9}
W(D) = -\frac{G(\phi)G(D)^2}{qG(\phi D^2)}\sum_{j \in \f } \no^4(j)R(D,j).$$ For brevity, let $Y(D)$ denote this sum on $j$. It remains to prove that $$\label{eq:5.10}
Y(D): = \sum_{j \in \f } \no^4(j)R(D,j)=
J_2(\no N M_8, \oD N)+ J_2(\no N M_8^5, \oD N).$$ Since the fourth powers in $\F$ are precisely the squares, it follows from definition that $$Y(D) = \sum_{j \in \f } \no(j^2)
\sum_{\substack{z \in \FF \\ N(z)=j^2}}
M_8(z)\oD N(1-z).$$ Thus $$\begin{split}
Y(D)=& \frac{1}{q-1}\sum_{j \in \f } \no(j^2)
\sum_{z \in \ff}M_8(z)\oD N(1-z)\sum_{\chi} \chi(N(z)/j^2)= \\
&\frac{1}{q-1}\sum_{\chi} J_2(\chi N M_8, \oD N)
\sum_{j \in \f} (\no \oc)^2(j).
\end{split}$$ The sum on $j$ on the right vanishes unless $\chi \in \{\no, \no \phi\}$, and so we obtain the desired result .
As interesting consequences of Theorem 5.3, we record the elegant double character sum evaluations – below.
For any character $\nu$ on $\F$, $$\label{eq:5.11}
\begin{split}
\sum_{j, x \in \f} &\phi \nu^4(j) \phi(x)\phi(1-x)\phi(x(j+1)^2+(j-1)^2) = \\
&J_2(\nu N M_8, \phi N) + J_2(\nu N M_8^5, \phi N).
\end{split}$$
This follows by putting $D=\phi$ in .
When $q \equiv 7 \pmod 8$, we have $$\label{eq:5.12}
\sum_{j, x \in \f} \phi(jx)\phi(1-x)\phi(x(j+1)^2+(j-1)^2)=2q.$$ When $q \equiv 3 \pmod 8$, we have $$\label{eq:5.13}
\sum_{j, x \in \f} \phi(jx)\phi(1-x)\phi(x(j+1)^2+(j-1)^2)=2u,$$ where $|u|$, $|v|$ is the unique pair of positive integers with $p \nmid u$ for which $q^2=u^2 + 2v^2$, and where the sign of $u$ is determined by the congruence $u \equiv -1 \pmod 8$. In particular, when $q=p \equiv 3 \pmod 8$, we have $$\label{eq:5.14}
\sum_{j, x \in \f} \phi(jx)\phi(1-x)\phi(x(j+1)^2+(j-1)^2)=4a_8^2 -2p,$$ where $p = a_8^2 + 2b_8^2$.
By with $\nu = \e$, the sum in equals $$J_2(M_8, \phi N) + J_2(M_8^5, \phi N).$$ First suppose that $q \equiv 7 \pmod 8$. Then $$G_2(M_8) = G_2(M_8^5), \quad
G_2(\phi N) = q$$ by [@BEW Theorem 11.6.1]. Thus each Jacobi sum above equals $q$, which proves .
Now suppose that $q \equiv 3 \pmod 8$. An application of shows that $J_2(M_8^5, \phi N)$ is the complex conjugate of $J_2(M_8, \phi N)$, so that the sum in equals $2 {\operatorname{Re}}J_2(M_8, \phi N)$.
First consider the case where $q$ is prime, i.e., $q=p$. Then $G_2(\phi N) = p$ and by [@BEW Theorems 12.1.1 and 12.7.1(b)], $$G_2(M_8) = G(\phi)\pi, \quad G_2(M_8^5)=G(\phi)\opi, \quad
J_2(M_8, \phi N) = \pi^2,$$ where $\pi=a_8 + i b_8 \sqrt{2}$ is a prime in $\Q(i\sqrt{2})$ of norm $p=\pi \opi=a_8^2 + 2b_8^2$. Note that $\pi^2 = u_1 + iv_1\sqrt{2}$, where $$u_1 = 2a_8^2 - p, \quad v_1= 2a_8b_8, \quad u_1^2 + 2v_1^2 = p^2,$$ so that $${\operatorname{Re}}J_2(M_8, \phi N) = u_1 = 2a_8^2 -p \equiv -1 \pmod 8.$$
In the general case where say $q = p^t$, the Hasse-Davenport lifting theorem [@BEW Theorem 11.5.2] yields $$J_2(M_8, \phi N) = (-1)^{t-1}\pi^{2t}=
(-1)^{t-1}(u_1 +iv_1\sqrt{2})^t=u+iv\sqrt{2},$$ for integers $u$, $v$ such that $q^2 = u^2 + 2v^2$. Since $u_1 \equiv -1 \pmod 8$, it is easily seen using the binomial theorem that $u \equiv -1 \pmod 8$. If $p=\pi\opi$ divided $u$, then $p$ would divide $v$, so that the prime $\opi$ would divide $\pi^{2t}$, which is impossible. Thus $p \nmid u$. For an elementary proof of the uniqueness of $|u|$, $|v|$, see [@BEW Lemma 3.0.1].
*Remark:* The sum in Theorem 5.5, namely $$\label{eq:5.15}
Z =\sum_{j \in \f} \phi(j) h(\phi, j),$$ can be evaluated when $q \equiv 1 \pmod 4$ as well. We have $Z=0$ when $q \equiv 5 \pmod 8$, which can be seen by applying [@Evans Lemma 5.1] with $\phi$ in place of $D$, and then replacing $j$ by $jI$, where $I$ is a primitive fourth root of unity in $\F$. More work is needed to evaluate $Z$ in the remaining case where $q \equiv 1 \pmod 8$. In this case $Z$ is equal to the sum $R_2$ in [@Evans (5.44)] with $\no = B_8$ and $A_4 = B_8^2$ for an octic character $B_8$ on $\F$. The proof of [@BEW Theorem 3.3.1] shows that $$J(B_8, \phi) = J(B_8^3, \phi) \in \Q(i\sqrt{2}).$$ Using this equality to evaluate the sum $R_2$, we have $$\label{eq:5.16}
Z= 2q + 2 {\operatorname{Re}}J(B_8, \phi)^2, \quad q \equiv 1 \pmod 8.$$ We will use to show that $$\label{eq:5.17}
Z=4q, \ \mbox{ when } p \equiv 5 \mbox{ or } 7 \pmod 8,$$ and $$\label{eq:5.18}
Z=4c^2, \ \mbox{ when } p \equiv 1 \mbox{ or } 3 \pmod 8,$$ where $c$ and $d$ are the unique pair of integers up to sign for which $$\label{eq:5.19}
q = c^2 + 2d^2, \quad p \nmid c.$$
First suppose that $p \equiv 7 \pmod 8$. Then $q = p^{2t}$ for some $t \ge 1$. If $t=1$, then $J(B_8, \phi)=p$ by [@BEW Theorem 11.6.1]. For general $t$, the Hasse-Davenport lifting theorem thus yields $J(B_8, \phi)=(-1)^{t-1}p^t$, so that $J(B_8, \phi)^2=q$. Thus $Z=4q$ by .
Now suppose that $p \equiv 5 \pmod 8$. Since $G(B_8)=G(B_8^p)=G(B_8^5)$ by [@BEW Theorem 1.1.4(d)], $J(B_8, \phi) = G(\phi)$. Thus $J(B_8, \phi)^2=q$, so again $Z=4q$. This completes the proof of .
Next suppose that $p \equiv 3 \pmod 8$. Then $q = p^{2t}$ for some $t \ge 1$. Since $-2$ is a square $\pmod p$, we have the prime splitting $p = \pi \opi$ in $\Q(i\sqrt{2})$. Assume first that $t=1$. Then $$\label{eq:5.20}
J(B_8, \phi) \oJ (B_8, \phi) = q = p^2 = \pi^2 \opi^2, \quad t=1.$$ We cannot have $J(B_8, \phi)= \pm p$, otherwise the prime ideal factorization of $J(B_8, \phi)$ in [@BEW Theorems 11.2.3, 11.2.9] would yield the contradiction that $p$ ramifies in the cyclotomic field $\Q(\exp(2\pi i/8))$. In view of and unique factorization in $\Q(i\sqrt{2})$, we may suppose without loss of generality that $J(B_8, \phi) = \pi^2$ when $t=1$. For general $t$, $$\label{eq:5.21}
J(B_8, \phi) = (-1)^{t-1} \pi^{2t} = c+ di\sqrt{2}$$ for some integers $c$ and $d$ such $c^2 + 2d^2 = q$. Note that $p$ cannot divide $c$, for otherwise $p$ also divides $d$ (since $c^2 + 2d^2 = q$), so that $p$ divides $\pi^{2t}$, yielding the contradiction that the prime $\opi$ divides $\pi$. By , $${\operatorname{Re}}J(B_8, \phi)^2 = c^2 -2d^2 = 2c^2 -q,$$ so that by , $Z = 4c^2$.
Finally, suppose that $p \equiv 1 \pmod 8$, and write $q=p^t$ for some $t \ge 1$. Since $-2$ is a square $\pmod p$, we have the prime splitting $p = \pi \opi$ in $\Q(i\sqrt{2})$. If $t=1$, then without loss of generality, $J(B_8, \phi) = \pi$. For general $t$, $$J(B_8, \phi) = (-1)^{t-1} \pi^t = c+ di\sqrt{2}$$ for some integers $c$ and $d$ such $c^2 + 2d^2 = q$. Arguing as in the case $p \equiv 3 \pmod 8$, we again obtain $Z=4c^2$ for $c$ as in . This completes the proof of .
Proof of Katz’s identities (1.5)
================================
When $jk=0$, both sides of vanish, by and . We thus assume that $jk \ne 0$. It suffices to show that the Mellin transforms of the left and right sides of are the same for all characters, for then follows by taking inverse Mellin transforms. Thus it remains to show that $S=T$, where $S$ and $T$ are given in Theorems 3.1 and 5.1, respectively. These theorems show that $S$ and $T$ both vanish when $\chi_1$ or $\chi_2$ is even, so we may assume that and hold. For brevity, write $D = \mu \phi^i$, where $i \in \{0,1\}$. Then the equality $S=T$ is equivalent to $$\label{eq:6.1}
\begin{split}
&\frac{q }{G_2(\oD N)}
\{J_2(\nu_1 N M_8, \oD N) + J_2(\nu_1 N M_8^5, \oD N)\} = \\
&\frac{G(\phi D^2)}{G(\phi)}(W(D) + 2(q-1)\delta(D)).
\end{split}$$ Noting that $G_2(\oD N)= -G(\oD)^2$ by , and using the formula for $W(D)$ in Theorem 5.3, we easily see that holds. This completes the proof that $S=T$.
[abcdefg]{}
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Salerno, A: Counting points over finite fields and hypergeometric functions, Funct. Approx. Comment. Math. 49, 137–157 (2013).
Sussman, DM, Wootters, WK: Discrete phase space and minimum- uncertainty states. In: Proceedings of the Eighth International Conference on Quantum Communication, Measurement and Computing, Hirota, O, Shapiro, JH, Sasaki, M (eds.) NICT Press (2007). arXiv:0704.1277.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'The exotic baryon $\Theta^+$(1540 MeV) is visualized as an expected (iso) rotational excitation in the Chiral Soliton Model. It is also argued as a Pentaquark baryon state in a constituent quark model with strong diquark correlations. I contrast the two points of view; observe the similarities and differences between the two pictures. Collective excitation, characteristic of Chiral Soliton Model points toward small mixing of representations in the wake of $SU(3)$ breaking. In contrast, Constituent quark Models prefer near “ideal” mixing, similar to $\omega - \phi$ mixing.'
address:
- |
The Inter University Centre for Astronomy and Astrophysics, University of Pune Campus, Pune, 411007, India\
and
- |
Department of Physics, University of Pune Campus, Pune, 411007, India\
[[email protected], [email protected]]{}
author:
- 'R. Ramachandran'
title: 'Chiral Soliton Model vs Pentaquark Structure for $\Theta(1540)$'
---
1.5cm
0.5cm
Introduction
============
An exotic baryon $\Theta^+(1540 MeV)$ with the quantum numbers of $K^+n$ has been observed as a very narrow width state by several groups[@Spring]. It has hypercharge $Y = 2$, the third component of isospin $I_3 = 0$. Since such a state has not been seen so far in the $K^+p$ channel, $I=1$ is ruled out and so $\Theta^+ $ is an isosinglet. While this is yet to be confirmed by some other experimental groups [@negresult] that don’t see evidence for a narrow state so far, there is a consensus that there is enough evidence to warrant its inclusion in the 2004 edition of Particle Data Book [@pdg]. The minimal $SU(3)$ assignment for such a state is at the top ($Y=2,I=0$) of $\{\overline{10}\}_F$ representation with $Y=1\ I={1\ov 2}\ (N^0,N^+)$, $Y=0\ I=1
\ (\Sigma^-,\Sigma^0,\Sigma^+)$ and $Y=-1\ I={3\ov 2}\
(\Xi^{--},\Xi^-,\Xi^0,\Xi^+)$ as other members of the family.
The term exotic refers to the fact that such a state is not realized as usual three-quark composite, since positive strangeness for the baryon calls for an $\overline{s}$ quark in it. Minimal quark configuration is $udud\overline{s}$. Even though the spin and parity of $\Theta^+(1540)$ are yet to be determined experimentally, most of the theoretical analysis has carried a general prejudice that it is $J^P = {1 \ov 2}^+$ state.
The static (low energy domain) properties of the baryons are not easily derived from the underlying QCD on account of the non-perturbative features of the theory. It becomes necessary to look for other models that are inspired by QCD to throw further light on the structure and properties of hadrons (mesons and baryons) that are indeed color singlet composites of quarks and antiquarks. Even though the color degrees of freedom are hidden, it is useful to talk about the quark, anti-quark and gluon content of hadrons as revealed to an external electromagnetic or weak probe in a deep inelastic scattering by leptons ($e^\pm,\mu^\pm,\nu, \overline{\nu}$) or photons. These hadron structure functions (or more correctly their evolution as a function of resolution scale) are accessible to perturbative QCD. For other non-perturbative properties, one resorts to study QCD either on a lattice or use other effective theories, presumably derivable from QCD. Chiral Lagrangian Dynamics is one such formalism in which QCD is seen to express its global (flavour) symmetries through the pseudo scalar meson degrees of freedom in the large $N_c$ (where $N_c$ is number of colors) limit. Chiral Lagrangian with the octet of pseudoscalar mesons as primary fields admits a solitonic mode (Skyrmion)[@skyrme], which provides the baryon sector. While the ground state in this sector is an $SU(3)$ octet of baryons, of which the nucleon is the $Y=1\ I={1\ov 2}$ member, other excited states are rotational (in ordinary and internal $SU(3)$ flavour space) excitations. Quantization in the collective co-ordinates associated with the Skyrmion solution gives the spectrum of baryonic states.
It is possible to show that$ \{\overline{10}\}_F$ baryons, of which the $\Theta(1540)$ is a member as the next rotational excitation after the ground state $\{8\}$, which has the nucleon and the well known $\{10\}$ representation of which $\Delta$, the isospin quartet at 1232 Mev, is a prominent member. Indeed, Diakanov, Petrov and Polyakov [@diakanov] predicted the mass and the narrow width of the observed $ \Theta^+$ on the basis of the chiral soliton model. There is now a vast literature accumulated on the subject treating $\Theta^+$ in terms of the chiral soliton model on the one hand [@chiraltheta] and in terms of a constituent pentaquark model (with special correlations) on the other[@cqmtheta]. While both models can account for the presence of $\{\overline{10}\}_F$ states, they differ on what else is expected and what may be the consequence of $SU(3)_F$ symmetry breaking. We provide the similarities and contrasts of both points of view. There are also efforts to find pentaquark states in Lattice QCD[@lattice], which at the moment remains inconclusive.
0.5cm
Chiral Soliton Model
====================
0.3cm Effective Lagrangian embodies the chiral symmetry and is a function of $U(x)$, a unitary $3\times3$ matrix and $\pa_{\mu} U(x)$. The pseudo scalar octet of mesons are expressed through $U(x) \equiv exp\ ({i \ov F_\pi}
\lambda^a\phi^a(x)),\ a=1,2,..8$; $\lambda^a$ are Gell Mann matrices and $x$ denotes $(\vec{x}, t)$. $F_\pi$ is pion decay constant and provides a scale for the masses in the theory. The theory admits finite energy static solutions (for the classical equations of motion), that has the form: $$U(x) = U_0(\vec{x})=\pmatrix {
exp \ i f(r) \vec{\tau}\cdot \hat{x} & 0 \cr
0 & 1 }$$ The constraint that $$f(r)\to \matrix {\pi\ &\ \mbox{when } r\to 0 \cr
0 &\ \mbox{when } r\to \infty},$$ ensures that $U(x)$ is both well defined at the origin and can lead to finite energy configuration. Precise form of $f(r)$ can be obtained numerically from the radial equation of motion. Quantization is then carried out by identifying the collective co-ordinates, a set of parameters that label the transformations that will leave the classical solution invariant. Apart from the centre of mass of the Skyrmion, the translation of which will indeed yield the overall momentum of the state and hence the kinetic energy, rotations in space as well as internal space are the other collective co-ordinates. To get their dynamics, it is convenient to identify the collective co-ordinates ${\mathcal {A}}$ by defining U(,t) = [A]{}(t)U\_0()[A]{}\^[-1]{}(t); SU(3) Notice that the solution is left invariant under ${\mathcal A}\to L{\mathcal A}$; $L\in SU(3)$ denoting a $SU(3)$ flavour transformation and under ${\mathcal A} \to {\mathcal A}R$; $R \in SU(2)$, an element of rotation in space. ${\mathcal A}(t)$ constitute the relevant collective co-ordinates that embody the rotational and iso-rotational degrees of freedom for the Skyrmion. The wavefunction for the baryons in various allowed irreducible representations of $SU(3)$ are given by the equivalent of Wigner D- functions ${\mathcal{D}}^{\{R\}}_{\alpha,\beta}({\mathcal A})$, where $\{R\}$ stands for the $SU(3)_F$ representation of the state; $\alpha = (I,I_3\ Y)$ and $\beta =
(I'=J,I'_3=J_3 \ Y'=1)$ respectively denote the iso-rotational (isospin $I$,$I_3$ and hypercharge $Y$) and rotational (angular momentum $J,J_3$) state of the baryon.[@smr] [@guadagnini]. The relevant Hamiltonian has the form: H=M\_0+ [1 2\_1]{} \_[a=1]{}\^[3]{}J\_a\^2+[1 2\_2]{} \_[a=4]{}\^[7]{}J\_a\^2 + (1/N\_c ) where ${\mathcal{I}}_{1,2}$ are ‘moments of inertia’ for the rigid rotator model for the baryon. The interlocking of the spin and isospin is an essential feature of the soliton sector and generates a constraint on the states that is further made precise from the presence of the Wess- Zumino term in the effective Lagrangian for $SU(n_F),\ n_F > 3$. For the $SU(2)$ Skyrmion this is reflected by the fact that the allowed baryons have their $I = J$. For $SU(3)$, the constraint admits in the spectrum only those $SU(3)$ representations that have $Y = 1$ member and the spin $J$ for the set will be the same as the isospin values of all the $Y = 1$ members of the representation. For $n_F > 3$, the angular momentum of the state assumes value(s) of isospin of the $Y=1$, $SU(n_F-2)$ singlet(s)[@manohar].
Since every $SU(3)$ unitary irreducible representation is given by a pair of indices $(p,q)$ (the wave function has $p$ indices that transform like $\{3\}$ and $q$ indices like $\{\overline{3}\}$) and the second casimir operator $C_2(p,q) \equiv \sum_{a=1}^{8} J_a^2 = {1 \ov 3} (p^2+q^2+pq+3(p+q))$ and $J_8 = -{\sqrt{3} \ov 2} Y$, we can read off the mass spectrum (Wess-Zumino constraint implies $|p-q| =0\ \mbox{modulo}\ 3$) as:
$$E^J(p,q) =M_0 +{1 \ov 2{\mathcal{I}}_2} C_2(p,q) + ({1 \ov 2{\mathcal{I}}_1}-{1 \ov 2{\mathcal{I}}_2})J(J+1)-
{3 \ov 8{\mathcal{I}}_2}.$$
If ${\mathcal{I}}_1 > {\mathcal{I}}_2$, we will have the observed sequence of states; ground state $\{8\}_{1/2}$ followed by $\{10\}_{3/2}$ and the just discovered $\{\overline{10}\}_{1/2}$. From the central or average values of the masses in these multiplets, $1115$ MeV for $ \{8\}_{1/2}$, $1382$ MeV for $\{10\}_{3/2}$ and $1755$ MeV for $\{\overline{10}\}_{1/2}$, we may determine values of ${\mathcal {I}}_{1,2}$ and get the sequence of further excitations to be at(in MeV) 1784 $\{27\}_{3/2}$, 1967 $\{35\}_{5/2}$, 2155 $\{27\}_{1/2}$, 2570 $\{64\}_{5/2}$, 2588 $\{35\}_{3/2}$, 2707 $\{81\}_{7/2}$, 2959 $\{\overline{35}\}_{1/2}$ and so on.
An important observation made on the basis of the chiral soliton picture has to do with the narrow width of the baryons in $\{\overline{10}\}_{1/2}$. Diakanov [*et al*]{} [@diakanov] attribute this to the next to leading order (in $1/N_c$) correction for the meson baryon couplings. While the leading order for all transitions in the soliton sector is characterized by $G_0$, there are two further terms with strengths $G_1$ and $G_2$ in the next order. For example, the $(B_8B_8M_8)$ has two distinct $SU(3)$ symmetric couplings $D$ and $F$, usually given instead in terms of $g_{\pi NN} (= D +F)$ and $\alpha (= D/(D+F))$ that are expressed through g\_[NN]{} &=& [7 10]{} (G\_0 + [1 2]{}G\_1 + [1 14]{}G\_2)\
&=& [9 14]{} [G\_0 + [1 2]{}G\_1 - [1 6]{}G\_2 G\_0 + [1 2]{}G\_1+ G\_2]{}
In view of the fact that the experimental value of $\alpha = 0.65$ is very close to 9/14, we expect $G_2$ to be small and negligible.
The decouplet baryon decay couplings ($B_{10}B_{8}M_{8})$ are characterized by the factor $G_0 + {1 \ov 2}G_1$ and the antidecouplet decays ($B_{\overline{10}}B_{8}M_{8}$) are governed by the term $G_0 - G_1
-{1 \ov 2}G_2$. Diakanov estimates $G_1/G_0$ to be in the range 0.4 to 0.6 with the result $G_{\overline{10}}/G_{10} \sim 1/3\ \mbox{to}\ 1/5$. Indeed, with a bit of adjustment in the value of $G_1$ and $G_0$, considerable suppression for width of antidecouplet baryons can be realized.
When $SU(3)$ is explicitly broken there are two further consequences. There will be splitting within each $SU(3)$ representation resulting in the spectrum governed by Gell Mann Okubo mass relations. For octet baryons: M\_[8]{}(I, Y) = M\_8\^0 - bY +c(I(I+1)-Y\^2/4) and equal spacing of levels for both $\{10\}$ and $\{\overline{10}\}$ states: M\_[10]{}(Y) &=& M\_[10]{}\^0 - aY,\
M\_(Y) &=& M\_\^0 - a’Y.
A second related consequence of symmetry breaking is the mixing of various states with the same combination of ($I$, $Y$) states among different $SU(3)$ representations. In particular we expect $N_8$ and $N_{\overline{10}}$ will mix to yield $N(939)$ and some $N^*$ state. Similarly $\Sigma_8$ and $\Sigma_{\overline{10}}$ will mix. These mixings will induce a shift in masses from the above octet symmetry breaking relations for the mass eigenstates as well as cause suppression or enhancement of the decay amplitudes. 0.3cm
Mixing angle from the spectrum
------------------------------
0.2cm
If we identify $N_8$, $\Lambda (1115)$, $\Sigma_8$, $\Xi (1318)$ as octet states and $\Theta^+(1540)$, $N_{\overline{10}}$, $\Sigma_{\overline{10}}$ and $\Xi_{3/2}(1862)$ as antidecouplet states and assign $ N(939)$ and $N(1710)$ as nucleon-like mass eigenstates and similarly $\Sigma (1193)$ and $\Sigma
(1880)$ as $\Sigma$ -like states, we may find mixing angles $\theta$ as follows:
|N(939)> &=& cos \_N |N\_8> - sin \_N |N\_>\
|N(1710)> &=& sin \_N |N\_8> + cos \_N |N\_> and |(1193)> &=& cos \_ |\_[8]{}> - sin \_ |\_>\
|(1880)> &=& sin \_ |\_[8]{}> + cos \_ |\_>.
It is easily obtained that 939 +1710 &=& <N\_8|H|N\_8> +<N\_|H|N\_>\
&=& M\_8\^0-b+[c 2]{} + M\_\^0-a’\
&=& 1115 +1755 -a’ -b +[c 2]{}.
From the masses of $\Theta^+$ and $\Xi_{3/2},$ we get $a'=107$ MeV, yielding $b-{c \ov 2} =114 $ MeV. Using the masses of $\Lambda(1115)$ and $\Xi(1318)$ we find $b+{c \ov 2} = 203$ MeV. We further have cos 2 \_N(1710 -939)& = &<N\_|H|N\_> - <N\_8|H|N\_8>\
&=& M\_ - M\_8 -a’ +b - [c 2]{}\
&=& 647
This implies that $cos\ 2 \theta_N = 0.84$, which translates into $tan^2\ \theta_N =0.087$.
If we denote $<N_8|H|N_{\overline{10}}>= <\Sigma_8|H|\Sigma_{\overline{10}}> = \delta$, we will have $$2\delta = (M_{10}-M_8-a+(b-{c \ov 2}))\ tan\ \theta_N$$ yielding a value $\delta = 220$ MeV as signifying the extent of representation mixing.
We may carry out a similar analysis in the mass mixing in the $\Sigma$ - sector to find
1880 + 1193 &=& 1755 +1115 + 2c\
cos 2\_ (1880 - 1193) &=& 1755 - 1115 - 2c.
We obtain $cos\ 2\theta_\Sigma = 0.63 $ and corresponds to the representation mixing value of $\delta$ =264 MeV instead.
As a result of the mixing the expectation value of the $N$ and $\Sigma$ in the octet representation will be $M_{N_{8}} = M_8-b+ {c \ov 2} = 1001$ MeV and $M_{\Sigma_8} = M_8 + 2c = 1293$ MeV. Gell Mann Okubo relation for the octet will imply $$2 M_{N_8} - M_{\Sigma_8} = 3 M_{\Lambda} - 2 M_{\Xi} = 709 \ MeV$$
Before mixing is applied the left hand side of the expression will have $2\times 939
-1193 = 645$ MeV. When mixing is taken into account it is instead $2\times 1001$ -1293 = 709 MeV, much more precise. Indeed the nucleon (and $\Sigma$) needs $\{\overline{10}\}$ admixture to fit GMO formula better![[^1]]{} 0.3cm
Mixing angle from the decays
----------------------------
0.2cm
The mixing angle can also be deduced by studying the decay width of the states. The couplings are assumed to be preserving $SU(3)$, but in the computation of the width, the phase space is computed with the actual \[$SU(3)$ broken\] mass values. We have already observed that while for $\{10\}_{{3/2}^
+} \to
\{8\}_{{1/2}^+} + \{8\}_{0^-}$, the rates are proportional to $(G_0+{1 \ov 2}
G_1)^2$, the transition rates for $\{\overline{10}\}_{{1/2}^+} \to
\{8\}_{{1/2}^+} + \{8\}_{0^-}$ are governed by the factor $(G_0 - G_1
-{1 \ov 2} G_2)^2$. The partial decay widths for the antidecouplet baryons are given by
(\^+ KN) &=& [3 2]{} [(G\_0 - G\_1 -[1 2]{} G\_2)\^2 (M\_N +M\_)\^2]{} [M\_N M\_]{} p\^3\_[ KN]{} ||\^2\
(\^[–]{} \^-\^-) &=& [3 2]{} [(G\_0 - G\_1 -[1 2]{} G\_2)\^2 (M\_[\^[–]{}]{} +M\_[\^[-]{}]{})\^2]{} [M\_[\^[-]{}]{} M\_[\^[–]{}]{}]{} p\^3\_[\^[–]{} \^- \^-]{} ||\^2 with $$p_{B_1 \to B_2 M} = \sqrt{(M_{B_1}^2 - (M_{B_2} + M_M)^2) (M_{B_1}^2 -
(M_{B_2} - M_M)^2)} /2 M_{B_1}.$$
Notice that there is no $SU(3)$ symmetric coupling that will permit $N_{\overline{10}} \to \Delta + \pi $ (because $\{\overline{10}\} \{10\}\{8\}$ coupling does not exist.) However, since $N(1710) \to \Delta(1232) \pi$ has a partial width of about 5 MeV, this is a clear indication that this state can not be a pure antidecouplet. It is the octet part of the state that is responsible for the $\Delta \pi $ decay mode. Comparison of the partial width for $N(1710) \to
\Delta \pi$ with that of $\Delta \to N \pi$ will give us the measure of the admixture. Since $|N(1710> = cos\ \theta_N |N_{\overline{10}}> + sin\ \theta_N
|N_8>$, we get $$\Gamma(N(1710)\to \Delta \pi) = sin^2\ \theta_N {3 \ov 2\pi} {(G_0+ {1 \ov
2}G_1)^2 \ov (1710
+1232)^2} {1710 \ov 1232} p^3_{N^*\to\Delta\pi} {4 \ov 5} \sim 5 MeV.$$ Comparing this with $$\Gamma(\Delta(1232) \to N \pi) = cos^2 \ \theta_N {3 \ov 2\pi}
{(G_0+{1 \ov 2}G_1)^2 \ov (939
+1232)^2} {1232 \ov 939} p^3_{\Delta \to N\pi} {1 \ov 5} =120 MeV,$$ we find that $tan^2\ \theta_N = 0.0035$. Note a very small admixture is all that is needed to get a 5 MeV partial decay width for $N(1710)\to \Delta \pi$.
We conclude that mixing angle from both mass spectrum and decay data are reasonably small. The decay amplitudes appear to give a much smaller value of mixing parameter than the data on the mass spectrum of states.
It will be useful to contrast these results with other analysis. Pakvasa and Suziki [@pakvasa], who assumed the mixing octet to consist of N(1440), the old Roper resonance and $\Sigma(1660)$ instead of $N(939)$ and $\Sigma(1193)$ along with appropriate excited states for both $\Lambda$ and $\Xi$, find an even larger discrepancy in view of their choice for states. Since they use Roper resonance in place of the nucleon of the ground state baryon octet,they get a substantial mixing from the mass spectra and therefore can not reconcile with a much smaller mixing angle that is indicated from the decay amplitudes. Weigel [@weigel], who has made an extensive study of the spectrum of Skyrmion states, identifies such an octet state with the radially excited Skyrmion. It is not clear to us why $N_{\overline{10}}$ would prefer to mix with the radially excited state over the ground state. Perhaps we need to compute mixing with all the three levels. However in such an analysis, in the leading order, we expect the radially excited state to be orthogonal to the ground state and hence should not be giving any qualitatively different answers.
Mixing of states may arise also with higher iso-rotational levels in the Skyrmion sector, that we have listed. Analysis have been carried out by including $\{27\}_{1/2}$ and $\{\overline{35}\}_{1/2}$ (which are expected to be much and very much heavier respectively) for $J={1 \ov 2}$ states and $\{27\}_{3/2}$, $ \{35\}_{3/2}$ states with $\{10\}_{3/2}$ baryons for $J={3 \ov 2}$ levels.
We don’t have much experimental support for higher states of baryons as of now and so such an analysis is of academic interest only. The basic premise that $\{\overline{10}\}$ states have a small admixture of the ground state octet baryons appears to be born out by experimental data so far available. while mixing angles in the most basic interpretation are indeed small, the decay data appears to suggest much smaller mixing. We will see that an alternative explanation in terms of Constituent quark model that will keep track of the number of strange quarks in a state will imply substantial mixing of representations, when $SU(3)_F$ is broken. 0.5cm
Constituent Quark Model
=======================
0.3cm In constituent quark model framework, the hadrons are considered built up of dressed (valence) quarks much the same way nuclei are built using nucleons and model effective interaction among dressed quarks. In a naive uncorrelated quark model for pentaquark baryons, we expect the ground state to be a $J^P = 1/2^-$ state with flavour quantum number to be both $\{\overline{10}\}_F$ and $\{8\}_F$. They expect to be accompanied with states with $J^P=3/2^-$ with roughly the same difference in mass as between $N$ and $\Delta$. Jaffe and Wilczek argue that there is a lower energy state with positive parity, exploiting a strong correlation among quarks that picks diquark pairs that are antisymmetric in color, spin and flavour. Call these condensates ${\bf Q}$ that are $\{\overline{3}\}_C$ and $\{\overline{3}\}_F$ scalars. Pentaquark baryons are then ${\bf QQ}{\bar q}$ composites. For two such diquarks, along with an antiquark, to form a color singlet they must be antisymmetric in color. In order to form a $\{\overline{10}\}_F$, the diquark pair have to be in a flavour symmetric $\{\overline{6}\}_F$, which in turn implies orbital excitation and $l=1$. When combined with an antiquark, we have the pentaquark states in $\{\overline{10}\}_F$ and $\{8\}_F$ , expected to be degenerate, both having $J^P=1/2^+$ and $3/2^+$, with the spin orbit coupling providing the splitting as found in the separation between $N$ and $\Delta$. The nucleon like states and the $\Sigma$ -like states of the $\{\overline{10}\}$ and $\{8\}$, when subjected to $SU(3)$ breaking are expected to mix [*ideally*]{}, so that the lighter member has no strange quarks and the higher has a $s\overline{s}$ pair for the nucleon-like member. This is similar to the vector meson nonet with $\omega - \phi$ mixing, where $|\omega> ={1 \ov \sqrt{2}}(|u{\bar u}> -|d{\bar d}>)$ and $|\phi> = |s{\bar s}>$. Ideally mixed nucleon like states then are:
|N\_[ud]{}> &=& |N\_8> + |N\_>\
|N\_s> &=& - |N\_8> + |N\_> with $|p_{ud}> = |udud\overline{d}>$, $ |n_{ud}> = |udud\overline{u}>$, $|p_s>=
|udus\overline{s}>$ and $|n_s> = |udsd\overline{s}>$. For ideal mixing $tan\ \theta =
\sqrt{2}$.
Similarly the $\Sigma$ -like states are, indeed: |\_[ud]{}> &=& |\_8> + |\_>\
|\_s> &=& - |\_8> + |\_> with $|\Sigma^+_{ud}> = |udus\overline{d}>$, $|\Sigma^+_s>= |usus\overline{s}>$; $|\Sigma^0_{ud}> = {1 \ov \sqrt{2}}
|udus\overline{u}> + {1 \ov \sqrt{2}} |udds\overline{d}>$, $|\Sigma^0_s> = |usds\overline{s}>$; and $|\Sigma^-_{ud}> =
|udds\overline{u}>$, $|\Sigma^-_s> = |dsds\overline{s}>$.
Jaffe and Wilczek expect the mass spectrum to be governed by the number of $s$ quarks and $\overline{s}$ antiquark the baryon has. If $H=M_0+ (n_s+n_{\overline{s}})\alpha
+n_s \beta$, it will imply ideal mixing of $\{\overline{10}\}$ and $\{8\}$. We will then have the sequence of pentaquark baryons with $N^*_{ud}\ < \Theta^+\
< \Sigma^*_{ud}, \Lambda^* \ < \ N^*_s \ < \ \Xi^*_{1/2},\ \Xi^*_{3/2} \ < \
\Sigma^*_s$. The states identified by them to fit this sequence, apart from $\Theta^+(1540)$, are $N^*_{ud}(1440)$, $N^*_s(1710)$, $\Lambda^*(1600)$, $\Sigma^*_s(1880)$. They agree with the above sequence if we accept their analysis of the systematics for exotic $\Xi$ decays, where they argue that there are nearly degenerate $\Xi_{1/2}$ and $\Xi_{3/2}$ in 1855 - 1860 mass region[@jw2].
This large mixing angle is incompatible with the information from decay data. In particular, $\Gamma(N(1710) \to \Delta \pi)$ will turn out to be absurdly large, if $tan\ \theta = \sqrt{2}$. Further the narrow width for $\Theta^+$ and $\Xi^{--}_{3/2}$ will be in sharp conflict with the expected large width for $N_{ud}$ and $N_s$. While the broad Roper resonance fits the bill for $N_{ud}$, it is not clear whether there is another broad nucleon-like state in that region instead, if $N(1710)$ is to be discounted as the candidate for $N_S^*$.
Equally dramatic is the prediction that both $\{8\}_F$ and $\{\overline{10}\}_F$ will be nearly degenerate before $SU(3)_F$ is broken. This is indeed the argument for interpreting that the $\Xi$ resonances cover both $I = 1/2$ and $I = 3/2$ components in the same region.
In contrast against the Chiral Soliton Model, the Pentaquark model predicts in addition to $J^P = 1/2^+$ state, $J^P = 3/2^+$ states as well with an expected mass difference, which should be of the same order as $(M_{\Delta} - M_N)$. We expect many more baryonic states than what has been reported. We observe that there is no interlocking of spin and isospin that was characteristic in the soliton sector.
We need more definitive experimental evidence before we are able to rule in favour of either Chiral Soliton Model or any of the specific constrained Constituent Quark Models. There are other variants of Jaffe-Wilczek diquark correlations; for example the one by Lipkin and Karliner [@lipkin] postulates two separate clusters one made up of the same diquark and another ($qq\overline{q}$) cluster in which the two quarks are in color symmetric $\{6\}_C$ state.The observed narrow width for $\Theta^+$ is attributed to the fact that the clusters are kept apart due to angular momentum barrier.
Comparisons and conclusion
==========================
Both in chiral model as well as in constituent pentaquark models with diquark correlations folded in, the spin parity assignment favoured is $J^P ={1 \ov
2}^+$. However, while all rotational excitations in chiral soliton sector are necessarily of positive parity, there is no reason to exclude negative parity baryons in the pentaquark picture. The additional states in the CSM are are attributed to radial excitations, all of which will have positive parity, and same $SU(3)_F$ quantum numbers. In contrast, in CPQM we expect a more or less degenerate octet of states in addition to $\{{\bar 10}\}$ baryons, both of spin 1/2 and 3/2 variety and further (perhaps a bit more massive ) negative parity states. More detailed spectroscopy in this mass region can clarify whether such additional states are present.
In the CSM, Nucleon like states in the ground state octet and the exotic antidecouplet will indeed mix. These mixing angles remain small and generally found to be so with our assignment, which parallels Diakanov et al’s choice. Even so, the mixing angle as obtained from the decay widths appear further smaller than that obtained from mass spectrum. In contrast, when $SU(3)_F$ is broken in the CPQM we expect a large mixing of the nearly degenerate $\{8\}$ and$ \{\bar {10}\}$ multiplets to give it the nature of ideal mixing. This will lead to a large strangeness content of one of the nucleon like states, say $N(1710)$ and so it must have a significant branching ratio into $\Lambda K$ and $\Sigma K$ channels in order that Zweig rule is obeyed. Further since it is made up of a substantial component of octet, the coupling to baryons and mesons such as $\Delta \pi$, $\Lambda K$ etc. will be comparable to the strength of $\Delta NK$ coupling. This is in conflict with the observed branching ratio and small widths of this state. Admittedly more detailed analysis is called for before we can confirm $N(1710)$ as the candidate $uuds{\overline s}$ state.
Is there a deeper reason for the dramatic narrow width? We note that the degenerate octet and decouplet states arise from $(\bar 3,\bar 6) \oplus (\bar
6,\bar 3)$ of the underlying $SU(3)_L \times SU(3)_R$ symmetry. In the same scheme baryon octet belongs to $(1,8) \oplus (8,1)$. In the limit of exact chiral symmetry (left handed currents transforming like $(8,1)$ and right handed currents as $(1,8)$ ) there will be no coupling between pentaquark baryons and the usual triquark octet. Thus $\Theta^+ \to NK$ decays are inhibited on account of the underlying chiral symmetry[@ioffe]. In exact chiral symmetry, (limit in which pion masses vanish) we expect stable $\{\bar {10}\}$ baryon states[@beane].
Both view points have room for many additional states; in CSM radial excitations will give further states with same flavour quantum numbers and in CPQM many other permutations of correlated clusters are possible as well. Of course several of these features could be consequences of higher (${1/N_c}
$) order and hence may not be very reliable predictions of CSM. Future experiments[@futurexp] should nail many of the predictions of both pictures.
It is satisfying to note, that the chiral soliton model, that starts from an underlying chiral symmetry has dynamical ingredients to account for its narrow width. In the CPQM, in contrast, small width is due to non-overlapping clusters on account of centrifugal barrier. Perhaps we need some ingredient that signals approximate chiral symmetry in the CPQM to reflect more similarity with the soliton picture. This calls for the possibility that we may be able to describe hybrid models that have features of both Chiral Soliton picture on the one hand, while being legitimately pentaquark constituent structures in terms of relevant variables. Main reason for such a prospect has to do with the feature that the narrow width is very likely a consequence of underlying approximate chiral symmetry.
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[^1]: However the Guadagnini hybrid mass relation $8M_{N_8}+3M_{\Sigma_8}=11M_{\Lambda}+8M_{\Sigma^*}-8M_{\Xi^*}$, which is satisfied to an amazing $0.1\%$ before mixing loses a shade when mixing is taken into account to an agreement at about $2 \%$ level.
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---
author:
- 'Saurabh W. Jha'
bibliography:
- 'handbookIax.bib'
title: Type Iax Supernovae
---
-1.5in Author’s version of a review chapter in the *Handbook of Supernovae*,\
edited by Athem W. Alsabti and Paul Murdin, Springer International Publishing,\
[[DOI 10.1007/978-3-319-20794-0](http://dx.doi.org/10.1007/978-3-319-20794-0)]{}
0.7in
0.7in
Introduction {#sec:intro}
============
Type Iax supernovae (SN Iax) are a class of objects similar in some observational properties to normal type-Ia supernovae (SN Ia), but with clear differences in their light-curve and spectroscopic evolution. SN Iax are also called “02cx-like” supernovae, based on the exemplar SN 2002cx, which was described by @Li03 as “the most peculiar known” SN Ia. Later, @Jha06 presented other similar objects, making SN 2002cx “the prototype of a new subclass” of SN Ia. @Foley13 coined the SN Iax classification, arguing these objects should be separated from SN Ia as “a new class of stellar explosion.”
Here I summarize the properties of SN Iax, describing their observational properties in Section \[sec:obs\] and models in Section \[sec:models\]. I discuss analogues of SN 2002cx and well-studied examples like SN 2005hk [@Chornock06; @Phillips07; @Stanishev07; @Sahu08], SN 2008A [@McCully14], SN 2012Z [@Stritzinger15; @Yamanaka15], and SN 2014ck [@2016MNRAS.459.1018T] as well as more extreme members of the class like SN 2008ha [@Foley09; @Valenti09] and SN 2010ae [@Stritzinger14]. There may be some connection between SN Iax and other classes of peculiar white-dwarf supernovae [like SN 2002es-like objects; @Ganeshalingam12; @White15; @2016ApJ...832...86C] but I restrict my focus to SN Iax here; other peculiar objects are explored by @Taubenberger17.
Observations {#sec:obs}
============
In this section I discuss the identification and classification of SN Iax, followed by their photometric and spectral properties from early to late time. I also discuss the host environments of SN Iax, their rates, and pre- and post-explosion high-resolution imaging.
Identification and Classification {#sec:ident}
---------------------------------
Supernovae are traditionally classified by their maximum light optical spectra, and SN Iax are no exception. These objects have spectra very similar to some normal SN Ia, particularly the “hot” SN 1991T-like or SN 1999aa-like objects, with typically weak absorption and prominent lines [@Nugent95; @Li01pecrate]. The key discriminant for SN Iax is the expansion velocity; unlike typical SN Ia where the line velocity (usually measured with ) is $\sim$10,000 [km s$^{-1}$]{}, SN Iax have much lower line velocities anywhere from $6000-7000$ [km s$^{-1}$]{} (for objects like SN 2002cx, 2005hk, 2008A, 2012Z, and 2014dt) down to $2000$ [km s$^{-1}$]{} (for objects like SN 2009J). Figure \[fig:max-spectra\] shows maximum-light spectra of SN Iax with a range of expansion velocities, compared to normal SN Ia. Because line velocities are not always measured or reported when SN are classified and host redshifts are sometimes uncertain, on occasion SN Iax have been misclassified as normal SN Ia. Indeed, it is possible to identify unrecognized SN Iax in past observations, like SN 1991bj[^1] [@Stanishev07]. Table \[tab:list\] gives a list of 53 objects that have been classified as SN Iax.
![Near-maximum light spectra of SN Iax compared to normal and 91T/99aa-like SN Ia. The left panel shows the similarity of SN Iax to normal SN Ia (including the weak presence of lines sometimes considered hallmarks of thermonuclear SN), but also the lower expansion velocities (e.g., compare the locations of the lines). The right panel shows the range of SN Iax expansion velocities, from the lowest velocity SN 2009J ($v_{\rm exp} \approx 2200$ [km s$^{-1}$]{}) at the top, SN 2005cc ($v_{\rm exp} \approx
5000$ [km s$^{-1}$]{}) intermediate, and SN 2008A ($v_{\rm exp} \approx 6400$ [km s$^{-1}$]{}) at the bottom, also compared to the normal SN 1999aa below. Note the similarity in the spectra as the lower-velocity objects are smoothed and shifted to resemble the higher-velocity objects. Figure adapted from @McCully14 and @Foley13.[]{data-label="fig:max-spectra"}](fig1-max-spectra.pdf){width="\textwidth"}
$\,$ Supernova $\,$ Host Galaxy $z$ $M_{\rm peak}$ References
-------------------------- ----------------------------------- ----------------- ---------------- ---------------------------------------------
SN 1991bj IC 344 $\,$ 0.018 $\,$ $\, -15.3 \ $ @Stanishev07
SN 1999ax A140357+1551 0.023 $\,-18.3\,$ @Foley13
SN 2002bp UGC 6332 0.020 $-16.4$ @Silverman12
SN 2002cx CGCG 044$-$035 0.023 $-18.8$ @Li03
SN 2003gq NGC 7407 0.021 $-17.2$ @Jha06
SN 2004cs UGC 11001 0.014 $-16.1$ @Foley13
SN 2004gw CGCG 283$-$003 0.017 $-16.9$ @Foley09
SN 2005P NGC 5468 0.009 $-14.8$ @Jha06
SN 2005cc NGC 5383 0.007 $-17.6$ @2005ATel..502....1A
SN 2005hk UGC 272 0.013 $-18.5$ @Chornock06 [@Phillips07];
@Stanishev07 [@Sahu08]
SN 2006hn UGC 6154 0.017 $-18.9$ @Foley09
SN 2007J UGC 1778 0.017 $-17.2$ @2007CBET..926....1F
SN 2007ie SDSS J21736.67$+$003647.6 0.093 $-18.2$ @2011AA...526A..28O
SN 2007qd SDSS J20932.72$-$005959.7 0.043 $-16.2$ @McClelland10
SN 2008A NGC 634 0.016 $-19.0$ @McCully14
SN 2008ae IC 577 0.030 $-18.8$ $\,[email protected]$\,$
SN 2008ge NGC 1527 0.003 $-17.4$ @Foley10_ge
SN 2008ha UGC 12682 0.004 $-14.0$ @Foley09 [@Valenti09]
SN 2009J IC 2160 0.015 $-15.9$ @2009CBET.1665....1S
SN 2009ho UGC 1941 0.048 $-18.2$ @2009CBET.1889....1S
SN 2009ku A032953$-$2805 0.079 $-17.9$ @Narayan11
PTF 09ego SDSS J172625.23$+$625821.4 0.104 $-18.6$ @White15
PTF 09eiy 0.06 @White15
PTF 09eoi SDSS J232412.96$+$124646.6 0.042 $-16.7$ @White15
SN 2010ae ESO 162-G17 0.003 $-14.0$ @Stritzinger14
SN 2010el NGC 1566 0.005 $-13.0$ @2010CBET.2337....1B
PTF 10xk 0.066 $-17.1$ @White15
SN 2011ay NGC 2315 0.021 $-18.1$ @2015MNRAS.453.2103S
SN 2011ce NGC 6708 0.008 $-17.1$ @2011CBET.2715....1M
PTF 11hyh SDSS J014550.57$+$143501.9 0.057 $-18.7$ @White15
SN 2012Z NGC 1309 0.007 $-18.1$ $\,$@Stritzinger15 [@Yamanaka15] $\,$
PS1-12bwh CGCG 205$-$021 0.023 $-16.2$ @Magee17
LSQ12fhs 0.033 $-18.2$ @2012ATel.4476....1C
SN 2013dh NGC 5936 0.013 $-17.3$ @2013ATel.5143....1J
SN 2013en UGC 11369 0.015 $-17.9$ @2015MNRAS.452..838L
SN 2013gr ESO 114$-$G7 0.007 $-15.4$ @2013ATel.5612....1H [@2013CBET.3733....1H]
$\,$OGLE-2013-SN-130$\,$ 0.09 $-18.0$ @2013ATel.5620....1B
OGLE-2013-SN-147 0.099 $-19.3$ @2013ATel.5689....1L
iPTF 13an 2MASX J12141590$+$1532096 0.080 @White15
SN 2014ck UGC 12182 0.005 $-15.5$ @2016MNRAS.459.1018T
SN 2014cr NGC 6806 0.019 $-17.0$ @2014ATel.6302....1C
LSQ14dtt 0.05 $-18.0$ @2014ATel.6398....1E
SN 2014dt NGC 4303 0.005 $-18.6$ @Foley15 [@2016ApJ...816L..13F]
SN 2014ek UGC 12850 0.023 $-18.0$ @2014ATel.6611....1Z
SN 2014ey CGCG 048$-$099 0.032 $-18.1$ @2017arXiv170704270L
SN 2015H NGC 3464 0.012 $-17.7$ @Magee16
PS15aic $\,$2MASX J13304792$+$3806450$\,$ 0.056 $-17.9$ @2015ATel.7534....1P
SN 2015ce UGC 12156 0.017 $-17.9$ @2017TNSCR.381....1B
PS15csd 0.044 $-17.5$ @2015ATel.8264....1H
SN 2016atw 0.065 $-18.0$ @2016ATel.8810....1P
OGLE16erd 0.035 $-17.1$ @2016ATel.9660....1D
SN 2016ilf 2MASX J02351956$+$3511426 0.045 $-17.6$ @2016ATel.9795....1Z
iPTF 16fnm UGC 00755 0.022 $-15.0$ @2017arXiv170307449M
: Partial list of type Iax supernovae. Host galaxies, redshifts, and estimated peak absolute magnitudes are from the Open Supernova Catalog [@OSC]. The references listed present the classification and/or maximum-light data and are incomplete.
\[tab:list\]
Photometric properties {#sec:photprops}
----------------------
The optical light curves of SN Iax show a general similarity to SN Ia, though with more diversity. SN Iax typically have faster rises ($\sim$10 to 20 days) in all bands, with pre-maximum light curves showing significant variety [@Magee16; @Magee17]. The $B$ and $V$-band decline rates are similar to normal SN Ia, though generally also on the faster side [@Stritzinger15], and the optical color evolution in SN Iax (e.g., in $B-V$) has a shape roughly similar to normal SN Ia [@Foley13]. However, SN Iax have significantly slower declines in redder bands, e.g., $\Delta m_{15}(R) \simeq$ 0.2 to 0.8 mag, compared to normal SN Ia which have $\Delta m_{15}(R) \simeq$ 0.6 to 0.8 mag [@Magee16]. Faster rising SN Iax are generally faster fading as well, with some exceptions, like SN 2007qd [@McClelland10]. SN Iax do not show the prominent “second-peak” in the redder and near-infrared bands [@2014ApJ...795..142G] that characterize normal SN Ia. The second-maximum is especially strong in slowly-declining SN 1999aa or 1991T-like SN Ia; thus the SN Ia that are spectroscopically most similar to SN Iax have quite a different photometric behavior. Nonetheless, similar to SN Ia, weeks after maximum light SN Iax show only modest color evolution. Late-time colors may thus provide a useful diagnostic of host-galaxy reddening [@Lira96; @Foley13], which is otherwise difficult to determine for SN Iax. The very late-time optical light curves of typical SN Iax continue to show a decline slower than SN Ia until about 300 to 400 days past maximum light, after which SN Ia light curves also slow to similar decline rates as SN Iax: $0.01-0.02$ mag day$^{-1}$ [@McCully14].
The peak optical luminosity of SN Iax is lower than typical SN Ia and it spans a much wider range, from $M_V \simeq -19$ on the bright end to $M_V \simeq
-13$ for the faintest SN Iax known. Compared to the light curve decline rate, SN Iax fall well below the @Phillips93 relation, by anywhere between 0.5 and several magnitudes [@Foley13]. Certainly, as seen in Figure \[fig:phillips\], SN Iax do not show as tight a relation in this parameter space as normal SN Ia (even including the SN 1991T/1999aa and SN 1991bg extremes of the normal SN Ia distribution). @Magee16 [ see their Figure 5] suggest a stronger correlation may exist between peak luminosity and rise time, rather than decline rate.
![Absolute magnitude vs. decline rate relation for SN Iax (colored) compared to normal SN Ia (black) showing the @Phillips93 relation, in $B$-band (above) and $R$ (or $r$)-band below. These plots are adapted from @Stritzinger15 and @Magee16.[]{data-label="fig:phillips"}](fig2-phillips.pdf){width="90.00000%"}
Near-infrared light curves of SN Iax are limited, with SN 2005hk still providing the best data set [@Phillips07]. Continuing the trend with wavelength in the optical, the $YJH$ light curves of SN 2005hk show a broad, single peak that is delayed significantly ($\sim$10 to 15 days) relative to the $B$ peak. The NIR contribution to the quasi-bolometric “UVOIR” flux seems not too dissimilar to normal SN Ia [@Stritzinger15], and this fraction seems roughly consistent even for the faintest SN Iax like SN 2008ha and SN 2010ae [@Stritzinger14]. A major surprise, however, is SN 2014dt, which showed a significant near- and mid-infrared excess beginning $\sim$100 days past maximum and lasting for a few hundred days at least [@2016ApJ...816L..13F].
The near-UV photometric behavior of SN Iax is also interesting, with objects showing a faster evolution in near-UV minus optical color than SN Ia, and “crossing” the typically parallel tracks made by normal SN Ia in this space. SN Iax start bluer than normal SN Ia in the UV before maximum light but quickly redden (by $\sim$ 1.5 to 2 mag in Swift $uvw1 - b$) so that about ten days after maximum they are redder than normal SN Ia [@2010ApJ...721.1627M].
As with other thermonuclear supernovae, no SN Iax has been definitively detected in the radio [@2016ApJ...821..119C] or X-ray [@2014ApJ...790...52M].
Spectroscopic properties {#specprops}
------------------------
Beyond the defining spectroscopic features used for classification of these supernovae ( dominated spectra near maximum light and low velocity; see Sec. \[sec:ident\]), SN Iax show quite homogeneous spectral evolution, which generally matches the evolution of SN Ia over the period of a few months from maximum light, except with lower line velocities [@Jha06]. Like SN Ia, the early-time spectra show Fe-group and intermediate mass elements (including Si, S, and Ca). This similarity extends to the near-UV (with Fe-group line blanketing). In their near-infrared maximum-light spectra, SN Iax show remarkable similarity to SN Ia, with and lines, except at lower typical velocities, and most prominently, beautiful and unambiguous detections of in the $H$ and $K$ bands for SN 2010ae, 2012Z, and 2014ck [@Stritzinger14; @Stritzinger15; @2016MNRAS.459.1018T].
It is at late times that the spectra of SN Iax radically diverge from SN Ia, and indeed, almost all other supernovae of any type [@Jha06; @Sahu08; @Foley10; @Foley16]. SN Iax never truly enter a fully “nebular” phase in which broad forbidden lines dominate the optical spectrum (Figure \[fig:late\]). Rather, in optical spectra taken more than a year past maximum light, SN Iax still show permitted lines of predominantly , often with low velocities $<$ 2000 [km s$^{-1}$]{}, plus D and the IR triplet. Forbidden lines of \[\], \[\], and \[\] are also usually present, and in some cases with narrow widths down to $<$ 500 [km s$^{-1}$]{} [@McCully14; @Stritzinger15]. The linewidths and relative strengths of the forbidden and permitted lines seem to vary significantly among different SN Iax [@Yamanaka15; @Foley16]. The late-time linewidth variation, for example, approaches nearly an order of magnitude, from a few hundred to $\sim$3000 [km s$^{-1}$]{}. Spectra of SN Iax seem to show relatively little evolution from 200 to past 400 days after maximum [@Foley16].
![Late-time spectra of SN Iax (colored) compared to a normal SN Ia (black; left), showing a clear divergence. Numerous lines that look like “noise” are actually permitted Fe transitions, as seen with the Syn++ [@Thomas11_syn] spectrum synthesis model (black) of SN 2005hk (upper right). SN Iax show a diversity of line velocities and strengths in the \[Fe II\] $\lambda 7155$, \[Ca II\] $\lambda\lambda 7291,7324$, and \[Ni II\] $\lambda 7378$ lines at late times. Figure panels adapted from @McCully14 and @Foley16.[]{data-label="fig:late"}](fig3-late.pdf){width="\textwidth"}
The faintest SN Iax, like SN 2008ha and SN 2010ae, show similar spectra to more luminous counterparts, perhaps with a more rapid spectral evolution to lower velocities in the few weeks after maximum light [@Foley09; @Valenti09; @Stritzinger14]. Around 250 days past maximum, the spectra of SN 2002cx [a “bright” SN Iax; @Jha06] and SN 2010ae (one of the faintest) are nearly identical [see Figure 12 of @Stritzinger14].
@Chornock06 and @Maund10 obtained spectropolarimetric observations of SN 2005hk and report 0.2–0.4% continuum polarization, consistent with spectropolarimetry of normal SN Ia.
Two objects, SN 2004cs and 2007J, have been classified as SN Iax by @Foley13 and show clear evidence of emission in their post-maximum spectra, something never seen in normal SN Ia. @White15 argue that these objects may be type-IIb supernovae instead, though @Foley16 counter that claim and call PTF 09ego and PTF 09eiy into question as SN Iax. In the end, there are a handful of objects for which the classification may be ambiguous.
Photometric and spectroscopic correlations {#photspec}
------------------------------------------
SN Iax span a wide range of peak luminosities and line velocities, and it is natural to ask if these are related. @McClelland10 suggested a positive correlation between these two, with the lowest-velocity SN Iax also being the lowest luminosity. While such a correlation does seem to hold for the majority of SN Iax, there are clear counterexamples like SN 2009ku [@Narayan11], which had a low velocity (similar to SN 2008ha or SN 2010ae), but relatively high luminosity (like SN 2002cx or SN 2005hk). @2016MNRAS.459.1018T show that SN 2014ck is similarly an outlier to the velocity/luminosity correlation. Neither do the lowest-velocity SN Iax necessarily have the fastest optical decline rates: while SN 2008ha and SN 2010ae decline quickly, SN 2014ck has an intermediate decline rate similar to other higher-velocity SN Iax, while SN 2009ku in fact has a slower decline rate than higher-velocity SN Iax.
Environments and Rates {#environments}
----------------------
SN Iax have a dramatically different distribution of host galaxies than normal SN Ia. In all but a couple of cases [like SN 2008ge; @Foley10_ge], SN Iax are found in star-forming, late-type host galaxies (Figure \[fig:gallery\]). The host galaxy distribution of SN Iax is closest to those of SN IIP or SN 1991T/1999aa-like SN Ia [@Foley09; @Valenti09; @Perets10]. @Lyman13 [@2017arXiv170704270L] confirm and amplify this result based on H$\alpha$ imaging and integral-field spectroscopy of Iax locations and hosts: SN Iax must arise from a relatively young population. Qualitatively, based on SN Iax with high-resolution Hubble Space Telescope like SN 2008A, SN 2012Z, and SN 2014dt, it seems as if SN Iax prefer the “outskirts” of their star-forming hosts, but this needs further quantification, given the selection bias against finding these fainter SN on a bright galaxy background. SN Iax show no strong preference for high- or low-metallicity galaxies [@Magee17], though their explosion locations are more metal-poor than normal SN Ia [@2017arXiv170704270L].
![Sloan Digital Sky Survey images centered at the locations of 25 SN Iax in the SDSS footprint. Note the preponderance of late-type, star-forming host galaxies. Each image is 100 arcsec on a side.[]{data-label="fig:gallery"}](fig4-gallery.png){width="\textwidth"}
The host reddening distribution of SN Iax is uncertain because of the difficulty in disentangling extinction from the intrinsic photometric diversity in the class, but most known SN Iax have low reddening, up to $E(B-V) \simeq 0.5$ mag for SN 2013en [@2015MNRAS.452..838L]. Again, selection biases work against finding heavily extinguished members of this already intrinsically faint class of supernovae.
Because the luminosity function of SN Iax extends down to quite faint magnitudes, precisely estimating the rate of SN Iax is challenging. @Foley13 calculate the Iax rate to be 31$^{+17}_{-13}$% of the SN Ia rate in a volume-limited sample. Consistent with this, @2017arXiv170307449M found one SN Iax (iPTF16fnm; $M \simeq -15$ mag) and 4 SN Ia in a volume-limited survey. The overall SN Iax rate is dominated by the lower luminosity objects; the rate of brighter SN Iax (comparable in luminosity to SN 2002cx or SN 2005hk) is likely to be between 2 and 10% of the SN Ia rate [@2011MNRAS.412.1441L; @Foley13; @2017ApJ...837..121G]. SN Iax are the most numerous “peculiar” cousins to normal SN Ia.
Progenitors and Remnants
------------------------
A major breakthrough in understanding SN Iax came with the discovery of the progenitor system of SN 2012Z [@McCully14_12Z]. Nature was kind: SN 2012Z exploded in the nearby galaxy NGC 1309, which was also the host of the normal type-Ia SN 2002fk, a calibrator for the SN distance scale to measure $H_0$ [@Riess11]. As such, extremely deep, multi-epoch HST imaging of NGC 1309 (to observe Cepheids) covered the location of SN 2012Z before its explosion (Figure \[fig:12Z\]). This deep, high-resolution pre-explosion imaging revealed a source coincident with SN 2012Z, the first time a progenitor system has been discovered for a thermonuclear supernova. The detected source is luminous and blue ($M_V \simeq -5.3$ mag; $B-V \simeq -0.1$ mag), and @McCully14_12Z argue that it is a helium-star companion (donor) to an exploding white dwarf. Further images taken after the supernova faded reveal the source has not disappeared, consistent with the companion scenario (McCully et al., in preparation). This discovery marks a critical contrast for SN Iax compared to normal SN Ia: no such progenitor system has ever been seen for a SN Ia! Note, however, there are only two normal SN Ia, SN 2011fe [@Li11] and SN 2014J [@Kelly14], with pre-explosion limits that are as deep as the data for SN 2012Z.
![Discovery of the only known progenitor system for a white-dwarf (thermonuclear) supernova. The left panel shows the deep Hubble Heritage image of NGC 1309 (made from observations to detect and monitor Cepheids) taken in 2005 and 2006 with HST ACS. Upper panels (b) and (c) zoom in on the region where the type Iax SN 2012Z was to explode, revealing the luminous, blue progenitor system S1, believed to be a helium star donor to an exploding white dwarf. Lower panels (d) and (e) show the region after the SN explosion, allowing for a precise measurement of its position with HST/WFC3, coincident with the progenitor. This figure is adapted from @McCully14_12Z.[]{data-label="fig:12Z"}](fig5-12Z){width="\textwidth"}
@Foley14 detect a luminous ($M_I \simeq -5.4$ mag), red ($R-I \simeq
1.6 \pm 0.6$ mag) source consistent with the location of SN 2008ha in HST imaging taken about 4 years after the supernova explosion. At these epochs the SN ejecta flux should have faded well below this level, so @Foley14 suggest they may be observing a companion star to the supernova, or else a luminous “remnant” of the explosion.
Models {#sec:models}
======
In this section I discuss potential models for SN Iax, starting from general considerations and observational constraints, and then moving to specific scenarios that have been proposed in the literature.
SN Iax are likely thermonuclear, white dwarf supernovae
-------------------------------------------------------
SN Iax show undeniable spectroscopic similarities to normal SN Ia, particularly near and in the few months after maximum light, with lower velocities being the primary distinguishing factor. Given that spectroscopic observations probe different layers of supernova ejecta over time[^2], a natural starting point would be to suggest that SN Iax and SN Ia share commonalities in their progenitors and explosions.
Conversely, the primarily star-forming environments of SN Iax may point to a core-collapse, massive-star supernova origin. Indeed @Valenti09 argue that SN 2008ha has similarities to some faint core-collapse SN, and suggest a “fallback” massive-star supernova [@Moriya10] or an electron-capture supernovae [@2009ApJ...705L.138P] could explain SN 2008ha, though not without some difficulties [@2013MNRAS.436..774E]. This core-collapse model does not seem to be able to account for higher luminosity SN Iax, and so would require objects like SN 2008ha, 2010ae, and 2010el to be distinct from other SN Iax.
The preponderance of evidence suggests that SN Iax are thermonuclear explosions of white dwarfs. Spectra near maximum light show carbon, intermediate mass elements like sulfur and silicon (weakly), and strong features of iron group elements. The infrared lines [@Stritzinger15; @2016MNRAS.459.1018T] clearly point to a fraternity with normal SN Ia. Iron lines are seen at a wide range of velocities, implying efficient mixing of fusion products rather than a highly layered structure. The lack of star formation or luminous massive stars in pre-explosion imaging of SN 2008ge [@Foley10_ge] and SN 2014dt [@Foley15], and the non-disappearance of the progenitor system flux in SN 2012Z [@McCully14_12Z] argue against the explosion of massive luminous stars. The peak luminosities of SN Iax compared to their late-time photometry and modeling suggest a $^{56}$Ni $\rightarrow\;^{56}$Co $\rightarrow\;^{56}$Fe radioactively powered light curve [@McCully14].
Moreover, even fainter, lower velocity objects like SN 2008ha and 2010ae seem to connect to brighter SN Iax. @Foley10 show an early spectrum of SN 2008ha that features sulfur lines similar to those seen in normal SN Ia. @Stritzinger14 show that SN 2010ae has strong lines like other SN Iax and normal SN Ia, and its late-time spectrum is very similar to the SN Iax prototype, SN 2002cx.
General observational constraints
---------------------------------
The environments of SN Iax do indeed suggest they come from a young population, but this does not require a core-collapse origin. SN 1991T-like SN Ia have a quite similar host galaxy preference [@Foley09; @Perets10] to SN Iax, and those SN Ia are still nearly-universally construed as white dwarf explosions. In fact, the problem can be turned around: the requirement for a young population favors certain binary systems that can produce and explode white dwarfs quickly. HST observations of nearby stars in the field of SN 2012Z yield ages of 10–50 Myr [@McCully14_12Z], while for SN 2008ha the nearby population is $\lesssim$ 100 Myr [@Foley14]. Though young, these are still significantly older than the expected lifetimes of, for example, Wolf-Rayet stars that might yield hydrogen-poor core-collapse supernovae [@Groh13].
Short evolutionary times in a binary system suggest that SN Iax arise from more massive white dwarfs. If the explosions are occurring at the Chandrasekhar mass ($M_{\rm Ch}$), as supported by other evidence (see below), then the quickest binary channel for a carbon/oxygen white dwarf (C/O WD) is to accrete helium from a He star companion [@1999ApJ...519..314H; @2014LRR....17....3P]. The stable mass transfer rate can be high for helium accretion, and @Claeys14 show that this channel dominates the thermonuclear SN rate between 40 Myr (with no younger systems) and 200 Myr (above which double-degenerate and hydrogen-accreting single degenerate systems dominate). This has been seen in several binary population synthesis studies; these generally find no problem for the He star + C/O WD channel to produce the required fraction of SN Iax relative to normal SN Ia, but the total rates may not quite reach the observed values .
Of course the WD+He star channel is in good accord with observations of SN 2012Z, for which the putative companion is consistent with a helium star. This scenario may also explain the helium observed in SN 2004cs and SN 2007J. In fact, @Foley13 predicted this type of system for SN Iax before the SN 2012Z progenitor discovery. @Liu10 present a model (though intended to explain a different kind of system) that starts with a 7 [$M_\odot$]{} + 4 [$M_\odot$]{} close binary that undergoes two phases of mass transfer and common envelope evolution and results in a 1 [$M_\odot$]{} C/O WD + 2 [$M_\odot$]{} He star. As the He star evolves, it can again fill its Roche lobe and begin stable mass transfer onto the white dwarf (at a high accretion rate, $\sim 10^{-5}$ [$M_\odot$]{} yr$^{-1}$, for instance) that could lead to the SN Iax.
The low ejecta velocities of SN Iax imply lower kinetic energy compared to SN Ia (under the reasonable assumption that SN Iax do not have significantly higher ejecta mass). Their lower luminosity also points in this direction (though that depends specifically on how much $^{56}$Ni is synthesized). Moreover, there is a much larger *variation* in the kinetic energy in SN Iax. All of this points towards a deflagration (subsonic) explosion; pure deflagration models of $M_{\rm Ch}$ C/O WDs show convoluted structure from the turbulent flame propagation [e.g., @Gamezo03; @2007ApJ...668.1118T] that can produce a wide range of explosion energies. This contrasts with $M_{\rm
Ch}$ detonation scenarios that lead to more uniform energy release and a layered structure [@Gamezo04; @Gamezo05].
Deflagrations are thought to naturally occur in the onset of runaway carbon burning for $M_{\rm Ch}$ C/O WD progenitors [e.g., @2011ApJ...740....8Z; @2012ApJ...745...73N and references therein]. Indeed for years, a leading model to match observations of normal SN Ia has been the delayed detonation scenario, in which an initial deflagration transitions to a detonation after the WD has expanded to lower density [@Khokhlov91; @Gamezo05]. In SN Iax, one posits that this transition does not occur. Such a model matches many of the observations [@2007ApJ...668.1132R; @2013ApJ...771...58M]: lower yet varied energy release, well-mixed composition [this inhibits the secondary near-infrared maximum; @Kasen06], unusual velocity structure [e.g., Ca interior of Fe, something not seen in normal SN Ia; @Foley13], and the strength of \[\] in late-time spectra implying stable nickel, preferentially produced at high density [near $M_{\rm Ch}$; @2016IJMPD..2530024M]. Early on, @Branch04 suggested a pure deflagration model to match spectra of SN 2002cx. In these models, there is a prediction of significant unburned material (C/O), which may be confirmed: carbon features are nearly ubiquitous in SN Iax, more so than SN Ia [@Foley13], and there are hints of low-velocity oxygen in some late-time spectra of SN Iax [@Jha06]. There remain challenges to a pure-deflagration model of SN Iax; for example, @2015ApJ...805..150F find generically too-weak deflagrations for $M_{\rm Ch}$ progenitors because of non-central, buoyancy driven ignition. Furthermore, asymmetry in pure deflagrations may lead to polarization signatures in excess of what is observed for SN 2005hk [@Chornock06; @Maund10; @2017ApJ...841...62M].
The low-velocity late-time features, which have final velocities that can be much less than the typical escape velocity from the surface of a white dwarf, suggest that perhaps not all of the material was unbound. Similarly, the range of energies that deflagrations produce could include outcomes less than the binding energy. Thus, in this model, though SN Iax would be $M_{\rm Ch}$ explosions, the ejecta mass could be significantly less, and the explosions could leave behind a bound remnant. The luminous remnant would be super-Eddington, and could be expected to drive an optically thick wind. This scenario might explain the high densities inferred at late times in SN Iax spectra and why they do not become completely nebular [$n \sim 10^9$ cm$^{-3}$; @McCully14]. Different amounts of ejecta versus wind material could furthermore explain the diversity in line widths and strengths between permitted and forbidden lines at late times [@Foley16]. A wind “photosphere” may also explain the luminous, red source seen in post-explosion observations of SN 2008ha [@Foley14] and perhaps is involved in producing the infrared excess in SN 2014dt [@2016ApJ...816L..13F]. @2017ApJ...834..180S present an intriguing model for these radioactively powered winds and show good agreement with SN Iax observations.
Specific models for SN Iax
--------------------------
In some sense, the pure-deflagration $M_{\rm Ch}$ model is a “failed” SN Ia; indeed the primary reason these models were first explored was to explain normal SN Ia. However, first in 1-d and later in 3-d, it was shown these were unlikely to match observations of SN Ia (see references listed above). After the identification and rapid observational growth in the SN Iax class, it became clear that these “failures” might be successfully applied to SN Iax.
@Jordan12 and @Kromer13 presented 3-d simulations of pure-deflagration $M_{\rm Ch}$ C/O WD explosions that did not fully unbind the star (Figure \[fig:kromer\]), and connected these to observed properties of SN Iax, as described above. @Fink14 explore a range of initial conditions in this scenario varying the number and location of ignition spots and find that they can yield a wide range of total energy and ejecta masses from 0.1 [$M_\odot$]{} to $M_{\rm Ch}$ (i.e., complete disruption). This provides a natural way to explain the diversity observed in SN Iax, and @Magee16 show some success in matching a particular realization of this explosion model to observations of SN 2015H. @2014ApJ...789..103L also show qualitatively similar results in being able to reproduce SN Iax properties, but find the opposite sense in the relation between energy yield and the number of ignition spots, with high luminosity objects resulting from fewer ignition points.
![Snapshots of a partial 3-d deflagration of a Chandrasekhar-mass carbon-oxygen white dwarf that leaves a bound remnant. Note the lack of burned material in the white dwarf core in the middle panel (1.5 sec into the explosion). While the thermonuclear runaway in the outer parts of the white dwarf surrounds and engulfs the core at later times, the explosion energy is not sufficient to unbind the star. The model predicts the composition of the ejecta and remnant in the table shown. This figure is adapted from @Kromer13.[]{data-label="fig:kromer"}](fig6-kromer){width="\textwidth"}
A new wrinkle on the pure-deflagration $M_{\rm Ch}$ scenario is based the idea of “hybrid” C/O/Ne white dwarfs [@2013ApJ...772...37D; @2014MNRAS.440.1274C]. Uncertainties in convective mixing and carbon-flame quenching may allow for central carbon to exist in WDs as massive as 1.3 [$M_\odot$]{}, rather than the traditional $\sim$1.05 [$M_\odot$]{}boundary between C/O and O/Ne white dwarfs [@1984ApJ...277..791N; @2017arXiv170306895D]. Such massive white dwarfs come from more massive progenitors and require less accreted material to reach $M_{\rm Ch}$: both of these lead to shorter delay time between formation and explosion (perhaps as low as 30 Myr), and thus could be particularly relevant to SN Iax [@2014ApJ...789L..45M; @2014ApJ...794L..28W]. Furthermore, a range of masses for the C/O core could play a role in SN Iax diversity [@2015MNRAS.447.2696D; @2015MNRAS.450.3045K]. suggest that even delayed detonations of hybrid white dwarfs could explain SN Iax, though @2016ApJ...832...13W find the detonation phase makes these explosions more similar to normal SN Ia. @2015ApJ...808..138L argue that from a binary evolution point of view the companion star to SN 2012Z is best explained in a system with a C/O/Ne white dwarf primary. @2017arXiv170306895D note some concerns about the viability of the hybrid white dwarf scenario, including whether the carbon flame can be successfully quenched [@2016ApJ...832...71L] or if the central C/O region can survive without being mixed into the O/Ne layer above [@2017ApJ...834L...9B].
Another class of SN Iax models explores the recent resurgence in sub-$M_{\rm
Ch}$ double-detonation scenarios for normal SN Ia. In this case varying WD mass at explosion can lead to diversity, and perhaps explain both prompt SN Ia and the full range of SN Iax [@Wang13; @2014RAA....14.1146Z; @2016AA...589A..43N]. @Stritzinger15 argue that the ejecta mass for SN 2012Z is consistent with $M_{\rm Ch}$ and advocate a pulsational delayed detonation model [@1995ApJ...444..831H] for bright SN Iax. @2012MNRAS.419..827M and @2013ApJ...763..108F explore the potential of SN Iax to result from white dwarf plus neutron star mergers.
Conclusion {#sec:conc}
==========
Taken together, observations and theory point to a leading model that needs to be tested: *a type Iax supernova results from a C/O or (hybrid C/O/Ne) white dwarf that accretes helium from a He-star companion, approaches the Chandrasekhar mass, and explodes as a deflagration that does not necessarily completely disrupt the star.* It is quite possible, even likely, that one or more aspects of this model is wrong, but it nonetheless gives observers something to directly test and modelers a general framework to explore and pick apart. Even within this model, there are important questions: is $M_{\rm Ch}$ always required? Is it always a deflagration? Does varying the ejecta/remnant mass explain the diversity? Does the WD + He-star channel always lead to a SN Iax?
In addition to testing this and other models with a broad range of observations, we can look forward to some novel possibilities. For example, what should happen to the He-star companion of SN 2012Z [e.g., @2013ApJ...765..150S; @2013ApJ...778..121L]? What might we expect to observe from SN Iax bound remnants and what happens to them? Will future extremely-large-telescopes be able to spectroscopically confirm that the companion to SN 2012Z ($m \approx 27.5$) was actually a helium star?
What are the broader impacts of our understanding of SN Iax? Is there a connection to systems like the Galactic “helium nova” V445 Pup, a near $M_{\rm Ch}$ white dwarf accreting from a helium star [@Kato08; @Woudt09]? Is it a SN Iax precursor, or is some parameter different (e.g., the accretion rate) that leads to the nova outcome? Is there a connection between SN Iax and 2002es-like SN [@Ganeshalingam12]? Those are found in older environments, but also have a detection of a likely single-degenerate progenitor [@Cao15]. What is the relation of SN Iax the population of Ca-rich transients [@Perets10]? Are those C/O + He WD mergers?
Of course, one of the key reasons why understanding SN Iax is important is the insight that gives us about normal SN Ia. But what is that insight telling us? Does it point to sub-$M_{\rm Ch}$ or double degenerate progenitors for SN Ia? Is some form of detonation required in SN Ia? Does the single degenerate channel always lead to peculiar supernovae? One factor in explaining why the SN Ia progenitor/explosion problem has been with us for decades is the vast array of possibilities to explode white dwarfs. Given the enormous effort that has gone into explaining normal SN Ia, it is astounding to think that we may have a better understanding of their peculiar cousins, SN Iax.
Cross-References {#cross-references .unnumbered}
================
Combustion in Thermonuclear Supernova Explosions [@Ropke17]\
Evolution of Accreting White Dwarfs to the Thermonuclear Runaway [@Starrfield2016]\
Explosion Physics of Thermonuclear Supernovae and Their Signatures [@Hoeflich2017]\
Light Curves of Type I Supernovae [@Bersten2017]\
Low- and Intermediate-Mass Stars [@Karakas2017]\
Nucleosynthesis in Thermonuclear Supernovae [@Seitenzahl2017]\
Observational and Physical Classification of Supernovae [@Gal-Yam2017]\
Population Synthesis of Massive Close Binary Evolution [@Eldridge2017]\
Spectra of Supernovae During the Photospheric Phase [@Sim2017]\
Supernova Progenitors Observed with HST [@VanDyk2016]\
The Extremes of Thermonuclear Supernovae [@Taubenberger17]\
Type Ia Supernovae [@Maguire2016]\
Unusual Supernovae and Alternative Power Sources [@Kasen2017]\
I thank Ryan Foley and Curtis McCully for our close collaboration working on SN Iax and their assistance with this manuscript. I am also grateful to Mark Sullivan, Daniela Graf, and Kerstin Beckert, for their seemingly inexhaustible patience. This work was supported in part by US National Science Foundation award 161545 and benefited greatly from discussions at the Munich Institute for Astro- and Particle Physics Scientific Program “The Physics of Supernovae” and associated Topical Workshop “Supernovae: The Outliers.” I dedicate this review to the memory of my friend and colleague, Weidong Li, who started it all.
[^1]: One speculates how the history of supernova cosmology would have changed if the diversity in SN Ia were not only typified by the extremes SN 1991T and SN 1991bg, but SN 1991bj as well, a SN Iax that would fall off the luminosity/light-curve decline rate relationship!
[^2]: “Spectrum is truth.” –R. P. Kirshner
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: |
We investigate the pure infiniteness and stable finiteness of the Exel-Pardo $C^*$-algebras $\mathcal{O}_{G,E}$ for countable self-similar graphs $(G,E,\varphi)$. In particular, we associate a specific ordinary graph $\widetilde{E}$ to $(G,E,\varphi)$ such that some properties such as simpleness, stable finiteness or pure infiniteness of the graph $C^*$-algebra $C^*(\widetilde{E})$ imply that of $\mathcal{O}_{G,E}$. Among others, this follows a dichotomy for simple $\mathcal{O}_{G,E}$: if $(G,E,\varphi)$ contains no $G$-circuits, then $\mathcal{O}_{G,E}$ is stably finite; otherwise, $\mathcal{O}_{G,E}$ is purely infinite.
Furthermore, Li and Yang recently introduced self-similar $k$-graph $C^*$-algebras $\mathcal{O}_{G,\Lambda}$. We also show that when $|\Lambda^0|<\infty$ and $\mathcal{O}_{G,\Lambda}$ is simple, then it is purely infinite.
address: |
Department of Mathematics\
Faculty of Mathematical Sciences and Computer\
Shahid Chamran University of Ahvaz\
P.O. Box: 83151-61357\
Ahvaz\
Iran
author:
- Hossein Larki
title: 'A dichotomy for simple self-similar graph $C^\ast$-algebras'
---
Introduction
============
In [@exe17], Exel and Pardo introduced self-similar graph $C^*$-algebras ${\mathcal{O}_{G,E}}$ to give a unified framework like graph $C^*$-algebras for the Katsura’s [@kat08] and Nekrashevych’s algebras [@nek04; @nek05]. These $C^*$-algebras were initially considered in [@exe17] only for countable discrete groups $G$ acting on finite graphs $E$ with no sources, and then generalized in [@bed17; @exe18] for larger classes. Roughly speaking, Exel and Pardo attached an inverse semigroup $\mathcal{S}_{G,E}$ and the tight groupoid $\mathcal{G}_{tight} (\mathcal{S}_{G,E})$ to $(G,E,\varphi)$ such that ${\mathcal{O}_{G,E}}\cong C^*(\mathcal{G}_{\mathrm{tight}}(\mathcal{S}_{G,E}))$, and then describe amenability [@exe17 Corollary 10.18], minimality [@exe17 Theorem 13.6], and effectivity (or topological principality) [@exe17 Corollary 14.15] of $\mathcal{G}_{\mathrm{tight}}(\mathcal{S}_{G,E})$, and thus simplicity and pure infiniteness of ${\mathcal{O}_{G,E}}$ [@exe17 Section 16], among others. Although only finite graphs are considered in [@exe17], but many arguments and proofs work for countable row-finite graphs with no sources (see [@exe18]).
The initial aim of this note comes from a dichotomy for simple groupoid $C^*$-algebras [@rai18; @bon18]. According to [@rai18 Theorem 4.7] and [@bon18 Corollary 5.13], a simple reduced $C^*$-algebra $C^*_r(\mathcal{G})$ of ample groupoid $\mathcal{G}$ with an almost unperforated type semigroup is either purely infinite or stable finite. We explicitly describe this dichotomy for self-similar graph $C^*$-algebras ${\mathcal{O}_{G,E}}$ by the underlying graphical properties. Here, we consider countable row-finite source-free graphs $E$ over an amenable (countable) group $G$ [@bed17; @exe18]. However, our results may be generalized to any countable graph $E$ by the desingularization of [@exe18].
We begin in Section 2 by reviewing necessary background on groupoid and self-similar graph $C^*$-algebras. Then, in Section 3, we generalize the Exel-Pardo’s characterization of purely infinite ${\mathcal{O}_{G,E}}$ to countable self-similar graphs by the groupoid approach (for not necessarily simple cases). Moreover, for certain self-similar graphs $(G,E,\varphi)$, we show that the $C^*$-algebra ${\mathcal{O}_{G,E}}$ is purely infinite and simple if and only if the additive monoid of nonzero Murray-von Neumann equivalent projections in $M_\infty({\mathcal{O}_{G,E}})$ is a group.
In Section 4, we focus on the stable finiteness of ${\mathcal{O}_{G,E}}$. We attach a spacial graph $\widetilde{E}$ to $(G,E,\varphi)$ such that some properties of ${\mathcal{O}_{G,E}}$- such as simplicity, pure infiniteness, and stable infiniteness- can be derived from those of the graph $C^*$-algebra $C^*(\widetilde{E})$. Then using known results about the graph $C^*$-algebras, we show that a simple $C^*$-algebra ${\mathcal{O}_{G,E}}$ is stable finite if and only if the underlying $(G,E,\varphi)$ contains no $G$-circuits. In particular, we deduce a dichotomy: A simple ${\mathcal{O}_{G,E}}$ is purely infinite if $(G,E,\varphi)$ has a $G$-circuit; otherwise, it is stable finite.
As the $k$-graph version of Exel-Pardo $C^*$-algebras, Li and Yang introduced self-similar $k$-graphs $(G,{\Lambda})$ and associated $C^*$-algebras $\mathcal{O}_{G,{\Lambda}}$. Briefly, by a groupoid approach, they investigated their properties such as nuclearity [@li18 Theorem 6.6(i)], amenability [@li18 Theorem 5.9], and simplicity [@li18 Theorem 6.6(ii)]. In Section 5, We investigate the pure infiniteness of $\mathcal{O}_{G,{\Lambda}}$ for the nonsimple cases. In particular, we modify and extend [@li18 Theorem 6.13].
[**Acknowledgement.**]{} The author appreciates Enrique Pardo for reviewing the initial version of the article and his helpful comments; in particular, for noting a gap in the proof of Theorem \[thm3.7\].
Preliminaries
=============
Groupoid $C^*$-algebras
-----------------------
We give here a brief introduction to ample groupoids and associated $C^*$-algebras; for more details see [@ren80; @ana00] for example. A [*groupoid*]{} is a small category ${\mathcal{G}}$ with inverses. The [*unit space of ${\mathcal{G}}$*]{} is the set of identity morphisms, that is ${\mathcal{G}}^{(0)}:=\{\alpha^{-1}\alpha:\alpha\in {\mathcal{G}}\}$. For each $\alpha\in {\mathcal{G}}$, we may define the range $r(\alpha):=\alpha\alpha^{-1}$ and the source $s(\alpha):=\alpha^{-1}\alpha$, which satisfy $r(\alpha)\alpha=\alpha=\alpha s(\alpha)$. Hence, for $\alpha,\beta\in {\mathcal{G}}$, the composition $\alpha\beta$ is well-defined in ${\mathcal{G}}$ if and only if $s(\alpha)=r(\beta)$. The [*isotropy subgroupoid of ${\mathcal{G}}$*]{} is defined by $$\mathrm{Iso}({\mathcal{G}}):=\{\alpha\in {\mathcal{G}}:s(\alpha)=r(\alpha)\}.$$
We work usually with groupoids ${\mathcal{G}}$ endowed with a topology such that the maps $r,s:{\mathcal{G}}\rightarrow {\mathcal{G}}^{(0)}$ are continuous (in this case, ${\mathcal{G}}$ is called a [*topological groupoid*]{}). A subset $B\subseteq {\mathcal{G}}$ is called a [*bisection*]{} if both restrictions $r|_B$ and $s|_B$ are homeomorphisms. We say that ${\mathcal{G}}$ is [*ample*]{} in case ${\mathcal{G}}$ has a basis of compact and open bisections.
Let ${\mathcal{G}}$ be a topological groupoid. We say that ${\mathcal{G}}$ is [*effective*]{} if the interior of $\mathrm{Iso}({\mathcal{G}})$ is just ${\mathcal{G}}^{(0)}$. Moreover, ${\mathcal{G}}$ is called [*topologically principal*]{} if $\{u\in {\mathcal{G}}^{(0)}: s^{-1}(u)\cap r^{-1}(u)=\{u\}\}$ is dense in ${\mathcal{G}}^{(0)}$.
Note that, when ${\mathcal{G}}$ is second-countable, [@ren08 Proposition 3.3] implies that ${\mathcal{G}}$ is effective if and only if it is topologically principal. In this paper, we will work frequently with second-countable effective ample groupoids.
We now recall the definition of reduced $C^*$-algebra $C^*_r({\mathcal{G}})$. Let ${\mathcal{G}}$ be an ample groupoid. We write $C_c({\mathcal{G}})$ for the complex vector space consisting of compactly supported continuous functions on ${\mathcal{G}}$, which is an $*$-algebra with the convolution multiplication and the involution $f^*(\alpha):=\overline{f(\alpha^{-1})}$. For each unit $u\in {\mathcal{G}}^{(0)}$ and ${\mathcal{G}}_u:=s^{-1}(\{u\})$, let $\pi_u:C_c({\mathcal{G}})\rightarrow B(\ell^2({\mathcal{G}}_u))$ be the left regular $*$-representation defined by $$\pi_u(f)\delta_\alpha:=\sum_{s(\beta)=r(\alpha)}f(\beta) \delta_{\beta \alpha} \hspace{5mm} (f\in C_c({\mathcal{G}}),~\alpha\in {\mathcal{G}}_u).$$ Then the [*reduced $C^*$-algebra $C^*_r({\mathcal{G}})$*]{} is the completion of $C_c({\mathcal{G}})$ under the reduced $C^*$-norm $$\|f\|_r:=\sup_{u\in {\mathcal{G}}^{(0)}}\|\pi_u(f)\|.$$ Moreover, there is a full $C^*$-algebra $C^*({\mathcal{G}})$ associated to ${\mathcal{G}}$, which is the completion of $C_c({\mathcal{G}})$ taken over all $\|.\|_{C_c({\mathcal{G}})}$-decreasing representations of ${\mathcal{G}}$. Hence, $C^*_r({\mathcal{G}})$ is a quotient of $C^*({\mathcal{G}})$, and [@ana00 Proposition 6.1.8] shows that they are equal if the underlying groupoid ${\mathcal{G}}$ is amenable.
We say that a $C^*$-algebra $A$ is [*purely infinite*]{} if every nonzero hereditary $C^*$-subalgebra of $A$ contains an infinite projection.
The following is analogous to [@bro15 Theorem 4.1] without the minimality assumption.
\[prop2.3\] Let $\mathcal{G}$ be a second-countable Hausdorff ample groupoid and let $\mathcal{B}$ be a basis of compact open sets for $\mathcal{G}^{(0)}$. Suppose also that $\mathcal{G}$ is effective. Then $C^*_r(\mathcal{G})$ is purely infinite if and only if $1_V$ is infinite in $C^*_r(\mathcal{G})$ for every $V\in \mathcal{B}$ ($1_V$ is the characteristic function of $V$).
The “only if" implication is immediate. For the converse, suppose that every $1_V$ in $C^*_r(\mathcal{G})$ is infinite for $V\in \mathcal{B}$. Let $A$ be a nonzero hereditary $C^*$-subalgebra of $C^*_r(\mathcal{G})$ and take some positive element $0\neq a\in A$. Using the hereditary property, we may follow the proof of [@lar19 Proposition 5.2] to find a projection $p\in A$ and some $V\in \mathcal{B}$ such that $p\sim 1_V$ in the Murray-von Nuemann sense. Since the infiniteness is preserved under $\sim$, then $p$ is an infinite projection, concluding the result.
Graph $C^*$-algebras
--------------------
Let $E=(E^0,E^1,r,d)$ be a directed graph with the vertex set $E^0$, the edge set $E^1$, and the range and domain maps $r,d:E^1\rightarrow E^0$. We say that $E$ is [*row-finite*]{} if each vertex receives at most finitely many edges. A [*source in $E$*]{} is a vertex $v\in E^0$ which receives no edges, i.e. $d^{-1}(v)=\emptyset$. We will write by $E^*$ the set of finite paths in $E$, that is $$E^*:=\bigcup_{n\geq 0}E^n=\bigcup_{n\geq 0}\{\alpha=e_1\ldots e_n: e_i\in E^1, d(e_i)=r(e_{i+1})\}.$$ Then one may extend $r,d:E^*\rightarrow E^0$ by defining $r(\alpha)=r(e_1)$ and $d(\alpha)=d(e_n)$ for every path $\alpha=e_1\ldots e_n\in E^n$. Throughout the paper, we will consider only countable directed graphs.
Given a directed graph $E$, a [*Cuntz-Krieger $E$-family*]{} is a collection $\{p_v,s_e:v\in E^0,e\in E^1\}$ of pairwise orthogonal projections $p_v$ and partial isometries $s_e$ with the following relations
1. $s_e^*s_e=p_{d(e)}$ for every $e\in E^1$,
2. $s_es_e^*\leq p_{r(e)}$ for every $e\in E^1$, and
3. $p_v=\sum_{d(e)=v}s_e s_e^*$ for all vertices $v$ with $0<|d^{-1}(v)|<\infty$.
The [*graph $C^*$-algebra*]{} $C^*(E)$ is the universal $C^*$-algebra generated by a Cuntz-Krieger $E$-family $\{p_v,s_e\}$ [@rae05]. By the above relations, for $e_1,\ldots,e_n\in E^1$, $s_{e_1}\ldots s_{e_n}$ is nonzero if and only if $\alpha:=e_1\ldots e_n$ is a path in $E$; in this case, we write $s_\alpha:=s_{e_1}\ldots s_{e_n}$.
Self-similar graphs and their $C^*$-algebras
--------------------------------------------
Let $G$ be a countable discrete group. An [*action*]{} $G\curvearrowright E$ is a map $G\times (E^0\cup E^1)\rightarrow E^0\cup E^1$, denoted by $(g,a)\mapsto ga$, such that the action of each $g\in G$ on $E$ gives a graph automorphism.
A [*self-similar graph*]{} is a triple $(G,E,\varphi)$ such that
1. $E$ is a directed graph,
2. $G$ acts on $E$ by automorphisms, and
3. $\varphi:G\times E^1\rightarrow G$ is a 1-cocycle for $G\curvearrowright E$ satisfying $\varphi(g,e)v=gv$ for every $g\in G$, $e\in E^1$, and $v\in E^0$.
According to [@exe17 Proposition 2.4], we may extend inductively the action $G\curvearrowright E$ and the cocycle $\varphi$ on the finite path space $E^*$ satisfying the desired relations [@exe17 Equation 2.6]. Indeed, if $\alpha=\alpha_1\alpha_2\in E^*$, then we define $$g \alpha=(g \alpha_1)(\varphi(g,\alpha_1)\alpha_2) \hspace{5mm} \mathrm{and} \hspace{5mm} \varphi(g,\alpha)=\varphi(\varphi(g,\alpha_1),\alpha_2).$$
Let $(G,E,\varphi)$ be a (countable) self-similar graph. Then ${\mathcal{O}_{G,E}}$ is the universal $C^*$-algebra generated by $$\{p_v,s_e:v\in E^0, e\in E^1\} \cup \{u_g p_v:g\in G,v\in E^0\}$$ satisfying the following properties:
1. $\{p_v,s_e:v\in E^0,e\in E^1\}$ is a Cuntz-Krieger $E$-family.
2. $u:G\rightarrow \mathcal{M}({\mathcal{O}_{G,E}})$, $g\mapsto u_g$, is a unitary $*$-representation of $G$ on the multiplier algebra $\mathcal{M}({\mathcal{O}_{G,E}})$.
3. $u_gp_v=p_{gv}u_g$ for every $g\in G$ and $v\in E^0$.
4. $u_gs_e=s_{ge}u_{\varphi(g,e)}$ for every $g\in G$ and $e\in E^1$.
We usually use the notation ${\mathcal{O}_{G,E}}$ instead of $\mathcal{O}_{(G,E,\varphi)}$ for convenience. Also, we will write each $u_g p_v$ by $u_{gv}$. Then one may easily verify relations (b)-(e) of [@exe18 Definition 2.2].
[**Standing assumption.**]{} All self-similar graphs $(G,E,\varphi)$ considered in this paper will be countable, row-finite and source-free.
The groupoid associated to $(G,E,\varphi)$
------------------------------------------
In [@exe17 Section 4], Exel and Pardo associated an inverse semigroup $\mathcal{S}_{G,E}$ to a self-similar graph $(G,E,\varphi)$ with finite graph $E$. They then showed that ${\mathcal{O}_{G,E}}\cong C^*_{tight}(\mathcal{S}_{G,E})\cong C^*({\mathcal{G}_{G,E}})$ where ${\mathcal{G}_{G,E}}$ is the groupoid of germs for the action of $\mathcal{S}_{G,E}$ on $E^\infty$ [@exe17 Corollary 6.4 and Proposition 8.4]. Note that the constructions of $\mathcal{S}_{G,E}$ and ${\mathcal{G}_{G,E}}$ in [@exe17] may be extended for countable row-finite, source-free self-similar graphs $(G,E,\varphi)$ with small modifications. We give a brief review of it here for convenience. So, fix a row-finite self-similar graph $(G,E,\varphi)$ without sources. Define the $*$-inverse semigroup $\mathcal{S}_{G,E}$ as $$\mathcal{S}_{G,E}=\{(\alpha,g,\beta):\alpha,\beta\in E^*,g\in G, d(\alpha)=gd(\beta)\}\cup \{0\}$$ with the operations $$(\alpha,g,\beta)(\gamma,h,\delta):=\left\{
\begin{array}{ll}
(\alpha,g\varphi(h,\varepsilon),\delta h\varepsilon) & \mathrm{if} ~~\beta=\gamma \varepsilon \\
(\alpha g\varepsilon,\varphi(g,\varepsilon)h,\delta) & \mathrm{if} ~~ \gamma=\beta\varepsilon\\
0 & \mathrm{otherwise}
\end{array}
\right.$$ and $(\alpha,g,\beta)^*:=(\beta,g^{-1},\alpha)$.
Let $E^\infty$ be the space one-sided infinite paths of the form $$x=e_1e_2\ldots \hspace{4mm} \mathrm{such ~ that} \hspace{5mm} d(e_i)=r(e_{i+1}) ~~~ \mathrm{for} ~~~ i\geq 1.$$ By [@exe17 Proposition 8.1], there is a unique action $G\curvearrowright E^\infty$ as follows: for each $g\in G$ and $x=e_1e_2\ldots \in E^\infty$, there is a unique infinite path $gx=f_1f_2\ldots$ such that $$f_1f_2\ldots f_n=g(e_1e_2\ldots e_n) \hspace{5mm} (\mathrm{for ~ all}~~ n\geq 1).$$ Moreover, we may consider the action of each $(\alpha,g,\beta)\in \mathcal{S}_{G,E}$ on $x=\beta \hat{x}\in E^\infty$ by $(\alpha,g, \beta)\cdot x=\alpha(g\hat{x})$. Then ${\mathcal{G}_{G,E}}$ is the groupoid of germs of the action of $\mathcal{S}_{G,E}$ on $E^\infty$, that is $${\mathcal{G}_{G,E}}=\big\{[\alpha,g,\beta;x]: x=\beta\hat{x}\big\}.$$ Recall that two germs $[s;x],[t;y]$ in ${\mathcal{G}_{G,E}}$ are equal if and only if $x=y$ and there exists an idempotent $0\neq e\in \mathcal{S}_{G,E}$ such that $e\cdot x=x$ and $se=te$. The unit space of ${\mathcal{G}_{G,E}}$ is $${\mathcal{G}_{G,E}}^{(0)}=\{[\alpha,1_G,\alpha;x]:x=\alpha \hat{x}\},$$ which is identified with $E^\infty$ by $[\alpha,1_G,\alpha;x]\mapsto x$. Then, the range and source maps are defined by $$r([\alpha,g,\beta;\beta\hat{x}])=\alpha(g\hat{x}) \hspace{5mm} \mathrm{and} \hspace{5mm} s([\alpha,g,\beta;\beta\hat{x}])=\beta\hat{x}.$$ Following [@exe17 Section 10], we endow ${\mathcal{G}_{G,E}}$ with the topology generated by compact open bisections of the form $$\Theta(\alpha,g,\beta;Z(\gamma)):=\{[\alpha,g,\beta;y]\in {\mathcal{G}_{G,E}}: y\in Z(\gamma)\}$$ where $\gamma\in E^*$ and $Z(\gamma):=\{\gamma x: x\in s(\gamma)E^\infty\}$. Hence, ${\mathcal{G}_{G,E}}$ is an ample groupoid.
We say that $(G,E,\varphi)$ is [*pseudo free*]{} if for every $g\in G$ and $e\in E^1$, $$ge=e \hspace{2mm} \mathrm{and} \hspace{2mm} \varphi(g,e)=1_G ~~~ \Longrightarrow ~~g=1_G.$$
In the end of this section, we recall briefly the following results from [@exe17] for convenience. Although they are proved there for finite self-similar graphs with no sources, but we can obtain them for countable cases by a same way (see also [@exe18]).
\[prop2.7\] Let $(G,E,\varphi)$ be a pseudo free self-similar graphs without sources and let ${\mathcal{G}_{G,E}}$ be the associated groupoid as above. Then
- ${\mathcal{G}_{G,E}}\cong {\mathcal{G}}_{tight}(\mathcal{S}_{G,E})$ [@exe17 Theorem 8.19], ${\mathcal{G}_{G,E}}$ is Hausdorff [@exe17 Proposition 12.1], and ${\mathcal{O}_{G,E}}\cong C^*({\mathcal{G}_{G,E}})$ [@exe17 Theorem 9.6].
- If moreover $G$ is an amenable group, then ${\mathcal{G}_{G,E}}$ is an amenable groupoid in the sense of [@ana00]. In particular, we have ${\mathcal{O}_{G,E}}\cong C^*({\mathcal{G}_{G,E}})\cong C_r^*({\mathcal{G}_{G,E}})$ by [@ana00 Proposition 6.1.8].
\[prop2.8\] Let $(G,E,\varphi)$ be a pseudo free self-similar graph with no sources. Then ${\mathcal{G}_{G,E}}$ is effective[^1] if and only if the following properties hold:
- Every $G$-circuit in $E$ has an entry, and
- for every $v\in E^0$ and $1_G\neq g\in G$, the action of $g$ on $Z(v)$ is nontrivial (i.e., there is $x\in Z(v)$ such that $g.x\neq x$).
Purely infinite self-similar graph $C^\ast$-algebras
====================================================
In [@exe17 Corollary 16.3] and [@exe18 Corollary 4.7], it is shown that when ${\mathcal{O}_{G,E}}$ is simple and $(G,E,\varphi)$ contains a $G$-circuit, then ${\mathcal{O}_{G,E}}$ is purely infinite. In this section, we study purely infinite $C^*$-algebras ${\mathcal{O}_{G,E}}$ of countable self-similar graphs in the sense of [@ror02] without the simplicity assumption. Our main result here is a generalization of [@exe17 Theorem 16.2] to countable self-similar graphs. Note that there is another well-known notion of pure infiniteness from [@kirch00] which is equivalent to that of [@ror02] for the simple cases. Moreover, our results in this section may be generalized for the Kirchberg-R[ø]{}rdam’s notion using [@kirch00 Corollary 3.15] and the ideal structure [@lal19 Corollary 6.15].
\[thm3.1\] Let $(G,E,\varphi)$ be a pseudo free self-similar graph over an amenable group $G$. Suppose also that $(G,E,\varphi)$ satisfies conditions (1) and (2) of Proposition \[prop2.8\] (i.e., the groupoid ${\mathcal{G}_{G,E}}$ is effective). Then ${\mathcal{O}_{G,E}}$ is purely infinite if and only if every vertex projection $s_v$ is infinite in ${\mathcal{O}_{G,E}}$.
We must prove the “if" implication only. So suppose that for every $v\in E^0$, $s_v$ is infinite in ${\mathcal{O}_{G,E}}$. Let $\mathcal{G}={\mathcal{G}_{G,E}}$ be the groupoid associated to $(G,E,\varphi)$. By Proposition \[prop2.7\](2), $\mathcal{G}$ is amenable, so $C^*_r(\mathcal{G})=C^*(\mathcal{G})={\mathcal{O}_{G,E}}$. We know that the cylinders $\{Z(\alpha):\alpha\in E^*\}$ is a basis of compact open sets for the topology induced on $E^\infty=\mathcal{G}^{(0)}$. Moreover, Proposition \[prop2.8\] says that $\mathcal{G}$ is effective. Hence, Proposition \[prop2.3\] implies that ${\mathcal{O}_{G,E}}=C^*_r(\mathcal{G})$ is purely infinite if and only if $\{1_{Z(\alpha)}=s_\alpha s_\alpha^*:\alpha\in E^*\}$ are all infinite projections in ${\mathcal{O}_{G,E}}$. Now since $s_\alpha s_\alpha^*\sim s_\alpha^*s_\alpha=s_{d(\alpha)}$ and the infiniteness passes through Murray-von Neumann equivalence, we conclude the result.
Let $v,w\in E^0$. We say that $v$ receives a [*$G$-path from $w$*]{} or $w$ [*connects to $v$ by a $G$-path*]{}, say $v\gtrsim w$, if there exist $\alpha\in E^*$ and $g\in G$ such that $r(\alpha)=v$ and $d(\alpha)=gw$. By [@exe17 Proposition 13.2], this is equivalent to $$\exists \alpha\in E^*, ~~ \exists g\in G \hspace{3mm} \mathrm{such~ that} \hspace{3mm} r(\alpha)=gv ~~\mathrm{and} ~~ d(\alpha)=w.$$
\[lem3.3\] Let $(G,E,\varphi)$ be a self-similar graph. For $v,w\in E^0$ and $\alpha,\beta\in E^*$, we have
- If $v=gw$ for some $g\in G$, then $s_v\sim s_w$ in the Murray-von Neumann sense.
- If $v$ receives a $G$-path from $w$, then $s_v\succsim s_w$.
- If $\beta=g\alpha$ for some $g\in G$, then $s_\beta s_\beta^* \sim s_\alpha s_\alpha^*$.
(1). If $v=gw$, then we have $s_v=(u_gs_v)^*(u_g s_v)$ and $$(u_gs_v)(u_g s_v)^*=(s_{gv}u_g)(s_{gv}u_g)^*=s_wu_gu_g^* s_w=s_w,$$ concluding $s_v\sim s_w$.
For (2), suppose that there exist $\alpha\in E^*$ and $g\in G$ such that $r(\alpha)=v$ and $d(\alpha)=gw$. Then, by the Cuntz-Krieger relations, $$s_v\geq s_\alpha s_\alpha^* \sim s_\alpha^* s_\alpha=s_{d(\alpha)}=s_{gw}\sim s_w,$$ and consequently $s_v \succsim s_w$.
For (3), if $\beta=g\alpha$, then by part (1) we have $$s_\beta s_\beta^*\sim s_\beta^* s_\beta=s_{d(\beta)}=s_{g.d(\alpha)}\sim s_{d(\alpha)}=s_\alpha^* s_\alpha \sim s_\alpha s_\alpha^*,$$ giving $s_\beta s_\beta^*\sim s_\alpha s_\alpha^*$.
\[prop3.4\] Let $(G,E,\varphi)$ be a pseudo free self-similar graph over an amenable group $G$. Suppose that conditions (1) and (2) of Proposition \[prop2.8\] hold. Then
- If every $v\in E^0$ receives a $G$-path from a $G$-circuit, then ${\mathcal{O}_{G,E}}$ is purely infinite.
- If the graph $C^*$-algebra $C^*(E)$ is purely infinite, then so is ${\mathcal{O}_{G,E}}$.
(1). In view of Theorem \[thm3.1\], it suffices to prove that each $s_v$ is infinite in ${\mathcal{O}_{G,E}}$. So, fix some $v\in E^0$. By hypothesis, there is a $G$-circuit $\alpha$ connecting to $v$ by a $G$-path.
We first show that $s_{r(\alpha)}$ is infinite. For, let $\gamma$ be an entry for $\alpha$ by assumption. Since each of $\alpha$ nor $\gamma$ is not a subpath of the other, one may compute that $s_\alpha s_\alpha^*$ and $s_\gamma s_\gamma^*$ are orthogonal. Hence, the Cuntz-Krieger relations imply that $$s_{r(\alpha)}\geq s_\alpha s_\alpha^*+s_\gamma s_\gamma^*> s_\alpha s_\alpha^*\sim s_\alpha^* s_\alpha=s_{d(\alpha)}.$$ If $d(\alpha)=gr(\alpha)$, then $s_{d(\alpha)}\sim s_{r(\alpha)}$ by Lemma \[lem3.3\](1), and whence $s_{r(\alpha)}$ is infinite in ${\mathcal{O}_{G,E}}$ as claimed.
Now, because there is a $G$-path from $r(\alpha)$ to $v$, we have $s_v\succsim s_{r(\alpha)}$ by Lemma \[lem3.3\](2), and therefore $s_v$ is infinite as well. As $v\in E^0$ was arbitrary, Theorem \[thm3.1\] follows the result.
(2). If $C^*(E)$ is purely infinite, then each $s_v$ is infinite in $C^*(E)$, and so is in ${\mathcal{O}_{G,E}}$ as well. Now apply Theorem \[thm3.1\].
If $v\in E^0$ receives a $G$-path from a $G$-circuit with an entry but not a path from a circuit, then $s_v$ is infinite in ${\mathcal{O}_{G,E}}$ while not in $C^*(E)$. Therefore, the converse of Proposition \[prop3.4\](2) does not necessarily hold.
In the simple case we conclude the following.
\[cor3.6\] Let $(G,E,\varphi)$ be a pseudo free self-similar graph over an amenable group $G$. Suppose that ${\mathcal{O}_{G,E}}$ is simple. If $E$ contains a $G$-circuit, then ${\mathcal{O}_{G,E}}$ is purely infinite.
Note that the simplicity of ${\mathcal{O}_{G,E}}$ gives conditions (1) and (2) in Proposition \[prop2.8\] [@exe18 Theorem 4.5]. So, by Theorem \[thm3.1\], it suffices to show that $s_v$ is infinite for each $v\in E^0$.
Let $(g,\alpha)$ be a $G$-circuit in $E$. By [@exe17 Theorem 16.1], $(g,\alpha)$ has an entry, hence $s_{r(\alpha)}$ is infinite as seen in the proof of Proposition \[prop3.4\](1).
Fix an arbitrary $v\in E^0$. We may form the infinite path $\alpha^\infty=\alpha(g\alpha)(g^2 \alpha)\cdots$, which is well-defined because $$d(g^n\alpha)=g^n d(\alpha)=g^n g r(\alpha)=r(g^{n+1}\alpha).$$ Since $E$ is also weakly $G$-transitive by [@exe18 Theorem 4.5], there is a $G$-path from $r(g^n\alpha)$ to $v$ for sufficiently large $n$. Note that as $r(g^n\alpha)=g^n r(\alpha)$, $s_{r(g^n \alpha)}=s_{g^n r(\alpha)}$ is infinite by Lemma \[lem3.3\](1). Also, Lemma \[lem3.3\](2) implies that $s_v\succsim s_{r(g^n \alpha)}\sim s_{r(\alpha)}$, and consequently $s_v$ is infinite too. As $v\in E^0$ was arbitrary, Theorem \[thm3.1\] concludes that ${\mathcal{O}_{G,E}}$ is purely infinite.
The converse of above corollary will be proved in Theorem \[thm4.9\] (1) $\Longleftrightarrow$ (6).
The following result gives necessary and sufficient criteria for the purely infinite simple $C^*$-algebras by the monoiod of equivalent projections. It is new even for the ordinary graph $C^*$-algebras. Before that we recall the definition of $K_0$-group of a unital $C^*$-algebra and establish some notations. Let $A$ be a unital $C^*$-algebra and write by $\mathcal{P}(A)$ the collection of all projections in $M_\infty(A)=\bigcup_{n\geq 1}M_n(A)$. We say that two projections $p\in M_m(A)$ and $q\in M_n(A)$ are equivalent, denoted by $p\sim q$, if $$\exists~ v\in M_{m,n}(A) \hspace{2mm} \mathrm{such ~ that} \hspace{2mm} p=v^*v \hspace{2mm} \mathrm{and} \hspace{2mm} q=v^*v.$$ Note that, if $m\leq n$, then $p\sim q$ if and only if $p\oplus 0_{n-m}$ is Murray-von Neumann equivalent to $q$ in $M_n(A)$, where $x\oplus y:=\mathrm{diag}(x,y)$. Define $\mathcal{D}(A):=\mathcal{P}(A)/\sim=\{[p]:p\in\mathcal{P}(A)\}$, which is an abelian monoid with the operation $[p]+[q]:=[p\oplus q]$. Then $K_0(A)$ is [*the Grothendieck group*]{} of $\mathcal{D}(A)$ endowed with a universal Grothendieck map $\phi:\mathcal{D}(A)\rightarrow K_0(A)$. The image of $\mathcal{D}(A)$ under $\phi$ is denoted by $K_0(A)^+$. It is known that when $\mathcal{D}(A)\setminus \{0\}$ is a group, then $K_0(A)=\mathcal{D}(A)\setminus\{0\}$.
\[thm3.7\]
- Let $E$ be an arbitrary directed graph (non necessarily row-finite, source-free, or even countable) with $|E^0|<\infty$. Then $C^*(E)$ is purely infinite and simple if and only if $\mathcal{D}(C^*(E))\setminus \{0\}$ is a group (or equivalently, $\mathcal{D}(C^*(E))\setminus \{0\}=K_0(C^*(E))$).
- Let $(G,E,\varphi)$ be a pseudo free self-similar graph over an amenable group $G$. Suppose also that $|E^0|<\infty$ and conditions (1) and (2) of Proposition \[prop2.8\] hold. Then ${\mathcal{O}_{G,E}}$ is purely infinite simple if and only if $\mathcal{D}({\mathcal{O}_{G,E}})$ is a group.
Note that the “only if" implications hold for every unital purely infinite simple $C^*$-algebra. Indeed, if $A$ is a purely infinite simple $C^*$-algebra, then nonzero projections of $A$ are all infinite. Thus, combining Proposition 1.5 and Theorem 1.4 of [@cun81] implies that $\mathcal{D}(A)\setminus \{0\}$ is a group ($=K_0(A)$).
So it is enough to prove the “if" parts. We first show that every projection $p$ in $A$ is infinite for any unital $C^*$-algebra $A$ with $\mathcal{D}(A)\setminus \{0\}$ a group. Indeed, if $[f]$ is the identity of $\mathcal{D}(A)\setminus\{0\}$, then $$[p]=[p]+[f]=[p\oplus f],$$ thus we have $$p\sim p\oplus 0<p\oplus f\sim p,$$ where $0$ is a zero matrix in $M_\infty(A)$. Therefore, $p$ is an infinite projection in $A$, as claimed.
In the case of statement (1), this follows that $E$ satisfies Condition (L). In fact if there exists a circuit in $E$ with no entries, then $C^*(E)$ contains an ideal Morita equivalent to $C(\mathbb{T})$, hence it has a finite projection. Recall that by Condition (L) every ideal of $C^*(E)$ has a (vertex) projection. Now take a nonzero ideal $I$ of $C^*(E)$ and some projection $0\neq p\in I$. As $|E^0|<\infty$, write $1:=\sum_{v\in E^0}s_v$ the unit of $C^*(E)$. Then $[p]+[1-p]=[1]$ and we have $$[p]=[1]+[q]=[1\oplus q],$$ where $[q]$ is the inverse of $[1-p]$ in $\mathcal{D}(C^*(E))\setminus \{0\}$. Therefore, $p\sim 1\oplus q$ which says that there is $x=[x_1 x_2 x_3 \ldots]\in M_{1,\infty}({\mathcal{O}_{G,E}})$ such that $x^* p x=1\oplus q$. In particular, $1=x_1^* p x_1\in I$, concluding $I=C^*(E)$. Therefore $C^*(E)$ is simple.
For the pure infiniteness, let $B$ be a nonzero hereditary $C^*$-subalgebra of $C^*(E)$. Again, Condition (L) gives a nonzero projection $p$ in $B$. If $[f]$ is the identity of $\mathcal{D}(C^*(E))\setminus\{0\}$, then $$[p]=[p]+[f]=[p\oplus f],$$ and we have $$p\sim p\oplus 0<p\oplus f\sim p$$ where $0$ is a zero matrix in $M_\infty({\mathcal{O}_{G,E}})$, and consequently $p$ is infinite. Therefore, $C^*(E)$ is purely infinite.
For statement (2), note that ${\mathcal{G}_{G,E}}$ is effective by Proposition \[prop2.8\], and ${\mathcal{O}_{G,E}}\cong C^*_r({\mathcal{G}_{G,E}})$ by Proposition \[prop2.7\]. This implies that every ideal of ${\mathcal{O}_{G,E}}$ contains a projection (see [@exe10 Theorem4.4] for example). Now we may follow the proof of statement (1) to obtain the result.
Stable finiteness and a dichotomy
=================================
In this section, we associate a special graph ${\widetilde{E}}$ to any self-similar graph $(G,E,\varphi)$. We show that if the graph $C^*$-algebra $C^*({\widetilde{E}})$ is either simple, purely infinite, or stable finite then so is ${\mathcal{O}_{G,E}}$ respectively. Then we will conclude a dichotomy for simple self-similar graph $C^*$-algebras.
Let $\mathbb{K}$ denote the $C^*$-algebra of compact operators on a separable, infinite dimensional Hilbert space. A (simple) $C^*$-algebra $A$ is called [*stably finite*]{} if $A\otimes \mathbb{K}$ contains no infinite projections.
Fix a self-similar graph $(G,E,\varphi)$. In the following we define a graph $\widetilde{E}$ associated to $(G,E,\varphi)$. Define $\approx$ on $E^*=\bigsqcup_{n=0}^\infty E^n$ by $$\alpha\approx \beta \hspace{5mm} \Longleftrightarrow \hspace{5mm} \exists g\in G ~~ \mathrm{such ~ that} ~~ \beta=g\alpha,$$ which is an equivalent relation on each $E^n$ (and so on $E^*$). The vertex set of ${\widetilde{E}}$ is ${\widetilde{E}}^0:=E^0/\approx$ the collection of vertex classes. In each class $[v] \in {\widetilde{E}}^0$ pick exactly one vertex up and collect them in the set $\Omega$. Hence, ${\widetilde{E}}^0=\{[v]:v\in \Omega\}$, and we have $[v]\ne [w]$ for $v\ne w\in \Omega$. For every $v\in \Omega$ and $e\in r^{-1}(v)$ draw an edge $\tilde{e}$ from $[d(e)]$ to $[v]$. Hence we obtain the graph ${\widetilde{E}}$ so that $$\begin{aligned}
{\widetilde{E}}^0&:=\{[v]:v\in \Omega\}, ~\mathrm{and}\\
{\widetilde{E}}^1&:=\bigcup_{v\in\Omega}\widetilde{r^{-1}(v)}=\bigcup_{v\in\Omega}\{\widetilde{e}:r(e)=v\},\end{aligned}$$ with the range $\widetilde{r}(\widetilde{e})=[r(e)]$ and domain $\widetilde{d}(\widetilde{e})=[d(e)]$ for every $\widetilde{e}\in {\widetilde{E}}^1$.
\[ex4.2\] For $n\geq1$, let $\mathbb{Z}_{\mathrm{mod}n}$ be the additive group $\{1,2,\ldots, n\}$. Let $(\mathbb{Z}_{\mathrm{mod}n},E,\varphi)$ be a triple with the cyclic graph $E$
\(v) at (0,0) [$v$]{}; (w1) at (0,2) [$w_1$]{}; (w2) at (1.7,1) [$w_2$]{}; (w3) at (1.7,-1) [$w_3$]{}; (wn) at (-1.7,1) [$w_n$]{};
() at (.8,1.3) [$f_1$]{}; () at (-.8,1.4) [$f_n$]{}; () at (-1.1,2) [$g_n$]{};
() at (1.45,-1.35) [$\cdot$]{}; () at (1.35,-1.5) [$\cdot$]{}; () at (1.2,-1.6) [$\cdot$]{};
() at (-1.93,.6) [$\cdot$]{}; () at (-1.99,.4) [$\cdot$]{}; () at (-2.01,.15) [$\cdot$]{};
\(v) edge node\[above=-15pt\] [$e_1\hspace{4mm}$]{} (w1); (v) edge node\[above=-5pt\] [$\hspace{-4mm}e_2$]{} (w2); (v) edge node\[above=-5pt\] [$$]{} (w3); (v) edge node\[above=-15pt\] [$e_n$]{} (wn); (w1) edge\[bend right\] node\[right=-5pt\] [$$]{} (w2); (w2) edge\[bend right\] node\[right=1pt\] [$g_1$]{} (w1); (w2) edge\[bend right\] node\[right=-4pt\] [$f_2$]{} (w3); (w3) edge\[bend right\] node\[right=1pt\] [$g_2$]{} (w2); (wn) edge\[bend right\] node\[right=-5pt\] [$$]{} (w1); (w1) edge\[bend right\] node\[left=0pt\] [$$]{} (wn);
and the action $\mathbb{Z}_{\mathrm{mod}n}\curvearrowright E$ defined by $$kv:=v \hspace{5mm} \mathrm{and} \hspace{5mm} k\alpha_i:=\alpha_{k+i} \hspace{10mm} (1\leq k,i \leq n),$$ for every $\alpha_i\in \{w_i, e_i, f_i, g_i\}$. Since $w_i\approx w_j$, for any $1\leq i,j\leq n$, we may select $w_1$ of the class $[w_1]=\{w_1,\ldots, w_n\}$. As $r^{-1}(v)=\{e_1,\ldots,e_n\}$ and $r^{-1}(w_1)=\{f_1,g_n\}$, then the graph ${\widetilde{E}}$ would be
\(v) at (0,0) [$[v]$]{}; (w1) at (0,2) [$[w_1]$]{};
() at (-.2,1) [$\cdots$]{};
(w1) ..controls (.7,1.3) and (.7,.7) .. node\[right=0pt\] [$\widetilde{e_1}$]{} (v); (w1) ..controls (.2,1.3) and (.2,.7) .. node\[right=-3pt\] [$\widetilde{e_2}$]{} (v); (w1) ..controls (-.7,1.3) and (-.7,.7) .. node\[left=1pt\] [$\widetilde{e_n}$]{} (v);
(w1) ..controls (2,1) and (2,3) .. node\[right=1pt\] [$\widetilde{f_1}$]{} (w1); (w1) ..controls (-2,1) and (-2,3) .. node\[left=0pt\] [$\widetilde{g_n}$]{} (w1);
\[lem4.3\] Let $(G,E,\varphi)$ be a self-similar graph, and consider an associated graph ${\widetilde{E}}$ as above. Then
- If $E$ is row-finite, then so is ${\widetilde{E}}$.
- For each finite path $\widetilde{\alpha}=\widetilde{\alpha}_1\ldots \widetilde{\alpha}_n\in {\widetilde{E}}^n$, there is a path $\gamma=\gamma_1\ldots \gamma_n$ in $E^n$ such that $\gamma\approx \alpha_i$ for $1\leq i\leq n$. Conversely, if $\gamma=\gamma_1\ldots \gamma_n\in E^n$, then there exists $\widetilde{\alpha}=\widetilde{\alpha}_1\ldots \widetilde{\alpha}_n\in {\widetilde{E}}^n$ such that $\gamma\approx \alpha_i$ for $1\leq i\leq n$.
- If $\widetilde{\alpha}\in {\widetilde{E}}^n$ and $\gamma\in E^n$ are two paths as in statement (2), then $\widetilde{\alpha}$ is a circuit in ${\widetilde{E}}$ if and only if $\gamma$ is a $G$-circuit in $E$. Moreover, $\widetilde{\alpha}$ has an entry if and only if $\gamma$ does.
Statement (1) is clear by the definition of ${\widetilde{E}}$. For (2), let first $\widetilde{\alpha}=\widetilde{\alpha}_1\ldots \widetilde{\alpha}_n\in {\widetilde{E}}^n$ be a path in ${\widetilde{E}}$. Then, for each $1\leq i<n$, we have $$[d(\alpha_i)]=\widetilde{d}(\widetilde{\alpha}_i)=\widetilde{r}(\widetilde{\alpha}_{i+1})=[r(\alpha_{i+1})],$$ and so there exists $g_i\in G$ such that $d(\alpha_i)=g_i r(\alpha_{i+1})$. Now set $\gamma_1:=\alpha_1$ and $\gamma_i:=g_1\ldots g_{i-1}\alpha_i$ for every $2\leq i\leq n$. Then $$d(\gamma_i)=d(g_1\ldots g_{i-1}\alpha_i)=g_1\ldots g_{i-1}d(\alpha_i)=g_1\ldots g_{i-1} g_i r(\alpha_{i+1})=r(\gamma_{i+1}),$$ and hence $\gamma=\gamma_1\ldots \gamma_n$ is a desired path in $E$.
Conversely, let $\gamma=\gamma_1\ldots \gamma_n$ be a finite path in $E^n$. For each $1\leq i\leq n$, there is $v_i\in \Omega$ such that $v_i=g_i r(\gamma_i)$ for some $g_i\in G$. Hence, we have $\widetilde{\alpha}=(\widetilde{g_1 \gamma_1})\ldots (\widetilde{g_n \gamma_n})\in {\widetilde{E}}$ with $\alpha \approx \gamma$.
For statement (3), given $\widetilde{\alpha}$ and $\gamma$ as in part (2), we have $$\begin{aligned}
\widetilde{\alpha} ~ \mathrm{is ~ a ~ circuit ~ in ~ } E &\Longleftrightarrow ~ ~ [d(\alpha_n)]=[r(\alpha_1)]\\
&\Longleftrightarrow ~ ~ d(\alpha_n)\approx r(\alpha_1)\\
&\Longleftrightarrow ~ ~ d(\gamma_n)\approx d(\alpha_n)\approx r(\alpha_1) \approx r(\gamma_1)\\
&\Longleftrightarrow ~ ~ \gamma ~ \mathrm{is ~ a ~} G\mathrm{-circuit}.\end{aligned}$$ Moreover, since $|r^{-1}(r(\gamma_i))|=|\widetilde{r}^{-1}(\widetilde{r}(\widetilde{\alpha}_i))|$ for each $1\leq i\leq n$, we have $$\begin{aligned}
\gamma \mathrm{ ~ has ~ an ~ entry} \hspace{2mm} &\Longleftrightarrow \hspace{2mm} |r^{-1}(r(\gamma_i))|>1 ~~ \mathrm{for ~ some~ } 1\leq i\leq n\\
&\Longleftrightarrow \hspace{2mm} |\widetilde{r}^{-1}(\widetilde{r}(\alpha_i))|>1 ~~ \mathrm{for ~ some~ } 1\leq i\leq n\\
&\Longleftrightarrow \hspace{2mm} \widetilde{\alpha} \mathrm{~has ~ an ~ entry ~ in ~ } \widetilde{E} .\end{aligned}$$
Let $(G,E,\varphi)$ be a self-similar graph. Following [@exe17 Definition 3.4], we say that $E$ is [*weakly $G$-transitive*]{} if for every $v\in E^0$ and $x\in E^\infty$, there exists a path $\alpha$ such that $d(\alpha)=x(n,n)$ for some $n\geq 0$ and $r(\alpha)=g v$ for some $g\in G$. If we have an ordinary graph $E$ (with the trivial group action), we say simply that $E$ is [*weakly transitive*]{}. Note that the weakly transitive is called [*cofinal*]{} in [@rae05].
\[lem4.5\] Let $(G,E,\varphi)$ be a self-similar graph, and associate a graph ${\widetilde{E}}$ as above. Then
- Every $G$-circuit in $E$ has an entry if and only if every circuit in ${\widetilde{E}}$ does.
- $E$ is weakly $G$-transitive if and only if ${\widetilde{E}}$ is weakly transitive.
Statement (1) follows from items (2) and (3) of Lemma \[lem4.3\]. For (2), let ${\widetilde{E}}$ be transitive. Take an arbitrary infinite path $x\in E^\infty$ and some $v\in E^0$. By item (2) in Lemma \[lem4.3\], there is $\widetilde{y}\in {\widetilde{E}}^\infty$ such that $y(0,n)\approx x(0,n)$ for every $n\geq 0$. By transitivity, there exists $\widetilde{\gamma}\in {\widetilde{E}}^*$ such that $\widetilde{r}(\widetilde{\gamma})=[v]$ and $\widetilde{d}(\widetilde{\gamma})=[y(n,n)]$ for some $n$. Hence, $v\approx r(\gamma)$ and $d(\gamma)\approx y(n,n)\approx x(n,n)$. This follows that $E$ is $G$-transitive. The converse is analogous.
\[prop4.6\] Let $(G,E,\varphi)$ be a self-similar graph over an amenable group $G$, and let ${\widetilde{E}}$ be an associated graph.
- In case the groupoid ${\mathcal{G}_{G,E}}$ is Hausdorff (see [@exe18 Theorem 4.2]), then ${\mathcal{O}_{G,E}}$ is simple if and only if
- the graph $C^*$-algebra $C^*({\widetilde{E}})$ is simple, and
- for $v\in E^0$ and $g\in G$, if the action of $g$ on the cylinder $Z(v)$ is trivial (i.e., $g x=x$ for every $x\in Z(v)$), then $g$ is slack at $v$.
- Suppose that $(G,E,\varphi)$ is pseudo free and for any $v\in E^0$ and $1_G\ne g\in G$, the action of $g$ on $Z(v)$ is nontrivial. If $C^*({\widetilde{E}})$ is purely infinite, then so is ${\mathcal{O}_{G,E}}$.
Statement (1) follows from Lemma \[lem4.5\] and [@exe18 Theorem 4.5]. For (2), if the graph $C^*$-algebra $C^*({\widetilde{E}})$ is purely infinite, then every circuit in ${\widetilde{E}}$ has an entry and every vertex $[v]\in {\widetilde{E}}^0$ can be reached from a circuit. By Lemma \[lem4.5\], every $G$-circuit has an entry and every $v\in E^0$ receives a $G$-path from a $G$-circuit. Now, Proposition \[prop3.4\](1) concludes that ${\mathcal{O}_{G,E}}$ is purely infinite.
The graph ${\widetilde{E}}$ in Example \[ex4.2\] is weakly transitive and every circuit in ${\widetilde{E}}$ has an entry. Then $C^*({\widetilde{E}})$ is simple and purely infinite, and so is the $C^*$-algebra ${\mathcal{O}_{G,E}}$ by Proposition \[prop4.6\].
Let $(G,E,\varphi)$ be a self-similar graph. A [*graph trace*]{} on $E$ is map $T:E^0\rightarrow \mathbb{R}^+$ such that
1. $T(r(e))\geq T(d(e))$ for every $e\in E^1$, and
2. $T(v)=\sum_{r(e)=v}T(d(e))$ for every $v\in E^0$.
A [*graph $G$-trace*]{} in $E$ is a graph trace $T:E^0\rightarrow \mathbb{R}^+$ such that $T(v)=T(w)$ for every $v\approx w$ in $E^0$.
\[thm4.9\] Let $(G,E,\varphi)$ be a pseudo free self-similar graph over an amenable group $G$. Suppose that ${\mathcal{O}_{G,E}}$ is simple. Then the following are equivalent.
- ${\mathcal{O}_{G,E}}$ is stably finite.
- ${\mathcal{O}_{G,E}}$ is quasi diagonal.
- $(G,E,\varphi)$ has a nonzero graph $G$-trace.
- ${\widetilde{E}}$ has a nonzero graph trace.
- ${\widetilde{E}}$ contains no circuits.
- $E$ contains no $G$-circuits.
Statements (1) and (2) are equivalent by [@rai18 Corollary 6.6].
\(1) $\Rightarrow$ (6). If $E$ has a $G$-circuit, then ${\mathcal{O}_{G,E}}$ is purely infinite by Corollary \[cor3.6\]. In particular, ${\mathcal{O}_{G,E}}$ is not stably finite, a contradiction.
\(6) $\Rightarrow$ (5) follows from Lemma \[lem4.3\](3).
\(5) $\Rightarrow$ (4). Suppose that ${\widetilde{E}}$ has no circuits. Arrange ${\widetilde{E}}^0=\{[v_1],[v_2],\ldots\}$. For each $n\geq 1$, let $F_n$ be the full subgraph of ${\widetilde{E}}$ containing all $\bigcup_{i=1}^n \widetilde{r}^{-1}([v_i])$. Since $F_n$’s have no circuits, [@kum98 Corollary 2.3] implies that $C^*(F_1)\subseteq C^*(F_2) \subseteq \ldots$ is a sequence of finite dimensional $C^*$-subalgebras of $C^*({\widetilde{E}})$ such that $C^*({\widetilde{E}})=\lim C^*(F_n)$ (i.e., $C^*({\widetilde{E}})$ is AF). Thus there exist bounded traces $\tau_n:C^*(F_n)\rightarrow \mathbb{C}$ such that $\tau_n|_{C^*(F_i)}$ equals with $\tau_i$ for $i\leq n$. This induces a semifinite trace $\tau=\lim \tau_n$ on $C^*({\widetilde{E}})$. Therefore, if $C^*({\widetilde{E}})=C^*(t_e,q_{[v]})$, we obtain the nonzero graph trace $T:{\widetilde{E}}^0\rightarrow {\mathbb{R}}^+$, by $T([v])=\tau(q_{[v]})$, on ${\widetilde{E}}$.
\(4) $\Rightarrow$ (3). Suppose that $T$ is a nonzero graph trace on ${\widetilde{E}}$. Note that, since the action of $G$ on $E^1$ gives automorphisms respecting to the range and domain, for any $v\neq w\in E^0$ with $w=g v$, the map $e\mapsto g e$ is a bijection from $r^{-1}(v)$ onto $r^{-1}(w)$. In particular, $|r^{-1}(w)|=|r^{-1}(v)|$. Being this fact in mind, one may easily see that the map $T':E^0\rightarrow {\mathbb{R}}^+$, defined by $T'(v):=T([v])$, is a nonzero graph $G$-trace on $E$, as desired.
\(3) $\Rightarrow$ (1). By [@ren80 Proposition II.4.8], there exists a faithful conditional expectation $\pi:C^*({\mathcal{G}_{G,E}})\rightarrow C_0({\mathcal{G}_{G,E}}^{(0)})$ such that $\pi(f)=f|_{{\mathcal{G}_{G,E}}^{(0)}}$ for all $f\in C_c({\mathcal{G}_{G,E}}^{(0)})$. Note that the isomorphism $\psi:{\mathcal{O}_{G,E}}\rightarrow C^*({\mathcal{G}_{G,E}})$ in Proposition \[prop2.7\](1) maps the core ${\mathcal{O}_{G,E}}^0:=\overline{\mathrm{span}}\{s_\alpha s_\alpha^*:\alpha\in E^*\}$ onto $C_0({\mathcal{G}_{G,E}}^{(0)})$. Hence $\varphi:=\psi^{-1}\circ \pi\circ \psi$ is a faithful conditional expectation from ${\mathcal{O}_{G,E}}$ onto ${\mathcal{O}_{G,E}}^0$ such that $$\phi(s_\alpha u_g s_\beta^*)=\left\{
\begin{array}{ll}
s_\alpha s_\alpha^* & \beta=\alpha,~ g=1_G \\
0 & \mathrm{otherwise}
\end{array}
\right.$$ for every $\alpha,\beta\in E^*$ and $g\in G$.
Now suppose that $T$ is a nonzero graph $G$-trace on $E$. Define $t:{\mathcal{O}_{G,E}}^0\rightarrow {\mathbb{C}}$ by $t(s_\alpha s_\alpha^*)=T(d(\alpha))$, which is a linear functional on ${\mathcal{O}_{G,E}}^0$. So, we may easily verify that $\tau:=t\circ \phi$ is a semifinite trace on ${\mathcal{O}_{G,E}}$ such that $0<\tau(s_v)<\infty$ for all $v\in E^0$. Moreover, $\tau$ is faithful because ${\mathcal{O}_{G,E}}$ is simple. Thus [@rai18 Corollary 6.6] yields that ${\mathcal{O}_{G,E}}$ is stably finite.
Recall from [@exe17 Corollary 10.16] that if $G$ is amenable, then ${\mathcal{O}_{G,E}}$ is a nuclear $C^*$-algebra. So, combining Corollary \[cor3.6\] and Theorem \[thm4.9\] implies the following dichotomy for simple ${\mathcal{O}_{G,E}}$.
\[cor4.10\] Let $(G,E,\varphi)$ be a pseudo free self-similar graph over an amenable group $G$. Suppose that ${\mathcal{O}_{G,E}}$ is simple. Then
- If $E$ has a $G$-circuit, then ${\mathcal{O}_{G,E}}$ is purely infinite. In this case, ${\mathcal{O}_{G,E}}$ is a Kirchberg algebra, and we have $K_0({\mathcal{O}_{G,E}})= D({\mathcal{O}_{G,E}})\setminus \{0\}$ whenever $|E^0|<\infty$.
- Otherwise, ${\mathcal{O}_{G,E}}$ is stably finite. In this case, $(K_0({\mathcal{O}_{G,E}}),K_0({\mathcal{O}_{G,E}})^+)$ is an ordered abelian group (see [@ror00 Proposition 5.1.5(iv)]).
Note that in case ${\mathcal{O}_{G,E}}$ is stably finite, the embedding $\iota:C^*(E)\hookrightarrow {\mathcal{O}_{G,E}}$ of [@exe17 Section 11] induces an embedding $K_0(\iota):K_0(C^*(E))\hookrightarrow K_0({\mathcal{O}_{G,E}})$ defined by $K_0(\iota)([p]_0):=[\iota(p)]_0$, where the map $\iota$ is naturally extended on $M_\infty(C^*(E))$ into $M_\infty({\mathcal{O}_{G,E}})$. Indeed, if $p\in M_\infty(C^*(E))$ is a projection with $[\iota(p)]_0=0$, then we must have $\iota(p)=0$ because $M_\infty({\mathcal{O}_{G,E}})$ has no infinite projection, and hence $p=0$.
Pure infiniteness of self-similar $k$-graph $C^*$-algebras
==========================================================
In this section, we consider the pure infiniteness of self-similar $k$-graph $C^*$-algebras. Let us first recall the definitions of self-similar $k$-graphs and their $C^*$-algebras from [@li18]. Fix $k\in{\mathbb{N}}\cup\{\infty\}$ and let ${\Lambda}=({\Lambda}^0,{\Lambda},r,s)$ be a row-finite $k$-graph with no sources (we refer the reader to [@rae05] for basic definitions and concepts about $k$-graphs and associated $C^*$-algebras). Consider ${\mathbb{N}}^k$ as a category with a single object $0$ and the coordinatewise partial order $\leq$. Let $\Omega_k:=\{(p,q):p,q\in {\mathbb{N}}^k, p\leq q\}$. An [*infinite path in ${\Lambda}$*]{} is a morphism $x:\Omega_k\rightarrow{\Lambda}$ with the range $r(x):=x(0,0)$. We write by ${\Lambda}^\infty$ the set of infinite paths in ${\Lambda}$.
Let $G$ be a (discrete and countable) group. [*An action $G\curvearrowright {\Lambda}$*]{} is a map $G\times {\Lambda}\rightarrow {\Lambda}$, $(g,{\lambda})\rightarrow g {\lambda}$, which gives a graph automorphism preserving the degree map for every $g\in G$.
\[defn5.1\] A [*self-similar $k$-graph*]{} is a triple $(G,{\Lambda},\varphi)$, where ${\Lambda}$ is a $k$-graph, $G$ is a group acting on ${\Lambda}$, and $\varphi:G\times {\Lambda}\rightarrow {\Lambda}$ is a cocycle for $G\curvearrowright {\Lambda}$ with the property $$\varphi(g,{\lambda}).v=g v \hspace{5mm} (g\in G,v\in {\Lambda}^0,{\lambda}\in {\Lambda}).$$
Following [@li18], we consider only self-similar $k$-graphs $(G,{\Lambda},\varphi)$ for [**row-finite**]{} and [**source-free**]{} $k$-graphs with $|{\Lambda}^0|<\infty$. We will write $(G,{\Lambda},\varphi)$ by $(G,{\Lambda})$ for simplicity. Note that $\varphi$ was called the restriction map in [@li18] and each $\varphi(g,{\lambda})$ was denoted by $g|_{\lambda}$ there.
Let $(G,{\Lambda})$ be a self-similar $k$-graph. We say that
1. $(G,{\Lambda})$ is [*pseudo free*]{}, if $g {\lambda}={\lambda}$ and $\varphi(g,{\lambda})=1_G$ imply $g=1_G$.
2. $(G,{\Lambda})$ is [*$G$-aperiodic*]{} if for any $v\in {\Lambda}^0$, there exists $x\in v{\Lambda}^\infty$ such that $x(p,\infty)=g x(q,\infty)$ implies $g=1_G$ and $p=q$ for $p,q\in{\mathbb{N}}^k$ and $g\in G$.
3. $(G,{\Lambda})$ is [*$G$-cofinal*]{} if for every $x\in {\Lambda}^\infty$ and $v\in{\Lambda}^0$, there exist $p\in{\mathbb{N}}^k$, $\mu\in{\Lambda}$, and $g\in G$ such that $s(\mu)=x(p,p)$ and $r(\mu)=g v$.
Let $(G,{\Lambda})$ be a self-similar $k$-graph as in Definition \[defn5.1\] with $|{\Lambda}^0|<\infty$. The $C^*$-algebra ${\mathcal{O}_{G,\Lambda}}$ associated to $(G,{\Lambda})$ is the universal $C^*$-algebra generated by $\{s_{\lambda}:{\lambda}\in {\Lambda}\}$ and $\{u_g:g\in G\}$ such that
1. $\{s_{\lambda}:{\lambda}\in {\Lambda}\}$ is a Cuntz-Krieger ${\Lambda}$-family in the sense of [@kum00].
2. $u:G\rightarrow {\mathcal{O}_{G,\Lambda}}$, given by $g\mapsto u_g$, is a unitary $*$-representation of $G$.
3. $u_gs_{\lambda}=s_{g {\lambda}}u_{\varphi(g,{\lambda})}$ for every $g\in G$ and ${\lambda}\in {\Lambda}$.
Similar to the construction of ${\mathcal{G}_{G,E}}$ in Section 2.4, Li and Yang associated an ample groupoid $\mathcal{G}_{G,{\Lambda}}$ in [@li18 Section 5.1] such that $\mathcal{O}_{G,{\Lambda}}\cong C^*(\mathcal{G}_{G,{\Lambda}})\cong C_r^*(\mathcal{G}_{G,{\Lambda}})$ when $G$ is amenable and $(G,{\Lambda})$ is pseudo free [@li18 Theorem 5.9]. In particular, the unit space $\mathcal{G}_{G,{\Lambda}}^{(0)}$ is homeomorphic to ${\Lambda}^\infty$ endowed with the topology generated by cylinders $Z({\lambda}):=\{{\lambda}x:x\in{\Lambda}^\infty\}$.
Recall that a [*circuit*]{} in ${\Lambda}$ is a path $\alpha\in {\Lambda}$ with $r(\alpha)=s(\alpha)$. $\tau\in {\Lambda}$ is called an [*entry*]{} for $\alpha$ if $r(\tau)=r(\alpha)$ and there are no common extensions for $\alpha$ and $\tau$ (i.e., $\alpha\mu\neq\tau\nu$ for all $\mu,\nu\in{\Lambda}$).
Let $(G,\Lambda)$ be a pseudo free self-similar $k$-graph with $|\Lambda^0|<\infty$ over an amenable group $G$. If $\Lambda$ is $G$-aperiodic, then $\mathcal{O}_{G,\Lambda}$ is purely infinite. In particular, if $\Lambda$ is also $G$-cofinal, then $\mathcal{O}_{G,\Lambda}$ is a Kirchberg algebra.
Let $\mathcal{G}_{G,\Lambda}$ be the groupoid associated to $(G,\Lambda)$. Then $\mathcal{G}_{G,\Lambda}$ is amenable and effective [@li18 Proposition 6.5], and we thus have $C^*(\mathcal{G}_{G,\Lambda})=C^*_r(\mathcal{G}_{G,\Lambda})=\mathcal{O}_{G,\Lambda}$ by [@li18 Theorem 5.9]. We know that the cylinders $\{Z({\lambda}):{\lambda}\in \Lambda\}$ form a basis of compact open sets for the topology on $\Lambda^\infty=\mathcal{G}_{G,\Lambda}^{(0)}$. So, in light of Proposition \[prop2.3\], it suffices to prove that each $1_{Z({\lambda})}$ is an infinite projection for ${\lambda}\in \Lambda$. For this, since $$1_{Z({\lambda})}=s_{\lambda}s_{\lambda}^*\sim s_{\lambda}^* s_{\lambda}=s_{s({\lambda})},$$ we show all $s_v$’s are infinite in $\mathcal{O}_{G,\Lambda}$ for $v\in \Lambda^0$.
So fix an arbitrary $v\in \Lambda^0$. We claim that $v$ reaches from a circuit with an entry. To see this, take some $x\in v\Lambda^\infty$. For any $t\in \mathbb{N}$, write $\textbf{t}:=(t,0,0,\ldots)\in \mathbb{N}^k$. Since $\{x(\textbf{t},\textbf{t}):t\geq 1\}\subseteq \Lambda^0$ is finite, there are $t_1<t_2$ such that $x(\textbf{t}_1,\textbf{t}_1)=x(\textbf{t}_2,\textbf{t}_2)$. Hence $x(\textbf{t}_1,\textbf{t}_2)$ is a circuit in $\Lambda$, which connects to $v$ by $x(0,\textbf{t}_1)\in{\Lambda}$. Note that the $G$-aperiodicity yields clearly the periodicity of ${\Lambda}$. Hence, one may follow [@lar18 Lemma 6.1] to find an (initial) circuit $\alpha$ with an entry $\tau$ connecting to $v$, as claimed.
Since $\alpha$ and $\tau$ have no common extensions, one may compute that $s_\alpha s_\alpha^*$ and $s_\tau s_\tau^*$ are orthogonal (by applying [@rae05 Lemma 9.4]). Thus, by the Cuntz-Krieger relations we have $$s_{r(\alpha)}\geq s_\alpha s_\alpha^*+ s_\tau s_\tau^*>s_\alpha s_\alpha^*\sim s_\alpha^* s_\alpha=s_{s(\alpha)}=s_{r(\alpha)},$$ so $s_{r(\alpha)}$ is infinite. Moreover, if $\lambda$ connects $r(\alpha)$ to $v$, then $$s_v\geq s_\lambda s_\lambda^*\sim s_\lambda^*s_\lambda=s_{s(\lambda)}=s_{r(\alpha)},$$ which says that $s_v$ is an infinite projection in $\mathcal{O}_{G,\Lambda}$ as well. Since $v\in \Lambda^0$ was arbitrary, this deduces that $\mathcal{O}_{G,\Lambda}$ is purely infinite by Proposition \[prop2.3\].
For the last statement, if moreover $\Lambda$ is $G$-cofinal, then [@li18 Theorem 6.6] implies that $\mathcal{O}_{G,\Lambda}$ is nuclear and simple, which satisfies UCT. Hence, $\mathcal{O}_{G,\Lambda}$ is a Kirchberg algebra.
Let $(G,\Lambda)$ be a pseudo free self-similar $k$-graph with $|\Lambda^0|<\infty$ over an amenable group $G$. Whenever $\mathcal{O}_{G,\Lambda}$ is simple, then it is purely infinite too.
[99]{}
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[^1]: Note that the ‘effective’ property of groupoids is called [*essentially principal*]{} in [@exe17; @exe18].
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{
"pile_set_name": "ArXiv"
}
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---
abstract: '[ This paper deals with bathymetry-oriented optimization in the case of long waves with small amplitude. Under these two assumptions, the free-surface incompressible Navier-Stokes system can be written as a wave equation where the bathymetry appears as a parameter in the spatial operator. Looking then for time-harmonic fields and writing the bottom topography as a perturbation of a flat bottom, we end up with a heterogeneous Helmholtz equation with impedance boundary condition. In this way, we study some PDE-constrained optimization problem for a Helmholtz equation in heterogeneous media whose coefficients are only bounded with bounded variation. We provide necessary condition for a general cost function to have at least one optimal solution. We also prove the convergence of a finite element approximation of the solution to the considered Helmholtz equation as well as the convergence of discrete optimum toward the continuous ones. We end this paper with some numerical experiments to illustrate the theoretical results and show that some of their assumptions could actually be removed. ]{}'
address:
- '$^\star$ Université de La Réunion, Laboratoire PIMENT, 117 Avenue du Général Ailleret, 97430 Le Tampon, France'
- '$^\dagger$ CEREMADE, CNRS, UMR 7534, Université Paris-Dauphine, PSL University'
- '$^*$ INRIA Paris, ANGE Project-Team, 75589 Paris Cedex 12, France and Sorbonne Université, CNRS, Laboratoire Jacques-Louis Lions, 75005 Paris, France'
author:
- 'Pierre-Henri COCQUET$^\star$'
- 'Sebastián RIFFO$^\dagger$'
- 'Julien Salomon$^*$'
bibliography:
- 'bathymetry\_bib.bib'
title: '[Optimization of bathymetry for long waves with small amplitude]{}'
---
Introduction
============
Despite the fact that the bathymetry can be inaccurately known in many situations, wave propagation models strongly depend on this parameter to capture the flow behavior, which emphasize the importance of studying inverse problems concerning its reconstruction from free surface flows. In recent years a considerable literature has grown up around this subject. A review from Sellier identifies different techniques applied for bathymetry reconstruction [@Sellier2016 Section 4.2], which rely mostly on the derivation of an explicit formula for the bathymetry, numerical resolution of a governing system or data assimilation methods [@Honnorat2009; @Beach-Wizard].
[[ An]{}]{} alternative is to use the bathymetry as control variable of a PDE-constrained optimization problem, an approach used in coastal engineering due to mechanical constraints associated with building structures and their interaction with sea waves. For instance, among the several aspects to consider when designing a harbor, building defense structures is essential to protect it against wave impact. These can be optimized to locally minimize the wave energy, by studying its interaction with the reflected waves [@IAMB]. Bouharguane and Mohammadi [@MB2; @MB1] consider a time-dependent approach to study the evolution of sand motion at the seabed, which could also allow these structures to change in time. In this case, the proposed functionals are minimized using sensitivity analysis, a technique broadly applied in geosciences. From a mathematical point of view, the solving of these kinds of problem is mostly numerical. A theoretical approach applied to the modeling of surfing pools can be found in [@DB-KdV; @NDZ], where the goal is to maximize locally the energy of the prescribed wave. The former proposes to determine a bathymetry, whereas the latter sets the shape and displacement of an underwater object along a constant depth.
In this paper, we address the determination of a bathymetry from an optimization problem, where a reformulation of the Helmholtz equation acts as a constraint. Even though this equation is limited to describe waves of small amplitude, it is often used in engineering due to its simplicity, which leads to explicit solutions when a flat bathymetry is assumed.
To obtain such a formulation, [we rely]{} on two asymptotic approximations of the free-surface incompressible Navier-Stokes equations. The first one is based on a long-wave theory approach and reduces the Navier-Stokes system to the Saint-Venant equations. The second one [[ considers]{}]{} waves of small amplitude from which the Saint-Venant model can be approximated by a wave-equation involving the bathymetry in its spatial operator. [It is finally]{} when considering time-harmonic solution of this wave equation that we get a Helmholtz equation with spatially-varying coefficients. Regarding the assumptions on the bathymetry to be optimized, [we assume the latter to be]{} a perturbation of a flat bottom with a compactly supported perturbation which can thus be seen as a scatterer. Since we wish to be as closed to real-world applications as possible, [we also assume]{} that the bottom topography is not smooth and, for instance, can be discontinuous. We therefore end up with a constraint equation given by a time-harmonic wave equation, namely a Helmholtz equation, with non-smooth coefficients.
It is worth noting that our bathymetry optimization problem aims at finding some parameters in our PDE that minimize a given cost function and can thus be seen as a parametric optimization problem (see e.g. [@Kunish_book_2012], [@allaire2007conception], [@haslinger2003introduction]). Similar optimization problems can also be encountered when trying to identify some parameters in the PDE from measurements (see e.g. [@Chen_BV_1999],[@beretta2018reconstruction]). Nevertheless, all the aforementioned references deals with real elliptic and coercive problems. Since the Helmholtz equation is unfortunately a complex and non-coercive PDE, these results do not apply.
We also emphasize that the PDE-constrained optimization problem studied in the present paper falls into the class of so-called topology optimization problems. For practical applications involving Helmholtz-like equation as constraints, we refer to [@wadbro2010shape],[@bernland2018acoustic] where the shape of an acoustic horn is optimized to have better transmission efficiency and to [@jensen2005topology],[@christiansen2019acoustic],[@christiansen2019designing] for the topology optimization of photonic crystals where several different cost functions are considered. Although there is a lot of applied and numerical studies of topology optimization problems involving Helmholtz equation, there are only few theoretical studies as pointed out in [@Haslinger_2015 p. 2].
Regarding the theoretical results from [@Haslinger_2015], [the authors]{} proved existence of optimal solution to their PDE-constrained optimization problem as well as the convergence of the discrete optimum toward the continuous ones. It is worth noting that, in [[ this paper]{}]{}, [ a relative permittivity is considered as optimization parameter]{} and that the latter appears as a multiplication operator in the Helmholtz differential operator. Since in the present study the bathymetry is assumed to be non-smooth and is involved in the principal part of our heterogeneous Helmholtz equation, we can not rely on the theoretical results proved in [@Haslinger_2015] to study our optimization problem.
[This paper is organized as follows: Section \[sec:wave-model\] presents the two approximations of the free-surface incompressible Navier-Stokes system, namely the long-wave theory approach and next the reduction to waves with small amplitude, that lead us to consider a Helmholtz equation in heterogeneous media where the bathymetry]{} plays the role of a scatterer. Under suitable assumptions on the cost functional and the admissible set of bathymetries, in Section \[sec:optimization\] we are able to prove the continuity of the control-to-state mapping and the existence of an optimal solution, in addition to the continuity and boundedness of the resulting wave presented in Section \[sec:continuity\]. The discrete optimization problem is discussed in Section \[sec:disc\_pb\], studying the convergence to the discrete optimal solution as well as the convergence of a finite element approximation. Finally, we present some numerical results in Section \[sec:numerics\].
Derivation of the wave model {#sec:wave-model}
============================
We start from the Navier-Stokes equations to derive the governing PDE. However, due to its complexity, we introduce two approximations [@LM]: a small relative depth (*Long wave theory*) combined with an infinitesimal wave amplitude (*Small amplitude wave theory*). An asymptotic analysis on the relative depth shows that the vertical component of the depth-averaged velocity is negligible, obtaining the Saint-Venant equations. After neglecting its convective inertia terms and linearizing around the sea level, it results in a wave equation which depends on the bathymetry. Since a variable sea bottom can be seen as an obstacle, we reformulate the equation as a *Scattering problem* involving the Helmholtz equation.
From Navier-Stokes system to Saint-Venant equations
---------------------------------------------------
For $t\geq 0$, we define the time-dependent region $$\Omega_t = \{(x,z)\in \Omega\times{ \mathbb{R} }\; \vert\; -z_b(x) \leq z \leq \eta(x,t)\}$$ where $\Omega$ is a bounded open set with Lipschitz boundary, $\eta(x,t)$ represents the water level and $-z_b(x)$ is the bathymetry or bottom topography, a time independent and negative function. The water height is denoted by $h = \eta +z_b$.\
-2.5 Ł[10]{} (0,0) – (Ł,0) node \[below\] [$x$]{}; (0,) –(0,) node\[right\] [$z$]{};
plot \[domain=0:360\*2, samples=144, smooth\] (,[wave()]{}); at (Ł,0.8) ;
5.3 (,) – (,0); (,0) – (,); at (,/2)[$\eta(x,t)$]{}; at (,/2)[$-z_b(x)$]{};
; (,) – (,); (,) – (,); (,) – (,); at (,/2)[$h$]{};
(0,) – (,); (,) – (Ł,/3); at (Ł,0.75\*) ;
In what follows, we consider an incompressible fluid of constant density (assumed to be equal to 1), governed by the Navier-Stokes system $$\left\{
\begin{aligned}
{\frac{\partial \mathbf{u}}{\partial t}} + \left(\mathbf{u}\cdot \nabla\right)\mathbf{u}
&= { {\rm div}\left(\sigma_{T}\right) } +\mathbf{g}
&&\textrm{ in }\Omega_t,\\
{ {\rm div}\left(\mathbf{u}\right) }
&= 0
&&\textrm{ in }\Omega_t,\\
\mathbf{u}
&= \mathbf{u}_0
&&\textrm{ in }\Omega_0,
\end{aligned}\right. \label{Euler}$$ where $\mathbf{u} = (u,v,w)^{{\top}}$ denotes the velocity of the fluid, $\mathbf{g} = (0,0,-g)^{\top}$ is the gravity and $\sigma_{T}$ is the total stress tensor, given by $$\sigma_{T} = -p\mathbb{I} +\mu\left({\nabla \mathbf{u}}+ {\nabla \mathbf{u}}^{\top}\right)$$ with $p$ the pressure and $\mu$ the coefficient of viscosity.
To complete , we require suitable boundary conditions. Given the outward normals $$n_s = \dfrac{1}{\sqrt{1+{\left\vert {\nabla \eta} \right\vert}^2}}
\begin{pmatrix}-{\nabla \eta}\\1 \end{pmatrix},\;
n_b = \dfrac{1}{\sqrt{1+{\left\vert {\nabla z_b} \right\vert}^2}}
\begin{pmatrix} {\nabla z_b} \\ 1\end{pmatrix},$$ to the free surface and bottom, respectively, we recall that the velocity of the two must be equal to that of the fluid: $$\left\{
\begin{aligned}
{\frac{\partial \eta}{\partial t}} -\mathbf{u}\cdot n_s
&= 0
&&\textrm{on } (x,\eta(x,t),t),\\
\mathbf{u}\cdot n_b
&= 0
&&\textrm{on } (x,-z_b(x),t).
\end{aligned} \right.\label{flux}$$ On the other hand, the stress at the free surface is continuous, whereas at the bottom we assume a no-slip condition $$\left\{
\begin{aligned}
\sigma_{T}\cdot n_s &= -p_a n_s
&&\textrm{on } (x,\eta(x,t),t), \\
(\sigma_{T} n_b)\cdot t_b &= 0 &&\textrm{on } (x,-z_b(x),t),
\end{aligned}\right. \label{stress}$$ with $p_a$ the atmospheric pressure and $t_b$ an unitary tangent vector to $n_b$.
A long wave theory approach can then be developed to approximate the previous model by a Saint-Venant system [@Perthame]. Denoting by $H$ the relative depth and $L$ the characteristic dimension along the horizontal axis, this approach is based on the approximation $\varepsilon := \dfrac{H}{L}\ll 1$, leading to a hydrostatic pressure law for the non-dimensionalized Navier-Stokes system, and a vertical integration of the remaining equations. For the sake of completeness, details of this derivation in our case are given in Appendix. For a two-dimensional system , the resulting system is then $$\begin{aligned}
{\frac{\partial \eta}{\partial t}}\sqrt{1+(\varepsilon\delta)^2 {\left\vert {\frac{\partial \eta}{\partial x}} \right\vert}^2} +{\frac{\partial (h_{\delta}\overline{u})}{\partial x}}
&= 0 \label{mass-av2} \\
{\frac{\partial (h_{\delta}\overline{u})}{\partial t}} +\delta{\frac{\partial (h_{\delta}\overline{u}^2)}{\partial x}}
&= -h_{\delta}{\frac{\partial \eta}{\partial x}} +\delta u(x,\delta\eta,t){\frac{\partial \eta}{\partial t}}\bigg(\sqrt{1+(\varepsilon\delta)^2{\left\vert {\frac{\partial \eta}{\partial x}} \right\vert}^2}-1\bigg) \nonumber\\
&\qquad +{\mathcal{O}}(\varepsilon) +{\mathcal{O}}(\delta\varepsilon),\label{momentum-av2} \end{aligned}$$ where $\delta := \dfrac{A}{H}$, $h_{\delta} = \delta\eta +z_b$ and $\overline{u}(x,t) := \dfrac{1}{h_{\delta}(x,t)}\int_{-z_b}^{\delta\eta}u(x,z,t)dz$. If $\varepsilon\rightarrow 0$, we recover the classical derivation of the one-dimensional Saint-Venant equations.
Small amplitudes
----------------
With respect to the classical Saint-Venant formulation, passing to the limit $\delta\rightarrow 0$ is equivalent to neglect the convective acceleration terms and linearize the system (\[mass-av2\]-\[momentum-av2\]) around the sea level $\eta = 0$. In order to do so, we rewrite the derivatives as $${\frac{\partial (h_{\delta}\overline{u})}{\partial t}}
= h_{\delta}{\frac{\partial \overline{u}}{\partial t}} +\delta{\frac{\partial \eta}{\partial t}}\overline{u},\;
{\frac{\partial (h_{\delta}\overline{u})}{\partial x}}
= \delta{\frac{\partial (\eta\overline{u})}{\partial x}} +{\frac{\partial (z_b\overline{u})}{\partial x}},$$ and then, taking $\varepsilon,\delta\rightarrow 0$ in (\[mass-av2\]-\[momentum-av2\]) yields $$\left\{
\begin{aligned}
{\frac{\partial \eta}{\partial t}} +{\frac{\partial (z_b\overline{u})}{\partial x}}
&= 0, \\
-{\frac{\partial (z_b\overline{u})}{\partial t}} +z_b{\frac{\partial \eta}{\partial x}}
&= 0.
\end{aligned}\right.$$ Finally, after deriving the first equation with respect to $t$ and replacing the second into the new expression, we obtain the wave equation for a variable bathymetry. All the previous computations hold for the three-dimensional system . In this case, we obtain $$\label{wave-eq}
{\frac{\partial^2 \eta}{\partial t^2}} -{ {\rm div}\left(gz_b{\nabla \eta}\right) } = 0.$$
Helmholtz formulation
---------------------
Equation defines a time-harmonic field, whose solution has the form $\eta(x,t)={\operatorname{Re}\{\psi_{tot}(x){\mathrm{e}^{-{\mathrm{i}}\omega t}}\}}$, where the amplitude $\psi_{tot}$ satisfies $$\label{ampli-helmholtz1}
\omega^2 \psi_{tot} +{ {\rm div}\left(gz_b{\nabla \psi_{tot}}\right) } = 0.$$
We wish to rewrite the equation above as a scattering problem. Since a variable bottom $z_b(x):= z_0 +\delta z_b(x)$ (with $z_0$ a constant describing a flat bathymetry and $\delta z_b$ a perturbation term) can be considered as an obstacle, we thus assume that $\delta z_b$ has a compact support in $\Omega$ and that $\psi_{tot}$ satisfies the so-called Sommerfeld radiation condition. In a bounded domain as $\Omega$, we impose the latter thanks to an impedance boundary condition (also known as first-order absorbing boundary condition), which ensures the existence and uniqueness of the solution [@Nedelec p. 108]. We then reformulate as $$\left\{
\begin{aligned}
{ {\rm div}\left((1+q) {\nabla \psi_{tot}}\right) } +k_0^2\psi_{tot} &= 0 && \textrm{ in }\Omega,\\
{{\nabla (\psi_{tot}-\psi_{0})} \cdot \hat{n}} -{\mathrm{i}}k_0(\psi_{tot}-\psi_{0}) &= 0 && \textrm{ on }\partial\Omega,
\end{aligned}\right.
\label{ampli-helmholtz}$$ where [[ we have introduced the parameter $q(x)\vcentcolon=\frac{\delta z_b(x)}{z_0}$ which is assumed to be compactly supported in $\Omega$]{}]{}, $k_0 \vcentcolon=\frac{\omega}{\sqrt{gz_0}}$, $\hat{n}$ the unit normal to $\partial \Omega$ and $\psi_{0}(x)=\mathrm{e}^{{\mathrm{i}}k_0x\cdot \vec{d}}$ is an incident plane wave propagating in the direction $\vec{d}$ (such that $|\vec{d}|=1$).
Decomposing the total wave as $\psi_{tot} = \psi_{0} + \psi_{sc}$, where $\psi_{sc}$ represents an unknown scattered wave, we obtain the Helmholtz formulation $$\label{eq:ampli-helmholtz_sc}
\left\{
\begin{aligned}
{ {\rm div}\left((1+q) {\nabla \psi}_{sc}\right) } +k_0^2\psi_{sc} &= -{ {\rm div}\left(q {\nabla \psi_0}\right) } && \textrm{ in }\Omega,\\
{{\nabla \psi_{sc}} \cdot \hat{n}} -{\mathrm{i}}k_0\psi_{sc} &= 0 && \textrm{ on }\partial\Omega.
\end{aligned}\right.$$ Its structure will be useful to prove the existence of a minimizer for a PDE-constrained functional, as discussed in the next section.
Description of the optimization problem {#sec:optimization}
=======================================
We are interested in studying the problem of a cost functional constrained by the weak formulation of a Helmholtz equation. The latter intends to generalize the equations considered so far, whereas the former indirectly affects the choice of the set of admissible controls. These can be discontinuous since they are included in the space of functions of bounded variations. In this framework, we treat the continuity and regularity of the associated control-to-state mapping, and the existence of an optimal solution to the optimization problem.
Weak formulation {#sub:weak_form}
----------------
Let $\Omega\subset{ \mathbb{R} }^2$ be a bounded open set with Lipschitz boundary. We consider the following general Helmholtz equation $$\label{eq:Helmholtz}
\left\{
\begin{aligned}
-{ {\rm div}\left((1+q) {\nabla \psi}\right) } -k_0^2\psi &= { {\rm div}\left(q {\nabla \psi_0}\right) } && \textrm{ in }\Omega,\\
(1+q){{\nabla \psi} \cdot \hat{n}} -{\mathrm{i}}k_0\psi &= g -q{{\nabla \psi_0} \cdot \hat{n}} && \textrm{ on }\partial\Omega,
\end{aligned}\right.$$ where $g$ is a source term. We assume that $q\in L^{\infty}(\Omega)$ and that there exists $\alpha>0$ such that $$\label{eq:hyp_q}
\mathrm{a.e.}\ x\in\Omega,\ 1+q(x)\geq \alpha.$$
\[rem:Link\_general\_Helmholtz\_scatt\] Here we have generalized the models described in the previous section: if $q$ has a fixed compact support in $\Omega$, we have that the total wave $\psi_{tot}$ satisfying is a solution to (\[eq:Helmholtz\]) with $g={{\nabla \psi_{0}} \cdot \hat{n}} -{\mathrm{i}}k_0\psi_{0}$ and no volumic right-hand side; whereas the scattered wave $\psi_{sc}$ satisfying is a solution to with $g=0$. All the proofs obtained in this broader setting still hold true for both problems.
A weak formulation for is given by $$\label{eq:FV_Helmholtz}
a(q;\psi,\phi)=b(q;\phi),\ \forall \phi\in H^1(\Omega),$$ where $$\begin{aligned}
a(q;\psi,\phi)
&\vcentcolon=\int_\Omega \left((1+q)\nabla\psi \cdot \nabla\overline{\phi}-k_0^2 \psi\overline{\phi}\right)\, dx-{\mathrm{i}}k_0\int_{\partial\Omega} \psi\overline{\phi}\, d\sigma,\label{def:a}\\
b(q;\phi)
&\vcentcolon= -\int_\Omega q \nabla \psi_0\cdot \nabla\overline{\phi}\, dx +{\langle g,\overline{\phi} \rangle}_{H^{-1/2},H^{1/2}}. \nonumber\end{aligned}$$ Note that, thanks to Cauchy-Schwarz inequality, the sesquilinear form $a$ is continuous $$\begin{aligned}
|a(q;\psi,\phi)| &\leq C(\Omega,q,\alpha) (1+{\left\lVert q \right\rVert}_{L^\infty(\Omega)}) {\left\lVert \psi \right\rVert}_{1,k_0}{\left\lVert \phi \right\rVert}_{1,k_0},\nonumber\\
{\left\lVert \psi \right\rVert}^2_{1,k_0}&\vcentcolon=k_0^2{\left\lVert \psi \right\rVert}_{L^2(\Omega)}^2+\alpha{\left\lVert \nabla \psi \right\rVert}^2_{L^2(\Omega)},\nonumber$$ where $C(\Omega,q,\alpha)>0$ is a generic constant. In addition, taking $\phi=\psi$ in the definition of $a$, it satisfies a Gårding inequality $$\label{eq:Garding_ineq}
{\operatorname{Re}\{a(q;\psi,\psi)\}}+2k_0^2{\left\lVert \psi \right\rVert}^2_{L^2(\Omega)} \geq {\left\lVert \psi \right\rVert}^2_{1,k_0},$$ and the well-posedness of Problem follows from the Fredholm Alternative. Finally, uniqueness holds for any $q\in L^\infty(\Omega)$ satisfying owning to [@Graham_variable_2018 Theorems 2.1, 2.4].
Continuous optimization problem
-------------------------------
We are interested in solving the next PDE-constrained optimization problem $$\label{eq:Optim_pbm}
\begin{aligned}
\textrm{minimize } & J(q,\psi),\\
\textrm{subject to } & (q,\psi)\in U_\Lambda\times H^1(\Omega), \textrm{ where } \psi \textrm{ satisfies } (\ref{eq:FV_Helmholtz}).
\end{aligned}$$ We now define [[ the set $U_\Lambda$]{}]{} of admissible $q$. We wish to find optimal $q$ that can have discontinuities and we thus cannot look for $q$ in some Sobolev spaces that are continuously embedded into $C^0(\overline{\Omega})$, even if such regularity is useful for proving [[ existence of minimizers]{}]{} (see e.g. [@Kunish_book_2012 Chapter VI], [@Bastide_2018 Theorem 4.1]). To be able to find [[ an]{}]{} optimal $q$ satisfying (\[eq:hyp\_q\]) and having possible discontinuities, we follow [@Chen_BV_1999] and introduce the following set $$U_\Lambda=\left\{ q\in BV(\Omega)\ \left| \ \alpha-1\leq q(x)\leq \Lambda\ a.e. \ x\in\Omega \right. \right\}.$$ Above $\Lambda\geq \max\{\alpha -1,0\}$ and $BV(\Omega)$ is the set of functions with bounded variations [@BV_properties], that is functions whose distributional gradient [[ belongs]{}]{} to the set $\mathcal{M}_\mathrm{b}(\Omega,{ \mathbb{R} }^N)$ of bounded Radon [[ measures]{}]{}. Note that the piecewise constant functions over $\Omega$ belong to $U_\Lambda$.
Some useful properties of $BV(\Omega)$ can be found in [@BV_properties] and are recalled below for the sake of completeness. This is a Banach space for the norm (see [@BV_properties p. 120, Proposition 3.2]) $${\left\lVert q \right\rVert}_{BV(\Omega)}\vcentcolon={\left\lVert q \right\rVert}_{L^1(\Omega)} + |D q|(\Omega),$$ where $D$ is the distributional gradient and $$\label{def:Dq}
|D q|(\Omega)=\sup\left\{\int_\Omega q\, { {\rm div}\left(\varphi\right) }\, dx\ \left|\ \varphi\in \mathcal{C}^1_\mathrm{c}(\Omega,{ \mathbb{R} }^2)\ \mathrm{and}\ {\left\lVert \varphi \right\rVert}_{L^\infty(\Omega)}\leq 1 \right.\right\},$$ is the variation of $q$ (see [@BV_properties p. 119, Definition 3.4]).
The weak$^*$ convergence in $BV(\Omega)$, denoted by $$q_n\rightharpoonup q,\ \mathrm{weak}^* \ \mathrm{in}\ BV(\Omega),$$ means that $$q_n\rightarrow q\ \mathrm{in}\ L^1(\Omega)\ \mathrm{and}\ D q_n \rightharpoonup D q\ \mathrm{in} \ \mathcal{M}_\mathrm{b}(\Omega,{ \mathbb{R} }^N).$$ Also, in a two-dimensional setting, the continuous embedding $ BV(\Omega)\subset L^1(\Omega) $ is compact. We finally recall that the application $q\in BV(\Omega)\mapsto |Dq|(\Omega)\in { \mathbb{R} }^+$ is lower semi-continuous with respect to the weak$^*$ topology of $BV$. Hence, for any sequence $q_n\rightharpoonup q$ in $BV(\Omega)$, one has $$|Dq|(\Omega)\leq \liminf_{n\to+\infty} |Dq_n|(\Omega).$$
The set $U_{\Lambda}$ is a closed, weakly$^*$ closed and convex subset of $BV(\Omega)$. However, since its elements are not necessarily bounded in the $BV$-norm, [[ we add a penalizing distributional gradient term to the cost functional $J(q,\psi)$ to prove the existence of a minimizer to Problem . In this way, we introduce the set of admissible parameters]{}]{} $$U_{\Lambda,\kappa}=\left\{ q\in U_{\Lambda}\ | \ |Dq|(\Omega)\leq \kappa\right\}$$ which also possesses the aforementioned properties. [[ Note that choosing $U_{\Lambda}$ or $U_{\Lambda,\kappa}$]{}]{} affects the convergence analysis of the discrete optimization problem, topic discussed in Section \[sec:disc\_pb\].
\[rem:U\_epsilon\] In this paper, we are interested in computing either the total wave satisfying or the scattered wave solution to Equation . Since this [[ requires]{}]{} to work with $q$ having a fixed compact support in $\Omega$, we also introduce the following set of admissible parameters $$\widetilde{U}_{\varepsilon}\vcentcolon=\left\{ q\in U\ | q(x)=0\ \mathrm{for\ a.e}\ x\in\mathcal{O}_\varepsilon \right\},\ \mathcal{O}_\varepsilon=\left\{x\in\Omega\ |\ \mathrm{dist}(x,\partial\Omega)\leq \varepsilon \right\},$$ which is a set of bounded functions with bounded variations that have a fixed support in $\Omega$. We emphasize that [[ this set]{}]{} is a convex, closed and weak-$*$ closed subset of $BV(\Omega)$. As a [[ consequence]{}]{}, all the theorems we are going to prove also hold for this set of admissible parameters.
Continuity of the control-to-state mapping
------------------------------------------
In this section, [[ we establish]{}]{} the continuity of the application $q\in U\mapsto \psi(q)\in H^1(\Omega)$ where $\psi(q)$ satisfies Problem (\[eq:FV\_Helmholtz\]). We assume that $U\subset BV(\Omega)$ is [[ a given weakly$^*$ closed set satisfying]{}]{} $$\forall q\in U,\ \mathrm{a.e.\ }x\in\Omega,\ \alpha-1\leq q(x)\leq \Lambda.$$ Note that both $U_\Lambda$, $U_{\Lambda,\kappa}$ and $\widetilde{U}_{\varepsilon}$ (see Remark \[rem:U\_epsilon\]) also satisfy these two assumptions. The next result consider the dependance of the stability constant with respect to the optimization parameter $q$.
\[thm:Uniform\_H1\_bound\] Assume that $q\in U$ and $\psi\in H^1(\Omega)$. Then there exists a constant $C_{\mathrm{s}}(k_0)>0$ that does not depend on $q$ such that $$\label{eq:inf_sup_cont}
{\left\lVert \psi \right\rVert}_{1,k_0}\leq C_{\mathrm{s}}(k_0) \sup_{{\left\lVert \phi \right\rVert}_{1,k_0}=1}|a(q;\psi,\phi)|,$$ where the constant $C_{\mathrm{s}}(k_0)>0$ only depend on the wavenumber and on $\Omega$. In addition, if $\psi$ is the solution to (\[eq:FV\_Helmholtz\]) then it satisfies the bound $$\label{eq:Uniform_H1_bound}
{\left\lVert \psi \right\rVert}_{1,k_0}\leq C_{\mathrm{s}}(k_0)C(\Omega)\max\{k_0^{-1},\alpha^{-1/2}\}\left({\left\lVert q \right\rVert}_{L^\infty(\Omega)}{\left\lVert \nabla \psi_0 \right\rVert}_{L^2(\Omega)}+{\left\lVert g \right\rVert}_{H^{-1/2}(\partial\Omega)} \right),$$ where $C(\Omega)>0$ only depends on the domain.
[[ The existence and uniqueness of a solution to Problem follows from [@Graham_variable_2018 Theorems 2.1, 2.4].]{}]{}
The proof of proceed by contradiction assuming [[ this inequality to be]{}]{} false. Therefore, we suppose there exist sequences $(q_n)_n\subset U$ and $(\psi_n)_n\subset H^1(\Omega)$ such that ${\left\lVert q_n \right\rVert}_{BV(\Omega)}\leq M$, ${\left\lVert \psi_n \right\rVert}_{1,k_0}=1$ and $$\label{eq:Contradiction_stab}
\lim_{n\to+\infty}\sup_{{\left\lVert \phi \right\rVert}_{1,k_0}=1}|a(q_n;\psi_n,\phi)|=0.$$ The compactness of the embeddings $BV(\Omega)\subset L^1(\Omega)$ and $H^1(\Omega)\subset L^2(\Omega)$ yields the existence of a subsequence (still denoted $(q_n,\psi_n)$) such that $$\label{eq:Convergence_contradiction}
\psi_n\rightharpoonup \psi_\infty\ \mathrm{in}\ H^1(\Omega),\
\psi_n\rightarrow \psi_\infty\ \mathrm{in}\ L^2(\Omega)\ \mathrm{and}\ q_n\rightarrow q_\infty\in U\ \mathrm{in}\ L^1(\Omega).$$ Compactness of the trace operator implies that $\displaystyle{\lim_{n\to+\infty}\psi_n|_{\partial\Omega}=\psi_\infty|_{\partial\Omega}}$ holds strongly in $L^2(\partial\Omega)$ and thus, from we get $$\begin{aligned}
\lim_{n\to+\infty} \int_\Omega k_0^2 \psi_n\overline{\phi}\, dx+{\mathrm{i}}k_0\int_{\partial\Omega} \psi_n\overline{\phi}\, d\sigma
&=\!\int_\Omega k_0^2 \psi_\infty\overline{\phi}\, dx+{\mathrm{i}}k_0\int_{\partial\Omega} \psi_\infty\overline{\phi}\, d\sigma,\ \forall\ v\in H^1(\Omega), \\
\lim_{n\to+\infty}\int_\Omega \nabla \psi_n\cdot \nabla\overline{\phi}\, dx
&=\!\int_\Omega \nabla \psi_\infty\cdot \nabla\overline{\phi}\, dx.\end{aligned}$$ [[ We now pass to the limit in the term of $a$ that involves $q_n$, see ]{}]{}. We start from $$\begin{aligned}
(q_n\nabla \psi_n,\nabla\overline{\phi})_{L^2(\Omega)}-(q_\infty\nabla \psi_\infty,\nabla\overline{\phi})_{L^2(\Omega)}
&= ((q_n-q_\infty)\nabla \psi_n,\nabla \overline{\phi})_{L^2(\Omega)}\\
&\qquad +(q_\infty\nabla(\psi_n-\psi_\infty),\nabla\overline{\phi})_{L^2(\Omega)},\end{aligned}$$ and use Cauchy-Schwarz inequality to get $$\begin{aligned}
\int_\Omega q_n \nabla \psi_n\cdot & \nabla\overline{\phi}\, dx
-\int_\Omega q_\infty\nabla \psi_\infty\cdot \nabla\overline{\phi}\,dx \\ &\leq \left|((q_n-q_\infty)\nabla \psi_n,\nabla \overline{\phi})_{L^2(\Omega)} \right|
+\left|(q_\infty\nabla(\psi_n-\psi_\infty),\nabla\overline{\phi})_{L^2(\Omega)}\right| \\
&\leq {\left\lVert \sqrt{|q_n-q_\infty|}\nabla \phi \right\rVert}_{L^2(\Omega)}{\left\lVert \sqrt{|q_n-q_\infty|}\nabla \psi_n \right\rVert}_{L^2(\Omega)} \\
&\qquad+\left|(q_\infty\nabla(\psi_n-\psi_\infty),\nabla\overline{\phi})_{L^2(\Omega)}\right| \\
&\leq 2\dfrac{\sqrt{\Lambda}}{\sqrt{\alpha}}{\left\lVert \psi_n \right\rVert}_{1,k_0} {\left\lVert \sqrt{|q_n-q_\infty|}\nabla \phi \right\rVert}_{L^2(\Omega)}+\left|(\nabla(\psi_n-\psi_\infty),q_\infty\nabla\overline{\phi})_{L^2(\Omega)}\right|.\end{aligned}$$ The right term above goes to $0$ owning to $q_\infty\in L^\infty(\Omega)$ and . For the other term, since $q_n\to q_\infty$ strongly in $L^1$, we can extract another subsequence $(q_{n_k})_k$ such that $q_{n_k}\to q_\infty$ pointwise almost everywhere in $\Omega$. Also, $\sqrt{|q_n-q_\infty|}|\nabla \phi|^2\leq 2\sqrt{\Lambda}|\nabla \phi|^2\in L^1(\Omega)$ and the Lebesgue dominated convergence theorem then yields $$\lim_{k\to+\infty}{\left\lVert \sqrt{|q_{n_k}-q_\infty|}\nabla \phi \right\rVert}_{L^2(\Omega)}= 0.$$ This gives that (see also [@Chen_BV_1999 Equation (2.4)]) $$\label{eq:Convergence_Terme_ordre_2}
\lim_{k\to+\infty} (q_{n_k}\nabla \psi_{n_k},\nabla\overline{\phi})_{L^2(\Omega)}=(q_\infty\nabla \psi_\infty,\nabla\overline{\phi})_{L^2(\Omega)},\ \forall \phi\in H^1(\Omega).$$ Finally, gathering together with yields $$0=\lim_{k\to+\infty}a(q_{n_k};\psi_{n_k},\phi)=a(q_\infty,\psi_\infty,\phi),\ \forall \phi\in H^1(\Omega),$$ and the uniqueness result [@Graham_variable_2018 Theorems 2.1, 2.4] shows that $\psi_\infty=0$ thus the whole sequence actually converges to $0$. To get our contradiction, it remains to show that ${\left\lVert \nabla \psi_n \right\rVert}_{L^2(\Omega)}$ converges to $0$ as well. From the Gårding inequality , we have $${\left\lVert \psi_n \right\rVert}^2_{1,k_0}\leq {\operatorname{Re}\{a(q_n;\psi_n,\psi_n)\}}+2k_0^2{\left\lVert \psi_n \right\rVert}^2_{L^2(\Omega)}
\xrightarrow[n\to+\infty]{} 0,$$ where we used and the strong $L^2$ convergence of $\psi_n$ towards $\psi_\infty=0$. Finally one gets ${\raisebox{0.5ex}{\scalebox{0.8}{$\displaystyle \lim_{n\to+\infty}\;$}}}{\left\lVert \psi_n \right\rVert}_{1,k_0}=0$ which contradicts ${\left\lVert \psi_n \right\rVert}_{1,k_0}=1$ and gives the desired result.
Applying then to the solution to finally yields $$\begin{aligned}
{\left\lVert \psi \right\rVert}_{1,k_0}
&\leq C_{\mathrm{s}}(k_0) \sup_{{\left\lVert \phi \right\rVert}_{1,k_0}=1}|a(q;\psi,\phi)|
\leq C_{\mathrm{s}}(k_0) \sup_{{\left\lVert \phi \right\rVert}_{1,k_0}=1} |b(q;\phi)| \\
& \leq C_{\mathrm{s}}(k_0)\sup_{{\left\lVert \phi \right\rVert}_{1,k_0}=1}\left( {\left\lVert q \right\rVert}_{L^\infty(\Omega)}{\left\lVert \nabla \psi_0 \right\rVert}_{L^2(\Omega)}{\left\lVert \nabla \phi \right\rVert}_{L^2(\Omega)}+{\left\lVert g \right\rVert}_{H^{-1/2}(\partial\Omega)}{\left\lVert \phi \right\rVert}_{H^{1/2}(\partial\Omega)}\right) \\
& \leq C_{\mathrm{s}}(k_0)C(\Omega)\max\{k_0^{-1},\alpha^{-1/2}\}\left({\left\lVert q \right\rVert}_{L^\infty(\Omega)}{\left\lVert \nabla \psi_0 \right\rVert}_{L^2(\Omega)}+{\left\lVert g \right\rVert}_{H^{-1/2}(\partial\Omega)} \right),\end{aligned}$$ where $C(\Omega)>0$ comes from the trace inequality.
Let us consider a more general version of Problem , given by $$\left\{
\begin{aligned}
-{ {\rm div}\left((1+q) {\nabla \psi}\right) } -k_0^2\psi &= F && \textrm{ in }\Omega,\\
(1+q){{\nabla \psi} \cdot \hat{n}} -ik_0\psi &= G && \textrm{ on }\partial\Omega.
\end{aligned}\right.$$ We emphasize that the estimation of the stability constant $C_{\mathrm{s}}(k_0)$ with respect to the wavenumber have been obtained for $(F,G)\in L^2(\Omega)\times L^2(\partial\Omega)$ for $q=0$ in [@Hetmaniuk_2007] and for $q\in \mathrm{Lip}(\Omega)$ satisfying in [@Barucq_2017; @Graham_variable_2018; @Graham_variable_2018_2]. Since their proofs rely on Green, Rellich and Morawetz identities, they do not extend to the case $(F,G)\in \left(H^1(\Omega)\right)^\prime\times H^{-1/2}(\partial\Omega)$ but such cases can be tackled as it is done in [@Esterhazy_2012 p.10, Theorem 2.5]. The case of Lipschitz $q$ has been studied in [@Brown_Helmholtz_Lip]. As a result, the dependance of the stability constant with respect to $q$, in the case $q\in U$ and $(F,G)\in \left(H^1(\Omega)\right)^\prime\times H^{-1/2}(\partial\Omega)$, does not seem to have been tackled so far to the best of our knowledge.
\[rem:H\_1\_bounds\_total\_sc\_waves\] From Remark \[rem:Link\_general\_Helmholtz\_scatt\], we obtain that the total wave $\psi_{tot}$ and the scattered wave $\psi_{sc}$ are solutions to , with respective right hand sides $$b_{tot}(q;\phi)=\int_{\partial\Omega} ({{\nabla \psi_0} \cdot \hat{n}} -{\mathrm{i}}k_0\psi_0)\overline{\phi}\, d\sigma, \;\;
b_{sc}(q;\phi)=-\int_{\Omega}q\nabla\psi_{0}\cdot\nabla{\overline\phi}\, dx.$$ As a result of Theorem \[thm:Uniform\_H1\_bound\] and the continuity of the trace, we have $$\begin{aligned}
{\left\lVert \psi_{tot} \right\rVert}_{1,k_0}
&\leq C(\Omega) C_\mathrm{s}(k_0)k_0 \max\{k_0^{-1},\alpha^{-1/2}\}, \\
{\left\lVert \psi_{sc} \right\rVert}_{1,k_0}
&\leq C_\mathrm{s}(k_0)\alpha^{-1/2}{\left\lVert q \right\rVert}_{L^\infty(\Omega)} {\left\lVert \nabla \psi_0 \right\rVert}_{L^2(\Omega)}
\leq k_0C_\mathrm{s}(k_0)\alpha^{-1/2}{\left\lVert q \right\rVert}_{L^\infty(\Omega)} \sqrt{|\Omega|}.\end{aligned}$$
We can now prove some regularity for the control-to-state mapping.
\[thm:CTS\_mapping\] Let $(q_n)_n\subset U$ be a sequence satisfying ${\left\lVert q_n \right\rVert}_{BV(\Omega)}\leq M$ and whose weak$^*$ limit in $BV(\Omega)$ is denoted by $q_\infty$. Let $(\psi(q_n))_n$ be the sequence of weak solution to Problem (\[eq:FV\_Helmholtz\]). Then $\psi(q_n)$ converges strongly in $H^1(\Omega)$ towards $\psi(q_\infty)$. In other words, the mapping $$q\in (U_\Lambda,\mathrm{weak}^*)\mapsto \psi(q)\in (H^1(\Omega),\mathrm{strong}),$$ is continuous.
Since ${\left\lVert q_n \right\rVert}_{BV(\Omega)}\leq M$ and $(q_n)_n\subset U$, there exists $q_\infty$ such that $q_n\rightharpoonup q_\infty,\ \mathrm{weak}^* \ \mathrm{in}\ BV(\Omega)$. Using that $U$ is $\mathrm{weak}^*$ closed, we obtain that $q_\infty\in U$. Note that the sequence $(\psi(q_n))_n$ of solution to Problem satisfies estimate uniformly with respect to $n$. As a result, [[ there exists some $\psi_\infty\in H^1(\Omega)$ such that]{}]{} the convergences hold. [[ Using then]{}]{} , we get that $a(q_n;\psi(q_n),\phi)\to a(q_\infty;\psi_\infty,\phi)$.
Since $b(q_n,\phi)\to b(q_\infty,\phi)$ for all $\phi\in H^1(\Omega)$, this proves that $a(q_\infty;\psi_\infty,\phi)=b(q;\phi)$ for all $\phi\in H^1(\Omega)$. [[ Consequently]{}]{} $\psi_\infty=\psi(q_\infty)$ [[ owning to the uniqueness of]{}]{} a weak solution to and [[ we have also proved that]{}]{} $\psi(q_n)\rightharpoonup \psi(q_\infty)$ in $H^1(\Omega)$.
We now show that $\psi(q_n)\to \psi(q_\infty)$ strongly in $H^1$. To see this, we start by noting that, up to extracting a subsequence (still denoted [[ by]{}]{} $q_n$), we can use to get that $$\lim_{n\to+\infty} b(q_n;\psi(q_n))=b(q_\infty;\psi(q_\infty)).$$ Since $\psi(q_n),\psi(q_\infty)$ satisfy the variational problem (\[eq:FV\_Helmholtz\]), we infer $$\label{eq:Strong_limit_bilinear_form}
\lim_{n\to+\infty} a(q_n;\psi(q_n),\psi(q_n))=a(q_\infty;{{\color{black} \psi(q_\infty)}},\psi(q_\infty)),$$ where the whole sequence actually converges owing to the uniqueness of the limit. Using then that $\psi(q_n)\rightharpoonup \psi(q_\infty)$ in $H^1(\Omega)$ together with , one gets $$\begin{aligned}
{\left\lVert \sqrt{1+q_n}\nabla \psi(q_n) \right\rVert}^2_{L^2(\Omega)}
&= a(q_n;\psi(q_n),\psi(q_n))+k_0{\left\lVert \psi(q_n) \right\rVert}^2_{L^2(\Omega)} +{\mathrm{i}}k_0{\left\lVert \psi(q_n) \right\rVert}^2_{L^2(\partial\Omega)} \\
& \hspace{-1cm} \xrightarrow[n\to+\infty]{}
a(q_\infty;\psi(q_\infty),\psi(q_\infty))+k_0{\left\lVert \psi(q_\infty) \right\rVert}^2_{L^2(\Omega)} +{\mathrm{i}}k_0{\left\lVert \psi(q_\infty) \right\rVert}^2_{L^2(\partial\Omega)} \\
&= {\left\lVert \sqrt{1+q_\infty}\nabla \psi(q_\infty) \right\rVert}^2_{L^2(\Omega)}.\end{aligned}$$
To show that ${\raisebox{0.5ex}{\scalebox{0.8}{$\displaystyle \lim_{n\to+\infty}\;$}}}{\left\lVert \nabla \psi(q_n) \right\rVert}^2_{L^2(\Omega)}={\left\lVert \nabla \psi(q_\infty) \right\rVert}^2_{L^2(\Omega)}$, note that $$\nabla \psi(q_n)=\frac{\sqrt{1+q_n}\nabla \psi(q_n)}{\sqrt{1+q_n}}.$$ Using the same arguments as those to prove (\[eq:Convergence\_Terme\_ordre\_2\]), we have a subsequence (same notation used) such that $q_n\rightarrow q_\infty$ pointwise a.e. in $\Omega$ and thus $\sqrt{1+q_n}^{-1}\rightarrow \sqrt{1+q_\infty}^{-1}$ pointwise a.e. in $\Omega$. Due to Lebesgue’s dominated convergence theorem and $\sqrt{1+q_n}\nabla \psi(q_n)\rightarrow \sqrt{1+q_\infty}\nabla \psi(q_\infty)$ strongly in $L^2(\Omega)$, we have $$\nabla \psi(q_n)=\dfrac{\sqrt{1+q_n}\nabla \psi(q_n)}{\sqrt{1+q_n}}\rightarrow \dfrac{\sqrt{1+q_\infty}\nabla \psi(q_\infty)}{\sqrt{1+q_\infty}}=\nabla \psi(q_\infty) \textrm{ strong in}\ L^2(\Omega).$$
The latter, together with [[ the weak $H^1$-convergence]{}]{} show that $\psi(q_n)\to \psi(q_\infty)$ strongly in $H^1$.
Existence of optimal solution in $U_{\Lambda}$
----------------------------------------------
We are now in [[ a]{}]{} position to prove the existence of a minimizer to Problem .
\[thm:existence\_min\] Assume that the cost function $(q,\psi)\in U_\Lambda\mapsto J(q,\psi)\in { \mathbb{R} }$ satisfies:
- There exists $\beta>0$ [and $J_0$]{} such that $$J(q,\psi)=J_0(q,\psi)+\beta |D q|(\Omega),$$ where $|D q|(\Omega)$ is defined in .
- $\forall (q,\psi)\in U_\Lambda\times H^1(\Omega)$, $J_0(q,\psi)\geq m>-\infty$.
- $(q,\psi)\mapsto J_0(q,\psi)$ is lower-semi-continuous with respect to the (weak$^*$,weak) topology of $BV(\Omega)\times H^1(\Omega)$.
Then the optimization problem has at least one optimal solution in $U_\Lambda\times H^1(\Omega)$.
The existence of a minimizer to Problem can be obtained with standard technique by combining Theorem \[thm:CTS\_mapping\] with weak-compactness arguments as done in [@Chen_BV_1999 Lemma 2.1], [@Bastide_2018 Theorem 4.1] or [@Haslinger_2015 Theorem 1]. We still give the proof for the sake of completeness.
We introduce the following set $$\mathcal{A}=\left\{(q,\psi)\in U_\Lambda\times H^1(\Omega)\ \left|\ a(q;\psi,\phi)=b(q;\phi)\ \forall \phi\in H^1(\Omega) \right.\right\}.$$ The existence and uniqueness of solution to Problem (\[eq:FV\_Helmholtz\]) ensure that $\mathcal{A}$ is non-empty. In addition, combining [[ Assumptions]{}]{} $(A1)$ and $(A2)$, we obtain that $J(q,\psi)$ is bounded from below on $\mathcal{A}$. We thus have a minimizing sequence $(q_n,\psi_n)\in \mathcal{A}$ such that $$\lim_{n\to +\infty} J(q_n,\psi_n)=\inf_{(q,\psi)\in \mathcal{A}} J(q,\psi).$$ Theorem \[thm:Uniform\_H1\_bound\] and $(A1)$ then gives that the sequence $(q_n,\psi_n)\in BV(\Omega)\times H^1(\Omega)$ is uniformly bounded with respect to $n$ and thus admits a subsequence that converges towards $(q^*,\psi^*)$ in the (weak$^*$,weak) topology of $BV(\Omega)\times H^1(\Omega)$. Using now Theorem \[thm:CTS\_mapping\] and the weak$^*$ lower semi-continuity of $q\mapsto |Dq|(\Omega)$, we end up with $(q^*,\psi^*)\in \mathcal{A}$ and $$J(q^*,\psi^*)\leq \liminf_{n\to+\infty} J(q_n,\psi_n)=\inf_{(q,\psi)\in \mathcal{A}} J(q,\psi).$$
It is worth noting that [[ the penalization term $\beta{\left\lVert q \right\rVert}_{BV(\Omega)}$]{}]{} has been introduced only to obtain a uniform bound in the $BV$-norm for the minimizing sequence.
Existence of optimal solution in $U_{\Lambda,\kappa}$
-----------------------------------------------------
We show here the existence of optimal solution to Problem for $U=U_{\Lambda,\kappa}$. Note that any $q\in U_{\Lambda,\kappa}$ is actually bounded in $BV$ since $${\left\lVert q \right\rVert}_{BV(\Omega)}\leq 2\max(\Lambda,\kappa,|\alpha-1|).$$ With this property at hand, we can get a similar result to Theorem \[thm:existence\_min\] without adding a penalization term in the cost function, hence $\beta=0$.
\[thm:existence\_min\_U\_lambda\_kappa\] Assume that the cost function $(q,\psi)\in U_{\Lambda,\kappa}\mapsto J(q,\psi)\in { \mathbb{R} }$ satisfies $(A2)-(A3)$ given in Theorem \[thm:existence\_min\] and that $\beta=0$. Then the optimization problem with $U=U_{\Lambda,\kappa}$ has at least one optimal solution.
We introduce the following non-empty set $$\mathcal{A}=\left\{(q,\psi)\in U_{\Lambda,\kappa}\times H^1(\Omega)\ |\ a(q;\psi,\phi)=b(q;\phi)\ \forall \phi\in H^1(\Omega) \right\}.$$ From $(A2)$, $J(q,\psi)$ is bounded from below on $\mathcal{A}$. We thus have a minimizing sequence $(q_n,\psi_n)\in \mathcal{A}$ such that $$\lim_{n\to +\infty} J(q_n,\psi_n)=\inf_{(q,\psi)\in \mathcal{A}} J(q,\psi).$$ Since $(q_n)_n\subset U_{\Lambda,\kappa}$, it satisfies ${\left\lVert q_n \right\rVert}_{BV(\Omega)}\leq 2\max(\Lambda,\kappa,|\alpha-1|)$ and thus admits a convergent subsequence toward some $q\in U_{\Lambda,\kappa}$. Theorem \[thm:CTS\_mapping\] then gives that $\psi(q_n)\to \psi(q)$ strongly in $H^1(\Omega)$ and the proof can be finished as the proof of Theorem \[thm:existence\_min\].
Boundedness/Continuity of solution to Helmholtz problem {#sec:continuity}
=======================================================
In this section, we prove that even if the parameter $q$ is not smooth enough for the solution to to be in $H^s(\Omega)$ for some $s>1$, we can still have continuous solution. In order to prove such regularity for $\psi$, we are going to rely on the De Giorgi-Nash-Moser theory [@Trudinger Chapter 8.5], [@Ural_1968 Chapters 3.13, 7.2] and more precisely on [@Nittka_2011 Proposition 3.6] which reads
\[thm:Tool\_for\_boundedness\] Consider the elliptic problem associated with inhomogeneous Neumann boundary condition given by $$\left\{
\begin{aligned}
\mathcal{L} v\vcentcolon={ {\rm div}\left(A(x)\nabla v\right) }
&= f_0-\sum_{j=1}^N {\frac{\partial f_j}{\partial x_j}},\\
{{\nabla v} \cdot \hat{n}}
&= h+\sum_{j=1}^N f_j n_j,
\end{aligned}
\right.\label{eq:Elliptic_PDE}$$ where $A\in L^\infty(\Omega,{ \mathbb{R} }^{N\times N})$ satisfy the standard ellipticity condition $A(x)\xi\cdot\xi \geq \gamma|\xi|^2$ for a.e. $x\in\Omega$. Let $p>N$ and assume that $f_0\in L^{p/2}(\Omega)$, $f_j\in L^p(\Omega)$ for all $j=1,\cdots,N$ and $h\in L^{p-1}(\partial\Omega)$. Then the weak solution $v$ to satisfies $${\left\lVert v \right\rVert}_{C^0(\Omega)}\leq C(N,p,\Omega,\gamma)\left( {\left\lVert v \right\rVert}_{L^2(\Omega)} +{\left\lVert f_0 \right\rVert}_{L^{p/2}(\Omega)} +\sum_{j=1}^N {\left\lVert f_j \right\rVert}_{L^p(\Omega)}+{\left\lVert h \right\rVert}_{L^{p-1}(\partial\Omega)}\right).$$
$C^0$-bound for the general Helmholtz problem
---------------------------------------------
Using Theorem \[thm:Tool\_for\_boundedness\], we can prove some $L^\infty$ bound for the weak solution to Helmholtz equation with bounded coefficients.
\[thm:Boundedness\_Helmholtz\] Assume that $q\in L^\infty(\Omega)$ and satisfies and $g\in L^2(\partial\Omega)$. Then the solution to Problem satisfies $$\label{eq:Final_L_infty_bound}
{\left\lVert \psi \right\rVert}_{C^0(\Omega)}
\leq \widetilde{C}(\Omega)\widetilde{C}_\mathrm{s}(k_0,\alpha)\left({\left\lVert q \right\rVert}_{L^\infty(\Omega)} {\left\lVert \nabla \psi_0 \right\rVert}_{L^\infty(\Omega)} +{\left\lVert g \right\rVert}_{L^2(\partial\Omega)} \right),$$ where $$\widetilde{C}_\mathrm{s}(k_0,\alpha) = 1 +\left((1+k_0^2)k_0^{-1} +\alpha^{-1/2}\right)\max\{k_0^{-1},\alpha^{-1/2}\}C_{\mathrm{s}}(k_0),$$ and $\widetilde{C}(\Omega)>0$ does not depend on $k$ nor $q$.
We cannot readily apply Theorem \[thm:Tool\_for\_boundedness\] to the weak solution of Problem since it involves a complex valued operator. We therefore consider the Problem satisfied by $\nu={\operatorname{Re}\{u\}}$ and $\zeta={\operatorname{Im}\{u\}}$ which is given by $$\left\{
\begin{aligned}
-{ {\rm div}\left((1+q)\nabla \nu\right) }-k_0^2\nu
&={ {\rm div}\left(q\nabla {\operatorname{Re}\{\psi_0\}}\right) }
&&\textrm{ in } \Omega, \\
-{ {\rm div}\left((1+q)\nabla \zeta\right) }-k_0^2\zeta
&={ {\rm div}\left(q\nabla {\operatorname{Im}\{\psi_0\}}\right) }
&&\textrm{ in } \Omega, \\
(1+q){{\nabla \nu} \cdot \hat{n}}
&={\operatorname{Re}\{g\}}-k_0\zeta-q {{\nabla {\operatorname{Re}\{\psi_0\}}} \cdot \hat{n}},
&&\textrm{ on } \partial\Omega, \\
(1+q){{\nabla \zeta} \cdot \hat{n}}
&={\operatorname{Im}\{g\}}+k_0\nu-q{{\nabla {\operatorname{Im}\{\psi_0\}}} \cdot \hat{n}}
&&\textrm{ on }\partial\Omega.
\end{aligned}
\right.\label{eq:Helmholtz_real_imaginary}$$ Since Problem is equivalent to Problem (\[eq:Helmholtz\]), we get that the weak solution $(\nu,\zeta)\in H^1(\Omega)$ to (\[eq:Helmholtz\_real\_imaginary\]) satisfies the inequality (\[eq:Uniform\_H1\_bound\]). Assuming that $g\in L^2(\partial\Omega)$ and using the continuous Sobolev embedding $H^1(\Omega)\subset L^6(\Omega)$, the (compact) embedding $H^{1/2}(\partial\Omega)\subset L^2(\partial\Omega)$, that $q\in L^\infty(\Omega)$ satisfies and the fact that $\psi_0$ is smooth we get the next regularities $$\begin{aligned}
f_{0,1}&=k_0^2\nu\in L^6(\Omega),\
f_{j,1}=q{\frac{\partial {\operatorname{Re}\{\psi_0\}}}{\partial x_j}}\in L^\infty(\Omega),\
h_{1}={\operatorname{Re}\{g\}}-k_0\zeta\in L^2(\partial\Omega), \\
f_{0,2}&=k_0^2\zeta\in L^6(\Omega),\
f_{j,2}=q{\frac{\partial {\operatorname{Im}\{\psi_0\}}}{\partial x_j}}\in L^\infty(\Omega),\
h_{2}={\operatorname{Im}\{g\}}+k_0\nu\in L^2(\partial\Omega).\end{aligned}$$
Applying now Theorem \[thm:Tool\_for\_boundedness\] to twice with $p=3$ and $N=2$, one gets $C^0$ bounds for $\nu$ and $\zeta$ $$\begin{aligned}
{\left\lVert \nu \right\rVert}_{C^0(\Omega)}
&\leq C(2,3,\Omega,\gamma)\left( {\left\lVert \nu \right\rVert}_{L^2(\Omega)} +{\left\lVert f_{0,1} \right\rVert}_{L^{3/2}(\Omega)} +\sum_{j=1}^2 {\left\lVert f_{j,1} \right\rVert}_{L^3(\Omega)}+{\left\lVert h_1 \right\rVert}_{L^{2}(\partial\Omega)}\right),
\\
{\left\lVert \zeta \right\rVert}_{C^0(\Omega)}
&\leq C(2,3,\Omega,\gamma)\left( {\left\lVert \zeta \right\rVert}_{L^2(\Omega)} +{\left\lVert f_{0,2} \right\rVert}_{L^{3/2}(\Omega)} +\sum_{j=1}^2 {\left\lVert f_{j,2} \right\rVert}_{L^3(\Omega)}+{\left\lVert h_2 \right\rVert}_{L^{2}(\partial\Omega)}\right).\end{aligned}$$
Some computations with the Holder and multiplicative trace inequalities then give $$\begin{aligned}
({\left\lVert \nu \right\rVert}_{L^2(\Omega)} +{\left\lVert \zeta \right\rVert}_{L^2(\Omega)})
&\leq 2 {\left\lVert \psi \right\rVert}_{L^2(\Omega)},\\ {\left\lVert f_{0,1} \right\rVert}_{L^{3/2}(\Omega)} +{\left\lVert f_{0,2} \right\rVert}_{L^{3/2}(\Omega)}
&\leq k_0^2 {\left\lVert \psi \right\rVert}_{L^{3/2}(\Omega)} \leq |\Omega|^{1/6} k_0^2 {\left\lVert \psi \right\rVert}_{L^{2}(\Omega)}, \\{\left\lVert f_{j,l} \right\rVert}_{L^3(\Omega)}
&\leq {\left\lVert q \right\rVert}_{L^\infty(\Omega)} {\left\lVert \nabla \psi_0 \right\rVert}_{L^\infty(\Omega)},\ j=1,2, \\
{\left\lVert h_1 \right\rVert}_{L^2(\partial\Omega)} +{\left\lVert h_2 \right\rVert}_{L^2(\partial\Omega)}
&\leq {\left\lVert g \right\rVert}_{L^2(\partial\Omega)}+k_0{\left\lVert \psi \right\rVert}_{L^2(\partial\Omega)} \\
& \leq {\left\lVert g \right\rVert}_{L^2(\partial\Omega)}+k_0C(\Omega)\sqrt{{\left\lVert \psi \right\rVert}_{L^2(\Omega)}{\left\lVert \psi \right\rVert}_{H^1(\Omega)}}.\end{aligned}$$ Using then Young’s inequality yields $$\begin{aligned}
k_0\sqrt{{\left\lVert \psi \right\rVert}_{L^2(\Omega)}{\left\lVert \psi \right\rVert}_{H^1(\Omega)}}
&\leq C \left({\left\lVert \psi \right\rVert}_{H^1(\Omega)}+k_0^2{\left\lVert \psi \right\rVert}_{L^2(\Omega)}\right)\\
&\leq C \left({\left\lVert \nabla \psi \right\rVert}_{L^2(\Omega)}+k_0^2 {\left\lVert \psi \right\rVert}_{L^2(\Omega)}\right) $$ where $C>0$ is a generic constant. We obtain the bound $$\begin{aligned}
{\left\lVert \psi \right\rVert}_{C^0(\Omega)}
&= {\left\lVert \nu \right\rVert}_{C^0(\Omega)}+{\left\lVert \zeta \right\rVert}_{C^0(\Omega)} \\
&\leq \widetilde{C}(\Omega)\left( \left(1+k_0^2 \right){\left\lVert \psi \right\rVert}_{L^2(\Omega)}+{\left\lVert \nabla \psi \right\rVert}_{L^2(\Omega)} +{\left\lVert q \right\rVert}_{L^\infty(\Omega)} {\left\lVert \nabla \psi_0 \right\rVert}_{L^\infty(\Omega)} +{\left\lVert g \right\rVert}_{L^2(\partial\Omega)} \right). \end{aligned}$$
Using the definition of ${\left\lVert \psi \right\rVert}_{1,k_0}$ on the estimate above, we get $$\label{eq:L_infty_bound}
\begin{aligned}
{\left\lVert \psi \right\rVert}_{C^0(\Omega)}
&\leq \widetilde{C}(\Omega)\Big( \left((1+k_0^2)k_0^{-1} +\alpha^{-1/2}\right){\left\lVert \psi \right\rVert}_{1,k_0} \\
&\hspace{2cm} +{\left\lVert q \right\rVert}_{L^\infty(\Omega)} {\left\lVert \nabla \psi_0 \right\rVert}_{L^\infty(\Omega)} +{\left\lVert g \right\rVert}_{L^2(\partial\Omega)} \Big).
\end{aligned}$$ To apply the a priori estimate , we recall that the $H^{-1/2}$ norm can be replaced by a $L^2$ norm (since $g\in L^2(\partial\Omega)$) and then, $$\begin{aligned}
{\left\lVert \psi \right\rVert}_{1,k_0}
&\leq C(\Omega)\max\{k_0^{-1},\alpha^{-1/2}\}C_{\mathrm{s}}(k_0)\left({\left\lVert q \right\rVert}_{L^\infty(\Omega)}{\left\lVert \nabla \psi_0 \right\rVert}_{L^2(\Omega)}+{\left\lVert g \right\rVert}_{L^{2}(\partial\Omega)} \right) \\
&\leq C(\Omega)\max\{k_0^{-1},\alpha^{-1/2}\}C_{\mathrm{s}}(k_0)\max\{1,\sqrt{{\left\vert \Omega \right\vert}}\}\left({\left\lVert q \right\rVert}_{L^\infty(\Omega)}{\left\lVert \nabla \psi_0 \right\rVert}_{L^{\infty}(\Omega)}+{\left\lVert g \right\rVert}_{L^{2}(\partial\Omega)} \right) \end{aligned}$$
Finally, combining the [[ latter]{}]{} expression with Equation , we obtain that the weak solution to the Helmholtz equation satisfies $$\begin{aligned}
{\left\lVert \psi \right\rVert}_{C^0(\Omega)}
&\leq \widetilde{C}(\Omega)\left(1 +\left((1+k_0^2)k_0^{-1} +\alpha^{-1/2}\right)\max\{k_0^{-1},\alpha^{-1/2}\}C_{\mathrm{s}}(k_0)\right) \\
& \hspace{2cm} \times \left({\left\lVert q \right\rVert}_{L^\infty(\Omega)} {\left\lVert \nabla \psi_0 \right\rVert}_{L^\infty(\Omega)} +{\left\lVert g \right\rVert}_{L^2(\partial\Omega)} \right),\end{aligned}$$ where $\widetilde{C}(\Omega) > 0$.
1. For the one-dimensional Helmholtz problem, the a priori estimate and the continuous embedding $H^1(I)\subset C^0(I)$ directly gives the continuity of $u$ over [[ a give interval]{}]{} $I$ $${\left\lVert \psi \right\rVert}_{C^0(I)}\leq C {\left\lVert \psi \right\rVert}_{1,k_0}\leq C(k_0)\left({\left\lVert q \right\rVert}_{L^\infty(\Omega)}{\left\lVert \nabla \psi_0 \right\rVert}_{L^\infty(\Omega)}+{\left\lVert g \right\rVert}_{H^{-1/2}(\partial\Omega)} \right).$$ It is worth noting that we do not need to assume that $g\in L^2(\partial\Omega)$.
2. For the two-dimensional Helmholtz problem with $q=0$, we can get the above $\mathcal{C}^0$ estimate from the embedding $H^2(\Omega)\hookrightarrow \mathcal{C}^0(\overline{\Omega})$ since $${\left\lVert \psi \right\rVert}_{C^0(\Omega)}\leq C {\left\lVert \psi \right\rVert}_{H^2(\Omega)},$$ for a generic constant $C$. We can then see that the estimate [[ has actually]{}]{} the same dependance with respect to $k_0$ as the $H^2$-estimate [[ in]{}]{} [@Hetmaniuk_2007 p. 677, Proposition 3.6].
$C^0$-bounds for the total and scattered waves
----------------------------------------------
Thanks to Remark \[rem:Link\_general\_Helmholtz\_scatt\] and following the proof of Theorem \[thm:Boundedness\_Helmholtz\], these bounds can be roughly obtained by setting $g={{\nabla \psi_0} \cdot \hat{n}}-{\mathrm{i}}k_0\psi_0$ and omitting the $L^{\infty}$-norms in for the total wave $\psi_{tot}$, and simply by setting $g=0$ in the case the scattered wave $\psi_{sc}$. Using after the $H^1$-bounds from Remark \[rem:H\_1\_bounds\_total\_sc\_waves\], we actually get $$\begin{aligned}
{\left\lVert \psi_{tot} \right\rVert}_{\mathcal{C}^0(\Omega)}
&\leq \widetilde{C}(\Omega)k_0 \left(\left((1+k_0^2)k_0^{-1} +\alpha^{-1/2}\right)\max\{k_0^{-1},\alpha^{-1/2}\}C_\mathrm{s}(k_0)+ 1\right) \\
{\left\lVert \psi_{sc} \right\rVert}_{\mathcal{C}^0(\Omega)}
&\leq \widetilde{C}(\Omega)k_0 \left(\left((1+k_0^2)k_0^{-1} +\alpha^{-1/2}\right)\alpha^{-1/2}C_\mathrm{s}(k_0) +1\right){\left\lVert q \right\rVert}_{L^\infty(\Omega)}. \end{aligned}$$
We emphasize that the previous estimates show that the scattered wave $\psi_{sc}$ vanishes in $\Omega$ if $q\to 0$. This is expected since, if $q=0$, there is no obstacle to scatter the incident wave which [[ amounts]{}]{} to saying that $\psi_{tot}=\psi_0$.
Discrete optimization problem and convergence results {#sec:disc_pb}
=====================================================
This section is devoted to the finite element discretization of the optimization problem . We consider a quasi-uniform family of triangulations (see [@Ern_book_FE p. 76, Definition 1.140]) $\left\{\mathcal{T}_h\right\}_{h>0}$ of $\Omega$ and the corresponding finite element spaces $$\begin{aligned}
\mathcal{V}_h & =\left\{\phi_h\in \mathcal{C}(\overline{\Omega})\ |\ \phi_h|_{T}\in \mathbb{P}_1(T),\ \forall T\in\mathcal{T}_h\right\}. $$ Note that thanks to Theorem \[thm:Boundedness\_Helmholtz\], the solution to the general Helmholtz equation is continuous, which motivates to use continuous piecewise linear finite elements. We are going to look for [[ a]{}]{} discrete optimal design that [[ belongs]{}]{} to some finite element spaces $\mathcal{K}_h$ and we thus introduce the following set of discrete admissible parameters $$U_{h}=U\cap \mathcal{K}_h.$$ The full discretization of the optimization problem then reads $$\label{eq:Discrete_optim_pbm}
\textrm{Find } q^*_h\in U_{h}\ \textrm{such that }
\widetilde{J}(q^*_h)\leq \widetilde{J}(q_h),\ \forall q_h\in U_{h},$$ where $\widetilde{J}(q_h)=J(q_h,\psi_h(q_h))$ is the reduced cost-functional and $\psi_h\vcentcolon=\psi_h(q_h)\in \mathcal{V}_h$ satisfies the discrete Helmholtz problem $$\label{eq:Discrete_Helmholtz}
a(q_h;\psi_h,\phi_h)=b(q_h;\phi_h),\ \forall \phi_h\in \mathcal{V}_h.$$ The existence of solution to Problem [[ is going to be discussed in the next subsection.]{}]{} Before giving the definition of $\mathcal{K}_h$, we would like to discuss briefly the strategy for proving that the discrete optimal solution converges toward the continuous ones. To achieve this, we need to pass to the limit in inequality . Since $J$ is only lower-semi-continuous with respect to the weak$^*$ topology of $BV$, we can only pass to the limit on one side of the inequality and the continuity of $J$ is then going to be needed to pass to the limit on the other side to keep this inequality valid as $h\to 0$. We discuss first the case $U=U_\Lambda$ for which Theorem \[thm:existence\_min\] gives the existence of optimal $q$ but only if $\beta>0$. Since we have to pass to the limit in , we need that ${\raisebox{0.5ex}{\scalebox{0.8}{$\displaystyle \lim_{h\to 0}\;$}}}|D q_h|(\Omega)=|D q|(\Omega)$. Since the total variation is only continuous with respect to the strong topology of $BV$, we have to approximate any $q\in U_\Lambda$ by some $q_h\in U_{h}$ such that $$\lim_{h\to 0}{\left\lVert q-q_h \right\rVert}_{BV(\Omega)}=0.$$ However, from [@Bartels_2012 p. 8, Example 4.1] there [[ exists]{}]{} an example of a $BV$-function $v$ that cannot be approximated by piecewise constant function $v_h$ over a given mesh in such a way that ${\raisebox{0.5ex}{\scalebox{0.8}{$\displaystyle \lim_{h\to 0}\;$}}}|D v_h|(\Omega)=|D v|(\Omega)$. Nevertheless, if one consider an adapted mesh that depends on a given function $v\in BV(\Omega)\cap L^\infty(\Omega)$, we get the existence of piecewise constant function on this specific mesh that strongly converges in $BV$ toward $v$ (see [@Belik_2003 p. 11, Theorem 4.2]). As a result, when considering $U=U_\Lambda$, we use the following discrete set of admissible parameters $$\mathcal{K}_{h,1}=\left\{q_h\in L^\infty(\Omega)\ |\ q_h|_{T}\in \mathbb{P}_1(T),\ \forall T\in\mathcal{T}_h\right\}.$$ Note that, from Theorem [@Belik_2003 p. 10, Theorem 4.1 and Remark 4.2], the set $U_{h}=U_{\Lambda}\cap \mathcal{K}_{h,1}$ defined above has the required density property hence motivated its introduction as a discrete set of admissible parameter. In the case $U=U_{\Lambda,\kappa}$, we will not need the density of $U_h$ [[ for]{}]{} the strong topology of $BV$ but only for the weak$^*$ topology. The discrete set of admissible [[ parameters]{}]{} is then going to be $U_{h}=U_{\Lambda,\kappa}\cap \mathcal{K}_{h,0}$ with $$\mathcal{K}_{h,0}=\left\{q_h\in L^\infty(\Omega)\ |\ q_h|_{T}\in \mathbb{P}_0(T),\ \forall T\in\mathcal{T}_h\right\}.$$
We show below the convergence of discrete optimal solution to the continuous one for both cases highlighted above.
Convergence of the Finite element approximation
-----------------------------------------------
We prove here some useful approximations results for any $U_h$ defined above. We have the following convergence result whose proof can be found in [@Esterhazy_2012 p. 22, Lemma 4.1] (see also [@Graham_variable_2018 p. 10, Theorem 4.1]).
\[thm:Cv\_FE\] Let $q_h\in U_{h}$ and $\psi(q_h)\in H^1(\Omega)$ be the solution to the variational problem $$a(q_h;\psi(q_h),\phi)=b(q_h,\phi),\ \forall \phi\in H^1(\Omega).$$ Let $S^*:(q_h,f)\in U_{h}\times L^2(\Omega)\mapsto S^*(q_h,f)=\psi^*\in H^1(\Omega)$ be the solution operator associated to the following problem $$\mathrm{Find\ }\psi^*\in H^1(\Omega)\ \mathrm{such\ that\ } a(q_h;\phi,\psi^*)=(\phi,\overline{f})_{L^2(\Omega)},\ \forall \phi\in H^1(\Omega).$$ Denote by $C_a$ the continuity constant of the bilinear form $a(q_h;\cdot,\cdot)$, which does not depend on $h$ since $q_h\in U_{h}$, and define the adjoint approximation property by $$\delta(\mathcal{V}_h)\vcentcolon=\sup_{f\in L^2(\Omega)} \inf_{\phi_h\in\mathcal{V}_h}\frac{{\left\lVert S^*(q_h,f)-\phi_h \right\rVert}_{1,k_0}}{{\left\lVert f \right\rVert}_{L^2(\Omega)}}.$$ Assume that the spaces $\mathcal{V}_h$ satisfies $$\label{hyp:maillage}
2C_a k_0\delta(\mathcal{V}_h)\leq 1,$$ then the solution $\psi_h(q_h)$ to Problem (\[eq:Discrete\_Helmholtz\]) satisfies $${\left\lVert \psi(q_h)-\psi_h(q_h) \right\rVert}_{1,k_0}\leq 2 C_a \inf_{\phi_h\in\mathcal{V}_h} {\left\lVert \psi(q_h)-\phi_h \right\rVert}_{1,k_0}.$$
[[ We emphasize that the above error estimates in fact implies the existence and uniqueness of a solution to the discrete problem (\[eq:Discrete\_Helmholtz\]) (see [@lohndorf2011wavenumber Theorem 3.9]).]{}]{} In the case $q\in \mathcal{C}^{0,1}(\Omega)$ where $\Omega$ is a convex Lipschitz domain, [[ Assumption]{}]{} (\[hyp:maillage\]) has been discussed in [@Graham_variable_2018 p. 11, Theorem 4.3] and roughly [[ amounts]{}]{} to say that (\[hyp:maillage\]) holds if $k_0^2h$ is small enough. Since the proof rely on $H^2$-regularity for a Poisson problem, we cannot readily extend the argument here since we can only expect to have $\psi\in H^1(\Omega)$ and that $S^*$ also depend on the meshsize. We can still show that is satisfied for small enough $h$.
\[lem:delta\_V\_h\_0\] Assume that $q_h\in U_{h}$ weak$^*$ converges toward $q\in BV(\Omega)$. Then is satisfied for small enough $h$.
Note first that Theorem \[thm:CTS\_mapping\] also holds for the adjoint problem and thus $$\lim_{h\to0}{\left\lVert S^*(q_h,f)-S^*(q,f) \right\rVert}_{1,k_0}=0.$$ Using the density of smooth [[ functions]{}]{} in $H^1$ and the properties of the piecewise linear interpolant [@Ern_book_FE p. 66, Corollary 1.122], we have that $$\lim_{h\to0}\left(\sup_{f\in L^2(\Omega)} \inf_{\phi_h\in\mathcal{V}_h}\frac{{\left\lVert S^*(q,f)-\phi_h \right\rVert}_{1,k_0}}{{\left\lVert f \right\rVert}_{L^2(\Omega)}}\right)=0,$$ and thus a triangular inequality shows that holds for small enough $h$.
We can now prove a discrete counterpart to Theorem \[thm:CTS\_mapping\].
\[thm:CTS\_discrete\] Let $(q_h)_h\subset U_{h}$ be a sequence satisfying ${\left\lVert q_h \right\rVert}_{BV(\Omega)}\leq M$ and whose weak$^*$ limit in $BV(\Omega)$ is denoted by $q$. Let $(\psi_h(q_h))_h$ be the sequence of discrete solutions to Problem . Then $\psi(q_h)$ converges, as $h$ goes to $0$, strongly in $H^1(\Omega)$ towards $\psi(q)$ satisfying Problem .
For $h$ small enough, Lemma \[lem:delta\_V\_h\_0\] ensures that holds and a triangular inequality then yields $$\begin{aligned}
{\left\lVert \psi_h(q_h)-\psi(q) \right\rVert}_{1,k_0}
&\leq {\left\lVert \psi_h(q_h)-\psi(q_h) \right\rVert}_{1,k_0}+{\left\lVert \psi(q_h)-\psi(q) \right\rVert}_{1,k_0}\\
&\leq 2 C_a \inf_{\phi_h\in\mathcal{V}_h} {\left\lVert \psi(q_h)-\phi_h \right\rVert}_{1,k_0}+{\left\lVert \psi(q_h)-\psi(q) \right\rVert}_{1,k_0} \\
&\leq (1+2C_a) {\left\lVert \psi(q_h)-\psi(q) \right\rVert}_{1,k_0}+2 C_a \inf_{\phi_h\in\mathcal{V}_h} {\left\lVert \psi(q)-\phi_h \right\rVert}_{1,k_0}.\end{aligned}$$ Theorem \[thm:CTS\_mapping\] gives that the first term above goes to zero as $h\to0$. For the second one, we can use the density of smooth function in $H^1$ to get that it goes to zero as well.
Convergence of the discrete optimal solution: Case $U_{h}=U_{\Lambda}\cap \mathcal{K}_{h,1}$
--------------------------------------------------------------------------------------------
We are now in [[ a]{}]{} position to prove the convergence of a discrete optimal design towards a continuous one in the case $$U=U_\Lambda,\ U_h=U_{\Lambda}\cap \mathcal{K}_{h,1}.$$ Hence the set of discrete control is composed of piecewise linear function on $\mathcal{T}_h$.
\[thm:Cv\_discrete\_control\] Assume that $(A1)-(A2)-(A3)$ from Theorem \[thm:existence\_min\] hold and that the cost function $J_0:(q,\psi)\in U_\Lambda\times H^1(\Omega)\mapsto J_0 (q,\psi)\in { \mathbb{R} }$ is continuous with respect to the $(\mathrm{weak}^*,\mathrm{strong})$ topology of $BV(\Omega)\times H^1 (\Omega)$. Let $(q_h^*,\psi_h(q_h^*))\in U_{\Lambda,h}\times \mathcal{V}_h$ be an optimal pair of (\[eq:Discrete\_optim\_pbm\]). Then the sequence $(q^*_h)_h\subset U_\Lambda$ is bounded and there exists $q^*\in U_\Lambda$ such that $q^*_h\rightharpoonup q^*$ weakly$^*$ in $BV(\Omega)$, $\psi(q_h^*)\to \psi(q^*)$ strongly in $H^1(\Omega)$ and $$\widetilde{J}(q^*)\leq \widetilde{J}(q),\ \forall q\in U_{\Lambda}.$$ Hence any accumulation point of $(q_h^*,\psi_h(q_h^*))$ is an optimal pair for Problem .
Let $q_{\Lambda}\in U_{\Lambda,h}$ be given as $$q_{\Lambda}(x)=\Lambda,\ \forall x\in\Omega.$$ Then $D q_{\Lambda}=0$. Since $\psi_h(q_{\Lambda})$ is well-defined and converges toward $\psi(q_{\Lambda})$ strongly in $H^1$ (see Theorem \[thm:Cv\_discrete\_control\]), we have that $$\widetilde{J}(q_{\Lambda})=J(q_{\Lambda},\psi_h(q_{\Lambda}))=J_0(q_\Lambda,\psi_h(q_{\Lambda})) \xrightarrow[h\to 0]{} J_0(q_\Lambda,\psi(q_{\Lambda})).$$ As a result, using that $(q_h^*,\psi_h(q_h^*))$ is an optimal pair to Problem , we get that $$\beta |D(q_h^*)|(\Omega)
\leq -J_0(q_h^*,\psi_h(q_h^*))+J(q_{\Lambda},\psi_h(q_{\Lambda})) \leq -m +J_0(q_{\Lambda},\psi_h(q_{\Lambda})),$$ and thus the sequence $(q_h^*)_h\subset U_{\Lambda,h}\subset U_{\Lambda}$ is bounded in $BV(\Omega)$ uniformly with respect to $h$. [[ We can then assume that it converges and denote by]{}]{} $q^*\in U_\Lambda$ its weak$^*$ limit and Theorem \[thm:CTS\_discrete\] then shows that $\psi_h(q_h^*)\to \psi(q^*)$ strongly in $H^1(\Omega)$. The lower semi-continuity of $J$ ensures that $$J(q^*,\psi(q^*))
=\widetilde{J}(q^*)\leq \liminf_{h\to 0}\widetilde{J}(q_h^*)
=\liminf_{h\to 0}J(q_h^*,\psi_h(q_h^*)).$$ Now, let $q\in U_\Lambda$, using the density of smooth [[ functions]{}]{} in $BV$, one gets that there exists a sequence $q_h\in U_{\Lambda,h}$ such that ${\left\lVert q_h-q^* \right\rVert}_{BV(\Omega)}\to 0$ (see also [@Bartels_2012 p. 10, Remark 4.2]). From Theorem \[thm:CTS\_discrete\], one gets $\psi_h(q_h)\to \psi(q)$ strongly in $H^1(\Omega)$ and the continuity of $J$ ensure that $
\widetilde{J}(q_h)\rightarrow \widetilde{J}(q).
$ Since $\widetilde{J}(q_h^*)\leq \widetilde{J}(q_h)$ for all $q_h\in U_{\Lambda,h}$, one gets by passing to the inf-limit that $$\widetilde{J}(q^*)\leq \liminf_{h\to 0}\widetilde{J}(q_h^*)\leq \liminf_{h\to 0}\widetilde{J}(q_h) = \widetilde{J}(q),\ \forall q\in U_\Lambda,$$ and [[ the proof is complete.]{}]{}
Convergence of the discrete optimal solution: Case $U_{h}=U_{\Lambda,\kappa}\cap \mathcal{K}_{h,0}$
---------------------------------------------------------------------------------------------------
We are now in [[ a]{}]{} position to prove the convergence of discrete optimal design toward continuous one in the case $$U=U_{\Lambda,\kappa},\ U_h=U_{\Lambda,\kappa}\cap \mathcal{K}_{h,0}.$$ Hence the set of discrete control is composed of piecewise constant [[ functions]{}]{} on $\mathcal{T}_h$ that satisfy $$\forall q_h\in U_h,\ {\left\lVert q_h \right\rVert}_{BV(\Omega)}\leq 2\max(\Lambda,\kappa,|\alpha-1|).$$ We can compute explicitly the previous norm by integrating by parts the total variation (see e.g. [@Bartels_2012 p. 7, Lemma 4.1]). This reads $$\forall q_h\in U_h,\ |Dq_h|(\Omega)=\sum_{F\in \mathcal{F}^i} |F|| [q_h]|_{F}|,$$ where $\mathcal{F}^i$ is the set of interior faces and $| [q_h]|_{F}$ is the jump of $q_h$ on the interior face $F=\partial T_1\cap \partial T_2$ [[ meaning that $| [q_h]|_{F}=|q_h|_{T_1}-|q_h|_{T_2}$, where $|\cdot|_{T_i}$ denotes the value of the a finite element function on the face $T_i$]{}]{}. Note then that any $q_h\in U_h$ can only have either a finite number of discontinuity or jumps that are not too large.
\[thm:Cv\_discrete\_control\_U\_lambda\_kappa\] Assume that $\beta=0$ and $(A2)-(A3)$ from Theorem \[thm:existence\_min\] hold and that the cost function $J:(q,\psi)\in U_\Lambda\times H^1(\Omega)\mapsto J(q,\psi)\in { \mathbb{R} }$ is continuous with respect to the $(\mathrm{weak}^*,\mathrm{strong})$ topology of $BV(\Omega)\times H^1 (\Omega)$. Let $(q_h^*,\psi_h(q_h^*))\in U_{h}\times \mathcal{V}_h$ be an optimal pair of . Then the sequence $(q^*_h)_h\subset U_{\Lambda,\kappa}$ is bounded and there exists $q^*\in U_{\Lambda,\kappa}$ such that $q^*_h\rightharpoonup q^*$ weakly$^*$ in $BV(\Omega)$, $\psi(q_h^*)\to \psi(q^*)$ strongly in $H^1(\Omega)$ and $$\widetilde{J}(q^*)\leq \widetilde{J}(q),\ \forall q\in U_{\Lambda}.$$ Hence any accumulation point of $(q_h^*,\psi_h(q_h^*))$ is an optimal pair for Problem .
Since $(q_h^*)_h$ belong to $U_h$, it satisfies ${\left\lVert q_h \right\rVert}_{BV(\Omega)}\leq 2\max(\Lambda,\kappa,|\alpha-1|)$ and is thus bounded uniformly with respect to $h$. We denote by $q^*\in U_{\Lambda,\kappa}$ its weak$^*$ limit. Theorem \[thm:Cv\_discrete\_control\] then shows that $\psi_h(q_h^*)$ converges strongly in $H^1(\Omega)$ toward $\psi(q^*)$. Now, let $q\in U_{\Lambda,\kappa}$, using the density of smooth function in $BV$, one gets that there exists a sequence $q_h\in U_{h}$ such that $q_h\rightharpoonup q$ weak$^*$ in $BV(\Omega)$ (see also [@Bartels_2012 Introduction]). From Theorem \[thm:CTS\_discrete\], one gets $\psi_h(q_h)\to \psi(q)$ strongly in $H^1(\Omega)$ and the continuity of $J$ ensure that $
\widetilde{J}(q_h)\rightarrow \widetilde{J}(q).
$ The proof can then be done as in Theorem \[thm:Cv\_discrete\_control\].
Numerical experiments {#sec:numerics}
=====================
In this section, we tackle numerically the optimization problem , when it is constrained to the total amplitude $\psi_{tot}$ described by . We focus on two examples: a *damping problem*, where the computed bathymetry optimally reduces the magnitude of the incoming waves; and an *inverse problem*, in which we recover the bathymetry from the observed magnitude of the waves. In what follows, we consider an incident plane wave $\psi_{0}(x)=\mathrm{e}^{{\mathrm{i}}k_0x\cdot \vec{d}}$ propagating in the direction $\vec{d}=(0\;\; 1)^{{\top}}$, with $$k_0 = \dfrac{\omega_0}{\sqrt{g z_0}},\,
\omega_0 = \dfrac{2\pi}{T_0}, \, T_0=20, \, g=9.81, \, z_0=3.$$ For the space domain, we set $\Omega=[0,L]^2$, where $L = \frac{10\pi}{k_0}$. We also impose a $L^\infty$-constraint on the variable $q$, namely that $q\geq -0.9$.
Numerical methods
-----------------
We discretize the space domain by using a structured triangular mesh of 8192 elements, that is a space step of $\Delta x=\Delta y=8.476472$.
For the discretization of $\psi_{sc}$, we use a $\mathbb{P}^1$-finite element method. The optimized parameter $q$ is discretized through a $\mathbb{P}^0$-finite element method. Hence, on each triangle, the approximation of $\psi_{sc}$ is determined by three nodal values, located at the edges of the triangle, and the approximation of $q$ is determined by one nodal value, placed at the center of gravity of the triangle.
On the other hand, we perform the optimization through a subspace trust-region method, based on the interior-reflective Newton method described in [@algo1] and [@algo2]. Each iteration involves the solving of a linear system using the method of preconditioned conjugate gradients, for which we supply the Hessian multiply function. The computations are achieved with `MATLAB` (version 9.4.0.813654 (R2018a)).
We emphasize that the setting of our numerical experiments presented below does not meet all the assumptions of Theorems \[thm:Cv\_discrete\_control\] and \[thm:Cv\_discrete\_control\_U\_lambda\_kappa\] which state [the convergence of the optimum of the discretized/discete problem toward the optimum of the continuous one]{}. Indeed, regarding Theorem \[thm:Cv\_discrete\_control\], we do not consider discrete optimization parameters that are piecewise affine bounded functions and the cost functions considered does not have the regularization term $\beta|Dq|(\Omega)$ with $\beta>0$. Concerning Theorem \[thm:Cv\_discrete\_control\_U\_lambda\_kappa\] we look for $q_h$ that are bounded and piecewise constant but we did not demand that $|Dq_h|(\Omega)\leq \kappa$ for some $\kappa>0$. Nevertheless, [we have observed in our numerical experiments that $|Dq^*_h|(\Omega)$ remains bounded when $h$ varies. ]{} We can thus [conjecture]{} that Theorem \[thm:Cv\_discrete\_control\_U\_lambda\_kappa\] actually applies to the two test cases considered in this paper.
Example 1: a wave damping problem
---------------------------------
We first consider the minimization of the cost functional $$J(q,\psi_{tot})=\dfrac{\omega_0^2}{2} \int_{\Omega_0}|\psi_{tot}(x,y)|^2dxdy,$$ where $\Omega_0=[\frac{L}{6},\frac{5L}{6}]^2$ is the domain where the waves are to be damped. The bathymetry is only optimized on a subset $\Omega_q=[\frac{L}{4},\frac{3L}{4}]^2\subset\Omega_0$.
The results are shown in Figure \[topo\_q\] for the bathymetry and Figure \[topo\_u\] for the wave. We observe that the optimal topography we obtain is highly oscillating. In our experiments, this oscillation remained at every level of space discretization we have tested. This could be related to the fact that in all our results, $q\in BV(\Omega)$. Note also that the damping is more efficient over $\Omega_q$. This fact is coherent with the results of the next experiment.
[0.475]{} ![Optimal topography for a wave damping problem. The yellow part represents $\Omega_0$ and the red part corresponds to the nodal points associated with $q$. []{data-label="topo_q"}](opti_topography_top "fig:"){width="\linewidth"}
[0.475]{} ![Optimal topography for a wave damping problem. The yellow part represents $\Omega_0$ and the red part corresponds to the nodal points associated with $q$. []{data-label="topo_q"}](opti_topography_bottom "fig:"){width="\linewidth"}
[0.75]{} ![Numerical solution of a wave damping problem. The yellow part represents $\Omega_0$ and the red part corresponds to the nodal points associated with $q$.[]{data-label="topo_u"}](abs_sol.png "fig:"){width="\linewidth"}
[0.475]{} ![Numerical solution of a wave damping problem. The yellow part represents $\Omega_0$ and the red part corresponds to the nodal points associated with $q$.[]{data-label="topo_u"}](incident.png "fig:"){width="\linewidth"}
[0.475]{} ![Numerical solution of a wave damping problem. The yellow part represents $\Omega_0$ and the red part corresponds to the nodal points associated with $q$.[]{data-label="topo_u"}](real_sol.png "fig:"){width="\linewidth"}
Example 2: an inverse problem
-----------------------------
Many inverse problems associated to Helmholtz equation have been studied in the literature. We refer for example to [@Colton; @Dorn; @Thompson] and the references therein. Note that in most of these papers the inverse problem rather consists in determining the location of a scatterer or its shape, often meaning that $q(x,y)$ is assumed to be constant inside and outside it. On the contrary, the inverse problem we consider in this section consists in determining a full real valued function. Given the bathymetry $$q_{ref}(x,y) \vcentcolon= {\mathrm{e}^{-\tau\left(((x-\tfrac{L}{4})^2+(y-\tfrac{L}{4})^2\right)}}+{\mathrm{e}^{-\tau\left((x-\tfrac{3L}{4})^2+(y-\tfrac{3L}{4})^2\right)}},$$ where $\tau= 10^{-3}$, we try to reconstruct it on the domain $\Omega_q = [\frac{L}{8},\frac{3L}{8}]^2\cup[\frac{5L}{8},\frac{7L}{8}]^2$, by minimizing the cost functional $$J(q,\psi_{tot})=\dfrac{\omega_0^2}{2}\int_{\Omega_0}|\psi_{tot}(x,y)-\psi_{ref}(x,y)|^2dxdy,$$ where $\psi_{ref}$ is the amplitude associated with $q_{ref}$ and $\Omega_0 = [\frac{3L}{4}-\delta,\frac{3L}{4}+\delta]^2$, $\delta=\frac{L}{6}$. Note that in this case, $\Omega_q$ is not contained in $\Omega_0$.
In Figure \[inv\_prob\], we observe that the part of the bathymetry that does not belong to the observed domain $\Omega_0$ is not recovered by the procedure. On the contrary, the bathymetry is well reconstructed in the part of the domain corresponding to $\Omega_0$.
[0.75]{} ![Detection of a bathymetry from a wavefield. The yellow part represents $\Omega_0$ and the red part corresponds to the nodal points associated with $q$.[]{data-label="inv_prob"}](inv_topography_top_error.png "fig:"){width="\linewidth"}
\
[0.475]{} ![Detection of a bathymetry from a wavefield. The yellow part represents $\Omega_0$ and the red part corresponds to the nodal points associated with $q$.[]{data-label="inv_prob"}](inv_topography_top_ex.png "fig:"){width="\linewidth"}
[0.475]{} ![Detection of a bathymetry from a wavefield. The yellow part represents $\Omega_0$ and the red part corresponds to the nodal points associated with $q$.[]{data-label="inv_prob"}](inv_topography_top_approx.png "fig:"){width="\linewidth"}
Acknowledgments {#acknowledgments .unnumbered}
===============
The authors acknowledge support from ANR Ciné-Para (ANR-15-CE23-0019) and ANR Allowap.
|
{
"pile_set_name": "ArXiv"
}
|
---
author:
- |
[**R.N. Garifullin$^{1,2}$, I.T. Habibullin$^{1,2}$ and R.I. Yamilov$^1$**]{}\
$^1$ Ufa Institute of Mathematics, Russian Academy of Sciences,\
112 Chernyshevsky Street, Ufa 450008, Russian Federation\
$^2$ Bashkir State University,\
32 Zaki Validi Street, Ufa 450074, Russian Federation\
[*E-mails: [mailto:[email protected]](mailto:[email protected]),*]{}\
[*[mailto:[email protected]](mailto:[email protected]), [mailto:[email protected]](mailto:[email protected])*]{}\
[*URL: <http://matem.anrb.ru/garifullinrn>,*]{}\
[*<http://matem.anrb.ru/habibullinit>, <http://matem.anrb.ru/en/yamilovri>*]{}
title: '**Peculiar symmetry structure of some known discrete nonautonomous equations**'
---
Introduction
============
We demonstrate that the generalized symmetry structure of some nonautonomous equations may be quite unusual by example of three known equations.
The first equation reads: Here $u_{n,m}$ is an unknown function depending on two discrete variables $\ n,m\in \Z$, while $\alpha_n,\beta_m$ are the given functions depending on one discrete variable. Eq. (\[abs\_h1\]) is the H1 equation of the Adler-Bobenko-Suris list [@abs03]. In the autonomous case, it is nothing but the discrete potential KdV equation, which has been known much earlier together with its $L-A$ pair, see e.g. [@nc95].
The second equation is the well-known dressing chain studied, e.g., in [@sy90; @s92; @vs93]: Here the unknown function $u_n=u_n(x)$ depends on one continuous $x$ and one discrete $n$ variables. The third equation is a completely discrete analogue of the dressing chain: which has been introduced in [@ly09]. In the autonomous case see, e.g., [@nc95; @ht95].
The equations and belong to the following class of discrete equations on the square lattice: $$F_{n,m}(u_{n+1,m},u_{n,m},u_{n,m+1},u_{n+1,m+1})=0. \label{gF}$$ In autonomous case, when the function $F_{n,m}$ does not depend on $n,m$ explicitly, all known integrable equations of this form have two hierarchies of generalized symmetries, and this property can be used as a criterion of integrability. Generalized symmetries in the $n$-direction have the form where $k\geq1$, and the number $k$ can be called the order of such symmetry. Generalized symmetries of an order $l\geq1$ in the $m$-direction have the form
In most of autonomous integrable cases, the simplest generalized symmetries in both directions have the orders $k=l=1$, see e.g. [@lpsy08; @ly11; @x09]. These symmetries correspond to integrable Volterra type equations of a complete list obtained in [@y83], see also the review article [@y06]. There are a few examples with the simplest symmetries of orders $k=l=2$, see [@a11; @mx13; @shl14]. Up to now there has been known the only example with an essentially asymmetric structure of generalized symmetries. It has been found in [@gy12], see also [@gmy14]. In that example, the orders of simplest symmetries are different ($k=2$ and $l=1$), and examples we discuss in this article will be asymmetric in the same sense.
As for the nonautonomous case, the situation is different. We know nonautonomous examples of the form with two hierarchies of generalized symmetries [@xp09; @gy14]. However, there are some known integrable nonautonomous equations which have only one hierarchy of generalized symmetries or have no hierarchy at all. This is the case of nonautonomous equations of the Adler-Bobenko-Suris list. It has been hypothesized in [@rh07], and this will be confirmed in the present paper, that there is no generalized symmetry in the $n$-direction when $\alpha_n$ is an arbitrary $n$-dependent function and no generalized symmetry in the $m$-direction when $\beta_m$ is an arbitrary $m$-dependent function.
In this paper, instead of arbitrary functions $\alpha_n, \beta_m$ in eq. , we consider the concrete ones. We look for functions $\alpha_n, \beta_m$, such that the corresponding equation has two hierarchies of generalized symmetries. We prove that symmetries of the form and exist if and only if $\alpha_n$ and $\beta_m$ are the periodic functions. We do this for some low orders $k$ and $l$ only. The orders $k$ and $l$ of the simplest generalized symmetries and depend on the periods of $\alpha_n$ and $\beta_m$ and may be arbitrarily high as well as different.
For eq. the picture is similar. In case of eq. , the form of symmetries is different, but the results are quite analogous too.
In case of the periodic coefficients $\alpha_n$ and $\beta_m$ in eqs. (\[abs\_h1\],\[dress\],\[d\_dress\]), whose periods may be arbitrarily large, we demonstrate that two hierarchies of generalized symmetries and conservation laws can be constructed by using known nonautonomous $L-A$ pairs of these equations. We do that, using a method presented in [@HY13].
It seems highly probable that two hierarchies of conservation laws also exist if and only if the coefficients of eqs. (\[abs\_h1\],\[dress\],\[d\_dress\]) are periodic. This property has been confirmed in a sense in [@rh07_1], where it has been shown for eq. that so-called five-point conservation laws disappear when the coefficients $\alpha_n$ and $\beta_m$ become nonconstant.
As a result of our investigation we come to an opinion that eqs. (\[abs\_h1\],\[dress\],\[d\_dress\]) with the periodic coefficients are “more integrable”. In this case we can derive, in both directions, generalized symmetries and conservation laws from their $L-A$ pairs, while in general case those $L-A$ pairs seem to be much more inconvenient for use.
In Section \[sec\_h1\] we prove a few theorems showing that two hierarchies of generalized symmetries of eq. exist only in the case of periodic coefficients. In particular, we construct an interesting example with a simplest generalized symmetry of the second order in one direction and of the third order in the second one. In Sections \[sec\_dr\],\[sec\_ddr\] we prove analogues theorems for eqs. (\[dress\],\[d\_dress\]). In Section \[sec\_theory\] we explain how to construct two hierarchies of generalized symmetries and conservation laws for eqs. (\[abs\_h1\],\[dress\],\[d\_dress\]) with the periodic coefficients. Examples of generalized symmetries of low orders together with their $L-A$ pairs are constructed for such equations in Section \[ex\_sym\]. Conservation laws of low orders are given in Section \[sec\_claws\]. The nature of some generalized symmetries of eqs. (\[abs\_h1\],\[dress\],\[d\_dress\]) with the periodic coefficients is discussed in Section \[sec\_sys\].
H1 equation {#sec_h1}
===========
We study in this section the H1 equation for which we use a natural assumption: for any $ n,m \in \Z$. In opposite case the equation becomes degenerate in some points, see e.g. [@ly11] for the autonomous case and [@gy14] for the nonautonomous one.
In the autonomous case, generalized symmetries of the Adler-Bobenko-Suris equations and of the H1 equation, in particular, were constructed in [@rh07; @ttx07; @lp07; @lps07]. Here we look for generalized symmetries of the H1 equation in the nonautonomous case and obtain, as a result, some statements on the symmetry structure of this equation.
Due to the invariance of eq. with respect to the involution $n\leftrightarrow m, \ \alpha_n\leftrightarrow\beta_m$, we can restrict ourselves to generalized symmetries of the form . According to its definition, see e.g. [@ly09], the symmetry of eq. must satisfy the relation on the solutions of eq. and for all $n,m\in\Z.$
It is natural to assume for the symmetries of order $k$ that $\Phi_{n,m}$ depends on both $u_{n+k,m}$ and $u_{n-k,m}$ at at least one point $n,m$. We prove theorems below under a stronger nondegeneracy condition for such generalized symmetries: In all known cases, if an equation of the form has a generalized symmetry of an order $k\geq1$, then it has the nondegenerate symmetry of the same order.
We find generalized symmetries by using a scheme developed in [@ly11; @ggh11]. Some annihilation operators [@h05] play an important role in this scheme.
Eq. (\[abs\_h1\]) has the following point symmetry: where $\nu_1,\nu_2,\nu_3$ are arbitrary constants. We write down below generalized symmetries up to this point one.
First and second order generalized symmetries
---------------------------------------------
Here we get some theoretical results in case of the first and second order generalized symmetries. The following result has been obtained in [@rh07], and we present it below for completeness.
Eq. (\[abs\_h1\],\[h1\_con\]) has a first order nondegenerate generalized symmetry in the $n$-direction iff $\alpha_n\equiv\alpha_{n-1}.$\[th\_h1\_1\]
#### Sketch of proof.
The compatibility condition for eqs. (\[abs\_h1\]) and (\[symk\]) implies: and we get the first part of the theorem. On the other hand, eq. (\[abs\_h1\]) with $\alpha_n\equiv\alpha_{n-1}$ has, for any $\beta_m$, the generalized symmetry which is nondegenerate.
[**Remark.**]{} The same result can be obtained under a weaker assumption instead of the nondegeneracy condition with $k=1$. For example, we can assume that there exists $m$, such that for any $n$. In this case we also derive $\alpha_n\equiv\alpha_{n-1}$ from eq. (\[con\_symp\]).
In the case $\alpha_n\equiv\alpha_{n-1}\equiv\alpha$ and $\beta_{m}\equiv\beta_{m-1}\equiv\beta$, there is one more symmetry: Any symmetry of the order $k\leq 1$ of eq. (\[abs\_h1\],\[h1\_con\]) with $\alpha_n\equiv\alpha_{n-1}$ is, up to a point symmetry , the following linear combination with constant coefficients $\mu_1,\mu_2$:
Eq. is a known integrable equation of the Volterra type [@y83; @y06]. Eq. is its known master symmetry [@asy00]. It generates generalized symmetries not only for eq. but also for the discrete equation (\[abs\_h1\],\[h1\_con\]) with $\alpha_n\equiv \alpha$ and an arbitrary $\beta_m$. For example, it can be checked by direct calculation that the following equation, constructed in the standard way, is the second order generalized symmetry for both of these equations.
Below we consider the two-periodic case: $\alpha_{n+1}\equiv\alpha_{n-1}$. There we have two possibilities. First of them is $\alpha_0=\alpha_1$, hence $\alpha_n\equiv\alpha$ is a constant, and we are led to the previous one-periodic case. In the second case $\alpha_0\not=\alpha_{1}$, then $\alpha_{n+1}\not =\alpha_{n}$ for any $n$.
The following two statements take place:
1. If eq. (\[abs\_h1\],\[h1\_con\]) has a nondegenerate generalized symmetry of order $2$ in the $n$-direction, then $\alpha_{n+1}\equiv \alpha_{n-1}$.
2. If $\alpha_n $ in eq. (\[abs\_h1\],\[h1\_con\]) satisfies the conditions then in the $n$-direction there exists the nondegenerate second order symmetry and there is no symmetry of the first order.
#### Proof.
We can derive from the compatibility condition (\[com\_con\]) the following relations: which provide the first part of the theorem. In case we have the second order symmetry where $c_{n+2}\equiv c_n$ is an arbitrary two-periodic function, and $\gamma_n=\alpha_{n+1}-\alpha_{n}\neq 0$ for any $n$. This formula yields the nondegenerate symmetries of order 2 (e.g. if $c_n\equiv 1$ or $c_n=2+(-1)^n$).
In the case $\beta_{m}\equiv\beta$ we have an additional symmetry: Any symmetry of the order $k\leq2$ of eq.(\[abs\_h1\],\[h1\_con\],\[p2n1\]) is a linear combination of (\[sp\],\[sym\_h1\_2\],\[ms\_h1\_2\]). For this reason, there is no first order symmetry.
Let us note that if $\alpha_{n+1}\equiv\alpha_{n}$, then eqs. (\[sym\_h1\_2\],\[ms\_h1\_2\]) turn into (\[sym\_nond\],\[mas\_h1\_1\]). The formula provides two linear independent and commuting symmetries of the discrete equation.
Eq. should be the master symmetry for , providing generalized symmetries of even orders not only for but also for eq. (\[abs\_h1\],\[h1\_con\],\[p2n1\]) with an arbitrary $\beta_m$. We have checked that by direct calculation in the first step, constructing a fourth order generalized symmetry.
An example with asymmetric symmetry structure {#asH1}
---------------------------------------------
We consider here eq. (\[abs\_h1\],\[h1\_con\]) satisfying the conditions We also require: This provides that $$\alpha_{n+2}-\alpha_n \equiv \alpha_{n-1}-\alpha_{n}\neq0,\quad \beta_m\neq\beta_{m-1}$$ for all $n,m$. Taking into account the condition , we see that all the five numbers $\alpha_0, \alpha_1, \alpha_2, \beta_0, \beta_1$ must be different.
There is in the $n$-direction the following generalized symmetry: It is of the order $k=3$ and is nondegenerate in particular cases, e.g. $a_n\equiv 1$. There are here three linear independent and commuting generalized symmetries. Any symmetry in the $n$-direction of an order $k\leq 3$ is a linear combination of eqs. and . That is why there is no symmetry of the orders $k=1$ and $k=2$.
There is in the $m$-direction the following generalized symmetry of the form (\[syml\]): It is of the order $l=2$ and is nondegenerate in particular cases, e.g. $b_m\equiv 1$. We have here two linear independent and commuting symmetries. Any symmetry in the $m$-direction of an order $l\leq 2$ is a linear combination of eqs. and . For this reason there is no symmetry of the order $l=1$.
The results can be formulated as follows:
Eq. (\[abs\_h1\],\[h1\_con\],\[p3p2\],\[alphaa\]) has in the $n$-direction a nondegenerate generalized symmetry of the order $k=3$ and has no symmetry of the orders $k=1,2$. This equation possesses in the $m$-direction a nondegenerate symmetry of the order $l=2$ and has no symmetry of the order $l=1$.
Dressing chain {#sec_dr}
==============
In this section we discuss the dressing chain . From the viewpoint of the generalized symmetry properties, equations of the form are the discrete analogues of hyperbolic type equations: $$u_{xy}=f(x,y,u,u_x,u_y).$$ Eq. belongs to the class of equations which are, in the same sense, the semidiscrete analogues of hyperbolic type equations. In the autonomous case, all integrable equations of these three classes should have two hierarchies of generalized symmetries in two different directions. In the paper [@y90] a number of autonomous examples of the form , including eq. with the constant $\alpha_n$, have been presented together with two generalized symmetries in two different directions.
Eq. is a nonautonomous representative of the class . A hierarchy of generalized symmetries in the $x$-direction exists for any $\alpha_n$, and the simplest equation has the form For any fixed $n$ this equation is nothing but the well-known modified Korteweg–de Vries equation. As it will be shown below, symmetries of eq. in the $n$-direction disappear in the generic case, i.e. when $\alpha_n$ is an arbitrary function. We will search the functions $\alpha_n$, such that generalized symmetries in the $n$-direction exist.
Generalized symmetries of eqs. in the $n$-direction of an order $k\ge 1$ have the form It is natural to assume for such symmetries that $\Phi_{n}$ depends on both $u_{n+k}$ and $u_{n-k}$ in at least one point $n$. We prove theorems below under the following nondegeneracy condition for the generalized symmetries: In all integrable cases we know, if an equation has a generalized symmetry of an order $k\geq1$, then it has the nondegenerate symmetry of the same order.
The generalized symmetry of eq. must satisfy to the compatibility condition on any solution of eq. and for all $n$.
First order generalized symmetries
----------------------------------
Eq. has a first order nondegenerate generalized symmetry in the $n$-direction iff $\alpha_n\equiv\alpha_{n+1}$\[th\_dres\_1\].
#### Proof.
We can derive from the compatibility condition the relations which provide the first part of the theorem. On the other hand, eq. with $\alpha_n\equiv\alpha_{n+1}$ has the following nondegenerate generalized symmetry:
In the autonomous case, both symmetries (\[mkdv\]) and of eq. have been found in [@y90].
There is one more generalized symmetry: and any symmetry of an order $k\le 1$ of eq. with $\alpha_n\equiv\alpha_{n+1}$ is a linear combination of (\[sym\_dr\_1\],\[ms\_dr\_1\]). Eq. provides the master symmetry for eq. . For example, is the generalized symmetry not only for eq. but also for the autonomous semidiscrete equation .
Second order generalized symmetries {#sec_or_dr}
-----------------------------------
The following two statements take place:
1. If eq. has a nondegenerate generalized symmetry of the second order in the $n$-direction, then $\alpha_{n+1}\equiv \alpha_{n-1}$.
2. If $\alpha_n $ of eq. (\[dress\]) satisfies the conditions then there exists in the $n$-direction a nondegenerate second order symmetry and there is no symmetry of the first order.
#### Proof.
The compatibility condition implies and we get the first part of the theorem.
In the case all symmetries of orders $k\leq2$ are described as follows: where $\gamma_n=\alpha_{n+1}-\alpha_{n}\neq 0$ for all $n$. The function $a_n$ is given by $a_n=b_n+cn,$ where $b_n$ is an arbitrary two-periodic function and $c$ is an arbitrary constant.
There are here nondegenerate examples of the second order, e.g. $a_n\equiv 1$ or $a_n=2+(-1)^n$, but there is no symmetry of the first order.
In the case when $a_n$ is the two-periodic function, i.e. $a_n=b_n$, we have in (\[sym\_dr\_2\]) two linear independent and commuting generalized symmetries. The linear case $a_n=cn$ provides the master symmetry for eq. with $a_n=b_n$. This master symmetry generates symmetries not only for eq. with $a_n=b_n$ but also for the semidiscrete equation (\[dress\],\[p2n11\]). We have checked that on the first step, constructing a fourth order generalized symmetry.
Discrete dressing chain {#sec_ddr}
=======================
In this section we discuss eq. with $\alpha_n\neq0$ for any $n$. In [@ly09] a complete analogue of the dressing chain has been introduced in the following form: If $d_m\neq0$ for all $m$, then after using the involution $n\leftrightarrow m$ and an obvious rescaling of $v_{n,m}$ we obtain the discrete equation with $\alpha_n=d_n^2\neq0$ for any $n$. This form is more comfortable for further investigation.
For any $\alpha_n$ there exists a hierarchy of generalized symmetries of eq. in the $m$-direction, and its simplest representative reads [@ly09]: For any fixed $n$ it obviously is the modified Volterra equation. We will look for functions $\alpha_n$, such that generalized symmetries in the $n$-direction exist.
It should be remarked that an integrable generalization of eq. has been presented in [@gy14] together with one hierarchy of generalized symmetries and an $L-A$ pair.
Simplest case
-------------
The following result has been obtained in [@gy14], and we present it here for completeness.
Eq. with $\alpha_n\neq0$ for any $n$ has a first order nondegenerate generalized symmetry of the form iff $\alpha_n\equiv\alpha_{n-1}.$\[th\_dd\_1\]
In the case $\alpha_n\equiv\alpha_{n-1}$, the following nondegenerate symmetry is known [@ly09]: We just can add that there is one more generalized symmetry in the $n$-direction: and any symmetry of an order $k\leq1$ of eq. is a linear combination of and .
Eq. is the known master symmetry of eq. [@cy95]. It provides generalized symmetries not only for but also for eq. .
Second order generalized symmetries {#second-order-generalized-symmetries}
-----------------------------------
The following two statements take place:
1. If eq. with $\alpha_n\neq0$ for all $n$ has a nondegenerate generalized symmetry of the second order in the $n$-direction, then $\alpha_{n+1}\equiv \alpha_{n-1}$.
2. If $\alpha_n\neq0 $ for any $n$ in eq. (\[d\_dress\]) and it satisfies the conditions then there exists in the $n$-direction a nondegenerate second order symmetry of eq. (\[d\_dress\]) and there is no symmetry of the first order.
#### Proof.
The compatibility condition implies: and we get the first part of the theorem.
In the case , all symmetries of orders $k\leq2$ in the $n$-direction are described as follows: where $\beta_n=\alpha_{n+1}/\alpha_n\neq1$ for all $n$. The function $a_n$ is given by $a_n=b_n+cn$, where $b_n$ is an arbitrary two-periodic function and $c$ is an arbitrary constant.
There are in (\[sym\_ddr\_2\]) nondegenerate examples of the second order, e.g. $a_n\equiv 1$ or $a_n=2+(-1)^n$, but there is no symmetry of the first order.
In the case when $a_n$ is the two-periodic function, i.e. $a_n=b_n$, we have in (\[sym\_ddr\_2\]) two linear independent and commuting generalized symmetries. The linear case $a_n=cn$ provides the master symmetry for eq. with $a_n=b_n$, which generates generalized symmetries for this equation. Those generalized symmetries should be the symmetries of the discrete equation (\[d\_dress\],\[p2n13\]) too, as the Lie algebra of symmetries should be closed under the operation of commutation, but the verification of this property is difficult even on the first step.
Method of the construction of generalized symmetries and conservation laws {#sec_theory}
===========================================================================
A method has been developed in [@HY13] for the autonomous and weakly nonautonomous discrete and semidiscrete equations, which allows one to construct generalized symmetries and conservation laws by using the $L-A$ pairs. That method is based on the formal diagonalization of an $L-A$ pair in the neighborhood of a stationary singular point. In this section we generalize that method to the case of the nonautonomous equations with periodic coefficients.
Formal diagonalization
----------------------
Let us first discuss the formal diagonalization in the case of systems of the linear discrete equations.
We consider a discrete linear vector equation of the form $$\label{linearequation}
\Psi_{n+k}=f_n({\bf u}_n,\lambda)\Psi_n, \quad k\geq1,$$ where $\Psi_n$ is an unknown vector, the matrix potential $f_n({\bf u}_n,\lambda)\in {\C^{s\times s}}$ is a meromorphic function of $\lambda\in{\C}$, and the vector function ${\bf u}_n$ is a functional parameter. A point $\lambda=\lambda_0$ is called the point of singularity of eq. (\[linearequation\]) if it is either a pole of $f_n({\bf u}_n,\lambda)$ or a solution of the equation $\det f_n({\bf u}_n,\lambda)=0$. It is assumed that the set of roots of the equation $\det f_n({\bf u}_n,\lambda)=0$, as well as the set of poles of $f_n$, does not depend on $n$.
We suppose here that eq. (\[linearequation\]) with the singular point $\lambda_0$ is reduced to the following special form: $$\label{lin_equation2}
\Psi_{n+k}=P_n({\bf u}_n,\lambda)Z\Psi_n,$$ where $Z$ is a diagonal matrix $Z=\diag((\lambda-\lambda_0)^{\gamma_1}, (\lambda-\lambda_0)^{\gamma_2},\dots , (\lambda-\lambda_0)^{\gamma_s})$ with integer exponents $\gamma_j$, such that $\gamma_1<\gamma_2<... <\gamma_s.$
It should be remarked that there is no proof that eq. can be transformed into the form . However, there is a general scheme which provides, as a rule, a transition from eq. to . That scheme has been presented in [@HY13]. Besides, it will be explained in detail in Section \[s\_3\_ex\] for three examples under consideration how to get the representation .
Let us rewrite eq. (\[lin\_equation2\]) as $L\Psi_n=\Psi_n$ with $$\label{dis_oper}
L=D_n^{-k}P_n({\bf u}_n,\lambda)Z,$$ where $D_n$ is the shift operator acting by the rule $D_n:\, n\rightarrow\, {n+1}$. The following statement on the formal diagonalization of the operator $L$ takes place, see [@HY13; @H85].
\[diagon\] Assume that, for any integer $n$ and for ${\bf u}_n$ ranging in a domain, the function $P_n({\bf u}_n,\lambda)$ is analytic in a neighborhood of the point $\lambda_0$, and all the leading principal minors ${\det_j}$ of the matrix $P_n({\bf u}_n,\lambda_0)$ do not vanish: $$\det_jP_n({\bf u}_n,\lambda_0)\neq0 \quad\mbox{for }\quad j=1,2,...,s\quad \mbox{and for all}\quad n.$$ Then there exists a formal series $T_n=\sum_{i\geq0} T_n^{(i)}(\lambda-\lambda_0)^i$, $\det T_n^{(0)}\neq0$, with the matrix coefficients, such that the operator $L_0=T^{-1}_nLT_n$ is of the form $L_0=D_n^{-k}h_nZ$, where $h_n=\sum_{i\geq0} h^{(i)}_n(\lambda-\lambda_0)^i$ is a formal series with the diagonal coefficients $h_n^{(i)}$, $\det h_n^{(0)}\neq0$.
The series $T_n$ is defined up to multiplication by a formal series with the diagonal coefficients. The latter can be chosen so that all the coefficients $T_n^{(i)}$ and $h_n^{(i)}$ depend on some finite sets of dynamical variables in $\left\{{\bf u}_p\right\}^{p=\infty}_{p=-\infty}$, which in turn depend on $i$.
For $k=1$ Theorem \[diagon\] has been proved in [@HY13]. In general case $k>1$, it can be proved by almost verbatim repeating a proof of [@HY13]. It should be remarked that there is in that proof an algorithm for recurrent construction of the coefficients $T_n^{(i)}$, $h_n^{(i)}$.
Diagonalization of the $L-A$ pair and construction of conservation laws {#sec_d_law}
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In this section we apply Theorem \[diagon\] to operators defining the $L-A$ pairs of discrete or semidiscrete scalar equations like eqs. (\[abs\_h1\],\[dress\]). We also explain how to derive a hierarchy of conservation laws from so-diagonalized operators. The same procedure can be used for analogous systems of discrete or semidiscrete equations.
First we consider a discrete equation and suppose that it is represented as the consistency condition of the following system of linear discrete equations: $$\label{lin_eq3}
\Psi_{n+p,m}=P_{n,m}([{ u}_{n,m}],\lambda)Z\Psi_{n,m}, \quad \Psi_{n,m+q}=R_{n,m}([{ u}_{n,m}],\lambda)\Psi_{n,m}.$$ Here the symbol $[{ u}_{n,m}]$ indicates that the matrix functions $P_{n,m}$ and $R_{n,m}$ depend on the dynamical variable ${ u}_{n,m}$ and on a finite number of its shifts. Note that the first equation in (\[lin\_eq3\]) is of the form (\[lin\_equation2\]). Let us suppose that it satisfies all the conditions of Theorem \[diagon\]. Then, due to the theorem, the operator $L=D_n^{-p}P_{n,m}Z$ is diagonalized by the conjugation by an appropriate formal series $T_{n,m}$. We assume that the potential $R_{n,m}([{ u}_{n,m}],\lambda)$ rationally depends on $\lambda$. Evidently, the consistency condition for the system (\[lin\_eq3\]) is equivalent to the commutativity condition for the operators $L$ and $M=D_m^{-q}R_{n,m}$.
It has been proved in [@HY13] that the operator $M$ commuting with $L$ is diagonalized by the conjugation with the same series $T_{n,m}$. Therefore we have a diagonal operator $M_0$ with the following representation as a formal series: The commutativity condition $[L_0,M_0]=0$, where $L_0=T_{n,m}^{-1}LT_{n,m} = D_n^{-p}h_{n,m}Z$, gives rise to the equation $$\label{cons_laws2}
h_{n,m+q}S_{n,m}=S_{n+p,m}h_{n,m},$$ as $[Z,S_{n,m}]=[Z,h_{n,m}]=0$. Eq. (\[cons\_laws2\]) implies the relation $(D^q_m-1)\log h_{n,m}=(D^p_n-1)\log S_{n,m}$. Here and below the notations $\log h_{n,m}$, $\log S_{n,m}$ mean that we apply the logarithm to coefficients of the diagonal matrices. Therefore the matrix function $H=(D^{q-1}_m+D^{q-2}_m+\dots +1)\log h_{n,m}$ is the generating function for conservation laws. In this way we get an infinite sequence of conservation laws for the discrete equation (\[gF\]) whenever the system (\[lin\_eq3\]) satisfies the conditions of Theorem \[diagon\].
In a similar way one can consider the semidiscrete model . Assume that the equation admits an $L-A$ pair of the form $$\label{lin_eq4}
\Psi_{n+k}=P_n([{ u_n}],\lambda)Z\Psi_n, \quad \Psi_{n,x}=A_n([{ u_n}],\lambda)\Psi_n.$$ Here the symbol $[{ u}_n]$ indicates the dependence on the dynamical variable ${ u}_{n}$ and on a finite number of its shifts and $x$-derivatives. Let us suppose that the first of eqs. (\[lin\_eq4\]) satisfies the conditions of Theorem \[diagon\]. The compatibility condition for eqs. takes the form $[L,D_x-A_n]=0$, where $L$ is given by (\[dis\_oper\]), and it is equivalent to the semidiscrete equation . According to Theorem \[diagon\] the operator $L$ is diagonalized by the conjugation transform $L_0=T_n^{-1}LT_n$. As it has been proved in [@HY13], the second operator $D_x-A_n$ is diagonalized by the same conjugation $D_x-B_n=T_n^{-1}(D_x-A_n)T_n$, where $B_n=-T_n^{-1}T_{n,x}+T_n^{-1}A_nT_n$ is a formal series with the diagonal coefficients. The commutativity condition $[L_0,D_x-B_n]=0$ of the diagonal operators implies an equation of the form $$\label{cons_laws3}
D_x\log h_n=(D_n^k-1)B_n,$$ which generates an infinite sequence of conservation laws. The diagonal operator $h_n$ is defined in Theorem \[diagon\].
Three examples {#s_3_ex}
--------------
Here we apply the above method to nonautonomous discrete and semidiscrete models with periodic coefficients. We show how to derive from the known $L-A$ pairs of eqs. (\[abs\_h1\],\[dress\],\[d\_dress\]) the representation . We also check that the conditions of Theorem \[diagon\] are satisfied.
### H1 equation {#h1-equation}
We consider eq. (\[abs\_h1\],\[h1\_con\]) satisfying the restrictions $$\label{per_h1}
\alpha_n\equiv\alpha_{n+N},\quad \beta_m\equiv\beta_{m+K} ,$$ An $L-A$ pair for this equation is known [@abs03], see also [@nc95] for the autonomous case. It can be written in the form where
Theorem \[diagon\] cannot be applied directly to any of the linear equations because their singularity points $\lambda=\alpha_n$ and $\lambda=\beta_m$ vary with the discrete variables $n,m$. However, instead of the $L-A$ pair (\[lax\_h1\]), one can use a compound one, for instance: with the new potential $$f_{n,m}=L^{(1)}_{n+N-1,m}L^{(1)}_{n+N-2,m}\dots L^{(1)}_{n,m}.$$ It has the singularity set $\{\infty, \alpha_0, \alpha_{1},\dots,\alpha_{N-1}\}$ which does not depend on the variables $n,m$.
Let us transform the first of eqs. (\[lax\_h1\_N\]) into the special form (\[lin\_equation2\]). To this end we factorize the matrix $L^{(1)}_{n,m}$ as follows: $$L^{(1)}_{n,m}=\delta_{n,m}Z\rho_{n,m},$$ where $Z=\diag(1,\lambda-\alpha_n)$ is the diagonal matrix and $\delta_{n,m},\,\rho_{n,m}$ have a triangular structure: $$\delta_{n,m}=\matrixx{1&0\\ -u_{n,m}+\frac{\lambda-\alpha_n}{u_{n+1,m}}&-\frac{1}{u_{n+1,m}}}, \quad \rho_{n,m}=\matrixx{-u_{n+1,m}&-1\\ 0&1}.$$ Then we change the unknown vector function: $\Psi_{n,m}=\rho^{-1}_{n,m}\Phi_{n,m}$. As a result eqs. take the form where $$P_{n,m}=\rho_{n+N,m} L^{(1)}_{n+N-1,m}L^{(1)}_{n+N-2,m}\dots L^{(1)}_{n+1,m} \delta_{n,m},\quad R_{n,m}=\rho_{n,m+1}L^{(2)}_{n,m}\rho^{-1}_{n,m}.$$
[**Important remark.**]{} In Theorem \[diagon\] the singular point $\lambda_0$ should be a constant. In eqs. the singular point $\alpha_n$ is not constant, but it is invariant under the action of the shift $D_n^{-N}$. This property also enables us to apply the theorem and to diagonalize the $L-A$ pair . Let us explain why the first equation of the $L-A$ pair satisfies the conditions of Theorem \[diagon\]. Evidently $P_{n,m}(\lambda)$ is a polynomial of $\lambda$ and therefore it is analytical around the point $\lambda=\alpha_n$. Its determinant is explicitly evaluated: $$\det P_{n,m}(\lambda)=\frac{u_{n+N+1,m}}{u_{n+1,m}}(\lambda-\alpha_{n+1})(\lambda-\alpha_{n+2})\dots (\lambda-\alpha_{n+N-1}).$$ At the point $\lambda=\alpha_n$ it is correctly defined and different from zero if the restrictions are valid and $u_{n,m}\neq0 $ for all $n,m$.
The leading principal minor $\det_1 P_{n,m}({\bf u},\lambda=\alpha_n)$, which is located at the left upper corner of the matrix, is a rational function of the coordinates of the point ${\bf u}=(u_{n,m},u_{n+1,m},...,u_{n+N+1,m})\in{\bf C}^{N+2}$, i.e. it is a ratio of two polynomials $Q_1({\bf u}) / Q_2({\bf u})$. Thus there are two possibilities: $Q_1({\bf u})\equiv0$, and then $\det_1 P_{n,m}( {\bf u},\lambda=\alpha_n)\equiv0$, or $Q_1({\bf u})$ is not identically zero. The first case is not realized, as the leading principal minor under consideration does not vanish at ${\bf u}_{0}=(1,0,0,...,0,1)$. This is easily seen from the following explicit formula: $$\det_1 P_{n,m}({\bf u}_{0},\lambda=\alpha_n)=(-1)^N\prod^{[N/2]}_{j=1}(\alpha_n-\alpha_{n+2j-1}).$$ So the polynomial $Q_1({\bf u})$ is nontrivial, and therefore a complete set of its zeros constitutes a manifold $M_1$ of a dimension not greater than $N+1$. Let us denote by $M_2$ the set of zeros of the denominator $Q_2({\bf u})$. Then the function $\det_1 P_{n,m}({\bf u},\lambda=\alpha_n)$ is defined and different from zero in the open domain $\C^{N+2}\backslash(M_1\cup M_2).$
Hence the first equation of the $L-A$ pair satisfies the conditions of Theorem \[diagon\]. We have proved that if $\alpha_n$ satisfies (\[per\_h1\],\[noteq\]) and $\beta_m$ is an arbitrary function, then the equation (\[abs\_h1\],\[h1\_con\]) admits an infinite sequence of conservation laws. In a similar way one can prove that eq. (\[abs\_h1\],\[h1\_con\]) with an arbitrary $\alpha_n$ and $\beta_m$ satisfying (\[per\_h1\],\[noteq\]) also possesses an infinite sequence of conservation laws. Certainly, if the restrictions (\[per\_h1\],\[noteq\]) are true for both $\alpha_n$ and $\beta_m$, then the equation admits two different hierarchies of conservation laws.
### Dressing chain {#dressing-chain}
The second example is the dressing chain (\[dress\]) obeying the constraint $$\label{per_dress}
\alpha_n\equiv\alpha_{n+N},\quad \mbox{such that} \quad \alpha_k\neq\alpha_{l} \quad \mbox{for} \quad 0\leq k<l\leq N-1.$$ Recall that the dressing chain is the compatibility condition [@sy90] of the following system of equations: with the potentials As this equation is of the hyperbolic type, it may have two hierarchies of conservation laws. One of them has been found in [@HY13] by diagonalizing the second of eqs. (\[lax\_dress\]) around the singular point $\lambda=\infty$ and without imposing on $\alpha_n$ any restriction.
In order to find the second hierarchy, we have to use the first equation of (\[lax\_dress\]). This part of the problem is much more complicated because the singular point $\lambda=-\alpha_n$ depends on $n$. In our opinion, the second hierarchy does exist only under an additional constraint, for instance, (\[per\_dress\]). In case of , one can avoid difficulties by passing from (\[lax\_dress\]) to a combined Lax pair: with the potential $$f_{n}=L^{(1)}_{n+N-1}L^{(1)}_{n+N-2}\dots L^{(1)}_{n}.$$ Here the potential $f_n$ has the set of singularity points $\{\infty, -\alpha_{0},-\alpha_{1},\dots ,-\alpha_{N-1}\},$ which does not depend on $n$ due to the periodicity condition (\[per\_dress\]).
Let transform the first of eqs. (\[lax\_dress\_N\]) into the special form (\[lin\_equation2\]). We first factorize the matrix $L_n^{(1)}$ as follows: $$L^{(1)}_{n}=\delta_{n}Z\rho_{n},$$ where $Z=\diag(1,\lambda+\alpha_n)$ and $$\delta_{n}=\matrixx{1&0\\ -u_{n}-\frac{\lambda+\alpha_n}{u_{n}}&-\frac{1}{u_{n}}}, \quad \rho_{n}=\matrixx{-u_{n}&1\\ 0&-1}.$$ Then, by changing the unknown vector function: $\Psi_{n}=\rho^{-1}_{n}\Phi_n$, one rewrites the linear system (\[lax\_dress\_N\]) in the following special form: \[lax\_dress\_NP\] where $P_{n}=\rho_{n+N} L^{(1)}_{n+N-1}L^{(1)}_{n+N-2}\dots L^{(1)}_{n+1} \delta_{n}$ and $R_{n}=\rho_{n,x}\rho^{-1}_{n}+\rho_{n}Y_{n}\rho^{-1}_{n}$.
The matrix valued function $P_n(\lambda)$ is analytic around the point $\lambda=-\alpha_n$, and its determinant evaluated at $\lambda=-\alpha_n$ is easily found: $$\det P_n(-\alpha_n)=\frac{u_{n}}{u_{n+N}}(\alpha_n-\alpha_{n+N-1})(\alpha_n-\alpha_{n+N-2})\dots (\alpha_n-\alpha_{n+1}).$$ It is correctly defined and different from zero if $u_n\neq0$ for all $n$. The leading principal minor $\det_1P_n({\bf u},-\alpha_n)$ is a rational function of the $N+1$-dimensional variable ${\bf u}=(u_n,u_{n+1},...,u_{n+N})$. We evaluate $\det_1P_n({\bf u}_0,-\alpha_n)$ at ${\bf u}_0=(u_n,0,...,0,u_{n+N})$ and get $$\det_1P_n({\bf u}_0,-\alpha_n)=-u_{n+N}\prod_{j=1}^{k}(\alpha_{n+2j-1}-\alpha_n)-u_{n}\prod_{j=1}^{k}(\alpha_{n+2j}-\alpha_n)\quad \mbox{if}\quad N=2k+1$$ or $$\det_1P_n({\bf u}_0,-\alpha_n)=\prod_{j=1}^{k}(\alpha_{n+2j-1}-\alpha_n)+u_{n}u_{n+N}\prod_{j=1}^{k-1}(\alpha_{n+2j}-\alpha_n)\quad \mbox{if}\quad N=2k.$$ The last formulas convince us that the rational function $\det_1P_n({\bf u},-\alpha_n)$ does not equal to zero identically. Hence there is an open domain in ${\C}^{N+1}$ in which $\det_1P_n({\bf u},-\alpha_n)$ does not vanish. So, all the conditions of Theorem \[diagon\] are satisfied, and the dressing chain (\[dress\]) with the periodic coefficients (\[per\_dress\]) admits the second hierarchy of conservation laws.
### Discrete dressing chain {#discrete-dressing-chain}
The third example is the discrete dressing chain (\[d\_dress\]) admitting the following Lax pair [@ly09]: Let us assume that the same periodicity constraint (\[per\_dress\]) on $\alpha_n$ is imposed. We transform the $L-A$ pair as: where $$f_{n,m}=L^{(1)}_{n+N-1,m}L^{(1)}_{n+N-2,m}\dots L^{(1)}_{n,m}.$$
We rewrite the first equation of (\[lax\_d\_dress\_N\]) in the special form (\[lin\_equation2\]). To this end the matrix $L^{(1)}_{n,m}$ is factorized as follows: $$L^{(1)}_{n,m}=\delta_{n,m}Z\rho_{n,m},$$ where $Z=\diag(1,\lambda+\frac{1}{4\alpha_n})$ and $$\delta_{n,m}=\matrixx{1&0\\ -\frac{2}{u_{n,m}-1}&1}, \quad \rho_{n,m}=\matrixx{1&2\lambda\alpha_n(u_{n,m}+1)\\ 0&\frac{u_{n,m}+1}{u_{n,m}-1}(1+4\lambda\alpha_n)}.$$ Then we pass to $\Psi_{n,m}=\rho^{-1}_{n,m}\Phi_{n,m}$ in the linear system (\[lax\_d\_dress\_N\]) and finally get where $P_{n,m}=\rho_{n+N,m} L^{(1)}_{n+N-1,m}L^{(1)}_{n+N-2,m}\dots L^{(1)}_{n+1,m} \delta_{n,m}$ and $R_{n,m}=\rho_{n,m+1}L^{(2)}_{n,m}\rho^{-1}_{n,m}$.
It is easy to check that $P_{n,m}(\lambda)$ satisfies all the conditions of Theorem \[diagon\], as it is analytical around the point $\lambda=-\frac{1}{4\alpha_n}$. The leading principal minors $\det P_{n,m}({\bf u},\lambda)$ and $\det_1 P_{n,m}({\bf u},\lambda)$ at $\lambda=-\frac{1}{4\alpha_n}$ are rational functions of ${\bf u}=(u_{n},u_{n+1},...,u_{n+N})\in{\C}^{N+1}$ and are not identically zero. Therefore they do not vanish in a domain in ${\C}^{N+1}$. So, as in preceding examples, there exists a hierarchy of conservation laws. Another hierarchy for eq. (\[d\_dress\]) exists with no restriction on the coefficient $\alpha_n$. It has been constructed in [@HY13] by diagonalization of the second equation of the $L-A$ pair .
Construction of generalized symmetries {#sec_sym}
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Results of the method of formal diagonalization can be successfully used for the calculation of generalized symmetries together with their $L-A$ pairs, and we explain here how to do that. Then in Section \[ex\_sym\] we apply this procedure to the H1 equation (\[abs\_h1\], \[h1\_con\], \[per\_h1\],\[noteq\]) and to the dressing chain (\[dress\],\[per\_dress\]).
We present a scheme applicable to all the three equations under consideration as well as to analogous ones. We describe it by example of the first operator of the $L-A$ pairs (\[lax\_h1\],\[lax\_dress\],\[lax\_d\_dress\]), and this operator is denoted here by $L_n^{(1)}$, as the index $m$ is inessential. The corresponding composite operator, we will use, is Let us denote by $ T_k$ a formal series which diagonalizes the operator $ L$ in the neighborhood of $\alpha_{n+k}$ with $0\leq k\leq N-1$. Then is a diagonal operator with coefficients which are series in powers of $\lambda-\alpha_{n+k}$. Let us describe a class of formal series $B_{n+k,K}=\sum_{j=-K}^{\infty}(\lambda-\alpha_{n+k})^jB_{n+k,K,j}$ satisfying the equation $$\label{bn}
[L,B_{n+k,K}]=0.$$ It can be proved [@HY13] that $B_{n+k,K}^{(0)}=T_k^{-1}B_{n+k,K}T_k$ is a formal series with diagonal coefficients: $$\label{b0_cond}
B_{n+k,K}^{(0)}=\sum_{j=-K}^{\infty}(\lambda-\alpha_{n+k})^jB_{n+k,K,j}^{(0)}, \quad \mbox{such that} \quad D_n^{N}B_{n+k,K,j}^{(0)}=B_{n+k,K,j}^{(0)}.$$ The converse is also true, namely, the series $B_{n+k,K}=T_kB_{n+k,K}^{(0)}T_k^{-1}$ solves the equation (\[bn\]) for any $B_{n+k,K}^{(0)}$ satisfying (\[b0\_cond\]).
Notice that the operator $L$ can be diagonalized around any of the singular points $\alpha_0,\, \alpha_{1},\,...,\alpha_{N-1}$, and thus a formal series $B_{n+k,K}$ commuting with $L$ can be constructed around any of these points. Therefore there is a class of objects of the form $B_K=\sum_{k=0}^{N-1}B_{n+k,K}$ solving the equation $[L,B_K]=0$. Cutting off simultaneously all the infinite sums in $B_K$ and finding an appropriate $X_K$ which does not depend on $\lambda$, we construct a rational function: It is remarkable that this function can be constructed in such a way that the operator equation $$\label{combined_Lax}
\frac{d}{dt}L=[L,A_n^{(K)}]$$ defines a generalized symmetry of equations under consideration.
We note that there is no general algorithm of finding the operator $X_K$ defining $A^{(K)}_n$ of . However in examples below, in Section \[ex\_sym\], we manage to find it.
The first equation of an $L-A$ pair obtained is rather complicated due to the structure of $L$: $$\label{comp_Lax_pair}
L\Psi_n=\Psi_n, \quad \Psi_{n,t}=A^{(K)}_n\Psi_n.$$ Let us explain now why it can be drastically simplified.
Due to the formula for $L$, eq. (\[combined\_Lax\]) is represented as: We denote $W=L^{(1)}_{n+N-2}L^{(1)}_{n+N-3}\dots L^{(1)}_{n}$, then this equation takes the form $$\frac{dL^{(1)}_{n+N-1}}{dt}W+L^{(1)}_{n+N-1}\frac{dW}{dt}=L^{(1)}_{n+N-1}WA^{(K)}_n-A^{(K)}_{n+N}L^{(1)}_{n+N-1}W.$$ Evidently it implies $$(L^{(1)}_{n+N-1})^{-1}(\frac{d}{dt}L^{(1)}_{n+N-1}+A^{(K)}_{n+N}L^{(1)}_{n+N-1}-L^{(1)}_{n+N-1}A^{(K)}_{n+N-1})=$$ $$=-(\frac{d}{dt}W+A^{(K)}_{n+N-1}W-WA^{(K)}_{n})W^{-1}=C_{n+N-1},$$ and we have $$\label{Ln}
\frac{d}{dt}L^{(1)}_{n}=L^{(1)}_{n}A^{(K)}_n-A^{(K)}_{n+1}L^{(1)}_{n}-L^{(1)}_{n}C_n.$$
In order to specify the structure of $C_n$, let us deduce for it an equation. To this end we replace the derivatives at the left hand side of (\[combin\_Lax2\]) by using eq. (\[Ln\]). After some elementary simplifications, one gets $$\begin{aligned}
\nonumber
C_{n+N-1}L^{(1)}_{n+N-2}L^{(1)}_{n+N-3}\dots L^{(1)}_{n}+L^{(1)}_{n+N-2}C_{n+N-2}L^{(1)}_{n+N-3}\dots L_{n}^{(1)}+\dots\\
+L^{(1)}_{n+N-2}L^{(1)}_{n+N-3}\dots L^{(1)}_{n}C_{n}=0. \label{equat_K}\end{aligned}$$ Calculating the operator $\Omega_{n+1} L_n^{(1)}-L^{(1)}_{n+N-1}\Omega_n$, where $\Omega_n$ is the left hand side of , we are led to the relation $f_nC_n-C_{n+N}f_n=0$ which is equivalent to $[L,C_n]=0$. By applying the conjugation , we find $[L_{0,k},C_{0,n,k}]=0$, where $C_{0,n,k}=T_k^{-1}C_nT_k$ is a diagonal matrix such that $D_n^NC_{0,n,k}=C_{0,n,k}$. On the other hand, by construction, $C_n$ is a rational matrix function of $\lambda$ depending on a finite number of the dynamical variables. According to the standard linear algebra theory, it can be diagonalized by applying a conjugation matrix $R$ which also depends, unlike $T_k$, on a finite set of the dynamical variables: $R^{-1}C_nR=C_{0,n,k}$.
Let us prove that $C_{0,n,k}$ is a scalar matrix, i.e. it is proportional to the unity matrix. Suppose the contrary, then the commutativity relation $C_{0,n,k}R^{-1}T_k=R^{-1}T_kC_{0,n,k}$ implies that the product $\hat T_k=R^{-1}T_k$ is a diagonal matrix, therefore $T_k=R\hat T_k$. The relation $L_{0,k}=T_k^{-1}LT_k$ implies the equation $R^{-1}LR=\hat T_kL_{0,k}\hat T_k^{-1}$, which shows that the operator $L$ is diagonalized by a matrix $R$ depending on a finite set of dynamical variables. This contradicts Theorem \[diagon\], and thus $C_{0,n,k}=C_n=c(n,\lambda)E$ is a scalar matrix. The equation (\[equat\_K\]) immediately gives $$\label{condit_K}
c(n,\lambda)+c(n+1,\lambda)+\dots +c(n+N-1,\lambda)=0.$$ It follows from the equation $[L,c(n,\lambda)]=0$ equivalent to $c(n,\lambda)=c(n+N,\lambda)$ that $c(n,\lambda)$ does not depend on the dynamical variables $u_{n+j}$.
Evidently eq. (\[Ln\]) can be reduced to the usual Lax equation. Indeed, fixing the value $n=n_0$ and setting $\hat A^{(K)}_n=A^{(K)}_n+\sum_{k=n_0}^{n-1}c(k,\lambda)$, we get the equation $$\label{final_lax}
\frac{d}{dt}L^{(1)}_{n}=L^{(1)}_{n}\hat A^{(K)}_n- \hat A^{(K)}_{n+1}L^{(1)}_{n}.$$ This equation is nothing but the compatibility condition for the linear system $$\Psi_{n+1}=L^{(1)}_{n}\Psi_{n},\quad \frac{d}{dt}\Psi_{n}=\hat A^{(K)}_n\Psi_{n}.$$ Let us emphasize that due to eq. (\[condit\_K\]) the term $\sum_{k=n_0}^{n-1}c(k,\lambda)$ contains not more than $N-1$ summands.
We hypothesize that the function $c(n,\lambda)$ is always zero, as in all examples below, but we cannot prove this fact.
Examples of generalized symmetries and their $L-A$ pairs {#ex_sym}
========================================================
In this section we apply the diagonalization procedure of Section \[sec\_sym\] to the H1 equation (\[abs\_h1\],\[h1\_con\]) and the dressing chain (\[dress\]). In some cases the generalized symmetries are given in Sections \[sec\_h1\],\[sec\_dr\], and we just construct corresponding $L-A$ pairs. In one case we find a new generalized symmetry together with its $L-A$ pair.
H1 equation {#h1-equation-1}
-----------
Let us consider the H1 equation (\[abs\_h1\],\[h1\_con\]) with an arbitrary coefficient $\beta_m$ and a periodic $\alpha_n$ satisfying the restriction . Following Section \[sec\_sym\], we construct $L-A$ pairs of the form: Here $A_{n,m}^{(K,N)}$ corresponds to $\hat A_{n}^{(K)}$ of the previous section, $N$ is the period of $\alpha_n$, and $K=1$ indicates the lowest term of a hierarchy of generalized symmetries. It turns out that the additional term $X_K$ of is equal to zero here.
In case of the period $N=1$, i.e. when $\alpha_n$ is a constant, the matrix $A^{(1,1)}_{n,m}$ reads: The compatibility condition for the $L-A$ pair is equivalent in this case to the generalized symmetry , and this is one more way to construct this symmetry.
In case of the period $N=2$, we obtain for the generalized symmetry an $L-A$ pair of the form defined by the following matrix: where by $\times$ we denote the matrix product.
Dressing chain {#dressing-chain-1}
---------------
Here we consider the dressing chain with a periodic $\alpha_n$ satisfying the restriction . As in previous examples, we construct $L-A$ pairs of the form where $N$ is the period of $\alpha_n$. It turns out that, in case of the dressing chain, the additional term $X_K$ of is a matrix with a nonzero element in the lower left corner only.
In the case $N=2$ we have found in Section \[sec\_or\_dr\] the generalized symmetry . Putting $c=0$ to exclude from consideration the master symmetry, we construct the $L-A$ pair defined by the following matrix: where $v_n=u_{n+1}+u_n$.
In the case $N=3$, let us denote by $a_n$ an arbitrary three-periodic function, so that $a_{n+3}\equiv a_n$, and let us introduce the notations where $\gamma_n\neq0$ for any $n$ due to . We construct a matrix $A_{n}^{(1,3)}$ determining the $L-A$ pair , which can be expressed as follows:
The compatibility condition for this $L-A$ pair is equivalent to the equation This is a new third order generalized symmetry of the dressing chain with the three-periodic coefficient $\alpha_n$. This symmetry is the lowest term of a hierarchy in this case.
Examples of conservation laws {#sec_claws}
=============================
In this section we apply the diagonalization procedure of Section \[sec\_d\_law\] to three equations under consideration with periodic coefficients and write down for them a number of conservation laws. The structure of those conservation laws essentially differs from the standard one, cf. [@HY13; @gmy14; @gy14].
Dressing chain {#dressing-chain-2}
--------------
We consider the dressing chain with a two-periodic coefficient $\alpha_n$ satisfying . In case of an arbitrary $\alpha_n$, one hierarchy of conservation laws has been constructed in [@HY13]. Conserved densities in that hierarchy depend on the $x$-derivatives of $u_n$, and those conservation laws can be called ones in the $x$-direction. Here we construct some conservation laws in the $n$-direction, which can be represented in the form: The functions $p^{(j)}_n$ and $q_n^{(j)}$ depend on the shifts of $u_n$, and first two pairs of them read: where and $\gamma_n\neq0$ for all $n$ due to .
The conserved density $p^{(0)}_n$ depends on three variables, while $p^{(1)}_n$ depends on five ones. The following conditions take place: for all $n$. In accordance with a general theory of [@ly97], the number of variables of such functions cannot be reduced. This shows, in particular, that two conservation laws defined by are essentially different. These conservation laws can be called the three- and five-point ones, respectively.
Discrete dressing chain {#discrete-dressing-chain-1}
-----------------------
Let us consider the discrete dressing chain with a two-periodic coefficient $\alpha_n\neq0$ for any $n$. In case of an arbitrary $\alpha_n$, one hierarchy of conservation laws has been constructed in [@HY13]. Conserved densities in that hierarchy depend on the $m$-shifts of $u_{n,m}$, and those conservation laws can be called ones in the $m$-direction. Here we construct some conservation laws in the $n$-direction, and those can be represented in the form: The functions $p^{(j)}_{n,m}$ and $q_{n,m}^{(j)}$ will depend on the $n$-shifts of $u_{n,m}$.
We use the notation where $\beta_n=\alpha_{n+1}/\alpha_n\neq1$ for any $n$ due to . Two simplest conservation laws are given by:
The following conditions are satisfied for all $n,m$: In accordance with some theoretical remarks of [@gmy14], the number of variables of such functions cannot be reduced, in particular, two above conservation laws are essentially different three- and five-point ones.
Asymmetric H1 equation
----------------------
We construct here conservation laws for the asymmetric example of Section \[asH1\], i.e. for the H1 equation with three-periodic coefficient $\alpha_n$ and two-periodic coefficient $\beta_m$. Recall that, in the autonomous case, conservation laws for the H1 equation have been found in [@rh07_1].
The diagonalization procedure in the neighborhood of the singular point $\alpha_n$ gives us a hierarchy of conservation laws of the form First two of them are defined by the following functions: Here we use the notations where $v_{n,m}$ and $\gamma_n$ are given in eq. .
Standard relations take place for all $n,m$: i.e. these conservation laws are essentially different. We have the five- and eight-point conservation laws in the $n$-direction, the simplest of those we can get by this procedure.
The diagonalization procedure in the neighborhood of the singular point $\beta_m$ allows one to construct the second hierarchy of conservation laws of the form First two conservation laws are defined by: where $w_{n,m}$ and $\delta_m$ are given in .
Standard relations are satisfied for all $n,m$: i.e. we have four- and six-point conservation laws in the $m$-direction.
We can see that the structure of conservation laws in different directions is quite different as well as one of generalized symmetries.
The nature of generalized symmetries {#sec_sys}
====================================
In this section we briefly discuss the nature of second order generalized symmetries obtained in Sections \[sec\_h1\]-\[sec\_ddr\]. Recently, some examples of discrete equations of the form have been obtained, whose simplest generalized symmetries in at least one of directions are of the second order as well. Most of those generalized symmetries [@a11; @mx13; @shl14] are similar to the Ito-Narita-Bogoyavlensky lattice equation. In one of such examples, the second order generalized symmetry is of the relativistic Toda type [@gy12; @gmy14]. The second order symmetries obtained in this paper are of the relativistic Toda type too.
The discrete-differential nonautonomous scalar equations with discrete periodic coefficients can be rewritten as autonomous systems, see [@ly97]. For example, eq. for any fixed value of $m$ can be represented as an autonomous system of two equations, as it has the two-periodic coefficients $c_n$ and $\gamma_n$. Introducing the notations we obtain the following system:
Here we have two similar and commuting with each other systems: the first one is defined by $A=1,\ B=0$ and the second one by $A=0,\ B=1$. According to their symmetry structure, such systems are similar to relativistic Toda type systems, cf. [@asy00 Section 5.1]. The system is an analogue of the well-known Ablowitz–Ladik example which is a linear combination of two commuting systems of equations of the relativistic Toda type (see, e.g., Section 5.2 in [@asy00]). Two other generalized symmetries and with $c=0$ are of the same kind. The case $c\neq0$ is not periodic and corresponds to the master symmetry.
In case of the system , we can illustrate the same property in a more explicit way. Let us consider the system with $A=1$ and $B=0$. It can be checked by direct calculation that each of the functions $v_k$ and $w_k$ satisfies, up to rescaling the time, the following lattice equation: This is the well-known equation of the relativistic Toda type, see e.g. the review articles [@asy00 Section 4.2] and [@y06 Section 3.3.3]. The same is true for the system with $A=0$ and $B=1$.
Note that any solution $u_{n,m}$ of the second order symmetry is transformed into a solution $\hat u_{n}$ of an equation of the form by the following formula: More precisely, the function $\hat u_{n}$ satisfies eq. with $c=0$ and slightly changed $\gamma_n$ and $b_n$. This shows that the symmetries with $c=0$ and are almost the same and have the same nature.
Conclusions
===========
In Sections \[sec\_h1\]-\[sec\_ddr\] we have proved a number of theorems which allow us to formulate the following hypothesis:
[**Hypothesis 1.**]{} [*The generalized symmetries of eqs. (\[abs\_h1\],\[dress\],\[d\_dress\]) in the $n$-direction exist if and only if the coefficient $\alpha_n$ is periodic. If $\alpha_n$ has the period $N$, then the simplest generalized symmetries of these equations have the order $N$.*]{}
As for conservation laws, we assume that a similar picture takes place:
[**Hypothesis 2.**]{} [*A hierarchy of conservation laws for eqs. (\[abs\_h1\],\[dress\],\[d\_dress\]) in the $n$-direction exists if and only if the coefficient $\alpha_n$ is periodic.*]{}
The case of the $m$-direction for eq. is analogous. The first hypothesis is substantiated in Sections \[sec\_h1\]-\[sec\_ddr\] in case of the first and second order generalized symmetries. For equations under consideration, which have periodic coefficients with an arbitrarily large period, both hypotheses are partially confirmed in Section \[sec\_theory\].
In this section we develop a theory for the case of nonautonomous discrete equations, which allows one, in particular, to construct generalized symmetries and conservation laws for eqs. (\[abs\_h1\],\[dress\],\[d\_dress\]) with periodic coefficients. In Sections \[ex\_sym\],\[sec\_claws\] we apply this theory to construct some examples.
The picture for all nonautonomous equations of the Adler-Bobenko-Suris list should be the same as for eq. .
We also come to an opinion that eqs. (\[abs\_h1\],\[dress\],\[d\_dress\]) with periodic coefficients are integrable in the same sense as the autonomous equations possessing $L-A$ pairs. The case of nonperiodic coefficients seems to be much more difficult from the standpoint of integrability.
#### Acknowledgments.
This work has been supported by the Russian Foundation for Basic Research (grant numbers: 13-01-00070, 14-01-97008-r-povolzhie-a).
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|
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[Some identities in algebras obtained by the Cayley-Dickson process]{}
$$$$
Cristina FLAUT and $\ $Vitalii SHPAKIVSKYI [ ]{}$$$$
**Abstract.** [In this paper we will prove that the Hall identity is true in all algebras obtained by the Cayley-Dickson process and, in some conditions, the converse is also true. ]{}$$$$
[Key Words: Cayley-Dickson process; Clifford algebras; Hall identity.]{}
**2000** **AMS**[ Classification: 15A66, 17A05, 17A20, 17A35, 17A45. ]{}$$$$
**0.** **Introduction**
$$$$
In October 1843, William Rowan Hamilton discovered the quaternions, a $4$-dimensional algebra over $\mathbb{R}$ which is associative and noncommutative algebra. In December $1843,$ John Graves discovered the octonions, an $8$-dimensional algebra over $\mathbb{R}$ which is nonassociative and noncommutative algebra. These algebras were rediscovered by Arthur Cayley in $1845$ and are also known sometimes as the Cayley numbers. This process, of passing from $\mathbb{R}$ to $\mathbb{C}$, from $\mathbb{C}$ to $\mathbb{H}$ and from$\ \mathbb{H}$ to $\mathbb{O}$ has been generalized to algebras over fields and over rings. It is called the *Cayley-Dickson doubling process* or the *Cayley–Dickson process.* In 1878, W. K. Clifford discovered Clifford algebras defined to have generators $e_{1},$ $e_{2},...,e_{n}$ which anti-commute and satisfy $e_{i}^{2}=a_{i}\in \mathbb{R},$ for all $i\in \{1,2,...,n\}.$ These algebras generalize the real numbers, complex numbers and quaternions( see \[Le; 06 \])
Even if are old, quaternions, octonions and Clifford algebras have at present many applications, as for example in physics, coding theory, computer vision, etc. For this reasons these algebras are intense studied. In \[Ha; 43\], Hall proved that the identity $\left( xy-yx\right)
^{2}z=z\left( xy-yx\right) ^{2}$ holds for all elements $x,y,z$ in a quaternion algebra. This identity is called *Hall identity*. Moreover, he also proved the converse: if the Hall identity is true in a skew-field $F,\ $then $F$ is a quaternion division algebra. In \[Smi; 50\], Smiley proved that the Hall identity is true for the octonions and he also proved the converse: if the Hall identity is true in an alternative division algebra $A,\ $then $A$ is an octonion division algebra.
In this paper we will prove that the Hall identity is true in all algebras obtained by the Cayley-Dickson process and, in some conditions, the converse is true for split quaternion algebras and split octonion algebras. $$$$
**1. Preliminaries**$$$$
In this paper, we assume that $K$ is a commutative field with $charK\neq 2$ and $A$ is an algebra over the field $K.$ The *center* $C$ of an algebra $A$ is the set of all elements $c\in A$ which commute and associate with all elements $x\in $ $A.$ An algebra $A$ is a *simple* algebra if $A$ is not a zero algebra and $\{0\}$ and $A$ are the only ideals of $A.$ The algebra $A$ is called *central simple* if the algebra $A_{F}=F\otimes _{K}A$ is simple for every extension $F$ of $K.$ A central simple algebra is a simple algebra. An algebra $A$ is called *alternative* if $x^{2}y=x\left( xy\right) $ and $xy^{2}=\left( xy\right) y,$ for all $x,y\in A,$ * flexible* if $x\left( yx\right) =\left(
xy\right) x=xyx,$ for all $x,y\in A$ and *power associative* if the subalgebra $<x>$ of $A$ generated by any element $x\in A$ is associative. $\
$Each alternative algebra is$\ $a flexible algebra and a power associative algebra. In each alternative algebra $A,$ the following identities1) $a(x(ay))=(axa)y$2) $((xa)y)a=x(aya)$3) $(ax)(ya)=a(xy)a$hold, for all $a,x,y\in A.$ These identities are called the *Moufang identities.*
A unitary algebra $A\neq K$ such that we have $x^{2}+\alpha _{x}x+\beta
_{x}=0$ for each $x\in A,$ with $\alpha _{x},\beta _{x}\in K,$ is called a *quadratic algebra*.
In the following, we briefly present the *Cayley-Dickson process* and the properties of the algebras obtained. For details about the Cayley-Dickson process, the reader is referred to $\left[ \text{Sc; 66}\right] $ and \[Sc; 54\].
Let $A$ be a finite dimensional unitary algebra over a field $\ K$ with a *scalar* *involution* $\,$$$\,\,\,\overline{\phantom{x}}:A\rightarrow A,a\rightarrow \overline{a},$$$\,\,$ i.e. a linear map satisfying the following relations:$\,\,\,\,\,$$$\overline{ab}=\overline{b}\overline{a},\,\overline{\overline{a}}=a,$$$\,\,$and $$a+\overline{a},a\overline{a}\in K\cdot 1\ \text{for all }a,b\in A.\text{ }$$The element $\,\overline{a}$ is called the *conjugate* of the element $a,$ the linear form$\,\,$$$\,\,t:A\rightarrow K\,,\,\,t\left( a\right) =a+\overline{a}$$and the quadratic form $$n:A\rightarrow K,\,\,n\left( a\right) =a\overline{a}$$are called the *trace* and the *norm *of the element $a,$ respectively$.$ Hence an algebra $A$ with a scalar involution is quadratic. $\,$
Let$\,\,\,\gamma \in K$ be a fixed non-zero element. We define the following algebra multiplication on the vector space $$A\oplus A:\left( a_{1},a_{2}\right) \left( b_{1},b_{2}\right) =\left(
a_{1}b_{1}+\gamma \overline{b_{2}}a_{2},a_{2}\overline{b_{1}}+b_{2}a_{1}\right) .$$We obtain an algebra structure over $A\oplus A,$ denoted by $\left( A,\gamma
\right) $ and called the *algebra obtained from* $A$* by the Cayley-Dickson process.* $\,$We have $\dim \left( A,\gamma \right) =2\dim A$.
Let $x\in \left( A,\gamma \right) $, $x=\left( a_{1},a_{2}\right) $. The map $$\,\,\,\overline{\phantom{x}}:\left( A,\gamma \right) \rightarrow \left(
A,\gamma \right) \,,\,\,x\rightarrow \bar{x}\,=\left( \overline{a}_{1},\text{-}a_{2}\right) ,$$is a scalar involution of the algebra $\left( A,\gamma \right) $, extending the involution $\overline{\phantom{x}}\,\,\,$of the algebra $A.$ Let $$\,t\left( x\right) =t(a_{1})$$and$\,\,\,$ $$n\left( x\right) =n\left( a_{1}\right) -\gamma n(a_{2})$$be $\,\,$the *trace* and the *norm* of the element $x\in $ $\left( A,\gamma \right) ,$ respectively.$\,$
If we take $A=K$ and apply this process $t$ times, $t\geq 1,\,\,$we obtain an algebra over $K,\,\,$$$A_{t}=\left( \frac{\alpha _{1},...,\alpha _{t}}{K}\right) . \tag{1.1.}$$By induction in this algebra, the set $\{1,e_{2},...,e_{n}\},n=2^{t},$ generates a basis with the properties:$$e_{i}^{2}=\gamma _{i}1,\,\,_{i}\in K,\gamma _{i}\neq 0,\,\,i=2,...,n
\tag{1.2.}$$and $$e_{i}e_{j}=-e_{j}e_{i}=\beta _{ij}e_{k},\,\,\beta _{ij}\in K,\,\,\beta
_{ij}\neq 0,i\neq j,i,j=\,\,2,...n, \tag{1.3.}$$$\ \beta _{ij}$ and $e_{k}$ being uniquely determined by $e_{i}$ and $e_{j}.$
From \[Sc; 54\], Lemma 4, it results that in any algebra $A_{t}$ with the basis $\{1,e_{2},...,e_{n}\}$ satisfying relations $\left( 1.2.\right) $ and $\left( 1.3.\right) $ we have:
$$e_{i}\left( e_{i}x\right) =\gamma _{i}^{2}=(xe_{i})e_{i}, \tag{1.4.}$$
for all $i\in \{1,2,...,n\}$ and for every $x\in A$
It is known that if an algebra $A$ is finite-dimensional, then it is a division algebra if and only if $A$ does not contain zero divisors. (See \[Sc;66\])
Algebras $A_{t}$ of dimension $2^{t}\ $obtained by the Cayley-Dickson process, described above, are central-simple, flexible and* *power associative for all $t\geq 1$ and, in general, are not division algebras for all $t\geq 1$. But there are fields on which, if we apply the Cayley-Dickson process, the resulting algebras $A_{t}\ $are division algebras for all $t\geq 1.$ (See \[Br; 67\] and \[Fl; 12\] ). We remark that the field $K$ is the center of the algebra $A_{t},\ \ $for $t\geq 2.$(See \[Sc; 54\])
Let $K$ be a field containing $\omega \,,\ $a primitive $n-$th root of unity, and $A$ be an associative algebra over $K.$ Let $S=\{e_{1},...,e_{r}\}$ be a set of elements in $A$ such that the following condition are fulfilled: $e_{i}e_{j}=\omega e_{j}e_{i}$ for all $i<j$ and $e_{i}^{n}\in \{1,\omega ,\omega ^{2},...,\omega ^{n-1}\}$. A *generalized Clifford algebra* over the field $K,$ denoted by $Cl_{r}^{n}\left( K\right) ,$ is defined to be the polynomial algebra $K[e_{1},...,e_{r}].$ We remark that the algebra $Cl_{r}^{n}\left( K\right) $ is an associative algebra. For details about generalized Clifford algebra, the reader is referred to \[Ki, Ou; 99\], \[Ko; 10\] and \[Sm; 91\].
**Example 1.1.** 1) For $n=2,$ we obtain $Cl_{r}^{2}\left( K\right) $ with $\omega =-1,$ $e_{i}e_{j}=-e_{j}e_{i}$ for all $i<j$ and $e_{i}^{2}\in \{-1,1\}.$ If $\ r=p+q$ and $e_{1}^{2}=...=e_{p}^{2}=1,$ $e_{p+1}^{2}=..=e_{q}^{2}=-1,$ then the algebra $Cl_{r}^{2}\left( K\right) $ will be denoted $Cl_{p,q}\left( K\right) .$
2\) i) For $p=q=0$ we have $Cl_{0,0}\left( K\right) \simeq K;$
ii\) For $p=0,q=1,$ it results that $Cl_{0,1}\left( K\right) $ is a two-dimensional algebra generated by a single vector $e_{1}$ such that $e_{1}^{2}=-1$ and therefore $Cl_{0,1}\left( K\right) \simeq K\left(
e_{1}\right) $. For $K=\mathbb{R}$ it follows that $Cl_{0,1}\left( \mathbb{R}\right) \simeq \mathbb{C}.$
iii\) For $p=0,q=2,$ the algebra $Cl_{0,2}\left( K\right) $ is a four-dimensional algebra spanned by the set $\{1,e_{1},e_{2},e_{1}e_{2}\}.$ Since $e_{1}^{2}=e_{2}^{2}=(e_{1}e_{2})^{2}=-1$ and $e_{1}e_{2}=-e_{2}e_{1},$ we obtain that this algebra is isomorphic to the division quaternions algebra $\mathbb{H}$.
iv\) For $p=1,q=1$ or $p=2,q=0,$ we obtain the algebra $Cl_{1,1}\left(
K\right) \simeq Cl_{2,0}\left( K\right) $ which is isomorphic with a split quaternion algebra, called *paraquaternion algebra* or *antiquaternion algebra*. (See \[Iv, Za; 05\])
$$$$
**2. Main results**$$$$
Let $A$ be an algebra obtained by the Cayley-Dickson process with the basis $\{e_{0}:=1,e_{1},...,e_{n}\}$ such that,$\ e_{m}e_{r}=-e_{r}e_{m},$ $r\neq
m,e_{m}^{2}=\gamma _{m}\in K,m\in \{1,2,...,n\}.$ For elements $a=\sum\limits_{m=0}^{n}a_{m}e_{m},b=\sum\limits_{m=0}^{n}b_{m}e_{m}$ we define an element in $K$, denoted by $T\left( a,b\right) ,$ $T\left(
a,b\right) =\sum\limits_{m=0}^{n}e_{m}^{2}a_{m}b_{m}.\ \ $We denote by $\overrightarrow{A}$ the set the elements $\{\overrightarrow{a}~\mid ~\overrightarrow{a}=\sum\limits_{m=1}^{n}a_{m}e_{m},a_{m}\in K\}.$ It results that the conjugate of the element $a$ can be written as $\overline{a}=a_{0}-~\overrightarrow{a}.$ Obviously, $\overrightarrow{\left( ~\overrightarrow{a}\right) }=~\overrightarrow{a}$ and $\overrightarrow{e_{m}}=e_{m}.\bigskip $
**Lemma 2.1.** *Let* $A$ *be an algebra obtained by the Cayley-Dickson process. The following equalities are true:*
1\) $$T\left( a,b\right) =T\left( b,a\right) ,$$ for all $a,b\in A.$
2\) $$T\left( \lambda a,b\right) =\lambda T\left( a,b\right) ,$$ for all $\lambda \in K,~a,b\in A.$
3\) $$T\left( a,b+c\right) =T\left( a,b\right) +T\left( a,c\right) ,$$ for all $a,b,c\in A.$
4\) $$T\left( a,\overline{a}\right) =a\overline{a}=n\left( a\right) ,$$for all $a\in A$
5\) $$\overrightarrow{a}\overrightarrow{b}=2T\left( \overrightarrow{a},\overrightarrow{b}\right) -\overrightarrow{b}\overrightarrow{a}, \tag{2.1.}$$$$ab=ba-2\overrightarrow{b}\overrightarrow{a}+2T\left( \overrightarrow{a},\overrightarrow{b}\right) , \tag{2.2.}$$
$$\overrightarrow{\overrightarrow{a}\overrightarrow{b}}=-T\left(
\overrightarrow{a},\overrightarrow{b}\right) +\overrightarrow{a}\overrightarrow{b}. \tag{2.3.}$$
$$(\overrightarrow{a})^{2}\in K, \tag{2.4.}$$
for all $a,b\in A.\medskip $
**Proof.**
Relation from 1), 2), 3), 4) are obvious.
5\) For $\overrightarrow{a}=\sum\limits_{m=1}^{n}a_{m}e_{m},\overrightarrow{b}=\sum\limits_{m=1}^{n}b_{m}e_{m}$ we obtain
$$\overrightarrow{a}\overrightarrow{b}\text{=}\sum\limits_{m=1}^{n}a_{m}e_{m}\cdot \sum\limits_{m=1}^{n}b_{m}e_{m}\text{=}\sum\limits_{m=1}^{n}e_{m}^{2}a_{m}b_{m}\text{+}\alpha \text{=}T\left(
\overrightarrow{a},\overrightarrow{b}\right) \text{+}\alpha ,\alpha \in
\overrightarrow{A}. \tag{2.5.}$$Computing $\overrightarrow{b}\overrightarrow{a}$, it follows that $$\overrightarrow{b}\overrightarrow{a}=T\left( \overrightarrow{a},\overrightarrow{b}\right) -\alpha ,\alpha \in \overrightarrow{A}. \tag{2.6.}$$If we add relations $\left( 2.5.\right) $ and $\left( 2.6.\right) $, it results $\overrightarrow{a}\overrightarrow{b}+\overrightarrow{b}\overrightarrow{a}=2T\left( \overrightarrow{a},\overrightarrow{b}\right) ,$ therefore relation $\left( 2.1.\right) $ is obtained.
For $a=a_{0}+\overrightarrow{a}$ and $b=b_{0}+\overrightarrow{b},$ we compute $$ab=\left( a_{0}+\overrightarrow{a}\right) \left( b_{0}+\overrightarrow{b}\right) =a_{0}b_{0}+a_{0}\overrightarrow{b}+b_{0}\overrightarrow{a}+\overrightarrow{a}\overrightarrow{b}$$and$$ba=\left( b_{0}+\overrightarrow{b}\right) \left( a_{0}+\overrightarrow{a}\right) =b_{0}a_{0}+b_{0}\overrightarrow{a}+a_{0}\overrightarrow{b}+\overrightarrow{b}\overrightarrow{a}.$$
Subtracting the last two relations and using relation $\left( 2.1.\right) $, we obtain $ab-ba=$ $\overrightarrow{a}\overrightarrow{b}-\overrightarrow{b}\overrightarrow{a}=2T\left( \overrightarrow{a},\overrightarrow{b}\right) -2\overrightarrow{b}\overrightarrow{a},$ then relation $\left( 2.2.\right) $ is proved.
Relation $\left( 2.3.\right) $ is obvious.
$\bigskip $For $\overrightarrow{a}=\sum\limits_{m=1}^{n}a_{m}e_{m},$ it results that $(\overrightarrow{a})^{2}=\sum\limits_{m=1}^{n}(a_{m})^{2}\in
K.\Box $
For quaternion algebras, the above result was proved in \[Sz; 09\].
**Theorem 2.2.** *Let* $A$ *be an algebra obtained by the Cayley-Dickson process such that* $e_{m}^{2}=-1,$* for all* $m\in \{1,2,...n\}$*. If *$n-1\in K-\{0\},$ *then, for all* $x\in A,$ *we have* $$\overline{x}=\frac{1}{1-n}\underset{m=0}{\overset{n}{\sum }}e_{m}xe_{m}.$$
**Proof.** Let $x=\underset{m=0}{\overset{n}{\sum }}e_{m}x_{m}$. From Lemma 2.1 and relation $\left( 1.4.\right) ,$ we obtain$\underset{m=0}{\overset{n}{\sum }}e_{m}xe_{m}=x+\underset{m=1}{\overset{n}{\sum }}e_{m}xe_{m}=$ $=x+\underset{m=1}{\overset{n}{\sum }}e_{m}\left( e_{m}x-2e_{m}\overrightarrow{x}+2T\left( e_{m},\overrightarrow{x}\right) \right) =$$=x+\underset{m=1}{\overset{n}{\sum }}e_{m}^{2}x-2\underset{m=1}{\overset{n}{\sum }}e_{m}^{2}\overrightarrow{x}+2\underset{m=1}{\overset{n}{\sum }}e_{m}^{2}e_{m}x_{m}=$$=x-nx+2n\overrightarrow{x}-2\underset{m=1}{\overset{n}{\sum }}e_{m}x_{m}=$$=\left( 1-n\right) x-2\left( 1-n\right) \overrightarrow{x}=\left(
1-n\right) \left( x-2\overrightarrow{x}\right) =$$=\left( 1-n\right) \overline{x}.\Box \medskip $
For the real quaternions, the above relation is well known:
$$\overline{x}=-\frac{1}{2}\left( x+ixi+jxj+kxk\right) .$$
**Theorem 2.3.** * Let* $A$ *be an algebra obtained by the Cayley-Dickson process. Then for all* $x,y,z\in A,$ *it results that* $$\left( xy-yx\right) ^{2}z=z\left( xy-yx\right) ^{2}. \tag{2.7.}$$
**Proof.**
We will compute both members of the equality $\left( xy-yx\right) ^{2}z$=$z\left( xy-yx\right) ^{2}.$ Using relation $\left( 2.2.\right) $ from Lemma 1 and since $T\left( \overrightarrow{x},\overrightarrow{y}\right) \in K$, we obtain $\left( -2\overrightarrow{y}\overrightarrow{x}+2T\left( \overrightarrow{x},\overrightarrow{y}\right) \right) ^{2}z=z\left( -2\overrightarrow{y}\overrightarrow{x}+2T\left( \overrightarrow{x},\overrightarrow{y}\right)
\right) ^{2}\Rightarrow $$\Rightarrow \left[ 4\left( \overrightarrow{y}\overrightarrow{x}\right)
^{2}+4T^{2}\left( \overrightarrow{x},\overrightarrow{y}\right) -8\left(
\overrightarrow{y}\overrightarrow{x}\right) T\left( \overrightarrow{x},\overrightarrow{y}\right) \right] z=$$=z\left[ 4\left( \overrightarrow{y}\overrightarrow{x}\right)
^{2}+4T^{2}\left( \overrightarrow{x},\overrightarrow{y}\right) -8\left(
\overrightarrow{y}\overrightarrow{x}\right) T\left( \overrightarrow{x},\overrightarrow{y}\right) \right] \Rightarrow $$\Rightarrow 4\left( \overrightarrow{y}\overrightarrow{x}\right)
^{2}z+4T^{2}\left( \overrightarrow{x},\overrightarrow{y}\right) z-8T\left(
\overrightarrow{x},\overrightarrow{y}\right) \left( \overrightarrow{y}\overrightarrow{x}\right) z=$$=4z\left( \overrightarrow{y}\overrightarrow{x}\right) ^{2}+4T^{2}\left(
\overrightarrow{x},\overrightarrow{y}\right) z-8T\left( \overrightarrow{x},\overrightarrow{y}\right) z\left( \overrightarrow{y}\overrightarrow{x}\right) .$Dividing this last relation by $4$ and after reducing the terms, it results$\left( \overrightarrow{y}\overrightarrow{x}\right) ^{2}z-2T\left(
\overrightarrow{x},\overrightarrow{y}\right) \left( \overrightarrow{y}\overrightarrow{x}\right) z=z\left( \overrightarrow{y}\overrightarrow{x}\right) ^{2}-2T\left( \overrightarrow{x},\overrightarrow{y}\right) z\left(
\overrightarrow{y}\overrightarrow{x}\right) .$ We denote $$\begin{aligned}
E &=&\left[ \left( \overrightarrow{y}\overrightarrow{x}\right) ^{2}z-z\left(
\overrightarrow{y}\overrightarrow{x}\right) ^{2}\right] - \\
&&-\left[ 2T\left( \overrightarrow{x},\overrightarrow{y}\right) \left(
\overrightarrow{y}\overrightarrow{x}\right) z-2T\left( \overrightarrow{x},\overrightarrow{y}\right) z\left( \overrightarrow{y}\overrightarrow{x}\right) \right]\end{aligned}$$and we will prove that $E=0.$We denote $\ $$$E_{1}=\left( \overrightarrow{y}\overrightarrow{x}\right) ^{2}z-2T\left(
\overrightarrow{x},\overrightarrow{y}\right) \left( \overrightarrow{y}\overrightarrow{x}\right) z$$and $$E_{2}=z\left( \overrightarrow{y}\overrightarrow{x}\right) ^{2}-2T\left(
\overrightarrow{x},\overrightarrow{y}\right) z\left( \overrightarrow{y}\overrightarrow{x}\right) .\newline$$First, we compute $E_{1}.$ We obtain$E_{1}=[\left( \overrightarrow{y}\overrightarrow{x}\right) ^{2}-2T\left(
\overrightarrow{x},\overrightarrow{y}\right) \left( \overrightarrow{y}\overrightarrow{x}\right) ]z.$ From Lemma 2.1., relation $\left( 2.3.\right) ,$ we have $\overrightarrow{y}\overrightarrow{x}$= $T\left( \overrightarrow{y},\overrightarrow{x}\right) +$ $\overrightarrow{\overrightarrow{y}\overrightarrow{x}}.$ Then $\left( \overrightarrow{y}\overrightarrow{x}\right) ^{2}=T^{2}\left(
\overrightarrow{y},\overrightarrow{x}\right) +\left( \overrightarrow{\overrightarrow{y}\overrightarrow{x}}\right) ^{2}+2T\left( \overrightarrow{y},\overrightarrow{x}\right) $ $\overrightarrow{\overrightarrow{y}\overrightarrow{x}}.$ Therefore$E_{1}$=$[T^{2}\left( \overrightarrow{y},\overrightarrow{x}\right) $+$\left(
\overrightarrow{\overrightarrow{y}\overrightarrow{x}}\right) ^{2}$++$2T\left( \overrightarrow{y},\overrightarrow{x}\right) \overrightarrow{\overrightarrow{y}\overrightarrow{x}}$-$2T\left( \overrightarrow{x},\overrightarrow{y}\right) \left( \overrightarrow{y}\overrightarrow{x}\right)
]z$=$=[T^{2}\left( \overrightarrow{y},\overrightarrow{x}\right) $+$\left(
\overrightarrow{\overrightarrow{y}\overrightarrow{x}}\right) ^{2}$++$2T\left( \overrightarrow{y},\overrightarrow{x}\right) (\overrightarrow{\overrightarrow{y}\overrightarrow{x}}-\overrightarrow{y}\overrightarrow{x})]z.$Since $\overrightarrow{\overrightarrow{y}\overrightarrow{x}}-\overrightarrow{y}\overrightarrow{x}=-T\left( \overrightarrow{y},\overrightarrow{x}\right) ,$ it results that $[\left( \overrightarrow{y}\overrightarrow{x}\right) ^{2}-2T\left(
\overrightarrow{x},\overrightarrow{y}\right) \left( \overrightarrow{y}\overrightarrow{x}\right) ]$==$[\left( \overrightarrow{\overrightarrow{y}\overrightarrow{x}}\right)
^{2}-T^{2}\left( \overrightarrow{y},\overrightarrow{x}\right) ]$=$\alpha \in
K,$from Lemma 2.1., relation $\left( 2.4.\right) .$ Hence $E_{1}=\alpha z.$ Now, we compute $E_{2}.$ We obtain$E_{2}=z[\left( \overrightarrow{y}\overrightarrow{x}\right) ^{2}-2T\left(
\overrightarrow{x},\overrightarrow{y}\right) \left( \overrightarrow{y}\overrightarrow{x}\right) ]=$$=z\alpha =\alpha z$ since $\alpha \in K.$
It follows that $E=E_{1}-E_{2}=0,$ therefore relation $\left( 2.7.\right) $ is proved. $\Box $
**Remark 2.4.** 1) Identity $\left( 2.7.\right) $ is called the *Hall identity*. From the above theorem, we remark that Hall identity is true for all algebras obtained by the Cayley-Dickson process.
2\) Relation $\left( 2.7.\right) $ can be written: $\left[ x,y\right] ^{2}z=z\left[ x,y\right] ^{2}$ or $\left[ \left[ x,y\right] ^{2},z\right] =0,$ where $\left[ x,y\right] =xy-yx$ is the commutator of two elements. If $A=\mathbb{H},$ then the identity $\left( 2.7.\right) $ is proved by Hall in \[Ha; 43\]. If $A=\mathbb{H}$ and, for example, $y=i,z=j,$ we have a quadratic quaternionic equation for which any quaternion is a root:$$xixk\text{+}kxix\text{+}ixixj-jxixi\text{+}x^{2}j-jx^{2}-ix^{2}k-kx^{2}i\text{=}0.$$
**Proposition 2.5.** *Let* $A$ *be an arbitrary algebra over the field* $K$* such that the relation* $\left( 2.7.\right) $* holds for all* $x,y,z\in A.$ *Then we have* $$\left[ \left[ x,y\right] \left[ u,y\right] ,z\right] \text{+}[[x,y][x,v],z]\text{+}[[u,y][x,y],z]+[[x,v][x,y],z]\text{=}0,\newline
\tag{2.8.}$$$$\left[ \lbrack x,v][u,y],z\right] \text{+}\left[ [u,y][x,v],z\right] \text{+}\left[ [x,y][u,v],z\right] \text{+}\left[ [u,v][x,y],z\right] \text{=}0,\newline
\tag{2.9.}$$$$\lbrack \lbrack u,y][u,v],z]\text{+}[[x,v][u,v],z]\text{+}[[u,v][u,y],z]\text{+}[[u,v][x,v],z]\text{=}0\newline
\tag{2.10.}$$*for all* $x,y,z,u,v\in A.\medskip $
**Proof.** $~$We linearize relation $\left( 2.7.\right) .$ Let $x,y,z\in A$ be three arbitrary elements such that $\left( xy-yx\right)
^{2}z=z\left( xy-yx\right) ^{2}.$ For $x+\lambda u,y+\lambda v,z$ we obtain$[\left( x+\lambda u\right) \left( y+\lambda v\right) -\left( y+\lambda
v\right) \left( x+\lambda u\right) ]^{2}z=$$=z[\left( x+\lambda u\right) \left( y+\lambda v\right) -\left( y+\lambda
v\right) \left( x+\lambda u\right) ]^{2}.$It results$[xy-yx$+$\lambda (uy+xv-yu-vx)$+$\lambda ^{2}\left( uv-vu\right) ]^{2}z=$$=z\left[ xy-yx\text{+}\lambda (uy+xv-yu-vx)\text{+}\lambda ^{2}\left(
uv-vu\right) \right] ^{2}.$ We obtain $\left( xy-yx\right) ^{2}z$+$\lambda ^{2}[\left( uy-yu\right) $+$\left(
xv-vx\right) ]^{2}z+$$+\lambda ^{4}\left( uv-vu\right) ^{2}z+$$+\lambda \lbrack \left( xy-yx\right) \left( \left( uy-yu\right) \text{+}\left( xv-vx\right) \right) ]z+$$+\lambda \lbrack \left( \left( uy-yu\right) +\left( xv-vx\right) \right)
\left( xy-yx\right) ]z+$$+\lambda ^{2}[\left( uv-vu\right) \left( xy-yx\right) ]z+$$+\lambda ^{2}[\left( xy-yx\right) \left( uv-vu\right) ]z+$$+\lambda ^{3}[[\left( uy-yu\right) $+$\left( xv-vx\right) ]\left(
uv-vu\right) ]z+$$+\lambda ^{3}[\left( uv-vu\right) [\left( uy-yu\right) $+$\left(
xv-vx\right) ]]z=$$z\left( xy-yx\right) ^{2}$+$\lambda ^{2}z[\left( uy-yu\right) $+$\left(
xv-vx\right) ]^{2}+$$+\lambda ^{4}z\left( uv-vu\right) ^{2}+$$+\lambda z[\left( xy-yx\right) \left( \left( uy-yu\right) \text{+}\left(
xv-vx\right) \right) ]+$$+\lambda z[\left( \left( uy-yu\right) \text{+}\left( xv-vx\right) \right)
\left( xy-yx\right) ]+$$+\lambda ^{2}z[\left( uv-vu\right) \left( xy-yx\right) ]+$$+\lambda ^{2}z[\left( xy-yx\right) \left( uv-vu\right) ]+$$+\lambda ^{3}z[[\left( uy-yu\right) $+$\left( xv-vx\right) ]\left(
uv-vu\right) ]+$$+\lambda ^{3}z[\left( uv-vu\right) [\left( uy-yu\right) $+$\left(
xv-vx\right) ]],$ for all $x,y,z,u,v\in A.$ Since the coefficients of $\lambda $ are equal in both members of the equality, we obtain:$[\left( xy-yx\right) \left( \left( uy-yu\right) \text{+}\left( xv-vx\right)
\right) ]z+$$+[\left( \left( uy-yu\right) \text{+}\left( xv-vx\right) \right) \left(
xy-yx\right) ]z=$$=z[\left( xy-yx\right) \left( \left( uy-yu\right) \text{+}\left(
xv-vx\right) \right) ]+$$+z[\left( \left( uy-yu\right) +\left( xv-vx\right) \right) \left(
xy-yx\right) ].$ We can write this last relation under the form:$\{\left[ x,y\right] \left[ u,y\right] \}z$+$\{[x,y]\left[ x,v\right] \}z+$$+\{\left[ u,y\right] [x,y]\}z+\{\left[ x,v\right] \left[ x,y\right] \}z=$$=z\{\left[ x,y\right] \left[ u,y\right] \}+z\{[x,y]\left[ x,v\right] \}+$$+z\{\left[ u,y\right] [x,y]\}+z\{\left[ x,v\right] \left[ x,y\right] \}.$ It results$\left[ \left[ x,y\right] \left[ u,y\right] ,z\right] $+$[[x,y][x,v],z]$+$[[u,y][x,y],z]$+$[[x,v][x,y],z]$=$0$and we obtain relation $\left( 2.8.\right) .$Since the coefficients of $\lambda ^{2}$ are equal in both members of the equality, we obtain:$[\left( uy-yu\right) +\left( xv-vx\right) ]^{2}z+$$+[\left( uv-vu\right) \left( xy-yx\right) ]z+$$+[\left( xy-yx\right) \left( uv-vu\right) ]z=$$=z[\left( uy-yu\right) +\left( xv-vx\right) ]^{2}+$$+z[\left( uv-vu\right) \left( xy-yx\right) ]+$$+z[\left( xy-yx\right) \left( uv-vu\right) ].$It results that$[\left( uy-yu\right) \left( xv-vx\right) ]z$+$[\left( xv-vx\right) \left(
uy-yu\right) ]z+$$+[\left( uv-vu\right) \left( xy-yx\right) ]z$+$[\left( xy-yx\right) \left(
uv-vu\right) ]z=$$z[\left( uy-yu\right) \left( xv-vx\right) ]$+$z[\left( xv-vx\right) \left(
uy-yu\right) ]+$$+z[\left( uv-vu\right) \left( xy-yx\right) ]$+$z[\left( xy-yx\right) \left(
uv-vu\right) ].$ We can write this last relation under the form:$\left[ \left[ x,v\right] \left[ u,y\right] ,z\right] $+$\left[ \left[ u,y\right] \left[ x,v\right] ,z\right] $+$\left[ \left[ x,y\right] \left[ u,v\right] ,z\right] $+$\left[ \left[ u,v\right] \left[ x,y\right] ,z\right] $=$0$ and we obtain relation $\left( 2.9.\right) .$Since the coefficients of $\lambda ^{3}$ are equal in both members of the equality, we obtain:$[[\left( uy-yu\right) +\left( xv-vx\right) ]\left( uv-vu\right) ]z+$$+[\left( uv-vu\right) [\left( uy-yu\right) +\left( xv-vx\right) ]]z=$$=z[[\left( uy-yu\right) +\left( xv-vx\right) ]\left( uv-vu\right) ]+$$+z[\left( uv-vu\right) [\left( uy-yu\right) +\left( xv-vx\right) ]].$ We can write this last relation under the form:$[[u,y][u,v],z]$+$[[x,v][u,v],z]$+$[[u,v][u,y],z]$+$[[u,v][x,v],z]$=$0$ and we obtain relation $\left( 2.10.\right) .$ $\Box \bigskip $
**Remark 2.6.** 1) In \[Ti; 99\] and \[Fl; 01\] some equations over division quaternion algebra and octonion algebra are solved: in \[Fl; 01\] for general case, when $K$ is a commutative field with $charK\neq 2$ and $\gamma _{m}$ are arbitrary and in \[Ti; 99\] for $K=\mathbb{R},$ $\gamma
_{m}=-1,$ with $m\in \{1,2\}$ for quaternions and $m\in \{1,2,3\}$ for octonions. Let $A$ be such an algebra. For example, equation $$ax=xb,a,b,x\in A, \tag{2.11.}$$for $a\neq \overline{b}$ has general solution under the form $x=\overrightarrow{a}p+p\overrightarrow{b},$ for arbitrary $p\in A.$
2\) In \[Fl, Şt; 09\], authors studied equation $x^{2}a=bx^{2}+c,a,b,c\in
A, $ where $A$ is a generalized quaternion division algebra or an generalized octonion division algebra. If $A$ is an arbitrary algebra obtained by the Cayley-Dickson process and $\ a,b,c\in A$ with $a=b$ and $c=0,$ then, from Theorem 2.3., it results that this equation has infinity of solutions of the form $x=vw-wv,$ where $v,w\in A.\bigskip $
**Proposition 2.7.** *Let* $A$ *be a quaternion algebra or an octonion algebra. Then for all* $x,y\in A,$ *there are the elements* $z,w$ *such that* $(xy-yx)^{2}=\overrightarrow{z}w+w\overrightarrow{z}.\medskip $
**Proof.** Let $z$ be an arbitrary element in $A-K.$ From Theorem 2.3., we have that $\left( xy-yx\right) ^{2}z=z\left( xy-yx\right) ^{2},$ for all $x,y,z\in A.$ Since $z\neq \overline{z}$ and $\left( xy-yx\right) ^{2}$ is a solution for the equation $\left( 2.11.\right) ,$ from Remark 2.6., it results that there is an element $w\in A$ such that $(xy-yx)^{2}=\overrightarrow{z}w+w\overrightarrow{z}.\Box \bigskip $
**Proposition 2.8.** *Let* $A$ *be a finite dimensional unitary algebra with a scalar involution* $$\,\,\,\overline{\phantom{x}}:A\rightarrow A,a\rightarrow \overline{a},$$*such that for all* $x,y\in A,$ *the following equality holds:*
$$(x\overline{y}+y\overline{x})^{2}=4\left( x\overline{x}\right) \left( y\overline{y}\right) . \tag{2.12.}$$
*Then the algebra* $A$ *has dimension* $1.\medskip $
**Proof.** We remark that $x\overline{y}+y\overline{x}=x\overline{y}+\overline{x\overline{y}}\in K.$ First, we prove that $[x\overline{y}+y\overline{x}]^{2}=4\left( x\overline{x}\right) \left( y\overline{y}\right)
,\forall x,y\in A,$ if and only if $x=ry,$ $r\in K.$ If $x=ry,$ then relation $\left( 2.12.\right) $ is proved. Conversely, assuming that relation $\left( 2.12.\right) $ is true and supposing that there is not an element $r\in K$ such that $x=ry,$ then for each two non zero elements $a,b\in K,$ we have $ax+by\neq 0.$ Indeed, if $ax+by=0,$ it results $x=-\frac{b}{a}y,$ false. We obtain that $$\left( ax+by\right) \overline{\left( ax+by\right) }\neq 0. \tag{2.13.}$$Computing relation $\left( 2.13.\right) ,$ it follows $$a^{2}\left( x\overline{x}\right) +abx\overline{y}+bay\overline{x}+b^{2}y\overline{y}\neq 0. \tag{2.14.}$$If we put $a=y\overline{y}$ in relation $\left( 2.14.\right) $ and then simplify by $a,$ it results $$\left( y\overline{y}\right) \left( x\overline{x}\right) +bx\overline{y}+by\overline{x}+b^{2}\neq 0. \tag{2.15.}$$Let $b=-\frac{1}{2}\left( x\overline{y}+y\overline{x}\right) \in K,b\neq 0.$ If we replace this value in relation $\left( 2.15.\right) ,$ we obtain $4\left( x\overline{x}\right) \left( y\overline{y}\right) -(x\overline{y}+y\overline{x})^{2}\neq 0,$ which it is false. Therefore, there is an element $r\in K$ such that $x=ry.$
Assuming that the algebra $A$ has dimension greater or equal with $2,$ it results that there are two linearly independent vectors, $v$ and $w,$ respectively$.$ Since relation $\left( 2.12.\right) $ is satisfies for $v$ and $w$, we obtain that there is an element $s\in K$ such that $v=sw,$ which it is false. Hence $\dim A=1.\Box \bigskip $
**Proposition 2.9.** *Let* $A$ *be an alternative algebra over the field* $K$ *whose center is* $K.$ *If* $\left( xy-yx\right) ^{2}z=z\left( xy-yx\right) ^{2}$ * for all* $x,y,z\in A,$ *then* $A$ *is a quadratic algebra*.
**Proof.** Let $x,y\in A$ such that $xy\neq yx.$ If we denote $z=xy-yx,$ it follows that $z^{2}$ commutes with all elements from $A,$ then $z^{2}$ is in the center of $A.$ We obtain $z^{2}=\alpha \in K^{\ast }.$ For $t=x^{2}y-yx^{2}$ it results that $t^{2}=(x^{2}y-yx^{2})^{2}\in K$ and $t=\left( xy-yx\right) x+x\left( xy-yx\right) =zx+xz.$ We have $zt=z\left(
zx+xz\right) =z^{2}x+zxz=\alpha x+zxz$ and $tz=\left( zx+xz\right)
z=zxz+xz^{2}=\alpha x+zxz.$ Therefore $tz=zt.$ For $z+t=\left(
x^{2}+x\right) y-y\left( x^{2}+x\right) $ $\ $ we have that $\left(
z+t\right) ^{2}=\beta \in K,$ then $z^{2}+t^{2}+2tz=\beta ,$ hence $tz=\gamma \in K.$ Since $\ zx=x(yx)-\left( yx\right) x,$ it follows that $(zx)^{2}=\delta \in K.$ If we multiply the relation $\left( zx\right) \left(
zx\right) =\delta $ with $z$ in the left side, we obtain $z\left( \left(
zx\right) \left( zx\right) \right) =\delta z.$ Using alternativity and then flexibility, it results $\left( z^{2}x\right) \left( zx\right) =\delta z,$ therefore $\ \ \alpha \left( xzx\right) =\delta z,$ hence $xzx=\theta z,$ where $\theta =\alpha ^{-1}\delta .$ It follows that $z\left( xzx\right)
=\theta z^{2}=\theta \alpha \in K.$ Since $z\left( xzx\right) =\left(
zxz\right) x,$ from Moufang identities, we have that $\left( zxz\right)
x=\theta \alpha \in K.$ It results that $\gamma x=\left( tz\right) x=\left(
\alpha x+zxz\right) x=\alpha x^{2}+\left( zxz\right) x=\alpha x^{2}+\theta
\alpha ,$ hence $x^{2}=ax+b,$ where $a=\alpha ^{-1}\gamma ,b=-\theta .$ We obtain that $A$ is a quadratic algebra.$\Box $
When $A$ is a division associative algebra, this proposition was proved by Hall in \[Ha; 43\].
**Theorem 2.10.** *Let* $\ A$ *be a alternative algebra such that the center of* $A$ *is* $K$ *and* $\left(
xy-yx\right) ^{2}z=z\left( xy-yx\right) ^{2},$ * for all* $x,y,z\in
A.$
1\) *If* $A$ *is a division algebra*, *then* $A=K$ *or* $A=A_{t},t\in \{1,2,3\}$, *where* $A_{t}$ *is a division algebra obtained by the Cayley-Dickson process.*
2\) *If* $A$ *is not a division algebra* *and* *there are two elements* $y,z\in A$ *such that* $y^{2},z^{2}\in K,\
yz=-zy,$ *then* $A$ *is* *a generalized split quaternion algebra* * or* $A$ *is* *a generalized split octonion algebra.*
**Proof.** 1) From Proposition 2.9., it results that $A$ is a quadratic algebra, therefore, from \[Al; 49\], Theorem 1, we have $\dim A\in
\{ $ $1,2,4,8\}.$ If $\dim A=1,$ then $A=K.$ If $\dim A=2,$ since the center is $K,$ then we can find an element $x\in A-K$ such that $x^{2}\in K.$ It results that the set $\{1,x\}$ is a basis in $A,$ therefore $A=K\left(
x\right) $ is a quadratic field extension of the field $K.$ If $\dim A=4,$ from \[Al; 39\], p. 145, we have that there are two elements $x,y\in A$ such that $x^{2}=x+a$ with $\ 4a+1\neq 0,$ $xy=y\left( 1-x\right)
,y^{2}=b,a,b\in K.$ Denoting $z=x-\frac{1}{2},$ we obtain that $z^{2}=\left(
x-\frac{1}{2}\right) ^{2}=a-\frac{1}{4}\in K.$ and $\ zy=-yz.$ Since $zy=\left( x-\frac{1}{2}\right) y=xy-\frac{y}{2}=y-yx-\frac{y}{2}=\frac{y}{2}-yx$ and $yz=y\left( x-\frac{1}{2}\right) =yx-\frac{1}{2},$ we have $yz=-zy$ then $\left( yz\right) ^{2}\in K.$ It follows that in the algebra $A$ we can find the elements $\ y.z$ such that $y^{2},z^{2},\left( yz\right)
^{2}\in K$ and $yz=-zy.$ Therefore, from \[Al; 49\], Lemma 4, it results that $A$ is a generalized division quaternion algebra. If $A$ has dimension $8$, denoting with $Q$ the algebra $Q=K+yK+zK+yzK,$ from \[Al; 49\], Lemma 3, Lemma 4 and Lemma 5, it results that there are the elements $w\in A-Q,$ such that $w^{2},\left( yw\right) ^{2},\left( zw\right) ^{2}\in K,yw=-wy,zw=-wz,yz=-zy.$ It follows that $A=K+yK+zK+yzK+wK+wyK+wzK+w(yz)K\ $ is a generalized division octonion algebra.
2\) From the above, it results that $A=Q=K+yK+zK+yzK$ is a generalized quaternion algebra, which is split from hypothesis. or there is an element $w\in A-Q$ such that $A=K+yK+zK+yzK+wK+wyK+wzK+w(yz)K.\ $ In the last case, $A $ is a generalized split octonion algebra.$\Box \bigskip $
**Corollary 2.11.** *Let* $\ A$ *be a non-division associative algebra such that the center of* $A$ *is* $K.$ *If in algebra* $A$ *we have* $\ \left( xy-yx\right) ^{2}z=z\left(
xy-yx\right) ^{2}$ * for all* $x,y,z\in A$ *and* *there are two elements* $v,w$ *such that* $v^{2},w^{2}\in K,$ $vw=-wv, $ *then* $A$ *is* *a generalized split quaternion algebra.*$\Box \bigskip $
**Example 2.12.**
1\) Using notations given in Preliminaries, if in Theorem 2.10., we have $t=1$ and $\alpha _{1}=-1,$ it results that $A=Cl_{0,1}\left( K\right) $ is a quadratic field extension of the field $K.$ If $t=2$ and $\alpha _{1}=\alpha
_{2}=-1,$ we have that $A=Cl_{0,2}\left( K\right) $ is a quadratic division quaternion algebra$.$
2\) If we have $v^{2},w^{2}\in \{-1,1\}$ in Corollary 2.11$,$ then $A=Cl_{1,1}\left( K\right) \simeq Cl_{2,0}\left( K\right) .$$$$$
**Conclusions.** In this paper we proved that the Hall identity is true in all algebras obtained by the Cayley-Dickson process and that the converse is also true, in some particular conditions. As we can see in Remark 2.6., some identities in algebras obtained by the Cayley-Dickson process can be used to find solutions for some equations in these algebras or to solve them. This idea can constitute the starting point for a further research.
$$$$
$$$$
**References**$$$$
\[Al; 39\] Albert, A. A., *Structure of algebras*, Amer. Math. Soc. Colloquium Publications, vol. **24**, 1939
\[Al; 49\] Albert, A. A., *Absolute-valued algebraic algebras*, Bull. Amer. Math. Soc., **55**(1949), 763-768.
\[Br; 67\] Brown, R. B., *On generalized Cayley-Dickson algebras*, Pacific J. of Math.,** 20(3)**(1967), 415-422.
\[Fl; 12\] Flaut, C., *Levels and sublevels of algebras obtained by the Cayley-Dickson process*, 2011, submitted.
\[Fl; 01\] Flaut, C., *Some equations in algebras obtained by the Cayley-Dickson process*, An. St. Univ. Ovidius Constanta, 9(2)(2001), 45-68.
\[Fl, Şt; 09\] Flaut, C., Ştefănescu, M., *Some equations over generalized quaternion and octonion division algebras*, Bull. Math. Soc. Sci. Math. Roumanie, **52(4)**(100), 2009, 427–439.
\[Ha; 43\] Hall, M., *Projective planes*, Trans. Amer. Math. Soc. vol. **54**(1943), 229-277.
\[Iv, Za; 05\] Ivanov, S. Zamkovoy, S., *Parahermitian and paraquaternionic manifolds* , Differential Geometry and its Applications **23**(2005), 205–234
\[Le; 06 \] Lewis, D. W., *Quaternion Algebras and the Algebraic Legacy of Hamilton’s Quaternions*, Irish Math. Soc. Bulletin **57**(2006), 41–64.
\[Ki, Ou; 99\] El Kinani, E. H., Ouarab, A., *The Embedding of* $U_{q}(sl\left( 2\right) )$ *and Sine Algebras in Generalized Clifford Algebras*, Adv. Appl. Clifford Algebr., **9(1)**(1999), 103-108.
\[Ko; 10\] Koç, C., *C-lattices and decompositions of generalized Clifford algebras,* Adv. Appl. Clifford Algebr., **20(2)**(2010), 313-320.
\[Sc; 66\] Schafer, R. D., *An Introduction to Nonassociative Algebras,* Academic Press, New-York, 1966.
\[Sc; 54\] Schafer, R. D., *On the algebras formed by the Cayley-Dickson process,* Amer. J. Math., **76**(1954), 435-446.
\[Smi; 50\] Smiley, M. F., *A remark on a theorem of Marshall Hall,* Proceedings of the American Mathematical Society, **1**(1950), 342-343.
\[Sm; 91\] Smith T. L., *Decomposition of Generalized Clifford Algebras*, Quart. J. Math. Oxford, 42 (1991), pp. 105-112.
\[Sz; 09\] Szpakowski, V. S., *Solution of general quadratic quaternionic equations*, Bull. Soc. Sci. Lettres Łódź 59, Ser. Rech. Déform. **58**(2009), 45 – 58.
\[Ti; 99\] Tian, Y., *Similarity and cosimilarity of elements in the real Cayley-Dickson algebras*, Adv. Appl. Clifford Algebras, 9(1)(1999), 61-76.
Cristina FLAUT
[Faculty of Mathematics and Computer Science,]{}
[Ovidius University,]{}
[Bd. Mamaia 124, 900527, CONSTANTA,]{}
[ROMANIA]{}
[http://cristinaflaut.wikispaces.com/]{}
[http://www.univ-ovidius.ro/math/]{}
[e-mail:]{}
[[email protected]]{}
[cristina\[email protected]]{}$$$$
Vitalii SHPAKIVSKYI
[Department of Complex Analysis and Potential Theory]{}
[ Institute of Mathematics of the National Academy of Sciences of Ukraine,]{}
[ 3, Tereshchenkivs’ka st.]{}
[ 01601 Kiev-4]{}
[ UKRAINE]{}
[ http://www.imath.kiev.ua/complex/]{}
[ e-mail: [email protected]]{}
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We consider two aspects of the ground state of the $U=\infty$ Hubbard ladder: ferromagnetism and the metal-insulator transition at quarter-filling. First, we present rigorous results for the $U=\infty$ Hubbard ladder in the limit of the large inter-chain hopping ($t_{\perp}/t\rightarrow\infty$). In this limit, the total spin $S$ of the ground state is shown to be zero for the electron density $n\le 0.5$ and its maximum ($S=S_{\rm max}$) for $n>0.5$. The charge gap at quarter-filling is $2t_{\perp}$. We extend these results to finite $t_{\perp}/t$ by means of the density-matrix renormalization group method. We estimate the phase boundaries with respect to spontaneous magnetization and the charge gap at quarter-filling for finite $t_{\perp}/t$. Applying the extended Aharonov-Bohm method, we give numerical evidence that the critical ratio $t_{\perp}/t$, above which the charge gap opens, is less than 0.001. Ferromagnetism in the $t$-$J$ ladder is briefly discussed.'
address: |
Institute for Solid State Physics,\
University of Tokyo, Roppongi, Minato-ku, Tokyo 106, Japan
author:
- Masanori Kohno
title: 'Aspects of the ground state of the $U=\infty$ Hubbard ladder '
---
\#1[[$\backslash$\#1]{}]{}
Introduction {#Introduction}
============
Recently, a lot of attention has been given to the Hubbard or $t$-$J$ ladders. One of the reasons is that these models are believed to describe the essential features of various materials such as ${\rm (VO)}_2{\rm P}_2{\rm O}_7$[@VOPO] or ${\rm SrCu}_2{\rm O}_3$[@SrCuO]. Another reason is that some properties of the ladder models are closer to planar models than to single chain models. One such property is the enhancement of the $d_{x^2-y^2}$-like pairing correlation. In relation to high-$T_{\rm c}$ superconductivity[@HTC; @Anderson], this feature and the existence of the spin gap have been studied by various approaches such as bosonization[@Fabrizio; @Khveshchenko; @BalentsFisher], projector Monte Carlo[@Kuroki], exact diagonalization[@Troyer; @HaywardED; @Sano] and the density-matrix renormalization group (DMRG) method[@Noack; @Hayward].
Another interesting aspect of the ladder models is the existence of the ferromagnetic ground state. Nagaoka’s theorem[@Nagaoka; @Thouless; @Tasaki] holds for the ladders and the planar models, although not for the single chains. Hence, ferromagnetic ground states may exist in the strong-coupling regime near half-filling for the ladders and the planar models, but not for the single chains. For two dimensions, many workers have investigated the ferromagnetic ground state at $U=\infty$ for finite hole-density [@MTakaFerro; @KuboFerro; @Riera; @Yedidia; @Zhang; @Putikka; @Kusakabe1; @Liang], in order to clarify the origin of itinerant ferromagnetism from the electron-correlation view-point. In spite of such attempts, results are inconclusive, because of the lack of efficient methods. On the other hand, for the ladders, there is an efficient method, i.e. the density-matrix renormalization group (DMRG) method proposed by White[@White]. Liang and Pang[@Liang] applied this method to the $U=\infty$ Hubbard ladder for $t_{\perp}/t=1$, and obtained some indications of ferromagnetism. As for the (two-leg) ladder, the DMRG calculation in ref.[@Liang] suggests that the fully-polarized ferromagnetic state is one of the ground states for $\delta<0.22$ ($\delta$ : hole density) and that the ground state is a spin-singlet for $\delta\buildrel > \over \sim 0.4$. They have tried to extend these results to two dimensions by investigating multi-leg ladders. In this paper, we extend their results to various values of $t_{\perp}$ and $J$ for the $t$-$J$ ladder, in order to understand the ground-state properties of the Hubbard or $t$-$J$ ladders in the strong-coupling regime.
Another interesting feature of strongly correlated electron systems is the metal-insulator (MI) transition. For single chains and planar models, there have been a lot of works on the MI transition[@LiebWu; @Mott1D; @Furukawa; @Imada; @Assaad; @Kohno]. However, for ladder models, there are relatively few. One of the characteristic features of ladder models is the band structure. In the weak-coupling regime ($U\simeq 0$), the low-energy physics for $n\le 1$ is effectively described by the bonding band, if $t_{\perp}/t$ is large enough. Thus, the MI transition in this parameter regime is essentially the same as that of the single chains[@BalentsFisher]. This argument may be true for $t_{\perp}\gg U$. A natural question arising here is what happens in the opposite limit, i.e. $U\gg t_{\perp}$. In order to answer this question, we consider the MI transition in the $U=\infty$ Hubbard ladder for simplicity.
In the present paper, we mainly consider the $U=\infty$ Hubbard ladder (or $t$-ladder), which is defined as the $t$-$J$ ladder at $J=J_{\perp}=0$. Later, in Sec.\[DMRGFerro\], we briefly discuss ferromagnetism of the $t$-$J$ ladder. The Hamiltonian of the $t$-$J$ ladder is defined as follows : $$\begin{aligned}
{\cal H}_{tJ} &=& {\cal H}_t+{\cal H}_J, \\
{\cal H}_t &=& -t\sum_{i \sigma}
(\tilde{c}^{1 \dagger}_{i\sigma}\tilde{c}^{1}_{i+1\sigma}+\tilde{c}^{2 \dagger}_{i\sigma}\tilde{c}^{2}_{i+1\sigma}+{\mbox{h.c.}})\nonumber\\
&&-t_{\perp}\sum_{i \sigma}
(\tilde{c}^{1 \dagger}_{i\sigma}\tilde{c}^{2}_{i\sigma}+{\mbox{h.c.}}), \nonumber\\
{\cal H}_J &=& J\sum_{i}(\mbox{\boldmath $S^{1}_i\cdot S^{1}_{i+1}$}-\frac{1}{4}n^{1}_in^{1}_{i+1}+\mbox{\boldmath $S^{2}_i\cdot S^{2}_{i+1}$}-\frac{1}{4}n^{2}_in^{2}_{i+1})\nonumber\\
&&+J_{\perp}\sum_{i}(\mbox{\boldmath $S^{1}_i\cdot S^{2}_i$}-\frac{1}{4}n^{1}_in^{2}_i),\nonumber\end{aligned}$$ where $\tilde{c}^{\alpha\dagger}_{i\sigma}$ denotes a creation operator of an electron at site $i$ with spin $\sigma
(\sigma=\uparrow,\downarrow)$ in the $\alpha$-th chain ($\alpha=1,2$) with the constraint that no site is doubly occupied, i.e. $\tilde{c}^{\alpha\dagger}_{i\sigma}\equiv c^{\alpha\dagger}_{i\sigma}(1-n^{\alpha}_{i -\sigma})$. The number operator $n^{\alpha}_{i\sigma}$ is defined as $n^{\alpha}_{i\sigma}\equiv c^{\alpha\dagger}_{i\sigma}c^{\alpha}_{i\sigma}$, using the standard electron creation operator $c^{\alpha\dagger}_{i\sigma}$. The spin operator at site $i$ in the $\alpha$-th chain is defined as $\mbox{\boldmath
$S$}^{\alpha}_i\equiv\frac{1}{2}\sum_{\beta\gamma}c^{\alpha\dagger}_{i\beta}\mbox{\boldmath
$\sigma_{\beta\gamma}$}c^{\alpha}_{i\gamma}$, where $\mbox{\boldmath
$\sigma$ }$ is the vector of Pauli matrices. The number of electrons, the number of sites and the number of rungs are denoted by $N_e$, $N_s$ and $L(=N_s/2)$, respectively. The electron density $n$ and the hole density $\delta$ are defined as $n\equiv N_e/N_s$ and $\delta\equiv 1-n$, respectively. The maximum value of the total spin $S$ is denoted by $S_{\rm max}(=N_e/2)$.
This paper is organized as follows: In Sec \[Proof\], we present rigorous results on the ground state of the $t$-ladder in the limit $t_{\perp}/t\rightarrow\infty$. In Sec \[DMRGFerro\], we present numerical results on ferromagnetism in the $t$-ladder and the $t$-$J$ ladder. In Sec \[MItrans\], we discuss the metal-insulator transition at quarter-filling for the $t$-ladder. Section \[Summary\] is a summary.
$t$-ladder in the limit $t_{\perp}/t\rightarrow\infty$ {#Proof}
======================================================
In this section, we prove the following statements:
1. The ground state of the $t$-ladder in the limit $t_{\perp}/t\rightarrow\infty$ for $n\le 0.5$ is a spin-singlet ($S=0$) and is unique in finite-size clusters with an even number of electrons with open boundary conditions. \[Theorem1\]
2. The ground state of the $t$-ladder in the limit $t_{\perp}/t\rightarrow\infty$ for $n>0.5$ has the maximum total spin ($S=S_{\rm max}$) and is unique up to the trivial $(N_e+1)$-fold degeneracy in finite-size clusters with open boundary conditions. \[Theorem2\]
Before investigating the above properties, we consider the $t$-ladder at $t=0$. The ground states of this model can be written in the following form: $$%%%%%%
%|\Phi\rangle=\prod_{i}\otimes|\alpha\rangle_{i},
|\Phi\rangle=\bigotimes_{i=1}^L|\alpha\rangle_{i},
%%%%%%
\label{unperturb}$$ where $|\alpha\rangle_{i}$’s correspond to either of states (i)-(iii) defined in table \[BOD\] for $n\le 0.5$, and (ii)-(vii) for $n>0.5$. The degeneracy of the ground states, the energy $E$ and the chemical potential $\mu$($\equiv \partial E/\partial N_e$) are summarized in table \[t0\]. The charge gap $\Delta_c$ at quarter-filling ($n=0.5$) is $2t_{\perp}$.
Next, we consider the $t$-ladder in the limit $t/t_{\perp}\rightarrow 0$. Let us consider the cases $n\le 0.5$ and $n>0.5$, separately.
$n\le 0.5$
----------
Up to order $t$, the effective Hamiltonian is $${\cal H}^{\rm eff}_{t} = -t\sum_{i \sigma}
(\tilde{b}^{\dagger}_{i\sigma}\tilde{b}_{i+1\sigma}+{\mbox{h.c.}}),$$ where $\tilde{b}^{\dagger}_{i\sigma}$ denotes a creation operator of an electron in the bonding band at rung $i$ with spin $\sigma
(\sigma=\uparrow,\downarrow)$ with the constraint that no rung is doubly occupied, or $\tilde{b}^{\dagger}_{i\sigma}\equiv b^{\dagger}_{i\sigma}(1-n^1_i-n^2_i)$. Here, the creation operator $b^{\dagger}_{i\sigma}$ is defined as $b^{\dagger}_{i\sigma}\equiv(\tilde{c}^{1\dagger}_{i\sigma}+\tilde{c}^{2\dagger}_{i\sigma})/\sqrt 2$. This Hamiltonian is equivalent to the $U=\infty$ Hubbard chain. Thus, the ground states are degenerate with respect to the spin degrees of freedom. The charge part of the ground-state wavefunction is simply that of the spinless fermion model on a chain.
Next, we consider the effective Hamiltonian of order $t^2/t_{\perp}$. Here, we define the local Hamiltonian ${\cal H}^{\rm loc}_{i,i+1\sigma}$: $${\cal H}^{\rm loc}_{i,i+1\sigma} = -t
(\tilde{c}^{1 \dagger}_{i\sigma}\tilde{c}^{1}_{i+1\sigma}+\tilde{c}^{2 \dagger}_{i\sigma}\tilde{c}^{2}_{i+1\sigma}+{\mbox{h.c.}}).$$ Letting this local Hamiltonian operate on $|\alpha\rangle_{i}\otimes|\beta\rangle_{i+1}$, we obtain the following relations: $$\begin{aligned}
{\cal H}^{\rm loc}_{i,i+1\sigma}|B\sigma\rangle_{i}\otimes|B\sigma\rangle_{i+1}&=&0,\nonumber\\
{\cal H}^{\rm loc}_{i,i+1\sigma}|B-\sigma\rangle_{i}\otimes|B\sigma\rangle_{i+1}&=&\frac{\sigma t}{\sqrt 2}|S\rangle_{i}\otimes|00\rangle_{i+1},\nonumber\\
{\cal H}^{\rm loc}_{i,i+1\sigma}|B\sigma\rangle_{i}\otimes|B-\sigma\rangle_{i+1}&=&-\frac{\sigma t}{\sqrt 2}|00\rangle_{i}\otimes|S\rangle_{i+1},\nonumber\\
{\cal H}^{\rm loc}_{i,i+1\sigma}|S\rangle_{i}\otimes|00\rangle_{i+1}&=&\frac{\sigma t}{\sqrt 2}(|B-\sigma\rangle_{i}\otimes|B\sigma\rangle_{i+1}-|A-\sigma\rangle_{i}\otimes|A\sigma\rangle_{i+1}),\nonumber\\
{\cal H}^{\rm loc}_{i,i+1\sigma}|00\rangle_{i}\otimes|S\rangle_{i+1}&=&-\frac{\sigma t}{\sqrt 2}(|B\sigma\rangle_{i}\otimes|B-\sigma\rangle_{i+1}-|A\sigma\rangle_{i}\otimes|A-\sigma\rangle_{i+1}),\nonumber\end{aligned}$$ where $|S\rangle_{i}$ and $|A\sigma\rangle_{i}$ are defined as $\frac{1}{\sqrt 2}(\tilde{c}^{1\dagger}_{i\uparrow}\tilde{c}^{2\dagger}_{i\downarrow}-\tilde{c}^{1\dagger}_{i\downarrow}\tilde{c}^{2\dagger}_{i\uparrow})|00\rangle_{i}$ and $\frac{1}{\sqrt 2}(\tilde{c}^{1\dagger}_{i\sigma}-\tilde{c}^{2\dagger}_{i\sigma})|00\rangle_{i}$, respectively. Thus, the second-order perturbation energy $E_2$ is obtained as in table \[2nd\]. It is also shown that the following relations are satisfied: $$\begin{aligned}
n^{B}_{i\uparrow}n^{B}_{i+1\downarrow}|B\uparrow\rangle_{i}\otimes|B\downarrow\rangle_{i+1}&=&|B\uparrow\rangle_{i}\otimes|B\downarrow\rangle_{i+1},\nonumber\\
S^{B-}_{i}S^{B+}_{i+1}|B\uparrow\rangle_{i}\otimes|B\downarrow\rangle_{i+1}&=&|B\downarrow\rangle_{i}\otimes|B\uparrow\rangle_{i+1},\nonumber\end{aligned}$$ where $S^{B+}_{i}\equiv\tilde{b}^{\dagger}_{i\uparrow}\tilde{b}_{i\downarrow}$ and $n^{\rm B}_{i\sigma}\equiv \tilde{b}^{\dagger}_{i\sigma}\tilde{b}_{i\sigma}$. Thus, the effective Hamiltonian ${\cal H}^{\rm eff}$, up to order $t^2/t_{\perp}$, is written as follows: $$\begin{aligned}
\label{Th1eq}
{\cal H}^{\rm eff} &=& {\cal H}^{\rm eff}_{t}+{\cal H}^{\rm eff(1)}_{J}+{\cal H}^{\rm eff(2)}_{J},\\ \nonumber
{\cal H}^{\rm eff(1)}_{J} &=& J_{\rm eff}\sum_{i}
(\mbox{\boldmath $S^{B}_i\cdot S^{B}_{i+1}$}-\frac{1}{4}n^{B}_in^{B}_{i+1}),\\ \nonumber
{\cal H}^{\rm eff(2)}_{J} &=& \frac{J_{\rm eff}}{4}\sum_{i \sigma}
(\tilde{b}^{\dagger}_{i-1-\sigma}\tilde{b}^{\dagger}_{i\sigma}\tilde{b}_{i-\sigma}\tilde{b}_{i+1\sigma}-\tilde{b}^{\dagger}_{i-1\sigma}n^{B}_{i-\sigma}\tilde{b}_{i+1\sigma}+{\mbox{h.c.}}),\end{aligned}$$ where $\mbox{\boldmath $S^{B}_i$}$ is the spin operator in the bonding band at rung $i$ and $J_{\rm eff}=t^2/t_{\perp}$. Hence, it is shown that the effective Hamiltonian, up to order $t^2/t_{\perp}$ for $n\le 0.5$, has the same form as that of the $U\rightarrow\infty$ Hubbard chain, up to order $t^2/U$ for $n\le 1$[@MTakaU; @Hirsh; @Gros; @Ogata]. As a result, the ground-state properties of the $t$-ladder in the limit $t_{\perp}/t\rightarrow\infty$ for $n\le 0.5$ are the same as those of the $U\rightarrow\infty$ Hubbard chain for $n\le 1$[@Ogata]. This leads to a spin-singlet ground state by the Lieb-Mattis theorem[@LiebMattes].
$n>0.5$
-------
In this subsection, we consider the case $n>0.5$. The unperturbed ground states are written in the form of eq.(\[unperturb\]). The matrix elements of the local Hamiltonian eq.(\[localH\]) are summarized in table \[matrix\]. $$\label{localH}
{\cal H}^{\rm loc}_{i,i+1} = -t\sum_{\sigma}
(\tilde{c}^{1 \dagger}_{i\sigma}\tilde{c}^{1}_{i+1\sigma}+\tilde{c}^{2 \dagger}_{i\sigma}\tilde{c}^{2}_{i+1\sigma}+{\mbox{h.c.}}).$$ Considering the matrix elements in table \[matrix\], it is shown that the state $|B-\sigma\rangle_{i}\otimes|\sigma\sigma\rangle_{i+1}$ can reach the state $|\sigma\sigma\rangle_{i}\otimes|B-\sigma\rangle_{i+1}$ after successive multiplication by the local Hamiltonian ${\cal H}^{\rm loc}_{i,i+1}$ as follows: $|B-\sigma\rangle_{i}\otimes|\sigma\sigma\rangle_{i+1}\rightarrow|\sigma-\sigma\rangle_{i}\otimes|B\sigma\rangle_{i+1}\rightarrow|B\sigma\rangle_{i}\otimes|\sigma-\sigma\rangle_{i+1}\rightarrow|\sigma\sigma\rangle_{i}\otimes|B-\sigma\rangle_{i+1}$. Using this property, we can show that the following processes are possible by applying the local Hamiltonians ${\cal H}^{\rm loc}_{i-1,i}$ and ${\cal H}^{\rm loc}_{i,i+1}$ successively. $$\begin{aligned}
|B\sigma\rangle_{i-1}\otimes|\sigma\sigma\rangle_{i}\otimes|\sigma-\sigma\rangle_{i+1}&\leftrightarrow&|B\sigma\rangle_{i-1}\otimes|\sigma-\sigma\rangle_{i}\otimes|\sigma\sigma\rangle_{i+1},\nonumber\\
|B\sigma\rangle_{i-1}\otimes|\sigma\sigma\rangle_{i}\otimes|-\sigma\sigma\rangle_{i+1}&\leftrightarrow&|B\sigma\rangle_{i-1}\otimes|-\sigma\sigma\rangle_{i}\otimes|\sigma\sigma\rangle_{i+1},\nonumber\\
|B\sigma\rangle_{i-1}\otimes|-\sigma-\sigma\rangle_{i}\otimes|-\sigma\sigma\rangle_{i+1}&\leftrightarrow&|B\sigma\rangle_{i-1}\otimes|-\sigma\sigma\rangle_{i}\otimes|-\sigma-\sigma\rangle_{i+1},\nonumber\\
|B\sigma\rangle_{i-1}\otimes|-\sigma-\sigma\rangle_{i}\otimes|\sigma-\sigma\rangle_{i+1}&\leftrightarrow&|B\sigma\rangle_{i-1}\otimes|\sigma-\sigma\rangle_{i}\otimes|-\sigma-\sigma\rangle_{i+1},\nonumber\\
|B\sigma\rangle_{i-1}\otimes|\sigma\sigma\rangle_{i}\otimes|-\sigma-\sigma\rangle_{i+1}&\leftrightarrow&|B\sigma\rangle_{i-1}\otimes|-\sigma-\sigma\rangle_{i}\otimes|\sigma\sigma\rangle_{i+1},\nonumber\\
|B\sigma\rangle_{i-1}\otimes|\sigma-\sigma\rangle_{i}\otimes|-\sigma\sigma\rangle_{i+1}&\leftrightarrow&|B\sigma\rangle_{i-1}\otimes|-\sigma\sigma\rangle_{i}\otimes|\sigma-\sigma\rangle_{i+1}.\nonumber\end{aligned}$$ Thus, $|\uparrow\uparrow\rangle$, $|\uparrow\downarrow\rangle$, $|\downarrow\uparrow\rangle$ and $|\downarrow\downarrow\rangle$ can have their positions changed after successive multiplication by the Hamiltonian (${\cal H}_t$), if there exists at least one $|B\uparrow\rangle$ or $|B\downarrow\rangle$. Furthermore, the following processes are also possible: $$\begin{aligned}
|\sigma\sigma\rangle_{i-1}\otimes|B\sigma\rangle_{i}\otimes|B-\sigma\rangle_{i+1}&\leftrightarrow&|\sigma\sigma\rangle_{i-1}\otimes|B-\sigma\rangle_{i}\otimes|B\sigma\rangle_{i+1},\nonumber\\
|\sigma-\sigma\rangle_{i-1}\otimes|B\sigma\rangle_{i}\otimes|B-\sigma\rangle_{i+1}&\leftrightarrow&|\sigma-\sigma\rangle_{i-1}\otimes|B-\sigma\rangle_{i}\otimes|B\sigma\rangle_{i+1}.\nonumber\end{aligned}$$ Thus, $|B\uparrow\rangle$ and $|B\downarrow\rangle$ can have their positions changed, if there exists at least one $|\uparrow\uparrow\rangle$, $|\uparrow\downarrow\rangle$, $|\downarrow\uparrow\rangle$ or $|\downarrow\downarrow\rangle$. As a result, any unperturbed ground state in the form of eq.(\[unperturb\]) can be reached after successive multiplication by the Hamiltonian (${\cal H}_t$) for $0.5<n<1$ in the subspace of fixed total $S^z$ and fixed electron number. This property is usually called connectivity. Combined this property and the fact that the matrix elements of the Hamiltonian in this representation are all positive (table \[matrix\]), the Perron-Frobenius theorem ensures that the state of the largest eigenvalue is unique and the wavefunction is positive (nodeless) in this representation in each subspace.
Next, we consider the ground-state wavefunction in the subspace of the maximum total $S^z$ ($S^z=S_{\rm max}$). The Hamiltonian in this subspace has the same form as that of the spinless fermion model on a chain. Thus, the wavefunction of the largest eigenvalue in this subspace is nothing but that of the spinless fermion model ($|\Psi_{\rm SF}\rangle$). Applying the spin-lowering operator $S^-$ to $|\Psi_{\rm SF}\rangle$, we obtain the eigenstates of various total $S^z$’s. These states have all positive (nodeless) wavefunctions in the present representation. Thus, these states have finite overlap with the states of the largest eigenvalue in the corresponding subspaces. By use of the gauge transformation $\tilde{c}^{\alpha}_{i}\rightarrow(-1)^i\tilde{c}^{\alpha}_{i}, \alpha=1,2$, the sign of the hopping amplitude $t$ can be changed, i.e. $t\rightarrow -t$, with spin operators unchanged. As a result, it is shown that the ground state of the $t$-ladder in the limit $t_{\perp}/t\rightarrow\infty$ for $0.5<n<1$ has the maximum total spin ($S=S_{\rm max}$) and is unique up to the trivial $(N_e+1)$-fold degeneracy.
Remarks
-------
Here, we give some remarks on the above theorems.
1. Part of theorem \[Theorem1\] can be extended to higher dimensions. The effective Hamiltonian of double-layer $t$-models up to order $t^2/t_{\perp}$ for $n\le 0.5$ has the same form as that of the single-layer Hubbard models up to order $t^2/U$ for $n\le 1$[@MTakaU; @Hirsh; @Gros]. Thus, the ground-state properties of double-layer $t$-models in the limit $t_{\perp}/t\rightarrow\infty$ for $n\le 0.5$ are the same as those of the single-layer $U\rightarrow\infty$ Hubbard models for $n\le 1$[@Lieb].
2. The proof of theorem \[Theorem2\] is mathematically similar to that of Kubo’s theorem[@Kubo]. However, the physical situation is different. In Kubo’s theorem, the limit of the strong Hund-coupling is taken, i.e. $H_{\rm Hund}\equiv -J_{\rm H}\sum_i\mbox{\boldmath $S^1_i\cdot S^2_{i}$}$, $J_{\rm H}\rightarrow \infty$. Furthermore, almost degenerate bands are assumed. On the other hand, in theorem \[Theorem2\], we do not assume explicit ferromagnetic couplings. In contrast to Kubo’s case, the limit of the large band splitting is taken in the present case. The extension of Kubo’s theorem to $n\le 0.5$ shows that the ground state is also ferromagnetic[@Kusakabe2], which is contrasted with theorem \[Theorem1\]. The proof of theorem \[Theorem2\] is mathematically similar to that of ref.[@Kondo] for the one-dimensional Kondo-lattice model, too.
3. The restriction on the boundary condition can be relaxed such that the Hubbard chain has a unique spin-singlet ground state for theorem \[Theorem1\] and that the spinless fermion model on a chain has all positive matrix elements in the site representation for theorem \[Theorem2\]. For example, theorem \[Theorem2\] can be extended to the case of periodic boundary conditions with an odd number of electrons and an even number of rungs. Nagaoka’s theorem is recovered for the one-hole case[@Nagaoka; @Thouless; @Tasaki]
Ferromagnetism {#DMRGFerro}
==============
In this section, we present the numerical results on the $t$-ladder for finite $t_{\perp}/t$ and the $t$-$J$ ladder for small $J$ and $J_{\perp}$ obtained by the DMRG method (finite-size algorithm)[@White]. The DMRG calculation has been performed with open boundary conditions.
As a test of the DMRG calculation, we compare the ground-state energy obtained by the DMRG method with that of the exact diagonalization method. In Fig.\[diag\], the agreement of the data obtained by these two methods is good. The maximum error is about 0.01%. Next, we consider the truncation error, i.e. the error due to small $m$, where $m$ is the number of states kept in the superblock[@White]. The difference between $m=50$ and $m=100$ is very small as shown in Fig.\[m50m100\], suggesting that $m=50$ is sufficient. (See also Fig.\[SzJ0\].) Thus, we mainly report the results of $m=50$ hereafter.
Let us consider ferromagnetism of the $t$-ladder. In Fig. \[dEJ0\], we show the energy difference $\Delta E_{\rm F}$ between the ground-state energy in the subspace of $S^z=0$ and that of $S^z=S_{\rm max}$ as a function of filling. The data in various-size clusters are scaled to a single line, indicating that the finite-size error is small. (See also Fig.\[ChemJ0\].) From this plot, we find the region FF where the fully-polarized ferromagnetic state is one of the ground states. The phase boundary $n_{c_1}$ between FF and non-FF is estimated as shown in Fig. \[PDJ0\]. At $t_{\perp}/t=1$, the result in ref.[@Liang] ($n_{c_1}\simeq0.78$) is recovered. Qualitatively, the region FF becomes larger as $t_{\perp}/t$ increases. The phase boundary $n_{c_1}$ gets closer to 0.5 as $t_{\perp}/t$ increases. This is consistent with the rigorous results in Sec.\[Proof\].
Next, we consider the phase boundary $n_{c_2}$ between SS and non-SS, where SS is defined as the region in which the ground state is a spin-singlet. Figure \[SzJ0\] shows the ground-state energy as a function of the total $S^z$ at $t_{\perp}/t=1$ for $n=0.5625$, $0.625$ and $0.6875$. The ground state is a spin-singlet for $n=0.5625$ and not for $n=0.625$ and $0.6875$. Hence, the phase boundary $n_{c_2}$ is estimated as $n_{c_2}=0.59\pm0.03$, which is consistent with ref.[@Liang]($n_{c_2}\simeq 0.6$). In this way, the phase boundary $n_{c_2}$ is estimated for various $t_{\perp}/t$ as shown in Fig.\[PDJ0\]. The region PF shrinks as $t_{\perp}/t$ increases, where PF is defined as the region which is neither FF nor SS. This is also consistent with the rigorous results in Sec.\[Proof\].
Here, we present the numerical results on the $t$-$J$ ladder for small $J$ and $J_{\perp}$. For simplicity, we choose $t_{\perp}=t$ and $J_{\perp}=J$, and set $t=1$ as the energy unit. Figure \[dEJ\] shows the $n$-dependence of $\Delta E_{\rm F}$ at $J=0.00, 0.05, 0.07, 0.10$ and $0.15$. As shown in this figure, $\Delta E_{\rm F}$ becomes large near half-filling ($n=1$) due to antiferromagnetic correlation, and $\Delta E_{\rm F}$ seems to be the smallest near $n\simeq0.8$ for $n>0.5$. Figure \[dEJ\] also indicates that $J=0.05$ is enough to destroy the region FF.
Next, we consider the stability of PF against $J$. The ground-state energy as a function of the total $S^z$ near half-filling for $J=0.05, 0.07, 0.10$ and $0.15$ is shown in Fig. \[SzJ\]. This figure suggests that FF is surrounded by PF in the phase diagram of the $t$-$J$ ladder for finite $\delta$.
Metal-insulator transition at quarter-filling {#MItrans}
=============================================
Before discussing the metal-insulator transition, we consider the charge gap $\Delta_c$ at quarter-filling ($n=0.5$). Figure \[ChemJ0\] shows the $n$-dependence of the chemical potential $\mu$ ($\equiv\partial E/\partial N_e$). The chemical potential $\mu$ in a finite-size cluster is defined as $$\mu(\bar n)\equiv \frac{E(n_1)-E(n_2)}{(n_1-n_2)N_s},$$ where $E(n_i)$ denotes the ground-state energy at filling $n_i$, $i=1,2$, and $\bar n\equiv (n_1+n_2)/2$. We took $(n_2-n_1)N_s=2$. The charge gap $\Delta_c$ is defined as $\Delta_c\equiv\mu(n_c+0)-\mu(n_c-0)$, where $n_c$ is the critical electron-density ($n_c=0.5$ in the present case). It is expected that the charge gap opens at quarter-filling, if $t_{\perp}/t$ is large enough (Sec.\[Proof\]). Actually, for large values of $t_{\perp}/t$, the charge gap seems to open as shown in Fig.\[ChemJ0\](a). For smaller values of $t_{\perp}/t$, we cannot determine whether the charge gap opens from Fig.\[ChemJ0\]. Thus, we extrapolate the charge gap in a finite-size cluster \[eq.(\[gapeq\])\] as $a+b/L$, using the data for $L=12-24$, and estimate the charge gap as shown in Fig. \[Chargegap\]. $$\label{gapeq}
\Delta_c(N_e=N_s/2)\equiv \frac{E(N_e=N_s/2+2)+E(N_e=N_s/2-2)-2E(N_e=N_s/2)}{2}.$$
There are some possibilities for the critical value $t_{\perp c}/t$ above which the charge gap opens. One of the possibilities is that the critical value $t_{\perp c}/t$ is zero and that the gap is exponentially small in the limit $t_{\perp}/t\rightarrow 0$ as in the case of the Hubbard chain in the limit $U\rightarrow 0$[@LiebWu]. In order to determine the critical value $t_{\perp c}/t$, we adopt the extended Aharonov-Bohm (AB) method proposed by Kusakabe and Aoki[@KusakabeAB]. In the framework of this method, we investigate the extended spectral flow by introducing a Peierls phase as $$\tilde{c}^{\alpha \dagger}_{i\sigma}\tilde{c}^{\alpha}_{i+1\sigma}\rightarrow\exp({\rm i}\frac{2\pi\Phi}{L\Phi_0})\tilde{c}^{\alpha \dagger}_{i\sigma}\tilde{c}^{\alpha}_{i+1\sigma}.$$ It is expected that the period of the spectral flow reduces to $L\Phi_0/M$ if $M$-particle bound states are formed in the ground state. For example, $M=1$ for a metallic state, and $M=2$ for a BCS state. We apply this method to the $t$-ladder with a very small value of $t_{\perp}/t$ ($t_{\perp}/t=0.001$). As shown in Fig.\[ABflux1\], the spectral flow at quarter-filling has the minimum extended AB period, i.e. $\Phi=\Phi_0$, suggesting that the ground state is an $L$-particle bound state or, in this case, an insulator[@AritaAB]. This behavior is contrasted with the case off quarter-filling. For example, at $n=1/3$, the extended AB period is larger than $\Phi_0$ as shown in Fig.\[ABflux2\]. This implies that the ground state at quarter-filling is an insulator already for $t_{\perp}/t=0.001$. It is plausible to consider that the critical value $t_{\perp c}/t$ may be zero. A possible scenario may be the following: The perturbation of the small $t_{\perp}$-term produces the relevant Umklapp process which leads to an insulator, at the same time as the degeneracy with respect to the spin degrees of freedom is removed. The numerical results presented above are quite different from the case in the weak-coupling limit ($U\rightarrow+0$). In the weak-coupling limit, it is shown by bosonization that $t_{\perp c}/t$ is one[@BalentsFisher]. Thus, it is expected that $t_{\perp c}/t$ decreases from 1 to 0 as the interaction $U$ increases from 0 to $\infty$.
Now, let us consider the metal-insulator (MI) transition at quarter-filling. As discussed in Sec. \[Proof\], in the limit $t_{\perp}/t\rightarrow \infty$, the MI transition for $n\rightarrow 0.5-0$ is effectively described by the equivalent model to the $U\rightarrow\infty$ Hubbard chain, and the MI transition for $n\rightarrow0.5+0$ is described by the spinless fermion model on a chain. It is interesting to compare these features with those in the weak-coupling regime. In the weak-coupling regime ($U\rightarrow+0$), the charge gap is also expected at quarter-filling for $t_{\perp}/t>1$ because of the relevant Umklapp process[@BalentsFisher]. This MI transition is understood as the Mott transition which is described by the $U\rightarrow+0$ Hubbard model on a chain written in terms of the bonding-band operators. In both weak-coupling \[$U\ll t_{\perp}(>t)$\] and strong-coupling \[$U\gg t_{\perp} (\gg t$)\] regimes, as $n\rightarrow 0.5-0$, the MI transition is described by single chain effective Hubbard Hamiltonians. However, the value of the charge gap will have different energy scales. In the weak-coupling regime, the value of the charge gap would be determined mainly by $U$. On the other hand, in the strong-coupling regime, it would be determined mainly by $t_{\perp}$. This feature is similar to the two types of the MI transition for transition-metal compounds[@MottCharge], i.e. the Mott-Hubbard type and the charge-transfer type.
Summary {#Summary}
=======
In summary, two aspects of the ground state of the $U=\infty$ Hubbard ladder are investigated. One is ferromagnetism, the other is the metal-insulator (MI) transition. In the limit $t_{\perp}/t\rightarrow\infty$, it is rigorously shown that the ground state is a spin-singlet for $n\le 0.5$ and that the total spin is maximum for $0.5<n<1$. For finite $t_{\perp}/t$, we have estimated the phase boundaries, with respect to spontaneous magnetization, by the density-matrix renormalization group method. It is numerically shown that the region FF becomes larger and spreads down to quarter-filling as $t_{\perp}/t$ increases, which is consistent with the rigorous results presented in Sec.\[Proof\]. The rigorous results ($t_{\perp}/t\rightarrow\infty$) and the numerical results for finite $t_{\perp}/t$ support one another and confirm that the ground state can be ferromagnetic for the $U=\infty$ Hubbard ladder with finite hole-density. The numerical results for the $t$-$J$ ladder suggest that FF is surrounded by PF for finite $\delta$ in the small $J$ regime. We have also estimated the value of the charge gap at quarter-filling ($n=0.5$). Applying the extended Aharonov-Bohm method, we have obtained numerical evidence that the critical value $t_{\perp c}/t$, above which the charge gap opens, is less than 0.001. This is quite different from that of the weak-coupling limit ($U\rightarrow+0$) ($t_{\perp c}/t=1$)[@BalentsFisher].
The author would like to thank M. Takahashi, M. Ogata, K. Kusakabe and F.V. Kusmartsev for helpful discussions and useful comments. The author also thanks D. Lidsky for reading of the manuscript. The exact diagonalization program is partly based on the subroutine package “TITPACK Ver.2” coded by H. Nishimori. Part of the calculations were performed on the Fujitsu VPP500 at Institute for Solid State Physics, Univ. of Tokyo.
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Symbol Definition Energy
------- ------------------------------------ ---------------------------------------------------------------------------------------------------------- --------------
(i) $|00\rangle_{i}$ ${\rm vacuum}$ 0
(ii) $|B\uparrow\rangle_{i}$ $\frac{1}{\sqrt 2}(\tilde{c}^{1\dagger}_{i\uparrow}+\tilde{c}^{2\dagger}_{i\uparrow})|00\rangle_{i}$ $-t_{\perp}$
(iii) $|B\downarrow\rangle_{i}$ $\frac{1}{\sqrt 2}(\tilde{c}^{1\dagger}_{i\downarrow}+\tilde{c}^{2\dagger}_{i\downarrow})|00\rangle_{i}$ $-t_{\perp}$
(iv) $|\uparrow\uparrow\rangle_{i}$ $\tilde{c}^{1\dagger}_{i\uparrow}\tilde{c}^{2\dagger}_{i\uparrow}|00\rangle_{i}$ 0
(v) $|\downarrow\downarrow\rangle_{i}$ $\tilde{c}^{1\dagger}_{i\downarrow}\tilde{c}^{2\dagger}_{i\downarrow}|00\rangle_{i}$ 0
(vi) $|\uparrow\downarrow\rangle_{i}$ $\tilde{c}^{1\dagger}_{i\uparrow}\tilde{c}^{2\dagger}_{i\downarrow}|00\rangle_{i}$ 0
(vii) $|\downarrow\uparrow\rangle_{i}$ $\tilde{c}^{1\dagger}_{i\downarrow}\tilde{c}^{2\dagger}_{i\uparrow}|00\rangle_{i}$ 0
: Basis set.[]{data-label="BOD"}
$n<0.5$ $0.5<n<1$
-------------------------- ----------------------------------- -------------------------------------
Degeneracy $_{L}{\rm C}_{N_e}\times 2^{N_e}$ $_{L}{\rm C}_{N_e-L}\times 2^{N_e}$
Energy $E$ $-t_{\perp}\times N_e$ $-t_{\perp}\times (2L-N_e)$
Chemical Potential $\mu$ $-t_{\perp}$ $t_{\perp}$
: Ground states at $t=0$.[]{data-label="t0"}
$E_2$ between $|\alpha\rangle$ and $|\beta\rangle$ $|\alpha\rangle$ $|\beta\rangle$
---------------------------------------------------- --------------------------------------------------------------------------- -------------------------------------------------------------------------
0 $|B\sigma\rangle_{i}\otimes|B\sigma\rangle_{i+1}$ $|B\sigma\rangle_{i}\otimes|B\sigma\rangle_{i+1}$
$-t^2/(2t_{\perp})$ $|B\sigma\rangle_{i}\otimes|B-\sigma\rangle_{i+1}$ $|B\sigma\rangle_{i}\otimes|B-\sigma\rangle_{i+1}$
$t^2/(2t_{\perp})$ $|B\sigma\rangle_{i}\otimes|B-\sigma\rangle_{i+1}$ $|B-\sigma\rangle_{i}\otimes|B\sigma\rangle_{i+1}$
$t^2/(4t_{\perp})$ $|00\rangle_{i-1}\otimes|B\sigma\rangle_{i}\otimes|B-\sigma\rangle_{i+1}$ $|B\sigma\rangle_{i}\otimes|B-\sigma\rangle_{i}\otimes|00\rangle_{i+1}$
$-t^2/(4t_{\perp})$ $|00\rangle_{i-1}\otimes|B\sigma\rangle_{i}\otimes|B-\sigma\rangle_{i+1}$ $|B-\sigma\rangle_{i}\otimes|B\sigma\rangle_{i}\otimes|00\rangle_{i+1}$
: Second-order perturbation energy $E_2$.[]{data-label="2nd"}
$\langle\alpha|{\cal H}^{\rm loc}_{i,i+1}|\beta\rangle$ $|\alpha\rangle$ $|\beta\rangle$
--------------------------------------------------------- ---------------------------------------------------------- ---------------------------------------------------------- --
$t$ $|B\sigma\rangle_{i}\otimes|\sigma\sigma\rangle_{i+1}$ $|\sigma\sigma\rangle_{i}\otimes|B\sigma\rangle_{i+1}$
$|B\sigma\rangle_{i}\otimes|\sigma-\sigma\rangle_{i+1}$ $|\sigma-\sigma\rangle_{i}\otimes|B\sigma\rangle_{i+1}$
$|B\sigma\rangle_{i}\otimes|\sigma-\sigma\rangle_{i+1}$ $|\sigma\sigma\rangle_{i}\otimes|B-\sigma\rangle_{i+1}$
$t/2$ $|B\sigma\rangle_{i}\otimes|-\sigma\sigma\rangle_{i+1}$ $|-\sigma\sigma\rangle_{i}\otimes|B\sigma\rangle_{i+1}$
$|B\sigma\rangle_{i}\otimes|-\sigma\sigma\rangle_{i+1}$ $|\sigma\sigma\rangle_{i}\otimes|B-\sigma\rangle_{i+1}$
$|B\sigma\rangle_{i}\otimes|-\sigma-\sigma\rangle_{i+1}$ $|-\sigma\sigma\rangle_{i}\otimes|B-\sigma\rangle_{i+1}$
$|B\sigma\rangle_{i}\otimes|-\sigma-\sigma\rangle_{i+1}$ $|\sigma-\sigma\rangle_{i}\otimes|B-\sigma\rangle_{i+1}$
0 otherwise
: Matrix elements.[]{data-label="matrix"}
=3.5in Fig.\[diag\]\
\
=3.5in\
Fig.\[m50m100\]\
=3.5in Fig.\[dEJ0\](a)\
=3.5in Fig.\[dEJ0\](b) =3.5in Fig.\[dEJ0\](c) =3.5in\
Fig.\[SzJ0\]\
\
=3.5in\
Fig.\[dEJ\]\
=3.5in Fig.\[SzJ\](a)\
=3.5in Fig.\[SzJ\](b)\
=3.5in Fig.\[SzJ\](c) =3.2in Fig.\[ChemJ0\](a)\
=3.2in Fig.\[ChemJ0\](b)\
=3.2in Fig.\[ChemJ0\](c) =3.5in\
Fig.\[Chargegap\](a)\
\
=3.5in\
Fig.\[Chargegap\](b)\
\
=3.5in\
Fig.\[ABflux1\](a)\
\
=3.5in\
Fig.\[ABflux1\](b)\
\
=3.5in\
Fig.\[ABflux2\](a)\
\
=3.5in\
Fig.\[ABflux2\](b)\
\
=3.5in\
Fig.\[PDJ0\]
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: |
It is shown that the vacuum of QED$_2$ in Minkowski spacetime does not favour a periodic electric mean field. The projected effective action exhibiting a genuine dependence on the non-vanishing background field has been introduced. The functional dependence of the energy density of the vacuum on the assumed periodic vacuum expectation value of the vector potential is determined from the component $T^{00}$ of the energy-momentum tensor at one-loop order. Treating the background field non-perturbatively, the energy of the vacuum in the presence of a periodic mean field is found not be equal to the negative of the effective action.\
\
[*PACS:*]{} 12.20.Ds
---
[S. Nagy and K. Sailer]{}
[[*Department for Thoretical Physics, Kossuth Lajos University, Debrecen, Hungary*]{}]{}
Introduction {#Introduction}
============
It is well-known that the perturbative vacuum of QED is trivial, i.e. it is characterized by the vanishing expectation value of the electromagnetic vector potential. Our goal is to investigate the vacuum non-perturbatively in the framework of the continuum theory in Minkowski spacetime. As a working hypothesis we assume the presence of a periodic electric background field. The interaction of the fermions with the background field is treated exactly by integrating out the fermion fields, whereas the quantum fluctuations of the electromagnetic field are taken into account at one-loop order. The non-perturbative treatment of the interaction with the background introduces infinitely many non-renormalizable, i.e. irrelevant interaction vertices. It may happen that such vertices generate non-trivial, periodic (otherwise anti-ferromagnetic) vacuum structure. Similar examples are known for Yang-Mills theories on the lattice [@Fin97], where the existence of various anti-ferromagnetic vacua have been established due to irrelevant interaction terms.
Our problem setting can also be motivated by the following intuitive picture. In the presence of a background scalar potential ${\bar A}_0
= a \cos (\ell x)$ with overcritical amplitude $a >m$ (with the rest mass of the electron $m$) localized electrons of the Dirac sea may tunnel into localized electron states of the same energy above the mass gap (see Fig. \[tunnel\]). Therefore, the creation of a certain amount of electron-positron pairs with the spatial separation of $\pi/\ell$ is imaginable. This might stabilize itself with a periodic charge density (but zero net charge) and a periodic mean electric field. The expected effect might be destroyed due to the uncertainty principle. Localisation in an interval $\Pi/2\ell$ means a momentum and, consequently, an energy spread of order $\Delta E=\frac{p}{E}\Delta p $ and the effect may occur only if $a>>m+\frac{\Delta E}2$.
Projection method\[projection\]
===============================
It is not completely trivial to define an effective action depending on a background field ${\bar A}^\mu (x)$. The simple shift of the integration variable $A^\mu (x)$ to $\alpha^\mu (x) = A^\mu (x)$ - ${\bar A}^\mu (x)$ is not the answer requested, since a dependence on the background ${\bar A}^\mu (x)$ shall only occur due to terminating the loop expansion. Below we introduce an effective action exhibiting a genuine dependence on the background field.
Our method of defining the sector of QED for field configurations belonging to quantum fluctuations around a given vacuum expectation value $\langle A^\mu (x) \rangle$ consists of the following steps:
1. [*Projection to the sector with a given background field.*]{} The generating functional of QED in Minkowski spacetime has the form: $$\begin{aligned}
Z_{QED} &=&
\int {\cal D} {\bar \psi} {\cal D} \psi {\cal D} A
\exp \left\{ {\rm i} S_{em} \lbrack A, \xi \rbrack
+ {\rm i} S_D \lbrack A, {\bar \psi}, \psi \rbrack
\right\}\end{aligned}$$
with the vector potential $A^\mu (x)$ $(\mu = 0,1,2,3)$ and the electron-positron field $\psi (x)$, where $S_{em} \lbrack A, \xi \rbrack $ and $ S_D \lbrack A, {\bar \psi}, \psi \rbrack$ are the action of the electromagnetic field and the Dirac action, resp., in the covariant gauge with the gauge parameter $\xi$. Let the vector $n^\mu (x)$ be introduced in the space of the vector potential configurations. Multiply the integrand of the generating functional by the factor 1 written in the form $$\begin{aligned}
1 &=& \int dc \ \delta ( \int dx A_\mu n^\mu - c \Omega)\end{aligned}$$ with $\Omega = TV$ the spacetime volume. Then, we find $$\begin{aligned}
Z_{QED} &=& \int dc Z_{QED} ' \lbrack n,c \rbrack ,\end{aligned}$$ where the functional $$\begin{aligned}
Z_{QED} ' \lbrack n,c \rbrack &=&
\int {\cal D} {\bar \psi} {\cal D} \psi {\cal D} A
\exp \left\{ {\rm i} S_{em} \lbrack A, \xi \rbrack
+ {\rm i} S_D \lbrack A, {\bar \psi}, \psi \rbrack
\right\}
\nonumber\\
& &
\delta ( \int dx A_\mu n^\mu - c \Omega)\end{aligned}$$ can be concieved as the generating functional of the sector belonging to vector potential configurations in a hypersurface orthogonal to $n^\mu (x)$. The projected generating functional $ Z_{QED} ' \lbrack n,c \rbrack$ is gauge invariant for the choice of $n^\mu (x)$ satisfying the condition $\partial_\mu n^\mu (x) =0$. The projected effective action $S_{eff} \lbrack n,c \rbrack$ is defined by $$\begin{aligned}
\label{seffdf}
Z_{QED} ' \lbrack n,c \rbrack &=&
\exp \left\{ {\rm i} S_{eff} \lbrack n,c \rbrack
\right\} .\end{aligned}$$
For the choice $n^\mu (x) = {\bar A}^\mu (x)$ with a given background field ${\bar A}^\mu (x)$ satisfying the Lorentz condition $\partial_\mu {\bar A}^\mu (x) =0$, we obtain the effective action for the sector belonging to the given background ${\bar A}^\mu (x)$. Introducing the shifted integration variable $\alpha^\mu (x) =
A^\mu (x) - {\bar A}^\mu (x) $, the projected generating functional can be rewritten as $$\begin{aligned}
\label{zqedp}
Z_{QED} ' \lbrack {\bar A},c \rbrack &=&
\int d\lambda \int {\cal D} \alpha
\exp \left\{ {\rm i} S_{em} \lbrack {\bar A} + \alpha , \xi \rbrack
\right\}
\nonumber\\
& &
\exp \left\{
{\rm i} \lambda \left( \int dx {\bar A}^\mu {\bar A}_\mu + \int dx {\bar
A}^\mu \alpha_\mu - c \Omega \right)
\right\}
\nonumber\\
& &
\int {\cal D} {\bar \psi} {\cal D} \psi
\exp \left\{ {\rm i} S_D \lbrack {\bar A} + \alpha , {\bar \psi}
, \psi \rbrack \right\} .\end{aligned}$$
2. [*Identification of the background with the vacuum expectation value of the vector potential.*]{} As to the next, it is required that $$\begin{aligned}
\label{constraint}
\langle A^\mu (x) \rangle = {\bar A}^\mu (x)
, \qquad {\mbox {i.e.}} \qquad
\langle \alpha^\mu \rangle =0 .\end{aligned}$$ This condition is used to determine the constant $c$ as the functional of the vacuum expectation value of the vector potential $ \langle A^\mu (x) \rangle$.
For later use it is useful to introduce the external sources $j^\mu
(x)$, ${\bar \eta} (x)$, and $\eta (x)$ coupled to the quantum fluctuation $\alpha^\mu (x)$ of the vector potential, and to the fermion fields $\psi (x)$ and ${\bar \psi} (x)$, resp. Then, we find instead of Eq. (\[zqedp\]) the expression $$\begin{aligned}
\label{zqedpj}
Z_{QED} ' \lbrack {\bar A},c , j, \eta , {\bar \eta} \rbrack &=&
\int d\lambda \int {\cal D} \alpha
\exp \left\{ {\rm i} S_\lambda \lbrack {\bar A} + \alpha,
\xi, c \rbrack
+ \int dx j^\mu \alpha_\mu
\right\}
\nonumber\\
& &
Z_F \lbrack A, \eta, {\bar \eta} \rbrack\end{aligned}$$ with $$\begin{aligned}
S_\lambda \lbrack {\bar A} + \alpha, \lambda
,\xi, c \rbrack
&=&
S_{em} \lbrack {\bar A} + \alpha , \xi \rbrack
+ \lambda \left( \int dx {\bar A}^\mu {\bar A}_\mu + \int dx {\bar
A}^\mu \alpha_\mu - c \Omega \right) ,\end{aligned}$$ and $$\begin{aligned}
Z_F \lbrack A, \eta, {\bar \eta} \rbrack
&=&
\int {\cal D} {\bar \psi} {\cal D} \psi
\exp \left\{ {\rm i} S_D \lbrack {\bar A} + \alpha , {\bar \psi}
, \psi \rbrack \right\} .\end{aligned}$$
Projected effective action
==========================
The projected generating functional (\[zqedp\]) defines the projected effective action via Eq. (\[seffdf\]). We determine it in one-loop approximation, i.e. we replace the full vector potential $A^\mu (x)$ in the Dirac action $S_D$ by the background field ${\bar
A}^\mu (x)$. Thus, the generating functional is factored into an electromagnetic part and a fermionic part, $$\begin{aligned}
Z_{QED} ' \lbrack {\bar A},c , j, \eta , {\bar \eta} \rbrack
&=&
Z_{em} ' \lbrack {\bar A},c , j \rbrack
Z_F \lbrack {\bar A}, \eta, {\bar \eta} \rbrack .\end{aligned}$$ The explicit forms of the actions are given as $$\begin{aligned}
S_{em } \lbrack {\bar A}+\alpha , \xi \rbrack
&=&
\frac{1}{2} ( {\bar A} D^{-1} {\bar A} )
+ \frac{1}{2} ( \alpha D^{-1} \alpha )
+ ( \alpha D^{-1} {\bar A} ) ,\end{aligned}$$ and $$\begin{aligned}
S_D \lbrack {\bar A} \rbrack &=&
( {\bar \psi} G^{-1} \psi ) \end{aligned}$$ with the inverse of the photon propagator in Lorentz gauge, $$\begin{aligned}
D^{-1}_{\mu \nu} (x,y) &=&
\left\lbrack g_{\mu \nu} {\Box}_x +
\left( \xi^{-1} - 1 \right) \partial^x_\mu \partial^x_\nu
\right\rbrack \delta (x-y) ,\end{aligned}$$ and the inverse of the fermion propagator in the background field ${\bar A}^\mu (x)$, $$\begin{aligned}
G^{-1} (x,y) &=& \left( {\rm i} \gamma^\mu (\partial^x_\mu
- i \bar A_\mu (x) ) - m \right) \delta (x-y) \end{aligned}$$ with the Dirac matrices $\gamma^\mu$ $( \mu = 0,1,2,3)$. For the sake of simplicity, the notation $( f O g)= \int dx dy
f^a(x) O_{ab }(x,y) g^b (y)$ is used, where $a$ and $b$ are either Lorentz or spinor summation indices. In the one-loop approximation the path integrals are Gaussian ones and can be performed explicitly, leading to $$\begin{aligned}
\ln Z_{em} ' \lbrack {\bar A},c , j \rbrack
&=&
- \frac{1}{2} {\mbox { Tr }} \ln D^{-1}
- \frac{1}{2} \ln ( {\bar A} D {\bar A} )
+ \frac{ \rm i}{2} (jD_1 j) - (j {\bar A} )
\nonumber\\\
& &
+ \frac{i}{2} \frac{ c^2 \Omega^2}{ ( {\bar A} D {\bar A} ) }
+ c \Omega \frac{ (j D {\bar A} ) }{ ( {\bar A} D {\bar A} ) } ,\end{aligned}$$ and $$\begin{aligned}
\ln Z_F \lbrack {\bar A}, \eta, {\bar \eta} \rbrack
&=&
{\mbox { Tr }} \ln G^{-1} + {\rm i} ( {\bar \eta} G \eta )\end{aligned}$$ with the modified photon propagator $$\begin{aligned}
\label{modpro}
D_1^{\mu \nu} (x,y)
&=&
D^{\mu \nu} (x,y) - \frac{ \int du D^{\mu \rho}(y,u) {\bar A}_\rho
(u) \int dv D^{\nu \sigma}(x,v) {\bar A}_\rho (v)}{
( {\bar A} D {\bar A} ) } .\end{aligned}$$
Now we restrict our considerations to $1+1$ dimensional systems and time independent periodic backgrounds of the form $$\begin{aligned}
\label{ansatz}
{\bar A}^\mu = \delta ^{0\mu} a \cos (\ell x_1 )\end{aligned}$$ satisfying the Lorentz condition.
The constant $c$ can be determined by using the fact that $$\begin{aligned}
\label{cnull}
0 = \int {\cal D} \alpha \int d\lambda
\frac{\partial}{\partial\lambda}
\exp \left\{ {\rm i} S_\lambda \lbrack {\bar A} + \alpha,
\xi, c \rbrack
+ \int dx j^\mu \alpha_\mu
\right\}.\end{aligned}$$ From this we find that $$c_0\Omega=\int dx \bar A_\mu \bar A^\mu.$$
One establishes now for the one-loop effective action: $$\begin{aligned}
\label{effac}
{\rm i} S_{eff}& =&
\ln Z_{em} ' \lbrack {\bar A},c_0 , j=0 \rbrack
+ \ln Z_F \lbrack {\bar A}, \eta=0, {\bar \eta}=0 \rbrack
\nonumber\\
& = &
{\mbox { Tr }} \ln G^{-1} - \frac{1}{2}{\mbox { Tr }} \ln D^{-1}
+ \Omega \frac{\rm i}{4} a^2 \ell^2 - \frac{1}{2} \ln
\frac{a^2}{\ell^2} .\end{aligned}$$
In the infinite volume limit one finds $$\begin{aligned}
\label{actden}
- \Omega^{-1} S_{eff} & \sim &
- V^{-1} {\sum_{k}}^> \epsilon_k + V^{-1} \sum_k \frac{1}{2} \omega_k
- \frac{1}{4} a^2 \ell^2 .
\end{aligned}$$
Here the first and the second terms represent the energy density of the Dirac vacuum and that of the free electromagnetic field, resp. (see e.g. [@Raj89]). $\sum^>$ denotes the summation over all non-negative single fermion energies $\epsilon_k \ge 0$, being the energy eigenvalues of the Dirac equation in the external field ${\bar A}^\mu (x)$. The third term on the r.h.s. of Eq. (\[actden\]) is just the negative of the energy density of the periodic electric background field. The last term of the effective action (\[effac\]) gives a vanishing contribution to the action density in the infinite volume limit.
The meaning of the first two terms of Eq. (\[actden\]) might lead one to the false conclusion that the negative of the action density is equal to the energy density of the vacuum, as it would happen if the background field were constant [@Raj89]. Determining the energy density from the energy-momentum tensor we will show that the energy density does not equal the negative of the action density for inhomogeneous background field.
As 1+1 dimensional QED is a superrenormalizable theory, the action density is UV finite after subtracting the action density of the free vacuum $S_{eff,free}$, i.e. that of the vacuum in the absence of the background field. This difference $-\Gamma=\Omega^{-1}(-S_{eff}+S_{eff,free})$ has been calculated numerically by solving the Dirac equation for the single-fermion energies.
These energy eigenvalues show the same band structure plotted against the momentum in fig. \[felhasad\] as the electrons in the Kronig-Penney model [@Lan81] because of the periodic potential. Fig. \[effacden\] shows how $-\Gamma$ changes by modifying the amplitude $a$ and the wavelength $\ell$ of the potential. (The fermion rest mass is chosen for $m{=}1$.) One can recapitulate from fig. \[effacden\] that the surface $-\Gamma(a,\ell)$ has only a single stationary point at the origin of the parameter space $(a, \ell)$, i.e. the path integral defining the vacuum-vacuum transition amplitude is dominated by the trivial, identically vanishing field configuration $A^\mu (x) =0$.
Mean field equation
===================
The vacuum expectation value of the current can be written at one-loop order in the following form: $$\begin{aligned}
\label{aram}
\langle {\bar \psi} (x) \gamma^\mu \psi (x) \rangle
&=&
\int d\lambda \int {\cal D}\alpha
e^{ {\rm i} S_\lambda }
\frac{ \delta }{ {\rm i} \delta {\bar A}_\mu (x) } Z_F
\left/
\int d\lambda \int {\cal D}\alpha
e^{ {\rm i} S_\lambda } Z_F \right.
\nonumber\\
& = &
e^{ - {\rm i} S_{eff} } \left\lbrack
\frac{ \delta }{ {\rm i} \delta {\bar A}_\mu (x) }
e^{ {\rm i} S_{eff} }
- Z_F \frac{ \delta }{ {\rm i} \delta {\bar A}^\mu (x) }
Z_{em} ' \right\rbrack .\end{aligned}$$ The first term on the r.h.s. of Eq. (\[aram\]) is just the first functional derivative of the effective action $\delta S_{eff}/\delta
{\bar A}_\mu (x) $. In the infinite volume limit the second term gives: $$\begin{aligned}
&-& \exp \left\{ - \frac{ {\rm i} c_0^2}{ 2 ( {\bar A} D {\bar A} ) }
\right\}
\frac{ \delta }{ {\rm i} \delta {\bar A}_\mu (x) }
\exp \left\{ \frac{ {\rm i} c_0^2}{ 2 ( {\bar A} D {\bar A} ) }
\right\}
\nonumber\\
&=&
- \frac{ {\bar A}^\nu (x) }{ (D{\bar A} )^\nu (x) }
\left( \frac{ \delta c_0 }{ \delta {\bar A}_\mu (x) }
- {\bar A}^\mu (x) \right)
\nonumber\\
&=& - \ell^2 {\bar A}^\mu (x) = \partial_\nu {\bar F}^{\mu \nu}\end{aligned}$$ with the field strength tensor $ {\bar F}^{\mu \nu}$ evaluated from the background ${\bar A}^\mu$. Here we used that $$\begin{aligned}
\frac{ \delta c_0 }{ \delta {\bar A}_\mu (x) }
&=&
2 {\bar A}^\mu (x) \end{aligned}$$ due to Eq. (\[cnull\]). Thus, one obtains the equation $$\begin{aligned}
\label{mfeqact}
\langle {\bar \psi} (x) \gamma^\mu \psi (x) \rangle
&=&
\frac{ \delta S_{eff} }{ \delta {\bar A}_\mu (x) }
+ \partial_\nu {\bar F}^{\mu \nu}.\end{aligned}$$ For the background field configuration making the effective action extremum, one recovers the vacuum expectation value of the Maxwell equation. Thus, the effective action has an extremum at the background field configuration coinciding with the mean field solution. In our numerical search, this is the trivial extremum found at $a=\ell=0$.
Satisfying the necessary condition of the extremum of the effective action, Eq. (\[mfeqact\]) results in the Poisson’s equation for the mean field $\bar A^0$, which must be considered together with coupled set of operator equations for the quantum fields $\psi(x)$, $\bar\psi(x)$ and $\alpha^\mu(x)$. The latter equations should be solved and the result substituted in the l.h.s. of Eq. (\[mfeqact\]), in order to make the charge density explicit.
Energy density of the vacuum
============================
The symmetric energy momentum tensor is defined by [@Lan81]: $$\begin{aligned}
T^{\mu \nu} &=&
\frac{ \partial {\cal L} }{\partial F_{\kappa \mu} }
F_\kappa^{\; \nu}
+ \frac{ \partial {\cal L} }{\partial ( D_\mu \psi ) }
D^\nu \psi
+ D^{\nu \star}\psi \frac{ \partial {\cal L} }{ \partial
(D_\mu^\star {\bar \psi} ) }
- g^{\mu \nu} {\cal L}\end{aligned}$$ with the Lagrange density $$\begin{aligned}
{\cal L} &=& - \frac{1}{4} F_{\mu \nu} F^{\mu \nu} - \frac{1}{2\xi}
(\partial_\mu A^\mu )^2
+ \lambda A_\mu {\bar A}^\mu - \lambda c_0
\nonumber\\
& &
+ \frac{i}{2} \left( {\bar \psi} \gamma^\mu D_\mu \psi
- D_\mu^\star {\bar \psi} \gamma^\mu \psi \right) - m {\bar
\psi} \psi\end{aligned}$$ corresponding to the action $\int dx {\cal L} = S_\lambda + S_D$ and with the covariant derivative $D^\mu = \partial^\mu - {\rm i}
A^\mu $.
Substituting the ansatz (\[ansatz\]) one obtains for the component $T^{00}$ of the energy-momentum tensor: $$\begin{aligned}
T^{00} &=& T^{00}_{em,2} \lbrack \alpha \rbrack
+ T^{00}_{em,1} \lbrack {\bar A}, \alpha , \lambda \rbrack
+ T^{00}_{em,0} \lbrack {\bar A} \rbrack
+ T^{00}_\lambda \lbrack {\bar A}, c \rbrack
\nonumber\\
& &
+T^{00}_F \lbrack {\bar A}+\alpha , {\bar \psi}, \psi \rbrack\end{aligned}$$ where $T^{00}_{em, a}$ $(a=0,1,2)$ are the terms independent of the fermion field and being of the order $(\alpha )^a$. Due to the constraint (\[constraint\]), the expectation value of the first order term vanishes, therefore it can be neglected. The second order term $$\begin{aligned}
T^{00}_{em,2} \lbrack \alpha \rbrack &=&
- \partial_0 \alpha_\mu \partial^0 \alpha^\mu +
2 \partial_0 \alpha_\mu \partial^\mu \alpha^0
- \partial_\mu \alpha_0 \partial^\mu \alpha^0
\nonumber\\
& &
+ \frac{1}{2} \partial_\mu \alpha_\nu \partial^\mu \alpha^\nu
- \frac{1}{2} \partial_\mu \alpha_\nu \partial^\nu \alpha^\mu
+ \frac{1}{2\xi} \partial_\mu \alpha^\mu \partial_\nu \alpha^\nu\end{aligned}$$ is the expression for the free electromagnetic field, whereas the zeroth order term is given by $$\begin{aligned}
\label{fielden}
T^{00}_{em,0} \lbrack {\bar A} \rbrack
&=&
- \partial_\mu {\bar A}_0 \partial^\mu {\bar A}^0
+ \frac{1}{2} \partial_\mu {\bar A}_\nu
\partial^\mu {\bar A}^\nu
\nonumber\\
&=&
\frac{1}{2} \left( \nabla_j {\bar A}^0 \right)^{\; 2}
=
\frac{1}{2} \ell^2 a^2 \sin^2 (\ell x_1) \end{aligned}$$ and represents the energy density of the periodic background field. Due to the projection to a particular sector of the theory, the additional term $$\begin{aligned}
T^{00}_\lambda [{\bar A}, c]
&=&
( c_0 - {\bar A}^\mu (x){\bar A_\mu }(x) ) \lambda \nonumber\\
&=&
\biggl(( \Omega^{-1} \int du {\bar A}^\mu (u){\bar A_\mu }(u)
- {\bar A}^\mu (x){\bar A_\mu }(x) \biggr) \lambda
\nonumber\\
&=&
\frac{1}{2} a^2 ( 1 - 2 \cos^2 (\ell x_1 ) ) \lambda\end{aligned}$$ occurs. It is easy to see that this term does not contribute to the energy, but gives a non-vanishing periodic contribution to the energy density. Finally, the term $$\begin{aligned}
\label{kinen}
T^{00}_F &=& - \frac{i}{2}\partial^0 \left( {\bar \psi} \gamma^0 \psi
\right)
+ {\bar \psi} \gamma^0 {\tilde H_D} ( {\bar A} + \alpha )
\psi \end{aligned}$$ represents the contribution of the fermions to the energy density. The first term on the r.h.s. gives vanishing contribution to the total energy if charge conservation is required, whereas the second term accounts for the kinetic energy of the fermion system. At one-loop order, we have to substitute $\alpha^\mu =0$ in the ‘Hamilton operator’, i.e. write $$\begin{aligned}
{\tilde H}_D ( {\bar A} )
&=& - {\rm i} \gamma^0 \gamma^j \partial^x_j
- \gamma^0 \gamma^j {\bar A}_j (x)
+ \gamma^0 m .\end{aligned}$$
Since ${\bar A}_j(x) = 0$ in our case, we obtain ${\tilde H}_D ({\bar A}) =
- {\rm i} \gamma^0 \gamma^j \partial^x_j + \gamma^0 m$, i.e. the free ‘Hamilton operator’.
The vacuum expectation value of $T^{00}$ in the presence of the background field ${\bar A^{\mu}}$ is defined as
$$\begin{aligned}
\label{37}
\langle T^{00} (x) \rangle
&=&
\int d\lambda \int {\cal D} \alpha
{\cal D} {\bar \psi} {\cal D} \psi T^{00} (x)
e^{ {\rm i} \left( S_\lambda + S_D \right) }
\left/
\int d\lambda \int {\cal D} \alpha
{\cal D} {\bar \psi} {\cal D} \psi
e^{ {\rm i} \left( S_\lambda + S_D \right) }
\right.
\nonumber\\\end{aligned}$$
Similarly, the vacuum expectation value of $T^{00}_{free}$ in the absence of the background field $\langle T^{00}_{free} (x) \rangle_{0}$ is given by substituting ${\bar A}_{\mu} (x) =0$ in Eq.(\[37\]) and replacing $T^{00}(x)$ by $T^{00}_{free}(x)$. Then, the Casimir energy of the vacuum due to the background field ${\bar A}_{\mu} (x) \neq 0$ is $$\begin{aligned}
\label{39}
E_c &=& \int dx_1 \langle T^{00} (x_1) \rangle -
\int dx_1 \langle T^{00}_{free} (x_1) \rangle_{0}\nonumber\\
&=& \int dx_1 \left[(l^2 a^2/2) \sin^2(l x_1) +
\langle {\bar \Psi} \gamma^0 H_{D0} \Psi \rangle -
\langle {\bar \Psi} \gamma^0 H_{D0} \Psi \rangle_{0} \right]\end{aligned}$$ This expression of the Casimir-energy reminds one on the expression for the energy of a system of electric charges in classical electrodynamics. There, the energy is the sum of the energy of the electromagnetic field and the kinetic energy of the charges [@J]. In our case the Casimir energy is the sum of the background electric field and the change of the relativistic kinetic energy (including the rest mass) of the Dirac-sea due to the presence of the background field.
Eq.(\[37\]) can be rewritten by the help of the generating functional $Z'_{QED} \lbrack {\bar A}, c, j, \eta, {\bar \eta} \rbrack$ as $$\begin{aligned}
\langle T^{00} (x) \rangle
&=&
\left( 1/ Z'_{QED} \lbrack {\bar A}, c_0 \rbrack \right)
\left.
T^{00}_{op} (x)
Z'_{QED} \lbrack {\bar A}, c, j, \eta, {\bar \eta} \rbrack
\right|_{ j, {\bar \eta}, \eta =0; c=c_0 }\end{aligned}$$ where $ T^{00}_{op} (x)$ denotes the operator obtained from $T^{00}$ by replacing the fields $\alpha^\mu (x)$, $\psi (x)$, and ${\bar \psi} (x)$ by the operators $\delta/\delta j_\mu (x)$, $\delta / \delta {\bar \eta} (x) $, and $- \delta /\delta \eta (x)$ and the variable $\lambda$ by $ {\rm i} \Omega^{-1} \partial /\partial c$.
Then, we find $$\begin{aligned}
\langle T^{00}_\lambda \rangle &=&
\frac{1}{2} a^2 \ell^2 \left( 2 \cos^2 (\ell x_1 ) -1 \right) . \end{aligned}$$ Furthermore, the expectation value $\langle T^{00}_{em,2}
\lbrack \alpha \rbrack \rangle$ is equal to the energy density of the free electromagnetic field. Indeed, it holds for the second derivatives $$\begin{aligned}
\left.
\delta^2 Z_{em} ' / \delta j_\mu (y) \delta j_\nu (x)
\right|_{j=0, c_0}
&=&
\left.
{\rm i} D^{\mu \nu}_1 (x,y) Z_{em} '
\right|_{j=0, c_0} \end{aligned}$$ where $ D^{\mu \nu}_1 (x,y)$ tends to the free propagator $D^{\mu \nu}
(x,y)$ in the infinite volume limit, since the last term on the r.h.s. of Eq. (\[modpro\]) is of the order $\Omega^{-1}$. Consequently, the pure electromagnetic contribution to the Casimir energy density is given by $$\begin{aligned}
e_{em} (x) &=&
\langle
T^{00}_{em,0} \lbrack {\bar A} \rbrack
+ T^{00}_\lambda \lbrack {\bar A}, c_0 \rbrack
\rangle
\nonumber\\
&=&
\frac{1}{2} a^2 \ell^2 \cos^2 (\ell x_1 ) .\end{aligned}$$
It is more cumbersome to evaluate the fermionic contribution $e_F (x)
= \langle T^{00}_F \rangle - \langle T^{00}_{free} \rangle_{0} $ to the Casimir energy density, where the energy density of the free Dirac vacuum is subtracted. Since QED in dimensions $1+1$ is superrenormalizable, the difference turns out to be UV finite without further renormalizations. We perform the evaluation of the fermionic part of the Casimir energy in the second quantized formalism. Then we can write $e_F = \langle 0 | : T^{00}_F : | 0 \rangle$ where $ |0 \rangle$ is the ‘interacting’ vacuum (in the presence of the periodic background field) and $: \ldots :$ denotes normal ordering with respect to the normal vacuum (in the absence of the background field). The evaluation is performed in the following steps:
1. The fermion field $\psi$ is expanded in terms of the eigenspinors $f^{(ks)} (x_1)$ and $g^{(ks)} (x_1)$ of the Dirac Hamiltonian $H_D ( {\bar A} )$ belonging to the energy eigenvalues $\epsilon_{ks}$ and $-\epsilon_{ks}$, resp. Here the quasi-momentum $k \in \lbrack - \ell/2, \ell/2 \rbrack$ and the integer $s \ge 0$ enumerating the bands are introduced.
2. Then the creation and annihilation operators $a_{ks}^\dagger$, $ b_{ks}^\dagger$ and $a_{ks}$, $b_{ks}$ of these stationary single particle states are expressed as linear combinations of the creation and annihilation operators $A_{ks}^\dagger$, $B_{ks}^\dagger$ and $A_{ks}$, $B_{ks}$ of the free fermion states $F^{(kr)} (x_1)$ and $G^{(kr)} (x_1)$ of energies $\epsilon_{kr}^{(0)} = \sqrt{ m^2 + (k + \ell r)^2}$ and $-\epsilon_{kr}^{(0)}$, resp. In terms of the latter, the normal ordering is performed.
3. Finally, the normal ordered operator is reexpressed in terms of the operators $a_{ks}^\dagger$, $b_{ks}^\dagger$ and $a_{ks}$, $b_{ks}$ and the vacuum expectation value with respect to the vacuum in the presence of the background field is taken.
Thus, one arrives to the following expression $$\begin{aligned}
T^{00}_F (x)
&=& e_F (x) +
\sum_{\rho \rho'} \left\{
a_{\rho '}^\dagger (t) a_\rho (t) {\tilde B}_+^{\rho '}\cdot
B_-^\rho
+ b_{\rho }^\dagger (t) b_{\rho'} (t) {\tilde A}_+^{\rho '}\cdot
A_-^\rho
\right.
\nonumber\\
& &
\left.
+ b_{\rho '} (t) a_{\rho} (t) {\tilde A}_+^{\rho '}\cdot
B_-^\rho
+ a_{\rho '}^\dagger (t) b_{\rho}^\dagger (t) {\tilde B}_+^{\rho '}\cdot
A_-^\rho
\right\}\end{aligned}$$ where, with $\rho \equiv (ps)$, $\rho' \equiv (p's')$, $$\begin{aligned}
{\tilde A}_+^{\rho'} = {\tilde \alpha}_F^{\dagger \; \rho'}
+ {\tilde \alpha}_G^{\dagger \; \rho'},
\qquad
{\tilde B}_+^{\rho'} = {\tilde \beta}_F^{\dagger \; \rho'}
+ {\tilde \beta}_G^{\dagger \; \rho'},
\nonumber\\
{ A}_-^{\rho} = { \alpha}_F^{ \rho}
- { \alpha}_G^{\rho},
\qquad
{ B}_-^{\rho} = { \beta}_F^{ \rho}
- { \beta}_G^{\rho},\end{aligned}$$ and $$\begin{aligned}
\alpha_F^{ps} = \sum_{kr} \epsilon^0_{kr} \alpha_-^{pksr} F_{kr}
(x),
\qquad
\alpha_G^{ps} = \sum_{kr} \epsilon^0_{kr} \alpha_+^{pksr} G_{kr}
(x),
\nonumber\\
\beta_F^{ps} = \sum_{kr} \epsilon^0_{kr} \beta_-^{pksr} F_{kr}
(x),
\qquad
\beta_G^{ps} = \sum_{kr} \epsilon^0_{kr} \beta_+^{pksr} G_{kr}
(x),
\nonumber\\
{\tilde \alpha}_F^{* \; ps} = \sum_{kr} \alpha_-^{* \; pksr} F_{kr}^*
(x),
\qquad
{\tilde \alpha}_G^{*\; ps} = \sum_{kr} \alpha_+^{* \;pksr} G_{kr}^*
(x),
\nonumber\\
{\tilde \beta}_F^{* \; ps} = \sum_{kr} \beta_-^{* \;pksr} F_{kr}^*
(x),
\qquad
{\tilde \beta}_G^{* \; ps} = \sum_{kr} \beta_+^{* \; pksr} G_{kr}^*
(x),\end{aligned}$$ with the constant coefficients $$\begin{aligned}
\alpha_-^{pksr} = \int dx F^{* \; kr} (x) g^{ps} (x) ,
\qquad
\alpha_+^{pksr} = \int dx G^{* \; kr} (x) g^{ps} (x) ,
\nonumber\\
\beta_-^{pksr} = \int dx F^{* \; kr} (x) f^{ps} (x) ,
\qquad
\beta_+^{pksr} = \int dx G^{* \; kr} (x) f^{ps} (x) .\end{aligned}$$ The latter are the overlap integrals of the eigenspinors in the presence of the background and those in the absence of the background. Furthermore, the time dependent creation-annihilation operators are introduced: $$\begin{aligned}
a_{ps} (t) \equiv a_{ps} e^{ - {\rm i} \epsilon_{ps} t } ,
\qquad
b_{ps} (t) \equiv b_{ps} e^{ - {\rm i} \epsilon_{ps} t } .\end{aligned}$$ The constant term $$\begin{aligned}
\label{eF}
e_F (x) =
\sum_{ps} \left(
{\tilde \beta}_G^{* \; ps} \cdot \beta_G^{ps}
+ {\tilde \alpha}_F^{* \; ps} \cdot \alpha_F^{ps}
+ {\tilde \alpha}_G^{* \; ps} \cdot \alpha_F^{ps}
- {\tilde \alpha}_F^{* \; ps} \cdot \alpha_G^{ps}
\right) \end{aligned}$$ represents the Casimir energy density of the fermion vacuum. The last two terms of Eq. (\[eF\]) cancel.
Performing the calculation results in: $$\begin{aligned}
\label{enden}
e_F(x)&=&\sum_{%\begin{Sb}
k\in [-\frac{l}2,\frac{l}2]
rr^\prime\in N
}%\end{Sb}
\biggl[\epsilon^0_{-kr}
({\cal F}_1^{(krr^\prime)}+{\cal F}_2^{(krr^\prime)})
\cos((r-r^\prime)\ell x) \nonumber\\
&&+(\epsilon^0_{k-r^\prime}-\epsilon^0_{kr})
{\cal F}_3^{(krr^\prime)}
\cos((r+r^\prime)\ell x) \biggr],\end{aligned}$$ where we introduced the following notations: $$\begin{aligned}
{\cal F}_1^{(krr^\prime)}&=&\sum_{s\in Z}
U^{(-kr^\prime)}_\alpha U^{(-kr)}_\alpha
U^{(-kr^\prime)}_\beta v^{(-r^\prime k s)}_\beta
U^{(-kr)}_\gamma v^{(-r k s)}_\gamma\nonumber\\
{\cal F}_2^{(krr^\prime)}&=&\sum_{s\in Z}
V^{(-kr)}_\alpha V^{(-kr^\prime)}_\alpha
V^{(-kr)}_\beta u^{(-r k s)}_\beta
V^{(-kr^\prime)}_\gamma u^{(-r^\prime k s)}_\gamma\nonumber\\
{\cal F}_3^{(krr^\prime)}&=&\sum_{s\in Z}
U^{(-kr^\prime)}_\alpha V^{(kr)}_\alpha
U^{(-kr^\prime)}_\beta u^{(-r^\prime k s)}_\beta
V^{(kr)}_\gamma v^{(r k s)}_\gamma.\end{aligned}$$ $U^{(kr)}$ and $V^{(kr)}$ denotes the eigenspinors of the free Dirac-equation for the positive and negative energy eigenvalues, respectively.
It is straightforward to establish from Eq. (\[enden\]) that the volume integral of the energy density is not negative: $$E_F = \sum_{k r s}\biggl\{\epsilon^0_{-k r}\Bigl[
|U^{(-k r)}_\beta v^{(r k s)}_\beta|^2+
|V^{(-k r)}_\beta v^{(r k s)}_\beta|^2\Bigr]\biggr\}\geq 0.$$
By numerical calculations we were convinced (see Fig. \[thens\]) that this expression of the energy only vanishes in case of $a{=}\ell{=}0$. This means that the Casimir energy $E_c$ of the vacuum is always positive if a non-vanishing periodic electric background field is assumed. Thus, the vacuum of $QED_2$ does not favour a periodic mean field energetically with respect to the normal vacuum.
Necessary condition of energy minimum
=====================================
The vacuum expectation value ${\bar A}^0 (x)$ is defined by the minimum of the energy functional $$\begin{aligned}
T E \lbrack {\bar A}^0 (x) \rbrack
&=&
\int dx \langle T^{00} (x) \rangle
=
\int dx \left\langle \left( T^{00}_{em} (x) + T^{00}_{\lambda} (x)
+ T^{00}_F (x) \right)
\right\rangle .\end{aligned}$$ The second term vanishes due to the explicit value of $c_0$. Thus, the necessary condition of the energy minimum takes the form $$\begin{aligned}
\frac{\delta}{\delta {\bar A}^0 (x) }
\int dy \left( T^{00}_{em,0} \lbrack {\bar A}^0 (y) \rbrack
+
\left\langle T^{00}_F (y) \right\rangle
\right)=0 .\end{aligned}$$ The functional derivative of the first term gives $$\begin{aligned}
\frac{\delta}{\delta {\bar A}^0 (x) }
\int dy \left( T^{00}_{em,0} \lbrack {\bar A}^0 (y) \rbrack \right)
&=&
- \nabla^2 {\bar A}^0 (x) .\end{aligned}$$ At one-loop order the fermionic contribution to the energy can be rewritten as $$\begin{aligned}
TE_F &=&
\int dy \left\langle T^{00}_F (y) \right\rangle
\nonumber\\
&=&
Z_F^{-1} \lbrack {\bar A},0, 0 \rbrack \int dy T^{00}_{F,op} (y)
\left. Z_F \lbrack {\bar A}, \eta , {\bar \eta} \rbrack
\right|_{\eta={\bar \eta}=0} .\end{aligned}$$ Taking its functional derivative we find $$\begin{aligned}
\label{funcder}
\frac{\delta}{ \delta {\bar A}^0 (x) } TE_F
&=&
- \frac{\delta \ln Z_F \lbrack {\bar A}, 0,0\rbrack}{
\delta {\bar A}^0 (x) } TE_F \nonumber\\
&&
+ Z_F^{-1} \lbrack {\bar A},0, 0 \rbrack \int dy T^{00}_{F,op} (y)
\left. \frac{\delta Z_F \lbrack {\bar A}, \eta , {\bar \eta}
\rbrack
}{ \delta {\bar A}^0 (x) }
\right|_{\eta={\bar \eta}=0}
\nonumber\\
&=&
{\rm i} \int dy \biggl[
\frac{\rm i}{2} \partial_y^0 \biggl(
\frac{\delta }{ \delta \eta (y) } \gamma^0 \frac{\delta
}{\delta {\bar \eta} (y) } \biggr)\nonumber\\
&&
- \frac{\delta }{\delta \eta (y) } \gamma^0 H_{D0}^y
\frac{\delta }{ \delta {\bar \eta} (y) }
\biggr]
\left( {\bar \eta} \frac{ \delta G}{
\delta {\bar A}^0 (x) }
\eta \right)
e^{ {\rm i} ( {\bar \eta} G \eta )} \biggl|_0
\nonumber\\
&=&
- {\rm i} {\mbox { tr }} \left( \gamma^0 G(x,x) \right)
+ c^0 (x) \end{aligned}$$ with $$\begin{aligned}
c_0 (x) &=& \frac{1}{2} \int dy [
{\mbox { tr }} ( \gamma^0 G(y,x) \gamma^0 D^0_y
G(x,y) )\nonumber\\
&&
- {\mbox { tr }} ( \gamma^0 G(x,y) \gamma^0 D^{0*}_y
G(y,x) )] .\end{aligned}$$ Thus, we find the following equation for the field ${\bar A}^0 (x)$ minimizing the energy: $$\begin{aligned}
\partial_\nu {\bar F}^{0\nu}
&=&
\left\langle {\bar \psi} (x) \gamma^0 \psi (x) \right\rangle
+ c^0 (x) .
\label{mfeq}\end{aligned}$$ The first term on r.h.s. is the expectation value of the charge density $j^0=-i{\mbox{tr}}(\gamma^0G(x,x))$, determined via the propagator $G(x,x)$ as a given functional of $\bar A^0(x)$, and a similar statement holds for $c^0(x)$.
A similar equation appears in the classical case when a certain charge distribution moves in an electromagnetic field. Integrating out the effect of the charged particle distribution it will result in a polarisation charge density term beside the common charge density in the equation of motion for the scalar potential. In Eq. (\[mfeq\]) the term $c^0(x)$ corresponds to a polarisation charge density, so minimizing the energy with respect to $\bar
A^0(x)$ leads to an equation similar to that of Poisson’s equation in a polarised medium. Thus Eq. (\[mfeq\]) can be solved directly for $\bar A^0(x)$ in principle without the need for solving any other equations. On the contrary to this the mean field equation (\[mfeqact\]) obtained by extremizing the effective action does not take the polarisation of the vacuum into account, in order to do this we have to solve a system of operator equations as well.
Conclusions
===========
The energy density of $QED_2$ in the presence of a periodic mean field is determined from the energy-momentum tensor. For this purpose a projection method is worked out which is applied to treat the mean electromagnetic field self-consistently. The projected effective action and the energy density of the vacuum are derived at one-loop order, whereas the interaction of the electron-positron field with the periodic mean field is treated exactly. It is established, that the negative of the effective action must not be regarded as the energy of the system if the background field is not constant. It was shown that the necessary conditions of the extremum of the effective action and the minimum of the energy lead to different equations for the vacuum expectation value of $\bar A^0$. Eq. (\[mfeqact\]) obtained by extremizing the effective action is the vacuum expectation value of the Poisson’s equation that does not include the polarisation of the vacuum due to one-loop radiation corrections. Those are accounted for by separate operator equations for the quantum fluctuations of the electromagnetic field and for the Dirac field. On the other hand Eq. (\[mfeq\]) obtained by minimizing the energy functional includes the polarisation effects of the vacuum.
The expectation value of the component $T^{00}$ of the energy-momentum tensor was determined as the function of the amplitude $a$ and wave number $l$ of the static periodic scalar potential ${\bar A}^0 (x_1)
= a \cos(l x_1)$. It is found that the vacuum configuration with this periodic electric mean field is not favoured energetically compared to the normal vacuum. The volume integral of the energy density plotted against the amplitude $a$ in Fig. \[thens\] shows that the energy of the system increases with increasing $a$ monotonically.
The result obtained contradicts to our naive expectation discussed in the Introduction. Possibly, the reason is that the naive picture mentioned there does not take into account the uncertainty principle i. e. for a certain $\ell$ the energy spread of a wave packet localized in an interval of $\sim
1/\ell$ could be much larger than the amplitude of the potential and of course in this case our naive picture is not valid any more. We set now forth our work looking for periodic ground states at finite chemical potential.
Solution of the Dirac equation in periodic external field {#soldirac}
=========================================================
Relativistic Bloch waves
------------------------
To get the projected effective action at one-loop order in Eq. (\[effac\]) we must find the fermionic single-particle energies, so we have to solve the Dirac equation in the presence of the potential (\[ansatz\]): $$(i\gamma^{\mu}(\partial_{\mu}-ie\bar A_{\mu})-m)\psi=0.
\label{diracegy}$$ We look for the solution of Eq. (\[diracegy\]) in the form of Bloch waves corresponding to the energy eigenvalues $E$: $$\begin{aligned}
\label{megoldas}
\psi_{\alpha}=e^{-i\epsilon_{ps}t}
e^{ipx}\sum_{n=-\infty}^\infty u_{\alpha n}e^{inlx}
=\left\{\begin{array}{r@{,\quad \mbox{if}\quad}l}
e^{-i\epsilon_{ps}t}f^{ps}(x)& E=\epsilon_{ps}>0 \\
e^{i\epsilon_{ps}t}g^{ps}(x)& E=-\epsilon_{ps}>0
\end{array}\right.\end{aligned}$$ Inserting (\[megoldas\]) into (\[diracegy\]) we find $$\sum_{n=-\infty}^\infty\biggl(
u_{\alpha n}(\epsilon\gamma^0-(p-nl)\gamma^1-m{\bf 1}) \ + \ \frac{a}2 \gamma^0
(u_{\alpha n+1}+u_{\alpha n-1})\biggr)e^{i(p+nl)x-iEt}=0.
\label{matrix}$$ We get non-trivial solutions when the determinant of the matrix appearing next to the Dirac-spinors equals zero. If we had solved the Schrödinger equation in the presence of this sinusoidal potential, the form of the matrix would have been tridiagonal which means that the diagonal and the neighouring diagonal elements are nonzero. In relativistic case the matrix elements are replaced by $d$ dimensional $\gamma$ matrices but the structure of the matrix remains unchanged.
Eigenvalues, eigenspinors {#eigenval}
-------------------------
We cannot get the eigenvalues analitically because of the complicated structure of the matrix but we can determine them as precisely as we wish. In numerical calculations we work with matrices with finite dimension. Using the well-known identity [@numrec] $${\rm det}[A]={\rm det}\pmatrix{
P & Q \cr
R & S \cr
}={\rm det}[P] \ {\rm det}[S-RP^{-1}Q]\label{lemma}$$ we can reduce the problem of calculating the determinants of these $d(2n+1)\times d(2n+1)$ dimensional matrices to calculating four dimensional matrices by identifying the upper left matrix element of the matrix under investigation with $P$ in Eq. (\[lemma\]). ($n$ denotes the number of $u_{\alpha,i}$-s taken into account in Eq. (\[matrix\]).) Using identity (\[lemma\]) $2n$ times we get a product of $2n$ determinants of four dimensional matrices. We computed the determinant of Eq.(\[matrix\]) as the function of $E$ and determined its zeros, corresponding to the energy eigenvalues $E=\pm\epsilon$
In order to evaluate $-\Gamma$ we have to sum the eigenvalues $-\epsilon_{ps}<0$ and extract from it the sum of the negative eigenvalues of the free Dirac equation. This difference will depend on the accuracy of the eigenvalues, the number of the eigenvalues taken into account in the sum, the size of the chosen matrix and, of course, the parameters of the potential, the amplitude $a$ and the wave number $\ell$. We have to be convinced of the stability of the numerical calculation. We increased the number of members in the sum and the numerical accuracy of the determination of eigenspinors by choosing larger matrices as far as we have seen that the energy difference does not change significantly.
To get the expectation value of the component $T^{00}$ of the energy-momentum tensor we also have to determine the eigenspinors of the Dirac equation (\[diracegy\]). We calculated them with the help of the eigenvalues by solving a system of homogeneous linear equations (\[matrix\]) for the eigenspinors.
The authors would like to thank J. Polónyi for consulting this work and G. Plunien for the valuable discussions. S.N. thanks G. Soff for his kind hospitality. K.S. expresses his gratitude for the follow-up grant of the Alexander von Humboldt Foundation and W. Greiner for his kind hospitality. This work was supported by the projects OTKA T023844/97, DAAD-MÖB 27/1999 and NATO Grant PST.CLG.975722.
[99]{} -1mm J. Fingberg, J. Polonyi, Nucl. Phys. [**B486**]{} (1997) 315 R. Rajaraman, [*Solitons and Instantons*]{} (North-Holland, Amsterdam, 1989) L. Landau, M. Lifshitz, [*Course of Theoretical Phys. Vol. 4.: Relativistic Quantum Theory*]{} (Pergamon Press, Oxford, 1971) J. D. Jackson, [*Classical Electrodynamics* ]{} (Wiley, New York, 1975) W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Wetterling, [*Numerical Recipes. The Art of Scientific Computing*]{} (Cambridge University Press, Cambridge, 1986)
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We present first observations of a dome-shaped large-scale EUV coronal wave, recorded by the EUVI instrument onboard STEREO-B on January 17, 2010. The main arguments that the observed structure is the wave dome (and not the CME) are: a) the spherical form and sharpness of the dome’s outer edge and the erupting CME loops observed inside the dome; b) the low-coronal wave signatures above the limb perfectly connecting to the on-disk signatures of the wave; c) the lateral extent of the expanding dome which is much larger than that of the coronal dimming; d) the associated high-frequency type II burst indicating shock formation low in the corona. The velocity of the upward expansion of the wave dome ($v \sim 650$ km s$^{-1}$) is larger than that of the lateral expansion of the wave ($v \sim 280$ km s$^{-1}$), indicating that the upward dome expansion is driven all the time, and thus depends on the CME speed, whereas in the lateral direction it is freely propagating after the CME lateral expansion stops. We also examine the evolution of the perturbation characteristics: First the perturbation profile steepens and the amplitude increases. Thereafter, the amplitude decreases with r$^{-2.5 \pm 0.3}$, the width broadens, and the integral below the perturbation remains constant. Our findings are consistent with the spherical expansion and decay of a weakly shocked fast-mode MHD wave.'
author:
- 'A.M. Veronig'
- 'N. Muhr'
- 'I.W. Kienreich'
- 'M. Temmer'
- 'B. Vršnak'
title: 'First observations of a dome-shaped large-scale coronal EUV wave'
---
Introduction
============
Large-scale propagating disturbances in the solar atmosphere occuring in association with flares and coronal mass ejections (CMEs) have been first observed in chromospheric filtergrams [@moreton60; @athay61]. These “Moreton waves” propagate with typical velocities in the order of 1000 km s$^{-1}$, and have been interpreted as the ground-track of a dome-shaped MHD wave front propagating through the solar corona, which compresses and pushes the chromospheric plasma downward when sweeping over it [@uchida68]. The Extreme-ultraviolet Imaging Telescope [EIT; @delaboudiniere95] onboard SOHO for the first time imaged such wave-like disturbances in coronal emission lines [@moses97; @thompson98].
EIT waves have been initially interpreted as the coronal counterparts of the chromospheric Moreton waves as predicted in Uchida’s fast-mode coronal MHD wave model [@thompson99]. However, differences in morphology and propagation velocities of Moreton waves and EIT waves, which lie mostly in the range 200–400 km s$^{-1}$ [@klassen00; @thompson09] led to severe doubts of this interpretation and alternative models were put forward. Some of them question if the phenomenon is a wave at all, and instead suggest that it is a signature of the large-scale coronal restructuring due to the erupting CME causing plasma compression or localized energy release [e.g. @delanee99; @chen02; @attrill07]. For detailed discussions of the different models, we refer to the recent reviews by [@warmuth07], [@vrsnak08], and [@wills10].
There seems to be some consensus that Moreton waves are indeed shock waves, as is suggested by their high propagation velocities and perturbation characteristics. It was shown that the amplitude and velocity of Moreton waves decrease, and the width of the wave pulse broadens as it propagates, consistent with a large-amplitude wave or freely propagating shock wave that formed by steepening of a simple wave [e.g. @warmuth04b]. However, the situation is unclear for large-scale waves observed in the corona. There is rather qualitative insight that the wave fronts not only become more diffuse but also broaden as they propagate [@thompson99; @klassen00; @podladchikova05], which may be due to energy flux conservation or dispersion of the wave [e.g. @wills10]. On a statistical basis of different EIT waves, [@warmuth10] concludes that for larger distances, the perturbation amplitudes tend to become smaller and the width larger. However, a case study of the wave evolution in high-cadence TRACE images by [@wills06] revealed that the width of the wave pulse remained constant during the propagation.
Due to the low EIT cadence of $\sim$12–15 min it was not possible to detect fast EIT waves and to study the evolution of the wave pulse characteristics, which is important in constraining the physical processes. The Extreme Ultraviolet Imager [EUVI; @howard08] instruments onboard the twin spacecraft of the Solar-Terrestrial Relations Observatory [STEREO; @kaiser08] overcome these limitations and offer several advantages for the study of large-scale coronal waves, in particular due to their high cadence, large field-of-view (FoV), high sensitivity, and the simultaneous observations from two vantage points.
Since the launch of the STEREO satellites in October 2006, a variety of case studies of large-scale coronal waves have been carried out using EUVI data [@long08; @veronig08; @gopalswamy09; @patsourakos09; @patsourakos09b; @attrill09; @cohen09; @kienreich09; @ma09; @zhukov09; @dai10]. The typical propagation velocities that were derived from EUVI waves (that all occurred during solar minimum conditions and were not accompanied by Moreton waves) are in the range 200–350 km s$^{-1}$. For the kinematical evolution, the results so far are rather inconclusive: some of the waves studied with EUVI showed evidence for deceleration during their propagation, consistent with the decay and deceleration of a large amplitude wave to the fast-mode speed of the ambient corona [e.g. @long08; @veronig08], whereas others showed constant velocity [e.g. @ma09; @kienreich09]. It is worth noting that many of the EUVI wave studies revealed a close association of the wave and the erupting CME and its expanding flanks [e.g. @veronig08; @attrill09; @kienreich09; @patsourakos09b], whereas the associated flares were all very weak. Stereoscopic EUVI studies revealed that the wave signal is typically confined to about 1–2 coronal scale heights above the chromosphere [@patsourakos09; @patsourakos09b; @kienreich09].
In this letter, we present the first observations of the full dome of the wave observed in EUV, which is consistent with the three-dimensional expansion of a coronal shock front. We note that part of a wave dome had been observed in soft X-rays by Yohkoh/SXT for a fast coronal wave that occurred in association with a Moreton wave [@narukage04]. We also study the evolution of the wave pulse characteristics (velocity, amplitude, width) in high-cadence EUV imaging over a propagation distance of more than $1\,R_\odot$, and discuss the implications for the physical processes involved.
Data
====
The EUVI instrument is part of the Sun Earth Connection Coronal and Heliospheric Investigation [SECCHI; @howard08] instrument suite onboard the STEREO-A(head) and STEREO-B(ehind) spacecraft. EUVI observes the chromosphere and low corona in four spectral channels (He [ii]{} 304 [Å]{}: $T \sim 0.07$ MK, Fe [ix]{} 171 [Å]{}: $T \sim 1$ MK, Fe [xii]{} 195 [Å]{}: $T \sim 1.5$ MK, Fe [xv]{} 284 [Å]{}: $T \sim 2.25$ MK) out to 1.7$R_\odot$ with a pixel limited spatial resolution of 1.6$''$/pixel [@wuelser04]. On 2010 January 17, STEREO-B was 69.2$^\circ$ behind Earth on its orbit around the Sun, observing a large-scale coronal wave in its Eastern hemisphere. The EUVI-B imaging cadence was 2.5 min in the 171 [Å]{}, 5 min in the 195 [Å]{}, 2.5–5 min in the 284 [Å]{}, and 5 min in the 304 [Å]{} passband. The EUVI data were reduced using the secchi$\underline{~}$prep routine available within Solarsoft, and corrected for solar differential rotation before we derived base and running difference and ratio images, respectively. We also show white light observations from the COR1-B coronagraph, which has a FoV of 1.5–4 R$_\odot$, and observed with a cadence of 5 min during the event under study.
Results
=======
Figure \[fig1\] shows the evolution of the wave in EUVI 195 [Å]{} direct and running difference images, where we subtracted from each frame the frame taken 5 min before (see also the accompanying movie no. 1). The wave is best observed in the EUVI 195 [Å]{} filter, which has a broad temperature response peaking at $\sim$1.5 MK, but can be observed in all four wavelengths. In the 195 [Å]{} images at 03:56 and 04:01 UT the full dome of the wave is clearly observed, even in the direct images. The image sequence also reveals that the on-disk signatures of the wave perfectly connect to the wave dome observed above the limb. The sharp and very regular edges of the dome further suggest that we really observe the shock front of the wave. In EUVI 171 [Å]{} images, erupting CME loops are observed behind/inside the dome (cf. Fig. \[fig2\]).
In Fig. \[fig2\], we plot difference images taken almost simultaneously (between 03:56 and 03:57 UT) in all four EUVI channels. The wave dome can be identified in all four wavelengths, which indicates that structures at different temperatures are disturbed by the wave, covering at least the temperature range 1.0–2.3 MK. The fact that we can observe the wave dome also in the EUVI 304 [Å]{} filter actually indicates that Si [xi]{} is significantly contributing to the 304 [Å]{} emission in addition to the $10^4$ K emission due to He [ii]{} lines [see also @long08; @patsourakos09].
We also note that HIRAS (Hiraiso Radio Spectrograph) reported an associated high-frequency type II burst drifting from $\sim$310 MHz to $\sim$80 MHz during $\sim$03:51–03:58 UT. The wave center was derived at a meridional distance of 57$^\circ$ for STEREO-B (cf. Fig. \[fig3\]), which implies that it was located $36^\circ$ behind the Eastern solar limb for an Earth-based vantage point, corresponding to an occultation height of about 0.23 $R_\odot \sim 160$ Mm. Since the radio source was behind the solar limb when looking from the Earth, the observed emission has to be at the harmonic of the plasma frequency. Applying two- to ten-fold Saito density models this corresponds to the height range 0.11–0.35R$_\odot$, suggesting that the shock occurred relatively low in the solar corona. These shock formation heights are consistent with the heights of the wave dome observed in EUV (cf. Fig. \[fig1\]).
In the sixth panel of Fig. \[fig1\], the outer contours of the coronal dimming region as identified in 195 [Å]{} base ratio images (05:01UT/03:36UT) are overlaid. At the time, at which we extract the contours, the dimming was maximally developed and darkest; the contour lines plotted in Fig. \[fig1\] are at 80% of the pre-event intensity. It is evident that the North-South extent of the dimming region, which outlines the lateral extent of the CME structure low in the corona (CME flanks), is significantly smaller than that of the expanding wave dome. If the dome corresponded to the CME body, its extent should not exceed that of the coronal dimming. In Fig. \[fig3\] (panels b and c) we show composits of EUVI 195 [Å]{} and COR1 images. The wave dome observed in EUVI images smoothly extends to the white-light structure observed in COR1. This suggests that the outer edge observed in the white-light coronagraphic images correspond rather to the coronal shock ahead of the CME than to the CME leading edge itself [see also @vourlidas03].
Figure \[fig3\]a shows an EUVI-B 195 Å difference image together with all wavefronts determined in the 195 Å passband and the center obtained from circular fits to the earliest wavefronts in the 3D spherical plane [cf. @veronig06]. For each wavefront visually identified in the 195, 171, 284 and 304 [Å]{} difference images, we determined the mean distance from the derived center along the spherical solar surface. The top panel in Fig. \[fig4\] shows the resulting wave kinematics. The velocity of the wave obtained from the linear fit to the kinematical curve is $v \sim 283\pm 27$ km s$^{-1}$, and remains constant over the propagation distance up to 950 Mm. In the same panel we also plot the kinematics of the wave dome followed along its main propagation direction as observed in EUVI and COR1. The distance of the wave dome is measured against the plane-of-sky, and for the starting point we use the center derived for the on-disk EUVI wave measurements. The plot shows that at the beginning the distance of the on-disk wave and the height of the dome are roughly in agreement, but the upward movement of the wave dome is much faster ($v \sim 650$ km s$^{-1}$) than the lateral expansion of the wave observed on the solar disk ($v \sim 280$ km s$^{-1}$).
In Fig. \[fig5\] we plot the evolution of the intensity amplitudes of the wave, so-called “perturbation profiles”, determined from 195 Å ratio images, where we divided each frame by the frame recorded 10 min before. We calculated the intensity profiles over a 60$^\circ$ sector on the solar sphere (indicated in Fig. \[fig3\]a), where the signal of the wave is strongest, by averaging the intensities of all pixels in “rings” of increasing radius around the wave origin shown in Fig. \[fig3\] [cf. @podladchikova05; @muhr10]. Base ratio images would in principle be better suited to study the perturbation profiles. However, these are affected by changes in the quiet Sun as well as by brightenings induced by the wave front passage that are only slowly fading [“stationary brightenings”; @attrill07]. As a result of several tests, we use 10-min running ratio images since they well represent the propagating wave profile and ensure that the peak of the wave amplitude is not cut.
The propagation of the wave is well reflected in the perturbation profile evolution plotted in Fig. \[fig5\] (see also the accompanying movie no. 2). We also observe steepening and amplitude increase in the profiles 03:56 to 04:01 UT, where the highest amplitude of 1.45 is reached, corresponding to an enhancement of 45% above the pre-event level. Assuming that the intensity enhancement is primarily due to plasma compression rather than due to temperature changes, which is somewhat justified by the observations of the wave over a broad temperature range in the four EUVI channels, this intensity amplitude of $I/I_0=1.45$ in an optically thin emission line corresponds to a density ratio $n/n_0 \propto (I/I_0)^{1/2} \sim 1.2$. The perturbation amplitudes in the other EUVI channels are smaller than in 195 [Å]{} but clearly recognized; see the peak amplitudes in 171 [Å]{} and 284 [Å]{} plotted in the second panel of Fig. \[fig5\]. We also note that at the time of maximum amplitude, the outer edge of the wave front is steepest. In the subsequent evolution, the amplitude and the steepness of the wave front decrease until it can no longer be followed in the profiles at about 04:36 UT.
In the middle panel of Fig. \[fig4\] we plot the evolution of the intensity amplitude, which decreases continuously after the peak at 04:01 UT. We fitted the profile amplitude $A = (I-I_0)/I_0$ as a function of the propagation distance $d$ with a power-law $A = a \cdot d^{b}$, giving $b=-2.5 \pm 0.3$. We also measured the width of the wave and followed its evolution. Due to the problematics of the stationary brightenings behind the wave front (which masks the trailing part), we extract only the width of the frontal part, i.e. from the profile peak to the outer front. We derived the full width of the frontal part (defined from the peak down to 2% enhancement above the pre-event level), as well as the frontal width at half maximum. These are plotted in the bottom panel of Fig. \[fig4\] together with the integral of the perturbation profile (as shown in Fig. \[fig5\]) over the frontal part of the wave. The width of the wave pulse increases during its evolution by a factor of 3–4, whereas the integral remains basically constant (changes are $\lesssim$25%).
Discussion and Conclusions
==========================
There are four main arguments supporting the conclusion that the observed EUV structure is the wave dome and not the CME body: a) The dome appears spherically three-dimensional with a sharp outer edge; inside the dome erupting CME loops are observed. b) The wave signatures observed in the low corona above the limb perfectly connect to the wave signatures observed against the solar disk. c) The lateral extent of the expanding dome is much larger than that of the coronal dimming. d) The event was associated with a high-frequency type II burst indicating shock formation low in the corona, which is consistent with the observed EUV dome height.
The velocity derived for the upward expansion of the wave dome ($v \sim 650$ km s$^{-1}$) is larger than that of the lateral expansion of the wave observed on-disk ($v \sim 280$ km s$^{-1}$). There are at least two alternative explanations for this. In the freely propagating phase, the upward-lateral velocity difference can be due to differences of the fast magnetosonic velocities of the active region (AR) and the ambient quiet corona. The tip of the wave dome lies above the AR, and the magnetosonic speed is much higher above ARs than at low heights in its surroundings [e.g. @warmuth05b]. An alternative explanation is that the upward dome expansion is driven all the time, and thus depends on the CME speed, whereas in the lateral direction the wave is freely propagating as soon as the lateral expansion of the CME flanks has stopped, and its velocity is determined by the characteristic speed of the medium. The second interpretation is supported by the evolution of the dimming region in the low corona, which is fast expanding up to about 04:01–04:06 UT. Thereafter, the coronal dimming still gets darker until about 05:00 UT, indicating ongoing mass depletion, but with only little (or no) further expansion after 04:06 UT.
The high-cadence EUV observations together with the distinct wave signal observed in the low corona against the solar disk allowed us to study in detail the evolution of the perturbation characteristics during the coronal wave propagation. We find that the amplitude of the perturbation first increases and the perturbation profile steepens within the first 5 min of the event (i.e. until 04:01 UT), thereafter the amplitude $A$ decreases, following a power-law of the form $A \propto r^{-2.5 \pm 0.3}$. The width of the perturbation profile broadens during its evolution by a factor of 3–4. This broadening is observed for both the full width as well as for the width at half maximum. We stress that we only measured the frontal width of the wave pulse due to the uncertainties in the trailing part (mostly due to stationary brightenings induced by the wave passage). Since the perturbation profile is not necessarily symmetric, the evolution of the total width of the wave pulse may be different, but we do not expect that it would alter the main outcome of broadening, which is a quite pronounced effect. The integral below the frontal part of the perturbation profile basically remains constant. These characteristics of the wave pulse evolution are all consistent with the spherical expansion and decay of a large-amplitude fast-mode MHD wave.
The velocity of the lateral wave expansion ($v \sim 280$ km/s) lies in the range of the fast magnetosonic speed in the quiet solar corona during solar minimum condition [see discussion in @kienreich09]. From the observed EUV wave intensities at the highest amplitude reached at $\sim$04:01 UT, we obtained a rough estimate of the peak density jump at the wave front, $n/n_0 \sim 1.2$. This corresponds to a perpendicular fast magnetosonic wave number of $M_{fms} \sim 1.15$ [@priest82]. Thereafter, as the wave decays to $n/n_0=1.04$ (Fig. \[fig5\]; last profile at 04:36 UT), it approaches a linear regime since $M_{fms} \sim 1.03$. These numbers indicate the evolution of a weak shock, and may explain why we do not observe a significant deceleration of the free-propagation of the lateral wave expansion in association with the amplitude decay.
AMV, NM, and IWK acknowledge the Austrian Science Fund (FWF): P20867-N16. The European Community’s Seventh Framework Programme (FP7/2007-2013) under grant agreement no. 218816 (SOTERIA) is acknowledged (BV, MT). We thank the STEREO/SECCHI teams for their open data policy.
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|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We demonstrate that the lowest possible price change (tick-size) has a large impact on the structure of financial return distributions. It induces a microstructure as well as it can alter the tail behavior. On small return intervals, the tick-size can distort the calculation of correlations. This especially occurs on small return intervals and thus contributes to the decay of the correlation coefficient towards smaller return intervals (Epps effect). We study this behavior within a model and identify the effect in market data. Furthermore, we present a method to compensate this purely statistical error.'
address: 'Fakultät für Physik, Universität Duisburg-Essen, Germany'
author:
- 'Michael C. Münnix'
- Rudi Schäfer
- Thomas Guhr
bibliography:
- 'Manuskript\_arxiv\_v4.bib'
title: 'Impact of the tick-size on financial returns and correlations'
---
Financial correlations ,Epps effect ,Market emergence ,Covariance estimation ,Tick-size ,Market microstructure
Introduction
============
The lowest possible price change of a financial security, the so called *tick-size* or *minimum tick*, plays an important role in quantitative finance. All raw price information is discretized by the tick-size. Historically, the tick-size of most securities has been consecutively reduced resulting in tick-sizes of . This process is often referred to as decimalization. One reason for it was to aim at an enhanced market efficiency. In principle, small tick-sizes allow for a faster clearing of market arbritage. Nonetheless, it is controversial whether a smaller tick-size generally improves the market quality [@beaulieu04; @huang01; @declerck02; @chung01], e.g., in view of the fact that a larger tick-size ensures liquidity [@harris91]. Furthermore, a recent study indicates that in some cases only a fraction of the theoretically possible prices are used. Hence, prices cluster at certain multiples of the tick-size resulting in an effective tick-size [@onnela09].
However, a large tick-size can lead to erroneous data in financial indices due to rounding errors [@kozicki09]. The actual tick-size for stocks is typically \$0.01. This is the case for instance on the New York Stock Exchange (NYSE) and the National Association of Securities Dealers Automated Quotations (NASDAQ). However, some securities such as U.S. Government Securities are still quoted in of a dollar.
The tick-size certainly affects many fields in quantitative finance. In this study we want to focus on its impact on two of the most important observables: relative price changes (*financial returns*) and financial correlations. These elementary values are of particular importance for many applications, for example portfolio optimization [@markowitz52; @doran09] and risk management [@b_bouchaud00].
The article is organized as follows. In section \[s:returns\], we will study the influence of the tick-size on the microstructure of financial return distributions. The impact of the tick-size on the calculation of financial correlations will be discussed in section \[s:corr\]. The decay of the correlation coefficients towards small return intervals is of particular interest. This behavior is commonly referred to as the Epps effect [@epps79; @reno03]. The identified mechanism is solely caused by the discrete tick-size and therefore represents a statistical effect. Hence, we are able to develop a method for compensating this distortion. We summarize the results in section \[s:conclusion\].
Financial returns {#s:returns}
=================
Observations on financial data on very small time scales are usually referred to as market microstructure [@b_voit01micro]. In this study, we will first investigate the influence of the tick-size on the shape and the microstructure of the financial return distribution. For this purpose, we decompose the set of returns according to the absolute price changes and disclose its microstructure.
Subsequently we will demonstrate that this microstructure can alter the tail behavior of the return distribution compared to the underlying price change distribution. Accordingly, we will disclose a relation between the tail behavior of each microstructure return distribution and the overall return distribution.
Return microstructure {#s:returnmicro}
---------------------
A *financial return* describes the relative price change of a security between two points in time. The arithmetic return is defined as $$\label{eq:returndef}
r_{\Delta t}(t)=\frac{ \Delta S_{\Delta t}(t) }{S(t)}\ ,$$ where $S(t)$ denotes the price at time $t$ and $$\Delta S_{\Delta t}(t) = S(t+\Delta t) - S(t)
\label{eq:DeltaS}$$ is the (absolute) price change within the interval $[t,t+\Delta t]$.
\
As the price change $\Delta S_{\Delta t}$ can only take values that are multiples of the tick-size $q$, its histogram consists of equally spaced peaks as shown in Fig. \[fig:dS\]. In other words, the distribution of $\Delta S_{\Delta t}$ is discretized. At first glance, it is conceivable that the transition from absolute price changes $\Delta S_{\Delta t}$ to relative price changes $r_{\Delta t}$ removes this discretization from the distribution, since the returns are almost continuously distributed, as Fig. \[fig:r\] illustrates. However, a closer look at the center of the distribution in Fig. \[fig:rzoom\] reveals that the discretization effects are still visible. Despite its non-visibility, the discretization affects returns on any interval. We will discuss this point more detailed in section \[s:corr\].
For an analytical description of this discretization, we introduce the set of all returns
$$R_{\Delta t}=\left\{\frac{ \Delta S_{\Delta t}(t) }{S(t)} \,\Big|\,\Delta S_{\Delta t}(t) \in \left[N_{-}q,(N_{-}+1)q, \dots, (N_{+}-1)q,N_{+}q\right] \right\}\ ,$$
where $N_{-}q$ defines the lower and $N_{+}q$ the upper bound of the price change distribution that is discretized by the tick-size $q$.
The set of all returns $R$ can be separated into subsets for each price change $\Delta S_{\Delta t}$, $$R_{\Delta t}= \bigcup_{n=N_{-}}^{N_{+}} R^{(n)}_{\Delta t}\ ,$$ with $$R^{(n)}_{\Delta t}=\left\{\frac{ \Delta S_{\Delta t}(t) }{S^{(n)}(t)}\, \Big|\, \Delta S_{\Delta t}(t) = nq \right\}\ .$$ $S^{(n)}$ in the denominator refers to the subset of starting prices that increase (or decrease) by $nq_{S}$ in the interval $\Delta t$. Therefore, $R^{(n)}_{\Delta t}$ represents the returns that are based on the price change $nq$. Evidently, $R^{(n)}_{\Delta t}$ is bounded by
$$\min(R^{(n)}_{\Delta t}) = \frac{nq}{\max(S^{(n)})}
\quad,\quad
\max(R^{(n)}_{\Delta t}) = \frac{nq}{\min(S^{(n)})} \ .
\label{eq:bounds}$$
In our study, empirical data from the TAQ database [@TAQ] of the New York Stock exchange (NYSE) indicate that the approximations $\max(S^{(n)}) \approx \max(S)$ and $\min(S^{(n)})
\approx \min(S)$ are legitimate for small $|n|$.
Therefore, the interval between minimum and maximum return on a specific price change $$I(R^{(n)}) = \left[\min\left(R^{(n)}\right),\max\left(R^{(n)}\right)\right]
\label{eq:rinterval}$$ increases with $|n|$, while the distance $d$ between their centers remains almost constant $$d(R^{(n)}) = \frac{q_{S}}{2}\left(\frac{1}{\min(S^{(n)})}-\frac{1}{\max(S^{(n)})}\right) \approx \frac{q_{S}}{2}\left(\frac{1}{\min(S)}-\frac{1}{\max(S)}\right) = \mathrm{const.}
\label{eq:rdistance}$$ Thus, the intervals $I(R^{(n)})$ are increasingly overlapping for larger $|n|$. From this viewpoint the discretization is only “visible” for small $|n|$, that is, for small price changes. Fig. \[fig:rzoom\] illustrates the clustering of returns with an example where we compare the returns of the Apollo Group Inc. (APOL) share with the intervals $I(R^{(n)})$ calculated by equations (\[eq:rinterval\]) and (\[eq:rdistance\]). The calculated boundaries match with the empirical data.
Tail behavior of return and price change distribution {#s:tails}
-----------------------------------------------------
\
We will now investigate, how the composition of the returns changes the shape of their distribution compared to the distribution of price changes. In the framework of a model, we generate price changes that are, in a first scenario, Gaussian distributed and, in a second scenario, powerlaw distributed with a given tick-size. Afterwards, we calculate returns using uniformly distributed price values within the regions $S_{\mathrm{min}}$ and $S_{\mathrm{max}}$ (analogously to Figs.\[fig:rzoom\] and \[fig:r\]). In this manner, we generate a discrete price change distribution with a specific shape and then divide each set of equal price changes by uniformly distributed prices. The price distributions are generated individually for each subset.
To compare the shape of the obtained return distribution with the shape which we have chosen for the price change distribution, we normalize the distributions to zero mean and unit variance $$g^{(i)}_{\Delta t}(t)=\frac{r^{(i)}_{\Delta t}(t)-\langle r^{(i)}_{\Delta t} \rangle}{\sigma_{r^{(i)}_{\Delta t}}}
,\quad
\Delta\hat{S}^{(i)}_{\Delta t}(t)=\frac{\Delta S^{(i)}_{\Delta t}(t)-\langle \Delta S^{(i)}_{\Delta t} \rangle}{\sigma_{\Delta S^{(i)}_{\Delta t}}}\ ,$$ where $\langle \ldots \rangle$ denotes the mean value of a time series with length $T$ and where $\sigma$ refers to the standard deviation of the same time series. The index $i$ corresponds to the a certain security, e.g., a stock.
The results of this simple setup indicate that neither the tick-size nor the width of the price change distribution or the absolute sizes of $S_{\mathrm{min}}$ and $S_{\mathrm{max}}$ have an effect on the shape of the obtained return distribution. Only the microstructure of its center is affected, as discussed in the previous section. In general, the return distribution acquires stronger tails compared to the price change distribution. Surprisingly, the shape-change of the distribution only depends on the ratio of the minimum and maximum price.
Figure \[fig:tails\] shows the corresponding distributions for Gaussian and powerlaw distributed prices and for various price ranges. It turns out that the influence on the tail behavior is much stronger for a Gaussian price change distribution. For a powerlaw price change distribution, the return distribution retains almost the same powerlaw shape, except for the tails far out, while their center becomes slightly sharper.
Of course, the assumption of uniformly distributed prices on each price change is a rough approximation within this simple setup. In the market, there can be a strong relation between $\Delta S$ and $S$, which leads to a shape retaining of the price change distribution to the return distribution. This is because the prices which undergo a very large price change during the interval $\Delta t$ can be much more sparsely distributed than prices which change only slightly. Furthermore, the price range is usually not very high in a period of time, in which the price distribution is approximately uniform. In view of this and under the assumption of powerlaw distributed price changes, the situation in Figs. \[fig:tailsp1\] and \[fig:tailsp2\] may describe most stocks suitably. Put differently, the shape of the return distribution is almost retaining the shape of the price change distribution in most cases.
However, if the price of a stock covers a large range in a relatively short period of time, we actually can observe a change in the tail behavior. This is illustrated in Fig. \[fig:realtails\] for an ensemble of 50 stocks taken from the S&P 500 index (See Tab. \[tab:var\]). The stocks have been chosen to provide the highest ratio between their mean price and its standard deviation. Although the stock ensemble shows the expected behavior, it is difficult to make an accurate statement regarding the tails far out, as these events are very rare, even within this statistical ensemble.
Tail behavior of the return microstructure {#s:microtail}
------------------------------------------
![Comparison of the kurtosis of the return distributions on a specific price change $\Delta S$ compared to the complete return distribution. The negative peak at $\Delta S' = 0$ originates from the fact that this return subset “distribution” of $\Delta S'/S$ only contains returns with the value zero and therefore leads to a value of $-3/\mathrm{kurt}(\Delta S/S)$.[]{data-label="fig:kurtosis"}](joint_kurtosis_GOOG_5){width="48.00000%"}
Another question arises in this regard: If the return distribution is heavy tailed, how is this connected to the tail behavior of the subsets of returns? As we demonstrated in the previous section, the set of all returns can be divided into subsets that are corresponding to a certain price change. Now, do these subparts feature stronger or weaker tails than the complete return distribution?
In Fig. \[fig:kurtosis\], we compare the kurtosis of the return subsets normalized to the kurtosis of the overall return distribution. As it is difficult to perform a proper normalization of a stock ensemble in this graphical representation, we show the result for the Google Inc. (GOOG) share as an example.
We make two observations: First, there seems to be a connection between the tail behavior and the return interval for the return subset distributions. The return subset distributions feature stronger tails for smaller price changes. Second, surprisingly the return subset distributions exhibit a much smaller kurtosis than the complete return distribution. The strong tails of the complete distribution develop not until combining all the return subset distributions.
Financial correlations {#s:corr}
======================
We now turn to the impact of the tick-size on the calculation of correlations and analyze the influence on the decay of correlation coefficients towards smaller return intervals (Epps effect). Financial correlations are an important measure in economics. The knowledge of precise correlations is essential for quantifying and minimizing financial risk. As we will show, the discreteness of stock quotes can distort the calculated correlation coefficients.
A financial return is a compound observable value. Due to that fact, we develop the compensation method step by step. We start in section \[a:quantcor\], where we turn to the distortion of the correlation coefficient of value-discretized time series in general. We develop a compensation for the discretization error in the correlation between financial (absolute) price changes in section \[s:error\]. In section \[s:return\_correct\], we extend this formalism to financial returns.
It is a basic assumption in our model that we can statistically describe the discreteness in market prices by a discretization of a hypothetical underlying continuous price. This is not to say that market prices actually result from a discretization process. Individual traders are well aware of the finite tick size and may try to exploit it in their trading strategies. However, there is a large variety of trading strategies simultaneously acting on the market. These strategies involve a large scale of different investment horizons. Since the price formation results from the interaction of a large diversity of strategies, the price fluctuations on the level of the tick size can be viewed as purely statistical. This is the basis for our modeling ansatz.
Despite the interpolation of the price change distribution, neither parameter fixing nor calibration of the model is necessary, in contrast to many other compensation methods for the Epps effect [@zebedee09; @voev07; @griffin06; @corsi07; @zhang08; @barndorff08].
Calculation of the correlation coefficient for value-discretized time series {#a:quantcor}
----------------------------------------------------------------------------
Almost any time series of data is discretized. This can simply be caused by numerical reasons, such as a finite number of decimal places. But how can we measure the impact of the discretization or even compensate it? We will show, that this can simply be achieved by a decomposition of the correlation coefficient and a estimation of the average discretization errors.
Let $x_{1}$ and $x_{1}$ be two time series which are correlated. The correlation coefficient of $x_{1}$ and $x_{2}$ is given by $$\label{eq:basiccorr}
\mathrm{corr}(x_{1},x_{2})= \frac{\left\langle x_{1} x_{2} \right\rangle - \left\langle x_{1} \right\rangle \left\langle x_{2} \right\rangle}{\sigma_{1}\sigma_{2}}\ .$$ Now we consider the time series $\bar{x}_{1}$ and $\bar{x}_{2}$ which are the discretized values of $x_{1}$ and $x_{2}$ with tick-sizes $q_{1}$ and $q_{2}$, respectively. Thus we have $$\begin{aligned}
x_{1}(t)&=&\bar{x}_{1}(t)+\vartheta^{(1)}(t)\label{eq:discx} \\
x_{2}(t)&=&\bar{x}_{2}(t)+\vartheta^{(2)}(t)\label{eq:discy} \ ,\end{aligned}$$ where $\vartheta^{(1)}(t)$ and $\vartheta^{(2)}(t)$ are the discretization errors. We assume the discretization errors as uniformly distributed in the intervals $]-q_1/2,q_1/2]$ and $]-q_2/2,q_2/2]$. This seems natural, as a discretization is commonly caused by a rounding process.
Using equations (\[eq:discx\]) and (\[eq:discy\]) we can write the correlation coefficient (\[eq:basiccorr\]) as $$\begin{aligned}
\mathrm{corr}(x_{1},x_{2}) &=&
\frac{\left\langle (\bar{x}_{1}+\vartheta^{(1)})(\bar{x}_{2}+\vartheta^{(2)}) \right\rangle - \left(\left\langle \bar{x}_{1} \right\rangle+\left\langle \vartheta^{(1)} \right\rangle\right) \left(\left\langle \bar{x}_{2} \right\rangle+\left\langle \vartheta^{(2)} \right\rangle\right)}
{\sqrt{\mathrm{var}\left(\bar{x}_{1}+\vartheta^{(1)}\right)}\sqrt{\mathrm{var}\left(\bar{x}_{2}+\vartheta^{(2)}\right)}}\\
&=&\frac{\mathrm{cov}\left(\bar{x}_{1},\bar{x}_{2}\right)+\mathrm{cov}\left(\bar{x}_{1},\vartheta^{(2)}\right)+\mathrm{cov}\left(\bar{x}_{2},\vartheta^{(1)}\right)+\mathrm{cov}\left(\vartheta^{(1)},\vartheta^{(2)}\right)}
{\sqrt{\mathrm{var}\left(\bar{x}_{1}\right)+\mathrm{var}\left(\vartheta^{(1)}\right)+2\mathrm{cov}\left(\bar{x}_{1},\vartheta^{(1)}\right)}
\sqrt{\mathrm{var}\left(\bar{x}_{2}\right)+\mathrm{var}\left(\vartheta^{(2)}\right)+2\mathrm{cov}\left(\bar{x}_{2},\vartheta^{(2)}\right)}} \label{eq:errorcorr} \ .\end{aligned}$$ Apart from the terms $\mathrm{cov}\left(\bar{x}_{1},\bar{x}_{2}\right)$, $\mathrm{var}\left(\bar{x}_{1}\right)$ and $\mathrm{var}\left(\bar{x}_{2}\right)$ of expression (\[eq:errorcorr\]), which can be calculated with the discretized data, all other terms are lost in the discretization process. However, these terms can be estimated when the distributions $\varrho_{\bar{x}_{1}}$ and $\varrho_{\bar{x}_{2}}$ of $\bar{x}_{1}$ and $\bar{x}_{2}$ are known, as we will demonstrate. The continuous distributions $\varrho_{x_{1}}$ and $\varrho_{x_{2}}$ can be obtained by interpolating the distributions of the discretized values (we assume these distributions in the following context to be normalized). Sometimes, the shape of the distribution for a certain process is known (e.g. Gaussian). Therefore, the interpolated distribution function can be determined by a fit of the distributions of $\bar{x}_{1}$ and $\bar{x}_{2}$.
If the shape of the distribution is unknown, an interpolation can be performed section by section using e.g. polynomial or linear fits. The fitting processes cannot be performed as typically by minimizing the difference of values from the discrete distribution and the desired fit function. Rather the discretization process needs to be included. This gains particular importance when the level of discretization is high and thus the distribution is discretized only with a small range of values.
As the value that has been discretized to e.g. $x'_{1}$ can originate from region $x'_{1}-q_{1}/2$ to $x'_{1}+q_{1}/2$, the difference function $f$, which provides a measure for the residual between the fit and the empirical data is then given by
$$\begin{aligned}
f_{x_{1}}(\varrho_{x_{1}},\varrho_{\bar{x}_{1}})=\sum\limits_{n=N_-}^{N_+}
\left[\,
\int\limits_{q_1(n-\frac{1}{2})}^{q_1(n+\frac{1}{2})}\varrho_{x_{1}}(z)\,dz -
\varrho_{\bar{x}_{1}}(nq_1)
\right]\end{aligned}$$
for $x_{1}$ and analogously for $x_{2}$.
To compensate the overall discretization error, we first introduce the discretization errors that led to a certain discretized value. We call these errors conditional discretization errors. They are defined as
$$\begin{aligned}
\vartheta_{n}^{(1)}\left(\tilde{t}\right) &=& x_{1}(\tilde{t})-nq_{1}\ ,\ \tilde{t} \in \left\{t\:\: \big|\:\: |\:x_{1}(t)-nq_{1}|\le\frac{q_{1}}{2}\right\}\\
\vartheta_{m}^{(2)}\left(\tilde{t}\right) &=& x_{2}(\tilde{t})-mq_{2}\ ,\ \tilde{t} \in \left\{t\:\: \big|\:\: |x_{2}(t)-mq_{2}|\le\frac{q_{2}}{2}\right\}\\
\vartheta_{n,m}^{(1)}\left(\tilde{t}\right) &=& x_{1}(\tilde{t})-nq_{1}\ ,\ \tilde{t} \in \left\{t\:\: \big|\:\: |x_{1}(t)-nq_{1}|\le\frac{q_{1}}{2},|x_{2}(t)-mq_{2}|\le\frac{q_{2}}{2}\right\}\\
\vartheta_{m,n}^{(2)}\left(\tilde{t}\right) &=& x_{2}(\tilde{t})-mq_{2}\ ,\ \tilde{t} \in \left\{t\:\: \big|\:\: |x_{2}(t)-mq_{2}|\le\frac{q_{2}}{2},|x_{1}(t)-nq_{1}|\le\frac{q_{1}}{2}\right\}\ .\end{aligned}$$
Here, $\vartheta_{n}^{(1)}$ and $\vartheta_{m}^{(2)}$ are the discretization errors that resulted in a discrete value of $\bar{x}_{1}=nq$ and $\bar{x}_{2}=mq$ accordingly, where $n$ and $m$ are integers. Consequently, $\vartheta_{n,m}^{(1)}$ and $\vartheta_{m,n}^{(2)}$ are discretization errors that led to a value of $\bar{x}_{1}=nq$ and $\bar{x}_{2}=mq$, while the other (correlated) time series was simultaneously discretized to $\bar{x}_{2}=mq$ and $\bar{x}_{1}=nq$. In all cases, $\tilde{t}$ is the set of time points at which these actual discretizations occur.
Using the interpolated distribution functions $\varrho_{x_{1}}(x(t))$ and $\varrho_{x_{1}}(y(t))$ and the interpolated joint distribution function $\varrho_{x_{1},x_{2}}(x(t),y(t))$, the average discretization errors can be calculated as
$$\begin{aligned}
\left\langle \vartheta^{(1)}_{n} \right\rangle &=& \int_{q_{1}\left(n-\frac{1}{2}\right)}^{q_{1}\left(n+\frac{1}{2}\right)} (z-nq_{1})\varrho_{x_{1}}(z)\,dz
\,\Big/
\int_{q_{1}\left(n-\frac{1}{2}\right)}^{q_{1}\left(n+\frac{1}{2}\right)} \varrho_{x_{1}}(z)\,dz \\
\left\langle \vartheta^{(2)}_{m} \right\rangle &=& \int_{q_{2}\left(m-\frac{1}{2}\right)}^{q_{2}\left(m+\frac{1}{2}\right)} (z-mq_{2})\varrho_{x_{2}}(z)\,dz
\,\Big/
\int_{q_{2}\left(m-\frac{1}{2}\right)}^{q_{2}\left(m+\frac{1}{2}\right)} \varrho_{x_{2}}(z)\,dz \\
\left\langle \vartheta^{(1)}_{n,m} \right\rangle &=& \int_{q_{x}\left(n-\frac{1}{2}\right)}^{q_{1}\left(n+\frac{1}{2}\right)} (z-nq_{1})\varrho_{x_{1},y_{2}}(z,mq_{2})\,dz
\,\Big/
\int_{q_{1}\left(n-\frac{1}{2}\right)}^{q_{1}\left(n+\frac{1}{2}\right)} \varrho_{x_{1},x_{2}}(z,mq_{2})\,dz \\
\left\langle \vartheta^{(2)}_{m,n} \right\rangle &=& \int_{q_{2}\left(m-\frac{1}{2}\right)}^{q_{2}\left(m+\frac{1}{2}\right)} (z-mq_{2})\varrho_{x_{1},x_{2}}(nq_{1},z)\,dz
\,\Big/
\int_{q_{2}\left(m-\frac{1}{2}\right)}^{q_{2}\left(m+\frac{1}{2}\right)} \varrho_{x_{1},x_{2}}(nq_{1},z)\,dz \ ,\end{aligned}$$
where $$\begin{aligned}
\int\limits_{-\infty}^{+\infty} \varrho_{x_{1},x_{2}}(x_{1}(t),z)\,dz &=& \varrho_{x_{1}}(x_{1}(t))\quad\mathrm{and}\\
\int\limits_{-\infty}^{+\infty} \varrho_{x_{1},x_{2}}(z,x_{2}(t))\,dz &=& \varrho_{x_{2}}(x_{2}(t))\ .\end{aligned}$$ Therefore the overall average discretization errors can be written as $$\begin{aligned}
\left\langle \vartheta^{(1)} \right\rangle &\approx&
\frac{1}{T}\sum\limits_{n=N_{-}}^{N_{+}} T_{n} \left\langle \vartheta_{n}^{(1)}\right\rangle\\
\left\langle \vartheta^{(2)} \right\rangle &\approx&
\frac{1}{T}\sum\limits_{m=M_{-}}^{M_{+}} T_{m} \left\langle \vartheta_{m}^{(2)}\right\rangle\ , \label{eq:univary}\end{aligned}$$ where $T_{n}$ and $T_{m}$ are the number of values that have been discretized to $nq_1$ and $mq_2$.
Now we can calculate the discretization terms of equation (\[eq:errorcorr\]). We begin with: $$\begin{aligned}
\mathrm{cov}\left(\bar{x}_{1},\vartheta^{(2)}\right) &=&
\left\langle \bar{x}_{1} \vartheta^{(2)} \right\rangle - \left\langle \bar{x}_{1} \right\rangle \left\langle \vartheta^{(2)} \right\rangle \\
&=&
\frac{1}{T} \sum\limits_{n=N_{-}}^{N_{+}}\sum\limits_{m=M_{-}}^{M_{+}}\sum\limits_{\tilde{t}=0}^{T_{n,m}}\left(nq_{1} \vartheta_{m}^{(2)}(\tilde{t})\right)
- \left\langle \bar{x}_{1} \right\rangle
\left\langle \vartheta^{(2)} \right\rangle\\
&=&
\frac{q_{1}}{T} \sum\limits_{n=N_{-}}^{N_{+}} n \sum\limits_{m=M_{-}}^{M_{+}} T_{n,m}
\left\langle \vartheta_{m,n}^{(2)}\right\rangle
- \left\langle \bar{x}_{1} \right\rangle
\left\langle \vartheta^{(2)} \right\rangle\
\label{eq:cov_x_theta_y} \ .
%& \approx &
%\frac{q_{x}}{T} \sum\limits_{n=N_{-}}^{N_{+}} n \sum\limits_{m=M_{-}}^{M_{+}} T_{n,m}
%\left\langle \vartheta_{m,n}^{(y)}\right\rangle \end{aligned}$$ Here, $q_{1}N_{-}$ represents the minimum of the discretized time series $\bar{x}_{1}(t)$. $q_{1}N_{+}$ is its maximum. $T$ is the length of the whole time-series, while $T_{n,m}$ is the number of synchronous pairs of both time series, which are discretized to $nq_{1}$ and $mq_{2}$. We index these pairs with $\tilde{t}$ referring to these certain point in time.
Analogously, the other discretization terms of equation (\[eq:errorcorr\]) can be calculated as $$\begin{aligned}
\mathrm{cov}\left(\bar{x}_{2},\vartheta^{(1)}\right)
&= &
\frac{q_{2}}{T} \sum\limits_{m=M_{-}}^{M_{+}} m \sum\limits_{n=N_{-}}^{N_{+}} T_{n,m}
\left\langle \vartheta_{n,m}^{(1)}\right\rangle \label{eq:cov_y_theta_x}
- \left\langle \bar{x}_{2} \right\rangle
\frac{1}{T}\sum\limits_{n=N_{-}}^{N_{+}} T_{n} \left\langle \vartheta_{n}^{(x)}\right\rangle
\\
\mathrm{cov}\left(\bar{x}_{1},\vartheta^{(1)}\right) &=&
\frac{q_1}{T}\sum\limits_{n=N_{-}}^{N_{+}} T_{n} n \left\langle \vartheta_{n}^{(1)}\right\rangle
- \left\langle \bar{x}_{1} \right\rangle
\frac{1}{T}\sum\limits_{n=N_{-}}^{N_{+}} T_{n} \left\langle \vartheta_{n}^{(1)}\right\rangle
\\
\mathrm{cov}\left(\bar{x}_{2},\vartheta^{(2)}\right) &=&
\frac{q_2}{T}\sum\limits_{m=M_{-}}^{M_{+}} T_{m} m \left\langle \vartheta_{m}^{(2)}\right\rangle
- \left\langle \bar{x}_{2} \right\rangle
\frac{1}{T}\sum\limits_{m=M_{-}}^{M_{+}} T_{m} \left\langle \vartheta_{m}^{(2)}\right\rangle
\\
\mathrm{var}\left(\vartheta^{(1)}\right) &\approx& \frac{q_1^2}{12} \label{eq:var_x}
\\
\mathrm{var}\left(\vartheta^{(2)}\right) &\approx& \frac{q_2^2}{12} \label{eq:var_y}\end{aligned}$$ The terms (\[eq:var\_x\]) and (\[eq:var\_y\]) are estimated under the assumption that the discretization errors are uniformly distributed. Usually, the remaining term $\mathrm{cov}(\vartheta^{(1)}_{n},\vartheta^{(2)}_{m})$ cannot be calculated with the distribution functions as it contains the correlation between the discretization errors. This value is not necessarily connected to the correlation of the whole time series either. Yet, we will show in the next section, that this term is negligible in the present context.
Thus, we have shown that the error caused by the discretization can be estimated by decomposing the correlation coefficient and approximating the mean discretization errors by interpolating the discrete distributions.
Distortion of price change correlations {#s:error}
---------------------------------------
We now turn to the specific situation on the stock market. The situation differs, when applying the method from the previous section to stock price changes. Here, the discretization process does not take place on the actual observable. Instead the price change $\Delta S$ is a difference for two prices $S(t)$ and $S(t+\Delta t)$ that are discretized by the tick-size $q$.
Therefore, the discretization error on a specific price difference $\Delta S'$ can be in the range from $-q$ to $q$. However, the probability that a certain value is from a price difference within this range is not constant. It is described by a triangular-shaped distribution (See Fig. \[fig:tri\]). This is evident, as the distribution error is the difference of two uniformly distributed discretization errors. The normalized triangular distribution $\varrho_{\mathrm{Tri}}$ around a certain price change $\Delta S'$ vanishes at $\Delta S'-q$ and $\Delta S'+q$ and has the value $1/q$ at its maximum at $\Delta S'$. It reads as
$$\begin{aligned}
\varrho_{\mathrm{Tri}}(x,\Delta S') =
\begin{cases}
\frac{x-\Delta S'+q}{q^2} & (\Delta S' -q) \leq x < \Delta S'\\
\frac{-x+\Delta S'+q}{q^2} & (\Delta S' +q) \geq x \geq \Delta S'\\
0 & \mathrm{else}\ .
\end{cases}
\label{eq:tri}\end{aligned}$$
The average discretization errors have now to be calculated with the product of the triangular distribution $\varrho_{\mathrm{Tri}}$ and the interpolated price change distributions $\varrho_{\Delta S_{1}}$, $\varrho_{\Delta S_{2}}$ (and proper normalization). Thus,
$$\begin{aligned}
\left\langle \vartheta^{(1)}_{n} \right\rangle &=& \int_{q_1\left(n-1\right)}^{q_1\left(n+1\right)} (z-nq_{1})\varrho_{\Delta S_{1}}(z)\varrho_{\mathrm{Tri}}(z,nq_{1})\,dz
\,\Big/
\int_{q_{1}\left(n-1\right)}^{q_{1}\left(n+1\right)} \varrho_{\Delta S_{1}}(z)\varrho_{\mathrm{Tri}}(z,nq_{1})\,dz
\label{eq:meantriplaw}\ ,\\
\left\langle \vartheta^{(1)}_{n,m} \right\rangle &=& \int_{q_1\left(n-1\right)}^{q_1\left(n+1\right)} (z-nq_{1})\varrho_{\Delta S_{1},\Delta S_{2}}(z,mq_{2})\varrho_{\mathrm{Tri}}(z,nq_{1})\,dz
\,\Big/
\int_{q_1\left(n-1\right)}^{q_1\left(n+1\right)} \varrho_{\Delta S_{1},\Delta S_{2}}(z,mq_{2})\varrho_{\mathrm{Tri}}(z,nq_{1})\,dz\ .\label{eq:meantriplawv}\\end{aligned}$$
$\left\langle \vartheta^{(2)}_{m} \right\rangle$ and $\left\langle \vartheta^{(2)}_{m,n} \right\rangle$ are analogously defined.
Fig. \[fig:triplaw\] shows exemplarily the product of a triangular distribution and a power law distribution. The denominator in equation (\[eq:meantriplaw\]) refers to the area under this curve.
The triangular distribution also needs to be included in the fitting process. Thus, the difference function becomes
$$\begin{aligned}
f_{\Delta S}(\varrho_{\Delta S},\varrho_{\Delta \bar{S}})&=&\sum\limits_{n=N_-}^{N_+}
\left[\,
\int\limits_{q_{S}(n-1)}^{q_{S}(n+1)}\varrho_{\mathrm{Tri}}(z,nq_{S})\left[\varrho_{\Delta S}(z) -
\varrho_{\Delta \bar{S}}(nq_{S})\right]\,dz
\right]\\
&=&\sum\limits_{n=N_-}^{N_+}
\left[\,
\int\limits_{q_{S}(n-1)}^{q_{S}(n+1)}\varrho_{\mathrm{Tri}}(z,nq_{S})\varrho_{\Delta S}(z)\,dz -
\varrho_{\Delta \bar{S}}(nq_{S})
\right]\ .\end{aligned}$$
Where $\varrho_{\Delta \bar{S}}$ refers to the discretized distribution. $\varrho_{\mathrm{Tri}}$ acts like a weighting function in the residual measure. It provides a weight corresponding to the probability that the difference of the originating discretization errors result in the value $z$.
Now, we are able to estimate the correlation discretization error with the previously defined equations (\[eq:cov\_x\_theta\_y\]) to (\[eq:var\_y\]).
Distortion of return correlations {#s:return_correct}
---------------------------------
When calculating the correlation of financial returns as defined in equation (\[eq:returndef\]) the situation becomes more complex. Here, we also have to take the prices into account. The correlation coefficient (\[eq:errorcorr\]) for two return time series $r_1$ and $r_2$ now reads as $$\mathrm{corr}(r_1,r_2)
\frac{\mathrm{cov}\left(\bar{r}_1,\bar{r}_2\right)+
\mathrm{cov}\left(\frac{\Delta \bar{S}_1}{S_1},\frac{\vartheta^{(2)}}{S_2}\right)+\mathrm{cov}\left(\frac{\Delta \bar{S}_2}{S_2},\frac{\vartheta^{(1)}}{S_1}\right)+
\mathrm{cov}\left(\frac{\vartheta^{(1)}}{S_1},\frac{\vartheta^{(2)}}{S_2}\right)}
{\sqrt{\mathrm{var}\left(\bar{r}_1\right)+\mathrm{var}\left(\frac{\vartheta^{(1)}}{S_1}\right)+2\mathrm{cov}\left(\frac{\Delta \bar{S}_1}{S_1},\frac{\vartheta^{(1)}}{S_1}\right)}
\sqrt{\mathrm{var}\left(\bar{r}_2\right)+\mathrm{var}\left(\frac{\vartheta^{(2)}}{S_1}\right)+2\mathrm{cov}\left(\frac{\Delta \bar{S}_1}{S_1},\frac{\vartheta^{(1)}}{S_1}\right)}}\ .\label{eq:errorcorr_2}$$ Here, $\bar{r}_1$ and $\bar{r}_2$ refer to the discretized return time-series. Analogously to the correlation between price changes, the individual terms can be estimated, but in addition, the starting prices $S_{1}$ and $S_{2}$ need to be parameterized. We use the variables $k$ and $l$ for this. $q_{1}K_{-}$ represents the minimum price within the observed time series, while $q_{1}K_{+}$ represents the maximum price. $T_{n,m,k,l}$ represents the number of pairs whose returns equal $(q_{1}n)/(q_{1}k) =n/k$ and $m/l$. Similar to that, $T_{n,k}$ refers to the number of returns (from a single time-series) that are equal to $n/k$. Thus, we obtain $$\begin{aligned}
\mathrm{cov}\left(\frac{\Delta \bar{S}_1}{S_1},\frac{\vartheta^{(2)}}{S_2}\right)
&\approx& \frac{q_{1}}{T}
\sum\limits_{n=N_{-}}^{N_{+}} n
\sum\limits_{m=M_{-}}^{M_{+}} q_{2}
\sum\limits_{k=K_{-}}^{K_{+}}
\sum\limits_{l=L_{-}}^{L_{+}}
T_{n,m,k,l}
\frac{\left\langle \vartheta_{m,n}^{(2)}\right\rangle}{kl}
- \left\langle\frac{\Delta \bar{S}_1}{S_1}\right\rangle
\left\langle\frac{ \vartheta^{(2)} }{S_{2}}\right\rangle
\label{eq:compstart}
\\
\mathrm{cov}\left(\frac{\Delta \bar{S}_2}{S_2},\frac{\vartheta^{(1)}}{S_1}\right)
&\approx& \frac{q_{2}}{T}
\sum\limits_{m=M_{-}}^{M_{+}} m
\sum\limits_{n=N_{-}}^{N_{+}} q_{1}
\sum\limits_{k=K_{-}}^{K_{+}}
\sum\limits_{l=L_{-}}^{L_{+}}
T_{n,m,k,l}
\frac{\left\langle \vartheta_{n,m}^{(1)}\right\rangle}{kl}
- \left\langle\frac{\Delta \bar{S}_2}{S_2}\right\rangle
\left\langle\frac{ \vartheta^{(1)} }{S_{1}}\right\rangle
\\
\mathrm{cov}\left(\frac{\Delta \bar{S}_1}{S_1},\frac{\vartheta^{(1)}}{S_1}\right) &\approx&
\frac{q_{1}}{T}\sum\limits_{n=N_{-}}^{N_{+}} \sum\limits_{k=K_{-}}^{K_{+}} T_{n,k} \frac{n}{k^2} \left\langle \vartheta_{n}^{(1)}\right\rangle
- \left\langle\frac{\Delta \bar{S}_1}{S_1}\right\rangle
\left\langle\frac{ \vartheta^{(1)} }{S_{1}}\right\rangle
\label{eq:compmid}
\\
\mathrm{cov}\left(\frac{\Delta \bar{S}_2}{S_2},\frac{\vartheta^{(2)}}{S_2}\right) &\approx&
\frac{q_{2}}{T}\sum\limits_{n=N_{-}}^{N_{+}} \sum\limits_{k=K_{-}}^{K_{+}} T_{n,k} \frac{n}{k^2} \left\langle \vartheta_{n}^{(2)}\right\rangle
- \left\langle\frac{\Delta \bar{S}_2}{S_2}\right\rangle
\left\langle\frac{ \vartheta^{(2)} }{S_{2}}\right\rangle
\label{eq:compmid2}
\\
\mathrm{var}\left(\frac{\vartheta^{(1)}}{S_{1}}\right) &\approx& \frac{q_{1}^2}{6}
\left\langle \frac{1}{S_1^2}\right\rangle
\\
\mathrm{var}\left(\frac{\vartheta^{(2)}}{S_{2}}\right) &\approx& \frac{q_{2}^2}{6}
\left\langle \frac{1}{S_2^2}\right\rangle\ .\label{eq:compend}\end{aligned}$$ The terms $\left\langle \vartheta^{(1)}/ S_{1}\right\rangle$ and analogously $\left\langle \vartheta^{(2)}/ S_{2}\right\rangle$ in equations (\[eq:compstart\]) to (\[eq:compmid2\]) can be estimated as $$\begin{aligned}
\left\langle\frac{ \vartheta^{(1)} }{S_{1}}\right\rangle \approx
\frac{1}{T}
\sum\limits_{n=N_{-}}^{N_{+}} q_{1}
\sum\limits_{k=K_{-}}^{K_{+}}
T_{n,k}
\frac{\left\langle \vartheta_{n}^{(1)}\right\rangle}{k}\ .\end{aligned}$$ We note that the correlation between $\Delta S$ and $S$ is neglected in this approximation. Also the discretization of the prices in the denominator of the return is not compensated. However, the model results in the next section demonstrate that this simplification only induces a minor error.
Also the impact of specific trading strategies can be calculated using the presented modeling. Here, the distortion of correlation coefficients, the distribution of discretization errors (equation (\[eq:tri\])) needs to be chosen in a suitable manner.
\
\
![Benchmark of the error estimation: Comparison between real and estimated discretization errors within the model setup.[]{data-label="fig:meanbench"}](pricediffs_1000_04_1000_4_meandiv_a){width="48.00000%"}
Results / Impact on the Epps effect {#s:epps}
-----------------------------------
After we developed a method for compensating the discretization error in the calculation of correlations, we verify it in a model setup and apply it to empirical data. We perform the presented compensation for different time intervals, in order to examine wether there is also a connection to the Epps effect.
The Epps effect refers to the decay of the correlation coefficient towards small return intervals. Therefore financial correlations on returns which are based on intervals below a certain limit (e.g. 30 minutes) are unreliable. The ability to calculate the correlation structure on small return intervals is equivalent to an improved statistical significance or the gain of more recent information. In previous studies the asynchrony of the time series has been identified as a major cause for the Epps effect [@muennix09b; @hayashi05]. The following demonstrates that the price discretization can result in a sizable contribution to the Epps effect as well.
As the mean price change per return interval decreases with the length of the interval [@bouchaud04], the width of the price change distribution decreases as well. While the tick-size remains constant, the discretization error increases. Hence, the tick-size should also have an impact on the Epps effect - especially for stocks which are traded at low prices.
### Model results
Before applying the method to estimate the discretization error in empirical data, we evaluate it in a model setup. In addition, we will use the model to analyze the impact of each term from the decomposed correlation coefficient on the compensation.
We begin with generating an underlying correlated time series using the *Capital Asset Pricing Model* (CAPM) [@sharpe64], which is in a one-factor form known as Noh’s model [@noh00] in physics, $$r^{(i)}(t) = \sqrt{c}\,\eta(t) + \sqrt{1-c}\,\varepsilon^{(i)}(t)\ .$$ Here $r^{(i)}$ stands for the return of the $i$-th stock at time $t$ and $c$ is the correlation coefficient. The random variables $\eta$ and $\varepsilon ^{(i)}$ are taken from standard normal distributions. Two return time series $r^{(1)}$ and $r^{(2)}$ are generated representing two correlated stocks. The lengths of these time series is chosen as $7.2\cdot10^{6}$, corresponding to a return interval $\Delta t$ of 1 second during 1 trading year.
\
\
\
\
Using these returns, we generate two price time series $S^{(1)}$ and $S^{(2)}$ that perform a geometric Brownian motion with zero drift and a standard deviation of $10^{-3}$ per time step. The initial starting prices $S_{t=0}$ were set to $1000$ and $10000$. In the next step, we round the prices to integer values. An integer price of for example 1000 then corresponds to a price of 10 and a tick-size of 0.01.
Now we are able to construct the discretized return time series $\bar{r}^{(i)}$ from these discretized prices using return intervals from 60 data points (corresponding to 1 minute) to 1800 data points (corresponding to 30 minutes).
As we know the actual discretization errors in the model, we can use it to evaluate our error estimates. A comparison of the estimated average discretization errors with the actual discretization errors is shown in Fig. \[fig:meanbench\]. The estimated values show an excellent agreement with the original values. We restrict the interpolation to a single Gaussian fit, as we know the type of the price change distributions in this case. Thus, we can verify the scope of the estimation itself, not the suitability of the interpolation.
Before we perform the compensation, we want to see how much impact each correction term (equations (\[eq:compstart\]) to (\[eq:compend\])) has. We quantify the impact by calculating equation (\[eq:errorcorr\_2\]) and subtract the value of this expression with the regarded term set to zero. By this method we can see how the correlation coefficient changes, if a certain term of the discretization compensation is neglected (set to zero). Figure \[fig:termsize\] illustrates the results of this analysis for different start prices and correlation coefficients. It turns out that only equations (\[eq:compmid\]) to (\[eq:compend\]) provide a sizable contribution to the compensation. Therefore, we restrict our compensation to the calculation of these terms. This implies that the distortion of the correlation coefficient is mainly caused by an improper normalization of the returns, as the terms (\[eq:compmid\]) to (\[eq:compend\]) only appear in the correction of the standard deviations of each return.
Thereby, we are able to compensate the discretization effects. We first focus on the correlations between price changes. As shown in Fig. \[fig:e11\] and \[fig:e110\], the correlation coefficient decays towards smaller price change intervals. Therefore, this effect is also a cause of the Epps effect. This effect becomes especially relevant when the ratio of the price to the tick-size is sufficiently small. It is remarkable that this scaling behavior is observed even though the time series are synchronous. The effect vanishes in our simulation, when both prices start with a value of 10000, as Fig. \[fig:e1010\] illustrates.
When applying the compensation method to return time series as illustrated in Fig. \[fig:res\_r\], we are also able to correct the discretization error almost completely. The slight decay of the corrected correlation coefficient on very small return intervals is due to approximations, as stated at the end of section \[s:return\_correct\]. These are the negligence of the correlation between price changes and prices. In addition, even though the discretization of price changes is corrected, the price discretization in the denominator of the return is neglected. A further improvement of the compensation could be achieved by including these effects. However this would require further assumptions on the price process and would increase the necessary computing time dramatically. Thus, we restrict ourselves to the presented compensation.
### Empirical results
How large is the contribution of the discretization effect to the Epps effect? To answer this, we apply the compensation to empirical data from the NYSE TAQ database [@TAQ]. Here, we use a powerlaw approach for the interpolation of the price change distribution, as the model results indicate that the discretization effects are mainly relevant for small return intervals. On small return intervals, powerlaw tails can describe the distribution satisfactory [@gopikrishnan99]. We perform a least squares fit of $a$ and $b$ in $\varrho_{\Delta S} = ax^{-|b|}$ for each value of the (discrete) distribution and their next two left and right neighbors individually. For the very central part of the distribution, a Gaussian fit was performed.
It is particularly important that stock splits must not be corrected in order to maintain the correct tick-size. Of course, therefore overnight returns have to be excluded. To analyze the impact of the discretization effect, we construct two ensembles (See Tab. \[tab:high1\] and \[tab:high2\]) of stocks from the S&P 500 index. The first ensemble consists of stocks that are averagely priced between \$0.01 and \$10.00. The second ensemble consists of stocks that are on average priced between \$10.01 and \$20.00. Both ensembles are composed of 25 stock pairs providing the highest correlation during the year 2007 (based on daily data).
As figure \[fig:realepps\] demonstrates, we are able to compensate the impact of the tick-size on the correlation coefficient in empirical market data. Certainly, the decay can not be corrected completely with the presented method, as the discretization effect superimposes with other causes of the Epps effect such as asynchronous [@muennix09b] or lagged [@reno03; @toth09] time series. However, we were able to quantify the contribution of this particular effect to the Epps effect. Our results show, that the discretization effect can be responsible for up to 40% of the Epps effect, which we define as the difference between the correlation coefficient at a given time and its saturation value. The contribution is particularly large for stocks that are traded at low prices.
Conclusion {#s:conclusion}
==========
We demonstrated the impact of the tick-size on the microstructure of financial returns. This structure can lead to a change in the shape of the distributions of returns and price changes. If a stock exhibits a large price change in the observed period of time, the composition of the return distribution can lead to heavier tails. We also showed that the return distribution consists of return subset distributions that are more sparsely distributed than the complete distribution.
Furthermore, we demonstrated that the discretization effects can distort the calculation of correlation coefficients, especially if the stocks are traded at low prices. We showed that the erroneous correlation coefficient is mainly caused by an improper normalization of the returns. This distortion depends on the impact of the discretization, which grows for small return intervals. Therefore the observed behavior contributes to the Epps effect.
We developed a method to compensate these discretization effects, which we validated in a model setup. The compensation is only based on the tick-size. Despite the interpolation of the price change distribution, the compensation is parameter-free. This method was also applied to market data. We were able to identify and compensate the impact of the tick-size on the correlation coefficient. The results indicate that the discretization error makes a sizable contribution to the Epps effect for stocks that are traded at low prices.
Acknowledgements {#acknowledgements .unnumbered}
================
M.C.M acknowledges financial support from Studienstiftung des deutschen Volkes.
Stock ensembles
===============
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'There are many different ways that the exponents of Weyl groups of irreducible root systems have been defined and put into practice. One of the most classical and algebraic definitions of the exponents is related to the eigenvalues of Coxeter elements. While the coefficients of the height root when expressed as a linear combination of simple roots are combinatorial objects in nature, there are several results asserting relations between these exponents and coefficients. This study was conducted to give a uniform and fairly elementary proof of the fact that the second smallest exponent of the Weyl group is one or two plus the largest coefficient of the highest root of the root system depending upon a simple condition on the root lengths. As a consequence, we obtain a necessary and sufficient condition for a root system to be of type $G_2$ in terms of these numbers.'
address: 'Department of Mathematics, Hokkaido University, Sapporo 060-0810, Japan'
author:
- Tan Nhat Tran
bibliography:
- 'references.bib'
title: The largest coefficient of the highest root and the second smallest exponent
---
Introduction
============
Assume that $V={\mathbb{R}}^\ell$ with the standard inner product $(\cdot,\cdot)$. For $\alpha \in V$, $\beta \in V\setminus\{0\}$, denote $\langle \alpha,\beta \rangle := \frac{2( \alpha,\beta ) }{( \beta , \beta)}$. Let us denote by $\Phi$ an irreducible crystallographic root system in $V$. Let $\Phi^+$ be a set of positive roots. With the notation $\Delta=\{\alpha_1, \ldots ,\alpha_\ell\}$, we have the simple root system of $\Phi$ with respect to $\Phi^+$. For any $\alpha, \beta \in \Phi$, the number $\langle \alpha,\beta \rangle$ takes values in $\{0, \pm1,\pm2,\pm3\}$. For $\alpha = \sum_{i=1}^\ell d_i \alpha_i \in \Phi^+$, the *height* of $\alpha$ is defined by $ {\rm ht}(\alpha) :=\sum_{i=1}^\ell d_i$. Define the partial order $\le$ on $\Phi^+$ such that $\beta \le \alpha$ if $\alpha-\beta \in \sum _{i=1}^\ell {\mathbb{Z}}_{\ge0} \alpha_i$ for $\alpha, \beta\in \Phi^+$. Let $\theta:= \sum_{i=1}^\ell c_i\alpha_i$ be the highest root of $\Phi$ with respect to the partial order, and we call $c_i$’s the coefficients of $\theta$. Denote by $c_{\max}:=\max \{c_i\mid1 \le i\le \ell\}$ the largest coefficient. Let $W$ be the Weyl group of $\Phi$ and let $m_1, m_2, \ldots, m_\ell$ with $m_1 \le m_2 \le \ldots \le m_\ell$ be the exponents of $W$.
The exponents of the Weyl group may have been originally defined in terms of the eigenvalues of Coxeter elements [@C51]. In addition, they can be defined as the degrees of the basic polynomial invariants of the Weyl group [@C55]. The multiset of the exponents and its subsets also have led to many important results and applications in study of Weyl arrangements, which are important examples of free arrangements ([@OS83], [@OST86], [@OT92 Chapter 6]). All of these above-mentioned definitions and applications are purely algebraic. Shapiro (unpublished), Steinberg [@R59], Kostant [@K59], Macdonald [@M72] and most recently also Abe-Barakat-Cuntz-Hoge-Terao [@ABCHT16] have found and shown that there is another possibility to obtain the exponents, namely the dual partition of the height distribution of $\Phi^+$. This latter approach not only gives a particularly simple way of determining the exponents in the individual cases but also reveals connections between the exponents and the other combinatorial objects of the root system. There are also many results in the literature asserting relations between the exponents and the coefficients of the highest root. The most fundamental one is known that the largest exponent is equal to the sum of the highest root coefficients i.e., $m_\ell = \sum_{i=1}^\ell c_i$. A complete description of the exponents in terms of $c_{\max}$ found by a case-by-case check is mentioned in [@Bu09 Theorem 3.2]. What most interests us is the following interesting relation between $c_{\max}$ and $m_2$, which we describe in a uniform way.
\[thm:uniform\] Assume that $\ell\ge 2$. Set ${{\mathcal{U}}}:=\{\theta_i\in \Phi^+ \mid {\rm ht}(\theta_i) >m_{\ell-1}\}$, and $m:=|{{\mathcal{U}}}|$. Suppose that $\xi_i:=\theta_i-\theta_{i+1} \in\Delta$ for $1 \le i \le m-1$. If there is an integer $t$ such that $1 \le t \le m-1$ and ${\langle}\theta_t,\xi_t {\rangle}= 3$, then $c_{\max}=m_2-2$. For otherwise, $c_{\max}=m_2-1$.
By a “uniform" way we mean the proof does not rely on the Classification Theorem of root systems [@H72 Chapter III, 11.4, Theorem] except the fact that the Dynkin graph of $\Phi$ is a tree. As a consequence, we obtain a criterion for a root system to be of type $G_2$ in terms of $c_{\max}$ and $m_2$.
\[cor:criterion\] $\Phi$ is of type $G_2$ if and only if $c_{\max}=m_2-2$.
The aim of this paper is to provide a uniform and fairly elementary proof of Theorem \[thm:uniform\]. We build two isomorphic sets whose cardinalities are expressed in terms of $c_{\max}$ and $m_2$, respectively. One is described by a graph-theoretical property of the Dynkin graph, while the other is described by a combinatorial property of the root poset. The isomorphism between these sets is proved according to the condition on the root lengths (Theorem \[thm:iso\]).
The rest of this paper is organized as follows. In §\[sec:pre\] we review some fundamental definitions and results about root systems and Weyl groups. §\[sec:properties\] is intended to motivate our investigation on combinatorial and graph-theoretical properties of positive roots. §\[sec:proof\] contains the proofs of Theorem \[thm:uniform\] and Corollary \[cor:criterion\].
Preliminaries {#sec:pre}
=============
Our standard references for root systems and their Weyl groups are [@B68] and [@H72 Chapter III]. Let $V= {\mathbb{R}}^\ell $. Let $\Phi$ be an irreducible (crystallographic) root system spanning $V$ with the standard inner product $(\cdot,\cdot)$. We fix a positive system $\Phi^+$ of $\Phi$. We write $\Delta:=\{\alpha_1, \ldots ,\alpha_\ell\}$ for the simple system (base) of $\Phi$ with respect to $\Phi^+$. For $\alpha \in V$, denote $\|\alpha \|:=\sqrt{(\alpha ,\alpha)}$. Note that at most two root lengths can occur in $\Phi$ [@H72 Chapter III, 10.4, Lemma C].
A *reflection* in $V$ with respect to a nonzero vector $\alpha \in V$ is a mapping $s_{\alpha}: V \to V$ defined by $s_\alpha (x) := x -\langle x,\alpha\rangle \alpha$. The *Weyl group* $W:=W(\Phi)$ of $\Phi$ is a group generated by the set $\{s_{\alpha}\mid \alpha \in \Phi\}$. An element of the form $c=s_{\alpha_1}\dots s_{\alpha_\ell}\in W$ is called a *Coxeter element*. Since all Coxeter elements are conjugate [@B68 Chapter V, $\S$6.1, Proposition 1], they have the same order, characteristic polynomial and eigenvalues. The order ${\rm h}:={\rm h}(W)$ of Coxeter elements is called the *Coxeter number* of $W$. For a fixed Coxeter element $c\in W$, if its eigenvalues are of the form $\exp (2\pi\sqrt{-1}m_1/{\rm h}),\ldots, \exp (2\pi\sqrt{-1}m_\ell/{\rm h})$ with $0< m_1 \le \ldots \le m_\ell<{\rm h}$, then the integers $m_1,\ldots, m_\ell$ are called the *exponents* of $W$. The following facts can be found in [@B68 Chapter V, $\S$6.2 and Chapter VI, $\S$1.11].
\[exponents\] For any irreducible root system $\Phi$ in ${\mathbb{R}}^\ell$,
(i) $m_j + m_{\ell+1-j}={\rm h}$ for $1 \le j \le \ell$,
(ii) $1=m_1 < m_2 \le \ldots \le m_{\ell-1} <m_\ell={\rm h}-1$,
(iii) ${\rm h}={\rm ht}(\theta)+1$, where $\theta$ is the highest root.
Let $\Theta^{(r)} \subseteq \Phi^+$ be the set consisting of positive roots of height $r$. The *height distribution* of $\Phi^+$ is defined as a multiset of positive integers: $$\{t_1, \ldots , t_r, \ldots , t_{{\rm h}-1}\},$$ where $t_r := \left|\Theta^{(r)}\right|$. The *dual partition* ${{\mathcal{DP}}}(\Phi^+)$ of the height distribution of $\Phi^+$ is given by a multiset of nonnegative integers: $${{\mathcal{DP}}}(\Phi^+) := \{(0)^{\ell-t_1},(1)^{t_1-t_2},\ldots ,({\rm h}-2)^{t_{{\rm h}-2}-t_{{\rm h}-3}},({\rm h}-1)^{t_{{\rm h}-1}}\},$$ where notation $(a)^b$ means the integer $a$ appears exactly $b$ times.
\[thm:dual\] The exponents of the Weyl group are given by ${{\mathcal{DP}}}(\Phi^+)$.
Graph-theoretical and combinatorial properties of roots {#sec:properties}
=======================================================
In the remainder of the paper, we assume that $\ell\ge 2$. We denote by ${{\mathcal{D}}}(\Phi)$ the *Dynkin graph* and by $\widetilde{{{\mathcal{D}}}}(\Phi)$ the extended Dynkin graph of $\Phi$. A vertex of a graph is called a *terminal* vertex (resp., a *ramification point*) if it is adjacent to at most one other vertex (resp., to at least three other vertices). A graph is a *simple chain of length $n\ge0$* if it is isomorphic to the Dynkin graph of a root system of type $A_{n+1}$. Two adjacent vertices $\alpha, \beta$ of $\widetilde{{{\mathcal{D}}}}(\Phi)$ are joined by a single (resp., double or triple) edge if $\|\alpha\|=\|\beta\|$ (resp., $\|\alpha\|=\sqrt2\|\beta\|$ or $\|\alpha\|=\sqrt3\|\beta\|$).
We start with a construction of a set whose cardinality is equal to $c_{\max}$. It is inspired by a graph-theoretical interpretation [@MT11 Lemma B.27, Appendix B] of the highest root coefficients and the associated simple roots, which was originally formulated and proved in terms of coroots in [@R75 Lemma 1.5]. Other related results can be found in [@Bu07 Proposition 2.1]. There was an unfortunate error in the proof of [@MT11 Lemma B.27, Appendix B] and the proof itself was not completely correct. However, arguments used there can be well carried to restate the result correctly. We provide here a detailed edition for the reader’s convenience.
\[lem:coes\] Let $\Phi$ be an irreducible root system in ${\mathbb{R}}^\ell$. Let $\theta$ be the highest root of $\Phi$, and denote $\lambda_0 := -\theta$, $c_{\lambda_0}:=1$. Suppose that the elements of a fixed base $\Delta:=\{\lambda_1, \ldots, \lambda_\ell\}$ are labeled so that $\Lambda:=\{\lambda_0, \lambda_1, \ldots, \lambda_q\}$ is a set of minimal cardinality such that $c_{\max}=c_{\lambda_q}$ and $(\lambda_s,\lambda_{s+1}) < 0$ for $0 \le s \le q -1$.
(i) Then $c_{\lambda_s}=s + 1$ for $0\le s \le q$ and $|\Lambda|=c_{\max}$.
(ii) Assume that $c_{\max}\ge 2$. Then $(\lambda_0,\lambda_1, \ldots, \lambda_{q-1})$ is a simple chain of $\widetilde{{{\mathcal{D}}}}(\Phi)$ connected to the other vertices only at $\lambda_{q-1}$.
If $c_{\max}=1$, obviously, $\Lambda=\{\lambda_0\}$. Now assume that $c_{\max}\ge 2$. $$\label{many}
\begin{aligned}
2={\langle}\theta,\theta{\rangle}& = \sum_{s=1}^\ell c_{\lambda_s}{\langle}\lambda_s,\theta{\rangle}\ge c_{\lambda_1}{\langle}\lambda_1,\theta{\rangle}, \\
{\langle}\theta,\lambda_1{\rangle}& = \sum_{s=1}^\ell c_{\lambda_s}{\langle}\lambda_s,\lambda_1{\rangle}\le 2c_{\lambda_1} +c_{\lambda_2}{\langle}\lambda_2,\lambda_1{\rangle}, \\
0 \le {\langle}\theta,\lambda_j{\rangle}& \le c_{\lambda_{j-1}}{\langle}\lambda_{j-1},\lambda_j{\rangle}+2c_{\lambda_j} +c_{\lambda_{j+1}}{\langle}\lambda_{j+1},\lambda_j{\rangle}\quad (2 \le j \le q-1).
\end{aligned}$$ By definition of $\Lambda$, ${\langle}\lambda_1,\theta{\rangle}=1$. Thus $$\label{equas}
\begin{aligned}
2-c_{\lambda_1} & \ge 0, \\
\quad 2c_{\lambda_1} -c_{\lambda_2}- {\langle}\theta,\lambda_1{\rangle}& \ge 0,\\
- c_{\lambda_{j-1}}+2c_{\lambda_j} -c_{\lambda_{j+1}} & \ge 0.
\end{aligned}$$
Adding up the inequalities in yields $$2- {\langle}\theta,\lambda_1{\rangle}\ge c_{\lambda_q}-c_{\lambda_{q-1}} .$$
If ${\langle}\theta,\lambda_1{\rangle}=2$, by the minimality, we must have $q=1$, $\Lambda=\{\lambda_0, \lambda_1\}$, and $c_{\max}=c_{\lambda_1}=2$.
If ${\langle}\theta,\lambda_1{\rangle}=1$, by the minimality, $1+ c_{\lambda_{q-1}}= c_{\lambda_q}$. Thus equality occurs here and also in each of the inequalities used above. We obtain a recurrence relation defined by $c_{\lambda_0}=1$, $c_{\lambda_1}=2$, $c_{\lambda_{j+1}}= 2c_{\lambda_j} -c_{\lambda_{j-1}}$ $(1 \le j \le q-1)$. Thus $c_{\lambda_s}=s + 1$ for $0 \le s\le q$. Additionally, from ${\langle}\lambda_{j-1},\lambda_j{\rangle}={\langle}\lambda_{j},\lambda_{j-1}{\rangle}=-1$ $(1 \le j \le q-1)$, we get $\|\lambda_0\|=\|\lambda_1\|=\ldots=\|\lambda_{q-1}\|$. Thus $(\lambda_0,\lambda_1, \ldots, \lambda_{q-1})$ is a simple chain of $\widetilde{{{\mathcal{D}}}}(\Phi)$ connected to the other vertices only at $\lambda_{q-1}$.
\[rem:only1\] If $c_{\max}=1$, ${{\mathcal{D}}}(\Phi)$ contains only single edges (i.e., all roots of $\Phi$ have the same length). In addition, if $\ell \ge 2$, $-\theta$ is connected only to the terminal vertices of ${{\mathcal{D}}}(\Phi)$. Furthermore, the equation ${\langle}\theta,\theta{\rangle}=2$ implies that ${{\mathcal{D}}}(\Phi)$ has exactly two terminal vertices. In this case, we know explicitly that ${{\mathcal{D}}}(\Phi)$ must be a simple chain. If $c_{\max}\ge 2$, by Proof of Proposition \[lem:coes\], $-\theta$ is connected only to one vertex of ${{{\mathcal{D}}}}(\Phi)$.
\[cor:not-important\] Assume that $c_{\max}\ge 2$. Either ${\langle}\lambda_{q-1},\lambda_{q}{\rangle}\in\{-2,-3\}$ or $\lambda_{q}$ is a ramification point of $\widetilde{{{\mathcal{D}}}}(\Phi)$.
Assume that $c_{\max}= 2$ i.e., $q=1$. Suppose that ${\langle}\lambda_0,\lambda_1{\rangle}=-1$, and $\lambda_1$ is connected only to one vertex of $\widetilde{{{\mathcal{D}}}}(\Phi)$ apart from $\lambda_0$, say $\lambda_2$. Thus $\lambda_1$ is long and ${\langle}\lambda_2,\lambda_1{\rangle}=-1$. From ${\langle}\lambda_0,\lambda_1{\rangle}=-1$, we get $c_{\lambda_2}=3$, which is absurd. The case $c_{\max}\ge 3$ i.e., $q \ge 2$ is treated similarly by using ${\langle}\lambda_0,\lambda_q{\rangle}=0$ in place of ${\langle}\lambda_0,\lambda_1{\rangle}=-1$.
Next, we prove several combinatorial properties of positive roots according to their locations on the root poset (with respect to the partial order $\le$).
\[rem:k=2\] Assume that $\beta \in \Phi^+$, $\alpha \in \Delta$, and ${\langle}\beta, \alpha {\rangle}=k \in \{2,3\}$. Then there exists $\alpha' \in \Delta\setminus\{\alpha\}$ such that $\beta- (k-1)\alpha -\alpha'\in \Phi^+$.
By the assumption, $s_\alpha (\beta)= \beta -k \alpha \in \Phi$. Thus ${\rm ht}(\beta)\ge k+1$ and $\| \beta \| = \| \beta -k \alpha \| > \| \alpha \|$. In addition, $\beta- i\alpha \in \Phi^+$ for all $0 \le i\le k$ since the $\alpha$-string through $\beta$ is unbroken [@H72 Chapter III, 9.4]. If ${\langle}\beta- (k-1)\alpha, \alpha {\rangle}\ge 1$, then $\|\beta -k \alpha\| \le \| \beta -(k-1) \alpha\|$ i.e., $ \beta -(k-1) \alpha$ is a long root. We then have ${\langle}\beta- (k-1)\alpha, \alpha {\rangle}\ge 2$. But it implies that $\|\beta -k \alpha\| < \| \beta -(k-1) \alpha\|$, a contradiction. Thus $(\beta- (k-1)\alpha, \alpha)\le 0$. Suppose that $(\beta-(k-1)\alpha, \alpha' )\le 0$ for all $ \alpha' \in \Delta\setminus\{\alpha\}$. By [@H72 Chapter III, 10.1, Theorem$^\prime$(3)], $\{\beta- (k-1)\alpha\}\cup \Delta$ is a linearly independent set, which is absurd. There exists $\alpha' \in \Delta\setminus\{\alpha\}$ such that $(\beta- (k-1)\alpha, \alpha')> 0$ hence $\beta- (k-1)\alpha -\alpha' \in \Phi^+$.
\[lem:3roots\] Suppose $\beta_1, \beta_2, \beta_3 \in \Phi$ with $\beta_1+ \beta_2+ \beta_3 \in \Phi$ and $\beta_i+ \beta_j \ne 0$ for $i \ne j$. Then at least two of the three partial sums $\beta_i+ \beta_j$ belong to $ \Phi$.
See, for example, [@LN04 §11, Lemma 11.10].
Recall the notation $\Theta^{(r)}=\{\alpha \in \Phi^+ \mid {\rm ht}(\alpha)=r\}$. It follows from Theorem \[thm:dual\] that $|\Theta^{(r)}|=1$ if $m_{\ell-1}<r\le m_{\ell}$.
\[cor:G2\] Assume that $\beta \in \Phi^+$, $\alpha \in \Delta$, ${\langle}\beta, \alpha {\rangle}=3$ and $\{\alpha\}=\{\alpha_i \in \Delta \mid \beta-\alpha_i \in\Phi^+\}$. Then there is no $\alpha' \in \Delta\setminus\{\alpha\}$ such that $\beta-\alpha -\alpha'\in \Phi$. In particular, the statement holds true if the last assumption is replaced by ${\rm ht}(\beta)\ge m_{\ell-1}+2$.
Suppose that there exists $\alpha' \in \Delta\setminus\{\alpha\}$ such that $\gamma:=\beta-\alpha -\alpha'\in \Phi$. By Lemma \[lem:3roots\], $\alpha +\alpha' \in \Phi^+$. Thus ${\langle}\alpha' ,\alpha {\rangle}\in \{ -1, -2, -3\}$. Moreover, ${\langle}\gamma ,\alpha {\rangle}+ {\langle}\alpha' ,\alpha {\rangle}=1$. This contradicts to the fact that at most two root lengths occur in $\Phi$.
Recall the notation ${{\mathcal{U}}}=\{\theta_i\in \Phi^+ \mid {\rm ht}(\theta_i) >m_{\ell-1}\}$, and $m=|{{\mathcal{U}}}|=m_{\ell}-m_{\ell-1}=m_{2}-1$. Suppose that the elements of ${{\mathcal{U}}}$ are labeled so that $\theta_1$ denotes the highest root, and $\xi_i=\theta_i-\theta_{i+1} \in\Delta$ for $1 \le i \le m-1$. We also adopt a convention $\xi_0:=-\theta_1$. Set $\Xi:=\{\xi_i \mid 0 \le i \le m-1\}$. Note that $\Xi$ is a multiset, not necessarily a set.
\[cor:2\] Suppose that $m \ge 2$. Then the simple roots $\xi_0,\ldots,\xi_{m-2}$ all are non-ramification points of $\widetilde{{{\mathcal{D}}}}(\Phi)$.
By Remark \[rem:only1\], the condition $m \ge 2$ ensures that $\xi_1$ is the unique vertex of ${{{\mathcal{D}}}}(\Phi)$ connected to $\xi_0$. Suppose that $m \ge 3$. Fix $\xi_i \in \Xi$, $1 \le i\le m-2$ and let $\alpha$ be an adjacent vertex to $\xi_i$ on ${{\mathcal{D}}}(\Phi)$. We have $(\xi_i+\alpha)+\theta_{i+1}-\alpha= \theta_i \in\Phi^+$. By Lemma \[lem:3roots\], either $\theta_i+\alpha\in\Phi^+$ or $\theta_{i+1}-\alpha \in\Phi^+.$ If $i=1$ then $\alpha=\xi_2$. If $i > 1$ then $\alpha \in \{\xi_{i-1}, \xi_{i+1}\}$. Thus $\xi_i$ is not a ramification point.
\[lem:inner-prod\] Suppose $\beta_1, \beta_2 \in \Phi$ and $\beta_1- \beta_2\in \Phi$. If at least one of $\beta_1, \beta_2$ is a long root, then $(\beta_1, \beta_2)>0$.
Straightforward.
\[prop:lengths\] Suppose that $m \ge 3$.
1. If there is an integer $t$ such that $1 \le t \le m-1$ and ${\langle}\theta_t,\xi_t {\rangle}= 3$, then $t=m-2$. As a consequence, $m\ge 4$ and $\|\theta_1\|=\ldots = \|\theta_{m-2}\|=\|\xi_1\|=\ldots = \|\xi_{m-3}\|$.
2. If there is no such $t$, then $\|\theta_1\|=\ldots = \|\theta_{m-1}\|=\|\xi_1\|=\ldots = \|\xi_{m-2}\|$.
We only give a proof for (i). Proof of (ii) follows from a similar argument. It follows from Proof of Proposition \[lem:coes\] that ${\langle}\theta_1,\xi_1 {\rangle}\in \{1,2\}$. By Lemma \[rem:k=2\], we must have ${\langle}\theta_1,\xi_1 {\rangle}=1$. Thus $\|\theta_{2}\|=\|\xi_1\|=\|\theta_{1}\|$. The first statement of (i) follows from Lemma \[rem:k=2\] and Corollary \[cor:G2\]. One can use Lemma \[lem:inner-prod\] to prove inductively that ${\langle}\theta_i,\xi_i {\rangle}=1$ and $\theta_i,\xi_i$ all are long roots for $1 \le i \le m-3$ ($\theta_{m-2}$ is a long root as well), which proves the second statement.
*Convention*: For simplicity, in the remainder of the paper, let us call the case “there is an integer $t$ such that $1 \le t \le m-1$ and ${\langle}\theta_t,\xi_t {\rangle}= 3$" Case $1$, and its negation Case $2$.
Our candidate for a set whose cardinality is expressed in terms of $m_2$, and isomorphic (actually equal) to the set $\Lambda$ in Proposition \[lem:coes\] will be introduced below. For a finite multiset $S=\{(a_1)^{b_1},\ldots, (a_n)^{b_n}\}$, we write $\overline{S}$ for the base set of $S$ i.e., $\overline{S}=\{a_1,\ldots, a_n\}$.
\[prop:b-a\]
1. If Case 1 occurs, then $\Xi=\{\xi_0, \xi_1, \ldots, (\xi_{m-2})^2\}$ with $\xi_i \ne \xi_j$ for $0 \le i < j \le m-2$. As a result, $|\overline{\Xi}|=m_2-2$. Moreover, $(\xi_0,\xi_1,\ldots,\xi_{m-3})$ is a simple chain of $\widetilde{{{\mathcal{D}}}}(\Phi)$ connected to the other vertices only at $\xi_{m-3}$, and ${\langle}\xi_{m-3},\xi_{m-2}{\rangle}=-3$.
2. If Case 2 occurs, then $\Xi=\{\xi_0, \xi_1, \ldots, \xi_{m-1}\}$ with $\xi_i \ne \xi_j$ for $0 \le i < j \le m-1$. As a result, $|\Xi|=m_2-1$. If ${\langle}\theta_1,\xi_1 {\rangle}=2$, then $\Xi=\{\xi_1\}$ and $m=2$. If ${\langle}\theta_1,\xi_1 {\rangle}=1$ and $m \ge 3$, then $(\xi_0,\xi_1,\ldots,\xi_{m-2})$ is a simple chain of $\widetilde{{{\mathcal{D}}}}(\Phi)$ connected to the other vertices only at $\xi_{m-2}$.
We only give a proof for (i). Proof of (ii) follows from a similar argument. Obviously, $\xi_0\ne \xi_i$ for all $1 \le i \le m-2$ by a reason of heights, and $\xi_{m-2} \ne \xi_i$ for all $0 \le i \le m-3$ by a reason of lengths. Suppose to the contrary that $\xi_i=\xi_j$ for some $1 \le i<j \le m-3$. Choose indexes $i,j$ so that $j-i$ is minimal. By Proposition \[prop:lengths\], $j>i+1$. If $j=i+2$, then $\theta_i=\theta_{i+3}+2\xi_{i}+\xi_{i+1}$. This cannot happen since ${\langle}\theta_i,\xi_{i}{\rangle}=1$, ${\langle}\theta_{i+3},\xi_{i} {\rangle}\ge -1$ and ${\langle}\xi_{i+1},\xi_{i} {\rangle}\ge -1$. Then $j>i+2$ and $\{\xi_i ,\xi_{i+1}\}, \{\xi_{i+1},\xi_{i+2}\}, \ldots, \{\xi_{j-1},\xi_{j}\}$ are connected subgraphs of ${{\mathcal{D}}}(\Phi)$. By the choices of $i,j$, the simple roots $\xi_i ,\xi_{i+1},\ldots,\xi_{j-1}$ are mutually distinct, the condition $\xi_i=\xi_j$ implies that ${{\mathcal{D}}}(\Phi)$ contains a cycle. This contradiction proves the first statement. The remaining statements follow immediately.
\[cor:differences\]
1. If Case 1 occurs, then $\theta_i-\theta_j \in \Phi^+$ for $1 \le i<j \le m$, $\{i,j\} \ne \{m-2,m\}$, and $\theta_{m-2}-\theta_m \in 2\Delta$.
2. If Case 2 occurs, then $\theta_i-\theta_j \in \Phi^+$ for $1 \le i<j \le m$.
We only give a proof for (i). Obviously, $\theta_{m-2}-\theta_m=2 \xi_{m-2} \in 2\Delta$. By [@B68 Chapter VI, §1.6, Corollary 3(b)], $\theta_i-\theta_{j} =\xi_i +\xi_{i+1}+\ldots+\xi_{j-1}\in \Phi^+$ for $1 \le i<j \le m-1$. Note that $\theta_i-\theta_m =(\theta_i-\theta_{m-1})+\xi_{m-2}$ for all $1 \le i \le m-3$. Thus $\theta_i-\theta_m\in\Phi^+$ because $\theta_i-\theta_{m-1}\in\Phi^+$ as above and $$\begin{aligned}
(\theta_i-\theta_{m-1},\xi_{m-2}) &= (\xi_i +\ldots+\xi_{m-3}+\xi_{m-2},\xi_{m-2}) \\
& =(\xi_{m-3}+\xi_{m-2},\xi_{m-2})<0.\end{aligned}$$
\[rem:related\] Corollary \[cor:differences\] is related to a property of a *chain* in the poset in [@H16 Lemma 5.1]. However, for the particular chain ${{\mathcal{U}}}$, Corollary \[cor:differences\] is a bit more explicit and the proof does not need to go through the classification whether the root system is of type $G_2$ or not.
Proof of the main result {#sec:proof}
========================
Theorem \[thm:uniform\] is a consequence of the following:
\[thm:iso\] With the notations as in Propositions \[lem:coes\] and \[prop:b-a\], $\overline{\Xi}=\Lambda$.
The proof will be proceeded in three steps. In what follows, $\theta$ and $\theta_1$ both denote the highest root.
Step $1$. If $c_{\max}=1$, by Remark \[rem:only1\], all roots of $\Phi$ have the same length. So the problem falls in Case 2. It is easily seen that $\Xi=\Lambda=\{-\theta\}$, and $m_2=c_{\max}+1=2$. Note also that $c_{\max}=1$ if and only if $m=1$.
Step $2$. Now consider $c_{\max}\ge 2$ and $m=2$. This implies that $\xi_1 \equiv \lambda_1$ is the unique vertex of ${{{\mathcal{D}}}}(\Phi)$ connected to $-\theta$. By Proof of Proposition \[lem:coes\], ${\langle}\theta,\xi_1 {\rangle}\in \{1,2\}$. So the problem falls in Case 2. Hence $\Xi=\{-\theta, \xi_1\}$ and $m_2=m+1=3$. If ${\langle}\theta,\lambda_1 {\rangle}=2$, by Proposition \[lem:coes\], $\Lambda=\{-\theta, \lambda_1\}$ and $c_{\max}=2$. Now consider ${\langle}\theta,\lambda_1 {\rangle}=1$, and suppose that $|\Lambda| \ge 3$ i.e., $\Lambda$ contains a simple root other than $\lambda_1$, say $\lambda_2$. Recall the notation $\Theta^{(r)}=\{\alpha \in \Phi^+ \mid {\rm ht}(\alpha)=r\}$. Since $m=2$, we may assume that $\Theta^{(m_{\ell-1})} \supseteq \{\mu:=\theta-\lambda_1-\lambda_2,\mu' :=\theta-\lambda_1-\lambda'_2\}$ with $\lambda'_2\ne\lambda_2$. If $\lambda_1 =\lambda'_2$, then ${\langle}\mu', \lambda_1{\rangle}=-3$, which is absurd because $\lambda_1$ is long. If $\lambda_1 \ne\lambda'_2$, by Lemma \[lem:3roots\], $\lambda'_2+\lambda_1 \in \Phi^+$. Thus $\lambda_{1}$ is a ramification point of $\widetilde{{{\mathcal{D}}}}(\Phi)$, a contradiction. In either case, $\Xi=\Lambda=\{-\theta, \lambda_1\}$ and $m_2=c_{\max}+1=3$.
Step $3$. Now consider $c_{\max}\ge 2$ and $m \ge 3$. The condition $m \ge 3$ ensures that ${\langle}\theta_1,\lambda_1{\rangle}=1$.
Firstly, we prove that $\overline{\Xi}\supseteq\Lambda$. By Proposition \[lem:coes\](ii), we have $$\lambda_1+ \ldots+ \lambda_i\in \Phi^+ \mbox{ and } (\theta_1,\lambda_1+ \ldots+ \lambda_i)=(\theta_1,\lambda_1)>0 \quad (1\le i \le q).$$ Set $\eta_1:=\theta_1$, and for $2 \le p \le q+1$ set $$\eta_p:=\theta_1 - (\lambda_1+ \ldots+ \lambda_{p-1}) \in \Phi^+, \mbox{ then } \eta_{p}=\eta_{p-1}-\lambda_{p-1}.$$ One can use Lemma \[lem:inner-prod\] and the fact that $\lambda_1, \ldots, \lambda_{q-1}$ are long roots from Proposition \[lem:coes\](ii) to prove inductively that ${\langle}\eta_i, \lambda_i{\rangle}=1$ for $1 \le i \le q-1$ and $\|\eta_1\|=\ldots = \|\eta_{q-1}\|= \|\eta_{q}\|$. We claim that $$\label{eq:claim}
\Theta^{(m_l-p+1)}=\{\eta_p\} \mbox{ for } 1 \le p \le q+1.$$ It is clearly true when $1 \le p\le2$. Suppose to the contrary that we can choose the smallest $p$ such that $3 \le p \le q+1$ and $|\Theta^{(m_l-p+1)}|>1$. In particular, $m_{l-1}=m_l-p+1$. To obtain a contradiction, we use a very similar argument to that used in Step $2$. Assume that $\{\eta_p,\eta'_p\} \subseteq \Theta^{(m_l-p+1)}$ with $\eta_p\ne\eta'_p$. There exists $\lambda'_{p-1} \in \Delta$ such that $\lambda_{p-1}\ne\lambda'_{p-1}$ and $\eta_{p-2}=\eta'_p+\lambda'_{p-1}+\lambda_{p-2}$. If $\lambda_{p-2} =\lambda'_{p-1}$, then ${\langle}\eta'_p,\lambda_{p-2}{\rangle}=-3$, which is absurd since $\lambda_{p-2}$ is long. If $\lambda_{p-2} \ne \lambda'_{p-1}$, by the minimality of $p$ and Lemma \[lem:3roots\], $\lambda'_{p-1}+\lambda_{p-2} \in \Phi^+$. If $p=3$, $\lambda_{1}$ is connected to three different roots: $-\theta_1, \lambda_{2}, \lambda'_{2}$, which is absurd. Suppose henceforth that $p \ge 4$. Since $\lambda_{p-2}$ is connected only to $\lambda_{p-3}, \lambda_{p-1}$, we must have $\lambda_{p-3} =\lambda'_{p-1}$. Thus $\eta_{p-3}=\eta'_p+2\lambda_{p-3}+\lambda_{p-2}$, and ${\langle}\eta'_p,\lambda_{p-3}{\rangle}=-2$. This is impossible since $\lambda_{p-3}$ is long. We complete the proof of the claim . Therefore, $$\Lambda=\{\lambda_{p-1} \mid 2 \le p \le q+1\} =\{\eta_{p-1}-\eta_{p} \mid 2 \le p \le q+1\} \subseteq \overline{\Xi}.$$
Secondly, we prove that $\overline{\Xi}\subseteq\Lambda$. The proofs for Case 1 and Case 2 are similar, we only give a proof for Case 1. For the occurrence of Case 1, we can assume that $m \ge 4$. We need to prove that starting from $-\theta$, going along vertices of ${{{\mathcal{D}}}}(\Phi)$, the elements of $\overline{\Xi}$ produce a correct path to reach a first simple root associated to $c_{\max}$. To this end, we show that $c_{\xi_0} < c_{\xi_1} < \ldots < c_{\xi_{m-2}}$. Note that ${\langle}\xi_{m-2},\xi_{m-3}{\rangle}=-1$ since $\xi_{m-3}$ is a long root. Using Proposition \[prop:b-a\] and working out the equations ${\langle}\theta_1,\theta_1{\rangle}=2, {\langle}\theta_1,\xi_1{\rangle}=1, {\langle}\theta_1,\xi_i{\rangle}=0$ for $2 \le i \le m-3$, we obtain $$c_{\xi_0}=1, \, c_{\xi_1}=2, \, c_{\xi_{i+1}}= 2c_{\xi_i} -c_{\xi_{i-1}} \quad (1 \le i \le m-3).$$ Thus $c_{\xi_i}=i+ 1$ for $0 \le i \le m-2$, which proves the claim.
Corollary \[cor:criterion\] is a consequence of the following:
\[thm:criterion-G2\] The following statements are equivalent:
(i) The Dynkin graph of $\Phi$ has the form
\(1) at (0,0) ; (2) at (1,0) ; (1) edge\[tedge\] (2);
(ii) The extended Dynkin graph of $\Phi$ has the form
\(0) at (-1,0) ; (1) at (0,0) ; (2) at (1,0) ; (0) edge (1) ; (1) edge\[tedge\] (2);
(iii) $c_{\max}=m_2-2$.
\(i) $\Leftrightarrow$ (ii) is clear. (ii) $\Rightarrow$ (iii) It is easy to calculate from the graph that $c_{\max}=c_{\xi_2}=3$. By Theorem \[thm:uniform\], $m_2 \ge 4$. Note that $\theta_2 := \theta- \xi_1$ is the unique root of height $m_\ell -1$. Moreover, ${\langle}\theta_2, \xi_2{\rangle}=-{\langle}\xi_1,\xi_2{\rangle}=3$. So the problem falls in Case 1. Thus $c_{\max}=m_2-2$. (ii) $\Leftarrow$ (iii) The condition $c_{\max}=m_2-2$ ensures that $c_{\max}\ge 2$ i.e., $m_2 \ge 4$. Theorem \[thm:uniform\] implies that the problem falls in Case 1. In particular, $m_2 \ge 5$ by Proposition \[prop:lengths\](i). Moreover, by Proposition \[lem:coes\] and Theorem \[thm:iso\], the equations ${\langle}\theta, \xi_{m-2}{\rangle}=0$, $c_{\xi_{m-2}}=c_{\max}$, $c_{\xi_{m-3}}=c_{\max}-1$ yield $c_{\max} \le 3$. So we must have $c_{\max}= 3$ i.e., $m_2=5$, and there are no vertices of ${{{\mathcal{D}}}}(\Phi)$ connected to $\xi_2$ other than $\xi_1$. This completes the proof.
Acknowledgements {#acknowledgements .unnumbered}
================
This paper originates from the author’s Master’s thesis, written under the supervision of Professor Hiroaki Terao at the Hokkaido University. The author wishes to express his sincere thanks to Professor Terao for many stimulating conversations. The author also gratefully acknowledges the support of Master’s scholarship program of the Japanese Ministry of Education, Culture, Sports, Science, and Technology (MEXT) under grant number 142506.
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---
abstract: 'M. Saito [@MS] proved that the jumping numbers of a hyperplane arrangement depend only on the combinatorics of the arrangement. However, a formula in terms of the combinatorial data was still missing. In this note, we give a formula and a different proof of the fact that the jumping numbers of a hyperplane arrangement depend only on the combinatorics. We also give a combinatorial formula for part of the Hodge spectrum and for the inner jumping multiplicities.'
address: 'Department of Mathematics, The University of Notre Dame, IN 46556, USA'
author:
- Nero Budur
date: '[September 6, 2008]{}'
title: Jumping numbers of hyperplane arrangements
---
Introduction
============
Jumping numbers are numerical measures of the complexity of the singularities of a variety (see section \[sec. review\]). M. Saito [@MS] proved that the jumping numbers of a reduced hyperplane arrangement depend only on the combinatorics of the arrangement, answering a question of M. Mustaţă [@Mu]. The method of his proof was by reduction to the corresponding statement about the Hodge spectrum. His proof extends to non-reduced arrangements as well by taking into account the multiplicities along the hyperplanes in the arrangement. However, a formula in terms of the combinatorial data was still missing.
In this note, we give a formula and a different proof of the fact that the jumping numbers of a hyperplane arrangement depend only on the combinatorics and the multiplicities along hyperplanes. We also give a combinatorial formula for part of the Hodge spectrum and for the inner jumping multiplicities. Combinatorial formulas for those jumping numbers which change the support of the multiplier ideals have been obtained in [@Mu]- Example 2.3, for reduced arrangements, and refined by [@Te]- Remark 3.2.
Let ${{\mathcal A}}$ be a central hyperplane arrangement in ${{\mathbb C}}^n$. Denote the intersection lattice of ${{\mathcal A}}$ by $L({{\mathcal A}})$, that is the set of subspaces of ${{\mathbb C}}^n$ which are intersections of subspaces $V\in{{\mathcal A}}$. We consider the corresponding arrangement of projective hyperplanes in $Y={{\mathbb P}}^{n-1}$ given by ${{\mathbb P}}(V)$ for $V\in {{\mathcal A}}$. Let $D$ be an effective divisor on $Y$ supported on $Supp \,(D)=\cup_{V\in
{{\mathcal A}}}{{\mathbb P}}(V)$. We assume that $Supp\,(D)$ is the compactification of a central hyperplane arrangement in some ${{\mathbb C}}^{n-1}\subset Y$. For our purposes, the general case can be reduced to this particular case.
We will give a combinatorial criterion, in terms of $L({{\mathcal A}})$ and the multiplicities of $D$, for a positive rational number to be a jumping number of $D$ in $Y$. It is known that $1$ is trivially a jumping number of $D$ and that $c>1$ is a jumping number if and only if $c-1$ is. Thus it is enough to determine which $c\in (0,1)$ are jumping numbers of $D$.
Let ${{\mathcal G}}'\subset L({{\mathcal A}})-\{{{\mathbb C}}^n\}$ be a building set (see [@DP]-2.4 or [@Te]-Definition 1.2). Let ${{\mathcal G}}={{\mathcal G}}'\cup\{0\}$. For simplicity, one can stick with the following example for the rest of the article: ${{\mathcal G}}=L({{\mathcal A}})\cup\{0\}-\{{{\mathbb C}}^n\}$, when ${{\mathcal G}}'$ is chosen to be $L({{\mathcal A}})-\{{{\mathbb C}}^n\}$. The advantage of considering smaller building sets is that computations might be faster (see [@Te]-Example 1.3-(c)).
For $V\in {{\mathcal G}}$, define $r(V)={\hbox{\rm codim}\,}(V)$, $\delta(V)=\dim V$, and $$s(V)=\sum_ {V\subset
W\in {{\mathcal A}}}\text{mult} _{{{\mathbb P}}(W)}(D).$$ Set $d=\sum_{V\in{{\mathcal A}}}\text{mult}_{{{\mathbb P}}(V)}(D)$ and $$a_0=\max\left\{d-n+1,\sum_ {W\in {{\mathcal G}}-\{0\}}\max\{0, s(W)-r(W)\}\right\}.$$ For any finite set ${{\mathcal S}}$, set $|{{\mathcal S}}|$ to be the number of elements of ${{\mathcal S}}$. For a rational number $c$ let $${{\mathcal S}}_c=\{\ V\in {{\mathcal G}}-\{0\}\ |\ cs(V)\in{{\mathbb Z}}\ \}.$$ For $V\in{{\mathcal G}}$ let $$a_V(c)=\left\{
\begin{array}{ll}
r(V)-1-{\llcorner {cs(V)} \lrcorner} & \text{ if }V\in {{\mathcal G}}-\{0\}, \notin {{\mathcal S}}_c,\\
r(V)-cs(V) & \text{ if }V\in {{\mathcal G}}-\{0\}, \in{{\mathcal S}}_c,\\
-a_0 & \text{ if }V=0,
\end{array}\right.$$
For a nonempty nested subset ${{\mathcal S}}$ of ${{\mathcal G}}-\{0\}$ and for $V\in{{\mathcal S}}\cup\{{{\mathbb C}}^n\}$, denote by $V_{{\mathcal S}}$ the subspace $\sum W$ where the sum is over $W\varsubsetneq V$ such that $W\in{{\mathcal S}}$. In other words, $V_{{\mathcal S}}$ is the maximal element of ${{\mathcal S}}$ which is $\varsubsetneq V$. Set $V_{{\mathcal S}}=0$ if there is no such maximal element. Let $Q(x)=x/(1-\exp(-x))$ considered as an element of the formal power series ring ${{\mathbb Q}}[[x]]$.
\[def. poly P\] Let ${{\mathcal S}}$ be a nonempty nested subset of ${{\mathcal G}}-\{0\}$ and let $V\in{{\mathcal S}}\cup\{{{\mathbb C}}^n\}$. For $W\in{{\mathcal G}}$ with $V_{{\mathcal S}}\subset W\varsubsetneq V$ define a formal power series $P_W^{{{\mathcal S}}, V}\in{{\mathbb Q}}[[c_{W'}]]_{W'\in {{\mathcal G}}}$ as follows. If $W=V_{{\mathcal S}}$ set $$P_W^{{{\mathcal S}}, V}= Q(-\mathop{\sum_{W'\subset
V_{{\mathcal S}}}}_{ \{W'\}\cup{{\mathcal S}}\subset{{\mathcal G}}\text{ nested} }
c_{W'})^{\delta(V)-\delta(V_{{\mathcal S}})}.$$ If $W\ne V_{{\mathcal S}}$ define $$\begin{aligned}
P_W^{{{\mathcal S}}, V} & = Q(-\mathop{\sum_{W'\varsubsetneq W}}_{
\{W'\}\cup{{\mathcal S}}\subset{{\mathcal G}}\text{ nested}}
c_{W'})^{-(\delta(V)-\delta(W))} \cdot Q(c_W)\cdot \\
& \cdot Q(-\mathop{\sum_{ W'\subset
W}}_{ \{W'\}\cup{{\mathcal S}}\subset{{\mathcal G}}\text{ nested}
}c_{W'})^{\delta(V)-\delta(W)}.\end{aligned}$$
\[def. T\_j\^S\] Let ${{\mathcal S}}$ be a nonempty nested subset of ${{\mathcal G}}-\{0\}$. Let $0\le j\le n-1-|{{\mathcal S}}|$. Define the polynomial $T_j^{{\mathcal S}}\in {{\mathbb Q}}[c_V]_{V\in{{\mathcal G}}}$ to be the homogeneous part of degree $j$ of the formal power series $$T^{{\mathcal S}}:=\prod_{V\in{{\mathcal S}}\cup\{{{\mathbb C}}^n\}} \ \ \mathop{\prod_{V_{{\mathcal S}}\subset W\varsubsetneq
V}}_{W\in{{\mathcal G}}} P^{{{\mathcal S}},V}_W.$$
Let $I\subset{{\mathbb Z}}[c_V]_{V\in{{\mathcal G}}}$ be the ideal of [@DP]-5.2 (for the projective case, see Remark \[rem. isom\]). Recall that $I$ depends only on ${{\mathcal G}}$ and that ${{\mathbb Z}}[c_V]_{V\in{{\mathcal G}}}/ I$ is isomorphic to the cohomology ring of the canonical log resolution in terms of ${{\mathcal G}}$ of $(Y,D)$, i.e. the wonderful model of [@DP]. More precisely, $I$ is generated by two types of polynomials: $$\label{eq, type 1}
\prod_{V\in {{\mathcal H}}}c_V$$ if ${{\mathcal H}}\subset{{\mathcal G}}$ is not a nested subset, and by $$\label{eq, type 2}
\prod_{V\in {{\mathcal H}}}c_V\left ( \sum_{W'\subset W} c_{W'} \right
)^{d_{{{\mathcal H}},W}},$$ where ${{\mathcal H}}\subset {{\mathcal G}}$ is a nested subset, $W\in{{\mathcal G}}$ is such that $W\varsubsetneq V$ for all $V\in{{\mathcal H}}$, and $d_{{{\mathcal H}},W}=\delta(\cap_{V\in{{\mathcal H}}} V) -\delta (W)$. In (\[eq, type 2\]), one considers ${{\mathcal H}}=\emptyset$ to be nested, in which case (\[eq, type 2\]) is defined for every $W\in{{\mathcal G}}$ by setting $\delta (\emptyset)=n$.
\[thm. jumping numbers\] With the notation as above, a rational number $c\in (0,1)$ is a jumping number of $D\subset Y$ if and only if $$\mathop{\sum_{\text{nested}}}_{{\emptyset}\ne{{\mathcal S}}\subset {{\mathcal S}}_c}
\sum_{0\le j}^{n-1-|{{\mathcal S}}|} \frac{(-1)^{|{{\mathcal S}}|+1}}{j!} \left (
\sum_{V\in{{\mathcal G}}} a_V(c)c_V \right )^{j} T_{n-1-|{{\mathcal S}}|-j}^{{\mathcal S}}\prod_{V\in{{\mathcal S}}} c_V$$ does not belong to the ideal $I\subset {{\mathbb Q}}[c_V]_{V\in{{\mathcal G}}}$.
Since we are assuming that $D$ is the compactification of a central hyperplane arrangement in ${{\mathbb C}}^{n-1}$, let $x\in Y$ be the point corresponding to the origin of ${{\mathbb C}}^{n-1}$. As for jumping numbers, the method of the proof of Theorem \[thm. jumping numbers\] gives a formula in terms of combinatorics for the inner jumping multiplicities $n_{c,x}(D)$ of a positive rational number $c$ along $D$ at the point $x$ (see section \[sec. review\]).
\[thm. inner jumping numbers\] With the notation as above, let $c$ be a positive rational number. Then the inner jumping multiplicity of $c$ along $D$ at $x$ is $0$ if there are no subspaces $V\in{{\mathcal G}}$ with $\delta (V)=1$ or if $cd\not\in {{\mathbb Z}}$. Otherwise, let $V_x\in{{\mathcal G}}$ be the only subspace with $\delta=1$, that is ${{\mathbb P}}(V_x)=\{x\}$. Then $$n_{c,x}(D)= \sum_{0\le j\le n-2}\frac{1}{j!}\left (
\sum_{V\in{{\mathcal G}}-\{0\}} a_V(c)c_V \right
)^{j}T_{n-2-j}^{\{V_x\}}c_{V_x},$$ where the right-hand side is viewed as a number via identification of the degree $n-1$ homogeneous part of ${{\mathbb Q}}[c_V]_{V\in{{\mathcal G}}}/I$ with ${{\mathbb Q}}\cdot (-c_0)^{n-1}$.
By a result of [@Bu] (see also [@BS]), for $c\in (0,1]$ the inner jumping multiplicities $n_{c,x}(D)$ are the multiplicities of $c$ in the Hodge spectrum of $D$ at $x$ ([@St]). Thus we have a combinatorial formula for the beginning part of the Hodge spectrum of a central hyperplane arrangement.
In section 2 we review multiplier ideals and intersection theory. In section 3 we set the problem into global setting, in preparation for using the Hirzebruch-Riemann-Roch theorem. In section 4, we prove Theorems \[thm. jumping numbers\] and \[thm. inner jumping numbers\] via Hirzebruch-Riemann-Roch on wonderful models. In the last section we give examples illustrating how Theorems \[thm. jumping numbers\] and \[thm. inner jumping numbers\] work.
In this article, inclusion of sets is denoted by $\subset$ and strict inclusion of sets is denoted by $\subsetneq$.
We would like to thank M. Saito, to whom we are indebted for the proof of Lemma \[lema linear forms\], for sharing with us the results of his preprint [@MS] which was the inspiration behind this article, and for many comments. We also thank M. Mustaţă, M. Schulze, and Z. Teitler for useful discussions. The author was supported by the NSF grant DMS-0700360.
[Review of multiplier ideals, intersection theory]{}\[sec. review\]
The notation of the current section is independent of the rest of the article.
[**Multiplier ideals.**]{} We review some basic facts from the theory of multiplier ideals (see [@La]- Chapter 9). Let $Y$ be a smooth complex variety. Let $D$ be an effective ${{\mathbb Q}}$-divisor on $Y$. Let $\rho: Y' {\rightarrow}Y$ be a log resolution of $(Y,D)$ and let $K_{Y'/Y}$ be the relative canonical divisor. The [*multiplier ideal*]{} of $D$ is the ideal sheaf $${{\mathcal J}}(D) = \rho_* {{\mathcal O}}_{Y'}(K_{Y'/Y}-{\llcorner {\rho^* D} \lrcorner})\ \ \ \subset
{{\mathcal O}}_Y.$$ The choice of log resolution does not matter in the definition of the ${{\mathcal J}}(D)$ and one can extend the definition by allowing, instead of $D$, any finite formal linear combination of subschemes of $Y$ with positive coefficients. A positive rational number $c$ is called a [*jumping number*]{} of $D$ if ${{\mathcal J}}(c\cdot
D)\ne {{\mathcal J}}((c-\epsilon)\cdot D)$ for all $0< \epsilon \ll 1$. It is known that a positive rational number $c$ is a jumping number if and only if $c+1$ is a jumping number ([@La]- Example 9.3.24). Let $x$ be a point in the support of $D$ and let $c>0$. The [*inner jumping multiplicity*]{} of $c$ along $D$ at $x$ ([@Bu]- Definition 2.4) is defined as $$n_{c,x}(D)=\dim_{{\mathbb C}}\frac{{{\mathcal J}}((c-\epsilon)D)}{{{\mathcal J}}((c-\epsilon)D+\delta
\{x\})}\ ,$$ where $0<\epsilon\ll\delta\ll 1$. By [@Bu]-Proposition 2.8, if the inner jumping multiplicity of $c$ is nonzero then $c$ is a jumping number.
\[thm local vanishing\] (Local vanishing, [@La]- Theorem 9.4.1). With the notation as above, $$R^j\rho_*{{\mathcal O}}_{Y'}(K_{Y'/Y}-{\llcorner {\rho^* D} \lrcorner}) =0\ \ \text{ for
}j>0.$$
\[thm Nadel vanishing\] (Nadel vanishing, [@La]- Theorem 9.4.9). With the notation as above, assume in addition that $Y$ is projective. Let $L$ be any integral divisor such that $L-D$ is nef and big. Then $$H^i(Y,{{\mathcal O}}_Y(K_Y+L)\otimes{{\mathcal J}}(D))=0\ \ \ \text{ for }i>0.$$
[**Intersection theory.**]{} We recall some facts about intersection theory (see [@Fu]). Let $Y$ be a smooth projective complex variety. For a vector bundle, or locally free ${{\mathcal O}}_Y$-module of finite rank, ${{\mathcal E}}$ on $Y$, we denote by $c_j({{\mathcal E}})$ the image of the $j$-th Chern class of ${{\mathcal E}}$ in $H^{2j}(Y,{{\mathbb Z}})$. The total Chern class is defined to be $c({{\mathcal E}})=\sum_j c_j({{\mathcal E}})$ in the cohomology ring $H^*(Y,{{\mathbb Z}})$. The roots $x_i$ of ${{\mathcal E}}$ are formal symbols satisfying the formal decomposition $\sum_j c_j({{\mathcal E}})t^j=\prod_i (1+x_it)$. Then one defines $ch({{\mathcal E}})=\sum _i\exp (x_i)$, and writes $ch ({{\mathcal E}})=\sum_j
ch_j({{\mathcal E}})$ with $ch_j({{\mathcal E}})\in H^{2j}(Y,{{\mathbb Q}})$. The Todd class of ${{\mathcal E}}$ is defined as $td ({{\mathcal E}})=\prod Q(x_i)$, where $Q(x)=x/(1-\exp(-x))$. The Todd class of $Y$ is denoted by $Td(Y)$ and is defined as the Todd class of the tangent bundle of $Y$. One writes $Td(Y)=\sum_j Td_j(Y)$ where $Td_j(Y)\in H^{2j}(Y,{{\mathbb Q}})$.
\[thm HRR\] (Hirzebruch-Riemann-Roch, [@Fu]- Corollary 15.2.1) Let ${{\mathcal E}}$ be a vector bundle on a smooth projective complex variety $Y$. Then $\chi (Y,{{\mathcal E}})$ is the intersection number $\sum_{i+j=\dim Y} ch_i({{\mathcal E}})\cdot Td_j(Y)$.
Let $X_1,\ldots X_t$ be disjoint smooth subvarieties of $Y$ of codimension $d$. Let $\rho:{\widetilde}{Y}{\rightarrow}Y$ be the blow up of $\coprod X_i$. Let $E_i$ be the exceptional divisor on ${\widetilde}{Y}$ corresponding to $X_i$. Let $[E_i]\in H^2({\widetilde}{Y},{{\mathbb Z}})$ be the cohomology class of $E_i$. Let $N_i$ be the normal bundle of $X_i$ in $Y$. Suppose there exist $c_{k,i}\in
H^{2k}(Y,{{\mathbb Z}})$ such that the Chern classes $c_k(N_i)$ are the restriction of $c_{k,i}$ to $X_i$. The following computes the total Chern class of ${\widetilde}{Y}$ and follows from [@Fu]-Example 15.4.2.
\[prop. chern classes blow up\] With the notation as above, $$c({\widetilde}{Y})=\rho^* c(Y)\prod_{1\le j\le t}\left \{ \left ( \sum_{0\le k\le d} \rho^*c_{k,j} \right )^{-1}(1+[E_j])
\left ( \sum_{0\le i\le d} (1-[E_j])^i\rho^* c_{d-i,j} \right )
\right \}.$$
[Uniform bound for jumps in multiplier ideals]{}
[**Affine case.**]{} Let ${{\mathcal A}}'$ be a central hyperplane arrangement in ${{\mathbb C}}^{n-1}$. Let $D'$ be an effective divisor on ${{\mathbb C}}^{n-1}$ with support ${{\mathcal A}}'$. Let $L({{\mathcal A}}')$ be the intersection lattice of ${{\mathcal A}}'$. For $V\in L({{\mathcal A}}')$, define $r'(V)={\hbox{\rm codim}\,}(V)$ and $s'(V)=\sum_ {V\subset
W\in {{\mathcal A}}' }\text{mult} _W(D')$. Let ${{\mathcal G}}'\subset L({{\mathcal A}}')-\{{{\mathbb C}}^{n-1}\}$ be a building set. Recall the following result of M. Mustaţă [@Mu]-Corollary 2.1 for the case of reduced arrangements, and refined by Teitler [@Te]-Theorem 1.4.
\[prop characterization of jump. nos.\] If $D'$ is an effective divisor supported on a central hyperplane arrangement in ${{\mathbb C}}^{n-1}$, then $${{\mathcal J}}(cD')=\bigcap_{W\in{{\mathcal G}}'}I_W^{\ {\ulcorner {cs'(W)} \urcorner}-r'(W)}.$$ Moreover, $c$ is a jumping number of $D'$ if and only if there are $V\in{{\mathcal G}}'$ and $m\in{{\mathbb N}}$ such that $c=\frac{r'(V)+m}{s'(V)}$ and such that $$\bigcap_{V\subset W\in{{\mathcal G}}'} I_W^{\ {\ulcorner {cs'(W)} \urcorner}-r'(W)}\not\subset I_V^{m+1}.$$
The following lemma will allow us to bound the degrees of the polynomials at which we need to look to detect a jump of multiplier ideals. We have conjectured the statement, proved some cases, and M. Saito proved it in general.
\[lema linear forms\] For $1\le
i\le s$, let $I_i\subset {\bf{C}}[x_1,\ldots,x_n]$ be ideals generated by linear forms. Suppose $I_1^{a_1}\cap \ldots\cap
I_s^{a_s}\not\subset I_1^{a_1+1}$ for some positive integers $a_i$. Then there exists $f$ in $I_1^{a_1}\cap \ldots\cap I_s^{a_s}$ but not in $I_1^{a_1+1}$ of degree at most $a_1+\ldots +a_s$.
The following short and elementary proof of this lemma is due M. Saito who kindly allowed us to reproduce it here. After a change of coordinates, we can assume that $I_1=(x_1,\ldots ,x_m)$ for some $m\le n$. After reordering of indices, we can assume that there is $r\in\{1,\ldots ,s\}$ such that $I_i\subset I_1$ for $1\le i\le r$ and $I_i\not\subset I_1$ for $r<i\le s$. Let $J_i=I_i\cap{{\mathbb C}}[x_1,\ldots ,x_m]$. Then $$\bigcap_{1\le i\le r}I_i^{a_i}=\bigcap_{1\le i\le r}J_i^{a_i}\;\cdot\;{{\mathbb C}}[x_1,\ldots ,x_n].$$ Since $\cap_{1\le i\le r}I_i^{a_i}\not\subset I_1^{a_1+1}$, we have that $\cap_{1\le i\le r}J_i^{a_i}\not\subset J_1^{a_1+1}$. The ideals $J_i$ are homogeneous. Hence we can find a homogeneous polynomial $u$ in $\cap_{1\le i\le r}J_i^{a_i}$ which does not belong to $J_1^{a_1+1}=(x_1,\ldots ,x_m)^{a_1+1}$. Then the degree of $u$ must be $a_1$. For $r<i\le s$, take $v_i\in I_i$ but $\not\in I_1$ to be a linear form. Let $f=u\prod_{r<i\le s}v_i^{a_i}$. Then $f\in\cap_{1\le i\le s}I_i^{a_i}$, but $f\not\in I_1^{a_1+1}$, and the degree of $f$ is $a_1+a_{r+1}+\ldots +a_s$.
Let $a_0'=\sum_{W\subset{{\mathcal G}}'}\max \{0,s'(W)-r'(W)\}$. By Proposition \[prop characterization of jump. nos.\] and Lemma \[lema linear forms\], we have:
\[cor. affine case\] If $D'$ is an effective divisor supported on a central hyperplane arrangement in ${{\mathbb C}}^{n-1}$, then $c\in (0,1)$ is a jumping number of $D'$ if and only if there exists $f\in{{\mathbb C}}[x_1,\ldots ,x_{n-1}]$ of degree at most $a_0'$ with $f\in{{\mathcal J}}((c-\epsilon)D')$ for $0<\epsilon\ll 1$, but $f\not\in {{\mathcal J}}(cD')$.
[**Projective case.**]{} Let ${{\mathcal A}}$ be a central hyperplane arrangement in ${{\mathbb C}}^n$. Denote the intersection lattice of ${{\mathcal A}}$ by $L({{\mathcal A}})$. We consider the corresponding arrangement of projective hyperplanes in $Y={{\mathbb P}}^{n-1}$ given by ${{\mathbb P}}(V)$ for $V\in {{\mathcal A}}$. Let $D$ be an effective divisor on $Y$ supported on $\cup_{V\in {{\mathcal A}}}{{\mathbb P}}(V)$. Assume that the support of $D$ is the compactification of a central hyperplane arrangement in some ${{\mathbb C}}^{n-1}\subset Y$. Let ${{\mathcal G}}'\subset L({{\mathcal A}})-\{{{\mathbb C}}^n\}$ be a building set and let ${{\mathcal G}}={{\mathcal G}}'\cup\{0\}$. For $c$ a positive real number, let ${{\mathcal J}}(c D)$ be the multiplier ideal of $cD$ in $Y$. Let ${{\mathcal G}}(c D)={{\mathcal J}}((c-\epsilon) D)/{{\mathcal J}}(c D)$ for $0<\epsilon\ll 1$. Thus $c$ is a jumping number of $D$ if and only if ${{\mathcal G}}(cD)\ne 0$. Recall that we defined in the introduction, for $V\in {{\mathcal G}}-\{0\}$, the numbers $r(V)$ and $s(V)$. Let $a_0$ be defined as in the introduction. By Corollary \[cor. affine case\], we have:
\[cor. reduction to global invariants\] For all $c\in (0,1)$, $${{\mathcal G}}(c D)\ne 0 \Leftrightarrow H^0(Y,{{\mathcal O}}_Y(a_0)\otimes{{\mathcal G}}(c D))\ne 0.$$
[Intersection theory on canonical log resolutions.]{}
[**The canonical log resolution.**]{} Let ${{\mathcal A}}$ be a central hyperplane arrangement in ${{\mathbb C}}^n$. We consider the corresponding arrangement of projective hyperplanes in $Y={{\mathbb P}}^{n-1}$ given by ${{\mathbb P}}(V)$ for $V\in {{\mathcal A}}$. Let $D$ be an effective divisor on $Y$ supported on $\cup_{V\in {{\mathcal A}}}{{\mathbb P}}(V)$. We assume also that the support of $D$ is the compactification of a central hyperplane arrangement in some ${{\mathbb C}}^{n-1}\subset Y$. Let ${{\mathcal G}}'\subset L({{\mathcal A}})-\{{{\mathbb C}}^n\}$ be a building set. Let ${{\mathcal G}}={{\mathcal G}}'\cup\{0\}$. For example, ${{\mathcal G}}=L({{\mathcal A}})\cup\{0\}-\{{{\mathbb C}}^n\}$.
We consider the canonical log resolution $\rho:{\widetilde}{Y}{\rightarrow}Y$ of $D$ obtained from succesive blowing ups of the (disjoint) unions of (the proper transforms) of ${{\mathbb P}}(V)$ for $V\in {{\mathcal G}}-\{0\}$ of same dimension. This is the so-called wonderful model of [@DP]- section 4. More precisely, $\rho$ and ${\widetilde}{Y}$ are constructed as follows.
The following notation is taken from [@MS]- section 2. Let $Y_0=Y$. Let $C_0$ be ${{\mathbb P}}(V)$ for $V\in{{\mathcal G}}-\{0\}$ with $\delta (V)=1$ (there is at most one such $V$, by assumption). Let $\rho_0:Y_1{\rightarrow}Y_0$ be the blow up of $C_0$. Then $\rho_i$ and $Y_{i+1}$ are constructed inductively as follows. Let $C_i\subset Y_{i}$ be the disjoint union of the proper transforms, under the map $\rho_{i-1}$, of ${{\mathbb P}}(V)$ for $V\in {{\mathcal G}}-\{0\}$ with $\delta(V)=i+1$. Let $\rho_i:Y_{i+1}{\rightarrow}Y_i$ for $0\le i <n-2$ be the blow up of $C_i$. Define ${\widetilde}{Y}=Y_{n-2}$ and $\rho$ as the composition of the $\rho_i$.
We need some more notation, also from [@MS]- section 2. Let $C_{V,0}={{\mathbb P}}(V)\subset Y_0$. For $V\in {{\mathcal G}}-\{0\}$ with $\delta(V)=i+1$, $C_{V,j}$ denotes the proper transform of $C_{V,0}$ in $Y_j$ for $1\le j\le i$. Let $E_{V,i+1}$ be the exceptional divisor in $Y_{i+1}$ corresponding to $C_{V,i}$. Let $E_{V,j}$ be the proper transform of $E_{V,i+1}$ in $Y_j$ for $i+1<j\le n-2$. On ${\widetilde}{Y}$, let $E_V=E_{V,n-2}$ if $\delta (V)<n-1$, and $E_V=C_{V,n-2}$ if $\delta (V)=n-1$. Also let $E_{0,i}$ ($0\le i\le n-2)$, and $E_0$, denote the proper transform in $Y_i$, respectively in ${\widetilde}{Y}$, of a general hyperplane of $Y={{\mathbb P}}^{n-1}$. Denote by $[E_V]$ the cohomology class of $E_V$ on ${\widetilde}{Y}$, where it will be clear from context what coefficients (integral, rational) we are considering.
For any subset ${{\mathcal S}}$ of ${{\mathcal G}}-\{0\}$, set $E^{{{\mathcal S}}}=\cup_{V\in
{{\mathcal S}}}E_V$ and $E_{{{\mathcal S}}}=\cap_{V\in{{\mathcal S}}}E_V$. For a rational number $c$, recall the definitions of ${{\mathcal S}}_c$, $a_0$, and $a_V(c)$ from the introduction. Also define $a_V'(c)$ to equal $a_V(c)$ for $c\ne 0$ and, otherwise, $a_V'(0)=a_0$.
\[lemma reduction to euler char\] With the notation as above, $$H^0(Y,{{\mathcal O}}_Y(a_0)\otimes{{\mathcal G}}(c D))=\chi\left ({{\mathcal O}}_{E^{{{\mathcal S}}_c}}\left (\sum_{V\in{{\mathcal G}}}a_V'(c)E_V\right )\right ).$$
We have that $K_{{\widetilde}{Y}/Y}=\sum_{V\in
{{\mathcal G}}-\{0\}}(r(V)-1)E_V$ and $\rho^*(D)=\sum_{V\in
{{\mathcal G}}-\{0\}}s(V)E_V$ ([@Te]-Lemma 2.1). Then, from the definition of multiplier ideals and Theorem \[thm local vanishing\], we have $$\begin{aligned}
{{\mathcal G}}(cD) &=\rho_*({{\mathcal O}}_{E^{{{\mathcal S}}_c}}(\sum_{V\in{{\mathcal G}}-\{0\}}a_V(c)E_V)),\text{ and}\\
0 &=R^i\rho_*({{\mathcal O}}_{E^{{{\mathcal S}}_c}}(\sum_{V\in{{\mathcal G}}-\{0\}}a_V(c)E_V)) \text{ for }i>0.\end{aligned}$$ We can rewrite ${{\mathcal O}}_Y(a_0)$ as $\omega_Y\otimes{{\mathcal O}}_Y(a_0+n)$. By definition, $a_0+n>d$. Hence Theorem \[thm Nadel vanishing\] applies and we have $$\begin{aligned}
H^0(Y,{{\mathcal O}}_Y(a_0)\otimes{{\mathcal G}}(c D)) &=\chi\left ({{\mathcal O}}_{{\widetilde}{Y}}(a_0E_0)\otimes{{\mathcal O}}_{E^{{{\mathcal S}}_c}}(\sum_{V\in{{\mathcal G}}-\{0\}}a_V(c)E_V)\right )\\
&= \chi\left ({{\mathcal O}}_{E^{{{\mathcal S}}_c}}\left (\sum_{V\in{{\mathcal G}}}a_V'(c)E_V\right
)\right ).\end{aligned}$$
\[lema after mayer-vietoris\] With the notation as in Lemma \[lemma reduction to euler char\], a rational number $c\in (0,1)$ is a jumping number of $D$ if and only if $$\mathop{\sum_{\emptyset \ne{{\mathcal S}}\subset{{\mathcal S}}_c}}_{\text{nested}}(-1)^{|{{\mathcal S}}|+1}\chi\left ({{\mathcal O}}_{E_{{{\mathcal S}}}}\left (\sum_{V\in{{\mathcal G}}}a_V'(c)E_V\right ) \right
)\ne 0.$$
Follows from Lemma \[lemma reduction to euler char\] and Corollary \[cor. reduction to global invariants\] via the Mayer-Vietoris exact sequence $$0{\rightarrow}{{\mathcal O}}_{E^{{{\mathcal S}}_c}}{\rightarrow}\mathop{\bigoplus_{{{\mathcal S}}\subset{{\mathcal S}}_c}}_{
|{{\mathcal S}}|=1}{{\mathcal O}}_{E_{{\mathcal S}}}{\rightarrow}\mathop{\bigoplus_{{{\mathcal S}}\subset{{\mathcal S}}_c}}_{
|{{\mathcal S}}|=2}{{\mathcal O}}_{E_{{\mathcal S}}}{\rightarrow}\ldots{\rightarrow}{{\mathcal O}}_{E_{{{\mathcal S}}_c}}{\rightarrow}0.$$ The intersection $E_{{\mathcal S}}$ is nonempty if and only if ${{\mathcal S}}$ is nested ([@MS]-2.7, [@DP]-4.2).
Next goal is to compute $\chi\left ({{\mathcal O}}_{E_{{{\mathcal S}}}}\left
(\sum_{V\in{{\mathcal G}}}a_V'(c)E_V\right )\right )$ via Hirzebruch-Riemann-Roch.
\[rem. isom\] Let $I\subset{{\mathbb Z}}[c_V]_{V\in{{\mathcal G}}}$ be the ideal of [@DP]-5.2 described in the introduction. By loc. cit. there is an isomorphism $$\begin{aligned}
\label{eq. isom cohomology}
{{\mathbb Z}}[c_V]_{V\in{{\mathcal G}}} / I\ & \mathop{\longrightarrow}^{\sim}\
H^*({\widetilde}{Y},{{\mathbb Z}})\ \mathop{\longleftarrow}^{\sim}\
{{\mathbb Z}}[[c_V]]_{V\in{{\mathcal G}}} / I \\
\notag & 1\mapsto [{\widetilde}{Y}],\\
\notag & c_V \mapsto [E_V]\ \ \ \ \text{ if }V\ne 0,\\
\notag & c_0 \mapsto -[E_0].\end{aligned}$$ Indeed, this follows from [@DP]-5.2 Theorem, [@DP]-4.1 Theorem, part (2), and [@DP]-4.2 Theorem, part (4). The only case left out by [@DP]-4.2 Theorem, part (4) is the one corresponding with $E_0$ in our notation. But this follows from the fact that, in their notation, the linear equivalence class of $D_{V^*}$ restricted to $D_{V^*}$ is the negative of the class of the proper transform in $D_{V^*}$ of a general hyperplane in the exceptional divisor of the blowup of the origin of $V$. The objects $V$ and $D_{V^*}$ of [@DP] correspond to ${{\mathbb C}}^n$ and, respectively, ${\widetilde}{Y}$, in our notation. The exceptional divisor of the blowup of the origin of $V$ is, in our notation, ${{\mathbb P}}^{n-1}$, the ambient space of our projective arrangement of hyperplanes.
\[lem Computation of chi E\_S\] With the notation as in Theorem \[thm. jumping numbers\], let $\emptyset\ne{{\mathcal S}}$ be a nested subset of ${{\mathcal G}}-\{0\}$, and $c$ a rational number. Then
$$\begin{aligned}
\chi\left ({{\mathcal O}}_{E_{{{\mathcal S}}}} \left
(\sum_{V\in{{\mathcal G}}}a_V'(c)E_V\right )\right ) &=\\
= \sum_{0\le j}^{n-1-|{{\mathcal S}}|} \frac{1}{(n-1-|{{\mathcal S}}|-j)!} & \left (
\sum_{V\in{{\mathcal G}}} a_V(c)c_V \right )^{n-1-|{{\mathcal S}}|-j} T_j^{{\mathcal S}}\prod_{V\in{{\mathcal S}}} c_V,\end{aligned}$$
where the right-hand side is viewed as an intersection number via the isomorphism (\[eq. isom cohomology\]).
[**Proof of Theorem \[thm. jumping numbers\].**]{} It follows from Lemma \[lema after mayer-vietoris\] and Lemma \[lem Computation of chi E\_S\]. $\Box$
Before we prove Lemma \[lem Computation of chi E\_S\], we need some preliminary results.
Write $Y={{\mathbb P}}({{\mathbb C}}^n)$ and ${\widetilde}{Y}={{\mathbb P}}({{\mathbb C}}^n)^{{\mathcal G}}$. This notation makes sense if one replaces ${{\mathbb C}}^n$ and ${{\mathcal G}}$ by any vector space with a finite set of proper vector subspaces which is closed under intersections and contains $\{0\}$. For a nested subset ${{\mathcal S}}\subset
{{\mathcal G}}-\{0\}$ and $V\in{{\mathcal S}}\cup \{{{\mathbb C}}^n\}$, let $V_{{\mathcal S}}$ be as in introduction. Define ${{\mathbb C}}^{{\mathcal S}}_V=V/V_{{\mathcal S}}$ and set $${{\mathcal G}}^{{\mathcal S}}_V=\{\ W'\subset{{\mathbb C}}^{{\mathcal S}}_V\ |\ W' \text{ is the image of }W\text{ in }{{\mathbb C}}^{{\mathcal S}}_V\text{ for some }W\in{{\mathcal G}}, W\subsetneq V \}.$$ We have the following description of $E_{{\mathcal S}}$ ([@MS]-2.7, [@DP]-4.3):
\[prop. decomposition of canonical resolution\] With the notation as above, let ${{\mathcal S}}\subset {{\mathcal G}}-\{0\}$ be a nested subset. Then $$E_{{\mathcal S}}=\prod_{V\in{{\mathcal S}}\cup\{{{\mathbb C}}^n\}}{{\mathbb P}}({{\mathbb C}}^{{\mathcal S}}_V)^{{{\mathcal G}}^{{\mathcal S}}_V}.$$
By [@Fu]- Example 15.2.12, the Todd class of $E_{{\mathcal S}}$ is also a product:
\[lem. todd class of E\_S\] With the notation as in Proposition \[prop. decomposition of canonical resolution\], $$\label{eq. todd class of E_S}
Td(E_{{\mathcal S}})=\prod_{V\in{{\mathcal S}}\cup\{{{\mathbb C}}^n\}}Td({{\mathbb P}}({{\mathbb C}}^{{\mathcal S}}_V)^{{{\mathcal G}}^{{\mathcal S}}_V}).$$ More precisely, $Td(E_{{\mathcal S}})$ is the product of the pullbacks of $Td({{\mathbb P}}({{\mathbb C}}^{{\mathcal S}}_V)^{{{\mathcal G}}^{{\mathcal S}}_V})$ under the projections associated to the decomposition in Proposition \[prop. decomposition of canonical resolution\].
For every $V\in{{\mathcal G}}-\{0\}$ define a formal power series $F_V\in{{\mathbb Z}}[[c_V]]_{V\in{{\mathcal G}}}$ by $$F_V:=(1-\mathop{\sum_{ W\varsubsetneq
V}}_{W\in{{\mathcal G}}}c_W)^{-(n-\delta(V))}(1+c_V)(1-\mathop{\sum_{
W\subset V }}_{W\in{{\mathcal G}}} c_W)^{n-\delta (V)}.$$ Also, set $F_0=(1-c_0)^n$.
\[prop. chern class canonical resolution\] With the notation as above, the total Chern class $c({\widetilde}{Y})$ is the image in $H^*({\widetilde}{Y},{{\mathbb Z}})$ of $\prod_{V\in{{\mathcal G}}}F_V$ under the map (\[eq. isom cohomology\]).
For $V\in {{\mathcal G}}-\{0\}$ with $\delta(V)=i+1$, let $N_{V,i}$ be the normal bundle of $C_{V,i}$ in $Y_i$. Let $$L_{V,i}={{\mathcal O}}_{Y_i} (E_{0,i}-\mathop{\sum _{0\ne W\varsubsetneq
V}}_{W\in{{\mathcal G}}} E_{W,i})).$$ By [@DP]-5.1 (the statement in [*loc. cit.*]{} needs to be adjusted for the projective case as in Remark \[rem. isom\]), $$N_{V,i}\cong {{L_{V,i}}^{\oplus n-1-i}}_{| C_{V,i}}.$$ We want to apply Proposition \[prop. chern classes blow up\]. One of the quantities we need is $$\begin{aligned}
\left [\sum_{0\le k\le n-1-i}\rho_i ^* c_k({L_{V,i}}^{\oplus (
n-1-i)} )\right ]^{-1} &=\rho_i^*c(L_{V,i})^{-(n-1-i)} \\
&=( 1+ [E_{0,i+1}]- \mathop{\sum _{0\ne W\varsubsetneq
V}}_{W\in{{\mathcal G}}} [E_{W,i+1}])^{-(n-1-i)}.\end{aligned}$$ Also, we have $$\begin{aligned}
&\sum_{0\le j\le n-1-i} (1-[E_{V,i+1}])^j \rho_i ^*
c_{n-1-i-j}(L_{V,i}^{\oplus\ n-1-i})=&\\
&=\sum_{0\le j\le n-1-i} (1-[E_{V,i+1}])^j {n-1-i \choose n-1-i-j}
\rho_i^* c_1(L_{V,i})^{n-1-i-j} \\
&= (1-[E_{V,i+1}] + [E_{0,i+1}] - \mathop{\sum _{0\ne
W\varsubsetneq V}}_{W\in{{\mathcal G}}} [E_{W,i+1}])^{n-1-i}\end{aligned}$$ By Proposition \[prop. chern classes blow up\], $$\begin{aligned}
c(Y_{i+1}) = \rho _i ^* c(Y_i)\mathop{\prod _{V\in{{\mathcal G}}}}_{\delta
(V)=i+1}\left \{ ( 1+ [E_{0,i+1}]- \mathop{\sum _{0\ne
W\varsubsetneq V}}_{W\in{{\mathcal G}}}
[E_{W,i+1}])^{-(n-1-i)} (1+[E_{V,i+1}]) \cdot \right. \\
\left. (1-[E_{V,i+1}] + [E_{0,i+1}] - \mathop{\sum _{0\ne
W\varsubsetneq V}}_{W\in{{\mathcal G}}} [E_{W,i+1}])^{n-1-i} \right \}\end{aligned}$$ Since ${\widetilde}{Y}=Y_{n-2}$, the Proposition follows from the last formula.
Let $Q(x)=x/(1-\exp (-x))$. For every $V\in{{\mathcal G}}-\{0\}$ define a formal power series $G_V^{{\mathcal G}}\in{{\mathbb Q}}[[c_V]]_{V\in{{\mathcal G}}}$ by $$G_V^{{\mathcal G}}:=Q(-\mathop{\sum_{ W\varsubsetneq
V}}_{W\in{{\mathcal G}}}c_W)^{-r(V)}Q(c_V)Q(-\mathop{\sum_{ W\subset
V}}_{W\in{{\mathcal G}}} c_W)^{r(V)}.$$ Also, set $G_0^{{\mathcal G}}=Q(-c_0)^n=Q(-c_0)^{r(0)}$. Recall from introduction that the codimension function $r$ depends only ${{\mathcal G}}$, a fact which is suppressed from the notation. Since the Todd class, as the total Chern class, is multiplicative on exact sequences of vector bundles, by Proposition \[prop. chern class canonical resolution\] we have:
\[cor. todd class canonical resolutions\] With the notation as in Proposition \[prop. chern class canonical resolution\], the Todd class $Td({\widetilde}{Y})$ is the image in $H^*({\widetilde}{Y},{{\mathbb Q}})$ of $\prod_{V\in{{\mathcal G}}}G_V^{{\mathcal G}}$ under the map induced by (\[eq. isom cohomology\]) after $\otimes_{{\mathbb Z}}{{\mathbb Q}}$.
Replacing, in Corollary \[cor. todd class canonical resolutions\], ${\widetilde}{Y}={{\mathbb P}}({{\mathbb C}}^n)^{{\mathcal G}}$ with ${{\mathbb P}}({{\mathbb C}}_V^{{\mathcal S}})^{{{\mathcal G}}^{{\mathcal S}}_V}$, we obtain:
\[cor. todd class special can res\] With the notation as in Proposition \[prop. decomposition of canonical resolution\] and Corollary \[cor. todd class canonical resolutions\], let ${{\mathcal S}}\subset{{\mathcal G}}-\{0\}$ be a nested subset and let $V\in{{\mathcal S}}\cup\{{{\mathbb C}}^n\}$. The Todd class $Td({{\mathbb P}}({{\mathbb C}}_V^{{\mathcal S}})^{{{\mathcal G}}^{{\mathcal S}}_V})$ is the image in $H^*({{\mathbb P}}({{\mathbb C}}_V^{{\mathcal S}})^{{{\mathcal G}}^{{\mathcal S}}_V},{{\mathbb Q}})$ of $$\prod_{W'\in\;{{\mathcal G}}^{{\mathcal S}}_V}G_{W'}^{{{\mathcal G}}_V^{{\mathcal S}}}\ \ \ \in{{\mathbb Q}}[[c_{W''}]]_{W''\in {{\mathcal G}}_V^{{\mathcal S}}}$$ under the map $c_{W''}\mapsto [E_{W''}]$ ($W''\ne 0$) and $c_0\mapsto -[E_0]$.
Next lemma puts together some computations from [@DP]-4.3, [@MS]-Propositions 2.8 and 2.9:
\[lem. pullback of divisors\] With the notation as in Proposition \[prop. decomposition of canonical resolution\], let $\emptyset\ne{{\mathcal S}}\subset{{\mathcal G}}-\{0\}$ be a nested subset. For $V\in{{\mathcal S}}\cup \{{{\mathbb C}}^n\}$, let $p_V$ be the projection of $E_{{\mathcal S}}$ onto the factor ${{\mathbb P}}({{\mathbb C}}^{{\mathcal S}}_V)^{{{\mathcal G}}^{{\mathcal S}}_V}$ associated to the decomposition in Proposition \[prop. decomposition of canonical resolution\]. Let $W'\in{{\mathcal G}}^{{\mathcal S}}_V$ with corresponding divisor $E_{W'}'$ in ${{\mathbb P}}({{\mathbb C}}_V^{{\mathcal S}})^{{{\mathcal G}}^{{\mathcal S}}_V}$.
\(a) If $W'\ne 0$, then $p_V^* E_{W'}' \sim E_W\,_{|E_{{\mathcal S}}}$, where $W$ is the unique element of ${{\mathcal G}}$ nested between $V$ and $V_{{\mathcal S}}$ whose image in ${{\mathbb C}}^{{\mathcal S}}_V=V/V_{{\mathcal S}}$ is $W'$.
\(b) If $W'=0$, then $$p_V^* E_0' \sim \left (E_0 - \mathop{\sum_{0\ne W\subset V_{{\mathcal S}}, W\in{{\mathcal G}}}}_{\{W\}\cup S\text{ nested}} E_W\right )_{|E_{{\mathcal S}}}.$$
For ${{\mathcal S}}$ having only one element, this is [@MS]- Proposition 2.8. For the rest, one iterates as in [@MS]- Proposition 2.9 or, equivalently, as in the last paragraph of the proof of the theorem of [@DP]-4.3.
\[prop. todd class of E\_S\] With the notation as in Proposition \[prop. decomposition of canonical resolution\] and Definition \[def. poly P\], $Td(E_{{\mathcal S}})$ is the image of the formal power series $$T^{{\mathcal S}}:=\prod_{V\in{{\mathcal S}}\cup\{{{\mathbb C}}^n\}}\ \ \mathop{\prod_{V_{{\mathcal S}}\subset
W\varsubsetneq V}}_{W\in{{\mathcal G}}} P_W^{{{\mathcal S}}, V}\ \ \ \ \ \in
{{\mathbb Q}}[[c_W]]_{W\in{{\mathcal G}}}$$ in $H^*(E_{{\mathcal S}},{{\mathbb Z}})$ under the map $l_{{\mathcal S}}: 1\mapsto
[{\widetilde}{Y}]_{|E_{{\mathcal S}}}, c_{W}\mapsto [E_{W}]_{|\, E_{{\mathcal S}}}$ ($W\ne 0$), and $c_0\mapsto -[E_0]_{|\,E_{{\mathcal S}}}$.
For $V\in{{\mathcal S}}\cup \{{{\mathbb C}}^n\}$, let $p_V$ be the projection of $E_{{\mathcal S}}$ onto the factor ${{\mathbb P}}({{\mathbb C}}^{{\mathcal S}}_V)^{{{\mathcal G}}^{{\mathcal S}}_V}$ associated to the decomposition in Proposition \[prop. decomposition of canonical resolution\]. Define a map of ${{\mathbb Q}}$-algebras $$\begin{aligned}
p_V^* : {{\mathbb Q}}[[c_{W'}]]_{W'\in{{\mathcal G}}_V^{{\mathcal S}}} \longrightarrow
{{\mathbb Q}}[[c_W]]_{W\in{{\mathcal G}}},\end{aligned}$$ as follows. For $W'\ne 0$, let $c_{W'} \mapsto c_W$, where $W=\pi^{-1}(W')$ and $\pi: V \twoheadrightarrow {{\mathbb C}}_V^{{\mathcal S}}$. For $W'=0$, let $$c_0 \mapsto \mathop{\sum_{ W\subset V_{{\mathcal S}}, W\in{{\mathcal G}}}}_{
\{W\}\cup S \text{ nested} } c_W.$$ By Lemma \[lem. pullback of divisors\], we have a commutative diagram of ${{\mathbb Q}}$-algebras
$$\xymatrix{
{{\mathbb Q}}[[c_{W'}]]_{W'\in{{\mathcal G}}_V^{{\mathcal S}}} \ar[r]^{p_V^*} \ar[d] & {{\mathbb Q}}[[c_W]]_{W\in{{\mathcal G}}} \ar[d]^{l_{{\mathcal S}}} \\
H^*({{\mathbb P}}({{\mathbb C}}^{{\mathcal S}}_V)^{{{\mathcal G}}_V ^{{\mathcal S}}},{{\mathbb Q}}) \ar[r]^{\ p_V^*}&
H^*(E_{{\mathcal S}},{{\mathbb Q}}).}$$
For $W'\in {{\mathcal G}}_V^{{\mathcal S}}$, denote by $\bar{W'}$ the subspace $\pi^{-1}(W')$ of $V$, where $\pi:V \twoheadrightarrow {{\mathbb C}}^{{\mathcal S}}_V$. Then $p_V^*G_{W'}^{{{\mathcal G}}^{{\mathcal S}}_V}=P_{\bar{W'}}^{{{\mathcal S}},V}$. Then the Proposition follows from Lemma \[lem. todd class of E\_S\] and Corollary \[cor. todd class special can res\].
[**Proof of Lemma \[lem Computation of chi E\_S\].**]{} Let ${{\mathcal E}}$ be the invertible sheaf ${{\mathcal O}}_{E_{{{\mathcal S}}}}
(\sum_{V\in{{\mathcal G}}}a_V'(c)E_V )$. By definition, for $i\ge 0$, $ch_i({{\mathcal E}})=(1/i!)(\sum_{V\in{{\mathcal G}}} a_V'(c)[E_V]_{|\,E_{{\mathcal S}}} )^i$. Theorem \[thm HRR\] allows us to write $$\begin{aligned}
\chi(E_{{\mathcal S}},{{\mathcal E}}) = \sum_{i+j=n-1-|{{\mathcal S}}|}\frac{1}{i!}\left (\sum_{V\in{{\mathcal G}}} a_V'(c)[E_V]_{|\,E_{{\mathcal S}}}
\right )^i\cdot Td_j(E_{{\mathcal S}}).\end{aligned}$$ The Lemma follows from the map (\[eq. isom cohomology\]) and Proposition \[prop. todd class of E\_S\]. Remark that the map $l_{{\mathcal S}}$ from Proposition \[prop. todd class of E\_S\], factors on homogenous polynomials of degree $n-1-|{{\mathcal S}}|$ via: multiplication of the map (\[eq. isom cohomology\]) with $\prod_{V\in{{\mathcal S}}}c_V$.
$\Box$
[**Proof of Theorem \[thm. inner jumping numbers\].**]{} By [@Bu]- Proposition 2.7 (ii), $$n_{c,x}(D)=\chi({\widetilde}{Y}, {{\mathcal O}}_{E^{{{\mathcal S}}_{c,x}}}(K_{{\widetilde}{Y}/Y}-{\llcorner {(c-\epsilon)\rho^*
D} \lrcorner})),$$ for $0<\epsilon \ll 1$, where ${{\mathcal S}}_{c,x}$ is empty unless $cd\in{{\mathbb Z}}$ and there exists a divisor on ${\widetilde}{Y}$ mapping onto $\{x\}$. In the later case, ${{\mathcal S}}_{c,x}=\{V_x\}$. Thus $$n_{c,x}(D)=\chi({{\mathcal O}}_{E^{{{\mathcal S}}_{c,x}}}(\sum_{V\in{{\mathcal G}}-\{0\}}a_V(c)E_V),$$ and the Theorem follows by replacing $a_0$ with $0$ in the proof of Lemma \[lem Computation of chi E\_S\].
$\Box$
[Examples]{}
The following examples illustrate how Theorems \[thm. jumping numbers\] and \[thm. inner jumping numbers\] work.
\(a) Let $D$ be the union of three distinct lines passing through one point in ${{\mathbb P}}^2$. Let ${{\mathcal A}}=\{V_1,V_2,V_3 \}$, $V_i\subset{{\mathbb C}}^3$ mutually distinct subspaces of dimension 2, with $V_1\cap V_2\cap V_3=L$ where $\delta (L)=1$. Then $D={{\mathbb P}}(V_1)+{{\mathbb P}}(V_2)+{{\mathbb P}}(V_3)$ as a divisor in ${{\mathbb P}}({{\mathbb C}}^3)={{\mathbb P}}^2$.
Take ${{\mathcal G}}=\{0,L,V_1,V_2,V_3\}$. By (\[eq, type 2\]), $c_0+c_L+c_{V_i}$ ($i=1,2,3$) belongs to the ideal $I$. We can eliminate thus the variables $c_{V_i}$ ($i=1,2,3$) and have $${{\mathbb Q}}[c_V]_{V\in{{\mathcal G}}}/I\cong
{{\mathbb Q}}[c_0,c_L]/(c_0^3,c_0c_L,(c_0+c_L)^2),$$ and, by (\[eq. isom cohomology\]), this is isomorphic with the cohomology ring of ${\widetilde}{Y}$, the blow up of ${{\mathbb P}}^2$ at ${{\mathbb P}}(L)$.
The only $c\in (0,1)$ for which ${{\mathcal S}}_c\ne \emptyset$ are $c=1/3,
2/3$. For both cases, ${{\mathcal S}}_{c}=\{L\}$; call this set ${{\mathcal S}}$. We have $$T^{{{\mathcal S}}}=P_0^{{{\mathcal S}},L}P_L^{{{\mathcal S}},{{\mathbb C}}^3}\prod_{1\le i\le
3}P_{V_i}^{{{\mathcal S}},{{\mathbb C}}^3}.$$ From the fact that $Q(x)=1+\frac{1}{2}x
+\text{(degree }\ge 2\text{ terms)}$, we get $$T^{{\mathcal S}}=1+(-\frac{3}{2}c_0-c_L)+ \text{(degree }\ge 2\text{ terms)}.$$ It follows by Theorem \[thm. jumping numbers\] that $c=1/3$ is a jumping number for $D$ if and only if $-\frac{5}{2}c_0c_L$ does not lie in the ideal $I\subset{{\mathbb Q}}[c_V]_{V\in{{\mathcal G}}}$. Also, $c=2/3$ is a jumping number if and only if $-\frac{5}{2}c_0c_L+c_L^2$ does not lie in $I$. Therefore $c=\frac{2}{3}$ is the only jumping number of $D$ in the interval $(0,1)$. By Theorem \[thm. inner jumping numbers\], the inner jumping multiplicity at $x={{\mathbb P}}(L)$ of $c=2/3$ is given by writing $-\frac{3}{2}c_0c_L-c_L^2\in{{\mathbb Q}}[c_V]_{V\in{{\mathcal G}}}/I$ in terms of $c_0^2$. Thus $n_{x,\frac{2}{3}}(D)=1$. Also, $n_{x,1}(D)$ is given by $-\frac{3}{2}c_0c_L-2c_L^2$, hence $n_{x,1}(D)=2$. This gives the initial part of the spectrum of $D$ at $x$, and in fact (in this case by symmetry) the whole spectrum is $t^{2/3}+2t+t^{4/3}$.
\(b) Consider the central hyperplane arrangements in ${{\mathbb C}}^3$ given by $$(x^2-y^2)(x+z)(x+2z),$$ $$(x^2-y^2)(x^2-z^2).$$ They are combinatorially equivalent. To apply the algorithm of this article, we consider the completion of these arrangements to ${{\mathbb P}}^3$. Here ${{\mathcal A}}=\{A_i\subset{{\mathbb C}}^4\ |\ i=1,\ldots , 4\}$, and ${{\mathcal G}}=L({{\mathcal A}})-\{{{\mathbb C}}^4\}$ is given by $$\{0,C,B_1,\ldots ,B_6,A_1,\ldots, A_4\},$$ where $C, B_j, A_i$ have codimension $3, 2,$ resp. $1$, $C$ is included in all $B_j$, and $B_j\subset A_i$ if $(i,j)$ lies in $$M:=\{
(1,1),(1,2),(1,3),(2,2),(2,5),(2,6),(3,1),(3,4),(3,6),(4,3),(4,4),(4,5)
\}.$$ The ideal $I$ is generated by $c_{A_i}+\sum_{(i,j)\in
M}c_{B_j}+c_C+c_0$, $c_0c_C$, $c_{B_j}c_{B_{j'}}$ with $j\ne j'$, $c_{B_j}(c_0+c_C)$, and $c_{B_j}^2+c_0^2+c_C^2$. The only nonempty ${{\mathcal S}}_c$ for $c\in (0,1)$ are ${{\mathcal S}}_{1/4}=\{C\}$, ${{\mathcal S}}_{2/4}=\{C,B_1,\ldots,B_6\}$, ${{\mathcal S}}_{3/4}=\{C\}$. Then, modulo $I$, we have $$T^{\{C\}}=-\frac{2}{3}c_0^3+c_C^2+\frac{11}{6}c_0^2+\frac{1}{4}(c_{B_1}+\ldots
+c_{B_6})c_0-\frac{1}{2}(c_{B_1}+\ldots
+c_{B_6})-\frac{3}{2}c_C-2c_0+1,$$ $$T^{\{C,B_j\}}=-\frac{5}{8}c_0^3+\frac{11}{2}c_C^2+\frac{1}{2}c_{B_j}c_0+\frac{7}{4}c_0^2-c_{B_j}-
\frac{3}{2}c_C-2c_0+1,$$ $$T^{\{B_j\}}=-c_0^3+\frac{1}{4}c_C^2+c_{B_j}c_0+\frac{7}{4}c_0^2-c_{B_j}-c_C-2c_0+1.$$ One computes using Theorem \[thm. jumping numbers\] that $1/4$ and $2/4$ are not jumping numbers, but $3/4$ is the only jumping number in $(0,1)$. Using Theorem \[thm. inner jumping numbers\], one computes that the inner jumping multiplicities of $1/4$ and $2/4$ are $0$, whereas the inner jumping multiplicities of $3/4$ and $1$ are $1$, and resp. $3$. By [@Bu], these are the same as the spectrum multiplicities. We used Macaulay 2 for some of the computations above.
The jumping numbers in this case can be computed directly from [@Te] - Lemma 2.1 (see also Lemma \[lemma reduction to euler char\] here) and the result agrees with ours. The spectrum in this case can be computed by [@St] -Theorem 6.1 which treats the case of homogeneous polynomials with 1-dimensional critical locus, and the beginning part agrees with what we have found. Remark that there is a shift by multiplication by $t$ between the definition of spectrum of [@St] and that of [@Bu].
[EMS]{}
N. Budur, On Hodge spectrum and multiplier ideals. Math. Ann. 327 (2003), no. 2, 257–270.
N. Budur, and M. Saito, Multiplier ideals, $V$-filtration, and spectrum. J. Algebraic Geom. 14 (2005), no. 2, 269–282.
C. De Concini, and C. Procesi, Wonderful models of subspace arrangements. Selecta Math. (N.S.) 1 (1995), no. 3, 459–494.
A. Dimca, Singularities and topology of hypersurfaces. Universitext. Springer-Verlag, New York, 1992. xvi+263 pp.
W. Fulton, Intersection Theory. Second edition. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics \[Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics\], 2. Springer-Verlag, Berlin, 1998. xiv+470 pp.
R. Lazarsfeld, Positivity in algebraic geometry. II. Positivity for vector bundles, and multiplier ideals. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics \[Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics\], 49. Springer-Verlag, Berlin, 2004. xviii+385 pp.
M. Mustaţă, Multiplier ideals of hyperplane arrangements. Trans. Amer. Math. Soc. 358 (2006), no. 11, 5015–5023.
M. Saito, Jumping coefficients and spectrum of a hyperplane arrangement. Preprint.
J.H.M. Steenbrink, The spectrum of hypersurface singularities. Actes du Colloque de Théorie de Hodge (Luminy, 1987). Astérisque No. 179-180 (1989), 11, 163–184.
Z. Teitler, A note on Mustaţă’s computation of multiplier ideals of hyperplane arrangements. Proc. Amer. Math. Soc. 136 (2008), 1575-1579.
|
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---
abstract: 'We analyze transmission electron microscopy (TEM) images of self-assembled quasicrystals, composed of binary systems of nanoparticles. We use an automated procedure that identifies the positions of dislocations and determines their topological character. To achieve this we decompose the quasicrystal into its individual density modes, or Fourier components, and identify their topological winding numbers for every dislocation. This procedure associates a Burgers function with each dislocation, from which we extract the components of the Burgers vector after choosing a basis. The Burgers vectors that we see in the experimental images are all of lowest order, containing only 0’s and 1’s as their components. We argue that the density of the different types of Burgers vectors depends on their energetic cost.'
author:
- Liron Korkidi
- Kobi Barkan
- Ron Lifshitz
bibliography:
- 'LironBib.bib'
title: 'Analysis of dislocations in quasicrystals composed of self-assembled nanoparticles'
---
Dislocations in self-assembled soft-matter quasicrystals
========================================================
Self-assembled soft-matter quasicrystals have been observed in recent years in a wide variety of different systems, in all cases but one with dodecagonal (12-fold) point-group symmetry. First discovered by [@zeng04] in liquid crystals made of micelle-forming dendrimers, self-assembled soft-matter quasicrystals have since appeared in other systems such as ABC-star polymers [@hayashida07], in binary systems of nanoparticles [@TalpainNature], in block co-polymer micelles [@18fold], and in mesoporous silica [@xiao12]. These newly-realized systems not only provide exciting platforms for the fundamental study of the physics of quasicrystals [@barkan11], they also hold the promise for new and exciting applications, especially in the field of photonics. An overview of soft matter quasicrystals, including many references relevant to these systems, is given by [@LD07] as well as by Ungar et al. ([-@ungar05], [-@ungar11]) and [@Dotera11; @Dotera12].
Here we concentrate on the systems of nanoparticles studied by [@TalpainNature], consisting typically of two types of particles, such as PbS, Au, Fe$_{2}$O$_{3}$, and Pd, with different diameters. These binary systems of particles, when placed in solution, self-assemble into structures with long-range order, including 12-fold symmetric quasicrystals. The dimensions of the particles—typically a few nanometers in diameter—are such that they can be imaged directly using a transmission electron microscope (TEM). This allows one to study effects that are inaccessible with atomic-scale quasicrystals. Here we present a quantitative analysis of the dislocations that are naturally formed in these quasicrystals as they self-assemble.
In periodic crystals in $d$-dimensions one can usually identify the position of a dislocation rather easily by the termination of a plane of atoms in three dimensions, or a line of atoms in two dimensions. One then chooses a basis for the periodic lattice; encircles each dislocation with a Burgers loop, or a Burgers circuit, of basis vectors; and counts the accumulated difference between the number of steps taken forward and backward in the direction of each of the $d$ basis vectors. The $d$ integers thus obtained define the Burgers vector which encodes the topological character of the dislocation. A similar real-space procedure can be used on a quasiperiodic crystal by overlaying it with a quasiperiodic tiling of rank $D>d$ (for a definition, see below), yielding a $D$-component Burgers vector. A tiling-based analysis of binary systems of nanoparticles was indeed recently carried out by [@TalpainDislocations]. Here we propose an alternative approach for analyzing dislocations in Fourier space that we believe is useful when dealing with aperiodic crystals.
Density modes, winding numbers, and the Burgers function
========================================================
Let us describe the density of nanoparticles in a self-assembled crystal by a function $\rho({\mathbf{r}})$. The Fourier expansion of such a function is given by $$\label{eq:DensityModes}
\rho({\mathbf{r}})=\underset{{\mathbf{k}}\in L}{\sum}\rho({\mathbf{k}})e^{i{\mathbf{k}}\cdot{\mathbf{r}}},$$ where the (reciprocal) lattice $L$ is a finitely generated ${\mathbb{Z}}$-module, which means that it can be expressed as the set of all integral linear combinations of a finite number $D$ of $d$-dimensional wave vectors, ${\mathbf{b}}^{(1)},\ldots,{\mathbf{b}}^{(D)}$. In the special case where the smallest possible $D$, called the *rank* of the crystal, is equal to the physical dimension $d$, the crystal is periodic. More generally, for quasiperiodic crystals $D\geq d$, and we refer to all quasiperiodic crystals that are not periodic as *quasicrystals* [see @lifshitz03; @lifshitz07].
As explained elsewhere [@RonIsraelChem], the topological nature of a dislocation is related to the fact that it cannot be made to disappear by local structural changes. For this to be the case, as one follows a loop around the position of a dislocation and returns to the point of origin, one sees a crystal that is everywhere only-slightly distorted from the perfectly ordered state, except near the core of the dislocation. In particular, the complex amplitudes $\rho({\mathbf{k}})$ of the density modes maintain their magnitudes along the loop, each accumulating at most a phase, which upon return to the point of origin must be an integer multiple of $2\pi$. The collection of all such integers, or so-called *winding numbers*, for a given dislocation defines a linear function ${\mathcal{N_B}}({\mathbf{k}})$ from the lattice $L$ to the set of integers ${\mathbb{Z}}$, which we call the *Burgers function*.
The Burgers function of a given dislocation associates a particular winding number ${\mathcal{N_B}}({\mathbf{k}})$ with every wave vector ${\mathbf{k}}\in L$. Because this function is linear, after choosing a basis $\{{\mathbf{b}}^{(i)}\}$ for the lattice, it is uniquely specified by a set of only $D$ integers $n_i\equiv {\mathcal{N_B}}({\mathbf{b}}^{(i)})$, forming the *Burgers vector* $(n_1,\ldots,n_D)$. Thus, $$\label{eq:BG}
\forall {\mathbf{k}}= \sum_{i=1}^D a_{i} {\mathbf{b}}^{(i)} \in L:\quad {\mathcal{N_B}}({\mathbf{k}})
=\sum_{i=1}^D a_{i} {\mathcal{N_B}}({\mathbf{b}}^{(i)}) = \sum_{i=1}^D a_{i} n_i,$$ where $a_i\in{\mathbb{Z}}$. This implies that in order to fully characterize a dislocation in an experimental image it suffices to isolate the $D$ density modes associated with a chosen basis, and obtain their corresponding winding numbers. This is the basis of the approach presented below for analyzing dislocations [for more detail, see @Gilad; @Freedman06; @Freedman07].
Analysis of the dislocations in a quasicrystal of nanoparticles
===============================================================
![\[fig:The-basis-vectors\] Schematic representation of the three strongest rings in the Fourier transform of our dodecagonal quasicrystal. The inner ring is the strongest, containing the four basis vectors ${\mathbf{b}}^{(1)}\ldots{\mathbf{b}}^{(4)}$. The second strongest ring is the outer one, obtained from all the sums of two adjacent vectors in the inner ring, as indicated by solid arrows. The third strongest ring lies in between, obtained from all sums of vector pairs in the inner ring that are separated by 90 degrees, as indicated by dashed arrows.](fig2){width="40.00000%"}
We begin with a high-resolution TEM image of one of the dodecagonal quasicrystals grown by [@TalapinTEM], a section of which is shown in Fig. \[fig:Procedure\](a). This particular quasicrystal is self-assembled from 11.2nm PbS and 5.2nm Au nanoparticles, and contains a distribution of dislocations that are formed naturally during the self assembly. We Fourier transform the TEM image, to obtain the diffraction image shown in Fig. \[fig:Procedure\](b), and then choose four of the Bragg peaks in the 12-fold ring containing the strongest reflections as a basis ${\mathbf{b}}^{(1)},\ldots,{\mathbf{b}}^{(4)}$ for the reciprocal lattice. These are labeled in the schematic representation of the lattice in Fig. \[fig:The-basis-vectors\].
For each of the four pairs of Bragg peaks, associated with the chosen basis vectors and their negatives, we then carry out the following procedure:
1. We filter out small regions around the two opposite Bragg peaks, as indicated by a pair of red circles in Fig. \[fig:Procedure\](b) for the case of the density mode associated with the wave vectors $\pm{\mathbf{b}}^{(1)}$.
2. We inverse Fourier transform the filtered regions resulting in a real-space image of a single density mode. Dislocations appear as discontinuities in the stripes. We use a routine that identifies all the discontinuities and marks their positions, as shown in Fig. \[fig:Procedure\](c) for this density mode.
3. For each dislocation, a second routine then extracts the $i^{\rm
th}$ component $n_i={\mathcal{N_B}}({\mathbf{b}}^{(i)})$ of the Burgers vector. This is done by enclosing a counter-clockwise loop around its position and calculating the accumulated phase in units of $2\pi$. Practically what we do is count the number of stripes crossed moving in the direction of the wave vector ${\mathbf{b}}^{(i)}$ on one side of the dislocation, and subtract the number of stripes crossed moving against the direction of ${\mathbf{b}}^{(i)}$ while returning on the other side, as demonstrated in Fig. \[fig:dislocations\].
Finally, we verify the correctness of the calculation by extracting the values ${\mathcal{N_B}}({\mathbf{k}})$ for additional wave vectors ${\mathbf{k}}$ and checking that they satisfy the linearity requirement given by Eq. .
Results and discussion
======================
We typically find a density of a few dozen dislocations per $\mu
m^{2}$ in the nanoparticle quasicrystals of @TalapinTEM. All of these dislocations are of lowest order in the sense that $n_{i} = 0$, $1$ or $-1$. To understand the topological nature of these dislocations it is useful to classify them by dividing the four basis vectors into two hexagonal subsets—$\{{\mathbf{b}}^{(1)},{\mathbf{b}}^{(3)}\}$ and $\{{\mathbf{b}}^{(2)},{\mathbf{b}}^{(4)}\}$ (see Fig. \[fig:The-basis-vectors\]). By doing so we find that the density of dislocations with non-zero components in only one of the subsets, which we call *single-subset* dislocations, is five times larger than that of dislocations with non-zero components in both subsets, which we call *dual-subset* dislocations. Examples of the two types of dislocations are shown in Fig. \[fig:dislocations\].
To try and explain these observations, consider the free energy of the self-assembled crystal as an expansion in products of density mode amplitudes $\rho({\mathbf{k}})$ [see @RonIsraelChem], $$\label{eq:free}
{\cal F}\{\rho\} = \sum_{n=2}^\infty \sum_{{\mathbf{k}}_1\ldots{\mathbf{k}}_n}
A({\mathbf{k}}_1,\ldots{\mathbf{k}}_n) \rho({\mathbf{k}}_1)\cdots\rho({\mathbf{k}}_n),$$ where one can show that $A({\mathbf{k}}_1,\ldots{\mathbf{k}}_n)=0$ unless ${\mathbf{k}}_1 +
\ldots + {\mathbf{k}}_n = 0$. We argue that products in the sum that contain high-intensity modes with large winding numbers have a greater contribution to increasing the free energy away from its minimum value in the perfect crystal. Accordingly, high-intensity modes tend to exhibit smaller winding numbers. Indeed, we find that all the winding numbers associated with the two brightest rings (see Fig. \[fig:Procedure\](b) and Fig. \[fig:The-basis-vectors\]) are either 0 or $\pm1$, whereas it is only on the 3$^{\rm rd}$ ring that we begin to see winding numbers that are either 0, $\pm1$, or $\pm2$. Moreover, owing to the linearity of the Burgers function, the fact that the winding numbers on the second brightest ring are at most of magnitude 1 prevents two adjacent peaks from the different subsets in the first ring from having non-zero winding numbers of the same sign. Because the ring of Bragg peaks, obtained by adding pairs of wave vectors separated by 60 degrees, is extremely weak \[see Fig. \[fig:Procedure\](b)\], there is no such constraint on the winding numbers belonging to the same subset of basis vectors. The fact that this constraint applies only to dual-subset dislocations, reduces their possible combinations and overall relative density.
\
A word of caution is in order regarding our approach for analyzing dislocations. Because the density of the dislocations is relatively high the Bragg peaks are not point-like but are rather spread as can be seen in Fig. \[fig:Procedure\](b). This means that some of the information about the dislocations may lie between Bragg peaks and may be lost if the filters are too small. Therefore, our approach is sensitive to the shape and size of the filter that we use around each Bragg peak. As we increase the filter size we obtain more information and potentially find more dislocations, as demonstrated in Fig. \[fig:FilterSize\]. We thus try to optimize the filter by gradually enlarging its size until the number of dislocations stops increasing substantially.
Our approach for analyzing dislocations should be easily adapted to other systems even when the density of the dislocations is quite large, as one may expect for soft matter systems. Moreover, for dynamical systems that can be imaged in real time one can use our automated method to follow and quantitatively analyze the dynamics of the dislocations.
We are very grateful to Dmitri Talapin for providing the TEM images. This research is supported by the Israel Science Foundation through grant No. 556/10.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We present synthetic spectra for circumstellar disks that are heated by radiation from a central brown dwarf. Under the assumption of vertical hydrostatic equilibrium, our models yield scaleheights for brown dwarf disks in excess of three times those derived for classical T Tauri (CTTS) disks. If the near-IR excess emission observed from brown dwarfs is indeed due to circumstellar disks, then the large scaleheights we find could have a significant impact on the optical and near-IR detectability of such systems. Our radiation transfer calculations show that such highly flared disks around brown dwarfs will result in a large fraction of obscured sources due to extinction of direct starlight by the disk over a wide range of sightlines. The obscured fraction for a ’typical’ CTTS is less than 20%. We show that the obscured fraction for brown dwarfs may be double that for CTTS, but this depends on stellar and disk mass. We also comment on possible confusion in identifying brown dwarfs via color-magnitude diagrams: edge-on CTTS display similar colors and magnitudes as a face-on brown dwarf plus disk systems.'
date: Released 2003 Xxxxx XX
title: THE STRUCTURE OF BROWN DWARF CIRCUMSTELLAR DISKS
---
\[firstpage\]
circumstellar matter — infrared: stars — low mass, brown dwarfs — stars: pre-main-sequence
Introduction
============
Accumulating observational evidence indicates the presence of circumstellar disks around brown dwarfs, including near-IR (Oasa et al. 1999; Muench et al. 2001; Liu et al. 2003) and mid-IR excess emission (Comerón et al. 1998; 2000), and H$\alpha$ signatures of accretion (Muzerolle et al. 2000). The existence of circumstellar disks around brown dwarfs is an important discovery, since it may suggest that brown dwarfs form in a similar fashion to more massive T Tauri stars (Shu, Adams, & Lizano 1987). At the same time, data indicating significant masses and extents of circumstellar material may cause problems for brown dwarf formation scenarios where the low mass object is formed and subsequently ejected from multiple systems (Reipurth & Clarke 2001).
The observed spectral energy distributions (SEDs) and IR excess emission of some brown dwarfs have been modelled using flat and flared reprocessing disks (e.g., Natta & Testi 2001; Testi et al. 2002; Liu et al. 2003). The SED models suggest brown dwarf disks are similar to those around CTTS. Models for the SEDs and scattered light images of CTTS require flared disks (e.g., Kenyon & Hartmann 1987; D’Alessio et al. 1999; Whitney & Hartmann 1992; Burrows et al. 1996). For highly inclined flared disks direct starlight is blocked by the optically thick disk resulting in a fraction of sources that will appear very faint in the optical and near-IR. Such faint or “optically obscured” sources may escape detection in magnitude limited surveys. The degree of disk flaring depends on the disk temperature structure and the mass of the central star, with the disk scaleheight $h\propto (T_d/M_\star)^{1/2}$ (Shakara & Sunyaev 1973). Therefore, low mass brown dwarfs may have more vertically extended disks than those around CTTS and, depending on the disk mass, the obscured fractions may be larger. We will show that this may lead to confusion in discriminating between brown dwarfs and edge-on CTTS.
In this paper we adopt the same working hypothesis as Natta & Testi (2001) that brown dwarf disks are in vertical hydrostatic equilibrium with dust and gas well mixed throughout the disk. Such disk models have been very successful in explaining the observed scattered light images and SEDs of CTTS. We extend our Monte Carlo radiative equilibrium code to calculate the structure of passively heated brown dwarf disks in vertical hydrostatic equilibrium. Our Monte Carlo radiation transfer technique naturally includes scattered light and the inclination dependence of the SED, which allows us to investigate the effects of highly flared disks. We construct synthetic spectral energy distributions and colors for disks of different sizes and masses surrounding brown dwarfs of different masses and luminosities. Our model SEDs enable us to determine to what extent observations in various spectral regions can diagnose disk parameters. Deriving disk parameters for large numbers of sources may help to discriminate brown dwarf formation mechanisms and whether they are different for dense and sparse star forming regions.
The layout of the paper is as follows: §2 outlines the ingredients of our models and the radiation transfer/disk density calculation, §3 presents disk structure models derived with our iterative technique, §4 presents our model SEDs and color-color diagrams, §5 compares our models with currently available observations of brown dwarf disks, and we summarize our results in §6.
Model Ingredients
=================
This study implements a number of extensions to the original Monte Carlo radiative equilibrium technique of Bjorkman & Wood (2001). These include a crude estimate of the inner disk radius (assumed to be at the dust destruction radius) and an improved temperature structure calculation. Our disks are not vertically isothermal or two-layered (Natta & Testi, 2001); 2D disk temperature structure is calculated in the MC simulation based on the technique described by Lucy (1999). Our code self-consistently determines the density structure of a passively heated disk in vertical hydrostatic equilibrium. The extensions to the radiation transfer technique are described in greater detail in the Appendix.
Disk Structure Calculation
--------------------------
Model SEDs are computed for a flared disk that is heated by radiation from a central brown dwarf. We only consider passive disks, since disk heating from viscous accretion is negligible compared to stellar heating in low accretion rate systems (Muzerolle et al. 2000; D’Alessio et al. 1999). Our disks extend from the dust destruction radius to an outer radius of 100AU. The disk is truncated sharply at its inner edge and there is no material between the inner edge and the star, equivalent to assuming material in this region is optically thin. In our previous modelling of CTTS SEDs (Wood et al. 2002a, b; Schneider et al. 2003; Grosso et al. 2003) we adopted the following flared disk density structure (e.g., Shakara & Sunyaev 1973) $$\rho=\rho_0 \left ({R_\star\over{\varpi}}\right )^{\alpha}
\exp{ -{1\over 2} [z/h( \varpi )]^2 }
\; ,$$ where $\omega$ is the radial coordinate in the disk midplane and the scaleheight increases with radius, $h=h_0\left ( {\varpi /{R_\star}} \right )^\beta$. With the disk structure fixed we then calculate the temperature structure and emergent SED using the Monte Carlo radiative equilibrium technique of Bjorkman & Wood (2001).
In this paper we adopt an iterative scheme to determine the disk density structure. Having calculated the disk temperature structure via our Monte Carlo radiative equilibrium technique, we impose vertical hydrostatic equilibrium and solve $${{dP}\over{dz}} = -\rho g_z \; .$$ Here, $P=\rho c_s^2$ is the gas pressure, $c_s$ is the isothermal sound speed, and $g_z=\frac{GM_\star z}{\varpi^3}$ is the vertical component of gravity in the disk. We make the usual thin disk assumptions and assume the disk is non self-gravitating (Pringle 1981).We impose the boundary condition that the disk surface density $\Sigma\sim \varpi^{-1}$, in accordance with the detailed disk structure models of D’Alessio et al. (1999). Our simulations begin with the disk structure given by equation 1 with $\alpha=2.25$, $\beta=1.25$, and we iterate to derive a self-consistent vertical density structure. The density converges within three iterations.
In hydrostatic disk models, the disk scaleheight scales with radius as $h/ \varpi=c_s/v_c$, where $c_s^2=kT/\mu m_H$ and $v_c^2=GM_\star/ \varpi$ are the isothermal sound speed and circular velocity at $\varpi$ (e.g., Shakara & Sunyaev 1973; Lynden-Bell & Pringle 1974). For CTTS, $h(100 {\rm AU})$ is in the range 7AU to 20AU as found from radiative and hydrostatic equilibrium models (D’Alessio et al. 1999) and from fitting SEDs and scattered light images of disks using equation 1 with $h_0$ as a free parameter (Burrows et al. 1996; Stapelfeldt et al. 1998; Grosso et al. 2003; Schneider et al. 2003). However, for disks around brown dwarfs the scaleheights may be larger due to the smaller circular velocity of these low mass objects. If brown dwarf disks are indeed more vertically extended, then there may be a larger fraction of obscured brown dwarfs compared with CTTS. Our SED calculations enable us to address this issue.
Dust Parameters and Model Atmospheres
-------------------------------------
The circumstellar dust opacity and scattering properties are taken to be those of the dust size distribution we adopted for modelling the SEDs of HH 30 IRS and GM Aur (Wood et al. 2002a; Schneider et al. 2003; Rice et al. 2003). This dust model has a larger average grain size and a shallower wavelength dependent opacity than ISM dust models (e.g., Mathis, Rumpl, & Nordsieck 1977; Kim, Martin, & Hendry 1994). There is much observational evidence for large grains and a shallow wavelength dependent opacity in T Tauri disks (e.g., Beckwith et al. 1990; Beckwith & Sargent 1991; D’Alessio et al. 2000; Cotera et al. 2001; Wood et al. 2002a). The larger grain dust model we adopt does not exhibit strong silicate features (see Wood et al. 2002a).
The input stellar spectra for the brown dwarf models are the BD\_Dusty model atmospheres presented by Allard et al. (2001), with $\log g = 3.5$ and effective temperatures of $T_\star = 2200$ K, 2600 K and 2800 K. For CTTS models we use a 4000 K Kurucz model atmosphere (Kurucz 1994).
Parameter Space
---------------
We construct radiative and hydrostatic equilibrium models for brown dwarf systems with the range of stellar and circumstellar disk parameters given in Table 1. The stellar mass range of $0.01M_\odot\le M_\star \le 0.08M_\odot$ covers objects from the hydrogen burning limit down to the lower limit for brown dwarfs as identified via color-magnitude diagrams by Muench et al. (2001). The corresponding stellar radii and temperatures yield models representative of 1Myr old systems from the evolutionary tracks of Baraffe et al. (2002). For each set of stellar parameters, disk to star mass ratios of log($M_d/M_\star$)=-1, -2 and -3 are initially considered. The disk mass $M_d$ refers to the total disk mass of dust and gas. As with our previous work, this mass does not include very large particles such as rocks or planetesimals and is therefore a lower limit. We compare our resulting brown dwarf disk structures with those of disks around a typical CTTS with $M_\star = 0.5M_\odot$, $R_\star =
2R_\odot$, and $T_\star = 4000$ K (e.g., Kenyon & Hartmann 1995; D’Alessio et al. 1999).
Disk Structure Models
=====================
At the end of our iterative procedure (described in the Appendix) the outputs of our code are the disk density and temperature structure and the emergent SED. All our brown dwarf models have Toomre parameter, Q, $>$ 1, throughout their disks, so the thin disk assumption implicit in our models is still valid (e.g. D’Alessio et al. 1999). Fig. 1 shows scaleheights and mid-plane temperatures for disks of various masses illuminated by stars of different mass. The full disk structure is now calculated, however we choose to define scaleheight using the mid-plane temperature (see Appendix). For comparison we show the scaleheight and mid-plane temperature for a CTTS illuminating disks of the same mass ratio as in the brown dwarf models. The scaleheights of the CTTS disks are $h(100\,{\rm AU})\sim 15$ AU, in agreement with the simulations of D’Alessio et al. (1999). The brown dwarf disks have scaleheights significantly in excess of those obtained for CTTS, with $h(100\,{\rm AU})$ ranging from just over 20 AU for $M_\star=0.08M_\odot$ to almost 60 AU for $M_\star=0.01M_\odot$.
Our temperature calculations in Fig. 1 for brown dwarf disks show that $T (100\,{\rm AU}) \sim 10$ K with little variation among the models. The stellar mass therefore predominantly controls the disk scaleheights. The brown dwarf models show disk scaleheights up to three times larger than for comparative disks illuminated by a CTTS. Such large scaleheights will result in a large range of viewing angles for which direct starlight will be extincted by the disk. The effects of large scaleheights on the SED and colors are discussed in \$4.
The extended nature of the brown dwarf disks is also clear in Fig 2 which shows K-band scattered light images of disks viewed at an inclination of $85^o$ from face-on. As with CTTS models (Wood et al. 1998), the dust lane narrows with decreasing disk mass and the central source becomes increasingly more visible. Hence it seems that the detection of low mass disks via scattered light may only be possible for edge-on systems or if coronographic techniques are used to block the starlight.
Model Spectra and Colors
========================
In addition to calculating the disk structure, our radiation transfer code outputs the SED for a range of viewing angles. This section shows SEDs and color-color diagrams that illustrate the main features of our models. With the Monte Carlo technique it is straightforward to determine the contributions to the SED of stellar, scattered, and thermally reprocessed photons (Wood et al. 2002a). We utilize this capability to determine the relative importance of the scattered light contribution to the SEDs and colors.
SEDs of Face-On Disks
---------------------
Our brown dwarf model SEDs have similar spectral characteristics to those of CTTS disks (e.g., Wood et al. 2002b). Fig. 3 shows SEDs of face-on disks for a range of star and disk parameters. Face-on covers 0 $\rightarrow$ $18^o$ due to binning of the photons in the Monte Carlo code. The dependence of SED on disk mass is readily evident and, as with CTTS, observations at long wavelengths provide the best diagnostics of disk mass.
As commented by Natta & Testi (2001) it is difficult to produce significant near-IR excesses for brown dwarfs because the stellar spectrum peaks at longer wavelengths than CTTS and can therefore dominate the disk thermal emission. At longer wavelengths however Fig. 3 shows that our brown dwarf models are capable of producing varying degrees of IR excess emission. As stellar mass decreases, scaleheights increase allowing the disk to intercept, scatter, and thermally reprocess more stellar radiation, which in turn gives rise to increasingly large IR excesses.
Fig. 4 shows the relative contribution of stellar, scattered, and thermal disk radiation for the highly flared $M_\star=0.01M_\odot$, $T_\star = 2200$ K brown dwarf disk system with $\log(M_d/M_\star)=-1$ and includes a CTTS model for comparison. Scattered light makes little contribution to face-on models, but it can account for up to 90% of K-band flux as disks become more inclined (see Wood et al. 2002b, Fig. 9). The importance of including scattered light will be highlighted in §4.3.
Recent work on brown dwarf formation suggests that many brown dwarfs are ejected from multiple systems and that any circumstellar disks that survive the ejection will be very small. In the numerical simulations of Bate et al. (2003), no disks survive around ejected brown dwarfs down to their simulation resolution of $\sim 10$ AU. We have computed SEDs for disks of constant mass, but varying $R_d$ in the range 10AU - 200AU. Because $M_d$ was held constant in these models, smaller $R_d$ yields larger optical depths. The SEDs are mostly unaltered as $R_d$ changes apart from some variation at far-IR/sub-mm wavelengths. We conclude that it would be very difficult to determine disk radii from SED data alone and more stringent tests of the small disks prediction of Bate et al. (2003) will require high resolution imaging to resolve the disks via their scattered light and thermal emission (see also, Beckwith et al. 1990; Chiang et al. 2001).
Near-IR Color-Color Diagrams
----------------------------
By far the most popular technique of identifying circumstellar disks is to identify sources with near-IR excess emission in color-color diagrams (e.g., Lada & Adams 1992; Rebull et al 2002). It was through near-IR color-magnitude and color-color diagrams that Meunch et al. (2001) and Liu et al. (2003) identified many candidate brown dwarfs that exhibit the tell-tale IR excess emission indicative of circumstellar disks.
All colors we present are relative to Vega and are computed using 2MASS JHK and UKIRT L filter transparency curves. The BD\_Dusty model atmospheres that we use have near-IR colors that are bluer than observations of the corresponding spectral type (e.g., Bessell & Brett 1988; Kirkpatrick et al. 2000). What is important is the relative color of our models (e.g., $[H-K] - [H-K]_\star$) and the underlying stellar spectrum does not affect this. As we ultimately compare our models with observations, we have used a similar approach to Liu et al. (2003) and applied a color offset to the models so that the model stellar colors match observations. We adopt spectral types of M9.5, M8.5 and M6 for our 2200, 2400 and 2800 K models respectively. There is no well defined temperature scale for M dwarfs and so classifications were chosen on consideration of observations and discussion by Luhman (1999), Pavlenko et al (2000) and Dahn et al (2002). We shift the stellar colors of our models to match the field M dwarf locus taken from Bessell & Brett (1988) and average colors from Kirkpatrick et al. (2000). This results in the following offsets for the M9.5, M8.5 and M6 fits: $\Delta(J-H) = 0.34,0.23,0.10$, $\Delta(H-K) = 0.12,0.06,0.00$. No shift in K-L is applied.
Fig. 5 shows $JHK$ and $JHKL$ color-color diagrams for our model disks viewed face-on, and following the afore mentioned adjustments. In general, excess emission is more readily detected at long wavelengths (e.g. Haisch, Lada, & Lada 2000, Natta & Testi 2001) and this is again seen here with models showing larger excesses at $K-L$ than at $J-H$ or $H-K$. The trend of our models is that the more massive and more flared disks exhibit the largest IR excesses. Inclination effects yield a spread in color-color diagrams and we explore this in the next section.
Inclination, Scattered Light, and Obscured Fractions
----------------------------------------------------
For highly inclined CTTS, direct starlight is blocked by the optically thick disk and such systems will be very faint in the optical and near-IR (e.g., D’Alessio et al. 1999; Wood et al. 2002a, b). Compared to CTTS, the larger disk scaleheights we derive for the brown dwarf models will result in a larger fraction of viewing angles over which the central starlight is blocked by the disk. For the purpose of this study we define an “obscured source” to be one where the near-IR flux is at least three magnitudes fainter than the corresponding face-on source. The obscured fraction therefore depends on the disk size, mass, and scaleheight. For CTTS, the obscured fraction is around 20% (D’Alessio et al. 1999; Wood et al. 2002b) for disks of $M_d\sim 10^{-3}M_\odot$.
Figure 6 shows the inclination effect on the SEDs for various brown dwarf disk models. The SEDs are shown for ten viewing angles evenly spaced in $\cos i$, so that each curve represents 10% of sources by number if we assume sources are randomly distributed in inclination angle. We find obscured fractions from $20\%$ to $60\%$ with highly inclined sources only detected in the near-IR via scattered light and weak thermal emission. The largest obscured fraction occurs for the lowest stellar mass of $M_\star=0.01M_\odot$ with log($M_d/M_\star$)=-1 and $T_\star = 2200$ K. The smallest obscured fraction occurs for the highest stellar mass of $M_\star=0.08M_\odot$ with log($M_d/M_\star$)=-3 and $T_\star=2800$ K.
Studies of the initial mass function in Trapezium, $\rho$ Ophiucus and IC348 show a relatively flat distribution over the range $0.08 \le M_\star(M_\odot) \le 0.04$ and then a sharp fall off below this (Luhman 2000; Muench et al. 2002). If the IMF is flat and the fall-off due to small number statistics then within a young cluster population up to 55% of brown dwarf candidates, as defined by our parameter range, may be obscured. This is an upper limit produced using maximum obscuration fractions for each stellar mass assuming a disk to stellar mass ratio of log($M_d/M_\star$)=-1. For a declining IMF, and a distribution of disk masses, the obscured fraction will be less. Within our parameter range a minimum of $20\%$ of sources are likely to be obscured regardless of stellar mass distribution and assuming disk to stellar mass ratios of log($M_d/M_\star$)=-3.
The relatively low luminosity of brown dwarfs and the increased obscuration due to highly flared disks may present detection problems. At a distance of 150pc (as used in Fig. 6) it would be possible to detect some obscured sources in the K band assuming a sensitivity limit of 16.5 mags. In the absence of high resolution imaging however these sources may be incorrectly identified as low luminosity systems. A three fold increase in distance would be sufficient to make all obscured sources undetectable at this sensitivity limit.
Figure 7 shows the inclination dependence in the brown dwarf $JHKL$ color-color diagrams. Relative colors are plotted for ten inclinations with the change in color at each inclination indicated by an arrow. Similar to the behaviour observed by Kenyon et al. (1993) and Whitney et al. (1997) we see a loop in the color-color plane with inclination. Starting from face-on, the sources generally get redder with increasing inclination and then loop around and end up with edge-on sources being slightly bluer than face-on, but still redder than the intrinsic stellar colors. Edge-on sources are seen almost entirely via scattered light. Note that these are slightly redder than the star because the scattered light, which is relatively blue, suffers extinction and becomes somewhat reddened. This trend is seen in all of the models.
Figure 8 contains data for the same model as in Fig. 7, but also shows the change in color with inclination if scattered light is ignored. The removal of scattered light makes the colors much redder, with the effect being particularly significant at moderate to high inclinations. This emphasizes the importance of including scattering when creating and studying models of such systems.
CTTS/Brown Dwarf Confusion
--------------------------
When only unresolved photometry is available our models show that edge-on CTTS could be mistaken as brown dwarfs. CTTS have edge-on flux levels that are comparable to face-on brown dwarfs and similar colors. Muench et al. (2001) identified sources within the Trapezium cluster with $13.5\la H\la 17.5$ as candidate brown dwarfs. They note that 21 of their 109 brown dwarf candidates are coincident with optically resolved proplyds (Bally, O’Dell, & McCaughrean 2000; O’Dell & Wong 1996) and 21% of the candidates that exhibit IR excess, indicative of circumstellar disks, are represented by these proplyds. In the absence of high resolution imaging the task of identifying faint sources such as brown dwarfs may be problematic. If no central star is seen then these sources could be edge-on CTTS that happen to have the same magnitude and colors as a pole-on brown dwarf and disk. This confusion could lead to an overestimation of brown dwarf numbers.
Comparison to Observations
==========================
This section compares our synthetic models and published observations of suspected brown dwarf disks. For the Chameleon cluster the SED data is taken from Comerón et al (2000) and Apai et al (2002); $\rho$ Ophiucus data comes from Barsony et al. (1997), Comerón et al (1998), Bontemps et al (2001) and Natta et al (2002). JHKL data is taken from the above papers along with Kirkpatrick et al (2000) and Liu et al.(2003). All near-IR photometry has been converted to the 2MASS system (Carpenter, 2001).
Spectral Energy Distributions
-----------------------------
Figure 9 shows flared disk model fits to the SED data for candidate brown dwarfs in the $\rho$ Ophiucus and Chameleon star clusters. Table 2. contains details of the model parameters used to produce the fits. We used stellar parameters from Natta & Testi (2001) and Natta et al (2002) as starting points for each of our models.
Natta & Testi (2001) modelled Cha H$\alpha$1, 2, & 9 using a flared disk model. Their models produced a successful fit in the MIR region of the spectrum and predicted a strong 10$\mu$m silicate emission feature. Apai et al (2002) later made observations of Cha H$\alpha$2 at 9.8 and 11.9 $\mu$m and did not detect the silicate feature. They presented an optically thick flat disk model which produced no silicate feature. The SEDs of $\rho$ Oph sources have been modelled by Natta et al. (2002) and they found indications that as many as eight of these stars may have flat disks.
Our models open up the possibility that the absence of a silicate feature may be explained with larger circumstellar dust grains. In addition, low mass disks may fit SEDs previously modelled with flat disks. As Fig. 9 demonstrates, it is possible to fit the observed data for all sources with a flared disk geometry. The use of larger grains naturally suppresses the silicate feature which has been shown to be missing from the Chameleon data and low mass disks of $10^{-5}M_\odot$ and $10^{-7}M_\odot$ allow us to fit the IR data of the candidates where flat disks were previously suspected. We note that many of these fits have disk to stellar mass ratios outside the typical range of $-1 \le log(M_d/M_\star) \le -3$ (Natta et al. 2000; Klein et al. 2003) and flat disks (Natta et al. 2002) remain a possibility.
Another alternative, testable with long wavelength observations, is that steeper surface density profiles can also be used to fit the data with higher mass disks. In Fig. 10, ISO\#030 has been modelled using both surface density $\Sigma\sim \varpi^{-1}$ and $\Sigma\sim \varpi^{-2}$. Using $\Sigma\sim \varpi^{-2}$ allows us to fit the data with a disk eight times more massive disk than used in the $\Sigma\sim \varpi^{-1}$ case. Both models fit the data well in the NIR/MIR, but are quite different in the FIR. Long wavelength observations would help to discriminate between flat disk, low mass flared disks and steeper surface density disk models.
If lower mass flared models are representative of disks in brown dwarf populations, as opposed to higher mass disks, then problems with obscuration may not be as significant as suggested in §4.3. Equally flat disks do not result in severe obscuration of the central star unless at very high inclinations.
Figure 11 shows the derived scaleheights for the disks that we used to model the observed SEDs of Fig. 9 and Fig. 10. This illustrates the range of disk structures that can produce fits to the observed data. In each plot the scaleheight of a model CTTS of corresponding disk to stellar mass ratio is presented as a comparison. For these models we find scaleheights up to three times that of the corresponding CTTS.
JHKL colors
-----------
Figure 12 shows $JHKL$ plots of our face-on models and published data. Following the adjustments discussed in §4.2, Fig. 11 shows that our models (if reddening were included) can reproduce the observed spread in colors of suspected brown dwarf disk systems. Including all inclinations (Fig. 13) allows for the redder colors of inclined disks and produces a spread in the $JHKL$ plots that is in very good agreement with the observed colors.
Figure 13 also shows the CTTS locus taken from Meyer et al. (1997). This again demonstrates that there is an overlap between CTTS and brown dwarf colors which may lead to incorrect identification of sources if only color-magnitude data is available.
Summary
=======
We have presented model SEDs and color-color diagrams for brown dwarf disks. The main assumptions in our models are that the disks are in vertical hydrostatic equilibrium with dust and gas well mixed throughout. Our models are self-consistent and employ an iterative procedure to determine the hydrostatic density structure for passively heated disks. Compared to CTTS, brown dwarf disks have larger scaleheights due to the lower mass of the central star. In some cases the scaleheights of brown dwarf disks are more than three times larger than for the same disk to stellar mass ratio for a CTTS. The larger scaleheights result in more inclinations over which the direct stellar radiation is blocked or obscured by the flared disk. The fraction of optically obscured systems depends on the stellar mass and disk optical depth and in our models is in the range $20\%\le f_{\rm obs}\le 60\%$. For a typical CTTS about 20% of sources will be optically obscured.
If, as our models suggest, brown dwarf disks are highly flared, detection of brown dwarf disk systems will be biased towards face-on systems. We also show that without direct imaging or spectroscopic identification, it will be difficult to distinguish between edge-on CTTS and face-on brown dwarfs. Color-color diagrams show that edge-on sources, which are only detected in the optical/near-IR via scattered light, have similar colors to face-on sources. This may lead to incorrect identification of sources. In particular, we find that an edge-on CTTS will have similar near-IR magnitudes and colors as face-on brown dwarf disk systems.
We compare our synthetic models to SED and color-color observations of suspected brown dwarfs and show that flared disks of varying mass can account for the observed SEDs and colors. Our adopted circumstellar dust model naturally suppresses the 10$\mu$m silicate feature that is absent in the observations of Cha H$\alpha$2. Long wavelength observations are required to discriminate between our flared disk models and alternative flat disk models that have been proposed for some sources.
We acknowledge financial support from a UK PPARC Studentship (CW); UK PPARC Advanced Fellowship (KW); NASA’s Long Term Space Astrophysics Research Program, NAG5 8412 (BW), NAG5 8794 (JEB); the National Science Foundation, AST 9909966 (BW), AST 9819928 (JEB).
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APPENDIX {#appendix .unnumbered}
========
MONTE CARLO RADIATION TRANSFER CODE DEVELOPMENTS {#monte-carlo-radiation-transfer-code-developments .unnumbered}
------------------------------------------------
A1. DENSITY STRUCTURE {#a1.-density-structure .unnumbered}
=====================
In this study we determine the structure of circumstellar disks based on the assumption that the disk is in vertical hydrostatic equilibrium with dust and gas well mixed. We therefore solve the hydrostatic equilibrium equation $${{dP}\over{dz}} = -\rho g_z\; ,$$ where $P$ is the pressure, $\rho$ is the density, and $g_z$ is the vertical component of gravity in the disk. Conservation of mass in the disk is enforced by keeping the radial dependence of the surface density.
The hydrostatic equation has an analytic solution if the disk temperature is assumed to be vertically isothermal at any given cylindrical radius, $\varpi$. Using the mid-plane temperature, $T(\varpi)$, the disk density has a Gaussian distribution about the midplane with scaleheight $$h(\varpi)=\left(\frac{k\, T(\varpi)\,\varpi^3}
{GM_\star \mu\, m_H}\right)^{\frac{1}{2}}\; ,$$ where $k$ and $G$ are the Boltzmann and Newton constants, $m_{H}$ is the mass of hydrogen and $\mu$ is the molecular weight of disk material and is taken to be $\mu=2.3$ for a molecular hydrogen/helium combination.
In order to solve for the density numerically, we approximate the integral of equation A-1 to a sum of finite contributions. Using the equation of state, $P=\rho c_s^2$, where $c_s^2={kT}/{\mu m_h}$ is the local sound speed squared, this leads to $$\ln\left({{\rho}\over{\rho_0}}\right) =
{-\sum{\frac{1}{T}\left(\frac{dT}{dz}+
\frac{g_{z} \mu\, m_{H}}{k}\right){\Delta{z}}}}$$ which can be solved using $${{dT}\over{dz}}=\cos\theta\frac{dT}{dr}-\frac{\sin\theta}{r}\frac{dT}{d\theta}$$ In the discretization of the disk density we use a spherical polar grid (Whitney & Wolf 2002) throughout which we calculate $g_z$ at the midpoint of each cell. The cell temperature determined from our radiative equilibrium calculation is assumed to be uniform within each cell and $\Delta$z is the incremental distance through each cell which lies directly below grid centre $(r,\theta,\phi)$. It is therefore possible to obtain values of ${\rho}/{\rho_{0}}$ for each grid cell.
We assume the disk surface density has the form, $\Sigma(r)=\Sigma_0(\varpi/{R})^{-1}$, which agrees with the disk structure models of D’Alessio et al. (1999). Since total disk mass is given by $$M_d=\int_{Rmin}^{Rmax}\Sigma(\varpi)2\pi\,\varpi\, {\rm d}\varpi\; ,$$ we can solve for $\Sigma_0$ and in turn get $\Sigma(\varpi)$. We then normalize $\rho_{0}$ so that surface density is a constant, $$\rho_0(\varpi)=\frac{\Sigma_0\left({R}/{\varpi}\right)}
{\sum{\rho}/{\rho_0}\,{\Delta z}}$$ For each cell ${\rho}/{\rho_{0}}$ and the cylindrical radius, $\varpi$, are known, so we may therefore determine the density in each cell.
A2. TEMPERATURE STRUCTURE {#a2.-temperature-structure .unnumbered}
=========================
In order to determine the density structure we require an accurate calculation of the disk temperature structure. While the Bjorkman & Wood (2001) technique yields accurate SEDs, we found that the temperature calculation was too noisy for use in our density calculation. Increasing the number of photon energy packets yields a smoother temperature structure at the cost of a large increase in CPU time. We have therefore implemented the temperature calculation technique of Lucy (1999), which is based on using an estimator for the mean intensity of the radiation field. Integrating this technique into our Monte Carlo code leads to a higher signal-to-noise in the disk temperature determination with fewer photon packets. What follows is an outline of the procedure we use and we refer the reader to Lucy (1999) for more details.
Provided a system is in radiative equilibrium, the rate at which matter absorbs energy from the radiation field is balanced by the rate at which matter emits energy, $\dot{A}=\dot{E}$ or, $$4\pi \int_0^{\infty}\rho(1-a_\nu)\kappa_{\nu}J_{\nu}\,{\rm d}\nu=
4\pi \int_0^{\infty}\rho(1-a_\nu)\kappa_{\nu}B_{\nu}\,{\rm d}\nu$$ As discussed by Lucy (1999) the mean intensity, $J_\nu$ and therefore heating is proportional to the photon path lengths, $l$, through the cells yielding, $${\dot{A}}=\frac{\epsilon}{\delta{t}\delta{V}}
\sum{l}\rho(1-a_\nu)\kappa_{\nu}\; ,$$ where $\delta{t}$ is the cell simulation time, $\delta{V}$ is the cell volume, $a_\nu$ is the scattering albedo, and $\kappa_\nu$ is the total opacity in $cm^2/g$. The energy of each photon packet is $\epsilon=L\,\delta t/N$, where $L$ is the source luminosity and $N$ is the number of Monte Carlo photon packets used in our simulation. The expression for the rate at which matter emits energy can also be simplified to: $$\dot{E}=4\pi\rho\kappa_P\,B(T)$$ where $\kappa_P$ is the Planck mean absorption coefficient and $B(T)=\sigma\,T^4/\pi$ is the integrated Planck function. Equating A-7 and A-8 leads to the following expression for temperature: $$T^4=\left(\frac{\dot{A}}{4\kappa_{p}(T)\sigma}\right)\; .$$ Since $\kappa_{P}(T)$ is a function of temperature we solve this equation iteratively using pre-tabulated values of $\kappa_{P}(T)$.
Our code uses the Bjorkman & Wood (2001) technique for reprocessing photon packets and our modification of Lucy’s (1999) pathlength technique to determine the cell temperature. Since we assume the opacity is not a function of temperature, we do not need to iterate to determine the temperature structure, as discussed in Bjorkman & Wood (2001).
A3. DUST DESTRUCTION {#a3.-dust-destruction .unnumbered}
====================
Model disks used in this study extend from a sharply cut-off inner radius out to a specified distance. The inner radius is defined by the dust destruction temperature which we take to be 1600 K. We make the simplifying assumption that the dust destruction radius is independent of latitude in the disk. The determination of the dust destruction radius is carried out after the temperature calculation and involves a nested loop that counts how many grid cells at each radius have temperatures below 1600K. If this number is outside some specified range then the inner radius is shifted either towards or away from the central star. The disk density is then re-gridded and the temperature calculation/dust destruction radius determination repeated. This continues until a stable radius is established. Once the inner radius is fixed the program starts to iteratively solve for density as described above.
A4. ITERATIVE PROCEDURE {#a4.-iterative-procedure .unnumbered}
=======================
The program self-consistently solves for density using an iterative procedure. For the first iteration the analytic density structure of Eq. 1 is assumed and this allows an initial temperature structure to be found. On the next iteration or at the point at which a suitable dust destruction radius has been established and a new grid set-up, this temperature structure is used to determine a density structure using the numerical density technique. This density replaces the analytic density structure and the next iteration begins. The procedure continues until the temperature and density structure converges, typically within three iterations.
[@cccc]{}\
$M_\star$&$T_\star$&$R_\star$&$L_\star$\
($M_\odot$)&(K)&($R_\odot$)&($L_\odot$)\
\
\
0.01&2200&0.25&0.0013\
0.04&2600&0.50&0.0038\
0.08&2800&0.90&0.044\
\
[@ccccccc]{}\
Object&$T_\star$&$R_\star$&$M_\star$&$M_d$&$A_v$&Inclination\
&(K)&($R_\odot$)&($M_\odot$)&($M_\odot$)&(mags)&(deg)\
\
\
ISO\#023&2600&0.95&0.04&$10^{-5}$&8&0\
ISO\#030&2600&1.2&0.08&$10^{-5}$&2&0\
ISO\#032&2600&1.2&0.08&$10^{-5}$&3&0\
ISO\#033&2200&0.63&0.01&$10^{-3}$&7&0\
ISO\#102&3000&1.17&0.08&$10^{-7}$&3.5&0\
ISO\#160&2600&0.95&0.08&$10^{-7}$&6&0\
ISO\#164&2600&1.36&0.08&$10^{-3}$&4&63\
ISO\#176&3000&1.17&0.08&$10^{-7}$&7&60\
ISO$\#$193&3000&1.5&0.08&$10^{-5}$&7&78\
CHA H$\alpha$1&2600&0.5&0.01&$10^{-5}$&0.3&0\
CHA H$\alpha$2&2600&1.05&0.04&$10^{-5}$&1.1&37\
CHA H$\alpha$9&2600&0.95&0.08&$10^{-5}$&3.2&72\
\[lastpage\]
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: |
The holomorphicity property of ${\cal N}=1$ superpotentials or of ${\cal N}=2$ F-terms involving vector multiplets is generalized to the case of ${\cal N}=4$ 1/2-BPS effective operators defined in harmonic superspace. The resulting [*harmonicity*]{} equations are shown to control the moduli dependence of the couplings of higher dimensional operators involving powers of the ${\cal N}=4$ Weyl superfield, computed by ${\cal N}=4$ topological amplitudes. These equations can also be derived on the string side, exhibiting an anomaly from world-sheet boundary contributions that leads to recursion relations for the [*non-analytic*]{} part of the couplings.\
author:
- |
\
I. Antoniadis[^1] [^2] , S. Hohenegger[^3] , K.S. Narain[^4] , E. Sokatchev[^5]
title: |
[CERN-PH-TH/2007-132\
LAPTH-1201/07]{}
**[Harmonicity in ${\cal N}=4$ supersymmetry and its quantum anomaly]{}**
---
Department of Physics, CERN – Theory Division\
CH-1211 Geneva 23, Switzerland\
High Energy Section,\
The Abdus Salam International Center for Theoretical Physics,\
Strada Costiera, 11-34014 Trieste, Italy\
Laboratoire d’Annecy-le-Vieux de Physique Théorique LAPTH,\
B.P. 110, F-74941 Annecy-le-Vieux, France\
Introduction {#intro}
============
An important property of ${\cal N}=2$ F-terms involving vector multiplets is holomorphicity, implying that the corresponding couplings are holomorphic functions of the vector moduli. This applies, for instance, to the couplings $F_g$ of the higher dimensional F-terms $W^{2g}$, where $W$ is the self-dual (chiral) Weyl superfield, appearing in the string effective action [@Antoniadis:1993ze]. On the other hand, the couplings $F_g$ are computed by the topological partition function of the ${\cal N}=2$ twisted Calabi-Yau $\sigma$-model associated to the six-dimensional compactification manifold of type II string theory in four dimensions [@Bershadsky:1993cx; @Antoniadis:1993ze]. It turns out, however, that there is a holomorphic anomaly, related to a violation of the conservation of the BRST current in the topological theory, implying that antichiral fields do not decouple at the quantum level [@Cecotti:1992qh; @Bershadsky:1993ta; @Bershadsky:1993cx]. The anomaly arises from boundary contributions and takes the form of an equation that amounts to a recursion relation for the non-holomorphic part of the couplings $F_g$. From the point of view of the string effective action, it arises from the quantum integration over massless states that is unavoidable when computing on-shell physical amplitudes [@Antoniadis:1993ze].
These couplings were generalized recently to 1/2-BPS terms of ${\cal N}=4$ compactifications, involving powers of the (superdescendant of the) ${\cal N}=4$ chiral Weyl superfield $K^{++}=D_-D_-W$, where $D_-$ are particular $SU(4)$ projections of the spinor derivatives. We recall that the ${\cal N}=4$ gravity multiplet contains, besides the graviton and the four gravitini, six graviphotons, one complex graviscalar and four spin-1/2 Weyl fermions [@Antoniadis:2006mr]. Moreover, there is an $SU(4)$ R-symmetry, transforming the gravitini in the fundamental and the graviphotons in the vector representation. The (linearized on-shell) superfield $K^{++}$ satisfies 1/2-BPS shortening conditions. Its lowest component is the (self-dual) graviphoton field strength and its next bosonic components are the (self-dual) Riemann tensor and the second derivative of the graviscalar. Another basic 1/2-BPS superfield in the ${\cal N}=4$ theory is the (linearized on-shell) vector multiplet $Y^{++}$. Its lowest component are the scalar moduli transforming in the vector representation of $SU(4)$, like the graviphoton field strengths.
In [@Antoniadis:2006mr] two series of 1/2-BPS couplings were found: ${\cal F}_g^{(1)}{\bar K}^2K^{2g}$ and ${\cal F}_g^{(3)}K^{2(g+1)}$. Here ${\cal F}_g^{(1)}$ and ${\cal F}_g^{(3)}$ are functions of the ${\cal N}=4$ moduli vector multiplets $Y^{++}$ and of the $SU(4)$ harmonic variables that can again be computed by topological amplitudes on $K3\times T^2$ of genus $g$ and $g+1$, respectively. Actually, in six dimensions there is also the series $F_g^{(6d)}W_{6d}^{4g}$, where $W_{6d}$ is a similar Weyl superfield of the six-dimensional gravity multiplet and $F_g^{(6d)}$ is given by a topological theory on $K3$ [@Berkovits:1994vy].
In this work, we study the question of what is the analog of ${\cal N}=2$ holomorphicity for such 1/2-BPS ${\cal N}=4$ couplings. The main novelty of the generalization is that the relevant R-symmetry group becomes non-Abelian, transforming non-trivially the superfields $K^{++}$ and $Y^{++}$. As a consequence, the notion of chirality of the ${{\cal N}}=2$ theory is replaced by Grassmann analyticity (or 1/2-BPS shortness). The natural framework for studying this problem and covariantizing the expressions is then harmonic superspace [@Galperin:1984av; @Galperin:1984bu; @Hartwell:1994rp]. By introducing $SU(4)$ harmonic variables one can define $K^{++}$ as a particular harmonic projection of the sixplet of superfields $K_{ij}= -K_{ji} = D_iD_jW$, associated to a corresponding 1/2-BPS subspace of the full ${\cal N}=4$ superspace. Supersymmetry then implies that the coupling coefficients ${\cal F}_g$ are functions of the same harmonic projected vector superfields $Y^{++}$ living in the same 1/2-BPS subspace. Thus, ${\cal F}_g(Y^{++})$ is independent of the five remaining projections of the sixplet of the scalar moduli. This defines a notion of analyticity that naturally generalizes ${\cal N}=2$ holomorphicity for the chiral ${{\cal N}}=2$ vector multiplets.
In this work, we show that the above property of analyticity can be formulated in terms of a set of differential constraints on the couplings ${\cal F}_g$ of ${\cal N}=4$ 1/2-BPS effective operators. They express the property of the analytic functions ${\cal F}_g(Y^{++})$ that, when expanded in powers of the harmonic variables and the scalar fields, the coefficients should form symmetric traceless tensors of $SO(6)$. This yields two non-trivial equations. The first requires one scalar and one harmonic derivative to vanish and coincides with the so-called ‘harmonicity’ equation found previously in string computations, up to an anomaly [@Berkovits:1994vy; @Ooguri:1995cp; @Antoniadis:2006mr]. The second involves two scalar derivatives and gets modified in supergravity by an additive constant term due to the curvature of the scalar kinetic terms. Both equations are checked by an explicit string computation for ${\cal F}_g^{(3)}$, which receives one-loop corrections on the heterotic side for all $g$, and are found to be corrected by anomaly terms due to world-sheet boundary contributions that spoil the naive expectation of analyticity, in a way similar to the holomorphic anomaly equation of the ${\cal N}=2$ $F_g$’s. The resulting equations are reduced again to recursion relations for the non-analytic part of the moduli-dependent couplings.
We finally extend the above results to six-dimensional ${{\cal N}}=(1,1)$ supersymmetry, where the R-symmetry group is $SO(4)$. In particular, we consider the decompactification limit of ${\cal F}_g^{(3)}$ that gives rise to a new six-dimensional series of 1/2-BPS couplings of the form ${\cal F}_g^{\rm dec}W_{6d}^{2(g+1)}$. These couplings, although not exactly topological in six dimensions (the space-time part is not decoupled), become topological upon compactification to four dimensions on a two-torus. Despite this fact, ${\cal F}_g^{\rm dec}$ satisfy the same analyticity condition as $F_g^{(6d)}$ of $W_{6d}^{4g}$ since they are both 1/2-BPS. We then derive the corresponding analyticity equations, together with the anomaly terms.
The paper is organized as follows. In Section \[global\], we describe the analyticity conditions of the 1/2-BPS couplings in the case of global ${\cal N}=4$ supersymmetry. We first introduce the $SU(4)$ harmonic variables and the harmonic projected vector and Weyl superfields, $Y^{++}$ and $K^{++}$ respectively. We then derive the differential equations for the couplings ${\cal F}_g$ of the higher-derivative operators involving powers of $(K^{++})^2$, as described above. In Section \[coset\], we study the effects of the curvature of the scalar manifold that parametrizes the coset $SO(6,n)/SO(6)\times SO(n)$, where $n$ is the number of vector multiplets. We show in particular that the second-order derivative equation in the scalar fields gets modified by an additional term proportional to the Weyl weight of the operator. In Section \[css\], we go to curved superspace in the framework of ${\cal N}=4$ conformal supergravity and derive the fully covariantized final expressions of the higher-derivative couplings. In Section \[Sect:Review\], we give a brief review of the ${\cal N}=4$ topological amplitudes in string theory and recall the expression for ${\cal F}_g^{(3)}$ obtained from a one-loop string computation on the heterotic side. In Section \[Sect:FirstOrder\], we derive the harmonicity relation which is first order in the scalar field derivatives, exhibiting a boundary anomaly that invalidates the expected vanishing result. In Section \[Sect:SecondOrderRelation\], we obtain the second-order constraint which is also modified by an anomaly. In Section \[sixdim\], we generalize our analysis to six dimensions. We first introduce the $SO(4)$ harmonic variables and derive the harmonicity equations for the couplings of the 1/2-BPS terms. We then consider the decompactification limit of ${\cal F}_g^{(3)}$ and compute the two analyticity equations modified by the anomalous terms. Finally, Section \[Sec:Concl\] contains some concluding remarks.
Global ${{\cal N}}=4$ supersymmetry {#global}
===================================
$SU(4)$ harmonic variables
--------------------------
We consider ${{\cal N}}=4$ supersymmetry in four dimensions whose automorphism group is $SU(4)$. We introduce harmonic variables [@Galperin:1984av; @Galperin:1984bu; @Hartwell:1994rp] on the coset $SU(4)/S(U(2)\times U(2))$ in the form of matrices $(u^{+a}_i, \, u^{-{{\dot a}}}_i) \in SU(4)$. They have an index $i=1\ldots4$ transforming under the fundamental irrep of $SU(4)$ and indices $a, {{\dot a}}=
1,2$ of $SU(2)\times SU(2)$ as well as $U(1)$ charges $\pm 1$. Together with their complex conjugates ${\bar u}^i_{+a} = \overline{(u^{+a}_i)}, \, {\bar u}^i_{-{{\dot a}}} =
\overline{(u^{-{{\dot a}}}_i)}$ they satisfy the unitarity conditions $$\begin{aligned}
&&u^{+a}_i\, {\bar u}^i_{+b} = \delta^a_b\,, \quad u^{-{{\dot a}}}_i\, {\bar u}^i_{-{{\dot b}}} = \delta^{{\dot a}}_{{\dot b}}\,, \quad u^{+a}_i\, {\bar u}^i_{-{{\dot b}}} = u^{-{{\dot a}}}_i\, {\bar u}^i_{+b} = 0 {\nonumber}\\
&&u^{+a}_i\, {\bar u}^j_{+a} + u^{-{{\dot a}}}_i\, {\bar u}^j_{-{{\dot a}}} = \delta^j_i\label{12'}\end{aligned}$$ and the unit determinant condition $$\label{12}
{\epsilon}^{ijkl} u^{+a}_i u^{+b}_j u^{-{{\dot a}}}_k u^{-{{\dot b}}}_l = {\epsilon}^{ab}{\epsilon}^{{{\dot a}}{{\dot b}}}$$ (with ${\epsilon}^{1234} = {\epsilon}^{12}= {\epsilon}^{\dot 1\dot 2} = -{\epsilon}_{12}= -{\epsilon}_{\dot 1\dot 2} = 1$).
The harmonic functions have harmonic expansions homogeneous under the action of the subgroup $S(U(2)\times U(2))$. The harmonic expansions are organized in irreps of $SU(4)$, keeping the balance of projected indices so that the overall $SU(2)\times
SU(2)$ indices and the $U(1)$ charge are always the same. In what follows we shall often make use of functions depending on vector-like combinations of $SU(4)$ harmonics (i.e., with harmonics on $SO(6)/SO(4)\times SO(2)$) of the type $u^M_{ij} = -u^M_{ji}$, $M= (++,--, a{{\dot a}})$ (and their conjugates ${\bar u}_{M}^{ij} = \overline{u^{M}_{ij}}$) $$\begin{aligned}
&& u^{++}_{ij} = u^{+a}_i {\epsilon}_{ab} u^{+b}_j {\nonumber}\\
&& u^{--}_{ij} = u^{-{{\dot a}}}_i {\epsilon}_{{{\dot a}}{{\dot b}}} u^{-{{\dot b}}}_j \label{00} \\
&& u^{a{{\dot a}}}_{ij} = u^{+a}_{[i} u^{-{{\dot a}}}_{j]}\ , {\nonumber}\end{aligned}$$ where $[ij]$ denotes weighted antisymmetrization. They form $SO(6)$ matrices $u^M_N = \Gamma_N^{ij}u^M_{ij}$ where $\Gamma^M$ are the gamma matrices of $SO(6)$. The vector-like harmonics satisfy algebraic conditions expressing the fact that $u^M_N \in SO(6)$ and following from the conditions on the underlying $SU(4)$ harmonics: $$\begin{aligned}
&&u^{++}_{ij} {\epsilon}^{ijkl} u^{--}_{kl} = 4 {\nonumber}\\
&&u^{a{{\dot a}}}_{ij} {\epsilon}^{ijkl} u^{b{{\dot b}}}_{kl} = {\epsilon}^{ab}{\epsilon}^{{{\dot a}}{{\dot b}}} \label{vchac}\\
&&u^{++}_{ij} {\epsilon}^{ijkl} u^{++}_{kl} = u^{--}_{ij} {\epsilon}^{ijkl} u^{--}_{kl} = u^{++}_{ij} {\epsilon}^{ijkl} u^{a{{\dot a}}}_{kl} = u^{--}_{ij} {\epsilon}^{ijkl} u^{a{{\dot a}}}_{kl}= 0 {\nonumber}\\
&&u^{++}_{ij} u^{--}_{kl} + u^{--}_{ij} u^{++}_{kl} - 2\, u^{a{{\dot a}}}_{ij}{\epsilon}_{ab} {\epsilon}_{{{\dot a}}{{\dot b}}} u^{b{{\dot b}}}_{kl}= {\epsilon}_{ijkl} {\nonumber}\end{aligned}$$ An example of a harmonic function which we shall frequently encounter is $\phi^{++}(u) = \phi^{ij} u^{++}_{ij} + \phi^{ij\, kl}_{mn}
u^{++}_{ij} u^{++}_{kl} u^{mn}_{++} + \cdots$. The first component in this expansion is a of $SU(4)$ (or a vector of $SO(6)$) $\phi^{ij} = - \phi^{ji}$. The higher components give rise to higher-dimensional irreps, but we shall not need them here.
The harmonic derivatives can be viewed as the covariant derivatives on the harmonic coset $SU(4)/S(U(2)\times U(2))$, or equivalently, as the generators of the $SU(4)$ algebra written down in an $S(U(2)\times U(2))$ basis (see Section \[coset\]). This means that they are invariant under the left action of the group $SU(4)$, but covariant under the right action of the subgroup $S(U(2)\times U(2))$. They can be split into generators of the subalgebra $S(U(2)\times U(2))$: $$\begin{aligned}
D_{+a}{}^{+b} &=& \bigl{(} u^{+b}_i \frac{{\partial}}{{\partial}u^{+a}_i } - {\bar u}_{+a}^i \frac{{\partial}}{{\partial}{\bar u}_{+b}^i}\bigr{)} - \mbox{trace} {\nonumber}\\
D_{-{{\dot a}}}{}^{-{{\dot b}}} &=& \bigl{(}u^{-{{\dot b}}}_i \frac{{\partial}}{{\partial}u^{-{{\dot a}}}_i } - {\bar u}_{-{{\dot a}}}^i \frac{{\partial}}{{\partial}{\bar u}_{-{{\dot b}}}^i}\bigr{)} - \mbox{trace} \label{subhd}\\
D_0 &=& \bigl{(}u^{+a}_i \frac{{\partial}}{{\partial}u^{+a}_i } - {\bar u}_{+a}^i \frac{{\partial}}{{\partial}{\bar u}_{+a}^i} \bigr{)}- \bigl{(}u^{-{{\dot a}}}_i \frac{{\partial}}{{\partial}u^{-{{\dot a}}}_i } - {\bar u}_{-{{\dot a}}}^i \frac{{\partial}}{{\partial}{\bar u}_{-{{\dot a}}}^i} \bigr{)}{\nonumber}\end{aligned}$$ and of the coset: $$\begin{aligned}
D_{+a}{}^{-{{\dot b}}} &=& u^{-{{\dot b}}}_i \frac{{\partial}}{{\partial}u^{+a}_i } - {\bar u}_{+a}^i \frac{{\partial}}{{\partial}{\bar u}_{-{{\dot b}}}^i} {\nonumber}\\
D_{-{{\dot a}}}{}^{+b} &=& u^{+b}_i \frac{{\partial}}{{\partial}u^{-{{\dot a}}}_i } - {\bar u}_{-{{\dot a}}}^i \frac{{\partial}}{{\partial}{\bar u}_{+b}^i} \ . \label{cosethd}\end{aligned}$$ The harmonic derivatives are differential operators preserving the defining algebraic constraints (\[12’\]), (\[12\]).
The derivatives (\[subhd\]) act homogeneously on the harmonic functions. For instance, the function $\phi^{++}(u)$ above has no $SU(2)\times SU(2)$ indices, but has $U(1)$ charge, hence $$\label{exhf}
D_{+a}{}^{+b} \phi^{++}(u) = D_{-{{\dot a}}}{}^{-{{\dot b}}} \phi^{++}(u) = 0\,, \qquad D_0 \phi^{++}(u) = 2 \phi^{++}(u)\ .$$ The harmonic expansion of this function defines an infinitely reducible representation of $SU(4)$. It can be made irreducible by requiring that the raising operator $D_{-{{\dot a}}}{}^{+b}$ annihilate the function: $$\label{ropcon}
D_{-{{\dot a}}}{}^{+b} \phi^{++}(u) = 0\ \quad \Rightarrow \quad \phi^{++}(u) = \phi^{ij} u^{++}_{ij} \ .$$ In other words, such a function is a highest-weight state of the of $SU(4)$. The irreducibility condition (\[ropcon\]) is also called a condition for harmonic (H-) analyticity.
Grassmann analytic on-shell superfields
---------------------------------------
The introduction of harmonic variables allows us to define ‘1/2 BPS short’ or Grassmann (G-) analytic[^6] superfields.[^7] They depend only on half of the Grassmann variables which can be chosen to be ${\theta}^{+a}_{\alpha}= {\theta}^i_{\alpha}\, u_i^{+a}$ and ${\bar\theta}^{{\dot\alpha}}_{-{{\dot a}}} =
{\bar u}^i_{-{{\dot a}}}\,{\bar\theta}^{{\dot\alpha}}_i $. One such superfield is the [*linearized on-shell*]{} vector multiplet $$\begin{aligned}
Y^{++}(x^\mu,{\theta}^+,{\bar\theta}_-,u) &=& \phi^{ij} u^{++}_{ij} + {\theta}^{+a}_{\alpha}\,{\epsilon}_{ab}\, \psi^{{\alpha}\, i}\,u^{+b}_i +{\bar u}^i_{-{{\dot a}}}\,\bar\psi_{{{\dot\alpha}}\,i}\, {\epsilon}^{{{\dot a}}{{\dot b}}}\, {\bar\theta}^{{\dot\alpha}}_{-{{\dot b}}} \label{01} \\
&+& {\theta}^{+a}\sigma^{{\mu}{\nu}}{\theta}^{+b}{\epsilon}_{ab}\, F_{(+){\mu}{\nu}} + {\bar\theta}_{-{{\dot a}}}\tilde\sigma^{{\mu}{\nu}}{\bar\theta}_{-{{\dot b}}}{\epsilon}^{{{\dot a}}{{\dot b}}}\, F_{(-){\mu}{\nu}} + \mbox{derivative terms} \ . {\nonumber}\end{aligned}$$ Here $\phi^{ij}= \frac{1}{2}{\epsilon}^{ijkl} \bar\phi_{kl}$ are the six real scalars, $\psi^{\alpha}_i$ are the four Majorana gluinos and $F_{(\pm){\mu}{\nu}}$ is the (anti)self-dual part of the gluon field strength. To exhibit manifest G-analyticity, one has to choose the appropriate analytic basis in superspace, $$\label{chbashss}
x^\mu\ \to \ x^\mu + i {\theta}^{+a}\sigma^\mu\bar{\theta}_{+a} + i {\theta}^{-{{\dot a}}}\sigma^\mu\bar{\theta}_{-{{\dot a}}}\ ,$$ analogous to the familiar chiral basis. Note that the harmonic dependence here is cut down to linear in the vector-like and fundamental harmonics. This is typical for on-shell multiplets which, in addition to the G-analyticity condition, also satisfy the H-analyticity condition $$\label{hansf}
D_{-{{\dot a}}}{}^{+b} Y^{++}({\theta}^+,{\bar\theta}_-,u) = 0\ .$$ Here the harmonic derivative is supersymmetrized by going to the manifestly G-analytic superspace coordinates (\[chbashss\]). One can show that the ‘ultrashort’ on-shell superfield (\[01\]) is the solution to the simultaneous conditions for G- and H-analyticity [@Hartwell:1994rp; @Andrianopoli:1999vr].
Another example of a G-analytic superfield is the [*linearized*]{} on-shell Weyl multiplet. It is obtained from the off-shell chiral Weyl superfield [@Bergshoeff:1980is] $$\label{14}
W({\theta}_{\alpha}^i) = \Phi + {\theta}_{\alpha}^i {\theta}_{\beta}^j (\sigma_{{\mu}{\nu}}^{{\alpha}{\beta}} T_{(+)[ij]}^{{\mu}{\nu}} + {\epsilon}^{{\alpha}{\beta}} S_{(ij)}) + \cdots \ .$$ Here $\Phi$ is the physical scalar and $T_{(+)}$ is the self-dual part of the sixplet of graviphoton field strengths, while $S$ is an auxiliary field. On shell the latter must vanish, hence the additional constraint $$\label{15}
{\epsilon}_{{\alpha}{\beta}} D_i^{\alpha}D_j^{\beta}\ W = 0\,.$$ Now, define the superfield (a superdescendant of $W$) $$\label{16}
K^{++}_{{\mu}{\nu}} = (\sigma_{{\mu}{\nu}})_{{\alpha}{\beta}}D_{-{{\dot a}}}^{\alpha}D_{-{{\dot b}}}^{\beta}{\epsilon}^{{{\dot a}}{{\dot b}}}\ W$$ where we have projected the $SU(4)$ indices of $D^{\alpha}_i D^{\beta}_j$ with the harmonics ${\bar u}_{-{{\dot a}}}^i {\bar u}_{-{{\dot b}}}^j$ . This superfield is annihilated by half of the spinor derivatives and hence is 1/2 BPS short. Indeed, this is true for the projections $\bar
D^{+a}_{{{\dot\alpha}}}$ since $\{\bar D^{+a}, D_{-{{\dot b}}}\}=0$ and $\bar D^i_{{{\dot\alpha}}}W=0$ (chirality). Further, hitting (\[16\]) with $D_{-{{\dot a}}}^\gamma$ we obtain zero as a consequence of the projection of the on-shell constraint (\[15\]) with ${\bar u}_{-{{\dot a}}}^i {\bar u}_{-{{\dot b}}}^j$. We conclude that $K^{++}_{{\mu}{\nu}}$ satisfies the G-analyticity constraints $$\label{16'}
D_{-{{\dot a}}}^{\alpha}K^{++}_{{\mu}{\nu}} = \bar D^{+a}_{{{\dot\alpha}}} K^{++}_{{\mu}{\nu}} = 0$$ which imply that it depends on half of the ${\theta}$’s (bosons only): $$\label{17}
K^{++}_{{\mu}{\nu}}({\theta}^+,{\bar\theta}_-,u) = T_{(+){\mu}{\nu}}^{ij} u^{++}_{ij} + {\theta}^{+a}\sigma^{{\lambda}\rho}{\theta}^{+b}{\epsilon}_{ab}\, R_{(+){\mu}{\nu}{\lambda}\rho} + {\bar\theta}_{-{{\dot a}}}\tilde\sigma^{\lambda}\sigma_{{\mu}{\nu}} \sigma^\rho {\bar\theta}_{-{{\dot b}}}{\epsilon}^{{{\dot a}}{{\dot b}}}\, {\partial}_{\lambda}{\partial}_\rho \Phi + \cdots\ .$$ In addition, the harmonic dependence of $K^{++}_{{\mu}{\nu}}$ is restricted to be linear. As in (\[hansf\]), this follows from the condition for H-analyticity $$\label{hwconK}
D_{-{{\dot a}}}{}^{+b} K^{++}_{{\mu}{\nu}} = 0 \ .$$ This is another example of an ultrashort superfield. Note, however, that it is not a primary object but rather a superdescendant of the chiral on-shell Weyl multiplet.
Repeating the same steps, but this time starting with the antichiral superfield $\bar
W({\bar\theta})$ we obtain the other half of the on-shell Weyl multiplet. It is again described by an ultrashort superfield of the same type, $$\label{18}
\bar K^{++}_{{\mu}{\nu}}({\theta}^+,{\bar\theta}_-,u) = T_{(-){\mu}{\nu}}^{ij} u^{++}_{ij} + {\bar\theta}_{-{{\dot a}}} \tilde\sigma^{{\lambda}\rho}{\bar\theta}_{-{{\dot b}}}{\epsilon}^{{{\dot a}}{{\dot b}}}\, R_{(-){\mu}{\nu}{\lambda}\rho} + {\theta}^{+a}\sigma^{\lambda}\tilde\sigma_{{\mu}{\nu}} \tilde\sigma^\rho {\theta}^{+b}{\epsilon}_{ab} \, {\partial}_{\lambda}{\partial}_\rho \bar\Phi + \cdots$$ Note that in the ${{\cal N}}=4$ G-analytic superspace there exists a special conjugation $\
\widetilde{}\ $ combining complex conjugation with a reflection on the harmonic coset, such that G-analyticity is preserved. In this sense $Y^{++} = \widetilde{Y^{++}}$ and $\bar K^{++} = \widetilde{K^{++}}$, which implies, in particular, the reality condition on the six scalars in $Y$.
Higher-derivative couplings
---------------------------
After having defined the G-analytic superfields (\[01\]) and (\[17\]), we now want to construct the corresponding effective action couplings. Recently, by studying special higher loop scattering processes in the gravitational sector of type II superstring theory compactified on $K3\times T^2$ (or the corresponding dual formulation of heterotic string on $T^6$, as we will review in Section \[Sect:Review\]), the following two terms were found in [@Antoniadis:2006mr]: $$\begin{aligned}
S_1 &=&\int \ d^4x \ du \ d^4{\theta}^+ d^4{\bar\theta}_{-} \ (\bar K^{++}_{{\mu}{\nu}} \bar K^{++\, {\mu}{\nu}})\, (K^{++}_{\rho\sigma} K^{++\, \rho\sigma})^{g}\, F_1^{-4(g-1)}(Y^{++}_A,u)\ , \label{19}\\
S_2 &=& \int \ d^4x \ du \ d^4{\theta}^+ d^4{\bar\theta}_{-} \ (K^{++}_{{\mu}{\nu}} K^{++\, {\mu}{\nu}})^{g+1}\, F_2^{-4(g-1)}(Y^{++}_A,u)\ , \label{19'}\end{aligned}$$ where $A=1\cdots n$ is an $SO(n)$ vector index labeling the coordinates of the coset of physical scalars (see Section \[coset\]). In fact, if considered as $g$- and $(g+1)$-loop contributions respectively, both of these terms lead to so-called [*topological amplitudes*]{}, that is the corresponding physical amplitudes are computed by correlation functions of the ${{\cal N}}=4$ topological string on $K3\times T^2$. However, unlike the ${{\cal N}}=2$ case (see [@Antoniadis:1993ze]) these correlation functions are not simply the topological partition function, but differ from it by additional operator insertions in the twisted version of the theory. Actually, for convenience and notational simplicity, we changed the notation of the two couplings from [@Antoniadis:2006mr]: $F_1^{-4(g-1)}$ corresponds to ${\cal F}_g^{(1)}$, while $F_2^{-4(g-1)}$ corresponds to ${\cal F}_g^{(3)}$, with the upper index denoting the $U(1)$ charge.
It is important to stress upon two points concerning the effective action terms (\[19\]) and (\[19’\]):
1. The Grassmann measure is G-analytic, i.e. it involves only half of the projected ${\theta}$’s, and so must be the integrand, otherwise supersymmetry will be broken. This is why we have to use the [*linearized on-shell*]{} superfields $Y^{++}$ and $K^{++}$ which are G-analytic like the measure.
2. The harmonic integral should produce an $SU(4)$ invariant, i.e. it picks out the $SU(4)$ singlet part of the integrand. This is only possible if the latter is a [*chargeless*]{} harmonic function. For example, $f(u) = f_0 + f^{ij}_{kl}u^{++}_{ij}
u^{kl}_{++} + \cdots $ integrates to $\int du \, f(u) = f_0$, but a charged function like $f^{++} = f^{ij}u^{++}_{ij} + \cdots$ will have a vanishing integral. Notice that for this reason the harmonic integral should always be done [last]{}, after the Grassmann integrals, since the latter are charged.
In our case (\[19\]), (\[19’\]) the functions $F_{1,2}$ carry $U(1)$ charge $-4(g-1)$ needed to compensate that of the factor $K$ $(+4(g+1))$ and of the Grassmann measure $(-8)$. Given the fact that the argument $Y^{++}$ of $F$ has a positive charge, we have to introduce a set of [*constant*]{} $SU(4)$ multispinors $$\label{20'}
\xi^{-2p}(u) \equiv \xi_{(i_1 \cdots i_p) (j_1 \cdots j_p)}\ {\bar u}_{++}^{i_1 j_1} \cdots {\bar u}_{++}^{i_p j_p} + \ldots$$ thus explicitly breaking $SU(4)$.[^8] The dots denote higher-order terms in the harmonic expansion of the coefficients $\xi(u)$ which will not be of interest for us, see below. Note that the product of vector-like harmonics forms an irreducible representation of $SO(6)$, a symmetric traceless tensor of rank $p$ (recall that $({\bar u}_{++})^2 =0$, see (\[vchac\])). In $SU(4)$ notation this means that the indices $i$ and $j$ of the coefficients $\xi$ are separately symmetrized, but antisymmetrized between $i$ and $j$, i.e. we are dealing with the irrep $(0p0)$. In what follows this fact will be of crucial importance. So, we consider the potential ($m=2(g-1)$; the $SO(n)$ index $A$ and the labels $1,2$ are suppressed) $$\label{20}
F^{-2m}(Y^{++},u) = \sum_{n=0}^\infty \ \xi^{{-2(m+n)}}(u) \ (Y^{++})^n\ .$$
The factors $K$ in (\[19\]) and (\[19’\]) contribute, among others, the terms $$\label{21}
({\theta}^+)^4({{\bar\theta}}_-)^4\ R^2_{(+)} \ R^2_{(-)}\
(T^{++}_{(+)})^m\,, \qquad ({\theta}^+)^4({{\bar\theta}}_-)^4\ R^2_{(+)}({\partial}{\partial}\Phi)^2 \
(T^{++}_{(+)})^m\,,$$ respectively. The ${\theta}$’s saturate the superspace measure and are integrated out. The remainder has a harmonic charge, $$\label{22}
(T^{++})^{m} = T^{(i_1(j_1} \cdots T^{i_{m}) j_{m})}\ u^{++}_{i_1 j_1} \cdots u^{++}_{i_m j_m}$$ which is compensated by the factor $F$ in order to have a non-vanishing harmonic integral (i.e., an $SU(4)$ singlet). Clearly, (\[22\]) is an irrep of $SO(6)$, a symmetric traceless tensor of rank $m$. This can be reformulated as the highest-weight condition (cf. (\[ropcon\])) $$\label{hwcon}
D_{-{{\dot a}}}{}^{+b} (T^{++})^{m} = 0 \ .$$ A similar condition holds for the entire effective action expressions (\[19\]) and (\[19’\]) of the graviphoton field strength superfield. The singlet needed for the harmonic integral is obtained by combining (\[22\]) with the matching irrep in $F$. Consider the harmonic structure of $F$ (all ${\theta}=0$): $$\begin{aligned}
F^{-2m}(\phi^{++},u) &=& \sum_{n=0}^\infty \ \xi_{(i_1 \cdots i_{m+n}) (j_1 \cdots j_{m+n})}\ {\bar u}_{++}^{i_1 j_1} \cdots {\bar u}_{++}^{i_{m+n} j_{m+n}} \nonumber\\
& \times& \phi^{(k_1(l_1} \cdots \phi^{k_{n}) l_{n})} u^{++}_{k_1 l_1} \cdots u^{++}_{k_n l_n} \ .\label{23}\end{aligned}$$ Here we have restricted the harmonic expansion (\[20’\]) of the coefficient function $\xi^{{-2(m+n)}}(u)$ to the lowest-rank $SO(6)$ irrep. The higher-rank terms are irrelevant due to the gauge invariance of the couplings (\[19\]), (\[19’\]). Indeed, consider adding a total supersymmetrized harmonic derivative $D_{-{{\dot a}}}{}^{+b} \Lambda^{(-4g+2) }{}_b^{{{\dot a}}}({\theta}^+,{\bar\theta}_-,u)$ to the potential $F^{-4(g-1)}$. After integration by parts (the G-analytic measure allows this), $D_{-{{\dot a}}}{}^{+b}$ annihilates the on-shell superfield $K^{++}$ (recall (\[hwconK\])), hence the gauge invariance of (\[19\]), (\[19’\]) with the G-analytic parameter $\Lambda$. By examining the harmonic expansion of $\Lambda(0,0,u)$ one can show that all the omitted terms in (\[23\]) can be gauged away.
The gauge-fixed function (\[23\]) satisfies two differential conditions. The first one expresses the fact that it is a function only of the projection $\phi^{++}$ of the $SO(6)$ vector of physical scalars: $$\label{firstcon}
\frac{{\partial}}{{\partial}\phi^{--}} F^{-2m} = \frac{{\partial}}{{\partial}\phi^{a{{\dot a}}}} F^{-2m} = 0\ .$$ This is yet another kind of analyticity condition (S-analyticity), this time with respect to the scalars (which in fact are the coordinates on the curved manifold $SO(6,n)/SO(6)\times SO(n)$, see Section \[coset\]). The second one restricts the harmonic dependence $$\label{secocon}
D_{+a}{}^{-{{\dot b}}} F^{-2m} = {\epsilon}_{ab} \phi^{b{{\dot b}}}\frac{{\partial}}{{\partial}\phi^{++}} F^{-2m}\ .$$ Note that if the right-hand side in (\[secocon\]) vanished, this would be a condition defining a lowest-weight state of $SU(4)$ (or an $SO(6)$ tensor of rank $m$). However, the dependence on the scalars makes the harmonic structure in (\[23\]) reducible.
From (\[23\]) we have to extract the irreducible harmonic structure ${\bar u}_{++}^{i_1 j_1} \cdots {\bar u}_{++}^{i_{m} j_{m}}$ needed to match the conjugate structure in (\[22\]). It is obtained by contracting all the $u^{++}$ in (\[23\]) with a subset of the ${\bar u}_{++}$, using ${\bar u}_{++}^{ij}\; u^{++}_{kl} = -1/3\,
\delta^{[i}_k\delta^{j]}_l + \ldots$ (see (\[12’\])). This confirms that the omitted terms in the harmonic expansion of $\xi$ in (\[23\]) cannot contribute - they contain higher-rank $SO(6)$ tensors. The result is the [*relevant part*]{} of the function $F$, or the [*reduced*]{} function $$\label{24}
{\cal F}^{-2m} = \sum_n \xi_{(i_1 \cdots i_{m+n}) (j_1 \cdots j_{m+n})}\ {\bar u}_{++}^{i_1 j_1}
\cdots {\bar u}_{++}^{i_{m} j_{m}} \ \phi^{i_{m+1}j_{m+1}} \cdots \phi^{i_{m+n} j_{m+n}}\ .$$ Notice the full symmetrization of the $i$ and $j$ indices of the $\xi$ tensor inherited from (\[23\]). As required, the reduced function is manifestly H-analytic (i.e., $SU(4)$ irreducible), $$\label{mansu4}
D_{+a}{}^{-{{\dot b}}} {\cal F}^{-2m} = 0\ .$$ However, now the manifest S-analyticity (i.e., the dependence only on $\phi^{++}$) of (\[23\]) is lost.
It should be made clear that (\[24\]) is just a rearrangement of the harmonic expansion of the gauge-fixed function $F^{-2m}$. The information contained in this function is encoded in the fact that the coefficients $\xi_{(i_1 \cdots i_{m+n}) (j_1 \cdots j_{m+n})}$, which are the same in (\[23\]) and (\[24\]), form the $SU(4)$ irrep $(0,m+n,0)$ [^9]. This information can be translated into two types of differential constraints on the function ${\cal F}$. In general, the harmonic and scalar factors in (\[24\]) form the reducible representation $(0m0)\, \otimes\, \prod_{p=1}^n\otimes\, (010)\ \rightarrow \ (0,m+n,0) + \ldots$. The relevant projection $(0,m+n,0)$ is obtained by symmetrizing all the $i$ and separately all the $j$ indices (the antisymmetry of the $i$’s with the $j$’s is automatic). Any other irrep in this tensor product will have a subset of the $i$’s (and of the $j$’s) antisymmetrized. The product of two $u$’s is irreducible, $(010)\otimes(010) \
\rightarrow \ (020)$ as follows form the commuting nature of the $SU(4)$ harmonics ${\bar u}^i_{+a}$. The antisymmetrization of indices carried by the ${\bar u}$’s and the $\phi$’s is ruled out by the so-called [*harmonicity*]{} condition: $$\label{25}
{\epsilon}^{pqrs}\frac{{\partial}}{{\partial}{\bar u}^q_{+a}}\ \frac{{\partial}}{{\partial}\phi^{rs}_A} {\cal F} = 0\, ,$$ where we have restored the $SO(n)$ index $A$ and suppressed the $U(1)$ charge superscript ${-2m}$. The above equation forbids the decomposition $(100)\otimes(010) \ \rightarrow \ (001)$. This constraint involves partial derivatives with respect to ${\bar u}_{+}$. Strictly speaking, such an operation is illegal in the harmonic formalism, since the variables $u$ are not independent, as can be seen from (\[12’\]), (\[12\]). However the above equation can be rewritten using covariant harmonic derivatives introduced in (\[subhd\]) and (\[cosethd\]) as $$\label{25prime}
{\epsilon}^{pqrs}\left(u^{+b}_q D_{+b}{}^{+a}+ \frac{1}{2} u^{+a}_q D_0 +{\bar u}^{-\dot{b}}_q D_{-\dot{b}}{}^{+a}\right)
\ \frac{{\partial}}{{\partial}\phi^{rs}_A} {\cal F} = 0\ .$$ Indeed, it is easy to see that this equation reduces to (\[25\]) since our function ${\cal F}$ explicitly involves only ${\bar u}_{+}$ harmonics. The $D_0$ term in (\[25prime\]) is just to remove the contribution from the trace parts in $D_{+b}{}^{+a}$ as defined in (\[subhd\]) which measures the total $U(1)$ charge $-2m$ of ${\cal F}$. In the following however we will continue to write the formula using partial derivatives with respect to ${\bar u}_{+}$.
Further, the antisymmetrization of indices carried by the $\phi$’s is ruled out by the constraint $$\label{25'}
{\epsilon}^{pqrs} \frac{{\partial}}{{\partial}\phi^{pq}_A} \ \frac{{\partial}}{{\partial}\phi^{rt}_B}\ {\cal F} = 0$$ which forbids the decomposition $(010)\otimes(010) \ \rightarrow \ (101)\oplus (000)$. In $SO(6)$ (vector) notation (\[25’\]) reads $$\begin{aligned}
&& \frac{{\partial}}{{\partial}\phi^{[M}_A} \ \frac{{\partial}}{{\partial}\phi^{N]}_B}\ {\cal F} = 0\ , \label{trivial}\\
&& \frac{{\partial}}{{\partial}\phi^{M}_A} \ \frac{{\partial}}{{\partial}\phi^{M}_B}\ {\cal F} =
0\ . \label{25''}\end{aligned}$$ Here we do not take into account the fact that the physical scalars $\phi$ parametrize a curved manifold and hence the derivatives in (\[25’\]) should be considered covariant with respect to the metric of the manifold. In Section \[coset\] we show that this leads to a modification of (\[25”\]) by a term proportional to $\delta_{AB}$.
The coset of physical scalars {#coset}
=============================
The coset $SO(6,n)/SO(6)\times SO(n)$ {#subcoset}
-------------------------------------
Here we briefly recall why the scalars of ${\cal N}=4$ Poincaré supergravity describe the coset space $SO(6,n)/SO(6)\times SO(n)$ [@Cremmer:1977tt; @Bergshoeff:1980is; @de; @Roo:1984gd].
${\cal N}=4$ Poincaré supergravity is obtained by coupling the [*off-shell*]{} Weyl multiplet to $6+n$ free vector multiplets. The first six are compensating multiplets (i.e. their kinetic terms have the wrong sign), the remaining $n$ are physical. Each vector multiplet supplies 6 scalars, so the total number is $6(6+n)$. We denote the first $6\times 6$ by $\varphi^M_N$ and the other $6\times n$ by $\phi^M_A$, where $M,N =
1\ldots 6$ and $A=1\ldots n$ are $SO(6)$ and $SO(n)$ vector indices, respectively.
The Weyl multiplet contains an auxiliary field $D_{MN} = D_{NM}$, $D_{MM}=0$ in the of $SO(6) \sim SU(4)$. It serves as a Lagrange multiplier for the following quadratic combination of scalars: $$\label{26}
D_{MN}\ (\varphi^M_K \varphi^N_K - \phi^M_A \phi^N_A)\ .$$ So, it imposes an algebraic constraint which eliminates 20 of the scalars. In addition, one makes a Weyl (dilatation) gauge choice for the trace of the quadratic form in (\[26\]), thus fixing yet another scalar. So, the resulting condition is (up to normalization) $$\label{27}
\varphi^M_K \varphi^N_K - \phi^M_A \phi^N_A = \delta^{MN}\ .$$ Notice that this condition is invariant under [*local*]{} $SO(6)$ which allows to gauge away 15 additional scalars. Altogether 36 scalars are eliminated and the remaining $6n$ do indeed parametrize the coset $SO(6,n)/SO(6)\times SO(n)$.
Conditions (\[27\]) can be solved by first fixing an $SO(6)$ gauge such that the $6\times 6$ matrix $\varphi^M_N$ becomes symmetric, $\varphi = \varphi^T$, after which one can write down $$\label{28}
\varphi = \sqrt{\mathbb{I} + \phi \phi^T}\ .$$ We can say that the $6n$ physical scalars $\phi$ are the [*unconstrained*]{} coordinates on the coset $SO(6,n)/SO(6)\times SO(n)$.
Harmonic description
--------------------
The higher-derivative terms (\[19\]), (\[19’\]) involve the function (potential) $F$ defined on the coset of physical scalars. The peculiarity of this function is that it depends only on a single projection $Y^{++}_A = \phi^{++}_A(x,u) + \ldots$ of the six-vectors of coset coordinates, obtained with the help of the $SU(4)$ harmonic variables. This is a typical example of an [*analytic harmonic realization*]{} of a coset space. Another, very similar example is that of the ${{\cal N}}=4$ superconformal group $PSU(2,2/4)$ realized on the Grassmann analytic superfields (\[01\]) (see Section \[css\]). Here we explain this coset construction, following closely the case of ${{\cal N}}=2$ superconformal symmetry and Poincaré supergravity [@Galperin:1985zv; @Galperin:1987ek; @Galperin:2001uw] and of ${{\cal N}}=2$ quaternionic sigma models [@Bagger:1987rc; @Galperin:1992pj; @Ivanov:1999vg].
We start by writing down the algebra of $SO(6,n)$ in a basis suitable for the forthcoming introduction of the harmonic variables (\[12’\]). The $SO(n)$ generators are $M_{AB}=-M_{BA}$ and the $SO(6)$ ones are written in an $S(U(2)\times U(2))$ basis. Thus, the $S(U(2)\times U(2))$ generators are $$\label{29}
Z_{+a}{}^{+b}\ (Z_{+a}{}^{+a}=0)\,, \quad Z_{-{{\dot a}}}{}^{-{{\dot b}}}\ (Z_{-{{\dot a}}}{}^{-{{\dot a}}}=0)\,, \quad Z_0 \ ,$$ and the remaining generators of $SU(4)$ are $Z_{+a}{}^{-{{\dot b}}}$ and $Z_{-{{\dot a}}}{}^{+b}$. Finally, the generators of the coset $SO(6,n)/SO(6)\times SO(n)$ are $L_{A\, a{{\dot a}}}$, $L_{A\, ++}$ and $L_{A\, --}$. Then the algebra of $SO(6,n)$ takes the form [$$\begin{aligned}
[L_{A\, a{{\dot a}}},L_{B\, b{{\dot b}}}] &=& \delta_{AB}\left({\epsilon}_{{{\dot a}}{{\dot b}}} {\epsilon}_{(ac} Z_{+b)}{}^{+c} + {\epsilon}_{ab} {\epsilon}_{({{\dot a}}\dot c} Z_{-\dot b)}{}^{-\dot c}\right) + {\epsilon}_{ab}{\epsilon}_{{{\dot a}}{{\dot b}}} M_{AB} {\nonumber}\\
{[}L_{A\, a{{\dot a}}},L_{B\, ++}{]} &=& \delta_{AB} {\epsilon}_{{{\dot a}}{{\dot b}}}Z_{+a}{}^{-{{\dot b}}}{\nonumber}\\
{[}L_{A\, a{{\dot a}}},L_{B\, --}{]} &=& \delta_{AB}{\epsilon}_{ab} Z_{-{{\dot a}}}{}^{+b} {\nonumber}\\
{[}L_{A\, ++},L_{B\, --}{]} &=& \delta_{AB}Z_0 + M_{AB} {\nonumber}\\
{[}Z_{+a}{}^{+b} , L_{A\, c\dot c}{]} &=& \delta^b_c L_{A\, a\dot c} - \frac{1}{2} \delta^b_a L_{A\, c\dot c}{\nonumber}\\
{[}Z_{-{{\dot a}}}{}^{-{{\dot b}}} , L_{A\, c\dot c}{]} &=& \delta^{{{\dot b}}}_{\dot c} L_{A\, c\dot a} - \frac{1}{2} \delta^{{{\dot b}}}_{\dot a} L_{A\, c\dot c} {\nonumber}\\
{[}Z_0 , L_{A\, \pm\pm}{]} &=& \pm 2 L_{A\, \pm\pm} \label{algebra}\\
{[}Z_{+a}{}^{-{{\dot b}}}, L_{A\, --}{]} &=& {\epsilon}^{{{\dot b}}\dot c} L_{A\, a\dot c} {\nonumber}\\
{[}Z_{-{{\dot a}}}{}^{+b}, L_{A\, ++}{]} &=& {\epsilon}^{b c} L_{A\, c\dot a} {\nonumber}\\
{[}Z_{+a}{}^{-{{\dot b}}}, L_{A\, c\dot c}{]} &=& {\epsilon}_{ac} \delta^{{{\dot b}}}_{\dot a} L_{A\, ++} {\nonumber}\\
{[}Z_{-{{\dot a}}}{}^{+b}, L_{A\, c\dot c}{]} &=& {\epsilon}_{{{\dot a}}\dot c} \delta^{b}_{c} L_{A\, --}{\nonumber}\\
{[}Z_{+a}{}^{+b} , Z_{+c}{}^{+d} {]} &=& \delta^b_c Z_{+a}{}^{+d} - \delta^d_a Z_{+c}{}^{+b} {\nonumber}\\
{[}Z_{-{{\dot a}}}{}^{-{{\dot b}}} , Z_{-\dot c}{}^{-\dot d} {]} &=& \delta^{{{\dot b}}}_{\dot c} Z_{-\dot a}{}^{-\dot d} - \delta_{{{\dot a}}}^{\dot d} Z_{\dot c}{}^{-\dot b} {\nonumber}\\
{[}Z_0 , Z_{+a}{}^{+b} {]} &=& 2 Z_{+a}{}^{+b} {\nonumber}\\
{[}Z_0 , Z_{-{{\dot a}}}{}^{+b} {]} &=& -2 Z_{-{{\dot a}}}{}^{+b} {\nonumber}\end{aligned}$$ ]{} Now, we want to realize this algebra on a coset of the group $SO(6,n)$. The standard coset $SO(6,n)/SO(6) \times SO(n)$ is obtained by putting all the generators $M$ and $Z$ in the coset denominator and leaving all the $L$’s in the coset with associated $6n$ coordinates $\phi$: $$\label{30'}
\frac{SO(6,n)}{(M,Z)}\ \sim\ \{ \phi^{++}_A\,,\phi^{--}_A\,, \phi^{a{{\dot a}}}_A\}\ .$$
We wish to have an alternative [*S-analytic*]{} coset involving only the coordinates $\phi^{++}_A$ associated with the generators $L_{A\, ++}$. To this end we have to move the generators $L_{A\, a{{\dot a}}}$, $L_{A\, --}$ to the coset denominator. In doing this we encounter a problem: The $SO(6)$ generator $Z_{+a}{}^{-{{\dot b}}}$ converts $L_{A\, a{{\dot a}}}$ into the coset generator $L_{A\, ++}$. In order to avoid this, we proceed to the ‘harmonization’ of the coset. This means to introduce an additional group $\widehat{SU(4)}$ which we treat as independent of the $SO(6)$ from the coset denominator. Let us denote its generators by $T_{+a}{}^{+b}$, $T_{-{{\dot a}}}{}^{-{{\dot b}}}$, $T_0$, $T_{+a}{}^{-{{\dot b}}}$ and $T_{-{{\dot a}}}{}^{+b}$, in complete analogy with $SO(6)$. We assume that this extra $\widehat{SU(4)}$ acts as an external automorphism of (\[algebra\]), i.e. $[T,Z] = Z$, $[T,L] = L$. Then it is clear that the combination $Z_{+a}{}^{-{{\dot b}}} - T_{+a}{}^{-{{\dot b}}}$ commutes with the generators of (\[algebra\]), in particular, with $L_{A\, a{{\dot a}}}$. So, to avoid the above problem, we replace $Z_{+a}{}^{-{{\dot b}}}$ in the coset denominator by this combination. The group $\widehat{SU(4)}$ is itself realized on the harmonic coset $\widehat{SU(4)}/S(\widehat{U(2)}\times \widehat{U(2)})$, which means that we have to add the generators of the automorphism subgroup $S(\widehat{U(2)}\times \widehat{U(2)})$ to the coset denominator. The result is a particular [*S-analytic*]{} realization of the coset $$\label{30}
\frac{SO(6,n)\subset\hskip-11pt\times\ \widehat{SU(4)}}{(M,L_{a{{\dot a}}}, L_{--}, Z_+{}^+, Z_-{}^-, Z_0, Z_-{}^+,
Z_+{}^--T_+{}^-, T_+{}^+, T_-{}^-, T_0)}\ \sim (\phi^{++}_A, w_i^{+a}, w_i^{-{{\dot a}}})$$ parametrized by the coordinates $\phi^{++}_A$ associated with the $SO(6,n)$ generators $L_{A\, ++}$ and by harmonics $w_i^{+a}, w_i^{-{{\dot a}}}$ (the latter differ from the usual ${SU(4)}$ harmonics $u$ (\[12’\]), as explained below).
This coset is analytic in the sense that we consider functions $F(\phi^{++}_A,w)$ on it which are annihilated by the generators $L_{A\, a{{\dot a}}}$ and $L_{A\, --}$. Then the algebra (\[algebra\]) implies $$\label{31}
L_{A\, a{{\dot a}}}F = L_{A\, --} F = 0 \ \ \Rightarrow \ \ M_{AB}F = Z_{+a}{}^{+b}F = Z_{-{{\dot a}}}{}^{-{{\dot b}}} F = Z_{-{{\dot a}}}{}^{+b} F =0\,,$$ i.e., $F$ cannot carry $SO(n)\times SU(2)\times SU(2)$ indices, but can have $U(1)$ charges under both $Z_0$ and $T_0$. In addition, we impose the coset defining constraint $$\label{32}
(Z_{+a}{}^{-{{\dot b}}} - T_{+a}{}^{-{{\dot b}}}) F = 0 \ .$$ It leads to a particular mixing of the coordinates associated with the $SO(6)$ generators $Z$ and with the $\widehat{SU(4)}$ generators $T$. For this reason (\[30\]) is a semi-direct product (denoted by $\subset\hskip-11pt\times$ in (\[30\])) of the two cosets $SO(6,n)/SO(6) \times
SO(n)$ and $\widehat{SU(4)}/S(\widehat{U(2)}\times \widehat{U(2)})$.
The actual construction of the coset goes through the following steps. We first introduce a [*double harmonic space*]{} involving, in addition to the $\widehat{SU(4)}$ harmonic variables $u$, harmonics $\kappa_I{}^i$ on ${SU(4)} \sim SO(6)$ satisfying the defining conditions (cf. (\[12’\])) $$\label{33}
\kappa_I{}^i \bar\kappa_i{}^J = \delta^J_I\,, \quad \bar\kappa_i{}^I \kappa_I{}^j = \delta^j_i\,, \quad {\epsilon}^{IJKL}\kappa_I{}^i\kappa_J{}^j\kappa_K{}^k\kappa_L{}^l = {\epsilon}^{ijkl}\ .$$ They undergo $SU(4)$ transformations of two types: local (in the sense of ${SU(4)} \sim
SO(6)$ from the coset denominator) with parameter $\lambda$ and rigid with parameter $\sigma$: $$\label{34}
\delta \kappa_I{}^i = \lambda_I{}^J \kappa_J{}^i + \kappa_I{}^j \sigma_j{}^i \ .$$
Our task now will be to make a change of variables from $\kappa, u$ to $z, w$ which are inert under the rigid $SU(4)$ and have simple transformation properties under the local $SU(4)$. This will allow us to impose the coset constraint (\[32\]) in a covariant way. We start by projecting the harmonics $\kappa$ with $u, {\bar u}$: $$\begin{aligned}
&& \kappa_{+a}{}^{+b} = {\bar u}_{+a}{}^{I} \kappa_I{}^i u_i{}^{+b}\,, \quad \kappa_{+a}{}^{-{{\dot b}}} = {\bar u}_{+a}{}^{I} \kappa_I{}^i u_i{}^{-{{\dot b}}} {\nonumber}\\
&& \kappa_{-{{\dot a}}}{}^{+b} = {\bar u}_{-{{\dot a}}}{}^{I} \kappa_I{}^i u_i{}^{+b}\,, \quad \kappa_{-{{\dot a}}}{}^{-{{\dot b}}} = {\bar u}_{-{{\dot a}}}{}^{I} \kappa_I{}^i u_i{}^{-{{\dot b}}}\end{aligned}$$ and similarly for the conjugate matrix $\bar\kappa$. Next we make the following non-linear change of variables (to simplify the notation, we suppress the $SU(2)\times SU(2)$ indices; the position of the $U(1)$ charges allows to unambiguously restore them): $$\begin{aligned}
&& z_-{}^+ = \kappa_-{}^+ (\kappa_+{}^+)^{-1} = - (\bar\kappa_-{}^-)^{-1}\bar\kappa_-{}^+\,, \quad z_-{}^- = \bar\kappa_-{}^- {\nonumber}\\
&& z_+{}^- = \kappa_+{}^- (\kappa_-{}^-)^{-1} = - (\kappa_+{}^+)^{-1}\bar\kappa_+{}^-\,, \quad z_+{}^+ = \bar\kappa_+{}^+\ .\end{aligned}$$ These new variables satisfy an algebraic constraint following from the fact that $\kappa \in
SU(4)$, i.e. $\det \kappa =1$. It can be used to eliminate, e.g. $\det z_-{}^-$ while the remaining $z_0 \equiv \det z_+{}^+$ can be treated as the coordinate of the $U(1)$ factor in $S(U(2)\times U(2)) \subset SU(4)$.
It is then not hard to check that the new variables $z$ transform in the following way under the local $SU(4)$: $$\begin{aligned}
&& \delta z_-{}^+ = \hat\lambda_-{}^+ \,, \quad \delta z_-{}^- = z_-{}^-\hat\lambda_-{}^- \,, \quad \delta z_+{}^+ = \hat\lambda_+{}^+z_+{}^+ {\nonumber}\\
&& \delta z_+{}^- = \hat\lambda_+{}^+z_+{}^- + z_+{}^-\hat\lambda_-{}^- - \hat\lambda_+{}^-\,, \label{deltaz}\end{aligned}$$ where $\hat\lambda_\pm{}^\pm = \bar w_\pm{}^I \lambda_I{}^J w_J{}^\pm$ and we have introduced the [*new harmonics*]{} $$\begin{aligned}
&& w_i{}^{+a} = u_i{}^{+a} + u_i{}^{-{{\dot b}}}z_{-{{\dot b}}}{}^{+a}\,, \quad w_i{}^{-{{\dot a}}} = u_i{}^{-{{\dot a}}} {\nonumber}\\
&& \bar w_{+a}{}^i = \bar u_{+a}{}^i\,, \quad \bar w_{-{{\dot a}}}{}^i = \bar u_{-{{\dot a}}}{}^i - z_{-{{\dot a}}}{}^{+b} {\bar u}_{+b}{}^i \label{wharm}\end{aligned}$$ with transformation laws $$\begin{aligned}
&& \delta w_i{}^{+a} = w_i{}^{-{{\dot b}}}\hat\lambda_{-{{\dot b}}}{}^{+a}\,, \quad \delta w_i{}^{-{{\dot a}}} = 0{\nonumber}\\
&& \delta \bar w_{+a}{}^i = 0\,, \quad \delta \bar w_{-{{\dot a}}}{}^i = -\hat\lambda_{-{{\dot a}}}{}^{+b} \bar w_{+b}{}^i \ . \label{trw}\end{aligned}$$ We point out that these new harmonics are not unitary anymore (i.e., $\bar w$ is not the conjugate of $w$), but they still satisfy the same algebraic relations as the unitary harmonics $u$ (\[12’\]).
What we have achieved is that the new variables do not mix under the local $SU(4)$ transformations with parameters $\hat\lambda$. This allows us to eliminate all of the $z$ variables (with the exception of $z_0$) in a covariant way, which corresponds to imposing the $Z$ coset conditions from (\[31\]) and the $Z-T$ condition (\[32\]).
Covariant constraints on the function ${F}$
-------------------------------------------
Now we are able to see how the naive constraints (\[25\]), (\[trivial\]), (\[25”\]) are modified due to the curvature of the coset space (\[30\]) on which the reduced function ${\cal F}$ (\[24\]) lives. The origin of these constraints can be traced back to the S-analyticity conditions satisfied by the gauge-fixed function $F$ (\[23\]). On the curved manifold they become [covariant]{} constraints (cf. (\[31\])): $$\label{covsancon}
{\cal D}_{A\, a{{\dot a}}} F = {\cal D}_{A\, --} F = 0\ .$$ Here ${\cal D}_{A\, M}$ are covariant derivatives generalizing the flat derivatives ${\partial}/{\partial}\phi$. They satisfy the same $SO(6,n)$ algebra as the generators $L_{A\, M}$.
Let us start with the constraint (\[25”\]). The second-order derivative in it can be rewritten as follows: $$\begin{aligned}
{\cal D}_{A\, M}{\cal D}_{B\, M} F &=& ({\cal D}_{A++} {\cal D}_{B--} + {\cal D}_{A--} {\cal D}_{B++} -2 {\cal D}_{A a{{\dot a}}}{\epsilon}^{ab}{\epsilon}^{{{\dot a}}{{\dot b}}}{\cal D}_{Bb{{\dot b}}}) F {\nonumber}\\
&=& [{\cal D}_{A--} , {\cal D}_{B++}] F = -\delta_{AB} Z_0 F \,, \label{correctcons}\end{aligned}$$ where we have used the S-analyticity constraints (\[covsancon\]) and the algebra (\[algebra\]). The function $F^{-4(g-1)}$ has two independent $U(1)$ charges, one with respect to the generator $T_0$, $T_0 F^{-4(g-1)} = -4(g-1) F^{-4(g-1)}$ and the other for $Z_0$. For a reason which will become clear in the next section, the $Z_0$ charge takes a different value, $Z_0 F = -4(g+1) F$. Thus, we have $$\label{modconstr}
{\cal D}_{A\, M}{\cal D}_{B\, M} F = 4(g+1)\delta_{AB} F\ ,$$ or, in $SU(4)$ notation, $$\label{conts}
{\epsilon}^{pqrs} {\cal D}_{Apq} {\cal D}_{Brs}\ {F} = 32(g+1)\delta_{AB}\,{F}\ .$$
Further, the second-order derivative in (\[trivial\]) is replaced by ${\cal D}_{A\, [M}{\cal D}_{B\, N]} {F}$. Due to the constraints (\[covsancon\]), this operator has only two non-vanishing projections obtained by taking $M=++$ and $N=--$ or $N=a{{\dot a}}$. The first choice yields back the constraint (\[correctcons\]), while the second gives rise to the commutator $$\label{commu}
[{\cal D}_{Aa{{\dot a}}} , {\cal D}_{B++}] {F} = \delta_{AB}\, {\epsilon}_{{{\dot a}}{{\dot b}}} {\cal Z}_{+a}{}^{-{{\dot b}}} {F}$$ where ${\cal Z}$ is the covariant derivative replacing the generator $Z$. The effect of this is just a particular $SO(6)$ transformation of the coset coordinates, hence it is not really a constraint on the function.
Finally, in eq. (\[25\]) (or (\[25prime\])) the flat partial derivative with respect to scalars is replaced by a covariant derivative $${\epsilon}^{pqrs}\frac{{\partial}}{{\partial}{\bar u}^q_{+a}}\ {\cal D}_{Ars} {\cal F} = 0\,.\label{covfirstorderder}$$
We would like to point out that in the string theory analysis given in the following sections, the differential equations are obtained on functions ${\cal F}$ which is the relevant part of $F$ that survives the harmonic space integrals. Indeed string theory amplitudes directly see ${\cal F}$. The crucial step used in equation (\[correctcons\]) was that $F$ does not depend on 5 combinations of moduli as is expressed in the S-analyticity constraint (\[covsancon\]). It is easy to see that ${\cal F}$ does not satisfy this S-analyticity constraint since it is obtained by making a certain $SU(4)$ projection on $F$. Therefore the individual steps in this derivation cannot be applied to ${\cal F}$. However, the second order differential operators considered here are not sensitive to any particular $SU(4)$ projection of $F$ and therefore the final equations are still true on ${\cal F}$.
${{\cal N}}=4$ conformal supersymmetry and supergravity {#css}
=======================================================
Here we show that the realization of G-analytic superfields of the type (\[01\]) as functions on a particular coset of the ${{\cal N}}=4$ conformal superalgebra $PSU(2,2/4)$ is very similar to the bosonic coset construction of the preceding section. This algebra involves the generators of Lorentz transformations ($M_{{\mu}{\nu}}$), translations ($P_{\mu}$), conformal boosts ($K^{\mu}$), dilatation ($D$), R symmetry $SU(4)$ ($Z_i^j$), Poincaré supersymmetry ($Q^{\alpha}_i$ and $\bar Q_{{\dot\alpha}}^i$) and special conformal supersymmetry ($S_{\alpha}^i$ and $\bar S^{{\dot\alpha}}_i$) with anticommutation relations for the odd generators (schematically) $$\begin{aligned}
\{Q, \bar Q\} &=& P {\nonumber}\\
\{S, \bar S\} &=& K {\nonumber}\\
\{Q, S\} &=& M +D + Z \ .\end{aligned}$$ The standard superspace corresponds to the coset $$\label{44}
\frac{PSU(2,2/4)}{(M,K,D,S, \bar S, Z)}\ \sim (x^{\mu}, {\theta}_{\alpha}^i, {\bar\theta}^{{\dot\alpha}}_i)$$ involving all the 16 Grassmann variables associated with the supersymmetry generators. In order to obtain G-analytic superfields depending on half of these Grassmann variables, we add the $SU(4)$ harmonic projections of the $Q$ generators $Q^{\alpha}_{-{{\dot a}}} = {\bar u}^i_{-{{\dot a}}}\, Q^{\alpha}_i $ and $\bar Q_{{\dot\alpha}}^{+a} = \bar
Q_{{\dot\alpha}}^i\, u^{+a}_i $ to the coset denominator, thus leaving only the odd coordinates ${\theta}_{\alpha}^{+a}$ and ${\bar\theta}^{{\dot\alpha}}_{-{{\dot a}}}$ in the coset. However, exactly as in the bosonic case of Section \[coset\], the $SU(4)$ generator $Z_+{}^-$ converts $Q_-$ and $\bar Q^+$ from the coset denominator into the coset generators $Q_+$ and $\bar Q^-$. In order to avoid this, we introduce the external automorphism group $\widehat{SU(4)}$ with generators $T$. Then the combination $Z_+{}^--T_+{}^-$ commutes with all the $Q$’s and thus can be safely put in the coset denominator:[^10] $$\label{45}
\frac{PSU(2,2/4)\subset\hskip-11pt\times\ \widehat{SU(4)}}{(M,K,D,S, \bar S, Q_-, \bar Q^+, Z_+{}^+, Z_-{}^-, Z_0, Z_-{}^+, Z_+{}^--T_+{}^-, T_+{}^+, T_-{}^-, T_0)}\ \sim (x, {\theta}^+, {\bar\theta}_-, w)\ .$$ Here the harmonics $w$ are defined in exactly the same way as in Section \[coset\], eq. (\[wharm\]), replacing the $SO(6)$ harmonics $\kappa$ by R-symmetry $SU(4)$ harmonics. They transform as in (\[trw\]) with the parameter $\hat\lambda$ replaced by the G-analytic superparameter $$\label{46}
\Lambda_{-{{\dot b}}}{}^{+a}(x, {\theta}^+, {\bar\theta}_-, w) = \bar w_{-{{\dot b}}}{}^i \lambda_i{}^j w_j{}^{+a} + i{\theta}^{+a}\sigma^{\mu}{\bar\theta}_{-{{\dot b}}} k_{\mu}+ i \bar w_{-{{\dot b}}}{}^i \eta^{\alpha}_i {\theta}^{+a}_{\alpha}+ i {\bar\theta}_{-{{\dot b}}}^{{\dot\alpha}}\bar\eta_{{\dot\alpha}}^i w_i{}^{+a}$$ containing the parameters $\lambda$ of the R-symmetry $SU(4)$, $k$ of conformal boosts and $\eta$ of special conformal supersymmetry.
The basic G-analytic conformal superfield $Y^{++}(x, {\theta}^+, {\bar\theta}_-, w)$ (\[01\]) (with superconformal harmonics $w$ instead of $u$) describes the vector supermultiplet. It transforms with a G-analytic superconformal weight factor: $$\begin{aligned}
&& \delta Y^{++} = {Y^{++}}'(x',{\theta}',{\bar\theta}',w') - Y^{++}(x,{\theta},{\bar\theta},w) = \Lambda Y^{++} \label{47}\\
&& \Lambda(x, {\theta}^+, {\bar\theta}_-, w) = \rho +k_{\mu}x^{\mu}+ \bar w_{+a}{}^i \lambda_i{}^j w_j{}^{+a} + i \bar w_{+a}{}^i \eta^{\alpha}_i {\theta}^{+a}_{\alpha}+ i {\bar\theta}_{-{{\dot b}}}^{{\dot\alpha}}\bar\eta_{{\dot\alpha}}^i w_i{}^{+a} {\nonumber}\end{aligned}$$ where $\rho$ is the parameter of dilatations.[^11] The other G-analytic object we are discussing here is the descendant $K^{++}_{{\mu}{\nu}}$ (\[16\]) of the Weyl multiplet. It is superconformal covariant due to the on-shell constraint and transforms with weight two, according to its scaling dimension, $\delta K^{++} = 2 \Lambda K^{++}$.
The generalization to ${{\cal N}}=4$ conformal supergravity is done by replacing the parameters $\Lambda_{-{{\dot b}}}{}^{+a}$ and $\Lambda$ by arbitrary G-analytic superfields. Poincaré supergravity is obtained by coupling the Weyl multiplet to a set of six compensating vector multiplets (cf. (\[01\])) $$\label{48}
y^{++}_{ij}(x, {\theta}^+, {\bar\theta}_-, w) = - y^{++}_{ji} = \varphi^{kl}_{ij} w_k^{+a} w_l^{+b} {\epsilon}_{ab} + \mbox{${\theta}$ terms}\ .$$ Here we see the $6\times 6$ matrix of compensating scalars $\varphi^{kl}_{ij}$. Let us consider the following projections of $y^{++}_{ij}$ with the harmonics $w$: $$\label{49}
y^{++}_{a{{\dot a}}} = \bar w^i_{+a} \bar w^j_{-{{\dot a}}}\ y^{++}_{ij}\,, \qquad y_0 = {{{\textstyle}\frac{1}{2}}}{\epsilon}^{ab} \bar w^i_{+a}\bar w^j_{+b}\ y^{++}_{ij}\ .$$ It is easy to check that they transform as follows: $$\label{49'}
\delta y^{++}_{a{{\dot a}}} = {\epsilon}_{ab} \Lambda_{-{{\dot a}}}{}^{+b} \, y_0 + \Lambda\, y^{++}_{a{{\dot a}}}\,, \qquad \delta y_0 = \Lambda\, y_0\ ,$$ so their ratio transforms as a [*compensator*]{} for the local superconformal transformations: $$\label{50}
\delta \left(\frac{y^{++}_{a{{\dot a}}}}{y_0} \right) = {\epsilon}_{ab} \Lambda_{-{{\dot a}}}{}^{+b}\ .$$ Then, with the help of this compensator we can define new harmonics [*inert under the local superconformal transformations*]{} (notice the similarity with (\[wharm\]) and (\[trw\])): $$\begin{aligned}
&& v_i{}^{+a} = w_i{}^{+a} - w_i{}^{-{{\dot b}}}{\epsilon}^{ab}\frac{y^{++}_{b{{\dot b}}}}{y_0}\,, \quad v_i{}^{-{{\dot a}}} = w_i{}^{-{{\dot a}}} {\nonumber}\\
&& \bar v_{+a}{}^i = \bar w_{+a}{}^i\,, \quad \bar v_{-{{\dot a}}}{}^i = \bar w_{-{{\dot a}}}{}^i + {\epsilon}^{ab}\frac{y^{++}_{b{{\dot a}}}}{y_0} \bar w_{+a}{}^i \label{vharm}\\
&& \delta v = \delta \bar v = 0 \ . {\nonumber}\end{aligned}$$
The role of the compensators is to completely absorb the local superconformal transformations. This allows us to use the parameter $\Lambda_{-{{\dot a}}}{}^{+b}$ in (\[50\]) fix a gauge in which $y^{++}_{a{{\dot a}}}=0$, thus identifying the harmonics $v$ and $w$. This means, in particular, that the conformal $SU(4)$ (generators $Z$ in (\[45\])) is identified with $\widehat{SU(4)}$ (generators $T$ in (\[45\])). By the same logic, we can use the parameter $\hat\lambda_-{}^+ $ of local $SO(6)$ transformations in (\[deltaz\]) to gauge away the compensator $z_-{}^+$. This results in the identification of the harmonics $w$ with $u$. So, at the expense of manifest covariance, the different $SU(4)$ groups discussed above are reduced to a unique one, and the harmonics to the original ones (\[12’\]). This gauge fixing procedure establishes a bridge between the S-analytic coset (\[30\]) and the G-analytic coset (\[45\]).
Finally, we are ready for the superconformal covariantization of the higher-derivative terms (\[19\]), (\[19’\]). It is achieved in three steps. Firstly, we replace the explicit harmonics $u$ in $F(Y,u)$ by the new inert ones $v$ (however, the superfields $Y$ still depend on the conformal harmonics $w$). Secondly, we introduce weightless G-analytic superfields $Y/y_0$. In this way the potential $F(Y,v)$ becomes conformal invariant. Thirdly, we use the G-analytic density $y_0$ to compensate the weight $4(g+1)$ of the Weyl factor (the measure is weightless, as can be seen from its vanishing scaling dimension). The result is $$\begin{aligned}
S_1 &=& \int \, d^4x \, du \, d^4{\theta}^+ d^4{\bar\theta}_{-} \, (\bar K^{++}_{{\mu}{\nu}} \bar K^{++\, {\mu}{\nu}}) \, (y_0)^{-4(g+1)} \, (K^{++}_{\rho\sigma} K^{++\, \rho\sigma})^{g}\, F_1^{-4(g-1)}\left(\frac{Y^{++}_A}{y_0},v\right)\,,{\nonumber}\\
S_2 &=& \int \, d^4x \, du \, d^4{\theta}^+ d^4{\bar\theta}_{-} \, (K^{++}_{{\mu}{\nu}} K^{++\, {\mu}{\nu}})^{g+1} \, (y_0)^{-4(g+1)} \, F_2^{-4(g-1)}\left(\frac{Y^{++}_A}{y_0},v\right)\ . \label{19bis}\end{aligned}$$
The presence of the density $(y_0)^{-4(g+1)}$ in (\[19bis\]) explains why in (\[modconstr\]) we took the value $Z_0 F = -4(g+1) F$ of the charge $Z_0$, different from that of the charge $T_0$. This density should be viewed as part of the covariantized function $F$ discussed at the end of Section \[coset\]. Then, $F$ is a function of the G-analytic superfields $Y^{++}_A$ and $y^{++}_{ij}$ and hence is a G-analytic superconformal object itself. This means that it is annihilated by the supercharges $Q_-, \bar Q^+$ from the coset denominator in (\[45\]). This is compatible with the condition of superconformal primarity (that the object is annihilated by all the special superconformal charges $S$) only if the dilatation and $Z_0$ weights of the object coincide [@Galperin:2001uw; @Ferrara:2000eb]. Finally, the local $SU(4)$ gauge-fixing procedure (elimination of the compensators) results in the identification of the $Z_0$ charges from (\[30\]) and (\[45\]). The automorphism charge $T_0$ remains independent and, indeed, takes a different value.[^12]
Topological amplitudes - review {#Sect:Review}
===============================
In Sections \[global\] and \[coset\] it was argued from the general structure (\[24\]) of the harmonic expansion of the supergravity amplitudes $\mathcal{F}^{(1,3)}_g$ that they fulfill differential equations of first order (\[25\]) and second order (\[conts\]) in the moduli of the internal compactification manifold (i.e. $K3\times T^2$ for type II string theory). In this section, we would like to check these relations by applying them directly to the string amplitudes. Since, as we have already pointed out, the latter are captured by correlation functions of the topological string, it would be logical, to consider the twisted version of the theory. However, here we are facing the problem that some of the moduli involved in the $K3\times T^2$ compactification are in fact part of the Ramond-Ramond sector of the theory, for which we have at present no representation in terms of the ${\cal N}=4$ superconformal algebra, which is used to formulate the topological correlators. Besides that, the direct study of (\[25\]) and (\[conts\]) in the untwisted version of the type II string is quite cumbersome, since we would have to deal with (in principle) an arbitrary high number of loops.
Fortunately, as was found in [@Antoniadis:2006mr], the dual amplitudes of the couplings (\[19’\]) in the heterotic theory compactified on $T^6$ begin receiving corrections already at the 1-loop level, which are relatively simple to compute. Therefore, for the purpose of checking (\[25\]) and (\[conts\]), we will focus on this amplitude which we review below.
After performing explicitly the superspace integrals of the $1/2$-BPS $F$-type term (\[19’\]) we encounter among many different contributions a coupling of two self-dual Riemann tensors, two graviscalars and $2g-2$ graviphoton field strengths at $(g+1)$-loop order $$\begin{aligned}
S_2=\int d^4x\mathcal{F}_g^{(3)} R_{(+)}^2(\partial\partial\Phi)^2(T_{(+)}^{++})^{2g-2}\, ,\end{aligned}$$ where we remind that $\mathcal{F}_g^{(3)}$ corresponds in the supergravity context to the reduced part of $F_2^{-4(g-1)}$. The corresponding heterotic string 1-loop torus amplitude can be formulated as the following two-dimensional integral over the fundamental domain $\mathbb{F}$ of the world-sheet torus $$\begin{aligned}
{\mathcal{F}_g^{(\text{HET})}}=\int_{\mathbb{F}} \frac{d^2\tau}{\bar{\eta}^{24}}\tau_2^{2g-1} G_{g+1}\sum_{(P^L,P^R) \in \Gamma^{(6,22)}} \left( \frac{1}{2}\bar{u}_{++}^{ij}P^L_{ij} \right)^{2g-2}q^{\frac{1}{2}P_L^2}\bar{q}^{\frac{1}{2}P_R^2}.\label{hetamp}\end{aligned}$$ In this expression $\tau=\tau_1+i\tau_2$ is the Teichmüller parameter of the torus, while $q=e^{2\pi i\tau}$. Moreover, $\eta(\tau)$ is the Dedekind eta-function given by $$\begin{aligned}
\eta(\tau)=q^{\frac{1}{24}}\prod_{n=1}^\infty(1-q^n),\end{aligned}$$ and $G_{g+1}$ is defined via the following expansion of a generating functional for space-time correlation functions $$\begin{aligned}
G(\lambda,\tau,\bar{\tau})&\equiv \sum_{h=0}^\infty\frac{1}{(h!)^2}\left(\frac{\lambda}{\tau_2}\right)^{2h}
\langle\prod_{i=1}^h\int d^2x_iX^1\bar{\partial} X^2(x_i)\prod_{j=1}^{h}
\int d^2y_j\bar{X}^2\bar{\partial}\bar{X}^1(y_j)\rangle=\nonumber\\
&=\sum_{h=1}^\infty\lambda^{2h}G_{h}(\tau,\bar{\tau}).\end{aligned}$$ In [@Antoniadis:1995zn], this generating functional was calculated with the result $$\begin{aligned}
G(\lambda,\tau,\bar{\tau})=\left(\frac{2\pi i\lambda\bar{\eta}^3}{\bar{\vartheta}
(\lambda,\bar{\tau})}\right)^2
\text{exp}\left(-\frac{\pi\lambda^2}{\tau_2}\right),\end{aligned}$$ where $\vartheta$ is the usual odd theta-function defined by $$\begin{aligned}
\vartheta(z,\tau)=\sum_{n\in\mathbb{Z}}q^{\frac{1}{2}\left(n-\frac{1}{2}\right)^2}e^{2\pi i
\left(z-\frac{1}{2}\right)\left(n-\frac{1}{2}\right)}.\end{aligned}$$ The most important property of $G_g$ for our purposes is the fact that upon differentiation with respect to $\tau$ it becomes $G_{g-1}$: $$\begin{aligned}
&\frac{\partial}{\partial \tau} G_g= -\frac{i\pi}{2\tau_2^2} G_{g-1}.\label{Ggeq}\end{aligned}$$
In (\[hetamp\]), $\bar{u}_{++}^{ij}$ are precisely the harmonics of the coset $\frac{SU(4)}{S(U(2)\times U(2))}$, which appear in the reduced harmonic expansion of the ${\mathcal{F}_g^{(\text{HET})}}$ in (\[24\]). Finally, $P^L_{ij}$ and $P^R_A$ are the left- and right-moving momenta of a $\Gamma^{(6,22)}$ Narain-lattice describing the compactification of the heterotic string on the $T^6$ torus. They encode the full dependence of the amplitude on the corresponding $6\times 22=132$ moduli, which form the manifold $$\begin{aligned}
\mathcal{M}=\frac{SO(6,22)}{SO(6)\times SO(22)},\end{aligned}$$ as explained in Section \[subcoset\]. The exact parameterization of the lattice momenta, however, will be of no importance to our calculations and would involve the explicit construction of the world-sheet sigma model action, starting from the four-dimensional action of ${{\cal N}}=4$ supergravity coupled to 22 vector multiplets. The left-moving momenta $P^L_{ij}$ are formulated in a complex $SU(4)$ basis and their square is given by $$\begin{aligned}
(P^L)^2=\frac{1}{8}\epsilon^{ijkl}P^L_{ij}P^L_{kl},\end{aligned}$$ which is manifestly real and $SU(4)$ invariant. Moreover, in order to streamline our notation, we will also introduce the following projection of the momenta $$\begin{aligned}
{P^L_{++}}\equiv \frac{1}{2}{\bar{u}_{++}}^{ij}P^L_{ij}.\label{momentprojection}\end{aligned}$$
First-order harmonicity relation {#Sect:FirstOrder}
================================
With the above setting, we are now in a position to discuss the harmonicity equation (\[25\]) (or (\[covfirstorderder\])). In [@Antoniadis:2006mr], it was shown that ${\mathcal{F}_g^{(\text{HET})}}$ satisfy the following relation $$\begin{aligned}
\epsilon^{ijkl} \frac{\partial}{\partial {\bar{u}_{+1}}^i} \frac{\partial}{\partial {\bar{u}_{+2}}^j}D_{kl,A}{\mathcal{F}_g^{(\text{HET})}}=0,\label{WeakFirstOrder}\end{aligned}$$ up to an anomaly, which was calculated explicitly. The action of the differential $D_{ij,A}$ with respect to the moduli $\phi_{ij,A}$ can be analyzed in two different ways:
- From the world-sheet point of view, it amounts inserting the scalar vertex operator $$\begin{aligned}
V^{\text{mod.}}_{ij,A}=-\frac{1}{2\pi}\partial X_{ij}\bar{J}_A(z)e^{ip\cdot X},\label{scalarworldsheetvertex}\end{aligned}$$ into the correlation function, where $X_{ij}$ are the internal bosonic coordinates in an $SU(4)$ basis, satisfying the pseudo-reality condition $$\begin{aligned}
\overline{X^{ij}}=\frac{1}{2}\epsilon^{ijkl}X_{kl},\end{aligned}$$ and $\bar{J}_A$ are the right-moving (Abelian) currents.
This approach is rather cumbersome, since the correlator corresponding to e.g. (\[WeakFirstOrder\]) contains $(2g+3)$ vertices, for which all possible contractions need to be considered. We will therefore rather resort to the following approach.
- In terms of the $\Gamma^{(6,22)}$ lattice momenta, the differentials act as infinitesimal Lorentz boosts[^13] $$\begin{aligned}
&D_{ij,A}P^L_{kl}=\epsilon_{ijkl}P^R_A, &&D_{ij,A}P^R_B=\frac{\delta_{AB}}{2}P^L_{ij}.\label{naiveRules}\end{aligned}$$ These rules were proved in [@Antoniadis:2006mr] by an explicit world-sheet computation at the linearized level. It can be easily checked that they in fact reproduce the algebra (\[algebra\]), up to normalization factors. Moreover, they annihilate the $SO(6,22)$-square of the lattice vectors $$\begin{aligned}
D_{ij,A}\left((P^L)^2-(P^R)^2\right)=0.\end{aligned}$$
As we have seen in Section \[global\], the general harmonic expansion of ${\mathcal{F}_g^{(\text{HET})}}$ suggests that (\[WeakFirstOrder\]) is in fact merely a consequence of the stronger relation (\[25\]). The goal of this Section is to explicitly test the validity of (\[25\]) and to examine whether its right hand side is modified by an anomaly as it was the case for (\[WeakFirstOrder\]).
The computation is done in a straight-forward way using the differentiation rules (\[naiveRules\]) $$\begin{aligned}
E_1\equiv&\epsilon_{ab}\epsilon^{ijkl}\frac{\partial}{\partial {\bar{u}_{+b}}^j}D_{kl,A}\mathcal{F}_g^{(\text{HET})}=\label{weakstart}\\
&=\epsilon_{ab}\epsilon^{ijkl}\frac{\partial}{\partial {\bar{u}_{+b}}^j}D_{kl,A}\int \frac{d^2\tau}{\bar{\eta}^{24}}\tau_2^{2g-1}G_{g+1}\sum_{(P^L,P^R)}({P^L_{++}})^{2g-2}q^{\frac{1}{2}(P^L)^2}\bar{q}^{\frac{1}{2}(P^R)^2}=\nonumber\\
&=2\epsilon_{ab}\frac{\partial}{\partial {\bar{u}_{+b}}^j}\int \frac{d^2\tau}{\bar{\eta}^{24}}\tau_2^{2g-1}G_{g+1}\sum_{(P^L,P^R)}({P^L_{++}})^{2g-3}\bigg[(2g-2){\bar{u}_{++}}^{ij}-\pi \tau_2\epsilon^{ijkl}P^L_{kl}({P^L_{++}})\bigg]\cdot\nonumber\\
&\hspace{1cm}\cdot P_A^Rq^{\frac{1}{2}(P^L)^2}\bar{q}^{\frac{1}{2}(P^R)^2},\nonumber\end{aligned}$$ which was essentially already found in [@Antoniadis:2006mr]. Using the simple identity $$\begin{aligned}
\epsilon_{ab}\frac{\partial}{\partial {\bar{u}_{+b}}^j}P^L_{++}={\bar{u}_{+a}}^iP^L_{ij},\end{aligned}$$ we can easily calculate the harmonic partial derivative $$\begin{aligned}
E_1=2\int \frac{d^2\tau}{\bar{\eta}^{24}}&\tau_2^{2g-1}G_{g+1}\sum_{(P^L,P^R)}\bigg[3(2g-2){\bar{u}_{+a}}^i({P^L_{++}})-(2g-2)\pi \tau_2{\bar{u}_{+a}}^mP^L_{mj}\epsilon^{ijkl}P^L_{kl}({P^L_{++}})+\nonumber\\
&+(2g-3)(2g-2){\bar{u}_{+a}}^mP^L_{mj}{\bar{u}_{++}}^{ij}\bigg]({P^L_{++}})^{2g-4}P_A^Rq^{\frac{1}{2}(P^L)^2}\bar{q}^{\frac{1}{2}(P^R)^2}.\nonumber\end{aligned}$$ Using furthermore the trivial relation $$\begin{aligned}
&\epsilon^{inkl}P^L_{jn}P^{L}_{kl}=2\delta_j^i(P^L)^2,\label{momentidentity}\end{aligned}$$ we can further simplify the expression $$\begin{aligned}
E_1&=2(2g-2){\bar{u}_{+a}}^i\int \frac{d^2\tau}{\bar{\eta}^{24}}G_{g+1}\sum_{(P^L,P^R)}\left[2g\tau_2^{2g-1}-2\pi \tau_2^{2g}(P^L)^2\right]({P^L_{++}})^{2g-3}\cdot\nonumber\\
&\hspace{2cm}\cdot P_A^Rq^{\frac{1}{2}(P^L)^2}\bar{q}^{\frac{1}{2}(P^R)^2}=\nonumber\\
&=4i(2g-2){\bar{u}_{+a}}^i\int \frac{d^2\tau}{\bar{\eta}^{24}}G_{g+1}\sum_{(P^L,P^R)}\frac{\partial}{\partial\tau}\left[\tau^{2g}({P^L_{++}})^{2g-3}P_A^Rq^{\frac{1}{2}(P^L)^2}\bar{q}^{\frac{1}{2}(P^R)^2}\right].\nonumber\end{aligned}$$ At this point, we perform a partial integration in $\tau$ and use modular invariance together with the exponential suppression in the infra-red region $\tau_2\to\infty$, due to the presence of $P_L$ for $g > 1$, to conclude that there are no boundary terms we have to worry about.[^14] The only contribution therefore comes when the $\tau$-derivative acts on $G_{g+1}$. Using the identity (\[Ggeq\]) we get $$\begin{aligned}
E_1=-2(2g-2)\pi {\bar{u}_{+a}}^i\int \frac{d^2\tau}{\bar{\eta}^{24}}\tau_2^{2g-2}G_{g}\sum_{(P^L,P^R)}({P^L_{++}})^{2g-3}P_A^Rq^{\frac{1}{2}(P^L)^2}\bar{q}^{\frac{1}{2}(P^R)^2}.\label{resultweak2}\end{aligned}$$
This result has to be contrasted with the expression $$\begin{aligned}
&D_{++,A}\mathcal{F}_{g-1}^{(\text{HET})},\end{aligned}$$ where we have used the same projection as in (\[momentprojection\]) $$\begin{aligned}
D_{++,A}\equiv \frac{1}{2}\bar{u}_{++}^{ij}D_{ij,A}.\end{aligned}$$ The calculation follows much along the same lines as before and yields the result $$\begin{aligned}
&D_{++,A}\mathcal{F}_{g-1}^{(\text{HET})}=-2\pi\int \frac{d^2\tau}{\bar{\eta}^{24}}\tau^{2g-2}_2G_{g}\sum_{(P^L,P^R)}({P^L_{++}})^{2g-3}P_A^Rq^{\frac{1}{2}(P^L)^2}\bar{q}^{\frac{1}{2}(P^R)^2}.\end{aligned}$$ Comparing this result with (\[resultweak2\]), one concludes $$\begin{aligned}
\epsilon_{ab}\epsilon^{ijkl}\frac{\partial}{\partial {\bar{u}_{+b}}^j}D_{kl,A}{\mathcal{F}_g^{(\text{HET})}}=(2g-2){\bar{u}_{+a}}^iD_{++,A}\mathcal{F}_{g-1}^{(\text{HET})}.\label{weakharmcov}\end{aligned}$$ Since the $\mathcal{F}_{g-1}^{(\text{HET})}$, which appears on the right hand side is of lower order in $g$ than the initial one we considered on the left hand side, this term can be interpreted as an anomaly to the harmonicity relation. This is justified by comparison to the holomorphic anomaly equation [@Antoniadis:1993ze]-[@Bershadsky:1993ta], where (for the type II theory) the lower genus[^15] terms have their origin from boundary contributions in the moduli space of genus $g$ world-sheets.
As a trivial consistency check of this result, we can try to recover the weaker harmonicity relation presented in [@Antoniadis:2006mr], by applying a second partial differentiation with respect to ${\bar{u}_{+a}}^i$ to (\[weakharmcov\]) using the fact that it commutes with $\frac{\partial}{\partial {\bar{u}_{+b}}^j}D_{kl,A}$ $$\begin{aligned}
&\epsilon_{ab}\epsilon^{ijkl}\frac{\partial}{\partial {\bar{u}_{+a}}^i}\frac{\partial}{\partial {\bar{u}_{+b}}^j}D_{kl,A}{\mathcal{F}_g^{(\text{HET})}}=2(2g-2)(2g+1)D_{++,A}\mathcal{F}_{g-1}^{(\text{HET})},\label{strongharm}\end{aligned}$$ which is precisely the result found in [@Antoniadis:2006mr].
Second-order constraint {#Sect:SecondOrderRelation}
=======================
In the same way as equation (\[25\]), we can now check relation (\[25’\]) (or rather its counterparts (\[conts\]) and (\[commu\]) taking into account the curvature of the moduli space) by directly applying the corresponding differential operator to the topological amplitude ${\mathcal{F}_g^{(\text{HET})}}$. We use again the differentiation rules (\[naiveRules\]) to obtain [$$\begin{aligned}
E_2&\equiv\epsilon^{ijkm}D_{ij,A}D_{kl,B}{\mathcal{F}_g^{(\text{HET})}}=\nonumber\\
&=\epsilon^{ijkm}D_{ij,A}\int\frac{d^2\tau}{\bar{\eta}^{24}}\tau_2^{2g-1}G_{g+1}\sum_{(P^L,P^R)}\bigg[\frac{1}{2}(2g-2)\epsilon_{klpq}{\bar{u}_{++}}^{pq}-2\pi\tau_2P^L_{kl}({P^L_{++}})\bigg]({P^L_{++}})^{2g-3}\cdot\nonumber\\
&\hspace{1cm}\cdot P_B^Rq^{\frac{1}{2}(P^L)^2}\bar{q}^{\frac{1}{2}(P^R)^2}.\nonumber\end{aligned}$$ Taking now the second scalar derivative, one has $$\begin{aligned}
E_2&=\epsilon^{ijkm}\int\frac{d^2\tau}{\bar{\eta}^{24}}\tau_2^{2g-1}G_{g+1}\sum_{(P^L,P^R)}\bigg\{\left[-2\pi\tau_2\epsilon_{ijkl}({P^L_{++}})-\pi\tau_2P^L_{kl}\epsilon_{ijpq}{\bar{u}_{++}}^{pq}\right]({P^L_{++}})^{2g-3}+\nonumber\\
&\hspace{0.7cm}+\left[\frac{1}{2}(2g-2)\epsilon_{klpq}{\bar{u}_{++}}^{pq}-2\pi\tau_2P^L_{kl}({P^L_{++}})\right]\cdot\left[\frac{1}{2}(2g-3)\epsilon_{ijrs}{\bar{u}_{++}}^{rs}-2\pi\tau_2P^L_{ij}({P^L_{++}})\right]\cdot\nonumber\\
&\hspace{0.7cm}\cdot({P^L_{++}})^{2g-4}\bigg\}\cdot P^R_AP^R_Bq^{\frac{1}{2}(P^L)^2}\bar{q}^{\frac{1}{2}(P^R)^2}+\nonumber\\
&\hspace{0.5cm}+\frac{\delta_{AB}}{2}\epsilon^{ijkm}\int\frac{d^2\tau}{\bar{\eta}^{24}}\tau_2^{2g-1}G_{g+1}\sum_{(P^L,P^R)}P^L_{ij}\left[\frac{1}{2}(2g-2)\epsilon_{klpq}{\bar{u}_{++}}^{pq}-2\pi \tau_2P^L_{kl}({P^L_{++}})\right]\cdot\nonumber\\
&\hspace{0.7cm}\cdot({P^L_{++}})^{2g-3}q^{\frac{1}{2}(P^L)^2}\bar{q}^{\frac{1}{2}(P^R)^2}.\nonumber\end{aligned}$$]{} Regrouping the terms furthermore $$\begin{aligned}
E_2&=\int\frac{d^2\tau}{\bar{\eta}^{24}}\tau_2^{2g-1}G_{g+1}\sum_{(P^L,P^R)}\bigg\{-12\pi\tau_2\delta_l^m({P^L_{++}})^{2g-2}-4\pi(2g-2)\tau_2P^L_{kl}{\bar{u}_{++}}^{km}({P^L_{++}})^{2g-3}-\nonumber\\
&\hspace{1cm}-\pi\tau_2(2g-2)\epsilon_{klpq}\epsilon^{ijkm}{\bar{u}_{++}}^{pq}P^L_{ij}({P^L_{++}})^{2g-3}+4\pi^2\tau_2^2\epsilon^{ijkm}P^L_{kl}P^L_{ij}({P^L_{++}})^{2g-2}\bigg\}\cdot\nonumber\\
&\hspace{1cm}\cdot P^R_AP^R_Bq^{\frac{1}{2}(P^L)^2}\bar{q}^{\frac{1}{2}(P^R)^2}+\nonumber\\
&\hspace{0.5cm}+\frac{\delta_{AB}}{2}\left[(2g-2)\epsilon^{ijkm}\epsilon_{klpq}P^L_{ij}{\bar{u}_{++}}^{pq}-2\pi\tau_2\epsilon^{ijkm}P^L_{ij}P^L_{kl}({P^L_{++}})\right]({P^L_{++}})^{2g-3}q^{\frac{1}{2}(P^L)^2}\bar{q}^{\frac{1}{2}(P^R)^2}\nonumber\end{aligned}$$ and using the relations $$\begin{aligned}
&\epsilon^{klmi}P^L_{kl}P^L_{mj}=2\delta_j^i(P^L)^2,\\
&\frac{1}{2}\epsilon^{klmi}\epsilon_{mjpq}P^L_{kl}{\bar{u}_{++}}^{pq}+2P^L_{pj}{\bar{u}_{++}}^{pi}=2({P^L_{++}})\delta_j^i,\end{aligned}$$ we obtain $$\begin{aligned}
&E_2=\!-4\pi \int\frac{d^2\tau}{\bar{\eta}^{24}}\tau_2^{2g}G_{g+1}\sum_{(P^L,P^R)}\left[(2g+1)-2\pi\tau_2(P^L)^2\right]\delta_l^m({P^L_{++}})^{2g-2}P^R_AP^R_Bq^{\frac{1}{2}(P^L)^2}\bar{q}^{\frac{1}{2}(P^R)^2}+\nonumber\\
&\hspace{0.5cm}+\delta_{AB}\int\frac{d^2\tau}{\bar{\eta}^{24}}\tau_2^{2g-1}G_{g+1}\sum_{(P^L,P^R)}\left[2g-2\pi\tau_2(P^L)^2\right]\delta_m^l({P^L_{++}})^{2g-2}q^{\frac{1}{2}(P^L)^2}\bar{q}^{\frac{1}{2}(P^R)^2}-\nonumber\\
&\hspace{0.5cm}-\delta_{AB}\int\frac{d^2\tau}{\bar{\eta}^{24}}\tau_2^{2g-1}G_{g+1}\sum_{(P^L,P^R)}\left[2({P^L_{++}})\delta_m^l+(2g-2)P^L_{pm}u_{++}^{pl}\right]({P^L_{++}})^{2g-3}q^{\frac{1}{2}(P^L)^2}\bar{q}^{\frac{1}{2}(P^R)^2}\end{aligned}$$ At this point one can check that the first two lines as well as the last line, separately, are indeed modular invariant (for the last line, this follows mainly from the presence of the harmonics). Moreover, the first two lines can be written as differentiations with respect to the torus Teichmüller parameter: $$\begin{aligned}
&E_2=-8i\pi \delta_l^m\int\frac{d^2\tau}{\bar{\eta}^{24}}G_{g+1}\sum_{(P^L,P^R)}\frac{\partial}{\partial\tau}\left[\tau_2^{2g+1}({P^L_{++}})^{2g-2}P^R_AP^R_Bq^{\frac{1}{2}(P^L)^2}\bar{q}^{\frac{1}{2}(P^R)^2}\right]+\nonumber\\
&\hspace{0.5cm}+2i\delta_{AB}\delta_l^m\int\frac{d^2\tau}{\bar{\eta}^{24}}G_{g+1}\sum_{(P^L,P^R)}\frac{\partial}{\partial\tau}\left[\tau_2^{2g}({P^L_{++}})^{2g-2}q^{\frac{1}{2}(P^L)^2}\bar{q}^{\frac{1}{2}(P^R)^2}\right]-\nonumber\\
&\hspace{0.5cm}-\delta_{AB}\int\frac{d^2\tau}{\bar{\eta}^{24}}\tau_2^{2g-1}G_{g+1}\sum_{(P^L,P^R)}\left[2({P^L_{++}})\delta_m^l+(2g-2)P^L_{pm}u_{++}^{pl}\right]({P^L_{++}})^{2g-3}q^{\frac{1}{2}(P^L)^2}\bar{q}^{\frac{1}{2}(P^R)^2}.\end{aligned}$$ Since these terms are modular invariant, one is allowed to perform a partial integration, with $\frac{\partial}{\partial\tau}$ only hitting the factor $G_{g+1}$ $$\begin{aligned}
&E_2=4\pi^2\delta_l^m\int\frac{d^2\tau}{\bar{\eta}^{24}}\tau_2^{2g-1}G_g\sum_{(P^L,P^R)}({P^L_{++}})^{2g-2}P^R_AP^R_Bq^{\frac{1}{2}(P^L)^2}\bar{q}^{\frac{1}{2}(P^R)^2}-\nonumber\\
&\hspace{0.5cm}-\delta_{AB}\pi\delta_l^m\int\frac{d^2\tau}{\bar{\eta}^{24}}\tau_2^{2g}G_{g}\sum_{(P^L,P^R)}({P^L_{++}})^{2g-2}q^{\frac{1}{2}(P^L)^2}\bar{q}^{\frac{1}{2}(P^R)^2}-\nonumber\\
&\hspace{0.5cm}-\delta_{AB}\int\frac{d^2\tau}{\bar{\eta}^{24}}\tau_2^{2g-1}G_{g+1}\sum_{(P^L,P^R)}\left[2({P^L_{++}})\delta_m^l+(2g-2)P^L_{pm}u_{++}^{pl}\right]({P^L_{++}})^{2g-3}q^{\frac{1}{2}(P^L)^2}\bar{q}^{\frac{1}{2}(P^R)^2}.\nonumber\end{aligned}$$
This expression can be contrasted with $$\begin{aligned}
D_{++,A}D_{++,B}\mathcal{F}_{g-1}^{(\text{HET})},\end{aligned}$$ which can be computed using exactly the same rules as before $$\begin{aligned}
&D_{++,A}D_{++,B}\mathcal{F}_{g-1}^{(\text{HET})}=\nonumber\\
&=-2\pi D_{++,A}\int \frac{d^2\tau}{\bar{\eta}^{24}}\tau_2^{2g-2}G_{g}\sum_{(P^L,P^R)}({P^L_{++}})^{2g-3}P^R_Bq^{\frac{1}{2}(P^L)^2}\bar{q}^{\frac{1}{2}(P^R)^2}=\nonumber\\
&=4\pi^2\int \frac{d^2\tau}{\bar{\eta}^{24}}\tau_2^{2g-1}G_{g}\sum_{(P^L,P^R)}({P^L_{++}})^{2g-2}P^R_AP^R_Bq^{\frac{1}{2}(P^L)^2}\bar{q}^{\frac{1}{2}(P^R)^2}-\nonumber\\*
&\hspace{0.5cm}-\pi \delta_{AB}\int \frac{d^2\tau}{\bar{\eta}^{24}}\tau_2^{2g-2}G_{g}\sum_{(P^L,P^R)}({P^L_{++}})^{2g-2}q^{\frac{1}{2}(P^L)^2}\bar{q}^{\frac{1}{2}(P^R)^2}.\nonumber\end{aligned}$$ From this, we conclude for the second order constraint $$\begin{aligned}
&\epsilon^{ijkm}D_{ij,A}D_{kl,B}{\mathcal{F}_g^{(\text{HET})}}=\delta^m_lD_{++,A}D_{++,B}\mathcal{F}_{g-1}^{(\text{HET})}-2\delta^m_l\delta_{AB}{\mathcal{F}_g^{(\text{HET})}}-\nonumber\\
&\hspace{0.5cm}-(2g-2)\delta_{AB}\int\frac{d^2\tau}{\bar{\eta}^{24}}\tau_2^{2g-1}G_{g+1}\sum_{(P^L,P^R)}P^L_{pm}{\bar{u}_{++}}^{pl}({P^L_{++}})^{2g-3}q^{\frac{1}{2}(P^L)^2}\bar{q}^{\frac{1}{2}(P^R)^2}.\label{striresrot}\end{aligned}$$ Notice that the last two terms are not of the form of an anomaly but are generic “hard" contributions to the equation. As one can easily see, they correspond however to an $SU(4)$($\sim SO(6)$) rotation acting on the harmonics inside $\mathcal{F}_{g}^{(\text{HET})}$, which is exactly what one expects according to (\[commu\]).
On the other hand, (\[striresrot\]) is vastly simplified when we contract its free $SU(4)$ indices: $$\begin{aligned}
\epsilon^{ijkl}D_{ij,A}D_{kl,B}{\mathcal{F}_g^{(\text{HET})}}=4D_{++,A}D_{++,B}\mathcal{F}_{g-1}^{(\text{HET})}-4(g+1)\delta_{AB}{\mathcal{F}_g^{(\text{HET})}}.\label{secondorderstring}\end{aligned}$$ Comparing this result to (\[conts\]), we conclude that besides the anomalous term proportional to $\mathcal{F}_{g-1}^{(\text{HET})}$ the two relations indeed agree, up to an irrelevant normalization.\
Finally, let us mention in passing that a second order differentiation, which is antisymmetrized in the $SO(22)$ indices, is exactly vanishing $$\begin{aligned}
\frac{\partial}{\partial\phi_{[A}^{ij}}\frac{\partial}{\partial\phi_{B]}^{kl}}{\mathcal{F}_g^{(\text{HET})}}=0.\label{trivialantisymm}\end{aligned}$$ This can be seen most easily by representing the above expression as a correlator with two additional scalar vertices inserted and realizing that its right-moving part is given by $$\begin{aligned}
\langle\bar{J}_{[A}(\bar{z})\bar{J}_{B]}(\bar{w})\rangle,\label{rightmovcorr}\end{aligned}$$ which follows from the form of the scalar vertex operator (\[scalarworldsheetvertex\]). Since the right-moving currents are Abelian, it follows that expression (\[rightmovcorr\]) is identically zero. Note in particular that in this case there is not even an anomaly, and (\[trivialantisymm\]) remains in fact exact at the quantum level.
Harmonicity in six dimensions {#sixdim}
=============================
The origin of the harmonicity constraint {#Sect:Origin6dimensions}
----------------------------------------
In this subsection we summarize a few key points about six-dimensional harmonic superspace and derive the corresponding harmonicity constraint. The discussion closely follows that of the ${{\cal N}}=4$ case in four dimensions, therefore it is very brief.
We consider ${\cal N}=(1,1)$ supersymmetry in six dimensions whose automorphism group is $SU(2)_L\times SU(2)_R$. Let us introduce harmonic variables $v^I_a$ for $SU(2)_L$ and $v^{{\dot I}}_{{\dot a}}$ for $SU(2)_R$, together with their conjugates $v^a_I = (v^I_a)^*$ and $v^{{\dot a}}_{{\dot I}} = (v^{{\dot I}}_{{\dot a}})^*$. Here $a, {\dot a}$ are $SU(2)$ doublet indices while $I,{\dot I}=1,2$ are projections onto the subgroup $U(1)_L\times U(1)_R$. They satisfy the completeness conditions $$\label{1}
v^I_a\, v^a_J = \delta^I_J\,, \quad v^a_I\, v^I_b = \delta^a_b$$ (and similarly for $v^{{\dot I}}_{{\dot a}}$). Raising and lowering the indices with ${\epsilon}_{ab}$, ${\epsilon}_{IJ}$, etc., we can rewrite the non-trivial part of (\[1\]) as the unit determinant condition $$\label{1'}
{\epsilon}^{ab} v^1_a v^2_b = 1\ .$$ In fact, the harmonics can be viewed as matrices of the corresponding $SU(2)$ groups.
The harmonic functions are supposed to have harmonic expansions homogeneous under the action of the subgroup $U(1)_L\times U(1)_R$. For example, a function of unit $U(1)_L\times U(1)_R$ charges has the expansion $$\label{2}
\phi^{1{\dot 1}}(v) = \phi^{a{\dot a}} v^1_a v^{{\dot 1}}_{{\dot a}} + \phi^{(abc){\dot a}} v^1_a v^1_b v^2_c v^{{\dot 1}}_{{\dot a}} + \phi^{a({\dot a}{\dot b}\dot c)} v^1_a v^{{\dot 1}}_{{\dot a}} v^{{\dot 1}}_{{\dot b}} v^{{\dot 2}}_{\dot c} + \cdots\ ,$$ so that in each term the number of $v^1$ exceeds by one the number of $v^2$ (the same for $v^{\dot 1, \dot 2}$). Notice that due to the constraint (\[1’\]) each component is an irrep of $SU(2)_L\times SU(2)_R$ (i.e., only symmetrized indices appear). Effectively, such homogeneous functions live on the coset $S^2_L\times S^2_R = (SU(2)_L/U(1)_L)\times (SU(2)_R/U(1)_R)$.
The introduction of harmonic variables allows us to define G-analytic superfields which depend only on half of the Grassmann variables,[^16] e.g. on ${\theta}^1_{{\alpha}} = v^1_a {\theta}^a_{ {\alpha}}$, $ {\bar\theta}^{\alpha}_{ \dot 2} = v^{{{\dot a}}}_{\dot 2} {\bar\theta}^{\alpha}_{{{\dot a}}} = {\bar\theta}^{{\alpha}\dot 1}$. One such short superfield describes the (on-shell) vector multiplet $$ Y^{1{\dot 1}}({\theta}^1, {\bar\theta}_{\dot 2}, v) = \phi^{a{\dot a}} v^1_a v^{{\dot 1}}_{{\dot a}} + {\theta}^1_{\alpha}\bar\psi^{{{\alpha}}}_{{{\dot a}}} v_{{\dot 2}}^{{\dot a}}
+ {\bar\theta}_{{\dot 2}}^{\alpha}\psi_{\alpha}^a v^1_a + {\bar\theta}_{{\dot 2}}\sigma^{{\mu}{\nu}}{\theta}^1 \, F_{{\mu}{\nu}} + \cdots \ . \label{3}$$ Notice the conservation of the overall charges $1,{\dot 1}$ carried by the projected Grassmann variables or by the explicit harmonics projecting the component fields. This superfield is real in the sense $\widetilde{Y^{1{\dot 1}}} = Y^{1{\dot 1}}$, where $\widetilde{}\ $ is a combination of complex conjugation with a reflection on $S^2\times S^2$ preserving G-analyticity. In particular, this implies the reality of the first component, $(\phi^{a{\dot a}} )^* = {\epsilon}_{ab} {\epsilon}_{{\dot a}{\dot b}} \phi^{b{\dot b}} $.
Another short superfield of the same type describes the (on-shell) Weyl multiplet [@Berkovits:1994vy] $$\label{4}
(W^{1{\dot 1}})_{\alpha}{}^{\beta}({\theta}^1, {\bar\theta}_{\dot 2}, v) = (T^{a{\dot a}})_{\alpha}{}^{\beta}v^1_a v^{{\dot 1}}_{{\dot a}} + {\theta}^1_\gamma {\bar\theta}^\delta_{{\dot 2}} R_{{\alpha}\delta}{}^{{\beta}\gamma} + \cdots \ ,$$ where $(W^{1{\dot 1}})_{\alpha}{}^{\beta}$ is in the adjoint of $SU^*(4)$ ($(W^{1{\dot 1}})_{\alpha}{}^{\alpha}= 0$), $(T^{a{\dot a}})_{\alpha}{}^{\beta}= (T^{a{\dot a}})_{{\mu}{\nu}} (\sigma^{{\mu}{\nu}})_{\alpha}{}^{\beta}$ are the graviphoton field strengths and $R_{{\alpha}\delta}{}^{{\beta}\gamma} = R_{{\mu}{\nu}\lambda\rho} (\sigma^{{\mu}{\nu}})_{\alpha}{}^{\beta}(\sigma^{\lambda\rho})_\delta{}^\gamma$ is the curvature.
In [@Berkovits:1994vy] the following term of the six-dimensional effective action was considered: $$\begin{aligned}
\int d^6x \ dv \ d^4{\theta}^1 \ d^4{\bar\theta}_{\dot 2}\ &\left[{\epsilon}^{{\alpha}_1{\alpha}_2{\alpha}_3{\alpha}_4} {\epsilon}_{{\beta}_1{\beta}_2{\beta}_3{\beta}_4}(W^{1{\dot 1}})_{{\alpha}_1}{}^{{\beta}_1} (W^{1{\dot 1}})_{{\alpha}_2}{}^{{\beta}_2}(W^{1{\dot 1}})_{{\alpha}_3}{}^{{\beta}_3}(W^{1{\dot 1}})_{{\alpha}_4}{}^{{\beta}_4} \right]^g {\nonumber}\\
& \times F_{(1)^{4g-4} {({\dot 1})}^{4g-4}} ( Y^{1{\dot 1}}, v)\ . \label{5}\end{aligned}$$ In fact, what appears in (\[5\]) is the determinant of the $4\times4$ traceless matrix $(W^{1{\dot 1}})_{\alpha}{}^{\beta}$. This is a Lorentz invariant which breaks up into two independent invariants, $[{\rm Tr} (W^{1{\dot 1}})^2]^2$ and ${\rm Tr} (W^{1{\dot 1}})^4$. We could use anyone of them to construct an effective action term similar to (\[5\]). However, upon decompactification of the four-dimensional couplings (\[19\]) and (\[19’\]), one can show that only the first of the two invariants contributes. We will eventually study this case in the next subsection. The corresponding effective action term is $$\label{effact6}
\int d^6x \ dv \ d^4{\theta}^1 \ d^4{\bar\theta}_{\dot 2}\ \left[(W^{1{\dot 1}})_{{\alpha}}{}^{{\beta}} (W^{1{\dot 1}})_{{\beta}}{}^{{\alpha}} \right]^{g+1}\ F_{(1)^{2g-2} {({\dot 1})}^{2g-2}} ( Y^{1{\dot 1}}, v)\ .$$
The function $F_{(1)^{m} {({\dot 1})}^{m}}$ ($m=2g-2$) has to carry a ‘negative’ (i.e. indices $1,{\dot 1}$ downstairs) charges of each kind, in order to compensate that of the $K$ factor ($+4g$) and of the Grassmann measure $(-4)$. We consider functions of the type $$\label{6}
F_{(1)^{m} {({\dot 1})}^{m}} = \sum_{n=0}^\infty \ \xi_{(1)^{m+n} {({\dot 1})}^{m+n}}\ (Y^{1{\dot 1}})^n\ ,$$ where $$\label{6'}
\xi_{(1)^{p} {({\dot 1})}^{p}} = \xi_{(a_1\cdots a_p) ({\dot a}_1\cdots {\dot a}_p)} v_1^{a_1} \cdots v_1^{a_{p}}\ v_{{\dot 1}}^{{\dot a}_1} \cdots v_{{\dot 1}}^{{\dot a}_{p}}$$ introduces a set of [constant]{} $SU(2)_L\times SU(2)_R$ multispinors, thus explicitly breaking the symmetry.
Let us examine the coupling (\[effact6\]) in some detail. First of all, from the term $(W^{1{\dot 1}}W^{1{\dot 1}})^{g+1}$ we only consider contributions of the type $$\label{7}
({\theta}^1)^4({\bar\theta}_{\dot 2})^4 \ R^4\ (T^{1{\dot 1}})^{m} \ .$$ The Grassmann factor saturates the ${\theta}$ integrals. The harmonic dependence comes from the factor $$\label{8}
(T^{1{\dot 1}})^{m} = T^{(a_1({\dot a}_1} \cdots T^{a_{m}) {\dot a}_{m})} v^1_{a_1} \cdots v^1_{a_{m}}\ v^{{\dot 1}}_{{\dot a}_1} \cdots v^{{\dot 1}}_{{\dot a}_{m}}\ .$$ Notice that the projection with commuting harmonic variables forces symmetrization of the indices of the $T$’s. Thus, this term contributes an irrep of each $SU(2)$ of weight $m$. Since the harmonic integral in (\[effact6\]) only sees the singlet part of the integrand, we have to find a matching irrep in the $F$ sector, so that together they can form a singlet. Let us look at a term from (\[6\]) (where we replace the superfield $Y$ by its first component $\phi$), $$\begin{aligned}
\xi_{(1)^{m+n} {({\dot 1})}^{m+n}}\ (\phi^{1{\dot 1}})^n &=& \xi_{(a_1\cdots a_{m+n}) ({\dot a}_1\cdots {\dot a}_{m+n})} \ v_1^{a_1} \cdots v_1^{a_{m+n}}\ v_{{\dot 1}}^{{\dot a}_1} \cdots v_{{\dot 1}}^{{\dot a}_{m+n}} \nonumber\\
&& \times \phi^{(b_1({\dot b}_1} \cdots \phi^{b_{n}) {\dot b}_{n})} v^1_{b_1} \cdots v^1_{b_{n}}\ v^{{\dot 1}}_{{\dot b}_1} \cdots v^{{\dot 1}}_{{\dot b}_{n}}\ . \label{9}\end{aligned}$$ The first factor involves only harmonics with upper $SU(2)$ indices, the second only with lower indices. Such products of harmonics are reducible. Using the defining conditions (\[1\]), we can decompose a reducible product of $v^1_a$ with $v^b_1$ as follows: $v^1_a v^b_1 = 1/2\, \delta^b_a + v^1_{\{a} v^{b\}}_1$, where $\{\}$ denotes the traceless part. Contracting the indices of all $v^1$’s and $v^{{\dot 1}}$’s with those of a subset of the $v_1$’s and $v_{\dot{1}}$’s, we can eliminate the $v^1$’s and $v^{{\dot 1}}$’s from (\[9\]). The result is the irrep of weight $m$ of each $SU(2)$ needed to match that in (\[8\]); any traceless combination $v^1_{\{i} v^{j\}}_1$ will contribute to an irrep of higher isospin without a match in (\[8\]), thus irrelevant for the harmonic integral. So, we can reduce (\[6\]) to its relevant part $$\label{10}
{\cal F} = \sum_n \ \xi_{(a_1\cdots a_{m+n}) ({\dot a}_1\cdots {\dot a}_{m+n})} \ v_1^{a_1} \cdots v_1^{a_{m}}\ v_{{\dot 1}}^{{\dot a}_1} \cdots v_{{\dot 1}}^{{\dot a}_{m}} \ \phi^{a_{m+1}{\dot a}_{m+1}} \cdots \phi^{a_{m+n} {\dot a}_{m+n}}\ .$$
It is important to realize that the $\xi$ tensor in (\[10\]) has all its indices symmetrized. This is the origin of the harmonicity constraint $$\label{11}
{\epsilon}^{ab}\frac{{\partial}}{{\partial}v^a_1} \frac{{\partial}}{{\partial}\phi^{b\dot b}}\ {\cal F} = 0\ .$$ It involves a partial derivative with respect to $v_1$. Strictly speaking, such an operation is illegal in the harmonic formalism, since the variables $v_1$ and $v^1$ are not independent, as can be seen from (\[1\]). However, in (\[10\]) there are only $v_1$’s left, so we can [formally]{} take such a derivative. In fact, if needed, (\[11\]) can also be expressed using covariant harmonic derivatives as in (\[25prime\]).
In principle, we could go on and discuss the coset space $SO(4,n)/SO(4)\times SO(n)$ parametrized by the scalars $\phi$ of the vector multiplets (\[3\]) in a manner similar to that of Sect. \[coset\]. The conclusion would be a second-order constraint analogous to (\[correctcons\]). However, in six dimensions we do not have the setup of conformal supergravity of Sect. \[css\] which allowed us to fix the value of the charge $Z_0$ in (\[conts\]). Therefore, we can make a prediction for the structure of this constraint, but we cannot explain the precise value of the coefficient obtained from the string calculation, see (\[strire\]).
Decompactification of four-dimensional amplitudes
--------------------------------------------------
### Decompactification limit
In order to round up the six-dimensional discussion, let us now check the field theory predictions by direct string calculations for the decompactification of the topological amplitude (\[hetamp\]) from four to six dimensions, which corresponds to the coupling (\[effact6\]). Essentially, it was already shown in [@Antoniadis:2006mr] that upon decomposing $T^6$ into $T^4\times T^2$ and the subsequent reduction of the $\Gamma^{(6,22)}$ lattice into $$\begin{aligned}
\Gamma^{(6,22)}\to\Gamma^{(4,20)}\times \Gamma^{(2,2)},\label{latticeReduction}\end{aligned}$$ the weaker version of the first order harmonicity relation (\[strongharm\]) is reduced to a relation for type II string theory compactified on $K3$, proved in [@Ooguri:1995cp]. Below, we will check the stronger relation (\[11\]) and compute its corresponding quantum anomaly.
In order to perform the reduction (\[latticeReduction\]) we choose as in [@Antoniadis:2006mr] $P^L_{13}$ and its complex conjugate $P^L_{24}$ of $\Gamma^{(6,22)}$ to be entirely in $\Gamma^{(2,2)}$ and the remaining four $P^L_{12}$, $P^L_{14}$ and their complex conjugates $P^L_{34}$, $P^L_{23}$ to form the $\Gamma^{(4,20)}$. In this way, the group $SU(4)$ is reduced to its subgroup $SU(2)_L\times SU(2)_R$ where $SU(2)_L$ and $SU(2)_R$ are acting on the indices $(1,3)$ and $(2,4)$, respectively. In the decompactification limit, $P_{13}$ and $P_{24}$ decouple and are dropped from the correlation function. In this way, $\Gamma^{(4,20)}$ lattice vectors are denoted by: $$\begin{aligned}
P^{(4,20),L}_{a\dot{b}}\ ,\ P^{(4,20),R}_A\hspace{2cm}\text{with}\ a,\dot{b}=1,2\end{aligned}$$ and the index of the right-moving lattice momenta takes now the values $A=1,\ldots,20$. Moreover, the square of the left-moving momenta will be denoted by $$\begin{aligned}
(P^{(4,20),L})^2= \frac{1}{2}P^{(4,20),L}_{a_1\dot{b}_1}P^{(4,20),L}_{a_2\dot{b}_2}\epsilon^{a_1a_2}\epsilon^{\dot{b}_1\dot{b}_2}\, .\end{aligned}$$ In order to make contact with the six-dimensional harmonic coordinates introduced in Section \[Sect:Origin6dimensions\] we can assemble part of the $SU(4)$ harmonics ${\bar{u}_{+1}}$ and ${\bar{u}_{+2}}$ into the harmonics of $SU(2)_L\times SU(2)_R$ with the identification $$\begin{aligned}
&({\bar{u}_{+1}}^i, \bar{u}_{-1}^i)\rightarrow (v_1^{a}, v_2^{a}),\\
&({\bar{u}_{+2}}^i, \bar{u}_{-2}^i) \rightarrow (v_{\dot{1}}^{\dot{a}}, v_{\dot{2}}^{\dot{a}}),\end{aligned}$$ and we recall that the $SU(2)_L\times SU(2)_R$ harmonics satisfy the completeness condition (\[1\]) and the unit determinant condition (\[1’\]). Finally, the 1-loop heterotic amplitude (\[hetamp\]) was shown in [@Antoniadis:2006mr] to take the following form after the decompactification of the $T^2$ $$\begin{aligned}
\mathcal{F}^{\text{dec}}_g\sim\int \frac{d^2\tau}{\bar{\eta}^{24}}\tau_2^{2g-2} G_{g+1}(\tau,\bar{\tau})\sum_{(P^L,P^R)\in \Gamma^{(4,20)}}\left(v_1^aP^{L}_{a\dot{b}}v_{\dot{1}}^{\dot{b}}\right)^{2g-2}q^{\frac{1}{2}(P^L)^2}\bar{q}^{\frac{1}{2}(P^R)^2},
\label{fgdec}\end{aligned}$$ where from now on, we will drop the $(4,20)$ superscript of the lattice momenta and for further convenience, we define the following shorthand notation $$\begin{aligned}
{P^L_{1\dot{1}}}\equiv \left(v_1^aP^{L}_{a\dot{b}}v_{\dot{1}}^{\dot{b}}\right),\end{aligned}$$ similar to the four-dimensional definition (\[momentprojection\]).
### Harmonicity relation
We now study the six-dimensional harmonicity relation (\[11\]): $$\begin{aligned}
\epsilon^{\dot{a}\dot{b}}\frac{\partial}{\partial v_{\dot{1}}^{\dot{a}}}D_{a{{\dot b}},A}\mathcal{F}_g^{\text{dec}},\label{left6}\end{aligned}$$ where the covariant derivative $D_{a\dot{b},A}$ is with respect to the moduli forming the $\Gamma^{(4,20)}$ lattice. We can again apply simple rules for the differentials acting on the lattice momenta, similar to (\[naiveRules\]): $$\begin{aligned}
&D_{a{{\dot a}},A}P^L_{b{{\dot b}}}=\epsilon_{ab}\epsilon_{{{\dot a}}{{\dot b}}}P^R_A, &&D_{a{{\dot a}},A}P^R_B=\frac{\delta_{AB}}{2}P^L_{a{{\dot a}}}.\label{naiveRules6D}\end{aligned}$$ The computation can then be performed in the same straight-forward manner as in the four-dimensional case $$\begin{aligned}
E_1^{\text{dec}}\equiv&\epsilon^{\dot{a}\dot{b}}\frac{\partial}{\partial v_{\dot{1}}^{\dot{a}}}D_{a{{\dot b}},A}\int \frac{d^2\tau}{\bar{\eta}^{24}}\tau_2^{2g-2}G_{g+1}\sum_{(P^L,P^R)}({P^L_{1\dot{1}}})^{2g-2}q^{\frac{1}{2}(P^L)^2}\bar{q}^{\frac{1}{2}(P^R)^2}=\nonumber\\
&=\epsilon^{\dot{a}\dot{b}}\frac{\partial}{\partial v_{\dot{1}}^{\dot{a}}}\int \frac{d^2\tau}{\bar{\eta}^{24}}\tau_2^{2g-2}G_{g+1}\sum_{(P^L,P^R)}\bigg[(2g-2)v_1^cv_{\dot{1}}^{\dot{c}}\epsilon_{ac}\epsilon_{{{\dot b}}{\dot{c}}}+\pi i(\tau-\bar{\tau}){P^L_{1\dot{1}}}P^L_{a\dot{b}}\bigg]\cdot\nonumber\\
&\hspace{1cm}\cdot({P^L_{1\dot{1}}})^{2g-3}P_A^Rq^{\frac{1}{2}(P^L)^2}\bar{q}^{\frac{1}{2}(P^R)^2}.\nonumber\end{aligned}$$ The derivative with respect to the harmonic variable yields $$\begin{aligned}
&E_1^{\text{dec}}=\epsilon^{\dot{a}\dot{b}}\int \frac{d^2\tau}{\bar{\eta}^{24}}\tau_2^{2g-2}G_{g+1}\sum_{(P^L,P^R)}\bigg\{\bigg[(2g-2)v_1^c\epsilon_{ac}\epsilon_{{{\dot b}}{{\dot a}}}-2\pi \tau_2v_1^cP^L_{a{{\dot b}}}P^L_{c{{\dot a}}}\bigg]({P^L_{1\dot{1}}})^{2g-3}+\nonumber\\
&+(2g-3)\bigg[(2g-2)v_1^cv_{\dot{1}}^{{\dot{c}}}\epsilon_{ac}\epsilon_{{{\dot b}}{\dot{c}}}-2\pi\tau_2P^L_{a{{\dot b}}}({P^L_{1\dot{1}}})\bigg]v_1^cP^L_{c{{\dot a}}}({P^L_{1\dot{1}}})^{2g-4}\bigg\}P_A^Rq^{\frac{1}{2}(P^L)^2}\bar{q}^{\frac{1}{2}(P^R)^2}=\nonumber\\
&=-(2g-2)\int \frac{d^2\tau}{\bar{\eta}^{24}}\tau_2^{2g-2}G_{g+1}\sum_{(P^L,P^R)}({P^L_{1\dot{1}}})^{2g-3}\bigg[(2g-1)\epsilon_{ac}v_1^c+2\pi\tau_2v_1^cP^L_{c{{\dot a}}}\epsilon^{{{\dot a}}{{\dot b}}}P^L_{a{{\dot b}}}\bigg]\cdot\nonumber\\
&\hspace{0.5cm}\cdot P_A^Rq^{\frac{1}{2}(P^L)^2}\bar{q}^{\frac{1}{2}(P^R)^2}.\nonumber\end{aligned}$$ We can now make use of the identity $$\begin{aligned}
P^L_{b\dot{a}}\epsilon^{\dot{a}\dot{b}}P^L_{a\dot{b}}=-(P^L)^2\epsilon_{ab},\end{aligned}$$ which simplifies the expression to $$\begin{aligned}
&E_1^{\text{dec}}=\nonumber\\
&\!-(2g-2)\!\int\! \frac{d^2\tau}{\bar{\eta}^{24}}&\tau_2^{2g-2}G_{g+1}\!\!\sum_{(P^L,P^R)}\epsilon_{ac}v_1^c({P^L_{1\dot{1}}})^{2g-3}\bigg[(2g-1)-2\pi\tau_2(P^L)^2\bigg] P_A^Rq^{\frac{1}{2}(P^L)^2}\bar{q}^{\frac{1}{2}(P^R)^2}\nonumber\end{aligned}$$
The special form of this term allows for the following rewriting $$\begin{aligned}
E_1^{\text{dec}}=-2i(2g-2)\int \frac{d^2\tau}{\bar{\eta}^{24}}G_{g+1}\sum_{(P^L,P^R)}\epsilon_{ac}v_1^c\frac{\partial}{\partial\tau}\left[\tau_2^{2g-1}({P^L_{1\dot{1}}})^{2g-3} P_A^Rq^{\frac{1}{2}(P^L)^2}\bar{q}^{\frac{1}{2}(P^R)^2}\right]\nonumber\end{aligned}$$ while a partial integration in $\tau$ finally yields $$\begin{aligned}
E_1^{\text{dec}}=(2g-2)\pi\int \frac{d^2\tau}{\bar{\eta}^{24}}\tau_2^{2g-3}G_{g}\sum_{(P^L,P^R)}\epsilon_{ac}v_1^c({P^L_{1\dot{1}}})^{2g-3} P_A^Rq^{\frac{1}{2}(P^L)^2}\bar{q}^{\frac{1}{2}(P^R)^2},\label{result3}\end{aligned}$$ where we have once more made use of (\[Ggeq\]). We now confront this result with the following expression $$\begin{aligned}
&v_1^av_{\dot{1}}^{\dot{a}}D_{a{{\dot a}},A}\mathcal{F}_{g-1}^{\text{dec}}\equiv {D_{1\dot{1},A}}\mathcal{F}_{g-1}^{\text{dec}},\end{aligned}$$ which can be evaluated exactly in the same way as (\[left6\]) $$\begin{aligned}
&{D_{1\dot{1},A}}\int \frac{d^2\tau}{\bar{\eta}^{24}}\tau_2^{2g-4} G_{g}(\tau,\bar{\tau})\sum_{(P^L,P^R)}\left({P^L_{1\dot{1}}}\right)^{2g-4}q^{\frac{1}{2}(P^L)^2}\bar{q}^{\frac{1}{2}(P^R)^2}=\nonumber\\
&=-2\pi\int \frac{d^2\tau}{\bar{\eta}^{24}}\tau_2^{2g-3}G_{g}\sum_{(P^L,P^R)}({P^L_{1\dot{1}}})^{2g-3}P^R_Aq^{\frac{1}{2}(P^L)^2}\bar{q}^{\frac{1}{2}(P^R)^2}.\nonumber\end{aligned}$$ Comparing this expression to (\[result3\]), we conclude $$\begin{aligned}
\epsilon^{\dot{a}\dot{b}}\frac{\partial}{\partial v_{\dot{1}}^{\dot{a}}}D_{a{{\dot b}},A}\mathcal{F}_g^{\text{dec}}=-\frac{1}{2}(2g-2)\epsilon_{ab}v_1^b{D_{1\dot{1},A}}\mathcal{F}_{g-1}^{\text{dec}}.\label{decom1}\end{aligned}$$ Finally replacing the differentiation with respect to $v_{\dot{1}}^{{{\dot a}}}$ by one with respect to $v_1^a$, we can derive a similar equation $$\begin{aligned}
\epsilon^{ab}\frac{\partial}{\partial v_{1}^{a}}D_{b{{\dot a}},A}\mathcal{F}_g^{\text{dec}}=-\frac{1}{2}(2g-2)\epsilon_{{{\dot a}}{{\dot b}}}v_{\dot{1}}^{{{\dot b}}}{D_{1\dot{1},A}}\mathcal{F}_{g-1}^{\text{dec}}.\label{decom2}\end{aligned}$$ For both equations (\[decom1\]) and (\[decom2\]), the same considerations as in the four-dimensional case imply that the right hand side can be interpreted as an anomaly.
Notice that the left-hand side of (\[decom1\]) and (\[decom2\]) is exactly the harmonicity condition first derived in [@Berkovits:1994vy]. There, however, corrections to the equation by boundary terms of the Riemann surface as well as by certain contact terms in operator product expansions were neglected. In the later work [@Ooguri:1995cp], the missing of these extra contributions was pointed out and it was suggested that an additional contraction with harmonic coordinates would project out all extra terms. This was demonstrated by a careful analysis in the topological twisted theory. Indeed, if we project the free indices of (\[decom1\]) and (\[decom2\]) with $v_1$ and $v_{\dot{1}}$ respectively, we find $$\begin{aligned}
v_1^a\epsilon^{\dot{a}\dot{b}}\frac{\partial}{\partial v_{\dot{1}}^{\dot{a}}}D_{a{{\dot b}},A}\mathcal{F}_g^{\text{dec}}=-\frac{1}{2}(2g-2)v_1^a\epsilon_{ab}v_1^b{D_{1\dot{1},A}}\mathcal{F}_{g-1}^{\text{dec}}=0,\nonumber\\
v_{\dot{1}}^{{{\dot a}}}\epsilon^{ab}\frac{\partial}{\partial v_{1}^{a}}D_{b{{\dot a}},A}\mathcal{F}_g^{\text{dec}}=-\frac{1}{2}(2g-2)v_{\dot{1}}^{{{\dot a}}}\epsilon_{{{\dot a}}{{\dot b}}}v_{\dot{1}}^{{{\dot b}}}{D_{1\dot{1},A}}\mathcal{F}_{g-1}^{\text{dec}}=0,\nonumber\end{aligned}$$ in complete agreement with [@Ooguri:1995cp], serving as an additional check for our computation.
### Second order relation
Finally, we can also study the decompactification limit of the second order constraint (\[secondorderstring\]), whose left-hand side becomes the following differential operator: $$\begin{aligned}
\epsilon^{ab}\epsilon^{{{\dot a}}{{\dot b}}}D_{a{{\dot a}},A}D_{b{{\dot b}},B}\mathcal{F}_g^{\text{dec}}.\label{6dsecondexpression}\end{aligned}$$ Using again the differentiation rules (\[naiveRules6D\]), we can evaluate (\[6dsecondexpression\]) in a straight-forward way $$\begin{aligned}
&E_2^{\text{dec}}\equiv\epsilon^{ab}\epsilon^{{{\dot a}}{{\dot b}}}D_{a{{\dot a}},A}D_{b{{\dot b}},B}\mathcal{F}_g^{\text{dec}}=\nonumber\\
&=D_{a{{\dot a}},A}\int \frac{d^2\tau}{\bar{\eta}^{24}}\tau_2^{2g-2}G_{g+1}\sum_{(P^L,P^R)}({P^L_{1\dot{1}}})^{2g-3}\left[(2g-2)v_1^a{v_{\dot{1}}}^{{{\dot a}}}-2\pi\tau_2\epsilon^{ab}\epsilon^{{{\dot a}}{{\dot b}}}({P^L_{1\dot{1}}})P^L_{b{{\dot b}}}\right]\cdot\nonumber\\
&\hspace{0.8cm}\cdot P^R_Bq^{\frac{1}{2}(P^L)^2}\bar{q}^{\frac{1}{2}(P^R)^2}=\nonumber\\
&=-4\pi \int \frac{d^2\tau}{\bar{\eta}^{24}}\tau_2^{2g-2}G_{g+1}\sum_{(P^L,P^R)}\left[2g\tau_2-2\pi\tau_2^2(P^L)^2\right]P^R_AP^R_B({P^L_{1\dot{1}}})^{2g-2}q^{\frac{1}{2}(P^L)^2}\bar{q}^{\frac{1}{2}(P^R)^2}+\nonumber\\
&\hspace{0.3cm}+\delta_{AB}\int\frac{d^2\tau}{\bar{\eta}^{24}}\tau_2^{2g-2}G_{g+1}\sum_{(P^L,P^R)}\left[(2g-1)-2\pi\tau_2(P^L)^2\right]({P^L_{1\dot{1}}})^{2g-2}q^{\frac{1}{2}(P^L)^2}\bar{q}^{\frac{1}{2}(P^R)^2}-\nonumber\\
&\hspace{0.3cm}-g\delta_{AB}\mathcal{F}_g^{\text{dec}}.\nonumber\end{aligned}$$ Following the same steps as before, we can re-write the first two lines as total derivatives with respect to $\tau$, namely $$\begin{aligned}
E_2^{\text{dec}}&=8\pi i\int\frac{d^2\tau}{\bar{\eta}^{24}}G_{g+1}\sum_{(P^L,P^R)}\frac{\partial}{\partial\tau}\left[\tau_2^{2g}({P^L_{1\dot{1}}})^{2g-2}P^R_AP^R_Bq^{\frac{1}{2}(P^L)^2}\bar{q}^{\frac{1}{2}(P^R)^2}\right]+\nonumber\\
&\hspace{0.3cm}+2i\delta_{AB}\int\frac{d^2\tau}{\bar{\eta}^{24}}G_{g+1}\sum_{(P^L,P^R)}\frac{\partial}{\partial\tau}\left[\tau_2^{2g-1}({P^L_{1\dot{1}}})^{2g-2}q^{\frac{1}{2}(P^L)^2}\bar{q}^{\frac{1}{2}(P^R)^2}\right]-\nonumber\\
&\hspace{0.3cm}-g\delta_{AB}\mathcal{F}_g^{\text{dec}},\nonumber\end{aligned}$$ which after a partial integration become $$\begin{aligned}
E_2^{\text{dec}}&=4\pi^2\int\frac{d^2\tau}{\bar{\eta}^{24}}\tau_2^{2g-2}G_{g}\sum_{(P^L,P^R)}({P^L_{1\dot{1}}})^{2g-2}P^R_AP^R_Bq^{\frac{1}{2}(P^L)^2}\bar{q}^{\frac{1}{2}(P^R)^2}-\nonumber\\
&\hspace{0.3cm}-\pi\delta_{AB}\int\frac{d^2\tau}{\bar{\eta}^{24}}\tau_2^{2g-3}G_{g}\sum_{(P^L,P^R)}({P^L_{1\dot{1}}})^{2g-2}q^{\frac{1}{2}(P^L)^2}\bar{q}^{\frac{1}{2}(P^R)^2}-g\delta_{AB}\mathcal{F}_g^{\text{dec}}.\nonumber\end{aligned}$$ In order to simplify this result, we also evaluate the expression $$\begin{aligned}
&{D_{1\dot{1},A}}{D_{1\dot{1},B}}\mathcal{F}_{g-1}^{\text{dec}}=\nonumber\\
&=-2\pi {D_{1\dot{1},A}}\int \frac{d^2\tau}{\bar{\eta}^{24}}\tau_2^{2g-3}G_g\sum_{(P^L,P^R)}({P^L_{1\dot{1}}})^{2g-3}P^R_Bq^{\frac{1}{2}(P^L)^2}\bar{q}^{\frac{1}{2}(P^R)^2}=\nonumber\\
&=4\pi^2\int \frac{d^2\tau}{\bar{\eta}^{24}}\tau_2^{2g-2}G_g\sum_{(P^L,P^R)}({P^L_{1\dot{1}}})^{2g-2}P^L_{a{{\dot a}}}P^R_AP^R_Bq^{\frac{1}{2}(P^L)^2}\bar{q}^{\frac{1}{2}(P^R)^2}-\nonumber\\
&\hspace{0.3cm}-\pi\delta_{AB}\int \frac{d^2\tau}{\bar{\eta}^{24}}\tau_2^{2g-3}G_g\sum_{(P^L,P^R)}({P^L_{1\dot{1}}})^{2g-2}P^L_{a{{\dot a}}}q^{\frac{1}{2}(P^L)^2}\bar{q}^{\frac{1}{2}(P^R)^2}.\end{aligned}$$ We can thus obtain the relation $$\begin{aligned}
\epsilon^{ab}\epsilon^{{{\dot a}}{{\dot b}}}D_{a{{\dot a}},A}D_{b{{\dot b}},B}\mathcal{F}_g^{\text{dec}}={D_{1\dot{1},A}}{D_{1\dot{1},B}}\mathcal{F}_{g-1}^{\text{dec}}-g\delta_{AB}\mathcal{F}_g^{\text{dec}}. \label{strire}\end{aligned}$$ As already mentioned in Section \[Sect:Origin6dimensions\], the general structure of this equation can be anticipated from field theory, especially, the existence of the term proportional to $\delta_{AB}$ on the right hand side of (\[strire\]). However, due to the lack of the setup of conformal supergravity, we are not in a position to predict the exact coefficient $-g$, which is also different from the coefficient in the four-dimensional analog of the second order constraint (\[secondorderstring\]).
Note finally, that the six-dimensional couplings $\mathcal{F}^{\text{dec}}_g$ (\[fgdec\]) of the 1/2-BPS effective operator (\[effact6\]), although obtained by taking the decompactification limit of the ${{\cal N}}=4$ topological amplitudes ${\cal F}_g^{(3)}$, they are not given by the topological theory on $K3$. The reason is that in their exact genus $g+1$ type II expression, the $\det{\rm Im}\tau$ factors from the space-time coordinates do not cancel. Thus, these couplings are semi-topological, in the sense that string oscillator modes do not contribute, and upon compactification on a $T^2$ they become exactly topological.
Conclusions {#Sec:Concl}
===========
In conclusion, in this work, we generalized the holomorphicity property of the ${{\cal N}}=2$ supersymmetric couplings involving vector multiplets to the moduli dependence of the ${{\cal N}}=4$ couplings of 1/2-BPS operators defined in harmonic superspace. An example of such operators is provided by the two series found in [@Antoniadis:2006mr], involving powers of the (superdescendant of the) ${{\cal N}}=4$ chiral Weyl superfield $K$ whose coupling-coefficients are functions of the ${{\cal N}}=4$ vector moduli and are computed by the ${{\cal N}}=4$ topological string on $K3\times T^2$.
The resulting harmonicity or analyticity property is expressed in terms of two sets of differential constraints: the first requires the vanishing of one scalar and one harmonic derivatives, while the second imposes two scalar (covariant) derivatives to give back the same coupling up to a multiplicative constant proportional to its (super)conformal weight. We verified these equations on the string side using the explicit expressions for the couplings of one of the two series as 1-loop heterotic integrals on $T^6$.
We also extended the above analysis to ${{\cal N}}=2$ 1/2-BPS terms in six dimensions and we checked the resulting equations for the couplings obtained in the decompactification limit of the four-dimensional ${{\cal N}}=4$ topological amplitudes considered before. In principle, our analysis can be generalized in a straight-forward way to the couplings of any 1/2-BPS operator of extended supersymmetry in any space-time dimension.
Acknowledgements {#acknowledgements .unnumbered}
================
We have profited form enlightening discussions with N. Berkovits, B. de Wit, S. Ferrara and E. Ivanov. This work was supported in part by the European Commission under the RTN contract MRTN-CT-2004-503369, in part by the INTAS contract 03-51-6346 and in part by the French Agence Nationale de la Recherche, contract ANR-06-BLAN-0142. The work of S.H. was supported by the Austrian Bundesministerium für Wissenschaft und Forschung.
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[^1]: [email protected]
[^2]: On leave from CPHT (UMR CNRS 7644) Ecole Polytechnique, F-91128 Palaiseau
[^3]: [[email protected]]{}
[^4]: [[email protected]]{}
[^5]: [[email protected]]{}
[^6]: A more systematic derivation of the G-analytic superfields as functions on a coset of the ${{\cal N}}=4$ superconformal algebra $PSU(2,2/4)$ will be given in Section \[css\].
[^7]: The notion of Grassmann analyticity (with breaking of the R symmetry) was first proposed in [@Galperin:1980fg] in the context of the ${{\cal N}}=2$ hypermultiplet. Later on it was made R-symmetry covariant in the framework of ${{\cal N}}=2$ harmonic superspace in [@Galperin:1984av]. The harmonic superspace description of the ${{\cal N}}=3$ [*off-shell*]{} vector multiplet was given in [@Galperin:1984bu], and that of the ${{\cal N}}=4$ [*on-shell*]{} vector multiplet in [@Hartwell:1994rp].
[^8]: The other possibility, which we do not consider here, would be to use singular functions involving inverse powers of fields.
[^9]: $SU(4)$ irreps with Dynkin labels $(0p0)$ are equivalent to rank $p$ symmetric traceless tensor of $SO(6)$.
[^10]: Here we follow the formulation of ${{\cal N}}=2$ conformal supersymmetry of [@Galperin:1985zv; @Galperin:2001uw]. A somewhat different approach is proposed in [@Hartwell:1994rp].
[^11]: It can be shown that $\Lambda_{-{{\dot b}}}{}^{+a} = D_{-{{\dot b}}}{}^{+a}\Lambda$.
[^12]: This situation is different from ${{\cal N}}=2$ superconformal symmetry where the relevant G-analytic superfields, e.g. the hypermultiplet, have equal $Z_0$ and $T_0$ charges [@Galperin:2001uw]. This can be explained by the different properties of the G-analytic superspace measures – the ${{\cal N}}=2$ measure has a conformal weight while the ${{\cal N}}=4$ one does not.
[^13]: Note a factor of 2 misprint in eq. (10.7) of [@Antoniadis:2006mr] which had no effect in the subsequent analysis.
[^14]: Note that the equation is trivially fulfilled in the case $g=1$, since $\mathcal{F}_1^{(\text{HET})}$ is independent of the harmonic variables.
[^15]: In [@Bershadsky:1993cx] scattering amplitudes of two (self-dual) Riemann tensors and $(2g-2)$ graviphoton field strengths in type II theory compactified on Calabi-Yau threefolds were considered. In these amplitudes, the number $g$ corresponds to the genus of the world-sheet Riemann surface.
[^16]: We use $SU^*(4)\sim SO(1,5)$ chiral spinor notation with left-handed $\psi_{\alpha}$ and right-handed $\bar\psi^{\alpha}$ spinors.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'Existing black-box attacks on deep neural networks (DNNs) so far have largely focused on transferability, where an adversarial instance generated for a locally trained model can “transfer” to attack other learning models. In this paper, we propose novel black-box attacks for adversaries with query access to the target model’s class probabilities, which do not rely on transferability. We also propose strategies to decouple the number of queries required to generate each adversarial sample from the dimensionality of the input. An iterative variant of our attack achieves close to 100% adversarial success rates for both targeted and untargeted attacks on DNNs. We carry out extensive experiments for a thorough comparative evaluation of black-box attacks and show that the proposed attacks outperform all transferability based black-box attacks we tested on both MNIST and CIFAR-10 datasets, achieving adversarial success rates similar to well known, state-of-the-art white-box attacks. We also apply the attacks successfully against a real-world Content Moderation classifier hosted by Clarifai. Furthermore, we evaluate black-box attacks against state-of-the-art defenses. We show that the attacks are very effective even against these defenses.'
author:
- |
Arjun Nitin Bhagoji [^1]\
Department of Electrical Engineering\
Princeton University\
Warren He, Bo Li & Dawn Song\
EECS Department\
University of California, Berkeley\
bibliography:
- 'iclr2018\_conference.bib'
title: 'Exploring the Space of Black-box Attacks on Deep Neural Networks'
---
Introduction {#sec: intro}
============
Background and Evaluation setup {#sec: setup}
===============================
Query based attacks: attack {#sec: grad_est_attack}
===========================
Attacking defenses {#sec: defenses}
==================
Attacks on Clarifai: a real-world system {#sec: clarifai}
========================================
Existing black-box attacks {#sec: existing}
==========================
Conclusion {#sec: conclusion}
==========
[^1]: Work done while visiting UC Berkeley
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'Neglected tropical diseases (NTD), particularly vector-borne diseases (VBD), account for a large proportion of the global disease burden, and their control faces several challenges including diminishing human and financial resources for those distressed from such diseases. Visceral Leishmaniasis (VL), the second-largest parasitic killer in the world (after malaria) affects poor populations in endemic countries and causes considerable cost to the affected individuals and their society. Mathematical models can serve as a critical tool for understanding the driving mechanisms of a NTD such as VL. The WHO promotes integrated control programs for VL but this policy is not well supported by systematic quantitative and dynamic evidence and so potential benefits of the policy are limited. Moreover, mathematical models can be readily developed and used to understand the functioning of the VL system cheaply and systematically. The focus of this research is three-fold: (i) to identify non-traditional but critical mechanisms for ongoing VL transmission in resource limited regions, (ii) to review mathematical models used for other infectious diseases that have the potential to capture identified factors of VL, and (iii) to suggest novel quantitative models for understanding VL dynamics and for evaluating control programs in such frameworks for achieveing VL elimination goals.'
author:
- |
Swati DebRoy$^1$ , Olivia Prosper$^2$, Austin Mishoe$^1$, Anuj Mubayi$^{3}$[^1]\
\
$^{1}$Department of Mathematics and Computational Science, University of South Carolina-Beaufort, Beaufort, SC, USA\
$^{2}$Department of Mathematics, University of Kentucky, Lexington, KY, USA\
$^{3}$Simon A. Levin Mathematical, Computational, and Modeling Sciences Center, Arizona State University, Tempe, AZ, USA
title: |
Challenges in Modeling Complexity of Neglected Tropical Diseases:\
Assessment of Visceral Leishmaniasis Dynamics in Resource Limited Settings
---
Introduction {#Intro}
============
Visceral Leishmaniasis (VL) is a neglected vector-borne infectious disease that is transmitted to humans by infected sandflies and is the second-largest parasitic killer in the world after malaria [@chappuis2007visceral; @WHOFactSheet]. If left untreated, most cases result in death within two to three years of clinical manifestation. Most of the new cases (approximately 90%) occur in Bangladesh, Brazil, Ethiopia, India, Nepal, South Sudan, and Sudan. VL is identified as a Neglected Tropical Disease by the WHO because it is endemic in several poverty stricken regions of the world, although preventive measures and successful treatment is common in most developed countries. Many people living in these impoverished regions are daily-wage workers, for whom infection with a disease like VL restricts the bread-winners’ ability to provide livelihood for their families. Moreover, the cost of treatment and duration of stunted income pushes them into a vicious cycle of further hardship and irrecoverable financial deprivation. Although local government authorities and the WHO have devised several control programs to lower the burden of VL in these regions, the VL endemicity always creeps back after a brief period of relief. This ineffectiveness has been attributed to several factors, including severe under-reporting of cases and death due to VL, lack of clarity in the etiology of the disease, and limited estimation of reservoirs of the infection. Thus, the intensity and extent of the control programs were in conflict with the magnitude of the true VL burden. With limited resources available in many of the affected countries, mathematical modeling can help shed light on several of these challenges cheaply (including identifying cost-effective driving mechanisms), as it has done for other infectious diseases like malaria. Hence, immediate attention from the modeling community is in dire need.
In the past, the WHO has set several elimination target dates for VL, which could not be achieved in the Indian subcontinent. The primary reason for this shortcoming may be the ineffective implementation of policies in the face of a developing country’s infrastructure. Mathematical modeling approaches in conjunction with model guided additional field research in India could be a turning point for achieving optimal program implementation and may help to (1) quantify the “true" burden of VL in Bihar where it has proven to be particularly difficult to eliminate, (2) investigate the potential mechanisms for the spread of the Leishmania parasite, and (3) suggest optimal vector control programs that may help in achieving the WHO goal of elimination of VL by the year 2017 [@le2016feasibility]. The VL elimination program target is to reduce the annual incidence to less than 1 per 10,000 at the district or sub district level in South Asia by 2017 (WHO). Currently, the incidence is around 20 cases per 10,000 [@mondal2009visceral; @dhillon2008national; @chowdhury2016implication]. Understanding the mechanisms driving the transmission dynamics of VL may require the study of several factors, including complex interactions of multiple reservoir hosts, environment-dependent vector dynamics, changes in political and public health policy, spread of resistance to insecticides and drugs, and short and long term human migration patterns.
Since Sir Ronald Ross’ first paper using a mathematical model to study the transmission dynamics of malaria in 1906, there have been many modeling studies focusing primarily on infectious diseases; however, more studies are needed on neglected tropical diseases such as VL. One of the aims of this study is to suggest mathematical modeling approaches for capturing identified regional issues that may be critical in better evaluating control programs in resource limited settings, thereby, assisting in the development of cost effective elimination strategies. We discuss specific features of VL (including treatment availability, living conditions, effect of social status, and implementation cost of control programs) that should be incorporated into quantitative methods that can be effectively used to analyze control strategies for the disease. Carefully established and directly relevant mathematical models are urgently needed for VL control in an effort to develop a suitable tool to truly capture the complex dynamics of the disease within the given natural or man-made environments and to achieve elimination goals. The assessment of a tropical disease risk must be interpreted on the basis of local environment conditions, the effects of socioeconomic development, and its capability to effectively sustain control programs.
Challenges for Leishmaniasis in Resource-Limited Regions {#Risks}
========================================================
When considering the socio-economic challenges of a neglected tropical disease at the grassroots level, the depth and complexity appears overwhelmingly varied. So, to better comprehend the nature of obstacles, we classified some of the key issues in to the six categories viz., atmosphere, access, availability, awareness, adherence and accedence (6 A-s), which in turn can be traced back to lack of either inculcation or infrastructure. Each of these variables ultimately stems from the sheer poverty and its viscious cycle with diseases in the affected region (Figure \[fig:cycle\]). In this review we focus on the state of Bihar, which hosts 90% of India’s VL cases (WHO), and its neighboring countries of Nepal and Bangladesh where the disease dynamics is similar. The categories are briefly described below.
![A viscious cycle of socio-economic challenges and difficulty to access disease interventions[]{data-label="fig:cycle"}](Fig1){width="5in"}
**Atmosphere:** In the state of Bihar, the worst affected areas are in remote agricultural villages. Studies have revealed several living conditions positively correlated with higher prevalence of leishmaniasis. In two independent studies, factors like mud plastered houses, vegetation and bamboo near the house, and granary inside the house were found to significantly contribute to leishmaniasis [@dhiman1991epidemiology; @ranjan2005risk]. Other systematic studies in the subcontinent with similar geographical settings (Uttar Pradesh, India; West Bengal, India; Terai, Nepal; Mymensingh, Bangladesh) have found that living conditions with cracked mud walls, damp floors, and close proximity to a water body are risk factors for leishmaniasis. Also, a high density of occupants in a household with more than three people per room were found to increase transmission [@bern2007epidemiology; @nandy1987leishmanin; @barnett2005virgin; @bern2005risk; @bern2000factors; @ranjan2005risk; @schenkel2006visceral; @saha2009visceral; @rukunuzzaman2008epidemiological]. Models that have incorporated such features (house types, household size etc.) exists for other diseases [@yong2015agent; @kasaie2014timing; @perez2009agent; @akhtar2007chain; @noiva2014susceptible; @cohen2001modeling].
It is well known that sand-fly bites thrive during the warmer months (March - June, October), and late in the evening [@dinesh2001seasonal]. The role of climatic factors on transmission dynamics of vector-borne diseases has been thoroughly studied in the literature [@artzy2010transmission; @parham2010modeling]. The hot Indian summer in combination with lack of electricity often lead people to sleep outdoors, which increases the number of sand-fly bites and hence the risk of contracting leishmaniasis [@barnett2005virgin; @bern2000factors]. Understandably, proper use of bed-nets have been found to have a protective effect on people across several studies [@bern2007epidemiology; @barnett2005virgin; @bern2005risk; @schenkel2006visceral; @saha2009visceral; @rukunuzzaman2008epidemiological] and sleeping on a cot (versus on the floor) also demonstrated a protective effect. Models have shown that proximity to domesticated animals was found to play a complex role in containing, spreading and serving as a possible reservoir of the parasite [@gorahava2015optimizing]. For example, some studies found that proximity to livestock provided a protective effect against leishmaniasis [@bern2005risk; @bern2000factors], whereas in Uttar Pradesh, India, the risk of leishmaniasis was found to increase with increased numbers of cattle in the vicinity of a household [@barnett2005virgin].
**Access:** Currently, therapuetic interventions for Kala-azar (Indian VL) are significantly subsidised by the ministry of health in India (National Vector Bourne Disease Control Program’s Kala-azar Elimination Initiative under the Govt. of India). There are 38 District Health Societies (DHS) inside the state healthcare system (State Health Society) of Bihar. The DHS are further subdivided into a number of block-Primary Health Centers (PHCs); the number of PHCs per DHS varies for each DHS. Again, the PHC consists of Sub-Centers providing health care to a certain number of villages (e.g. the Muzaffarpur district (population 3.7 million) has fourteen Block PHCs while the Kanti Block (population 337,670, Census of India, 2001) has 48 Sub-Centers, each covering a population of roughly eight thousand individuals [@Singh2006serious]). PHCs, district hospitals and government medical colleges are the sources of reported cases (National Vector Borne Disease Control Programme, 2009). The private health sector includes not-for-profit and for-profit organizations. For-profit venues include corporations (e.g., private nursing homes), trusts, stand-alone specialist services, pharmacy shops, and self-appointed practitioners. Estimates suggest that 80% of the outpatients and 57% of the inpatients are handled in the private sector (The World Bank report, 2001 [@peters2001raising]). NGOs usually provide awareness and education programs, carryout research, and provide access to regular health services. Ninety percent of Bihar’s population lives in rural areas where less than 1% of health services are provided by not-for-profit/Non-Government Organizations (NGOs) (The World Bank, 2001 [@peters2001raising]). Patients in rural areas travel on average much further for treatment than patients in urban areas. Thus, access to healthcare can be tricky at present and efforts need to be made to encourage the set-up of temporary mobile clinics in harder to reach areas and to encourage people to seek out certified treatment. Bihar is the poorest state in India, where the “caste" (proxy for social standing) of a person is born into affects almost every aspect of the social conduct he/she receives their entire life. [@van2009decomposing]. Martinez [@Martinez2012] found that the people of lower caste are consistently being seen by a doctor at a more advanced stage of VL than those of a higher caste. In fact, most VL patients in the disadvantaged caste see a doctor more than eight weeks post symptom onset, which includes a larger spleen and lower hemoglobin level than normal. Thus, efforts need to focus more on the people of lower caste to diminish the disparity in healthcare; only then can planned control measures effectively reduce the overall burden of VL. Models considering underreporting of cases due to treatment of patients by non-reporting private healthcare clinics and patients’ healthcare seeking behaviors have been developed [@mubayi2010transmission; @medley2015health].
**Availability:** The WHO recommends the use of a single dose of Amphotericin B as the first line of treatment in the Indian sub-continent [@matlashewski2011visceral]. However, daily injections of Pentavalent Antimonials (SSG) for 20-30 days and 15 injections of Amphotericin B every other day are still more widely used in India (NVBDCP). The availability of drugs in a timely manner is dependent on several factors including the affordability of a drug by the government, reasonable forecasting of the quantity of drug required (to avoid shortage as well as waste), proper storage and distribution of the drugs throughout the lengthy route from the manufacturer to the affected people, avoiding cheaper counterfeit drugs and also drug legislation [@den2011leishmaniasis]. Mathematical models can capture each of these features [@moghadas2008population; @katouli2011worst] and study their role in the spread and transmission dynamics of VL.
**Awareness:** ‘Awareness’ can be defined as knowledge regarding the etiology of the disease which would help local individuals to prevent infection and to look out for VL symptoms and seek medical attention sooner rather than later. Figure \[fig:examples\_programs\] shows some social aspects for which awareness programs may be needed as a prevention for VL. Lack of awareness causes a disease which is curable upon treatment, to end up causing death. Even symptomatic VL-infected people mix in the population freely, thus considerably increasing the chances of transmission. In a study on Nepal (which shares a boarder with Bihar) by Rijal 2006 [*et al.*]{}, it was found that affected people from the poorest strata of the community preferred to visit a private doctor or local faith healer over public health clinics, leading to higher costs for these individuals [@rijal2006economic]. Moreover, debts aquired during this period, in addition to lack of income (the earning adult being ill), creates a major financial abyss which is almost impossible to recover from. Thus, it is not sufficient that the government provides free treatment to the people, it is also necessary to disseminate that information in an effective manner to every strata of people in the affected region. In a study in Brazil, significant awareness was spread in communities through educating school children, who in turn were assigned to discuss interventions mentioned in student’s homework assignments with their family members [@magalhaes2009dissemination]. Models have studied the role of awareness programs on transmission dynamics of diseases [@pittet2006evidence; @bhunu2011mathematical].
![Cartoon reflecting social aspects on which awareness programs to control spread of VL can be designed to reduce disease burden[]{data-label="fig:examples_programs"}](challenges_figure1){width="6in"}
**Adherence:** Non-adherence to treatment is a major factor contributing to the high development of resistance to pentavalent antimonials in the population exposed to VL in the Indian sub-continent [@den2011leishmaniasis], and subsequently resulted in it being discontinued as a first line treatment for VL. There are two major factors which contribute to non-adherence in the region: lack of knowledge about the consequences of incomplete treatment leads to patients stopping treatment once the symptoms are relieved, and the financial loss due to reduced days of productivity while on therapy is a major deterrent to continuing treatment. It has been well documented in a 2000-2010 cohort study in Nepal that another disease, Post Kala-azar Dermal Leishmaniasis (PKDL), a sequel and reservoir for VL, was more common in patients who were inadequately treated during VL versus the ones who adhered to the full course of treatment. The overall prevalence of PKDL in SSG treatment was 2.9%, 0.3% in supervised and 4.5% in unsupervised treatment [@uranw2011post]. As observed by Rijal 2006 [*et al.*]{}, loss of productivity implies no income at all for the poor families [@rijal2006economic]. Farmers are unable to attend to their fields, possibly during very important farming phases which results in lowered income for a considerable period of time. Alot of people in the poorest section of society in Bihar are also daily-wage earners and each missed day of work might present dire consequences for the entire family. To survive this period, the family takes out loans from private lenders at high interest, ultimately leading them to further poverty. Thus, it is important to inform the people that non-adherence would lead to relapses which are harder to treat; although the burden of missed work is difficult to accept in the immediate context, it is better ultimately for the family. Consequently, providing financial assistance to affected families while their primary earning members are under treatment may improve the outcome of VL control programs, by encouraging greater adherence to treatment. Although there is a need for better models that can capture irregular treatment adherence levels among patients, some simple models exist in the literature [@huang2008modeling; @smith2006adherence; @kalichman2010adherence; @fisman2014projected].
**Accedence:** ‘Accedence’ is defined as the acceptance to undergo testing and treatment for PKDL, which can occur in patients who have recovered from VL post-treatment caused by the same pathogen. As reported by Desjeux [*et al.*]{} in 2013, patients with PKDL serve as a reservoir of VL and it is unlikely that VL can be eradicated without addressing the issue of diagnosing and treating PKDL [@desjeux2013report]. In fact, aggressive measures are required to encourage people to consult a doctor in case of any persistent skin lesions, and once diagnosed these should be treated with Amphotericin B, which has been found effective in the Indian sub-continent. As reported by Thakur [*et al.*]{}, attempts were made to fast-track diagnosis and treatment of PKDL in several badly affected districts of Bihar and yielded a very positive outcome [@thakur2009newer]. However, in addition to the use of mathematical modeling methods, these efforts need the support of the local governmental authorities to continue and succeed where it is most needed.
It is evident that aggressive control measures are necessary to address every issue and to ultimately alleviate the burden of VL from Bihar. However, several of the issues arise from limited resources in the affected region and thus the control strategies should be carefully weighed by importance and cost-effectiveness. Moreover, it is not only important to take drastic measures using the one-time funds generated for this purpose, but it is essential that powerful and sustainable changes in the system are established through easy and systematic ways of approaching the difficulties, which can only be attained by identifying critical mechanisms of the system. Mathematical models are one of the best methods to cheaply and systematically identify driving factors.
Review of VL Mathematical Modeling Studies {#Modeling}
==========================================
Despite the incalculable harm and countless challenges leishmaniasis inflicts on populations around the globe, only a handful of publications address the problems from a mathematical modeling perspective. In fact, a recent review by Rock [*et al.*]{}, which tabulates all mathematical models of VL, found only twenty-four articles using mathematical models for VL, several of which used the same base model structure [@rock2015uniting]. Of these twenty-four articles, only seven addressed VL in the Indian Subcontinent. Arguably, one of the greatest modeling challenges is the limited understanding of the leishmania pathogen, the sandfly vector, and how disease manifests in humans. Dye [*et al.*]{} [@dye1988earthquakes] spear-headed the application of mathematical models to leishmania dynamics. The authors developed a simple discrete-time model with [*Susceptible*]{}, [*Infected*]{}, and [*Resistant*]{} humans to study the mechanism behind inter-epidemic periods observed in VL cases between 1875 and 1950 in Assam, India. Counter to the existing theory of the time, the model demonstrated that the observed inter-epidemic patterns could be explained by intrinsic factors in leishmania transmission. This modeling effort also stressed the significance of PKDL and sub-clinical infections in determining whether a region will have endemic or epidemic leishmaniasis. A few years later, Hasibeder [*et al.*]{} [@hasibeder1992mathematical] published a compartmental delay-differential equation model for canine leishmaniasis. This model accounts for two types of dogs: those that will develop symptoms, and those that will remain asymptomatic, following infection by a sandfly. The model also explicitly describes the infection dynamics of the sandfly vector and considers a fixed delay representing the extrinsic incubation period. The authors take a heuristic approach to derive a formula for the basic reproduction number $R_0$, the number of secondary sandfly infections resulting from a single infected sandfly, in an otherwise fully susceptible population. Although this model addresses two important aspects of the natural history of the disease that may be extended to human VL, namely the asymptomatic human and infected vector populations, the model does not consider the asymptomatic population to be an infectious reservoir, assumes constant human and vector population sizes, and omits the effects of seasonality. The model does, however, introduce heterogeneous biting, determined by a dog’s “occupation". The mathematics developed in [@hasibeder1992mathematical] was applied to age-structured serological data for the dog population in Gozo, Malta in [@dye_epidemiology_1992], and provided estimates for $R_0$. This modeling work was extended in [@dye1996logic] to include zoonotic transmission, that is, humans, dogs, and sandflies, were explicitly modeled. Dye conducted a sensitivity analysis to determine which of three control measures would be most effective in decreasing disease prevalence. Their results suggest vector control and vaccination of the human or dog population would be more effective than treating or killing infected dogs.
More recently, Stauch [*et al.*]{} developed a more comprehensive model of VL for the Indian subcontinent [@stauch2011visceral], and later extended it to include drug-resistant and drug sensitive [*L. donovani*]{} parasites, with a focus on Bihar, India [@stauch2012treatment]. The model proposed in [@stauch2011visceral] extended the basic SIR model structure for the human population by segmenting the infected stage into five distinct stages according to an individual’s infection status determined by the results of three diagnostic markers. These diagnostic markers were (1) PCR, the earliest infection marker, (2) DAT, which measures antibody response, and (3) LST, which can detect cellular immunity. The model also includes treatment of symptomatic VL cases, treatment failure, relapse characterized as PKDL, PKDL treatment, and HIV-co-infection (described in their Supplementary materials). The sandfly population is modeled using an SEI ([*Susceptible-Latent-Infectious*]{}) model. Treatment of VL is divided into first and second-line treatment, and treatment-induced mortality caused by drug-toxicity is considered. The model was fitted to data from the KalaNet trial using Maximum Likelihood. The authors explored several intervention strategies, including treatment, active case detection, and vector control. Although the authors warn that their model assumes homogeneous transmission, ignoring possible clustering of cases within affected households, their modeling approach and parameter estimation argues that the large asymptomatic reservoir precludes the ability for a treatment-only control program to attain the desired target of less than 1 case per 10,000 individuals annually. Vector-based control is much more promising, but the authors estimate it can only reasonably reduce VL incidence to 18.8 cases per 10,000. Consequently, the authors emphasize the need for active case detection, effective treatment of PKDL, and vector control to achieve VL elimination.
Based on the model in [@stauch2011visceral], and following up on their finding that treatment of VL does little to reduce transmisson, Stauch [*et al.*]{} investigated the uncertainty in their parameter estimates and explored the efficacy of different vector-based control measures in [@stauch2014model]. They estimated that $R_0$ for VL is approximately 4.71 in India and Nepal, and that reducing the sandfly population by 79% via reduction of breeding sites, or reducing the sandfly population by 67% through increasing sandfly mortality, are both sufficient to eliminate the [*L. donovani*]{} parasite in the human population. The authors argue that recent evaluations of IRS (indoor residual spraying) efficacy suggest that elimination should be possible, with the caveat that the situation may change if insecticide resistance emerges. However, vector management using LLIN’s (long-lasting insecticide-treated nets) or EVM (environmental management) would not be sufficient to achieve elimination. The authors emphasize the need to study infection rates, the parasite dynamics in both the human and vector population, animals serving as alternate hosts or potentially infection reservoirs, and the contribution of the asymptomatic population. Furthermore, Stauch [*et al.*]{} suggest extensions of the deterministic modeling framework to include heterogeneity and seasonality.
In [@stauch2012treatment], Stauch [*et al.*]{} extended the model in [@stauch2011visceral] to include both resistant and sensitive parasites. The authors considered five mechanisms by which the fitness of the resistant strain may differ from the sensitive strain: (1) increase probability of symptoms, (2) increase parasite load, (3) increase infectivity of asymptomatic humans, (4) increase transmission from symptomatic and asymptomatic host to vector, (5) increase transmission from vector to host. Simulations of this extended model indicate that a treatment failure rate over 60% is required to explain observations in Bihar. Furthermore, observations in Bihar cannot be explained without assuming an increase in fitness in resistant parasites. The authors explain that it is more likely that the necessary additional fitness is transmission-related rather than disease-related. Unfortunately, their results also suggest that once a more fit resistant parasite has been introduced, that parasite will eventually exclude the sensitive parasite, even in the absence of treatment.
The work of Mubayi [*et al.*]{} [@mubayi_transmission_2010] is the first to use a rigorous, and dynamic model to estimate underreporting of VL cases at the district level in Bihar, India. The authors designed a staged-progression model, composed of a system of nonlinear, coupled, ordinary differential equations. The stage-progression model exploits the fact that the sum of $n$ independent exponential distributions with rate parameter $\lambda$ is a gamma distribution with shape parameter $n$ and scale parameter $1/\lambda$ ($\Gamma(n,1/\lambda)$), and captures the observed variability in the incubation period, infectious period, and treatment duration. Furthermore, the authors address the differences between public and private health care facilities in their treatment and reporting practices by assuming a fraction of infected individuals $p$ seek treatment in public health care facilities, and the remaining proportion seek treatment in private clinics. Building an empirical distribution for this parameter $p$ and deriving a relationship between model parameters and monthly reported incidence data allowed the authors to estimate the degree of underreporting for each district for the years 2003 and 2005. This model analysis informed by incidence data revealed that districts previously designated as low-risk areas for VL are actually likely to be high-risk: the true burden masked by underreporting.
ELmojtaba [*et al.*]{} presented a more classical approach to modeling VL, with a focus on Sudan, in [@elmojtaba2010mathematical; @elmojtaba2010modelling; @elmojtaba2013vaccination]. Because leishmaniasis in the Sudan is zoonotic, the authors included the dynamics of an animal reservoir in [@elmojtaba2010mathematical]. This baseline model is extended in [@elmojtaba2010modelling] to address parasite diversity, and in [@elmojtaba2013vaccination] to explore the potential impact of mass vaccination in the presence of immigration. All of these modeling efforts have either contributed to our understanding of VL or highlighted the need for better data to construct and validate future models of VL. However, there are currently no models, to the best of our knowledge, that attempt to link socio-economic factors, like the 6 A’s discussed in Section \[Risks\], to disease transmission.
Bridging Socio-economic Factors and Mathematical Models
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In this section, we provide some examples of published mathematical modeling studies where researchers have attempted to incorporate some of the factors mentioned above and studied their role in the transmission dynamics of infectious and physiological diseases. Existing models of visceral leishmaniasis, though limited in number, have incorporated some of the biological complexity, contributing to a more developed understanding of the disease. However, to formulate applicable control measure recommendations with cost-estimation, models which can simultaneously incorporate the discussed risk-factors explicitly would be necessary. Many of the techniques to incorporate these factors individually can be drawn from the literature regarding heavily studied diseases like HIV, malaria, and tuberculosis.
In a simplistic mathematical model we can incorporate several risk factors associated with ambience implicitly through the interpretation and calculation of the model parameters. For example, the transmission parameter can be considered as a product of the predominant type of housing, density of vegetation around houses, number of domesticated animals, and number of inhabitants in a house.
Lipsitch [*et al.*]{} addressed adherence to treatment and its role in promoting drug resistance in a mathematical model for tuberculosis (TB) in the presence of bacterial heterogeneity [@lipsitch1998population]. To model non-compliance to drug therapy, the authors assumed that non-compliant patients adhere to the treatment regimen when bacterial loads are above a certain threshold, and will halt treatment when bacterial loads fall below this threshold, that is, $$Adherence\_level(t) = \begin{cases}
\frac{\alpha B(t)}{K+B(t)} & if B(t)\geq N_{min} \\
0 & if B(t)< N_{min}
\end{cases}$$ where $B(t)$ is the bacterial load at time $t$ and $N_{min}$ is the theshold minimum bacterial load under which drug treatment is discontinued. Simulation and analysis of their stochastic-deterministic hybrid model illustrated that non-compliance is one mechanism that can give rise to bacteria resistant to one or more drugs in a multi-drug therapy. Furthermore, the authors noted that the pattern of resistance driven by non-compliance more closely resembled observations of patients on multi-drug therapy, compared with the pattern of resistance promoted by bacterial heterogeneity. Consequently, the model suggests that non-compliance plays a larger role than heterogeneity of the bacteria population in promoting resistance during multi-drug therapy. The authors noted that an exception to this pattern occurred in HIV-positive TB patients. The modeling assumption for non-compliance in this TB model addresses one of the ‘adherence’ concerns discussed in Section \[Risks\], namely that patients often stop treatment once symptoms are relieved, suggesting a possible framework in which to study adherence to treatment, treatment failure, and if tied to a population-level model, the spread of drug resistant parasites in VL-endemic regions.
![General categories of modeling frameworks for studying dynamics of vector-borne diseases and types of data needed in such frameworks[]{data-label="figmathmodel"}](Math_fig){width="6in"}
Adherence to treatment is also a concern in diabetes patients, despite the fact that non-adherence increases the likelihood for stroke and other potentially fatal complications. Mason [*et al.*]{} [@mason2012optimizing] developed a Markov Decision process (MDP) model to study the timing of treatment initiation and drug-adherence in diabetes patients and the role these two factors play in determining a patients’ quality-adjusted life years (QALYs). Consistent with observations of adherence behavior in diabetes patients, the model assumed that a patient’s health status does not influence future adherence. This assumption may be relevant for some VL-endemic regions where non-adherence is a consequence of insufficient inculcation of the risks associated with improper treatment, or a result of the cost of treatment. The model also assumed, consistent with clinical practice, that if a patient or the patient’s physician had not already decided to begin treatment, treatment would be immediately initiated following a non-fatal complication. The reward function, dependent on adherence, included several important factors, including QALY, the cost of treatment and hospitalization, and disutility resulting from treatment side effects. The authors also developed a cost function, with the goal of optimizing treatment initiation, in the presence of different degrees of non-adherence, and compared the optimal timing for ‘uncertain adherence’ and ‘predictable adherers’. The authors quantified the benefit of treatment relative to the cost through a reward function $r_t(l,m)$, where $l$ denoted the patient’s health status, and $m$ equaled zero or one, depending on whether the patient was on treatment or not. $$r_{t}(l,m)=R(l,m)-C_{t}^{0}-(CF^{S}(l)+CF^{CHD}(l))-mC^{ST}(A)-(C^{S}(l)+C^{CHD}(l)),$$ for $t=1, \dots,T-1,l\in L,m\in M,$ where $$R(l,m)=R_0(d^{S}(l))(d^{CHD}(l))(md^{ST}(A)),~ l\in L,~ m\in M$$ describes the reward for 1 quality-adjusted life. The decrement factors $d^{S}, d^{CHD},$ and $d^{ST}$ denote the decrease in quality of life from a stroke (S), a coronary heart disease (CHD) event, or statins initiation (SI), respectively. The costs $C^{O}, C^{ST}, C^{S}$ and $C^{CHD}$, and $C^{FS}$ and $CF^{CHD}$ represent the cost of other health care for diabetes patients, cost of statin treatment, cost of initial hospitalization for stroke and CHD events, and cost of follow-up treatment for stroke and CHD events, respectively. This diabetes model suggested that initiation of treatment should be delayed in individuals predicted to have poor future adherence. Furthermore, the model predicted that over time, only 25% of patients will remain adherers for greater than 80% of the days during the study. The effect of change in disease dynamics due to behavioral change and educational awareness have been modeled using ordinary differential equations-based models in several studies. Hallett 2009 analyzed the effect of behavior change on the course of an HIV epidemic [@hallett2009assessing]. In their dynamic model, the behavior is incorporated by considering parameters such as mean rate of partner change and condom use in casual relationship as a step-function depending on time at which the change in behavior occurred and time it takes to reach a new value. In Mushayabasa 2012, an ordinary differential equation model was used to quantify the role of an educational campaign in controlling Hepatitis C among women in prison [@mushayabasa2012assessing]. Here, the effect of this campaign is reflected as an efficacy factor in conjunction to the parameters which represent the sharing of contaminated needles or syringes among the susceptible and exposed classes.
Ideas for VL modeling should also draw from modeling techniques used in economics in the context of social sciences to effectively optimize the cost and strategy in a resource-limited region. Fenichel [*et al.*]{} used an economic behavioral model in conjunction with the classical Susceptible-Infected-Recovered (SIR) model that explicitly models adaptive contact behavior [@fenichel2011adaptive]. The authors construct a utility function, an index which describes an individual’s well-being. The framework assumes that individuals make choices that maximize their utility. These decisions then impact disease risk, creating a feedback loop between disease risk and decisions made based on perceived disease risk. The authors demonstrate that fitting the classical SIR model to data generated by their new framework results in erroneous estimates of epidemiological parameters, because of its inability to jointly estimate behavioral and biological parameters. See Perrings [*et al.*]{} for a thorough review on the growing topic of economic epidemiology [@perrings2014merging]. A study by Gorahava [*et al.*]{} develops a dynamic optimization model to suggest novel ways of allocating insecticides in the districts based on sizes of both human and cattle populations while considering limited financial and resource constraints [@gorahava2015optimizing]. The model maximizes number of sandflies killed by insecticides intervention and minimizes number of human cases while identifying optimal number of houses and cattle sheds to be sprayed for a given budget.
The results of the above models addressing adherence to treatment, adaptive human behavior, and resource constraints emphasize the need to bridge the gap between socio-economic factors and existing VL modeling frameworks. The 6 A’s should be systematically incorporated into VL model frameworks to assess the sensitivity of model dynamics to these six socio-economic factors. Failing to address factors that result in significant changes in disease dynamics may result in models that do not effectively inform public health policy. Likewise, models that do not acknowledge resource constraints may lead to infeasible control policies.
Discussion {#Dis}
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Leishmaniasis continues to spread in every continent on the planet except Australia and Antartica, and VL is most common in the poorest regions of modern human civilization. The WHO has identified leishmaniasis as a neglected tropical disease owing to its endemicity in the under-developed tropical regions of the world, in spite of available treatment options in first world countries. The reason for the ongoing spread and failure to control lies mainly in underreporting of the disease burden [@mubayi2010transmission], poor infrastructure, lack of awareness, poverty and inadequate control measures. In this review we have presented some of the less highlighted, but nonetheless very important, factors which are key in one of the VL affected neglected regions, the Indian state of Bihar, and possibly in several other impoverished regions; these factors may also play a critical role in the transmission of other NTDs [@mubayi2010transmission].
Mathematical models have been used to understand disease dynamics in other parasitic infections and recommend control measures under different constraints. Thus, we propose mathematical modeling as a cheap and effective tool to devise meaningful control measures that will make the next WHO leishmaniasis elimination goal a reality. Most of the mathematical modeling research on leishmaniasis has focused on the development and choice of drugs and co-infection with other diseases [@hussaini2016mathematical; @rock2015uniting; @stauch2012treatment]. Although the importance of these topics cannot be overlooked, more attention needs to be focused on socio-economic issues leading to lack of infrastructure, inculcating awareness, and promoting healthier practices. In this article we identified key issues relating to these factors as observed and published in the literature. We also reviewed current mathematical models used for leishmaniasis and discussed some ways of explicitly incorporating these socio-economic issues into mathematical models. In light of the major financial constraints in the affected regions, a hybrid dynamic optimization model (an example framework is shown in Figure \[figmathmodel\]) will be necessary, which can calculate monetary (cost of interventions) and non-monterary (mortality and morbidity) factors related to VL for the specific region taking into consideration the socio-economic drawbacks. The building of such a model will require detailed quantification of every aspect of life in the regions, including non-tangible issues. Moreover, the execution of this model will require extensive sets of data on these varied aspects, which can be challenging considering the current dearth of data. However, in absence of data, projections can be made for different scenarios to elucidate an understanding of the magnitude of the problem, and to estimate the relative importance of different socio-economic factors to accurately predicting disease dynamics and informing effective public health policies.
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[^1]: Corresponding author, e-mail: [email protected]
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We propose a design for an universal absorber, characterized by a resonance frequency that can be tuned from visible to microwave frequencies independently of the choice of the metal and the dielectrics involved. An almost resonant perfect absorption up to $99.8 \%$ is demonstrated at resonance for all polarization states of light and for a very wide angular aperture. These properties originate from a magnetic Fabry-Perot mode that is confined in a dielectric spacer of $\lambda/100$ thickness by a metamaterial layer and a mirror. An extraordinary large funneling through nano-slits explains how light can be trapped in the structure. Simple scaling laws can be used as a recipe to design ultra-thin perfect absorbers whatever the materials and the desired resonance wavelength, making our design truly universal.'
author:
- Rafik Smaali
- Fatima Omeis
- Antoine Moreau
- Thierry Taliercio
- Emmanuel Centeno
title: Universal metamaterial absorber
---
Introduction
============
The control of light absorbance plays a fundamental role in today’s photonics technologies with strong impacts for solar energy harvesting or for light emitting and sensing components [@Collin:2014hd; @Liu:2010kw; @Park:2015gm]. Since according the Kirchhoff’s law, perfect absorbers and emitters are equivalent, significant efforts are pursued to realize compact artificial materials presenting an almost perfect absorption in a selective spectral range, for any polarization or incidence angle[@Shi:2014jb; @Hedayati:2014gf]. The targeted operating frequency usually imposes the choice of the materials constituting the absorbers and also strongly constraints the design: plasmonic absorbers have proven to be effective for realizing compact absorbers for visible and infrared radiations while metamaterials are preferably used from the terahertz to the microwaves [@Cui:2014bd; @Landy:2008gy]. Many designs have been proposed recently, a lot of them relying on the excitation of cavity resonances of some kind[@Kats:2012hr; @Kats:2012eb; @Yao:2014ii]. All these structures being cavities, a minimum size is required that is always larger than $\lambda/20$ despite strategies to reduce the effective wavelength of the mode responsible for the resonance[@Moreau:2012ub; @Collin:2007wp].
Here we propose a design for a very deeply subwavelength resonant absorber whose absorption frequency can be tuned from infrared to microwave frequencies by following simple scaling laws. This absorber is universal in the sense that its optical properties are independent from the choice of the metals and dielectrics involved for its realization and it presents an almost perfect absorption for incident angles up to $30^\circ$ and for any polarization of light. The absorption mechanism is based on a negative phase shift for the light which happens at the metamaterial interfaces and allows to built up a Fabry-Perot (FP) resonance into a near-zero dielectric thickness of $\lambda/100$. This resonance is activated by a funneling effect through slits of few nanometers wide, with a ratio of the period to the width of the slits that can easily be larger than 30,000.
In a first part, we concentrate on metamaterial absorber periodic in one direction and give the physical picture to explain its properties for one polarization. The resonance responsible for the absorption is demonstrated to be a FP resonance squeezed into a sub-wavelength thickness. In a second part, we show using the previous model that the design can be tuned to absorb almost perfectly for any frequency. Finally, these results are extended to universal absorbers periodic in two directions that present an absorption line for any polarization and a wide range of incidence angles.
Theory and design of super absorbers
====================================
The absorber consists of a deeply subwavelength grating made of nanometers slits etched in a thin metallic slab which is separated from a metallic back mirror by a dielectric spacer, Fig.1. The operating frequency range extends from the infrared to microwaves frequency when noble metals such as gold or silver are utilized (see supporting information). The upper frequency boundary is limited by the plasma frequency of the metallic medium that is typically located in the ultraviolet spectrum for noble metals but can be located in the mid-infrared for highly doped semiconductors. Without any loss of generality, we illustrate our results by considering InAsSb, a highly doped semiconductor whose plasma frequency can be tuned by playing with the doping concentration. This material is in addition compatible with CMOS technology and its relative permittivity is given by a Drude model $\epsilon_{InAsSb}=\epsilon_{\infty}(1-\omega_p^2/(\omega(\omega+i\gamma)))$ with $\epsilon_{\infty}=11.7$, $\omega_p= 351. 10^{12}\ rad.s^{-1}$ and $\gamma=10^{13}\ rad.s^{-1}$ [@NTsameGuilengui:2012jn; @Taliercio:2014fn]. The spacer, of thickness $g$, is filled with a GaSb insulator (that is assumed to be non-dispersive) of refractive index $n_d=3.7$. The absorbance $A$ is deduced from the energy reflection coefficient $R$ computed using the Rigorous Coupled-Wave Analysis (RCWA)[@Weiss:2009kx]. First, we consider a 1D periodic set of slits of width $f=10 \ nm$ etched in the x-direction, with a pitch $d=2\ \mu m$ and a thickness $h=320\ nm$ for the metamaterial layer. Two absorption lines can be observed in normal incidence on the computed absorption spectrum, for p-polarization case ([*i.e.*]{} a magnetic field along the slits), Fig. 2a. The lower resonant wavelength, $\lambda_s= 11 \ \mu m$, corresponds to a cavity-like mode localized into the slits, Fig. 2b-c. Similarly to the case of Extraordinary Optical Transmission (EOT) [@GarciaVidal:2010ed], a gap-plasmon having a high effective index $\bar{n}_{slit}=n_{slit}+i\kappa_{slit}$ is excited and reflected at the top and at the bottom of the metamaterial layer. The real part of the effective index is well approximated by $n_{slit}=\sqrt{1+2\delta_p/f}$ where $\delta_p=c/\omega_p$ is the penetration depth into the metal [@Collin:2007wp; @Pardo:2011cp]. The resonant condition for the slit reads $\lambda_s=2n_{slit} h+\lambda_{\Phi}$ where $\lambda_{\Phi}$ is a phase shift linked to the reflection coefficient of the gap-plasmon inside the metamaterial layer [@Moreau:2007jv; @Koechlin:2013wn]. The spatial extension of the magnetic field for the second absorption line at $\lambda_r=77 \ \mu m$ indicates that it can be assimilated to a symetric Fabry-Perot resonance localized inside the spacer of sub-wavelength thickness $g=850 \ nm$ (about $\lambda/100$) well below the common quarter-wavelength criterium, Fig. 2d-e.
{width="0.8\columnwidth"}
![(a) Absorbance for a 1D metamaterial absorber ($f=10 \ nm$, $d=2\ \mu m$, g=850 nm). The solid and dashed curves are respectively obtained with the exact electromagnetic simulation and with the equivalent dielectric model. (b) and (c) Maps of the modulus of the magnetic and electric fields corresponding to $\lambda_s= 11 \ \mu m$. (d) and (e) Maps of the modulus of the magnetic and electric fields corresponding to de FP resonance $\lambda_r= 77 \ \mu m$.[]{data-label="fig:fig2"}](fig2.eps){width="0.8\columnwidth"}
To explain this actual reduction of the FP cavity size, we derive an equivalent dielectric model assuming that the metallic grating is equivalent to a dielectric slab of complex refractive index $\bar{n}=n+i\kappa$ and thickness $\bar{t}$, Fig.1c. The optical property of such artificial dielectric layer is known to depend on the geometrical parameters of the grating and on the effective index of the gap plasmon by $\bar{n}=\bar{n}_{slit}d/(f+2\delta_p)$ [@Koechlin:2013wn]. The resonant wavelength of the gap plasmon mode is equivalently linked to the effective index and thickness by $\lambda_s=2\bar{n}\bar{t}$. With these definitions, the whole 1D absorber can be replaced by a much simpler equivalent system made of a dielectric spacer sandwiched between a back mirror assumed to be a perfect electric conductor (PEC) and an absorbing layer of complex index $\bar{n}$ corresponding to the grating layer, Fig. 1c. As seen on Fig. 2a, this equivalent dielectric system catches the global picture of the actual metamaterial since the FP absorption line centered at $\lambda_r= 77 \ \mu m$ is well reproduced. We try now to extract from this simpler structure, and analytical expression for the resonant wavelength, based on the different geometrical and physical parameters of the structure, which will allow us to better understand why the resonance can occur at arbitrarily large wavelengths and give us very simple design recipes for our metamaterial absorber. By taking into account the PEC back mirror, the magnetic field inside the spacer reads $H_z=B cos(k_0n_dy)$, with $k_0=2\pi/\lambda$. The electromagnetic continuity conditions applied at the interfaces lead to link by a **T** matrix the amplitude $B$ of the spacer mode to the amplitudes $I$ and $R$ of the incident and reflected waves: $$\begin{pmatrix} I \\ R \end{pmatrix}=\textbf{T} \begin{pmatrix} Bcos(k_0n_dg) \\ B\frac{k_0}{in_d}sin(k_0n_dg) \end{pmatrix}$$ from which the amplitude of the FP mode is expressed in a conventional formulation: $$B=\frac{2\tau_{eq}}{1-\Gamma_{eq} e^{2ik_0n_dg}}I
\label{eq:mode}$$ Here $\Gamma_{eq}=(1-\bar{n}_{eq})/(1+\bar{n}_{eq})$ is an equivalent reflection Fresnel coefficient determined by the equivalent index $\bar{n}_{eq}=\frac{n_d}{k_0}\frac{t_{1,1}}{t_{1,2}}$ related to the elements $t_{i,j}$ of the T-matrix. Remark that $\bar{n}_{eq}$ simply reduces to $n_d$ when the grating is removed leading for Eq. to the case of a single dielectric slab on top of a PEC mirror. The analytical expressions for $t_{1,2}$ and $t_{1,1}$ allows to write the equivalent index as $$\bar{n}_{eq}=\frac{n_d}{\bar{n}} f(\lambda)
\label{eq:exact}$$ where $f(\lambda)=(1+\rho e^{2i\pi\lambda_s/\lambda})/(1-\rho e^{2i\pi\lambda_s/\lambda})$ and $\rho=(\bar{n}-1)/(\bar{n}+1)$ designates the Fresnel coefficient at the air-dielectric interface for p-polarized light. For thin slits ($d \gg f$) $\rho \simeq 1$ since $\bar{n}$ takes very high values. Thus, in the long wavelength limit, when $\lambda \gg \lambda_s$, the function $f(\lambda)$ can be approximated by $i\lambda/(\pi\lambda_s)$. These simplifications leads to write the equivalent index in the following form: $$\bar{n}_{eq}=\frac{n_d}{n} \frac{\lambda}{\pi\lambda_s}(\frac{\kappa}{n}+i)
\label{eq:neff}$$ Introducing the figure of merit (FOM) $\mathcal{F}= n / \kappa$, the equivalent extinction coefficient $\kappa_{eq}=\frac{n_d}{n} \frac{\lambda}{\pi\lambda_s}$ is inversely linked to the equivalent refractive index by $\kappa_{eq}=\mathcal{F}n_{eq}$. The good agreement between the exact expression of the complex equivalent index of Eq. and that of the analytical one of Eq. is shown in Fig. 3a. Equipped with this complex equivalent index, two optical conditions (one for phase and a second for the modulus) can be extracted from the magnetic FP resonance condition: $$1-\Gamma_{eq} e^{2ik_0n_dg}=0
\label{eq:reson}$$ The first condition, $|\Gamma_{eq}| =1$, is satisfied for the trivial solution $n_{eq}=0$ whatever the value of the equivalent absorption $\kappa_{eq}$. In practice, almost perfect absorption higher than $98\%$ is achieved when a near-zero equivalent index condition is satisfied, which is the case when $n_{eq}=0.1$ for instance, see Fig.3a. From the definition of $n_{eq}$, we directly derive an analytical expression for the resonant wavelength: $$\lambda_r=n_{eq}\lambda_s \pi \frac{\mathcal{F}}{n_d} n_{slit}\frac{d}{(f+2\delta_p)}.
\label{eq:lam_r}$$ By considering the phase condition which implies that the FP mode is built up inside the spacer when the total phase is cancelled out, we obtain: $$\arg(\Gamma_{eq})+2k_dg=0 \ [2\pi]
\label{eq:phase}$$ The first term of Eq. can be written in terms of the equivalent refractive index and extinction coefficient as $\arg(\Gamma_{eq})=-2\kappa_{eq}/(1-n^2_{eq})$. As seen on Fig. 3b, it is well approximated by $\arg(\Gamma_{eff})=-2\kappa_{eq}$ when the near-zero equivalent index condition is satisfied. The ratio $\eta=g/\lambda_r$ of the spacer’s thickness over the wavelength thus given by the following equation: $$\eta= n_{eq}\frac{\mathcal{F}}{2\pi n_d}
\label{eq:gap}$$ In order to give rules of thumb for designing the metamaterial absorbers, we set $n_{eq}=0.1$. We have found that in the long wavelength limit and whatever the metal considered, the figure of merit reaches $\mathcal{F}_{\infty}= 8/\pi$ (see supporting information). With these parameters we arrive to $\eta=1.1/100$. This shows that the FP resonance is better excited when the dielectric layer playing the role of a cavity has a thickness that is roughly only one hundredth of a wavelength. This property originates from the negative phase, $-2\kappa_{eq}$, acquire by the electromagnetic waves when they are reflected by the equivalent high index dielectric layer. It turns out the RCWA allows to access this phase rigorously, by retrieving the actual reflection coefficient on the metamaterial layer (see Supporting Information), and that such a computation totally confirms the analytical results. Inserting this phase condition into Eq. and for nanometer slits ($2\delta \gg f$), we get a simple expression for the resonant wavelength of the FP mode: $$\lambda_r= \Lambda\frac{d}{\sqrt{f}}
\label{eq:lam_r2}$$ with $\Lambda=2\eta\pi^2\lambda_s/\sqrt{2\delta_p}$ that can be evaluated for 10-nanometer slits to $\Lambda=3.38$. Finally, Eq. and Eq. provide simple scaling laws for designing universal absorbers operating at arbitrary large wavelengths.
![(a) Equivalent refractive index and extinction coefficient as a function of the wavelength obtained with Eq. in solid curves and with the approximate formulation Eq.. (b) Phase terms Eq.: spacer phase $2k_dg$ (blue curve), $-\arg(\Gamma_{eq})$ (black curve) and $2\kappa_{eq}$ (dashed curve).[]{data-label="fig:fig3"}](fig3.eps){width="0.8\columnwidth"}
Perfect absorbers from infrared to microwave
============================================
These theoretical results are confirmed by the exact electromagnetic calculations of the absorbance $A$ deduced from the reflectivity and performed in normal incidence with the RCWA method where 100 Fourier modes are used for the largest pitches. On Figure 4a, the resonant wavelengths of the FP resonance are shown for grating periods $d$ ranging from $1 \ \mu m$ to $400 \ \mu m$. In agreement with our model, the resonant wavelength is seen to be linearly linked to the period by $\lambda_r\simeq 33d$ (for $d$ in microns). The use of Eq. for slits of $10 \ nm$ wide leads to a slope of 33.8, thus confirming the excellent accuracy of our analytical model. This means that the structure is able to absorb microwaves with a wavelength that is more than 6 orders of magnitude larger than the slits’ width (see Fig. 4a). The electric field associated to this FP magnetic resonance is actually squeezed in slits that are a million times smaller than the wavelength. This is where the absorption takes place, since the dielectric spacer is considered lossless.
![(a) Resonant wavelength $\lambda_r$ as a function to the pitch $d$ for a 1D absorber. The bold line corresponds to the theoretical scaling law $\lambda_r=33d$ and the dots to the exact electromagnetic computations. The inset represents the absorbance spectrum with a total absorption in the microwave range for $\lambda_r=11.2 \ mm$ when $d=350 \ \mu m$. (b) Ratio $\lambda_r/d$ with respect to the slit width, the dashed and solid curves are respectively obtained with the RCWA simulations and with Eq. for $d=1 \ \mu m$. (c) Ratio of the spacer over the wavelength with respect to the pitch. $g/\lambda$ remains constant for a period larger than $8 \ \mu m$.[]{data-label="fig:fig4"}](fig4.eps){width="0.8\columnwidth"}
As seen on Fig. 4b, the dimension of the slits have an important impact on the spectral position of the magnetic FP resonance. This is especially true when they are a few nanometers wide, as this dimension has a very large influence on the effective index of the gap-plasmon propagating in the slits. A quite good agreement is observed with the exact results obtained using the RCWA simulations and Eq. for $d=1 \ \mu m$. Beyond a pitch of $4 \ \mu m$ or equivalently for wavelengths larger than $100 \ \mu m$, the thickness of the spacer remains constant about $g/\lambda=1.3/100$, close to the theoretical limit $\eta=1.1/100$, Fig. 4c. For arbitrarily large wavelengths, the slits operate as antennas that funnel the incident waves into the spacer that constitutes the resonant cavity. The funneling factor, the ratio of the pitch to the slid width, is huge: it can be as large as 40,000 which is way above what usually happens for EOT when the resonance is located in the slits. This mechanism holds from the infrared to the microwave range despite the dispersive behavior of InAsSb and can be obtained for other metals such as silver (see supporting information). From the application point of view, realizing absorbers insensitive to the incident angle and to the polarization of light is a crucial issue. We address these problems by considering 2D metamaterial absorbers made of a square array of thin slits (width $f=10 \ nm$) separated by a pitch $d=2 \mu m$, Fig. 1a. We illustrate these properties for a targeted absorption line at $\lambda_r=70 \mu m$ leading to a spacer’s thickness $g=850 \ nm$, a pitch $d=2 \ \mu m$ and a grating’s thickness $h_r=320 \ nm$. More than $90 \%$ of the incident radiation is absorbed by the metamaterial for incident angles up to $50^{\circ}$ and the absorbance reaches $70 \%$ at grazing incidence for $70^{\circ}$, Fig. 5a. The efficiency of the absorber is also seen to be insensitive to the polarization of light: the structure can simply be seen as two crossed gratings, each one being responsive to one polarization only. In normal incidence, the absorbance thus remains constant whatever the polarization in normal incidence.
![(a) Absorbance with respect to the incident angle $\theta$ and for a polarization angle $\delta=0^{\circ}$. The inset represents the polar plot of the absorbance computed for the absorption line $\lambda_r=70 \ \mu m$. (b) Absorbance with respect to the polarization angle $\delta$ and for normal incidence (TM and TE polarizations cases are respectively defined by $\delta=0^{\circ}$ and $\delta=90^{\circ}$. In the inset shows the polar plot of the absorption line as a function of $\delta$.[]{data-label="fig:fig5"}](fig5.eps){width="0.8\columnwidth"}
Conclusion
==========
We have proposed a metamaterial resonant absorber whose absorption line can be chosen in any frequency range, from optics to microwaves, by following simple scaling laws, essentially. Our approach allows to design a perfect absorber that working for any frequency, independently of the materials that are considered. Almost perfect absorption can be obtained whatever the polarization over a broad incident angle range. This is why we think our design can be said to be [*universal*]{}. The metamaterial layer controlling the response of the structure allows to reduce to $\lambda/100$ the thickness of the spacing layer constituting a resonant cavity on which the device is based. The absorption takes place in slits that are no more than a few nanometers wide, despite wavelengths that are 1000,000 times larger. All the incoming radiation in funneled through these slits despite a ratio of 1 to 40000 between the slit width and the period. We have derived an analytical model thoroughly describing the electromagnetic response of the device that proved very accurate despite these extreme and unprecedented ratios. Universal absorbers constitute ultra-thin and flexible solutions for absorbing any kind of electromagnetic radiation especially at terahertz frequencies where absorbers, sensors and emitters are usually difficult to design. Integrating electric contacts into the device or making it dynamically tunable is something that can totally be envisaged, broadening even more the potential of our design.
This work is supported by the Agence Nationale de la Recherche of France, project “Physics of Gap-Plasmons” (ANR-13-JS10-0003).
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|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We calculate the Hilbert series of a quotient of the exterior algebra by a generic form of even degree, and give conjectures about the Hilbert series of other generic quotients.'
address:
- |
Laboratoire GAGE, École Polytechnique\
91128 Palaiseau Cedex\
France
- |
Department of Mathematics\
Linköping University\
SE-58183 Linköping\
SWEDEN
author:
- 'Guillermo Moreno-Socías'
- Jan Snellman
bibliography:
- 'journals.bib'
- 'snellman.bib'
- 'articles.bib'
title: Some conjectures about the Hilbert series of generic ideals in the exterior algebra
---
[^1]
[Introduction]{} In the symmetric algebra ${{{{K \lbrack {x}_{1}, \dots,
{x}_{n} \rbrack} }}}$, the set of Hilbert series coming from homogeneous quotients are classified by Macaulays theorem [@Macaulay:Enum; @Ebud:View; @Bigatti95]. There is an infinite number of possible series, but if we fix positive integers $d_1,\dots,d_r$, and restrict our study to quotients by homogeneous ideals $I$ of “type” or “numerical character” $(d_1,\dots,d_r)$, ie generated by forms of those prescribed degrees, then there are only finitely many Hilbert series. Furthermore, in the affine space parametrising these homogeneous ideals, there is a Zariski-open subset of ideals with the same Hilbert series, and the Hilbert series obtained on this open set is minimal [@Froeberg:OnHilb; @Froeberg:Inequality].
Unfortunately, even though we know the set of *all* Hilbert series, we do not know what Hilbert series arise from ideals of numerical character $(d_1,\dots,d_r)$. In fact, we do not even know the “generic” series, but it is conjectured [@Moreno:Revlex; @Froeberg:Inequality] that it is $\left \langle (1-t)^{-n} \prod_{i=1}^r (1-t^{d_i}) \right
\rangle$; the brackets mean “truncate before the first non-positive coefficient”.
In the exterior algebra $\bigwedge V_n$, we also know the set of all Hilbert series of homogeneous quotients, by the so-called Kruskal-Katona theorem [@Kruskal; @Katona; @CleLi; @Aramova:Gotzman]. Here, this set is finite, so one would think that it should be easy to find the subset of Hilbert series coming from quotients by ideals having a prescribed numerical character. In particular, it should be easy to find the generic value. However, very little is known.
In this article, we give one new result (the series for a quotient by *one* form of *even* degree) and several conjectures, supported by extensive computer calculations.
It is worthwhile to point out that the problem of determining the Hilbert series of quotients by generic *quadratic* forms is especially interesting, since it determines the Koszulness of the quadratic algebras in question. We refer to the recent article by Fröberg and Löfwall [@KosLie].
[Notation]{} Let $K$ be a field of characteristic 0. Then ${{\mathbb{Q}}}$ is the prime subfield of $K$. For any positive integer $n$, let $V=V_n$ be an $n$-dimensional vector space over $K$, with a distinguished basis $X_n={{\{x_1,\dots,x_n\}}}$. Let ${{{{K \lbrack {x}_{1}, \dots,
{x}_{n} \rbrack} }}}$ denote the symmetric algebra on $V_n$, and let $\bigwedge V_n$ denote the exterior algebra on $V_n$. We define ${\mathfrak{S}}(V_n)$, the *square-free* algebra on $V_n$, to be the commutative $K$-algebra generated by $X_n$, with the relations $x_i^2 = 0$; in other words, ${\mathfrak{S}}(V_n) =
\frac{{{{{K \lbrack {x}_{1}, \dots,
{x}_{n} \rbrack} }}}}{(x_1^2,\dots,x_n^2)}$. There is an isomorphism of graded vector spaces between $\bigwedge V_n$ and ${\mathfrak{S}}(V_n)$, but they are not isomorphic as $K$-algebras, since the exterior algebra is skew-commutative and ${\mathfrak{S}}(V_n)$ is commutative.
We shall need the following operations for formal power series.
Let $f(t) = \sum_{i=0}^\infty a_i t^i \in {{\mathbb{Z}}}[[t]]$, $g(t) =
\sum_{i=0}^\infty b_i t^i \in {{\mathbb{Z}}}[[t]]$. We say that $f \ge g$ if $a_i \ge b_i$ for all $i$. We define $$\begin{aligned}
\max(f(t),g(t)) & = \sum_{i=0}^\infty \max(a_i,b_i) \\
\left \langle f(t) \right \rangle & = \sum_{i=0}^\ell a_i t^i, \qquad
\ell = \max(\setsuchas{i}{a_j > 0 \text{ for } j \le i})
\\
\left \rangle f(t) \right \langle & = \sum_{i=\ell}^\infty a_i
t^i, \qquad \ell = \min(\setsuchas{i}{a_j > 0 \text{ for } j \ge i})
\qquad
\end{aligned}$$ We use the conventions $\max(\emptyset) = -1 = \min({{\mathbb{N}}})$, $\min(\emptyset) = +\infty = \max({{\mathbb{N}}})$.
Let $[X_n]$ denote the free abelian monoid on $X_n$, and denote by $Y_n$ the subset of square-free monomials. Then $Y_n$ is a $K$-basis for both $\bigwedge V_n$ and ${\mathfrak{S}}(V_n)$. We define the *degree* of a monomial in $[X_n]$ (and in $Y_n$) in the usual way, and denote by $[X_n]^d$ and $Y_n^d$ the subset of monomials (square-free monomials) of degree $d$.
A form $ \bigwedge \Kxn \ni f = \sum_{m \in [X_n]^d} c_m m
$ is said to be *generic* if the coefficients $c_m \in K$ fulfil the following conditions:
1. $c_m \not \in {{\mathbb{Q}}}$,
2. $m \neq m' \implies c_m \neq c_{m'}$,
3. The set of all $c_m$’s is algebraically independent over ${{\mathbb{Q}}}$.
A homogeneous ideal $I \subset {{{{K \lbrack {x}_{1}, \dots,
{x}_{n} \rbrack} }}}$ is called generic if it can be minimally generated by a finite set of generic forms, so that all of the occuring coefficients of the forms are different, and so that the set of all occuring coefficients is algebraically independent over ${{\mathbb{Q}}}$. If the forms have degrees $d_1,\dots,d_r$, then we say that $I$ has “numerical character” $(d_1,\dots,d_r)$. It is an important fact that any two generic ideals of the same numerical character have the same initial ideal and the same Hilbert series.
Now consider the affine space $V=\mathbf{A}^{\binom{n+d_1 -1}{d_1}}
\times \cdots \times \mathbf{A}^{\binom{n+d_r -1}{d_r}}$ parametrising the set of homogeneous ideals of numerical character $(d_1,\dots,d_r)$. Since there are countably many conditions to be fulfilled for an ideal to be generic, the subset of the parameter space corresponding to generic ideals is not open, but a countable intersection of open sets, hence dense. However, in $V$ there is a Zariski-open subset corresponding to ideals with the same Hilbert function, and the generic ideals are contained in this subset [@Froeberg:OnHilb].
We make similar definitions for the square-free algebra, and for the exterior algebra. Here, a generic form is a generic linear combination of *square-free* monomials of a certain degree. It is still true that the generic Hilbert series is attained on an open component of the parameter space, and that the generic ideals are contained in this component.
[Hilbert series for generic principal ideals in the symmetric and square-free algebra]{}
[Principal ideals in the symmetric algebra]{} If $f \in {{{{K \lbrack {x}_{1}, \dots,
{x}_{n} \rbrack} }}}$ is a non-zero form of degree $d$, not necessarily homogeneous, then clearly the Hilbert series of the quotient $\frac{{{{{K \lbrack {x}_{1}, \dots,
{x}_{n} \rbrack} }}}}{(f)}$ is $(1-t)^{-n} (1-t^d)$.
[Principal ideals in the square-free algebra]{} If $f \in
{\mathfrak{S}}(V_n)$ is a generic form of degree $d$, then there is a similar simple formula for $\frac{{\mathfrak{S}}(V_n)}{(f)}(t)$ (the Hilbert series of the quotient). To state the formula, we need some additional notation.
We denote the zero series by $0$, and define $$\begin{aligned}
\Delta_{n,d}(t) &= \left \rangle (t^d -1) (1+t)^n \right \langle \\
&=\sum_{v=\lceil (n-d)/2 \rceil}^n
\left(\binom{n}{v} - \binom{n}{v+d}\right) t^v \\
\delta_{n,d}(t) &= \left \langle (1+t)^n (1-t^{d}) \right \rangle \\
&= \sum_{v=0}^{\lfloor (n-d)/2 \rfloor} (\binom{n}{v+d}-
\binom{n}{v}) t^v
\end{aligned}$$
The following result is due to Fröberg [@Froeberg:HilbGenForm].
\[thm:principal\] Let $f \subset {\mathfrak{S}}(V_n)$ be a generic form of degree $d$. Then $$\label{eq:SQser}
\frac{{\mathfrak{S}}(V_n)}{(f)}(t) = \delta_{n,d}(t)$$
By considering the graded exact sequence $$\label{eq:exseqSQ}
0 \longrightarrow {\mathrm{ann}}(f)(-d) \longrightarrow {\mathfrak{S}}(V_n)(-d)
\xrightarrow{\cdot f} {\mathfrak{S}}(V_n) \longrightarrow
\frac{{\mathfrak{S}}(V_n)}{(f)} \longrightarrow 0$$ in each degree $r$, we see that holds if and only if multiplication by $f$, regarded as a linear map $\phi_r$ from ${\mathfrak{S}}(V_n)_r$ to ${\mathfrak{S}}(V_n)_{r+d}$, is injective when $\binom{n}{r} \le
\binom{n}{r+d}$, and surjective when $\binom{n}{r} \ge
\binom{n}{r+d}$.
Write $f=\sum_{m \in Y_n^d} c_m m$. For $0 \le r \le n-d$, $Y_n^r$ is a basis of ${\mathfrak{S}}(V_n)_r$, and $Y_n^{r+d}$ is a basis of ${\mathfrak{S}}(V_n)_{r+d}$. Thus, we must show that for each $r$, the matrix of $\phi_r$ in this basis has maximal rank. This matrix has rows indexed by $Y_n^{r+d}$ and columns indexed by $Y_n^{r}$. The entry at row $R$, column $C$ is $$\begin{cases}
0 & {{C \not \vert R}}\\
c_m & R=mC
\end{cases}$$ If we specialise this matrix, the rank can only decrease, so if we can prove that some specialised matrix has full rank, then we are done. Putting all $c_m=1$, we obtain the incidence matrix of $r$-subsets of $[n]$ into $r+d$-subsets of $[n]$, that is, the rows are indexed by $r$-subsets and the columns by $r+d$-subsets, with a $1$ at the $a,b$’th position iff $a
\subset b$, and 0 otherwise. It has been shown by combinatorialists that this matrix has full rank [@Wilson:design; @Kantor:incidence; @Graver:design].
[Principal ideals in the exterior algebra — the difference between even and odd degree]{}
Let $f \in \bigwedge V_n$ be a generic form of degree $d$. Denote the Hilbert series of $\frac{\bigwedge V_n}{f}$ by ${q_{n,d}(t)}$, that of the annihilator of $f$ by ${a_{n,d}(t)}$, and that of the principal ideal $(f)$ by ${p_{n,d}(t)}$. From the the graded exact sequence $$\label{eq:exseqWEDGE}
0 \longrightarrow {\mathrm{ann}}(f)(-d) \longrightarrow \bigwedge V_n(-d)
\xrightarrow{\cdot f} \bigwedge V_n \longrightarrow
\frac{\bigwedge V_n}{(f)} \longrightarrow 0$$ we get that $$\label{eq:dimsum}
\begin{split}
{q_{n,d}(t)}& = t^d {a_{n,d}(t)}- t^d(1+t)^n + (1+t)^n \\
&= t^d {a_{n,d}(t)}+ (1+t)^n(1-t^d) \\
{a_{n,d}(t)}&= t^{-d} \left( {q_{n,d}(t)}- (1+t)^n(1-t^d) \right)
\end{split}$$
If $d$ is even, we shall prove that the vector space map $$\label{eq:fmul}
\bigwedge^v V_n
\xrightarrow{\cdot f} \bigwedge^{v+d} V_n$$ is injective “when it can be”, ie when $\binom{n}{v} \le
\binom{n}{v+d}$, and surjective “when it can be”, ie when $\binom{n}{v} \ge
\binom{n}{v+d}$. This leads immediately to the formul[æ]{} $$\begin{split}
{q_{n,d}(t)}&=\left \langle (1+t)^n (1-t^{d})
\right \rangle =\delta_{n,d}(t) \\
{a_{n,d}(t)}&= t^{-d} \left( {q_{n,d}(t)}- (1-t^d)(1+t)^n \right) \\
&= t^{-d} \left( \delta_{n,d}(t) - (1-t^d)(1+t)^n \right) \\
& = t^{-d} \sum_{r=0}^n \Biggl[\max\left(0,\binom{n}{r+d} -
\binom{n}{r}\right) - \left(
\binom{n}{r+d} - \binom{n}{r}\right) \Biggr] t^r \\
& = t^{-d} \sum_{r=0}^n \max \left(0, -\binom{n}{r+d} + \binom{n}{r}
\right) t^r
\\
& = t^{-d} \Delta_{n,d}(t)
\end{split}$$ In particular, as $n \to \infty$, $(1+t)^{-n}{q_{n,d}(t)}\to
(1-t^d)$, and ${a_{n,d}(t)}\to 0$, with respect to the $t$-adic norm on ${{\mathbb{Z}}}[[t]]$.
If $d$ is odd, then we have that $f^2 = 0$, hence $fg=0$ whenever $g \in (f)$, hence ${\mathrm{ann}}(f) \supseteq (f)$, hence ${a_{n,d}(t)}\ge {p_{n,d}(t)}$. In other words, there is a graded complex $$\label{eq:infcomp}
\left(\bigwedge V\right)(-d) \xrightarrow{\cdot f}
\bigwedge V \xrightarrow{\cdot
f} \left(\bigwedge V\right)(d)$$ the graded homology of which determines ${a_{n,d}(t)}-
{p_{n,d}(t)}$. In the (not very interesting) case $d=1$, then we know from [@Aramova:Cohomology] that this homology vanishes. For odd $d > 1$, we guess that for a fixed degree $r$, and $n$ very large, this homology vanishes. Hence, in degree $r$, the “obstruction to injectivity” in is as small as possible. An equivalent formulation: consider the start of a minimal free graded $\bigwedge
V_n$-resolution of $\frac{\bigwedge V_n}{(f)}$, $$\frac{\bigwedge V_n}{(f)} \leftarrow \bigwedge V_n
\xleftarrow{\cdot f} \bigwedge V_n \leftarrow \bigoplus_{j=1}^r
(\bigwedge V_n)(-\beta_{2,i}),$$ where $\beta_{2,i}$ are the graded Betti numbers. Then we guess that as $n$ increases, and for a fixed $i \neq 2d$, $\beta_{2,i} = 0$. On the other hand, for sufficiently large $n$, we guess that $\beta_{2,2d} = 1$. Since $\beta_{2,i}$ is the dimension of the degree $i-d$ part of a certain Tor group, this conjecture can also be stated in terms of Cartan homology (see [@Aramova:Gotzman]).
We show the order (ie the smallest $\ell$ for which $t^\ell$ occurs with non-zero coefficient) of ${a_{n,d}(t)}- {p_{n,d}(t)}$ for small $n,d$ in Table 1.
\[tab:order\]
It would seem that the order of the difference grows linearly in $n$, so that ${a_{n,d}(t)}-
{p_{n,d}(t)}\to 0$ rather rapidly.
Let us turn to the consequences of this conjecture. We get that ${a_{n,d}(t)}\sim {p_{n,d}(t)}$ with respect to the $(t)$-adic filtration. It then follows from that $$\label{eq:oddconj}
{q_{n,d}(t)}\sim t^d {p_{n,d}(t)}+ (1+t)^n(1-t^d)$$ Substituting ${p_{n,d}(t)}= (1+t)^n - {q_{n,d}(t)}$ and solving for ${q_{n,d}(t)}$ we get that $${q_{n,d}(t)}\sim \frac{(1+t)^n t^d +
(1+t)^n(1-t^d)}{(1+t^d)} = \frac{(1+t)^n}{(1+t^d)},$$ hence $$\label{eq:oddagain}
\frac{\frac{\bigwedge V_n}{(f)}(t)}{\bigwedge(V_n)(t)}
= \frac{{q_{n,d}(t)}}{(1+t)^n} \to \frac{1}{1+t^d}
\qquad \text{ as } n
\to \infty.$$
[Principal ideals on generic forms of even degree in the exterior algebra]{} If $d=2$ then we can change coordinates on $V$ and replace $f$ with the form $x_1x_2 + x_3x_4 + \cdots$, as is demonstrated in [@Bourbaki:FormQ]. The Hilbert series of the quotient can now be easily calculated. We get that $\frac{\bigwedge(V_n)}{(f)}(t) = \left \langle (1+t)^n (1-t^2)
\right \rangle$, which is the same as the Hilbert series for the corresponding quotient in the square-free algebra.
It is *not true* that if $f_e = \sum_{1\le i<j \le n}
\alpha_{ij} x_i x_j$ is a non-generic quadratic form in $\bigwedge V_n$, and $f_s = \sum_{1\le i<j \le n} \alpha_{ij} x_i x_j$ is the corresponding form in ${\mathfrak{S}}(V_n)$, then $\frac{\bigwedge
V_n}{(f_e)}$ and $\frac{{\mathfrak{S}}(V_n)}{(f_s)}$ have the same Hilbert series. For an example, consider the form $x_1x_2 + x_1x_3 + x_1x_4 + x_3x_4$. The quotient of $\bigwedge V_4$ by this form has Hilbert series $ 5t^2 + 4t+1,
$ but the corresponding quotient of ${\mathfrak{S}}(V_4)$ has series $ t^3+5t^2 + 4t+1.
$
We next show that if the degree $d$ of $f$ is even, then the Hilbert series of the quotient $\frac{\bigwedge V_n}{(f)}$ is the same as for the square-free algebra. To this end, we need some combinatorial results, which we have collected in the appendix. With the aid of these, we can prove:
\[thm:evenform\] Let $f \in \wedge^d V$, with $d$ even, be a generic form. Then the linear transformation $$\label{eq:multbyf}
\wedge^r V \xrightarrow{f \cdot} \wedge^{r+d} V$$ is injective for $2r+d \le n$, and surjective for $2r+d \ge n$.
We put $k=r+d$. Suppose that $$\label{eq:fis}
f= \sum_{K \in \binom{[n]}{d}} c_K x_K$$ The matrix of the map is an $\binom{n}{r+d} \times \binom{n}{d}$ matrix, $\widetilde{M_{r,r+d,n}}$, where the rows are indexed by $(r+d)$-subsets $K$, and the columns by $d$-subsets $T$. The entry at position $(K,T)$ is $$\label{eq:pos}
\begin{cases}
0 & \text{ if } T \not \subseteq K \\
\sigma(T, K) c_T& \text{ if } T \subseteq K
\end{cases}$$ We must prove that this matrix has maximal rank. Clearly, the rank can not increase under specialisation, so if we prove that the matrix obtained by replacing each $c_T$ with 1 has maximal rank, then so does $\widetilde{M_{r,r+d,n}}$. However, the specialised matrix is nothing but the matrix $M_{r,r+d,n}$ of Theorem \[thm:fullrank\], so it has full rank.
\[thm:evenser\] Let $f \in \bigwedge V_n$ be a generic form of degree $d$, with $d$ even. Then $$\label{eq:evendegHser}
\frac{\bigwedge V_n}{(f)}(t) = \left \langle (1+t)^n (1-t^{d})
\right \rangle = \delta_{n,d}(t)$$
This follows from Theorem \[thm:evenform\], together with .
[Principal ideals on generic forms of odd degree in the exterior algebra]{} Let $d$ be an odd integer. Recall that we’ve conjectured that ${a_{n,d}(t)}- {p_{n,d}(t)}\to 0$ as $n \to \infty$, and that this conjecture leads to the conclusions that $p_{n,d}(t) \sim
(1+t)^n(1+t^d)^{-1}$. In this section, we shall try to guess the exact value of ${q_{n,d}(t)}$.
Since ${a_{n,d}(t)}\ge {p_{n,d}(t)}$, ${a_{n,d}(t)}\ge
\Delta_{n,d}(t)$, it follows that ${a_{n,d}(t)}\ge
\max({p_{n,d}(t)},\Delta_{n,d}(t))$. We tabulate the difference ${a_{n,d}(t)}-
\max({p_{n,d}(t)},\Delta_{n,d}(t))$ in Table \[tab:odd\] and Table \[tab:oddmac\].
n deg=3 5 7 9 11 13 15 17 19
---- ----------------------- ------- ----- ------- ----- ------- ----- ----- -----
3 0
4 0
5 t 0
6 0 0
7 0 t 0
8 0 0 0
9 $3t^3$ 0 $t$ 0
10 $t^4$ 0 0 0
11 $t^5$ $t^3$ 0 $t$ 0
12 $t^6+12t^5$ 0 0 0 0
13 $t^7+13t^6+t^5$ 0 0 0 $t$ 0
14 $t^8+14t^7+91t^6$ 0 0 0 0 0
15 $15t^8+105t^7$ 0 0 $t^3$ 0 t 0
16 $16t^9+120t^8+559t^7$ $t^6$ 0 0 0 0 0
17 0 0 0 $t$ 0
18 0 0 0 0
19 $t^3$ 0 $t$ 0
20 0 0 0
21 0 $t$
: Difference between true and predicted Hilbert series of the annihilator of a generic form of odd degree[]{data-label="tab:odd"}
Using the data of Table \[tab:oddmac\], we make the following conjecture:
\[conj:oddfive\] Let $d$ be an odd integer $> 3$. Then, putting $\tau_{n,d}(t)= {a_{n,d}(t)}-\max \left( {p_{n,d}(t)},
\Delta_{n,d}(t)
\right)$, $$\tau_{n,d}(t)=
\begin{cases}
t^{v(v-1)/2} & \exists v,s \in {{\mathbb{N}}}: \, v > 0, \, n-d = -1 +
\frac{5}{2}v + \frac{1}{2}v^2, \, d = 5 + 2vs \\
0 & \text{ otherwise}
\end{cases}$$
$n -d$ deg=5 7 9 11 13 15 17 19
-------- ------- ----- ------- ----- ------- ----- ----- -----
0 0 0 0 0 0 0 0 0
1 0 0 0 0 0 0 0 0
2 $t$ $t$ $t$ $t$ $t$ $t$ $t$ $t$
3 0 0 0 0 0 0 0
4 0 0 0 0 0 0 0
5 0 0 0 0 0 0 0
6 $t^3$ 0 $t^3$ 0 $t^3$
7 0 0 0 0
8 0 0 0
9 0 0
10 0
11 $t^6$
: Difference between true and predicted Hilbert series of the annihilator of a generic form of odd degree $>3$[]{data-label="tab:oddmac"}
This conjecture yields a formula for the Hilbert series, but since said formula is very complicated, we do not write it down; instead we show how to derive ${q_{n,d}(t)}$. From $$\label{eq:solveme}
\begin{split}
{a_{n,d}(t)}&= \tau_{n,d}(t) + \max \left(
{p_{n,d}(t)}, \Delta_{n,d}(t)
\right) \\
{q_{n,d}(t)}& = {a_{n,d}(t)}t^d + (1+t)^n(1-t^d) \\
{p_{n,d}(t)}& = (1+t)^n - {q_{n,d}(t)}\end{split}$$ we get $$\label{eq:sol}
\begin{split}
{p_{n,d}(t)}&= (1+t)^n - {q_{n,d}(t)}\\
&= (1+t)^n - {a_{n,d}(t)}t^d - (1+t)^n(1-t^d) \\
&= (1+t)^n - t^d\tau_{n,d}(t) - t^d\max \left( {p_{n,d}(t)},
\Delta_{n,d}(t) \right)
- (1+t)^n(1-t^d) \\
&= t^d \left( (1+t)^n - \tau_{n,d}(t)- \max({p_{n,d}(t)},
\Delta_{n,d}(t)) \right)
\end{split}$$ Hence, writing ${p_{n,d}(t)}= \sum_{i=0}^n a_i t^i$, with the $a_i$’s as undetermined coefficients, and denoting the $t^i$-coefficient of $\tau_{n,d}(t)$ by $b_i$, we get the equation $$\label{eq:iter}
a_\ell = \binom{n}{\ell - d} - b_{i-\ell} -\max(a_{\ell - d},
\binom{n}{\ell - d} -
\binom{n}{\ell})$$ which we can solve recursively, using the initial values $$a_0 = \cdots = a_{d-1} = 0, \qquad a_d = a_n = 1.$$
For the case $d=3$, we proceed differently: we tabulate $q_{n,3}(t) - w_{n,3}(t)$ in Table 4, and from that, make the following conjecture:
\[conj:deg3\] The Hilbert series of $\frac{\bigwedge V_n}{(f)}$, where $f$ is a generic cubic form, is given by $$\label{eq:pdeg3}
\begin{split}
p_{n,3}(t) &= \frac{t^d L_n(t) + (1+t)^n}{1+t^d} \\
L_n(t) &=
\begin{cases}
(3t)^{2\ell -1} (1+t)^2 & n = 4 \ell \\
c_1(n)t^{2\ell -1}(1+t)(1+(3^{c_2(n)}-1)t + t^2) & n = 4 \ell + 1\\
(3t)^{2\ell} (1+t)^2 & n = 4 \ell + 2\\
(3t)^{2\ell +1} (1+t) & n = 4 \ell + 3
\end{cases}
\end{split}$$ where $c_1(n),c_2(n)$ are some positive integers.
\[tab:anndeg3diff\]
n $q_n(t) - w_n(t)$
---- --------------------------
3 $ 3t(1+t) $
4 $ 3t(1+t)^2 $
5 $ t(1+t)(t^2+8t+1) $
6 $ 9t^2(1+t)^2 $
7 $ 27t^3(1+t) $
8 $ 27t^3(1+t)^2 $
9 $ 3t^3(1+t)(t^2+26t+1) $
10 $ 81t^4(1+t)^2 $
11 $ 243t^5(1+t) $
12 $ 243t^5(1+t)^2 $
13 $ t^5(1+t)(t^2+728t+1) $
14 $ 729t^6(1+t)^2 $
15 $ 2187t^7(1+t) $
16 $ 2187t^7(1+t)^2 $
: ${a_{n,d}(t)}- {p_{n,d}(t)}$ for a cubic generic form
[Hilbert series for generic non-principal ideals in the symmetric and square-free algebra]{} Let $I=(f_1,\dots,f_r)$ be a generic ideals in ${{{{K \lbrack {x}_{1}, \dots,
{x}_{n} \rbrack} }}}$, generated by forms of degree $d_1,\dots,d_r$. There is a famous conjecture [@Moreno:Revlex; @Froeberg:Inequality] for the Hilbert series of the quotient $\frac{{{{{K \lbrack {x}_{1}, \dots,
{x}_{n} \rbrack} }}}}{I_n}$.
\[conj:wk\] Let $I=(f_1,\dots,d_r) \subset {{{{K \lbrack {x}_{1}, \dots,
{x}_{n} \rbrack} }}}$ be a generic ideal, with ${\vertf_i\rvert} =d_1$ for $1 \le i \le r$. Then the Hilbert series of the graded algebra $\frac{{{{{K \lbrack {x}_{1}, \dots,
{x}_{n} \rbrack} }}}}{I_n}$ is given by $$\label{eq:moreno}
\left \langle (1-t)^{-n} \prod_{i=1}^r (1-t^{d_i}) \right \rangle$$
It is easy to see that if $r \le n$, the generators form a regular sequence, and hence that $$\label{eq:ci}
\frac{{{{{K \lbrack {x}_{1}, \dots,
{x}_{n} \rbrack} }}}}{I_n}(t) = (1-t)^{-n} \prod_{i=1}^r (1-t^{d_i}) , \qquad
\text{ for } n \ge r$$ In particular, the conjecture holds for $r \le n$. The conjecture is also know to be true for $r=n+1$.
We note that implies that $$\label{eq:analog}
\lim_{n \to \infty} \frac{ \frac{{{{{K \lbrack {x}_{1}, \dots,
{x}_{n} \rbrack} }}}}{I_n}(t)}{{{{{K \lbrack {x}_{1}, \dots,
{x}_{n} \rbrack} }}}(t)} =
\prod_{i=1}^r (1-t^{d_i})$$
Now suppose that $I=(f_1,\dots,f_r)$ is a generic ideal in the square-free algebra, and that $f_i$ is a generic form of degree $d_i$. Then $$\frac{{\mathfrak{S}}(V_n)}{(f_1,\dots,f_r)} \simeq
\frac{{{{{K \lbrack {x}_{1}, \dots,
{x}_{n} \rbrack} }}}}{(f_1',\dots,f_r',x_1^2,\dots,x_n^2)}$$ where $f_i'$ can be taken to be a generic form in ${{{{K \lbrack {x}_{1}, \dots,
{x}_{n} \rbrack} }}}$ which maps to $f_i$ under the canonical epimorphism ${{{{K \lbrack {x}_{1}, \dots,
{x}_{n} \rbrack} }}}\twoheadrightarrow {\mathfrak{S}}(V_n)$. It seems reasonable to assume that the Hilbert series of the quotient will not change if we replace the squares of variables with generic quadratic forms. Conjecture \[conj:wk\] then leads to the following:
\[conj:sqfree\] Let $r, n, d_1,\dots,d_r$, and let $I_n$ be a generic ideal i ${\mathfrak{S}}(V_n)$ with generators of degrees $d_1,\dots,d_r$. Then $$\label{eq:sqcase}
\frac{{\mathfrak{S}}(V_n)}{I_n}(t) = \left \langle (1+t)^n \prod_{i=1}^r
(1-t^{d_i}) \right \rangle$$
If this conjecture holds (our computations support this), then it follows that $$\label{eq:sqfr}
\lim_{n \to \infty}\frac{\frac{{\mathfrak{S}}(V_n)}{I_n}(t)}{{{\mathfrak{S}}(V_n)}(t)}
=
\prod_{i=1}^r (1-t^{d_i})$$ This is analogous to .
[Hilbert series for generic non-principal ideals in the exterior algebra]{} We now throw all caution to the wind to make some bold conjectures about the Hilbert series of non-principal generic ideals. Let $I_n=(f_1,\dots,f_r)$ be a generic ideal in $\bigwedge V_n$, with ${\vertf_i\rvert} = d_i$, and consider the exact sequence $$\label{eq:exseq2}
0 \longrightarrow {\mathrm{ann}}(f_r)(-d_r) \longrightarrow
\frac{\bigwedge V_n}{(f_1,\dots,f_{r-1})}(-d_r)
\xrightarrow{\cdot f_r} \frac{\bigwedge
V_n}{(f_1,\dots,f_{r-1})} \longrightarrow
\frac{\bigwedge V_n}{(I)} \longrightarrow 0$$ We denote the Hilbert series of $\frac{\bigwedge V_n}{(I)}$ by ${q_{n}(t)}$, that of $\frac{\bigwedge V_n}{(f_1,\dots,f_{r-1})}$ by $u_n(t)$, and that of ${\mathrm{ann}}(f_r)$ by ${a_{n}(t)}$. Then $$\label{eq:hsum}
{q_{n}(t)}= u_n(t) - t^{d_r} u_n(t) + t^d {a_{n}(t)}.$$ If $d_r$ is even, we conjecture that ${a_{n}(t)}\sim 0$, hence $$\label{eq:pmeven}
{q_{n}(t)}\sim (1-t^{d_r}) u_n(t)$$ If $d_r$ is odd, we conjecture that the annihilator of $f_r$ is “close” to the principal ideal on $f_r$, hence that ${a_{n}(t)}\sim (u_n(t) - {q_{n}(t)})$, which yields $$\label{eq:pmodd}
{q_{n}(t)}(1+t^d) \sim u_n(t)$$ By induction, we arrive at the following conjecture: $$\label{eq:hserguessnonp}
\lim_{n \to \infty} \frac{{q_{n}(t)}}{(1+t)^n} = \prod_{i=1}^r
\left(1-(-1)^{d_i} t^{d_i} \right)^{(-1)^{d_i}} \in {{\mathbb{Z}}}[[t]],$$ with respect to the $(t)$-adic topology.
One would be tempted to guess that if all $d_i$’s are even, the Hilbert series of $\frac{\bigwedge V_n}{(f_1,\dots,f_r)}$ should be *exactly* $$\label{eq:nottrue}
(1+t)^n \prod_{i=1}^r (1-t^{d_i})$$ However, this is not true, even for the simplest case $r=2$ and $d_1=d_2=2$. In Table \[tab:2gens\] we tabulate the difference between the true Hilbert series and .
n 2 3 4 5 6 7 8 9 10 11 12 13
------ --- --- --- ------- --- ------- ------- ------- --------- ----------- --------- ---------------
Diff 0 0 0 $t^3$ 0 $t^4$ $t^4$ $t^5$ $10t^5$ $t^6+t^5$ $64t^6$ $t^7 + 13t^6$
: Difference between the true Hilbert series and the “anticipated Hilbert series” for generic ideals generated by two quadratic forms[]{data-label="tab:2gens"}
[The signed incidence matrix has full rank when the difference in cardinality is even]{}
We prove a “signed version” of the well-known theorem that the incidence matrix of $r$-subsets of $[n]={{\{1,\dots,n\}}}$ into $d+r$-subsets have full rank. Our proof is a modification of the one by Wilson [@Wilson:design].
To begin, let us define the “signs” involved.
Let $[n]={{\{1,\dots,n\}}}$, and let ${\mathcal{C}}$ and ${\mathcal{R}}$ be two subsets of $[n]$, with $$\begin{split}
{\mathcal{C}}&={{\{t_1,\dots,t_a\}}}, \qquad t_1 < \dots < t_a \\
{\mathcal{R}}&={{\{k_1,\dots,k_b\}}}, \qquad k_1 < \dots < k_b
\end{split}$$ Then define $\sigma({\mathcal{C}},{\mathcal{R}})$ to be zero if ${\mathcal{C}}\not \subseteq {\mathcal{R}}$, and otherwise the sign of the permutation which sorts $[{\mathcal{C}},{\mathcal{R}}\setminus {\mathcal{C}}]$ in ascending order. In other words, if ${\mathcal{C}}\subseteq {\mathcal{R}}$ then $\sigma({\mathcal{C}},{\mathcal{R}})$ is the sign of the uniquely determined permutation $\gamma$ such that $$\begin{split}
t_{\gamma(i)} & = k_i, \qquad 1 \le i \le a \\
k_{\gamma(j)} & = k_{a+j}, \qquad 1 \le j \le b \\
\end{split}$$
\[def:rowsum\] Let $[n]={{\{1,\dots,n\}}}$, and let $A$, $B$ be two subsets of $[n]$, of cardinality $a$ and $b$, with $0 \le a < b$. For $a \le r < b$, we define $$\label{eq:sdik}
s_r(A,B,n) = \sum_{\substack{{\mathcal{C}}\in \binom{[n]}{r}\\
A \subseteq {\mathcal{C}}\subseteq B}} \sigma({\mathcal{C}},B)$$ For $0 \le d \le n$, we define $$\label{eq:sd}
s_{d,n} = \sum_{R \in \binom{[n]}{d}} \sigma(R,[n])
= s_d(\emptyset, [n] , n)$$
\[lemma:sindep\] With the notations of Definition \[def:rowsum\], put $d=b-r$. We have that $$\label{eq:sindep}
s_r(A,B,n) =
\begin{cases}
0 & A \not \subseteq B \\
(-1)^d s_{d,b-a} & A \subseteq B
\end{cases}$$
Put $d=b-r$. If $A \not \subseteq B$ then clearly $s_r(A,B,n)=0$. Suppose that $A \subseteq B$. Then $$s_r(A,B,n) = \sum_{\substack{{\mathcal{C}}\in \binom{[n]}{r}\\
A \subseteq {\mathcal{C}}\subseteq B}} \sigma({\mathcal{C}},B)
= \sum_{\substack{{\mathcal{C}}\in \binom{B}{r}\\ A \subseteq {\mathcal{C}}}}
\sigma({\mathcal{C}},B),$$ so the sum is independent of $n$. Furthermore, we can write $A \subseteq {\mathcal{C}}\in \binom{B}{r}$ as a disjoint union ${\mathcal{C}}= A \cup
({\mathcal{C}}\setminus A)$, hence the sum can be written $$\sum_{S \in \binom{B \setminus A}{r-a}} \sigma(S \cup A, B)
= \sum_{S \in \binom{B \setminus A}{r-a}} \sigma(S , B
\setminus A).$$ Now, since $S$ has cardinality $r-a$, the set $(B \setminus A)
\setminus S$ has cardinality $b-a - (r-a) = b-r =d$, so the permutation which transforms $[S, B \setminus A]$ to $[B
\setminus A, S]$ has cardinality $(-1)^d$. Hence, by substituting $R = (B \setminus A) \setminus S$, we get that the sum is equal to $$\begin{gathered}
(-1)^d \sum_{S \in \binom{B \setminus A}{v-a}} \sigma((B \setminus
A) \setminus S , B \setminus A)
= (-1)^d\sum_{R \in \binom{B \setminus A}{d}} \sigma(R, B
\setminus A) \\
= (-1)^d \sum_{R \in \binom{[b-a]}{d}} \sigma(R, [b-a]),
\end{gathered}$$ which is the desired result.
\[lemma:notzero\] Suppose that $0 < d \le n$, and that $d$ is even. Then $s_{d,n} > 0$.
The lemma is trivially true for $d=n$. If $d=2$, we note that $\sigma({{\{v,v+1\}}},[n]) = 1$ for $1 \le v < n$, since the permutation transforming $[v,v+1,1,2,\dots,v-1,v+2,v+3,\dots, n]$ to $[1,2,\dots,n]$ is even. Furthermore, the signs of $\sigma({{\{v,v+\ell\}}},[n])$ alternate in sign as $\ell$ goes from $1$ to $n-v$. Thus, for a fixed $v$, there are either as many positive as negative $\sigma({{\{v,v+\ell\}}},[n])$, or 1 more positive than negative, depending on the parity of $n-v$. By summing over all $v$, we conclude that there are always strictly more positive than negative signs.
Now suppose that we have shown that $s_{2k',n'} > 0$ for all $k',n'$ such that $k' < k$. We want to show that that $s_{2k,n}
> 0$. We have that $$s_{2k,n} = \sum_{R \in \binom{[n]}{2k}} \sigma(R,[n]),$$ and writing $R$ as a disjoint union of its first two element, and the remaining elements, this becomes $$\begin{gathered}
\sum_{1 \le k < \ell \le n-2} \sum_{R_2 \in \binom{{{\{\ell+1,
\ell+2,\dots, n\}}}}{2k-2}} \sigma({{\{k,\ell\}}} \cup R_2, [n]) \\
= \sum_{1 \le k < \ell \le n-2} \sum_{R_2 \in \binom{{{\{\ell+1,
\ell+2,\dots, n\}}}}{2k-2}}\sigma(R_2, {{\{\ell+1,
\ell+2,\dots, n\}}})
= \sum_{1 \le k < \ell \le n-2} s_{2k-2,n-\ell} > 0.
\end{gathered}$$
Next, we define the signed incidence matrix.
Let $0 < a < b \le n$ be integers. Then $M_{a,b,n}$ is the $\binom{n}{b}
\times \binom{n}{a}$ matrix where the rows are indexed by $b$-subsets of $[n]$, the columns by $a$-subsets of $[n]$, and where the entry in row $B$, column $A$ is $\sigma(A,B)$.
\[thm:fullrank\] Let $0 < a < b \le n$ be integers. If $d=b-a$ is even, then $M_{a,b,n}$ has full rank.
Denote the row indexed by ${\mathcal{R}}\in \binom{[n]}{b}$ by $\tau_{\mathcal{R}}$, then $\tau_{\mathcal{R}}$ can be regarded as an element in $V_a([n])$, the free ${{\mathbb{Q}}}$-vector space on the $a$-subsets of $[n]$. If we denote the basis element corresponding to a $a$-subset ${\mathcal{C}}$ by $\epsilon_{\mathcal{C}}$, then $$\tau_{\mathcal{R}}= \sum_{{\mathcal{C}}\in \binom{[n]}{a}}
\sigma({\mathcal{C}},{\mathcal{R}}) \epsilon_{\mathcal{C}}.$$
The number of rows in $M_{a,b,n}$ is $\binom{n}{b}$, and the number of columns is $\binom{n}{a}$. There are less rows than columns if $a+b > n$, as many rows as columns if $a+b = n$, and more rows than columns if $a+b < n$.
1. If $\mathrm{a+b \ge n}$, we must prove that the rows are linearly independent. Suppose that there is a linear relation among the $\tau_{\mathcal{R}}$’s, so that $$\label{eq:linjrel}
\sum_{{\mathcal{R}}\in \binom{[n]}{b}} a_{\mathcal{R}}\tau_{\mathcal{R}}= 0$$ for some numbers $a_{\mathcal{R}}$. We shall prove that all $a_{\mathcal{R}}= 0$.
Choose an $I \subset \binom{[n]}{i}$, $0 \le i \le a$, and define a linear functional $H_I: V_a([n]) \to {{\mathbb{Q}}}$ by $$\label{eq:linfunc}
f_I(\epsilon_{\mathcal{C}}) =
\begin{cases}
1 & I \subseteq {\mathcal{C}}\\
0 & I \not \subseteq {\mathcal{C}}\end{cases}$$ Then if ${\mathcal{R}}\in \binom{[n]}{b}$ we have that $$\begin{gathered}
\label{eq:calc1}
f_I(\tau_{\mathcal{R}}) = f_I \left( \sum_{{\mathcal{C}}\in \binom{[n]}{a}} \sigma({\mathcal{C}},{\mathcal{R}})
\epsilon_{\mathcal{C}}\right)
= \sum_{{\mathcal{C}}\in \binom{[n]}{a}} \sigma({\mathcal{C}},{\mathcal{R}}) f_I(\epsilon_{\mathcal{C}}) \\
= \sum_{I \subseteq {\mathcal{C}}\subseteq {\mathcal{R}}} \sigma({\mathcal{C}},{\mathcal{R}})
= s_a(I,{\mathcal{R}},n) =
\begin{cases}
s_{d,b-i} & I \subseteq {\mathcal{R}}\\
0 & I \not \subseteq {\mathcal{R}}\\
\end{cases}
\end{gathered}$$ The last step follows from Lemma \[lemma:sindep\]. Applying $f_I$ to we get that $$\begin{gathered}
\label{eq:calc2}
0 = f_I \left( \sum_{{\mathcal{R}}\in \binom{[n]}{b}} a_{\mathcal{R}}\tau_{\mathcal{R}}\right)
= \sum_{{\mathcal{R}}\in \binom{[n]}{b}} a_{\mathcal{R}}f_I(\tau_{\mathcal{R}}) \\
= \sum_{{\mathcal{R}}\in \binom{[n]}{b}} a_{\mathcal{R}}s_a(I,{\mathcal{R}})
= s_{d,b-i} \sum_{\substack{{\mathcal{R}}\in \binom{[n]}{b}
{\mathcal{R}}\supseteq I}} a_{\mathcal{R}}\end{gathered}$$ Since Lemma \[lemma:notzero\] tells us that $s_{d,b-i} \neq 0$, we conclude that $$\label{eq:subsetzero}
\sum_{{\mathcal{R}}\supseteq I} a_{\mathcal{R}}= 0$$
Now, for any $J \subset [n]$ we have, by exclusion-inclusion, that $$\label{eq:inex}
\sum_{{\mathcal{R}}\cap J = \emptyset} a_{\mathcal{R}}= \sum_{I \subset J} (-1)^{\lvert
I \rvert} \sum_{{\mathcal{R}}\supseteq I} a_{\mathcal{R}}$$ Fix ${\mathcal{R}}_0 \in \binom{[n]}{b}$ and put $J_0 = [n] \setminus
{\mathcal{R}}_0$. Since $\lvert J_0 \rvert = n - b \le a$ we have, using that $$\label{eq:a0}
a_{{\mathcal{R}}_0} = \sum_{{\mathcal{R}}\cap J_0 = \emptyset} a_{\mathcal{R}}= \sum_{I \subseteq
J_0} (-1)^{\lvert I \rvert} \sum_{{\mathcal{R}}\supseteq I} a_{\mathcal{R}}= 0$$ Since $a_{{\mathcal{R}}_0}$ was arbitrary, all $a_{\mathcal{R}}$ are zero. This shows that the $\tau_{\mathcal{R}}$ are linearly independent.
2. If $\mathrm{n = a+b}$, then $M$ is a square matrix. By the previous case, the vectors $\tau_{\mathcal{R}}$ are linearly independent, but since there are $\binom{n}{a} =
\binom{n}{b}$ such vectors, they form a basis of $V_a([n])$; in particular, they span this vector space.
3. Finally, let us consider the remaining case $\mathrm{n > a+b}$, so that there are more rows than columns. We must prove that the rows span $V_a([n])$. We prove this by induction over $n-a-b$. The case $n-a-b=0$ is already proved, and forms the basis of the induction. We assume $a,b$ fixed, and that the assertion has been proved for all $a+b \le n' < n$.
Let $\Gamma \in \binom{[n]}{a}$ be arbitrary. If we can express $\alpha=\epsilon_\Gamma$ as a linear combination of the $\tau_{\mathcal{R}}$’s, we are done. To this end, put $$\label{eq:ap}
\alpha' = \sum_{\substack{S \in \binom{[n-1]}{a-1}\\ S \cup
{{\{n\}}} = \Gamma}} \epsilon_S \in V_{a-1}([n-1])$$ Since $a-1 + b < n-1$, it follows by induction that there are scalars $\setsuchas{d_J}{J \in \binom{[n-1]}{a-1}}$ such that $$\label{eq:Js}
\alpha' = \sum_{J \in \binom{[n-1]}{a-1}} d_J \tau_J', \qquad
\tau_J' = \sum_{\substack{S \in \binom{[n-1]}{a-1}\\ S \subseteq
J}} \epsilon_S$$
For ${\mathcal{R}}\in \binom{[n]}{a}$, $n \in {\mathcal{R}}$, put $c_{\mathcal{R}}' = d_{\mathcal{R}}\setminus
{{\{n\}}}$. Define $$\label{eq:a00}
\alpha_0 = \sum_{\substack{{\mathcal{R}}\in \binom{[n]}{a}\\ n \in {\mathcal{R}}}}
c_{\mathcal{R}}' \tau_{\mathcal{R}}\in V_a([n])$$ If we write $$\alpha_0 = \sum_{{\mathcal{C}}\in \binom{[n]}{a}} a_{\mathcal{C}}' \epsilon_{\mathcal{C}}$$ we have that for ${\mathcal{C}}\in \binom{[n]}{a}$, $n \in {\mathcal{C}}$, that $$a_{\mathcal{C}}' =
\begin{cases}
1 & {\mathcal{C}}= \Gamma \\
0 & {\mathcal{C}}\neq \Gamma
\end{cases}$$ which implies that $$\alpha_0 =
\begin{cases}
\alpha & n \in \Gamma \\
0 & n \not \in \Gamma
\end{cases}$$ In either case, $\alpha - \alpha_0$ has coordinate 0 in component ${\mathcal{R}}\in \binom{[n]}{a}$, unless $n \in {\mathcal{R}}$. Hence, $\alpha-\alpha_0$ may be regarded as a vector in $V_a([n-1])$. By the induction hypothesis, there exist $c_{\mathcal{R}}''$ such that $$\label{eq:a0d}
\alpha-\alpha_0 = \sum_{{\mathcal{R}}\in \binom{[n-1]}{a}} c_{\mathcal{R}}'' \tau_{\mathcal{R}}$$ Defining $$c_{\mathcal{R}}=
\begin{cases}
c_{\mathcal{R}}' & n \in {\mathcal{R}}\\
c_{\mathcal{R}}'' & n \not \in {\mathcal{R}}\end{cases}$$ we get that $$\alpha = \alpha_0 + (\alpha-\alpha_0) =
\sum_{\substack{{\mathcal{R}}\in \binom{[n]}{a}\\ n \in {\mathcal{R}}}} c_{\mathcal{R}}'
\tau_{\mathcal{R}}+
\sum_{\substack{{\mathcal{R}}\in \binom{[n]}{a}\\ n \not \in {\mathcal{R}}}} c_{\mathcal{R}}'' \tau_{\mathcal{R}}= \sum_{{\mathcal{R}}\in \binom{[n]}{a}} c_{\mathcal{R}}\tau_{\mathcal{R}}$$
[Calculations]{} The computer calculations were done on the computers of the UMS Medicis, École Polytechnique, and on the computers at the Department of Mathematics, Stockholm University. We have used the programme Macaulay 2 [@MACAULAY2] to calculate Hilbert series and minimal free resolutions. To save time and memory, the calculations were performed in characteristic 31991. The holes in the tables show that there are limits to what we could calculate, even on a machine with 2 GB of memory.
[^1]: Snellman was supported grants from Svenska institutet and by grant n. 231801F from Centre International des Etudiants et Stagiaires while visiting École Polytechnique, and by grants from Svenska Institutet and Kungliga Vetenskapsakademin while visiting University of Wales, Bangor
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We present an equilibrium statistical mechanical theory of collisionless self-gravitational systems with isotropic velocity distributions. Compared to existing standard theories, we introduce two changes: (1) the number of possible microstates is computed in energy (orbit) space rather than phase space and (2) low occupation numbers are treated more appropriately than using Stirling’s approximation. Combined, the two modifications predict that the relaxed parts of collisionless self-gravitating systems, such as dark-matter halos, have a differential energy distribution $N(\varepsilon) \propto [\exp(\phi_0-\varepsilon) -1]$, dubbed “DARKexp”. Such systems have central power-law density cusps $\rho(r) \propto r^{-1}$, which suggests a statistical mechanical origin of cusps in simulated dark-matter halos.'
author:
- 'Jens Hjorth and Liliya L. R. Williams'
bibliography:
- 'smo.bib'
title: ' Statistical mechanics of collisionless orbits. I. Origin of central cusps in dark-matter halos '
---
Introduction\[introduction\]
============================
The apparent universal light distributions in elliptical galaxies with two-body collision relaxation time exceeding the age of the universe motivated @1957SvA.....1..748O and @1967MNRAS.136..101L to seek a fast relaxation process driving the systems toward equilibrium in a statistical mechanical sense. In a seminal paper, @1967MNRAS.136..101L introduced the process of ‘violent relaxation’ (collective energy exchange between rapid potential fluctuations and individual particles) as responsible for a short time scale for relaxation. In the same paper, he introduced a new kind of statistical mechanics, that of distinguishable particles subject to an exclusion principle because collisionless dynamics precludes two parcels of phase space from being superimposed. In the non-degenerate limit the theory predicts that isothermal spheres are the maximum entropy equilibrium states of the process, as also found by @1957SvA.....1..748O.
The predictions of the theory are, however, not entirely satisfactory. It predicts infinite mass systems despite being derived under the constraints of fixed energy and mass. It also predicts mass (phase-space density) segregation despite the dynamics being collisionless . The resulting isothermal sphere profile does not reproduce the light profiles of elliptical galaxies which was the original motivation. There is an arbitrariness in defining the initial states and whether to use a particle or phase element approach [@1978ApJ...225...83S; @1987ApJ...316..497M; @1987ApJ...316..502S; @1993MNRAS.265..237H; @1996ApJ...466L...1K; @1997ApJ...484...58K; @2008NewAR..52....1B], which makes it difficult to assess whether the degenerate limit may be relevant. And finally, it is not obvious how to extend the statistical mechanical approach to (spherical) systems with anisotropic velocity distributions.
Some of these issues have been addressed in terms of modifications of the extent of the relaxation process, so-called incomplete violent relaxation [@1987MNRAS.229...61S], relaxation in a finite volume [@1991MNRAS.253..703H], or explicit scattering processes [@1992ApJ...397L..75S]. Another approach has been to propose a change in the entropy functional to be optimized, applicable to non-extensive systems [@1988JSP....52..479T; @1993PhLA..174..384P], although this has been demonstrated not to work [@2007ApJ...655..847B; @2008PhRvE..77b2106F].
To address the infinite-mass problem, @1996MNRAS.280.1089M [and later @2006PhyA..367..269M; @2008AdAst2008E...3D] showed that appropriately dealing with small occupation numbers leads to finite-mass systems, similar to @1966AJ.....71...64K models, suitable for the description of globular clusters, which are driven by collisional relaxation.
The subject of the origin of universal collisionless self-gravitating structure has gained renewed attention in recent years with the demonstration that the end products of numerical simulations of cosmological structure formation and dark-matter halos have remarkably universal profiles, in density as well as in pseudo phase-space density, $\rho/\sigma^3$. A new aspect, not foreseen by Lynden-Bell’s theory or any modifications thereof, is that simulated dark-matter halos appear to have central density cusps, $\rho(r) \sim r^{-1}$, falling to $r^{-3}$ or $r^{-4}$, or even steeper, in the outer parts [@1997ApJ...490..493N; @2004MNRAS.349.1039N; @2005ApJ...624L..85M; @2010MNRAS.402...21N]. The origin of such cusps is unclear [see, e.g., @2006ApJ...653..894H]. There also appears to be a relation between the local density slope and the degree of radial velocity anisotropy [@2006NewA...11..333H].
In this paper, we explore a new avenue for implementing the collisionless nature of dark-matter halos. We suggest to partition state space in energy-per-unit mass shells rather than phase-space elements. Moreover, we implement the corresponding effect of small occupation numbers in this approach. We show that the resulting state is a finite-mass system with a cusp, $r^{-1}$, falling in the outer parts to $r^{-4}$ for a completely relaxed, isolated, isotropic system. This captures the overall properties of simulated halos. In companion papers we compare in detail the resulting states with numerically generated systems suitable for testing our predictions.
Statistical mechanics\[sm\]
===========================
We start out, following @1896gas.book.....B and @1957SvA.....1..748O, by defining the number of possible states: $$W = N! \prod_i \frac{ g_i^{n_i}}{ n_i!}$$ [as discussed in Section 4 we do not introduce an exclusion principle, cf. @1967MNRAS.136..101L; @1978ApJ...225...83S; @1987MNRAS.229...61S]. Here, $n_i$ is the occupation number in cell of size $g_i$ in state space ($\mu$ space).
It is customary to take the continuous limit ($n_i \to n$), identify $\Gamma(n+1) = n!$, and optimize $\ln W$ under constraints of fixed total energy and total number of particles using a variational approach. This yields $$\ln g - \psi(n+1)=\alpha+\beta E,
\label{var2}$$ where $\psi(n)\equiv d \ln \Gamma / d n$ is the digamma function, $\alpha$ and $\beta$ are Lagrange multipliers associated with the constraints of total number of particles and total energy, respectively, and $E=v^2/2+\Phi({\bf x})$ is the energy per unit mass, where $\Phi({\bf x})$ is the gravitational potential.
Next, one introduces Stirling’s formula $$\ln n! = \left ( n + \frac{1}{2} \right ) \ln n - n + \frac{1}{2} \ln (2\pi) + \frac{\theta(n)}{12 n}\ ; \ \ 0 < \theta(n) < 1$$ and uses Stirling’s approximation for large $n$ $$\ln n! = n \ln n - n \ ; \ \ n \gg 1
\label{stirling4}$$ which implies $$\psi(n+1)= \ln n\ ; \ \ n \gg 1.
\label{stirling5}$$ This leads to $$n = g \exp (-\alpha - \beta E).
\label{occupnum}$$ This is the classical finding for the non-degenerate limit, as we did not introduce an exclusion principle in Equation (1). In particular, identifying $\mu$ space with the 6-dimensional (${\bf x}$,${\bf v}$) phase space, the isothermal sphere is retrieved if phase space is divided up equally into equal-size phase-space cells $g$: $$f({\bf x},{\bf v}) = A \exp \left (-\beta\left[\frac{1}{2} v^2+\Phi({\bf x})\right ]\right ).
\label{classicalf}$$
Below we introduce two modifications to this standard approach, which, combined, give dramatically different structures, which turn out to be reminiscent of the end-products of numerical simulations of collisionless self-gravitating $N$-body systems.
Orbit Space Versus Phase Space\[orbits\]
----------------------------------------
The first major modification consists in noting that in an equilibrium collisionless system all particles retain their energies. This is not the case in an equilibrium collisional system, such as a classical gas, or a self-gravitating system dominated by two-body encounters, such as a globular cluster. The fundamental property of a collisionless system is that particles are distributed on orbits. In a relaxed system, once an energy is assigned to a particle it stays in a restricted portion of phase space, an energy cell. For an isotropic system, energy is the only isolating integral. Hence, we argue that in partitioning state space, the fundamental partition is energy space, not phase space [see also @Efthymiopoulos07].
This implies that the occupation number $n$, Equation (\[occupnum\]), should be interpreted as an indicator of the number of particles with a given energy, i.e., $$N(E) \propto \exp (-\beta E)
\label{DARKexp1}$$ and not as the number of particles in a parcel of classical phase space, as is usually assumed.
@1982MNRAS.200..951B [see also, e.g., @1990ApJ...356..359H; @1992ApJ...397L..75S; @1987gady.book.....B] found that elliptical galaxies obeying the $R^{1/4}$ law have energy distributions very similar to Equation (\[DARKexp1\]), however these authors’ motivation for using this form was largely empirical.
Given $N(E)$ one can, of course, recover the classical distribution function. @1987MNRAS.229...61S point out that the occupation numbers in the classical phase-space can be related to those in the energy space if the former are assigned non-equal a priori weights, inversely proportional to the volume of phase space with a given energy, i.e., $$f(E) \propto g(E)^{-1} \exp (-\beta E),$$ from which Equation (\[DARKexp1\]) is retrieved since $N(E)\equiv f(E)g(E)$ for isotropic systems. The relation between the phase-space density $f$ and the differential energy distribution $N$ is obtained assuming isotropy from $f({\bf x},{\bf v})d^3{\bf v} d^3{\bf x} = N(E) dE$ and the density of states $g(E) = d^3{\bf v} d^3{\bf x}/dE =
16 \pi^2 \int_0^{r_{\rm max}(E)} \sqrt{2(E-\Phi)} r^2 dr$ [@1987gady.book.....B].
Another interesting aspect of Equation (\[DARKexp1\]), as pointed out by @1982MNRAS.200..951B, is that for systems with less mass at the centers than in the outer parts, $\beta$, and hence the effective temperature, $T=\beta^{-1}$, need to be negative [see also @1989MNRAS.236..829M]. This is consistent with the self-gravitating nature of the systems, as these are characterized by negative heat capacity [@1987gady.book.....B].
Small occupation numbers\[occupation\]
--------------------------------------
### Integer approach
The second major modification is necessary because in systems with a finite potential depth the occupation numbers in energy cells with very bound energies can become small (Equation (\[DARKexp1\])). A similar problem is encountered when the classical phase-space distribution function, $f(E)$, takes on an exponential form (Equation (\[classicalf\])), but in this case, because the temperature is positive, the small occupation numbers are found at the near escape energies, i.e., in the outer regions of systems. @1996MNRAS.280.1089M pointed out that in this case the Stirling approximation, Equation (\[stirling4\]), breaks down. Following @1994AmJPh..62..515S, he argued that the appropriate form for Equation (\[occupnum\]) is $$n_i = [g_i \exp (-\alpha -\beta E_i)],
\label{occupnum2}$$ where \[$\cdot$\] means rounding down to the nearest integer.
Because the majority of particles in this latter case have energies near escape energies, the small occupation number modification has a dramatic effect on the resulting structures. @1996MNRAS.280.1089M showed that Equation (\[occupnum2\]) introduces a cutoff which results in finite-mass systems, similar to the @1966AJ.....71...64K models of globular clusters. In effect, @1996MNRAS.280.1089M analytically derived the well-known @1966AJ.....71...64K models, which were originally obtained as a simple heuristic modification of the isothermal sphere’s distribution function, Equation (\[classicalf\]).[^1]
For collisionless systems, where the temperature is negative, the cutoff occurs at energies close to that of the central potential value, i.e., close to the center of the system. Hence, the effect on the structures is not so dramatic in terms of total mass. But as we show below, it determines the inner density profile of the equilibrium systems.
### Continuous approach
Because a physical system is not expected to have a step-like differential energy distribution $N(E)$, a continuous version of Equation (\[occupnum2\]) is needed. We do this by introducing a superior approximation to Stirling’s formula which, unlike Equation (\[stirling5\]), is not limited to large occupation numbers.
We start by noting that when $n=0$, one has the exact result $\psi(1)=-\gamma$, where $\gamma\approx 0.57721566...$ is Euler’s constant. Combining this with the large $n$ limit, Equation (\[stirling5\]) leads to the approximation $$\psi(n+1)\approx \ln(n + \zeta),$$ with $\zeta= \exp(-\gamma) \approx 0.561459...$. While simple, this expression is a remarkably good approximation as shown in Figure 1 along with the classical large $n$ approximation (Equation (\[stirling5\]); $\zeta=0$). The deviation from the exact $\psi(n+1)$ is always positive, with a single maximum difference of 0.0237 at $n=0.680$. Using this in Equation (\[var2\]), one obtains a modification to Equation (\[occupnum\]), $$n = g \exp (-\alpha-\beta E) - \zeta.$$ Similar modifications have previously been proposed [@1954PNAS...40..149L; @1993hjorth.thesis.H; @2006PhyA..367..269M; @2008AdAst2008E...3D].
The Lagrange multiplier $\alpha$ is determined by requiring that $n=0$ at some energy $E'$. Thus, we get $$n = \zeta (\exp (-\beta [E - E'])-1),
\label{occupnum3}$$ thereby eliminating the cell size, $g$, which does not have a unique, physically meaningful value in gravitational systems.
For a system which vanishes at the escape energy, $E' = 0$, and so the classical phase-space occupation number function is $n = \zeta (\exp (-\beta E)-1)$. The systems we are interested in vanish at a finite central potential, $E'=\Phi_0$, and so the energy-space occupation number function becomes $n = \zeta (\exp (-\beta (E-\Phi_0))-1)$. This is the continuous version of Equation (\[occupnum2\]).
DARKexp models
--------------
Equation (\[occupnum3\]) with $E'=\Phi_0$ incorporates the two modifications we introduced in Sections \[orbits\] and \[occupation\]. Identifying the occupation number $n$ as being proportional to $N$, the differential energy distribution, one obtains $N(E) = A (\exp(-\beta [E-\Phi_0])-1)$, where $A$ is determined by the mass of the system. In this expression, $E$ and $\Phi_0$ have units of energy, and $\beta$ is the inverse of temperature. We convert this to dimensionless form, using $\varepsilon=\beta E$ and $\phi_0=\beta\Phi_0$ (note that both $\varepsilon$ and $\phi_0$ are positive quantities for bound systems), $$N(\varepsilon)=A(\exp[\phi_0-\varepsilon]-1).
\label{DARKexp}$$ We dub this expression the DARKexp. It represents our prediction for fully relaxed, collisionless, self-gravitating, isotropic systems. Because the arguments presented in this paper do not apply to non-isotropic systems, Equation (\[DARKexp\]) cannot be directly compared to the results of $N$-body simulations. Having said that, we note that $N(\varepsilon)$ is not very sensitive to anisotropy, as it depends primarily on the density profile and not on the dynamics of the system.
The detailed structure of the DARKexp systems, including their density and velocity dispersion profiles, will be considered in an accompanying paper [@WilliamsHjorth10]. In the next section, we discuss the limiting behavior of DARKexp models at small and large radii.
Resulting structures\[structures\]
==================================
Limiting Power-Law Behavior at Small Radii\[limit1\]
----------------------------------------------------
In a general spherically symmetric structure, the limiting form for the central potential can be assumed to be $$\phi=\phi_0-\phi_\alpha r^\alpha + \cdots$$ Possion’s equation $\nabla^2 \phi = 4 \pi G \rho$ yields a limiting power-law behavior of the central density, $$\rho(r)\propto r^{\alpha-2}. $$ For $\varepsilon \to \phi_0$ the distribution function then becomes $$f(\varepsilon)\propto(\phi_0-\varepsilon)^{-(4+\alpha)/2\alpha}$$ and the density of states becomes $$g(\varepsilon)\propto(\phi_0-\varepsilon)^{(6+\alpha)/2\alpha}.$$ For similar expressions, see @1995MNRAS.276..679H, @2000ApJS..131...39W, and @2004MNRAS.353...15A. The differential mass distribution $N(\varepsilon)=f(\varepsilon)g(\varepsilon)$ then becomes $$N(\varepsilon)\propto(\phi_0-\varepsilon)^{1/\alpha}.$$
In general, the value of $\alpha$ ranges from 0 for the singular isothermal sphere to $\alpha=1$ for Navarro–Frenk–White or Hernquist profiles, to shallower slopes, reaching a flat core for $\alpha=2$. Increasing $\alpha$ corresponds to increasingly steeper $\ln N(\varepsilon)$ versus $\varepsilon$ curves. For $\alpha\to\infty$, the system develops a hole in the central density profile, which is unphysical.
Limiting Power-Law Behavior at Small Radii for DARKexp Models
-------------------------------------------------------------
In the DARKexp model, Equation (\[DARKexp\]), $N(\varepsilon) \propto (\phi_0 - \varepsilon)$ as $\varepsilon \to \phi_0$. In other words, $\alpha = 1$ and we retrieve the central density cusps $$\rho(r) \propto r^{-1},$$ known from numerical simulations. The corresponding distribution function is $$f(\varepsilon) \propto (\phi_0-\varepsilon)^{-5/2}
$$ for $\varepsilon \to \phi_0$, which is the same as that of the @1990ApJ...356..359H model.
Thus our proposed differential energy distribution, the DARKexp, naturally predicts that the central density slopes of collisionless self-gravitating systems should asymptote to $-1$ at the centers of structures. Note that this slope is not the result of any specific dynamical process operating during the formation of halos, but a generic consequence of full relaxation.
The Appendix discusses the limiting behavior for more general cutoffs to $N(\varepsilon) \propto \exp(-\varepsilon)$.
Limiting Behavior at Large Radii
--------------------------------
While we are focusing on the inner parts of halos in this paper, we note that a full model for the entire system can be obtained by assuming that $N(\varepsilon)$ is finite at the escape energy and zero above (as plotted in Figure 2). The rationale behind this is that during violent relaxation the escape energy is no special location and $N(\varepsilon)$ is expected to be continuous at what will eventually become the escape energy [@1987IAUS..127..511J; @1991MNRAS.253..703H]. For such a model $f(\varepsilon) \propto \varepsilon^{5/2}$ and $\rho(r) \propto r^{-4}$ for $\varepsilon \to 0$ and $r \to \infty$, similar to the Hernquist model and broadly consistent with numerical simulations of dark-matter halos. However, if relaxation is not complete at radii where particles have near escape velocities, then the outer density profile slope may deviate from $-4$.
Discussion
==========
The approach to equilibrium
---------------------------
In this paper, we used statistical mechanics to predict the energy distribution function of relaxed systems. A key feature of statistical mechanics approaches is that they deal only with the final equilibrium states, and derive these by equating them to the most probable or maximum entropy states. We stress that our theory is therefore limited to the description of the final state of self-gravitating collisionless systems. Because the final state is an equilibrium state, it is the result of full relaxation and therefore does not rely on any additional assumptions.
Implicit in our derivation is the assumption of equal a priori probabilities in state space. While this can be accomplished through efficient mixing (ergodicity), our theory does not address specific physical mechanism for attaining full mixing. In a companion paper [@WilliamsHjorth10], we use the Extended Secondary Infall Model (ESIM) to test the DARKexp prediction, but stress that ESIM is a restricted physical model and is not equivalent to $N$-body simulations.
A straightforward way to evaluate the applicability of physical mechanisms for relaxation is to compare their end states. For example, the scattering model for violent relaxation introduced by @1992ApJ...397L..75S predicts $N(E) \propto [(E-\Phi_0)^{-2}+C(E-\Phi_0)^{-1/2}]^{-1} \exp (-\beta E)$ which is clearly inconsistent with the DARKexp final state. On the other hand, the final state of ESIM halos is well fit with DARKexp.
Collisional Versus Collisionless Systems
----------------------------------------
Our prediction for $N(\varepsilon)$ applies to the end states of collisionless dynamics. Collisionlessness has several different consequences, many of which can be used in one’s theory. @1967MNRAS.136..101L used the collisionless Boltzmann equation to introduce an exclusion principle to account for the incompressibility of the phase-space fluid. We use a different property; that collisionless dynamics implies the constancy of the energy per unit mass for each particle. Therefore, the collisionless nature of the problem is automatically included and there is no need for an exclusion principle.
In a system driven toward equilibrium by two-body relaxation, the situation is quite different. In this case, efficient relaxation implies that any particle can in principle end up anywhere in phase space and the usual partition of $\mu$ space is appropriate. In this case, taking into account low occupation numbers, one retrieves the @1966AJ.....71...64K $f(E)$ which is rounded down at near escape energies, while DARKexp’s collisionless $N(E)$ is rounded down at highly bound energies.
The density profiles of the two are clearly different, but both are very good approximations to what one sees in globular clusters and simulated dark-matter haloes, respectively.
Anisotropy\[anisotropy\]
------------------------
In this paper we dealt exclusively with isotropic systems for which the differential energy distribution is a function of energy only. The full simulated structures, however, are definitely anisotropic, and exhibit a correlation between the density slope and anisotropy [@2006NewA...11..333H].
In principle, our statistical mechanics formalism can be extended to spherical non-isotropic systems, and we plan to do so in a future publication. In a fully developed theory, the distribution function must depend explicitly on angular momentum, especially in the outer parts [e.g., @1987MNRAS.229...61S]. Whether the energy distribution of the full theory will be significantly affected by the introduction of angular momentum is yet unclear; numerical experiments of @2006ApJ...653...43M [Figure 9] seem to indicate that the DARKexp form is still a good description of systems that underwent radial orbit instability.
For now we note that even though the DARKexp prediction may not be expected to be an excellent fit to simulations at all energies, it should apply to the central regions of dark-matter halos, as these are isotropic. Therefore, our explanation of the central density cusp of $-1$ applies to simulated halos.
The Dark Cosmology Centre is funded by the Danish National Research Foundation. L.L.R.W. is very grateful for the hospitality of the Dark Cosmology Centre at the University of Copenhagen and the Institute for Theoretical Physics at the University of Zürich, where she spent the Fall of 2009 and the Spring of 2010, respectively.
Appendix {#appendix .unnumbered}
========
Limiting Behavior at Small Radii for Different Exponential Cutoffs {#limiting-behavior-at-small-radii-for-different-exponential-cutoffs .unnumbered}
==================================================================
One might accept that $N(\varepsilon) \propto \exp(-\varepsilon)$ (Section \[orbits\]), but not the proposed form of the cutoff motivated by the low occupation numbers (Section \[occupation\]), for example, because our proposed transition from the integer to the continuous form is more algebraic than physical. In this section we address possible alternatives to the cutoff shape.
First, is a cutoff needed at all? If there is no cutoff, then the differential energy distribution is of the form Equation (\[DARKexp1\]), which would lead to a singularity in the central potential. To avoid that, one needs a cutoff. One possible form is to simply truncate $N(\varepsilon)$ at some finite central value; $N(\varepsilon)=C$ at $\varepsilon=\phi_0$ and $N(\varepsilon)=0$ for $\varepsilon>\phi_0$. This would imply $\alpha=\infty$, and an infinite, positive central density slope, i.e., a hole (Section \[limit1\]) and divergent velocity dispersion, both of which are unphysical.
Therefore, we conclude that a smooth cutoff is required, i.e., that at some finite $\phi_0$, $N(\varepsilon=\phi_0)=0$. At energies near these highly bound energies, we can Taylor expand the non-truncated $N(\varepsilon)$. Using the abbreviation, $x=\phi_0-\varepsilon$: $\exp(x)\sim 1 + x + x^2/2 + \cdots$. Whatever the specific shape of the cutoff function, it too can be Taylor expanded, and then subtracted from that of $\exp(x)$. Three outcomes can result after this subtraction, depending on the exponent of $x$ of the surviving leading term, (1) $x^B$, where $B<1$ and $x$ and higher terms cancel out; (2) $x$; and (3) $x^2$, if the cutoff function goes exactly as $1+x$. In case (1) $\alpha>1$, and the central density slope will be shallower than $-1$. Case (2) leads to our main prediction of the central density slope of $-1$, while case (3) is the only way to get density slopes steeper than $-1$. In this case $\alpha=0.5$, and the central density slope is $-1.5$.
Since mathematically it is possible to get central density slopes shallower or steeper than $-1$, it is ultimately the physical arguments that will dictate the form of the cutoff in $N(\varepsilon)$ and hence the slope value.
[^1]: One can possibly improve upon @1996MNRAS.280.1089M’s Figure 1 by assigning different central potential values of the new models that match the old models. Then, the difference between the two will be confined to low density regions, where it belongs.
|
{
"pile_set_name": "ArXiv"
}
|
---
author:
- 'Jean-Yves Welschinger'
title: 'Optimalité, congruences et calculs d’invariants des variétés symplectiques réelles de dimension quatre'
---
[**Résumé:**]{}
Cet article fait suite à un précédent dans lequel étaient introduits une famille d’invariants par déformation $\chi^d_r$, $d \in H_2 (X ; \Z)$, $r \in \N$, des variétés symplectiques réelles fermées de dimension quatre $(X, \omega , c_X)$, invariants qui fournissent des bornes inférieures en géométrie énumérative réelle. Nous montrons ici par des méthodes de théorie symplectique des champs que ces bornes inférieures sont optimales lorsque $r \leq 1$ et le lieu réel de la variété contient une sphère, un tore ou un plan projectif réel (sous des hypothèses plus restrictives dans ce dernier cas). Nous montrons également qu’une puissance importante de deux divise $\chi^d_r$ pour des valeurs pas trop grandes de $r$ lorsque le lieu réel contient une sphère ou un plan projectif réel (sous les mêmes hypothèses plus restrictives dans ce dernier cas) et proposons enfin quelques calculs explicites dans le cas du plan projectif ou de la quadrique ellipsoïde ainsi que les formules générales permettant de les obtenir, lesquelles font intervenir des invariants relatifs précédemment définis.
Introduction {#introduction .unnumbered}
============
Le présent article fait suite au précédent [@Wels1] dans lequel étaient introduits une famille d’invariants par déformation des variétés symplectiques réelles fermées de dimension quatre. Une [*variété symplectique réelle*]{} est une variété symplectique équipée d’une involution anti-symplectique ; chaque variété projective réelle lisse en fournit un exemple. Ces invariants ont une propriété immédiate soulignée dans [@Wels1], ils fournissent des bornes inférieures en géométrie énumérative réelle. Comme son titre l’indique à présent, cet article poursuit trois objectifs ; le premier est de montrer l’optimalité de ces bornes inférieures, ce que l’on fera dans plusieurs situations (Théorèmes \[theoopt1\] et \[theoopt2\]), le second est de prouver des congruences satisfaites par ces invariants (Théorèmes \[theocong1\], \[theocong2\] et \[theocong3\]) et le dernier de présenter quelques calculs de ces invariants ainsi que les formules générales permettant de les obtenir (Théorèmes \[theocalcproj\], \[theocal2spher\] et \[theocal3spher\]). Remarquons en passant que les résultats d’optimalité en géométrie énumérative réelle se font rares et que les méthodes systématiques pour y aboutir sont, à ma connaissance, inexistantes. La méthode systématique que l’on utilise ici pour aboutir à nos résultats vient de la théorie symplectique des champs [@EGH].
Soit $(X, \omega , c_X)$ une variété symplectique réelle fermée de dimension quatre et soit $d \in H_2 (X ; \Z)$ une classe d’homologie satisfaisant la relation $(c_X)_* d = -d$. Choisissons une structure presque complexe auxiliaire $J$ aussi générale que possible parmi les structures $\omega$-positives qui rendent l’involution $c_X$ anti-holomorphe. Les [*courbes $J$-holomorphes rationnelles réelles*]{} homologues à $d$, c’est-à-dire les sphères $J$-holomorphes invariantes par $c_X$ homologues à $d$, forment alors un espace de dimension $c_1(X)d -1$, où $c_1(X)$ désigne la première classe de Chern de la variété $(X, \omega)$. Nous supposons dans ce travail comme dans [@Wels1] cette dimension positive ou nulle, puisque le cas contraire signifie que l’espace en question est vide, puis faisons chuter cette dimension à zéro en imposant quelques contraintes à ces courbes, à savoir de passer par une collection de $c_1(X)d -1$ points distincts. Ces derniers peuvent être choisis réels, c’est-à-dire fixés par $c_X$, ou bien complexes conjugués, c’est-à-dire échangés par $c_X$ ; nous noterons $r$ le nombre de points réels et $r_X$ le nombre de paires de points complexes conjugués, de sorte que $r+2r_X = c_1(X)d -1$. Seul un nombre fini de courbes $J$-holomorphes rationnelles réelles homologues à $d$ satisfont ces contraintes supplémentaires ; ce nombre dépend en général des choix auxiliaires de la structure presque complexe et de la configuration de points, essentiellement parce que le corps des réels n’est pas algébriquement clos. Toutefois, il ressort de [@Wels1] que si l’on compte ces courbes en fonction d’un signe, positif lorsqu’elles ont un nombre pair de points doubles réels isolés et négatif dans le cas contraire, alors l’entier $\chi_r^d$ que l’on obtient est indépendant des choix de la structure presque complexe $J$, de la configuration de points et même de la forme symplectique $\omega$ à l’intérieur de sa classe de déformation (voir le Théorème $2.1$ de [@Wels1]). Cet entier ne dépend que de la classe d’homologie $d$, du nombre $r$ de points choisis réels et de la répartition de ces points dans les différentes composantes connexes du lieu réel $\R X = \text{fix}(c_X)$ de la variété. En fait, la partie réelle d’une sphère holomorphe réelle étant connexe, cet invariant $\chi_r^d$ est contraint de s’annuler dès que ces points ne sont pas tous choisis dans une même composante $L$ du lieu réel. On adoptera la notation $\chi_r^d (L)$ pour indiquer que les $r$ points réels sont choisis dans $L$. Le nombre $R_d (\underline{x} , J)$ de courbes $J$-holomorphes rationnelles réelles homologues à $d$ qui contiennent l’ensemble $\underline{x}$ de points que l’on s’est donné se retrouve ainsi borné inférieurement par la valeur absolue de l’invariant $\chi_r^d (L)$ ; ce sont là les bornes inférieures en géométrie énumérative réelle que l’on a mentionnées plus haut. Ces bornes s’écrivent $$\begin{aligned}
\label{bornes}
\vert \chi_r^d (L) \vert \leq R_d (\underline{x} , J) \leq N_d,\end{aligned}$$ comme énoncées dans le Corollaire $2.2$ de [@Wels1], le membre $N_d$ désignant le nombre total de courbes $J$-holomorphes rationnelles satisfaisant ces conditions d’incidence (c’est un invariant de Gromov-Witten de genre zéro de la variété).
C’est à ce stade à peu près que nous a laissé [@Wels1] et que l’on reprend ici notre étude en appliquant un principe fondamental de la théorie symplectique des champs en présence d’une telle surface lagrangienne $L$ et d’une structure presque complexe $J$ : on allonge le [*cou*]{} de la structure presque-complexe au voisinage de $L$ pour lui conférer une longueur arbitrairement grande. Rappelons qu’un voisinage de $L$ dans $X$ est symplectomorphe à un voisinage de la section nulle dans son fibré cotangent $T^*L$, un résultat établi dans [@Wein]. Étant donnée une métrique riemannienne sur $L$, le fibré unitaire cotangent $S^*L = \{ (q,p) \in T^*L \; \vert \, \parallel p \parallel = 1 \}$ muni de la restriction de la forme de Liouville $\lambda$ est une variété de contact de dimension trois. Le complémentaire $T^*L \setminus L$ se trouve être symplectomorphe à la symplectisation $(S^*L \times \R , d(e^t \lambda))$ de cette variété. Ce que l’on appelle cou de longueur arbitrairement grande, c’est une portion arbitrairement grande $S^*L \times [-n, n]$ de cette symplectisation dans laquelle $J$ envoie le champ de Liouville $\partial / \partial t$ sur le champ de Reeb de $(S^*L , \lambda)$, préserve les plans de contact et est invariante par translation dans le second facteur, voir [@EGH] et la stratégie générale énoncée au §\[subsubsectstrat\].
Cette technique issue de la théorie symplectique des champs nous permet d’établir les résultats suivants. Lorsque $L$ est une sphère, un tore ou un plan projectif réel (mais dans ce dernier cas $(X, \omega , c_X)$ sera elle-même supposée symplectomorphe au plan projectif complexe éclaté en six boules complexes conjuguées au maximum) et lorsque $r_X$ est maximal ou en d’autres termes lorsque $r \leq 1$, les bornes inférieures (\[bornes\]) sont optimales, atteintes par les structures presque complexes au cou suffisamment long, voir les théorèmes \[theoopt1\] et \[theoopt2\]. Ainsi, lorsqu’on allonge le cou d’une structure presque complexe en préservant l’anti-holomorphicité de $c_X$, il arrive une longueur à partir de laquelle toutes les courbes rationnelles réelles sont comptées en fonction d’un même signe, toutes les éliminations possibles entre courbes s’étant réalisées au cours de l’allongement. Ce phénomène permet plus généralement d’éliminer parfois tous les disques $J$-holomorphes à bords dans une lagrangienne, même en l’absence de structure réelle. Nous le montrerons dans le cas de sphères lagrangiennes au paragraphe \[subsectmin\] qui fait office de digression, voir les Théorèmes \[theomin3\], \[theomin2\] et \[theoopt4\]. Tous ces résultats font l’objet de la première partie de cet article. Dans la seconde partie, on démontre qu’une puissance importante de deux divise l’invariant $\chi_r^d$ lorsque $r$ n’est pas trop grand et $L$ est une sphère ou un plan projectif réel, voir les Théorèmes \[theocong1\], \[theocong2\] et \[theocong3\], le fait que $S^*L$ est un fibré en cercles joue alors un rôle important. Dans la troisième partie de cet article, on présente quelques formules permettant le calcul de $\chi_r^d$ dans le plan projectif complexe ou l’ellipsoïde pour de faibles valeurs de $r$, voir les Corollaires \[corcalcproj\], \[corcalc2spher\] et \[corcalc3spher\]. Ces dernières sont obtenues en brisant la variété en deux morceaux, ce qui brise les courbes rationnelles réelles elle-même en deux morceaux et permet d’exprimer $\chi_r^d$ en fonction de deux ingrédients, l’un calculé à l’aide de courbes réelles dans $T^*L$ qui n’est autre qu’un invariant réel relatif à un diviseur réel sans lieu réel -conique imaginaire pure ou section hyperplane réelle disjointe de l’ellipsoïde- et l’autre à l’aide de paires de courbes complexes conjuguées dans $X \setminus L$. Les calculs d’invariants relatifs réalisés dans [@Vak] (voir aussi [@IoPar] et [@Katz]) permettent de maîtriser ce deuxième ingrédient. Or, plus $r$ est petit, plus le premier ingrédient est simple de sorte que pour les petites valeurs de $r$, on déduit de [@Vak] des formules de récurrence générales, voir les Théorèmes \[theocalcproj\], \[theocal2spher\] et \[theocal3spher\].
Ces résultats d’optimalité, de congruences et de calculs ont été annoncés dans la note [@Wels4] dans le cas du plan projectif ou de la quadrique de dimension deux ; ils ont été présentés la première fois en décembre $2005$ lors de l’atelier organisé en l’honneur de Dusa McDuff, à Banff au Canada.\
[**Remerciements :**]{}
Je remercie l’Agence nationale de la recherche pour son soutien ainsi que Y. Eliashberg pour ses encouragements à découper les variétés symplectiques en morceaux.
Optimalité
==========
Optimalité des bornes inférieures {#subsectopt}
---------------------------------
### Énoncés des résultats
Nous énonçons dans ce premier paragraphe les situations dans lesquelles nous sommes en mesure de montrer l’optimalité des bornes inférieures (\[bornes\]) en dimension quatre. Le paragraphe \[subsectmin\] tiendra lieu de digression en dimension supérieure.
\[theoopt1\] Soit $(X, \omega , c_X)$ une variété symplectique réelle fermée de dimension quatre et soit $d \in H_2 (X ; \Z)$ une classe d’homologie satisfaisant $(c_X)_* d = -d$. Supposons que le lieu réel de cette variété possède une sphère ou un plan projectif réel $L$. Dans ce dernier cas, supposons que $(X, \omega , c_X)$ est elle-même symplectomorphe au plan projectif complexe éclaté en six points complexes conjugués au maximum. Les bornes inférieures (\[bornes\]) sont sous ces hypothèses optimales dès que $0 \leq r \leq 1$, atteintes par les structures presque-complexes générales ayant un long cou au voisinage de $L$. Le signe de l’invariant $\chi_r^d (L)$ est en outre dans ce cas déterminé par l’inégalité $(-1)^{\frac{1}{2}(d^2 - c_1(X)d + 2)} \chi^d_r (L) \geq 0$.
La dernière partie du Théorème \[theoopt1\] signifie que le signe du coefficient de plus bas degré du polynôme $\chi^d (T)$ introduit dans [@Wels1] s’interprète comme la parité du genre lisse de la classe $d$. Le fait que ce signe puisse être négatif en degrés congrus à trois ou quatre modulo quatre dans le plan projectif complexe met en défaut la Conjecture $6$ de [@IKS1]. Nous montrerons en effet au §\[sectcalculs\] que cet invariant ne s’annule pas en degrés supérieurs à cinq, voir le Théorème \[theocalcproj\]
Soit $d$ une classe d’homologie de dimension deux du plan projectif complexe ou de la quadrique ellipsoïde et $0 \leq r \leq 1$. Les bornes inférieures (\[bornes\]) sont atteintes pour la structure complexe standard lorsque les points complexes conjugués sont choisis très proches d’une conique imaginaire pure dans le premier cas et d’une section hyperplane réelle disjointe de $L$ dans le second.
[**Démonstration :**]{}
Dans ces deux cas, la structure complexe standard de la variété possède un cou infiniment long au voisinage de $L$. Il s’agit d’un voisinage fibré en disques de la conique imaginaire pure ou de la section hyperplane réelle privé de la conique ou de la section elle même. Comme par ailleurs le plan projectif et la quadrique sont des surfaces convexes, l’hypothèse de généricité de la structure presque-complexe du Théorème \[theoopt1\] est satisfaite (voir les Théorèmes \[theocalcproj\] et \[theocal2spher\] pour un résultat plus général). Le Théorème \[theoopt1\] s’applique donc et fournit le résultat. $\square$
\[theoopt2\] Soit $(X, \omega , c_X)$ une variété symplectique réelle fermée de dimension quatre dont le lieu réel possède un tore $L$ et soit $d \in H_2 (X ; \Z)$ une classe d’homologie satisfaisant $(c_X)_* d = -d$. Les bornes inférieures (\[bornes\]) sont optimales lorsque $r=1$, atteintes par les structures presque-complexes générales ayant un long cou au voisinage de $L$. Lorsque le lieu réel est connexe -réduit au tore $L$-, l’invariant $\chi_1^d (L)$ est en outre positif. Dans le cas général, le signe de l’invariant $\chi_1^d (L)$ est déterminé par l’inégalité $(-1)^{\frac{1}{2}(d^2 - c_1(X)d + 2)} \chi^d_1 (L) \geq 0$ lorsque le lieu réel des courbes rationnelles ne s’annule pas dans $H_1 (L ; \Z /2\Z)$, tandis qu’il est déterminé par l’inégalité $(-1)^{\frac{1}{2}(d^2 - c_1(X)d + 2)} \chi^d_1 (L) \leq 0$ lorsque ce dernier s’annule.
Dans le cas particulier de la quadrique hyperboloïde, la positivité de $\chi_1^d (L)$ a déjà été observée dans [@IKS1] par d’autres méthodes.
### Stratégie générale {#subsubsectstrat}
On allonge le cou d’une structure presque-complexe générique jusqu’à briser la variété $(X, \omega , c_X)$ en deux, le fibré cotangent à $L$ d’une part et le complémentaire $X \setminus L$ de l’autre. Chacune de ces deux parties se retrouve munie d’une structure presque-complexe, notée $J_L$ et $J_X$ respectivement, qui rendent respectivement $c_L$ et $c_X$ antiholomorphes, où $c_L : (q,p) \in T^* L \mapsto (q, -p) \in T^* L$. De plus, en dehors d’un compact, ces structures sont cylindriques sur une structure $CR$ de la variété de contact $(S^* L , \lambda)$. Nous avons ici noté $S^* L$ le fibré unitaire cotangent de $L$ pour une métrique à courbure constante, de sorte que les orbites périodiques du flot du champ de vecteurs de Reeb $R_\lambda$ associé à la forme de Liouville $\lambda$, c’est-à-dire du flot géodésique, viennent en familles. Rappelons qu’une fois identifié le complémentaire d’un compact de $T^* L$ ou $X \setminus L$ avec une partie de la symplectisation $S^* L \times \R$ de $S^* L$, la structure presque complexe $J_L$ ou $J_X$ est définie en dehors de ce compact par la structure $CR$ et la relation $J \partial / \partial t = R_\lambda$. Nous réalisons cette scission de sorte que les $r_X$ paires de points complexes conjuguées que l’on s’est données se retrouvent dans $X \setminus L$. Le théorème de compacité de théorie symplectique des champs [@BEHWZ] permet de comprendre le devenir des courbes rationnelles réelles homologues à $d$ et qui passent par $\underline{x}$. Ces courbes se brisent en des courbes à deux étages, $J_L$-holomorphes (resp. $J_X$-holomorphes) pour celles habitant l’étage $T^* L$ (resp. $X \setminus L$), et asymptotes à des orbites périodiques de $R_\lambda$, la période pouvant être multiple de la période fondamentale. La réunion de ce nombre fini de composantes est invariante par l’involution $c_X$, de sorte que ces composantes sont organisées en paires de composantes complexes conjuguées de $T^* L$ ou $X \setminus L$ et d’une composante de $T^* L$ laissée invariante par $c_L$. Chaque courbe à deux étages limite $C$ peut donc être codée par un arbre $A_C$ ayant une racine $s_0$ et ses arêtes équipées de multiplicités entières strictement positives. Chaque sommet de cet arbre représente une composante du quotient de la courbe limite par l’action de $c_X$, composante qui se trouve dans l’étage $T^* L$ si ce sommet est à distance paire de $s_0$ et dans l’étage $X \setminus L$ sinon. Le sommet $s_0$ quant à lui représente l’unique composante laissée invariante par l’involution $c_L$ de $T^* L$. Le quotient de cette composante est une hémisphère pointée à bord dans $L$. Chaque arête adjacente à un sommet donné représente une asymptote de la composante correspondante à ce sommet et la multiplicité de l’arête n’est autre que la multiplicité de l’orbite de Reeb limite correspondante. Par exemple, l’arbre représenté par la figure \[figarbre\] représente une courbe rationnelle réelle à deux étages et neuf composantes. La composante racine est une sphère réelle dans $T^* L$ ayant deux paires de pointes complexes conjuguées asymptotes à deux paires d’orbites de Reeb, l’une de multiplicité deux, l’autre de multiplicité trois. L’étage $X \setminus L$ contient une paire de plans $J_X$-holomorphes complexes conjugués asymptotes à la paire d’orbites de Reeb de multiplicité deux précédente, cette paire est codée par la feuille de l’arbre adjacente à l’arête de multiplicité deux. Cet étage $X \setminus L$ contient également une paire de sphères $J_X$-holomorphes complexes conjuguées ayant trois pointes dont deux sont asymptotes à des orbites de Reeb simples et la troisième asymptote à la paire d’orbites de Reeb de multiplicité trois définie plus haut, cette paire de sphères est codée par le sommet trivalent. Enfin, l’étage $T^* L$ contient également deux paires de plans $J_L$-holomorphes complexes conjugués asymptotes aux paires d’orbites de Reeb simples précédentes, ces plans sont codés par les deux feuilles restantes de l’arbre.
\[figarbre\]
L’arbre $A_C$ vient de plus avec une fonction qui associe à chaque sommet à distance impaire de $s_0$ les classes d’homologies relatives de la paire de courbes correspondantes ainsi que les paires de points complexes conjugués de $\underline{x} $ que ces courbes contiennent.\
### Démonstration des Théorèmes \[theoopt1\] et \[theoopt2\] {#subsectoptdem}
Cette stratégie générale étant posée, remarquons que dans chacun des cas qui concernent le Théorème \[theoopt1\], $S^* L$ est un fibré en cercles. Par suite, le fibré normal de chaque courbe $C_s$ associé à un sommet $s$ de l’arbre $A_C$ est canoniquement trivialisé le long des orbites de Reeb asymptotes par le flot de Reeb. Notons $\mu_s$ le double de l’obstruction à étendre cette trivialisation sur $C_s$ tout entier. La dimension de l’espace des modules dans lequel habite $C_s$ s’exprime en fonction de cet indice de Maslov $\mu_s $, voir la Proposition \[propmaslov\] de notre formulaire donné au paragraphe \[subsectformulaire\]. Cette dimension vaut, pour peu que $C_s$ soit une courbe simple, c’est-à-dire ne soit pas un revêtement ramifié non-trivial d’une autre, $\mu_s + 2$ lorsque $s \neq s_0$ et $\frac{1}{2} \mu_s + 1$ lorsque $s = s_0$ puisque la courbe $C_s$ est alors contrainte d’être préservée par l’involution $c_X$ ce qui a pour effet de diviser la dimension par deux. Notons pour chaque sommet $s$ de l’arbre, sa valence par $v_s$ et la somme des multiplicités des arêtes adjacentes par $k_s$. L’indice de Maslov $\mu_s $ s’exprime pour les sommets à distance paire de $s_0$ en fonction de $v_s$ et $k_s$, voir la Proposition \[propcotangent\] de notre formulaire donné au paragraphe \[subsectformulaire\].
Supposons pour commencer que $L$ est une sphère et notons $S_1$ (resp. $S_2$) l’ensemble des sommets à distance impaire (resp. paire) de $s_0$. Lorsque $s \in S_2 \setminus \{ s_0 \}$, $\mu_s = 2k_s + 2v_s - 4$ tandis que l’indice de Maslov de l’hémisphère associée à $s_0$ vaut $\mu_{s_0} = 2k_{s_0} + 2v_{s_0} - 2$. Par suite, $$\sum_{s \in S_2} \mu_s = 2k + 2v - 4 \# S_2 + 2,$$ où $v$ désigne le nombre total d’arêtes de l’arbre et $k$ la somme de leurs multiplicités. Si l’on suppose que toutes les courbes de l’étage $X \setminus L$ sont simples, la généricité de $J_X$ impose la positivité de toutes les dimensions des espaces de modules intervenant, soit $\mu_s + 2 \geq 0$ pour tout $s \in S_1$. Lorsque la courbe $C_s$ contient $f_s$ points de notre configuration, cette condition d’incidence impose l’inégalité plus fine $\mu_s + 2 \geq 2f_s$. On déduit au total la minoration $$\sum_{s \in S_1} \mu_s \geq - 2 \# S_1 + 2r_X.$$ Le nombre d’arêtes d’un arbre diffère du nombre de sommets par un, soit $v = \# S_1 + \# S_2 - 1$, de sorte que l’indice de Maslov total de la courbe $C$ satisfait $\mu \geq 2k - 2 \# S_2 + 2r_X \geq 2r_X$. Or cet indice de Maslov total est par ailleurs majoré par $c_1 (X) d -2$, le degré du fibré normal d’une courbe rationnelle immergée homologue à $d$. Par hypothèse, ce degré vaut ici $2r_X$ puisque l’orientabilité de $L$ impose l’imparité de $r$. Les minorations précédentes sont par conséquent des égalités, de sorte que $k = \# S_2$. En particulier, toutes les orbites de Reeb intervenant sont simplement revêtues et tous les sommets de $S_2$ sont des feuilles. La courbe réelle codée par $s_0$ n’est autre qu’un cylindre réel sur une orbite de Reeb simple. Un tel cylindre est nécessairement plongé, voir le Lemme \[lemmepointsdoubles\] de notre formulaire. Le résultat en découle ; peu avant la brisure de la variété, toutes les courbes rationnelles réelles ont leurs parties réelles plongées, de sorte que les points doubles réels éventuels de ces courbes sont tous isolés. Ce nombre de points doubles est de même parité que le genre lisse de la courbe.
Il s’agit à présent d’aboutir à la même conclusion sans supposer que les courbes $C_s$ soient simples. L’indice de Maslov $\mu^l$ d’un revêtement de degré $l$ d’une courbe simple d’indice $\mu$ s’écrit $\mu^l = l \mu + 2R$ où $R$ est l’indice de ramification. Cet indice de Maslov peut être strictement plus petit que $\mu$ uniquement lorsque $\mu$ est négatif, donc égal à $-2$ et encore faut-il que la courbe revêtue ne soit pas plane. Cela ne concerne donc ni les courbes de l’étage $T^* L$, ni les courbes de $X \setminus L$ soumises à des conditions d’incidence. Notons $s_1 , \dots , s_j$ les sommets de $A_C$ correspondant à ces dernières et calculons la contribution à l’indice de Maslov total de chaque composante connexe de l’arbre privé des sommets $s_1 , \dots , s_j$. Pour ce faire, notons $S_1'$ (resp. $S_2'$) l’ensemble des sommets à distance impaire (resp. paire) de $s_0$ d’une telle composante connexe de $A_C \setminus \{ s_1 , \dots , s_j \}$. Comme précédemment, $\sum_{s \in S_2'} \mu_s = 2k + 2v - 4 \# S_2 + 2\delta$, où $\delta$ vaut un si la composante en question contient $s_0$ et zéro sinon, tandis que $$\begin{aligned}
\sum_{s \in S_1'} \mu_s & = & \sum_{s \in S_1'} (l_s \tilde{\mu}_s + 2R_s) \text{ o\`u } l_s \text{ d\'esigne le degr\'e du rev\^etement, } R_s \text{ l'indice de ramification} \nonumber \\
&& \text{et } \tilde{\mu}_s \text{ l'indice de Maslov de la courbe simple sous-jacente} \nonumber \\
& \geq & -2 \sum_{s \in S_1'} l_s + 2 \sum_{s \in S_1'} (l_s \tilde{\chi}_s - \chi_s) \text{ o\`u } \chi \text{ d\'esigne la caract\'eristique d'Euler,} \nonumber \\
& \geq & 2 \sum_{s \in S_1'} (l_s - l_s \tilde{v}_s + v_s) - 4 \# S_1' \text{ o\`u } v_s \text{ d\'esigne le nombre de pointes} \label{equ3} \\
& \geq & 2 \sum_{s \in S_1'} (l_s - k_s + v_s) - 4 \# S_1'. \label{equ1}\end{aligned}$$ Après sommation, on déduit $\sum_{s \in S_1' \cup S_2'} \mu_s \geq 2 \sum_{s \in S_1'} l_s + 2k' + 2v' - 4+ 2\delta$, où $v'$ et $k'$ désignent respectivement le nombre d’arêtes attachées à $s_1 , \dots , s_j$ et leur multiplicité totale. Les minorations $\sum_{s \in S_1' \cup S_2'} \mu_s \geq 2k'$ si $\delta$ vaut un et $\sum_{s \in S_1' \cup S_2'} \mu_s \geq 2(k' - 1)$ sinon en résultent. La contribution totale à l’indice de Maslov des sommets autres que $s_1 , \dots , s_j$ se trouve donc minorée par $2j$. La contribution des sommets $s_1 , \dots , s_j$ est quant à elle minorée par $2r_X - 2j$, de sorte qu’on aboutit à nouveau à la minoration $\mu \geq 2r_X$. On conclut donc comme précédemment.
Supposons à présent que $L$ est un plan projectif réel. D’après la Proposition \[propcotangent\] de notre formulaire donné au paragraphe \[subsectformulaire\], l’indice de Maslov d’un sommet $s \in S_2 \setminus \{ s_0 \}$ vaut $\mu_s = k_s + 2v_s - 4$ tandis que l’indice de Maslov de l’hémisphère associée à $s_0$ vaut $\mu_{s_0} = k_{s_0} + 2v_{s_0} - 2$ de sorte que $\sum_{s \in S_2} \mu_s = k + 2v - 4 \# S_2 + 2$. Les hypothèses faites sur la variété garantissent l’absence de courbes simples d’indices de Maslov strictement négatifs autres que des plans dans l’étage $X \setminus L$. En effet, cet étage est isomorphe au fibré en droites complexes de degré quatre sur la conique imaginaire pure éclaté en six points complexes conjugués au maximum. La classe d’homologie relative d’une courbe dans cet espace s’écrit $ne + kf - \sum_i \alpha_i E_i$, où $e$ est la section nulle du fibré, $f$ une fibre et $E_i$ les éventuels diviseurs exceptionnels. L’irréductibilité de la courbe $C_s$ impose les inégalités $\alpha_i \leq n$ dès que $n \geq 1$, ce que l’on obtient comme conséquence de la positivité d’intersection avec les courbes exceptionnelles $J_X$-holomorphes $E_i$ et $f - E_i$. L’indice de Maslov d’une telle courbe vaut $2(6n + 2k - \sum_i \alpha_i -2)$, il est positif dès que $k,n$ sont non nuls. Pour chaque sommet $s \in S_1$ l’inégalité $\mu_s + 2 \geq 0$ s’en déduit. Lorsque la courbe $C_s$ contient $f_s$ points de notre configuration, cette condition d’incidence impose l’inégalité plus fine $\mu_s + 2 \geq 2f_s$. De là la minoration $\sum_{s \in S_1} \mu_s \geq - 2 \# S_1 + 2r_X$ et finalement après sommation l’estimation de l’indice de Maslov total $\mu \geq k - 2 \# S_2 + 2r_X $. Il reste à remarquer que pour chaque $s \in S_2 \setminus \{ s_0 \}$, l’entier $k_s$ doit être pair puisque le noyau du morphisme $H_1 (S^* L ; \Z) \to H_1 ( L ; \Z)$ est engendré par une orbite de Reeb double. L’inégalité précédente se réécrit donc à présent $\mu \geq 2r_X $ si les parties réelles des courbes rationnelles que l’on considère sont non nulles dans $H_1 ( L ; \Z)$ et $\mu \geq 2r_X - 1$ sinon. Or cet indice de Maslov est par ailleurs majoré par le degré $c_1 (X) d -2$ du fibré normal à une courbe rationnelle immergée homologue à $d$, degré qui par hypothèse vaut ici $2r_X + r - 1$. Ainsi, toutes les minorations précédentes sont des égalités, de sorte que les sommets à distances paires de $s_0$ sont soit des cylindres sur des orbites simples, soit des plans sur des orbites de Reeb doubles. Le sommet $s_0$ quant à lui code un cylindre réel sur une orbite simple lorsque $r$ est nul, et soit un cylindre réel sur une orbite double, soit une sphère réelle ayant deux paires de pointes complexes conjuguées asymptotes à des orbites simples lorsque $r$ vaut un. Dans tous ces cas, une telle courbe est plongée, de par le Lemme \[lemmepointsdoubles\]. On conclut comme précédemment, ce qui achève la démonstration du Théorème \[theoopt1\].
Supposons enfin que $L$ soit un tore et munissons-le d’une métrique plate de sorte que son fibré unitaire cotangent $(S^* L , \lambda)$ soit un tore standard de dimension trois. Le flot de Reeb fournit à nouveau une trivialisation canonique du fibré normal aux courbes $C_s$ le long de leurs orbites de Reeb asymptotes. L’obstruction $\mu_s$ à étendre cette trivialisation sur $C_s$ tout entier vaut cette fois-ci $2v_s -4$ si $s \neq s_0$ est à distance paire de $s_0$ et $2v_s -2$ si $s = s_0$, voir la Proposition \[propcotangent\] de notre formulaire donné au paragraphe \[subsectformulaire\]. Si $s$ est au contraire à distance impaire de $s_0$, la dimension de l’espace des modules dans lequel habite $C_s$ s’écrit $\mu_s + 2 - v_s$, d’après la Proposition \[propmaslov\]. Contrairement aux cas précédents, la passage à un revêtement ramifié ne peut faire qu’augmenter cette dimension. On déduit donc de la parité de $\mu_s$ l’inégalité $\mu_s \geq 0$, inégalité stricte lorsque $v_s > 2$. Si la courbe est contrainte de passer par $f_s$ points de notre configuration, cette inégalité se trouve renforcée en $\mu_s \geq 2f_s$. En sommant les contributions de tous les sommets de l’arbre, on s’aperçoit donc que l’indice de Maslov total $\mu$ de la courbe est minoré par $2r_X$. Comme cet indice de Maslov est par ailleurs majoré par $c_1 (X) d -2$ et comme par hypothèse $r=1$, la minoration précédente est une égalité. Il en est par suite de même pour toutes les minorations faites, de sorte que toutes les composantes de l’étage $T^* L$ sont des cylindres. Les cylindres autres que celui associé à $s_0$ sont disjoints de $L$ pour un choix générique de $J_X$. Le cylindre réel associé à $s_0$ est un revêtement d’un cylindre plongé sur une orbite de Reeb simple. En effet, quitte à passer à un revêtement du fibré cotangent à $L$, on peut supposer le cylindre asymptote à une orbite simple. Un tel cylindre est, une fois l’orbite fixée, unique et plongé, ce qui est immédiat pour la structure complexe standard de $T^* L$ et est une propriété invariante par déformation de la structure presque-complexe. On en déduit que peu avant la brisure de la variété, toutes les courbes rationnelles réelles que l’on considère possédaient un nombre de points doubles réels non-isolés pair si le degré du revêtement est impair et impair sinon. En effet, la perturbation du revêtement $k$-uple d’une courbe simple du tore produit $k - 1$ points d’auto-intersection modulo deux. Le nombre de points doubles réels isolés de ces courbes rationnelles se trouve donc être de la même parité que le genre lisse de la courbe lorsque le lieu réel des courbes rationnelles est non-nul dans $H_1 (L ; \Z / 2\Z)$ et de la parité opposée lorsque celui-ci s’annule. Le Théorème \[theoopt2\] est démontré. $\square$
Minimisation du nombre de membranes $J$-holomorphes {#subsectmin}
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Soit $C$ une membrane $J$-holomorphe à bord dans une sous-variété lagrangienne $L$ d’une variété symplectique fermée $(X, \omega)$. Notons $\chi$ la caractéristique d’Euler de cette membrane, $d \in H_2 (X , L ; \Z)$ sa classe d’homologie relative et $\mu_{TX} \in H^2 (X , L ; \Z)$ la classe de Maslov de la paire $(X , L)$. La dimension attendue de l’espace des déformations de $C$ s’écrit $\langle \mu_{TX} , d \rangle + (n-3)\chi$. Cette dimension chute lorsque l’on impose à $C$ des contraintes supplémentaires. Si l’on impose par exemple à cette membrane de rencontrer $p$ cycles de codimensions $2 + q_1, \dots , 2 + q_p$, cette dimension attendue chute de la somme $q = q_1+ \dots + q_p$. Deux problèmes généraux sous-tendent nos résultats. Il s’agit d’une part de compter le nombre de membranes $J$-holomorphes homologues à $d$ soumises à de telles conditions d’incidence de sorte que ce comptage ne dépende pas de $J$ et ne dépende des conditions d’incidence qu’à homologie près. Il s’agit d’autre part de minimiser ce nombre de membranes. Si nous ne pouvons répondre au premier problème dans ce degré de généralité, il nous est par contre parfois possible de répondre au second sans même supposer l’égalité $q = \langle \mu_{TX} , d \rangle + (n-3)\chi$, lorsque le minimum en question est nul. Le présent paragraphe est consacré aux résultats que l’on a pu obtenir dans cette direction. Ici encore le minimum est atteint en allongeant le cou d’une structure presque complexe générale.
### En dimension supérieure
\[theomin3\] Soit $L$ une sphère lagrangienne dans une variété symplectique fermée $(X, \omega)$ satisfaisant $c_1 (X) = \lambda \omega$, $\lambda \leq 0$ et soit $E > 0$. Supposons la dimension de $X$ supérieure à cinq. Pour toute structure presque-complexe $J$ générale ayant un cou suffisamment long au voisinage de $L$, cette variété ne possède ni membrane $J$-holomorphe reposant sur $L$ ni courbe $J$-holomorphe rencontrant $L$ qui soit d’énergie inférieure à $E$. Ce résultat reste valable en dimension quatre pour les courbes ou membranes de genre nul.
Rappelons que l’énergie d’une courbe $C$ est par définition l’intégrale de la forme $\omega$ sur cette courbe. Les variétés projectives à fibré canonique nul ou ample, par exemple les intersections complètes de multidegrés $(d_1, \dots , d_k)$ de l’espace projectif de dimension $N$ dès lors que $\sum_{i=1}^k d_i \geq N+1$, satisfont les hypothèses du Théorème \[theomin3\]. Remarquons qu’une modification de ce dernier s’applique également aux variétés dont le fibré canonique est le produit d’un fibré ample et d’un fibré porté par un diviseur effectif disjoint de $L$. Le Théorème \[theomin3\] permet de définir l’homologie de Floer de deux sphères lagrangiennes proches dans les variétés symplectiques dont la première classe de Chern s’annule, j’espère développer ce résultat prochainement.
\[theomin2\] Soit $L$ une sphère lagrangienne dans une variété symplectique fermée semipositive $(X, \omega)$ de dimension $2n \geq 6$ et soit $d \in H_2 (X , L ; \Z)$. Écrivons $\langle \mu_{TX} , d \rangle + (n-3)\chi = q + r$ avec $q \in \Z$, $0 \leq r < 2 + (n-3)\chi$ et $\chi \leq 2$. Lorsque $q \geq 0$, choisissons $p$ cycles de $X \setminus L$ de codimensions $2 + q_1, \dots , 2 + q_p$ de sorte que $q = q_1+ \dots + q_p$. Dès que la structure presque complexe générale $J$ possède un cou suffisamment long au voisinage de $L$, cette variété ne contient aucune membrane $J$-holomorphe homologue à $d$, de caractéristique d’Euler $\chi$ qui rencontre ces $p$ cycles et repose sur $L$. Ce résultat reste valable pour des membranes de genre nul lorsque $n = 2$.
[**Exemple : la quadrique ellipsoïde.**]{}
Soit $X$ la quadrique ellipsoïde de dimension complexe $n \geq 3$ et $H$ une section hyperplane disjointe de $L$. Le groupe $H_2 (X , L ; \Z)$ est monogène, engendré par la classe $d_0$ satisfaisant $\langle H , d_0 \rangle = +1$. La première classe de Chern de $X$ vaut $n H$, d’où l’on déduit le calcul $\langle \mu_{TX} , l d_0 \rangle
= 2ln$ quel que soit l’entier $l$. Écrivons $l = (n-1)a + b$, le Théorème \[theomin2\] s’applique par exemple lorsque $n+1 \leq 2b < 2n$, les membranes sont des disques et lorsque toutes les conditions d’incidence sont ponctuelles.
\[theoopt4\] Soit $(X, c_X)$ une variété algébrique réelle convexe de dimension trois dont le lieu réel possède une sphère $L$. Supposons l’existence d’une classe d’homologie $d \in H_2 (X ; \Z)$ satisfaisant $c_1 (X) d = 2 \mod (4)$. L’invariant $\chi_1^d (L)$ est alors négatif et les bornes inférieures (\[bornes\]) sont optimales. Dans le cas de l’ellipsoïde, ces bornes sont atteintes pour la structure complexe algébrique lorsque les conditions d’incidence non réelles sont choisies suffisamment proches d’une section hyperplane réelle disjointe de $L$.
Il se peut que l’ellipsoïde de dimension trois soit en fait le seul exemple de variété satisfaisant les hypothèses du Théorème \[theoopt4\]. L’invariant $\chi_1^d (L)$ qui apparaît dans ce théorème a été construit dans [@Wels2]. Remarquons qu’en reprenant les notations du Théorème \[theomin2\], ce Théorème \[theoopt4\] traite du cas $r=n-1$ et montre ainsi en un sens l’optimalité des hypothèses faites dans ce Théorème \[theomin2\]. Nous montrerons en effet dans la troisième partie de cet article la non-trivialité de l’invariant $\chi_1^d (L)$ pour l’ellipsoïde de dimension trois et calculerons ce dernier, voir le §\[subsec3spher\].\
[**Démonstration des Théorèmes \[theomin3\], \[theomin2\] et \[theoopt4\] :**]{}
Nous suivons la stratégie générale énoncée au paragraphe \[subsubsectstrat\] en équipant la sphère lagrangienne d’une métrique à courbure constante. Les éventuelles membranes qui survivraient à l’allongement du cou de $J$ jusqu’à la brisure de la variété seraient cette fois-ci codés par des graphes $A_C$ ayant $b+1$ sommets marqués $s_0, \dots , s_b$ correspondant aux composantes ayant un bord dans $L$. Les sommets à distances paires de $s_0, \dots , s_b$ codent à nouveau les composantes de l’étage $T^* L$ et les sommets à distances impaires les composantes de l’étage $X \setminus L$. Le flot de Reeb trivialise le fibré normal de chaque composante $C_s$ associée au sommet $s$ d’un graphe $A_C$ le long de ses orbites de Reeb limites. Notons $\mu_s$ le double de l’obstruction à étendre cette trivialisation sur $C_s$ toute entière. Notons également, pour chaque sommet $s$ du graphe, sa valence par $v_s$, la somme des multiplicités des arêtes adjacentes par $k_s$ et la caractéristique d’Euler de la courbe qu’il code par $\chi_s$. L’indice de Maslov des composantes codées par les sommets à distances paires de $s_0, \dots , s_b$, c’est-à-dire des courbes $C_s$ de l’étage $T^* L$, s’exprime d’après la Proposition \[propcotangent\] de notre formulaire par la relation $\mu_s = 2(n-1)k_s - 2\chi_s$. Pour calculer la contribution totale des sommets à distances impaires de $s_0$, il faut tenir compte du fait que certaines des composantes associées peuvent revêtir des courbes simples. Notons pour chacun de ces sommets $l_s$ le degré du revêtement, $\tilde{\mu}_s$ l’indice de Maslov de la courbe simple sous-jacente et $\tilde{\chi}_s$ sa caractéristique d’Euler. D’après la Proposition \[propmaslov\] de notre formulaire, la dimension de l’espace des modules dans lequel habite cette courbe simple vaut $\tilde{\mu}_s + (n-1)(\tilde{\chi}_s + \tilde{v}_s)$. La généricité de la structure presque complexe assure donc la minoration $\tilde{\mu}_s \geq - (n-1)(\tilde{\chi}_s + \tilde{v}_s)$. Les courbes simples sous-jacente étant soumises à nos $p$ conditions d’incidence, cette dernière minoration peut après sommation être améliorée de $q$. Par conséquent, $$\begin{aligned}
\sum_{s \in S_1} \mu_s & = & \sum_{s \in S_1} \big( l_s (\tilde{\mu}_s + 2 \tilde{\chi}_s) - 2 \chi_s \big) \\
& \geq & q - (n-3)\sum_{s \in S_1} l_s (\tilde{\chi}_s + \tilde{v}_s) - 2\sum_{s \in S_1} (k_s + \chi_s) \; \text{Ê puisque } l_s \tilde{v}_s \leq k_s. \\\end{aligned}$$ Nous en déduisons $$2 \chi + \sum_{s \in S_1 \cup S_2} \mu_s \geq q + 2(n-2)k -(n-3)\sum_{s \in S_1} l_s (\tilde{\chi}_s + \tilde{v}_s),$$ où $k = \sum_{s \in S_1} k_s$. Lorsque $n \geq 3$, utilisant les majorations $\tilde{\chi}_s + \tilde{v}_s \leq 2$ et $l_s \leq k_s$, nous aboutissons à $\sum_{s \in S_1 \cup S_2} \mu_s + 2 \chi \geq q + 2$. Lorsque $n=2$, nos hypothèses imposent $\tilde{\chi}_s + \tilde{v}_s = 2$ de sorte qu’à nouveau $\sum_{s \in S_1 \cup S_2} \mu_s + 2 \chi \geq q + 2$. Le Théorème \[theomin2\] suppose la variété semipositive, les éventuelles composantes compactes de l’étage $X \setminus L$ ont donc un indice de Maslov positif. Par conséquent, l’indice de Maslov total satisfait la majoration $2 \chi + \sum_{s \in S_1 \cup S_2} \mu_s \leq \langle \mu_{TX} , d \rangle \leq q + r - (n - 3)\chi < q + 2$. Ces minoration et majoration étant incompatibles, aucune membrane ne peut survivre jusqu’à la brisure de la variété. Le Théorème \[theomin2\] est démontré. Dans le cas du Théorème \[theomin3\], $q=0$ et nous déduisons par recollement des composantes codées par le graphe $A_C$ une membrane symplectique $C$ de $(X,L)$ d’indice de Maslov $ \langle \mu_{TX} , [C] \rangle \geq 2$. Or par hypothèse, $\langle \mu_{TX} , [C] \rangle = 2 \langle c_1(X) , c \rangle = 2 \lambda \langle \omega , c \rangle \leq 0$, où $c \in H_2 (X ; \Z)$ relève $[C] \in H_2 (X , L ; \Z)$. Cette impossibilité démontre le Théorème \[theomin3\].
Le Théorème \[theoopt4\] correspond au cas où $r = n-1$. Dans, ce cas, les minoration et majoration précédentes coïncident, de sorte que toutes les inégalités sont des égalités. En particulier, $k=1$ de sorte que chaque graphe $A_C$ se trouve réduit à deux sommets reliés par une arête simple. La courbe réelle codée par $s_0$ est un cylindre sur une orbite de Reeb simple. L’état spinoriel de ces courbes se calcule comme suit. En perturbant le point réel dans toutes les directions dans $L$, on s’aperçoit que toutes ces courbes ont le même état spinoriel qu’une conique obtenue comme section plane réelle de la quadrique ellipsoïde réelle. Ce dernier vaut $-1$ comme on le vérifie en déformant l’équateur vers un parallèle proche d’un pôle de $L$. $\square$
### En dimension quatre
Nous noterons ${\cal M}_{g,b}$ l’espace des modules des structures complexes de la surface compacte connexe orienté de genre $g$ ayant $b$ composantes de bord.
\[propminsphere\] Soit $L$ une sphère lagrangienne dans une variété symplectique fermée de dimension quatre $(X, \omega)$. On suppose que cette dernière ne possède pas de sphère symplectique $S$ satisfaisant $\langle c_1 (X) , [S] \rangle > 0$. Soit $(d, g, b) \in H_2 (X , L ; \Z) \times \N \times \N^*$ et $K$ un compact de ${\cal M}_{g,b}$. Alors, pour toute structure presque-complexe générale ayant un cou suffisamment long au voisinage de $L$, la variété ne possède pas de membrane $J$-holomorphe homologue à $d$ à bord dans $L$ et conforme à un élément de $K$.
[**Démonstration de la Proposition \[propminsphere\] :**]{}
On poursuit la stratégie générale décrite au paragraphe \[subsubsectstrat\] précédent en équipant $L$ d’une métrique à courbure constante et en allongeant le cou d’une structure presque complexe générique jusqu’à briser la variété en deux morceaux. D’après le théorème de compacité de théorie symplectique des champs [@BEHWZ], les membranes que l’on considère se brisent en courbes à deux étages qui sont cette fois-ci codées par des graphes $A_C$ ayant $b$ sommets marqués $s_1, \dots , s_b$ correspondant aux $b$ composantes de bord. Les sommets à distances paires de $s_1, \dots , s_b$ codent à nouveau les composantes de l’étage $T^* L$ et les sommets à distances impaires les composantes de l’étage $X \setminus L$. Par hypothèse, l’étage $X \setminus L$ ne possède pas de courbe $J$-holomorphe rationnelle asymptote à des orbites de Reeb du fibré unitaire cotangent $S^* L$. En effet, une telle courbe $J$-holomorphe rationnelle simple $C$ aurait d’après la Proposition \[propmaslov\] un indice de Maslov $\mu \geq -2$. Notons $v \geq 1$ le nombre de pointes asymptotes à des orbites de Reeb de $S^* L$ et $\chi (C)$ la caractéristique d’Euler de $C$. En recollant à $C$ en chacune de ses pointes un plan $J$-holomorphe de $T^* L$, on obtient une sphère symplectique $S$ de $X$. Le fibré tangent à $X$ est trivialisé le long des pointes de $C$ par le flot de Reeb. D’après ce qui précède, le double de l’obstruction à étendre cette trivialisation le long de $C$ vaut $\mu + 2 \chi (C)$ alors qu’elle vaut deux le long de chaque plan de $T^* L$ d’après la Proposition \[propcotangent\]. Finalement, l’indice de Maslov de $S$ vaudrait $\mu + 4 \geq 2$, ce qui est exclu par les hypothèses. Remarquons à présent que chaque composante des courbes à deux étages est asymptote à une réunion de cylindres $J$-holomorphes sur les orbites de Reeb limites. Ces cylindres ont un module infini. On en déduit que peu avant la brisure de la variété $X$, lorsque $J$ possède un cou extrêmement long, les membranes $J$-holomorphes possèdent également des anneaux de grands modules dont les âmes sont homotopes aux orbites de Reeb codées par les arêtes de l’arbre $A_C$. Au moins un de ces anneaux ne borde pas de disque, lequel proviendrait nécessairement d’un plan de $T^* L$, puisque les membranes ont un bord dans $L$. Par suite, lorsque le cou de la structure presque-complexe $J$ est suffisamment allongé, les membranes $J$-holomorphes qui survivent à cet allongement ont une structure conforme n’appartenant pas au compact $K$. $\square$
\[propmin1\] Soit $L$ une surface lagrangienne orientable hyperbolique dans une variété symplectique fermée de dimension quatre $(X, \omega)$ et soit $d \in H_2 (X , L ; \Z)$. On note $N_d^g (\underline{x} , J)$ le nombre de courbes $J$-holomorphes homologues à $d$ à bords dans $L$, de topologie et de structure conforme données et qui passent par une configuration $\underline{x}$ de points distincts de $(X, \omega)$ de cardinal adéquat, pour $J \in {\cal J}_\omega$ générique. Ce nombre $N_d^g (\underline{x} , J)$ s’annule pour toute structure presque-complexe générale ayant un cou suffisamment long au voisinage de $L$.
[**Démonstration de la Proposition \[propmin1\] :**]{}
On équipe à nouveau $L$ d’une métrique à courbure constante et on allonge le cou d’une structure presque complexe générique au voisinage de $L$ jusqu’à briser la variété en deux, ceci de manière à ce que les points de la configuration $\underline{x}$ disjoints de $L$ se retrouvent dans l’étage $X \setminus L$. Notons $b+1$ le nombre de composantes connexes du bord des courbes que l’on considère. D’après le théorème de compacité de théorie symplectique des champs [@BEHWZ], ces dernières se brisent en courbes à deux étages qui sont cette fois-ci codées par des graphes $A_C$ ayant $b+1$ sommets marqués $s_0, \dots , s_b$ correspondant aux $b+1$ composantes de bord. Les sommets à distances paires de $s_0, \dots , s_b$ codent à nouveau les composantes de l’étage $T^* L$ et les sommets à distances impaires les composantes de l’étage $X \setminus L$. Les orbites de Reeb du fibré unitaire cotangent $S^* L$ sont cette fois-ci non-dégénérées, on fixe la trivialisation standard de $S^* L$ le long de ces orbites de Reeb, de sorte que leur indice de Conley-Zehnder soit nul, voir la Proposition $1.7.3$ de [@EGH]. La dimension de l’espace des modules d’une composante simple $C_s$ de l’étage $X \setminus L$ est donnée par le Théorème $2.8$ de [@HWZ], elle vaut $\mu_s^{CZ} + \chi_s$ où $\mu_s^{CZ}$ est l’indice de Conley-Zehnder total de la composante et $\chi_s$ sa caractéristique d’Euler. Un revêtement ramifié d’une courbe simple ne peut en particulier qu’augmenter cette dimension puisque les indices de Conley-Zehnder des orbites de Reeb ont ici la propriété de s’additionner sous de tels revêtements. Par suite, l’inégalité $\mu_s^{CZ} + \chi_s \geq 2n_s$ (resp. $\mu_s^{CZ} + \chi_s \geq n_s$) est satisfaite pour chaque sommet $s$ du graphe $A_C$ à distance impaire (resp. paire) de $s_0, \dots , s_b$, si $n_s \geq 0$ désigne le nombre de points de la configuration $\underline{x}$ par lesquels passe la composante codée par $s$. En sommant ces inégalités sur tous les sommets du graphe $A_C$, on déduit que la dimension totale attendue de la courbe $C$ se trouve minorée par $r + 2 r_X$ où $r$ est le cardinal de $\underline{x} \cap L$ et $r_X$ le cardinal de $\underline{x} \setminus L$. Comme par hypothèse cette dimension vaut $r + 2 r_X$, les inégalités précédentes sont des égalités. Il suit en particulier que toutes les courbes de l’étage $X \setminus L$ ont une dimension attendue paire ; elles ne peuvent par conséquent être planes. Comme par ailleurs $L$ ne possède pas de géodésique contractile, on vient de montrer que le graphe $A_C$ ne possède pas de feuille exception faite éventuellement des sommets $s_0, \dots , s_b$. Remarquons à présent que chaque composante de la courbe à deux étages est asymptote à une réunion de cylindres $J$-holomorphes sur les orbites de Reeb limites. Ces cylindres ont un module infini. On en déduit que les courbes comptées par $N_d^g (\underline{x} , J)$, lorsque $J$ possède un cou extrêmement long, possèdent également des anneaux de grands modules dont les âmes sont homotopes aux orbites de Reeb codées par les arêtes de l’arbre $A_C$. Or les courbes à bords comptées par $N_d^g (\underline{x} , J)$ sont supposées avoir une structure conforme fixée. Tout anneau dont le module est supérieur à une certaine quantité donnée par la structure conforme doit donc être contenu dans un disque. Par suite, lorsqu’on prive une telle courbe de la collection finie d’âmes de nos anneaux de grands modules codés par les arêtes de $A_C$, elle se trouve disconnectée en plusieurs composantes dont une au moins est un disque. Ce disque doit correspondre à une feuille du graphe $A_C$ distincte de $s_0, \dots , s_b$. Nous aboutissons ainsi à une impossibilité qui prouve que l’ensemble des courbes à deux étages sur lequel nous avons fondé notre raisonnement est vide, ce qu’il fallait démontrer. $\square$
\[propmintore\] Soit $(X, \omega , c_X)$ une variété symplectique réelle fermée de dimension quatre dont le lieu réel possède un tore lagrangien ou bien une surface hyperbolique lagrangienne $L$, orientable ou non. On suppose que $(X, \omega , c_X)$ ne possède pas de sphère symplectique réelle $S$ satisfaisant $\langle c_1 (X) , [S] \rangle > 1$ si $L$ est orientable et $\langle c_1 (X) , [S] \rangle > 0$ sinon. Soit $(d, g, b) \in H_2 (X , L ; \Z) \times \N \times \N^*$ et $K$ un compact de ${\cal M}_{g,b}$. Alors, pour toute structure presque-complexe générale ayant un cou suffisamment long au voisinage de $L$, la variété ne possède pas de membrane $J$-holomorphe homologue à $d$ à bord dans $L$ et conforme à un élément de $K$.
[**Démonstration de la Proposition \[propmintore\] :**]{}
On équipe à nouveau $L$ d’une métrique à courbure constante et on allonge le cou d’une structure presque complexe générique au voisinage de $L$ jusqu’à briser la variété en deux morceaux. D’après le théorème de compacité de théorie symplectique des champs [@BEHWZ], les membranes $J$-holomorphes homologues à $d$, de genre $g$ ayant $b$ composantes de bord dans $L$ qui survivent à cette déformation se brisent en courbes à deux étages codées par des graphes $A_C$ ayant $b$ sommets marqués $s_1, \dots , s_b$ correspondant aux $b$ composantes de bord. Les seules feuilles de ces arbres sont alors ces $b$ sommets $s_1, \dots , s_b$. En effet, ces feuilles coderaient sinon des plans $J$-holomorphes asymptotes à des orbites de Reeb du fibré unitaire cotangent $S^* L$. Ces orbites de Reeb n’étant pas contractiles dans $T^* L$, les plans $J$-holomorphes doivent être dans l’étage $X \setminus L$. La réunion d’un tel plan $P$, de son image par l’involution $c_X (P)$ et d’un cylindre $J$-holomorphe de $T^* L$ sur l’orbite de Reeb asymptote de $P$ fournit une sphère à deux étages. Cette sphère se recolle en une sphère symplectique $S$ de $(X, \omega)$ dont l’indice de Maslov vaut le double de l’obstruction à étendre la trivialisation canonique de $TX$ le long de l’orbite de Reeb à $S$ tout entier. Lorsque $L$ est orientable, cette obstruction est nulle le long du cylindre de $T^* L$ et supérieure à un le long de $P$ d’après la Proposition \[propmaslov\] et le Théorème $2.8$ de [@HWZ]. On en déduit que l’indice de Maslov de $S$ serait supérieur à quatre, ce qui contredit les hypothèses. De la même manière lorsque $L$ est non-orientable, l’indice de Maslov de $S$ vaut la somme des indices de Conley-Zehnder de $P$, $c_X (P)$ et du cylindre. Ces derniers sont supérieurs à leur caractéristique d’Euler puisqu’habitant des espaces de modules de dimensions attendues positives, voir le Théorème $2.8$ de [@HWZ]. Par sommation, l’indice de Maslov de $S$ devrait être supérieur à deux ce qui contredit à nouveau les hypothèses. Les seules feuilles des arbres codant les courbes à deux étages limites étant les $b$ sommets marqués $s_1, \dots , s_b$, on déduit comme dans la démonstration de la Proposition \[propmin1\] que peu avant la brisure de la variété, les membranes $J$-holomorphes homologues à $d$, de genre $g$ ayant $b$ composantes de bord dans $L$ possèdent un anneau de grand module au moins, d’âme voisine d’une orbite de Reeb de $S^* L$. En particulier, elles n’appartiennent pas au compact $K$ de ${\cal M}_{g,b}$, ce qu’il fallait démontrer. $\square$
Formulaire {#subsectformulaire}
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Désignons par $L$ une variété compacte de dimension $n$ homéomorphe à une sphère, un tore ou un espace projectif réel et munissons cette variété d’une métrique à courbure constante. Soit $C$ une courbe pseudo-holomorphe simple immergée d’énergie de Hofer finie dans un remplissage symplectique du fibré unitaire cotangent $(S^* L , \lambda)$ de $L$, où $\lambda$ désigne la restriction de la forme de Liouville. La caractéristique d’Euler de $C$ vaut $\chi = 2-2g - v$ si l’on note $g$ son genre et $v$ le nombre de ses pointes. Le flot de Reeb de $(S^* L , \lambda)$ trivialise le fibré normal de $C$ au voisinage de ses pointes ; nous noterons $\mu$ le double de l’obstruction à étendre cette trivialisation sur $C$ toute entière.
L’indice de Maslov $\mu$ est explicitement calculé dans la Proposition \[propcotangent\] lorsque $C$ est immergée dans le fibré cotangent $T^* L$. Nous calculons sous cette même hypothèse le nombre de points singuliers de $C$ dans le Lemme \[lemmepointsdoubles\] tandis que la Proposition \[propmaslov\] fournit une expression de la dimension de l’espace des déformations de $C$ en fonction des quantités $\chi$ et $\mu$. Les trois formules qui résultent de ces calculs se montrant bien utiles, nous leurs consacrons ce paragraphe.
\[propmaslov\] Soit $C$ une courbe pseudo-holomorphe simple d’énergie de Hofer finie dans un remplissage symplectique de dimension $2n$ du fibré unitaire cotangent d’une sphère, d’un tore ou d’un espace projectif réel à courbure constante $L$. Nous notons $g$ le genre de $C$, $v^-$ le nombre de ses pointes négatives, $\mu$ son indice de Maslov et $\chi^- = 2-2g-v^-$. La dimension de l’espace des déformations de $C$ vaut $\mu + (n-1)(2-2g)$ lorsque $L$ est une sphère ou un espace projectif réel et vaut $\mu + (n-1)\chi^-$ lorsque $L$ est un tore.
[**Démonstration :**]{}
Ces formules sont des conséquences de la formule d’indices calculée par Frédéric Bourgeois dans sa thèse [@Bour]. La courbe $C$ converge en ses pointes vers des orbites de Reeb qui appartiennent à des espaces de dimension $2(n-1)$ dans le premier cas et $n-1$ dans le second. Ces orbites contribuent donc à hauteur de $2(n-1)v$ dans le premier cas et $(n-1)v$ dans le second à la dimension que l’on calcule. Le reste de la contribution s’interprète comme l’indice de Fredholm de l’opérateur de Cauchy-Riemann associé à $C$ et perturbé par un facteur $\mp \frac{d}{p} Id$ en ses pointes, ce qui le rend non-dégénéré, voir la proposition $5.2$ de [@Bour]. Ce dernier est calculé par le Théorème $2.8$ de [@HWZ] et vaut $\mu^{CZ} + (n-1)\chi$ où $\mu^{CZ}$ désigne l’indice de Conley-Zehnder normal total de $C$. L’indice de Conley-Zehnder normal total se décompose ici en la somme de l’indice de Maslov $\mu$ et des indices de Conley-Zehnder des opérateurs de Cauchy-Riemann perturbés en chaque pointe de $C$ et calculés dans la trivialisation que l’on a fixé. Or ces indices de Conley-Zehnder des opérateurs de Cauchy-Riemann perturbés valent par définition $- (n-1)v$ dans le premier cas tandis qu’ils valent $0$ dans le second pour des pointes positives et $n-1$ pour des pointes négatives. Ces résultats sont établis dans le paragraphe $9.4$ de la thèse [@Bour]. Signalons toutefois une démonstration de ce dernier fait autre que celle proposée par Frédéric Bourgeois. Lorsqu’on allonge la structure complexe de $(\C P^1)^n$ au voisinage de $(\R P^1)^n$ jusqu’à briser la variété en deux morceaux, les fibres réelles de $(\C P^1)^n \to (\C P^1)^{n-1}$ se brisent en un cylindre sur un orbite simple de $T^* L$ et deux plans complexes conjugués de $(\C P^1)^n \setminus L$. La dimension de l’espace des déformations de chacun de ces morceaux vaut $n-1$ tandis que les indices de Maslov de ces composantes sont tous nuls. Confrontons ce résultat à ce qui précède. La dimension de l’espace des déformations du plan vaut $2(n-1)$ moins l’indice de l’opérateur de Cauchy-Riemann perturbé pour le plan, ce dernier vaut donc effectivement $n-1$. Elle vaut la moitié de $2(n-1)$ moins le double de l’indice de l’opérateur perturbé pour le cylindre, puisque le cylindre possède deux pointes et se voit contraint d’être préservé par l’antipodation dans les fibres de $T^* L$. On en déduit que l’indice de Conley-Zehnder des pointes positives s’annule. $\square$
\[propcotangent\] Soit $C$ une courbe pseudo-holomorphe simple d’énergie de Hofer finie dans le fibré cotangent d’une sphère, d’un tore ou d’un espace projectif réel de dimension $n$ à courbure constante $L$. Notons $\chi = 2-2g - v$ la caractéristique d’Euler de $C$ et $k$ la somme sur ses $v$ pointes des multiplicités de ses orbites de Reeb limites. L’indice de Maslov $\mu$ de $C$ vaut $2(n-1)k - 2\chi$ lorsque $L$ est une sphère, $(n-1)k - 2\chi$ lorsque $L$ est un espace projectif réel et $- 2\chi$ lorsque $L$ est un tore.
[**Démonstration :**]{}
Considérons le deux-cycle $C - c_L (C) - \sum_{i=1}^v \text{Cyl}_i$, où $c_L$ est l’antipodation dans les fibres de $T^* L$ et $\text{Cyl}_i$ les cylindres sur les orbites de Reeb limites de $C$. Ce deux-cycle se trouve renversé par $c_L$ de sorte qu’il est homologue à zéro. La première classe de Chern de $T^* L$ s’annule donc une fois évaluée contre ce cycle. Calculons cette dernière comme l’obstruction à trivialiser le fibré tangent en restriction à ce deux-cycle. La contribution de $C - c_L (C)$ vaut $2\chi + \mu$ où $\chi$. La contribution d’un cylindre $\text{Cyl}_i$ vaut l’opposé de son demi-indice de Maslov, soit $-k_i$ fois le demi-indice de Maslov du cylindre sur l’orbite simple sous-jacente si $k_i$ désigne la multiplicité de l’orbite.
Le demi-indice de Maslov d’un cylindre sur une orbite simple dans le cas d’une sphère vaut le degré du fibré normal d’une section plane de la quadrique ellipsoïde $Q^n$ puisque cette dernière est obtenue en recollant deux plans de fibrés normaux triviaux de $Q^n \setminus L$ au cylindre en question. Ce dernier vaut donc $2n - 2$, d’où la relation $2\chi + \mu - 2\sum_{i=1}^v (n-1)k_i = 0$.
Le demi-indice de Maslov d’un cylindre sur une orbite simple dans le cas d’un espace projectif réel vaut le degré du fibré normal d’une droite dans l’espace projectif complexe de dimension $n$ puisque cette dernière est obtenue en recollant deux plans de fibrés normaux triviaux de $\C P^n \setminus L$ au cylindre en question. Ce dernier vaut donc $n-1$, d’où la relation $2\chi + \mu - \sum_{i=1}^v (n-1)k_i = 0$.
Le demi-indice de Maslov d’un cylindre sur une orbite simple dans le cas d’un tore est trivial, d’où la relation $2\chi + \mu = 0$. $\square$
\[lemmepointsdoubles\] Soit $C$ une courbe pseudo-holomorphe d’énergie de Hofer finie immergée dans le fibré cotangent d’une sphère de dimension deux ou d’un plan projectif réel à courbure constante. Supposons que cette courbe soit simple, rationnelle, réelle et n’ayant que des points doubles transverses comme singularités. On note $v$ le nombre de paires de pointes complexes conjuguées de $C$ et $k$ la multiplicité totale des paires d’orbites de Reeb limites en ces pointes. Le nombre de points doubles de $C$ est majoré par $k^2 - 2k + 1$ dans le cas d’une sphère et $\frac{1}{2}(k^2 - 3k + 2)$ dans le cas d’un plan projectif réel.
[**Démonstration :**]{}
Considérons le deux-cycle $C - \sum_{i=1}^v \text{Cyl}_i$, où $\text{Cyl}_i$, $1 \leq i \leq v$, désignent les cylindres sur les orbites de Reeb limites de $C$. Ce deux-cycle se trouve renversé par $c_L$ de sorte qu’il est homologue à zéro. Choisissons un cylindre $\text{Cyl}$ sur une orbite de Reeb distincte des limites de $C$. L’indice d’intersection de $\text{Cyl}$ avec $C - \sum_{i=1}^v \text{Cyl}_i$ s’annule. Nous en déduisons que l’indice d’intersection de $\text{Cyl}$ avec $C$ vaut $2k$ dans le cas de la sphère de dimension deux et $k$ dans celui du plan projectif réel. Perturbons à présent $C$ en une courbe voisine $\widetilde{C}$ dont toutes les orbites de Reeb limites sont distinctes de celles de $C$. L’indice d’intersection de $\widetilde{C}$ avec $C - \sum_{i=1}^v \text{Cyl}_i$ s’annule. On déduit de ce qui précède que l’indice d’intersection de $\widetilde{C}$ avec $C$ se trouve majoré par $2k^2$ dans le cas de la sphère de dimension deux et $k^2$ dans celui du plan projectif réel. Cet indice est par ailleurs minoré par deux fois le nombre de points doubles de $C$ auquel s’ajoute la moitié de son indice de Maslov et le nombre de points d’intersection de $\widetilde{C}$ avec $C$ qui apparaissent au voisinage des pointes de $C$. Ces derniers sont au moins au nombre de $k_i -1$ au voisinage de chaque pointe convergeant vers une orbite de Reeb parcourue $k_i$ fois, ce qui découle du Théorème $1.5$ de [@HWZ1], soit $2(k-v)$ au total. L’indice de Maslov de $C$ est quant à lui donné par la Proposition \[propcotangent\], il vaut $4k + 4v - 4$ dans le cas de la sphère de dimension deux et $2k + 4v - 4$ dans celui du plan projectif réel. Ainsi, l’indice d’intersection de $\widetilde{C}$ avec $C$ se trouve minoré par $4k-2$ (resp. $3k - 2$) plus deux fois le nombre de points doubles de $C$ si $L$ est une sphère (resp. un plan projectif réel). Le résultat en découle. $\square$
\[reminters\] Nous avons établi au cours de la démonstration du Lemme \[lemmepointsdoubles\] la majoration $\vert \widetilde{C} \circ C \vert \leq 2k^2$ ou $k^2$ selon que $L$ est une sphère ou un plan projectif réel. Cette majoration nous sera utile au §\[sectcalculs\].
Congruences {#sectcong}
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Énoncés des résultats
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Étant donnée une classe d’homologie $d \in H_2 (X ; \Z)$ d’une variété symplectique réelle de dimension quatre $(X, \omega , c_X)$, nous noterons $g_d = \frac{1}{2} (d^2 - c_1 (X)d + 2)$ le genre lisse de $d$ et $c_d = c_1 (X)d -1$ le degré attendu du polynôme $\chi^d (T)$ défini dans [@Wels1].
\[theocong1\] Soit $(X, \omega , c_X)$ une variété symplectique réelle fermée de dimension quatre dont le lieu réel possède une composante connexe $L$ homéomorphe à une sphère. Soient $d \in H_2 (X ; \Z)$ et $r \in \N$. Lorsque $2r+1 < c_d$, la puissance $2^{\frac{1}{2} (c_d - 2r - 1)}$ divise $\chi^d_r (L)$.
Supposons en outre la connexité du lieu réel de la variété $(X, \omega , c_X)$. Alors,
a\) Lorsque $2r-1 < c_d$ et lorsque de plus $g_d$ et $\frac{1}{2} (r+1)$ sont de même parité, la puissance $2^{\frac{1}{2} (c_d - 2r + 1)}$ divise $\chi^d_r (L)$.
b\) Lorsque $2k < r+1 \leq \frac{1}{2} c_1 (X)d + 2$, la puissance $2^{\frac{1}{2} (c_d - 2r + 3)}$ divise $\chi^d_r (L)$, où $k$ désigne le maximum de l’ensemble $\{ j \in \N \, \vert \, j \neq g_d \mod(2) \text{ et } j \leq \vert d' \circ [L] \vert \text{ o\`u } $d’$ \text{ est effectif satisfaisant }\\ d' - c_X (d') = d \}.$
On entend ici par classe effective une classe d’homologie réalisable par un deux-cycle pseudo-holomorphe sur son deux-squelette.\
[**Exemple :**]{}
Le Théorème \[theocong1\] s’applique à l’ellipsoïde de dimension deux lorsque $d$ est un multiple positif, disons $\delta > 0$, d’une section plane réelle. Dans ce cas, $c_d = 4\delta - 1$ et $g_d = \delta^2 - 2 \delta + 1 = \delta + 1 \mod (2)$. Par conséquent, $2^{2 \delta - r - 1}$ divise $\chi^d_r (L)$ lorsque $r < 2 \delta - 1$, $2^{2 \delta - r}$ divise $\chi^d_r (L)$ lorsque de plus $r = 2 \delta + 1 \mod (4)$ et $\chi^d_{2 \delta - 3} (L) = 0 \mod (16)$.
\[theocong2\] Soient $(X, c_X)$ la quadrique ellipsoïde de dimension trois et $d$ un multiple positif, disons $\delta > 0$, d’une section hyperplane réelle. Lorsque $6r + 1 \leq 3 \delta$, la puissance $2^{\frac{3}{4} (\delta - 2r)}$ divise $\chi^d_r$.
\[theocong3\] Soit $(X, \omega , c_X)$ une variété symplectomorphe au plan projectif complexe éclaté en six boules complexes conjuguées au maximum. Soit $d \in H_2 (X ; \Z)$ une classe satisfaisant $c_d = c_1 (X)d -1 \geq 0$ et soient $r, r_X$ des entiers naturels satisfaisant la relation $r + 2r_X = c_d$. Lorsque $r+ 1 < r_X$, la puissance $2^{r_X - r - 1}$ divise $\chi^d_r (L)$. Lorsque $r < r_X$ et lorsque de plus $r = \langle d , h \rangle + 1 \mod (4)$, où $h$ est la classe d’une droite générique du plan, la puissance $2^{r_X - r }$ divise $\chi^d_r (L)$. Lorsqu’enfin la variété est le plan projectif complexe lui-même et $r + 1 < \langle d , h \rangle$, $\chi^d_{r} (L) = 0 \mod (64)$.
[**Exemple :**]{}
Le Théorème \[theocong3\] s’applique au plan projectif complexe où $d$ est un multiple positif, disons $\delta > 0$, d’une droite complexe. Dans ce cas, $8^{\frac{1}{2}(\delta - r - 1)}$ divise $\chi^d_r$ lorsque $r+1 < \delta $, $2^{\frac{1}{2}(3\delta - 3r - 1)}$ divise $\chi^d_r$ lorsque de plus $r = \delta + 1 \mod (4)$ et $\chi^d_{\delta - 3} = 0 \mod (64)$.
Démonstrations des Théorèmes \[theocong1\], \[theocong2\] et \[theocong3\]
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On suit la stratégie générale énoncée au paragraphe \[subsubsectstrat\] en équipant la sphère ou le plan projectif réel lagrangien d’une métrique à courbure constante. Les courbes $J$-holomorphes rationnelles réelles $C$ comptées par l’invariant $\chi^d_r (L)$ qui survivent à l’allongement du cou de $J$ jusqu’à la brisure de la variété sont codées par des arbres $A_C$, voir la figure \[figarbre\]. Ces derniers ont une racine $s_0$ qui code l’unique composante de la courbe à deux étages limite laissée invariante par l’involution $c_X$. Le fibré normal de chaque composante simple $C_s$ associé à un sommet $s$ d’un arbre $A_C$ est canoniquement trivialisé le long des orbites de Reeb asymptotes par le flot de Reeb et l’on note $\mu_s$ l’obstruction à étendre cette trivialisation sur $C_s$ tout entier. La dimension de l’espace des modules dans lequel habite $C_s$ s’exprime par la relation $\mu_s + 2(\text{dim}_\C X - 1)$ lorsque $s \neq s_0$ et $\frac{1}{2} \mu_s + \text{dim}_\C X - 1$ lorsque $s = s_0$ puisque la courbe $C_s$ est alors contrainte d’être préservée par l’involution $c_X$, voir la Proposition \[propmaslov\] de notre formulaire donné au paragraphe \[subsectformulaire\]. Notons $s_0 , \dots , s_j$ les sommets de $A_C$ qui codent les composantes $C_s$ soumises à des conditions d’incidence et $\widetilde{A}_C$ le sous arbre de $A_C$ obtenu en ne retenant que les sommets $s_0 , \dots , s_j$ et les arêtes reliant ces sommets entre eux.\
[**Démonstration du Théorème \[theocong1\] :**]{}
Nous allons commencer par minorer la contribution de chaque composante connexe de $A_C \setminus \widetilde{A}_C$ à l’indice de Maslov total de la courbe $C$. Lorsqu’une telle composante n’est pas connectée à $s_0$, cette contribution est minorée par $2(k' - 1)$ où $k'$ désigne la multiplicité totale des arêtes reliant cette composante connexe à $\widetilde{A}_C$. Ceci résulte de l’inégalité (\[equ1\]) établie au §\[subsectoptdem\]. Lorsqu’une telle composante est connectée à $s_0$, la Proposition \[propcotangent\] fournit l’estimation $\sum_{s \in S_2'} \mu_s = 2\sum_{s \in S_2'} (k_s + v_s) - 4 \# S_2'$ de la contribution des sommets à distance paire de $s_0$, où $S_1'$ (resp. $S_2'$) désigne l’ensemble des sommets à distance impaire (resp. paire) de $s_0$ de cette composante. Cette composante possède un unique sommet $s$ ayant la propriété d’être connecté à $s_0$. Notons $l^0$ le degré du revêtement de la courbe $C_s$ codée par ce sommet et $k^0$ la multiplicité de l’arête qui le joint à $s_0$. La minoration (\[equ3\]) établie au §\[subsectoptdem\] fournit, en reprenant les notations introduites dans ce paragraphe, $\sum_{s \in S_1'} \mu_s \geq 2 \sum_{s \in S_1'} ( l_s - l_s \tilde{v}_s + v_s) - 4 \# S_1'$, soit $\sum_{s \in S_1'} \mu_s \geq 2 \sum_{s \in S_1'} ( l_s - k_s + v_s) -
4 \# S_1' +2k^0 - 2l^0$. Nous en déduisons après sommation $\sum_{s \in S_1' \cup S_2'} \mu_s \geq 2 \sum_{s \in S_1'} l_s - 2l^0 + 2k' +2v' - 2$, où $v'$ (resp. $k'$) désigne le nombre d’arêtes (resp. leur multiplicité totale) reliant cette composante à un sommet $s_1 , \dots , s_j$. L’indice de Maslov d’une telle composante se trouve donc finalement minoré par $2v'$ excepté dans le cas où $v'$ est nul et cet indice vaut $-2$. Notons $c_{-2}$ le nombre de composantes de $A_C \setminus \widetilde{A}_C$ d’indice de Maslov total $-2$. De ces calculs résulte que la contribution de $A_C \setminus \widetilde{A}_C$ à l’indice de Maslov total de la courbe à deux étages codée par $A_C$ est minorée par $2a - 2c_{-2}$ si $a+1$ désigne le nombre de composantes connexes de $\widetilde{A}_C$.
La contribution des sommets $s_1 , \dots , s_j$ est quant à elle minorée par $2 r_X - 2j$. La courbe réelle que code le sommet $s_0$ se voit d’une part contrainte d’interpoler $r$ points de $L$ et d’autre part de converger en $c_{-2}$ paires complexes conjuguées de ses pointes vers $c_{-2}$ paires complexes conjuguées d’orbites de Reeb prescrites. En effet, les $c_{-2}$ sommets correspondant de $A_C \setminus \widetilde{A}_C$ adjacents à $s_0$ codent des courbes rigides. Enfin, chaque sommet $s_1 , \dots , s_j$ connecté à $s_0$ présente l’alternative suivante. Soit la minoration précédente $2 r_X - 2j$ est atteinte pour ce sommet et la courbe correspondante, avec ses conditions d’incidences, est rigide ; ce qui ajoute donc une contrainte supplémentaire pour une paire de pointes de $s_0$. Soit la minoration précédente $2 r_X - 2j$ n’est pas atteinte pour ce sommet et peut donc être améliorée de deux. Nous aboutissons dans tous les cas à la minoration $\sum_{s \in \widetilde{A}_C} \mu_s \geq 2 r_X - 2a + r - 1 +2c_{-2} $. L’indice de Maslov total $\mu$ de la courbe à deux étages codée par $A_C$ se trouve ainsi minoré par $r + 2 r_X - 1 = c_d - 1$. Comme cet indice est par ailleurs majoré par cette quantité qui n’est autre que le degré du fibré normal d’une courbe rationnelle irréductible immergée homologue à $d$, toutes nos inégalités doivent être égalités. Nous en concluons que les composantes connexes de $A_C \setminus \widetilde{A}_C$ qui ne sont pas connectées à $s_0$ sont réduites à un sommet codant un plan asymptote à une orbite de Reeb simple tandis que les composantes connectées à $s_0$ sont au nombre de $c_{-2}$ et leur indice de Maslov vaut $-2$. L’arbre $\widetilde{A}_C$ est en particulier connexe.
L’arbre $A_C$ vient avec une donnée combinatoire supplémentaire, une fonction qui associe à chaque sommet à distance impaire de $s_0$ les classes d’homologies relatives de la paire de courbes correspondantes codée par ce sommet ainsi que les paires de points complexes conjugués de $\underline{x} $ que ces courbes contiennent. Notons $r_1 , \dots , r_j$ le nombre de paires de points complexes conjugués de $\underline{x} $ associées à $s_1 , \dots , s_j$ respectivement, de sorte que leur somme vaille $r_X$. Il y a $2^{r_i - 1}$ partitions d’un ensemble de $r_i$ points complexes conjugués en deux ensembles complexes conjugués, soit ici $2^{r_X - j}$ partitions au total. Une fois attribués à chaque courbe $C_s$ l’ensemble de points qu’elle doit interpoler, certaines de ces courbes sont rigides et d’autres non. Notons $j^-$ le nombre de telles courbes rigides et $j^+ = j - j^-$. D’après ce qui précède, la courbe réelle codée par $s_0$, avec ses $j^- + c_{-2}$ paires d’asymptotes prescrites et ses $r$ points réels à interpoler, est rigide. La dimension $2k_{s_0} + 2v_{s_0} -1$ donnée par la Proposition \[propcotangent\] vaut donc en particulier $r + 2j^- + 2c_{-2}$. Par conséquent, les $j^+$ paires de courbes non-rigides précédentes héritent d’une contrainte supplémentaire, elles ont une paire d’asymptotes prescrites correspondant à une paire d’orbites de Reeb limites restées libres de la courbe $C_{s_0}$. Il y a deux bijections possibles entre une telle paire d’orbites de Reeb et une telle paire de courbes non-rigides, soit $2^{j^+}$ bijections au total. Ainsi, le nombre de courbes à deux étages ayant une combinatoire donnée par $A_C$ est divisible par $2^{r_X - j^-}$. Or d’après le théorème de recollement en théorie symplectique des champs [@Bour] et le Théorème \[theoinv\], la contribution à l’invariant $\chi^d_{r} (L)$ d’une courbe à deux étages ne dépend que de sa combinatoire, de sorte que $2^{r_X - j^-}$ divise $\chi^d_{r} (L)$. L’équation $$\label{relj1}
r + 2j^- + 2c_{-2} = 2k_{s_0} + 2v_{s_0} -1$$ impose l’inégalité $r+1 \geq 2k_{s_0}$. On en déduit $2j^- \leq 2v_{s_0} \leq 2k_{s_0} \leq r+1$ et le premier résultat énoncé dans le Théorème \[theocong1\].
Tous les arbres $A_C$ pour lesquels l’une des inégalités $2j^- \leq 2v_{s_0} \leq 2k_{s_0} \leq r+1$ est stricte satisfont $2j^- \leq r-1$ et le deuxième énoncé du Théorème \[theocong1\] est immédiat. Soit $A_C$ un arbre pour lequel $j^- = v_{s_0} = k_{s_0} = \frac{1}{2} (r+1)$. Toutes les courbes codées par les sommets de cet arbre sont simples puisque d’indices de Maslov positifs et ont des orbite de Reeb simples pour asymptotes. Considérons la courbe $C'$ formée de toutes les paires de courbes de l’étage $X \setminus L$ codées par $A_C$ et de paires de plans complexes conjugués de $T^* L$ convergeant vers $\partial C'$. Une telle courbe à deux étages se recolle en une courbe $J$-holomorphe réductible homologue à $d$ ayant un nombre pair de composantes irréductibles échangées par $c_X$. Le nombre de points doubles d’une telle courbe a la parité de $g_d + 1$, d’après la formule d’adjonction. Supposons le lieu réel de $X$ connexe, ce nombre de points double est alors également de la même parité que le nombre de points d’intersection avec $L$, c’est-à-dire que le nombre de paires de plans de $T^* L$ que l’on a introduit ou encore le nombre total d’arêtes $v$ de l’arbre $A_C$. Or par hypothèse, $g_d$ et $\frac{1}{2} (r+1)$ sont de même parité et $\frac{1}{2} (r+1) = v_{s_0}$. Par conséquent $v$ et $v_{s_0}$ ne sont pas de même parité ce qui impose l’existence d’un sommet de valence paire adjacent à $s_0$. La structure presque complexe de $X \setminus L$ étant générale, toutes les pointes de la courbe codée par ce sommet ont des asymptotes distinctes et cette dernière est rigide puisque $j^- = v_{s_0}$. Le résultat $a)$ découle à présent du fait qu’il y a un nombre pair de choix de la pointe de cette courbe à relier à $C_{s_0}$, ce qui permet d’améliorer d’une puissance de deux la divisibilité du nombre de courbes codées par ces arbres $A_C$.
Enfin, tous les arbres $A_C$ pour lesquels $2k_{s_0} < r+1$ ou $v_{s_0} + 1 < k_{s_0}$ satisfont $2j^- \leq r-3$, ce qui découle de (\[relj1\]). Or si $A_C$ est un arbre tel que $j^- = v_{s_0} = k_{s_0}$, le raisonnement que l’on vient de suivre fournit une paire de courbes $J$-holomorphes réductibles complexes conjuguées homologue à $d$. Notons $d'$ la classe d’homologie d’une telle courbe de sorte que $d' - c_X (d') = d$. Cette courbe peut être choisie de sorte que l’indice d’intersection $\vert d' \circ [L] \vert $ vaille la multiplicité totale $k$ des arêtes de $A_C$. Comme $k$ et $g_d$ ne sont pas de même parité, $k$ fait partie de l’ensemble défini en $b)$. L’hypothèse implique à présent $2k_{s_0} \leq 2k < r+1$, d’où le résultat $b)$ dans ce cas. Si en revanche $j^- = v_{s_0} = k_{s_0} -1$, l’inégalité $2j^- < r+1$ permet d’améliorer d’une puissance de deux le premier énoncé du Théorème \[theocong1\]. En outre, une arête adjacente à $s_0$ est de multiplicité deux, de sorte qu’une courbe $C_s$ se trouve connectée à $C_{s_0}$ par une orbite de Reeb double. Le théorème de recollement en théorie symplectique des champs [@Bour] garantit alors l’existence d’un nombre pair de courbes $J$-holomorphes convergeant vers une courbe à deux étages donnée codée par $A_C$. Cette parité provenant du paramètre de recollement associé à l’orbite double permet d’améliorer le premier énoncé du Théorème \[theocong1\] d’une puissance de deux supplémentaire. D’où le résultat. $\square$\
[**Démonstration du Théorème \[theocong2\] :**]{}
Le complémentaire $X \setminus L$ est isomorphe au fibré en droites de bidegré $(1,1)$ sur la quadrique $\C P^1 \times \C P^1$. Les courbes rationnelles irréductibles d’énergie de Hofer finie de ce complémentaire sont donc d’indice de Maslov positif. En effet, ce complémentaire se compactifie en le fibré en droites projectives $F$ obtenu à partir de la somme du fibré trivial et du fibré de bidegré $(1,1)$ sur $\C P^1 \times \C P^1$. Une courbe d’énergie de Hofer finie se compactifie en une courbe de $F$ dont la classe d’homologie s’écrit $e + kf$, où $e$ désigne une classe d’homologie effective de la section nulle $\C P^1 \times \C P^1$ du fibré et $f$ la classe d’une fibre. Or il suit de la formule d’adjonction que l’évaluation de la première classe de Chern de $F$ sur $e$ est positive et vaut deux sur $f$. Les indices de Maslov de ces courbes sont donc positifs et ne peuvent qu’augmenter par revêtements ramifiés. Il s’ensuit que toutes les courbes $C_s$ codées par les sommets $s$ de l’arbre sont des courbes simples pour peu que la configuration de points choisie soit suffisamment générale. La dimension $\mu_s + 4$ des espaces de modules associés est donc strictement positive et même supérieure au quadruple (resp. au double) du nombre de points de la configuration que doit interpoler $C_s$ si $s$ est à distance impaire (resp. paire) de $s_0$. Par suite, la contribution totale à l’indice de Maslov des sommets $s \in S_1$ de $A_C$ à distance impaire de $s_0$, lesquels codent les courbes de $X \setminus L$, se trouve minorée par $4 r_X - 4 \# S_1$. De même, la contribution totale à l’indice de Maslov des sommets $s \in S_2$ de $A_C$ à distance paire de $s_0$, lesquels codent les courbes de $T^* L$, se trouve minorée par $2r - 4 \# S_2 + 2$ puisque $s_0$ code une courbe réelle ayant $\frac{1}{2} \mu_{s_0} + 2$ degrés de liberté. En outre, chaque arête de l’arbre $A_C$ code une paire de pointes des courbes codées par les sommets adjacents et qui ont une même orbite de Reeb pour asymptote. Cette contrainte coûte quatre degrés de liberté supplémentaires par arête, soit au total $4 \# S_1 + 4 \# S_2 - 4$ degrés, puisque le nombre d’arêtes d’un arbre diffère du nombre de sommets par un. Ceci nous permet finalement de minorer l’indice de Maslov de notre courbe à deux étages codée par $A_C$ par $2r + 4r_X - 2 = \langle c_1 (X) , d \rangle -2$. Cet indice est par ailleurs majoré par le degré $\langle c_1 (X) , d \rangle -2$ du fibré normal de $C$, de sorte que toutes nos minorations sont des égalités. En particulier, tous les sommets adjacents à $s_0$ sont des feuilles qui doivent coder des courbes interpolant chacune au moins un point de la configuration, d’après la Proposition \[propcotangent\].
Notons à présent $r_1 , \dots , r_j$ le nombre de paires de points complexes conjugués de $\underline{x} $ associées aux sommets $s_1 , \dots , s_j$ adjacents à $s_0$ respectivement, de sorte que leur somme vaille $r_X$. Il y a $2^{r_i - 1}$ partitions d’un ensemble de $r_i$ points complexes conjugués en deux ensembles complexes conjugués, soit ici $2^{r_X - j}$ partitions au total. Une fois attribués à chaque courbe $C_s$ l’ensemble de points qu’elle doit interpoler, certaines de ces courbes sont rigides, d’autres conservent deux ou quatre degrés de liberté. Notons $j^-$ le nombre de telles courbes rigides, $j_1^+$ (resp. $j_2^+$) le nombre de celles qui conservent deux (resp. quatre) degrés de liberté. D’après ce qui précède, la courbe réelle codée par $s_0$, avec ses $j^- + j_1^+$ paires d’asymptotes prescrites et ses $r$ points réels à interpoler, est rigide. La dimension $4k_{s_0} + 2v_{s_0}$ donnée par la Proposition \[propcotangent\] vaut donc en particulier $2r + 4j^- + 2j_1^+$. Par conséquent, les $j_2^+$ paires de courbes non-rigides précédentes héritent d’une contrainte supplémentaire, elles doivent converger en une paire de pointes complexes conjuguées vers une paire prescrite d’orbites de Reeb limites de la courbe $C_{s_0}$. Il y a deux bijections possibles entre une telle paire d’orbites de Reeb et une telle paire de courbes non-rigides, soit $2^{j_2^+}$ bijections au total. Ainsi, le nombre de courbes à deux étages ayant une combinatoire donnée par $A_C$ est divisible par $2^{r_X - j^- - j_1^+}$. Or d’après le théorème de recollement en théorie symplectique des champs [@Bour] et le Théorème \[theoinv\], la contribution à l’invariant $\chi^d_{r}$ d’une courbe à deux étages ne dépend que de sa combinatoire, de sorte que $2^{r_X - j^- - j_1^+}$ divise $\chi^d_{r}$. L’équation $2r + 4j^- + 2j_1^+ = 4k_{s_0} + 2v_{s_0}$ impose l’inégalité $r \geq k_{s_0}$. On en déduit $j^- + j_1^+ \leq v_{s_0} \leq r$ de sorte que $2^{r_X - r}$ divise $\chi^d_{r}$. Or par hypothèse, $2r + 4r_X = 3\delta$ puisque la première classe de Chern de la quadrique de dimension trois est Poincaré duale au triple de la section hyperplane. Le Théorème \[theocong2\] en découle. $\square$\
[**Démonstration du Théorème \[theocong3\] :**]{}
D’après les hypothèses que l’on a faites, le complémentaire $X \setminus L$ est isomorphe au fibré en droites de degré quatre sur $\C P^1$ éclaté en six points complexes conjugués au maximum. Ce complémentaire ne contient par conséquent pas de courbes rationnelles irréductibles d’énergie de Hofer finie et d’indice de Maslov négatif ayant plus de deux pointes. En effet, ce complémentaire se compactifie en la surface réglée rationnelle de degré quatre $\Sigma_4$ éclatée en six points complexes conjugués au maximum. Une courbe d’énergie de Hofer finie se compactifie en une courbe dont la classe d’homologie s’écrit $ne + kf - \sum \alpha_i E_i$, où $e$ désigne la section nulle du fibré, $f$ une fibre et $E_i$ les diviseurs exceptionnels des éclatements. L’irréductibilité de la courbe force $0 \leq \alpha_i \leq n$ dès que $n \geq 1$ et la première classe de Chern de $\Sigma_4$ est duale à $2e - 2f- \sum E_i$, de sorte que son évaluation $6n + 2k - \sum \alpha_i$ sur la courbe soit minorée par $2k$ lorsque $n \geq 1$. L’indice de Maslov de telles courbes $C_s$ est donc positif, puisque minoré par $2k - 2 \chi (C_s)$. L’absence de courbes d’indice de Maslov $-2$ autres que planes a pour conséquence que pour tout arbre $A_C$, les courbes $C_s$ codées par les sommets $s$ de l’arbre sont des courbes simples pour peu que la configuration de points choisie soit suffisamment générale. La dimension $\mu_s + 2$ des espaces de modules associés est donc positive et même supérieure au double du nombre (resp. au nombre) de points de la configuration que doit interpoler $C_s$ si $s$ est à distance impaire (resp. paire) de $s_0$. Par suite, la contribution totale à l’indice de Maslov des sommets $s \in S_1$ de $A_C$ à distance impaire de $s_0$, lesquels codent les courbes de $X \setminus L$, se trouve minorée par $2 r_X - 2 \# S_1$. De même, la contribution totale à l’indice de Maslov des sommets $s \in S_2$ de $A_C$ à distance paire de $s_0$, lesquels codent les courbes de $T^* L$, se trouve minorée par $r - 2 \# S_2 + 1$ puisque $s_0$ code une courbe réelle ayant $\frac{1}{2} \mu_{s_0} + 1$ degrés de liberté. En outre, chaque arête de l’arbre $A_C$ code une paire de pointes des courbes codées par les sommets adjacents et qui ont une même orbite de Reeb pour asymptote. Cette contrainte coûte deux degrés de liberté supplémentaires par arête, soit au total $2 \# S_1 + 2 \# S_2 - 2$ degrés, puisque le nombre d’arêtes d’un arbre diffère du nombre de sommets par un. Ceci nous permet finalement de minorer l’indice de Maslov de notre courbe à deux étages codée par $A_C$ par $r + 2r_X - 1 = c_d-1$. Cet indice est par ailleurs majoré par le degré $c_d-1$ du fibré normal de $C$, de sorte que toutes nos minorations sont des égalités. En particulier, tous les sommets à distance paire de $s_0$ autre que $s_0$ lui-même codent soit des cylindres convergeant vers des orbites de Reeb simplement revêtues, soit des plans convergeant vers des orbites doublement revêtues puisque ce sont d’après la Proposition \[propcotangent\] les seules courbes de $T^* \R P^2$ rigides une fois leurs orbites de Reeb limites prescrites. Reprenons à ce stade la démarche suivie dans le troisième paragraphe de la démonstration du Théorème \[theocong1\]. On note $r_1 , \dots , r_j$ le nombre de paires de points complexes conjugués de $\underline{x} $ associées à $s_1 , \dots , s_j$ respectivement, de sorte que leur somme vaille $r_X$. Il y a $2^{r_i - 1}$ partitions d’un ensemble de $r_i$ points complexes conjugués en deux ensembles complexes conjugués, soit $2^{r_X - j}$ partitions au total. Une fois attribués à chaque courbe $C_s$ l’ensemble de points qu’elle doit interpoler, certaines de ces courbes sont rigides et d’autres non. Notons $j_1$ le nombre de telles courbes rigides non adjacentes à $C_{s_0}$ et $j^-$ le nombre de telles courbes rigides adjacentes à $C_{s_0}$. Les $j^+$ courbes restantes sont adjacentes à $C_{s_0}$ et gardent deux degrés de liberté une fois interpolés les $r_k$ points qu’elles doivent interpoler ; c’est la condition d’adjacence à $C_{s_0}$ qui les rigidifie. D’après ce qui précède, la courbe réelle codée par $s_0$, avec ses $j^- + c_{-2}$ paires d’asymptotes prescrites et ses $r$ points réels à interpoler, est rigide, où $c_{-2}$ désigne à nouveau le nombre de courbes rigides adjacentes à $C_{s_0}$ et autres que les $j^-$ courbes soumises à des conditions d’incidence. La dimension $k_{s_0} + 2v_{s_0} -1$ donnée par la Proposition \[propcotangent\] vaut donc en particulier $r + 2j^- + 2c_{-2}$. Par conséquent, les $j^+$ paires de courbes non-rigides précédentes héritent d’une contrainte supplémentaire, elles ont une paire d’asymptotes prescrites correspondant à une paire d’orbites de Reeb limites restées libres de la courbe $C_{s_0}$. Il y a deux bijections possibles entre une telle paire d’orbites de Reeb et une telle paire de courbes non-rigides, soit $2^{j^+}$ bijections au total. De même, les $j_1$ courbes rigides non adjacentes à $C_{s_0}$ sont codées par des sommets adjacents à au moins un sommet bivalent de l’arbre $A_C$ puisque ce dernier est connexe et que les autres sommets adjacents sont des feuilles. D’après ce qui précède, ce sommet bivalent code une paire de cylindres complexes conjugués de $T^* L$ reliant deux paires complexe d’orbites limites de deux paires complexes conjuguées de courbes rigides de $X \setminus L$. Il y a deux façons d’apparier ces orbites, soit $2^{j_1}$ bijections au total. Ainsi, le nombre de courbes à deux étages ayant une combinatoire donnée par $A_C$ est divisible par $2^{r_X - j^-}$. Or d’après le théorème de recollement en théorie symplectique des champs [@Bour] et le Théorème \[theoinv\], la contribution à l’invariant $\chi^d_{r} (L)$ d’une courbe à deux étages ne dépend que de sa combinatoire, de sorte que $2^{r_X - j^-}$ divise $\chi^d_{r} (L)$. Nous disposons cette fois-ci de la relation $r + 2j^- + 2c_{-2} = k_{s_0} + 2v_{s_0} -1$ qui impose l’inégalité $r+1 \geq k_{s_0}$. On en déduit donc $j^- \leq v_{s_0} \leq k_{s_0} \leq r+1$ et le premier résultat énoncé dans le Théorème \[theocong3\].
Tous les arbres $A_C$ pour lesquels l’une des inégalités $j^- \leq v_{s_0} \leq k_{s_0} \leq r+1$ est stricte satisfont $j^- \leq r$. Pour montrer le second résultat énoncé dans le Théorème \[theocong3\], on peut donc se restreindre aux arbres satisfaisant $j^- = v_{s_0} = k_{s_0} = r+1$. En particulier, la courbe codée par $s_0$ n’a que des orbites simples pour limites. Si un tel arbre possède une feuille à distance paire de $s_0$, on a vu qu’elle doit coder un plan asymptote à une orbite double. Le théorème de recollement en théorie symplectique des champs garantit alors qu’il y a deux façons de recoller ce plan au restant de la courbe. Ce paramètre de recollement permet donc à nouveau dans ce cas là d’améliorer d’une puissance de deux le premier énoncé du Théorème \[theocong3\]. On peut donc supposer que toutes les arêtes des arbres $A_C$ sont de multiplicité un. Notons $C_1$ la réunion des courbes codées par les sommets à distances impaires de $s_0$ et $\overline{C}_1$ sa compactifiée dans $\Sigma_4$. La classe d’homologie de $\overline{C}_1$ s’écrit $vf + g$, où $f$ est la classe d’une fibre de $\Sigma_4$, $v$ est le nombre d’arêtes de $A_C$ et $g \in H_2 (X \setminus L ; \Z)$. La classe d’homologie $d$ s’écrit alors $$\label{relv}
v(f + (c_X)_* f ) + g + (c_X)_* g = vh + g + (c_X)_* g,$$ d’où $\langle d , h \rangle = v \mod (4)$. On déduit donc des hypothèses faites que $v = r-1 \mod (4)$, puis que le nombre de sommets à distance paire de $s_0$, $s_0$ exclu, est impair puisque $v_{s_0} = r+1$. Ceci force l’existence d’un sommet $s$ de valence paire parmi les sommets à distance impaire de $s_0$. Or, le nombre de courbes à deux étages codées par un tel arbre est pair. En effet, si $s$ est adjacent à $s_0$, il y a parmi les pointes de $C_s$ un nombre pair de choix de celle reliée à $s_0$. Si $s$ n’est pas adjacent à $s_0$, il est relié à un nombre pair de sommets bivalents de $S_2$, eux-mêmes reliés à un nombre pair de sommets de $S_1$ de sorte que ces sommets bivalents ne font que connecter bijectivement ces derniers aux pointes de $C_s$. Le nombre de telles bijections étant pair, nous pouvons à nouveau dans ce dernier cas d’améliorer d’une puissance de deux le résultat précédent, ce qui démontre le second énoncé du Théorème \[theocong3\].
Enfin, dans le cas du plan projectif complexe, lorsque $r+1 = d-2$, on déduit aussi de la relation (\[relv\]) l’inégalité $v \leq d$, inégalité stricte dès que $\langle g , h \rangle \neq 0$. Or l’annulation $\langle g , h \rangle$ force les sommets adjacents à $s_0$ à être des feuilles, ce qui est exclus par la majoration $k_{s_0} \leq r+1 < d$. Par suite, $v \leq d - 4$, $k_{s_0} \leq d-4 < r+1$ et l’équation $r + 2j^- + 2c_{-2} = k_{s_0} + 2v_{s_0} -1$ impose $j^- < v_{s_0}$. Dans ce cas, on obtient donc $r_X - j^- =
d+1 - j^- \geq d + 2 - k_{s_0} \geq 6$, d’où le résultat. $\square$
Calculs {#sectcalculs}
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Invariants énumératifs réels de fibrés cotangents
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### Construction des espaces de modules {#subsubmodules}
Soit $L$ une sphère, un tore ou un espace projectif réel de dimension $n=2$ ou $3$. Le fibré cotangent de $L$ est équipé de sa forme de Liouville $\lambda$ et de l’involution $c_L$ définie par $(q,p) \in T^* L \mapsto (q,-p) \in T^* L$. Cette dernière satisfait $c_L^* \lambda = - \lambda$ de sorte que $(T^* L , d\lambda , c_L)$ est une variété symplectique réelle. Soit $g$ une métrique à courbure constante sur $L$, $U^* L$ l’ensemble des couples $(q,p) \in T^* L$ tels que $g(p,p) \leq 1$ et $S^* L$ le bord de $U^* L$. La restriction de $\lambda$ à $S^* L$ est une forme de contact et l’on note $R_\lambda$ le champ de Reeb associé. Le flot engendré par $R_\lambda$ n’est autre que le flot géodésique. Notons ${\cal J}_\lambda$ l’espace des structures presque-complexes positives pour $d \lambda$ et asymptotiquement cylindriques sur une structure $CR$ de $S^* L$. Plus précisément, le champ radial de $ T^* L$ identifie le complémentaire de la section nulle avec la symplectisation $(\mathbb{R} \times S^* L , d(e^\rho \lambda))$ de $(S^* L , \lambda)$. On note ${\cal J}_\lambda$ l’espace des structures presque-complexes $J$ positives pour $d \lambda$, de classe $C^l$, $l \gg 1$, qui satisfont $J(\frac{\partial}{\partial \rho}) = R_\lambda$ et préservent le noyau de $\lambda$ pour $\rho \gg 1$ et qui enfin sont invariantes par translation par $\rho$ au-delà d’un certain rang $\rho_0$. Nous notons alors $\mathbb{R} {\cal J}_\lambda \subset {\cal J}_\lambda$ le sous-espace des structures presque-complexes pour lesquelles $c_L$ est $J$-antiholomorphe. Ces espaces ${\cal J}_\lambda$ et $\Bbb{R} {\cal J}_\lambda$ sont tous deux des variétés de Banach séparable non-vides et contractiles.
Soit $S$ une sphère de dimension deux orientée et ${\cal J_S} $ l’espace des structures presque-complexes de classe $C^l$ sur $S$ qui sont compatibles avec son orientation. Soient $v_\C \in \N^*$ et $y_1, \dots , y_{v_\C}$ une collection de $v_\C$ points distincts sur $S$. Il suit de [@HWZ1] et du Corollaire $5.1$ de [@Bour] qu’il existe $0 < d << 1$ tel que pour tous $J_S \in {\cal J_S} $, $J \in {\cal J_\lambda} $ et toute application $J$-holomorphe propre $u : S \setminus \{y_1, \dots , y_{v_\C} \} \to T^* L$ d’énergie de Hofer finie, l’application $u$ a le comportement suivant au voisinage de chaque point $y_i$. Fixons un paramétrage local $J_S$-holomorphe de $S$ au voisinage de chaque point $y_i$, $1 \leq i \leq v_\C$ par l’anneau $\C_1 = \{ z \in \C \; \vert \; \vert z \vert \geq 1\}$, puis des coordonnées cylindriques $(s,t) \in \R^* \times [0 , 1] \mapsto e^{s + 2\pi it} \in \C_1$ de cet anneau. On en déduit un paramétrage de $S$ au voisinage de chaque point $y_i$ de la forme $\phi_i : (s,t) \in \R^* \times [0 , 1] \to S \setminus \{y_1, \dots , y_{v_\C} \}$, $1 \leq i \leq v_\C$. Pour $1 \leq i \leq v_\C$, notons $u \circ \phi_i = (\rho_i , \tilde{u}_i)$, où $\rho_i : (s,t) \in \R^+ \times [0 , 1] \to \R$ et $\tilde{u}_i : (s,t) \in \R^+ \times [0 , 1] \to S^* L$. Alors, pour $1 \leq i \leq v_\C$, il existe $s_i \in \R$, $k_i \in \N^*$ et des orbites $\gamma_i$ du flot de Reeb tels que les fonctions distances $\vert \rho_i (s,t) -(k_i A)s - s_i \vert$ et $d \big( \tilde{u} (s,t) - \gamma_i ((k_i A)t) \big)$ appartiennent à l’espace fonctionnel $L^{k,p}_d = \{ f : \R^+ \times [0 , 1] \to \R^+ \; \vert \; f(s,t) e^{ds} \in L^{k,p} (\R^+ \times [0 , 1] , \R) \}$ et ceci quel que soient $1 << k << l$ et $2 < p < + \infty$, où $A$ désigne l’intégrale de $\lambda$ sur l’orbite de Reeb simple sous-jacente et $L^{k,p} (\R^+ \times [0 , 1] , \R)$ désigne l’espace des fonctions ayant $k$ dérivées dans $L^p$. On note $L^{k,p}_d (S \setminus \{y_1, \dots , y_{v_\C} \} , T^*L)$ l’espace des fonctions propres $u : S \setminus \{y_1, \dots , y_{v_\C} \} \to T^* L$ ayant cette propriété. C’est une variété de Banach séparable. Soient à présent $v_\C^- \in \N^*$, $k_1, \dots , k_{v_\C} \in \N^*$ et $\gamma_1, \dots , \gamma_{v_\C^-}$une collection d’orbites disjointes du flot de Reeb, de sorte que lorsque $v_\C^-$ s’annule, $\Gamma = \{ \gamma_1, \dots , \gamma_{v^-_\C}\}$ soit vide. Soient $r_\C \in \N$, $z_1 , \dots , z_{r_\C} \in
S \setminus \{y_1, \dots , y_{v_\C} \} $ et $x_1 , \dots , x_{r_\C} \in T^* L$ une collection de points distincts, de sorte qu’à nouveau, lorsque $r_\C$ s’annule, $ \{ x_1, \dots , x_{r_\C}\}$ soit vide. Soit alors $${\cal P} (\Gamma) = \{ (u, J_S , J) \in L^{k,p}_d (S \setminus \{y_1, \dots , y_{v_\C} \} , T^*L) \times {\cal J_S} \times {\cal J_\lambda} \; \vert \; du + J \circ du \circ J_S = 0 , \; u (z_i) = x_i,$$ $$\lim_{z \to y_i^-} u(z) = k_i \gamma_i \text{ et } \lim_{s \to +\infty} \int (u \circ \phi_i)^* \lambda = k_i A \},$$ et ${\cal P}^* (\Gamma) \subset {\cal P} (\Gamma)$ le sous-espace des applications pseudo-holomorphes non-multiples. Soit ${\cal D}iff^+ (S , \underline{z} , \underline{y})$ le groupe des difféomorphismes de classe $C^{l+1}$ de $S$ qui préservent l’orientation et fixent $ \underline{z} , \underline{y}$. Ce groupe agit sur ${\cal P}^* (\Gamma)$ par $(\phi , (u , J_S , J)) \in {\cal D}iff^+ (S , \underline{z} , \underline{y}) \times {\cal P}^* (\Gamma) \mapsto (u \circ \phi^{-1} ,
\phi^* J_S , J) \in {\cal P}^* (\Gamma) $, où $\phi^* J_S = d \phi \circ J_S \circ d\phi^{-1}$. On note ${\cal M}_{r_\C}^{v_\C} (\Gamma , \underline{x})$ le quotient ${\cal P}^* (\Gamma) / {\cal D}iff^+ (S , \underline{z} , \underline{y})$ et $\pi : {\cal M}_{r_\C}^{v_\C} (\Gamma , \underline{x}) \to {\cal J}_\lambda $ la projection induite par $(u , J_S , J) \in {\cal P}^* (\Gamma) \to J \in {\cal J}_\lambda $
Fixons une métrique $g_L$ sur $T^* L$ préservée par $c_L$ et invariante par translation par $\rho$ pour $\rho$ assez grand. Elle induit une connexion $\nabla$ sur $TT^* L$ et tous les fibrés associés. Si $(u , J_S , J) \in {\cal P}^* (\Gamma)$, on note $D$ l’opérateur de Gromov $v \in L^{k,p}_d (S \setminus \{y_1, \dots , y_{v_\C} \} , T^*L)
\mapsto \nabla v + J \circ \nabla v \circ J_S + \nabla_v J \circ du \circ J_S \in L^{k-1,p}_d (S \setminus \{y_1, \dots , y_{v_\C} \} , \Lambda^{0,1} S \otimes u^*TT^*L)$. Ce dernier induit un opérateur $\overline{D} : L^{k,p}_d (S \setminus \{y_1, \dots , y_{v_\C} \} , u^*TT^*L) / du(L^{k,p}_d (S \setminus \{y_1, \dots , y_{v_\C} \} , TS)
\to L^{k-1,p}_d (S \setminus \{y_1, \dots , y_{v_\C} \} , \Lambda^{0,1} S \otimes u^*TT^*L) / du(L^{k-1,p}_d (S \setminus \{y_1, \dots , y_{v_\C} \} , \Lambda^{0,1} S \otimes TS))$ (voir la formule $1.5.1$ de [@IShev] et le paragraphe $1.4$ de [@Wels1]). On note $H^0_{\overline{D}} ( S ; {\cal N}_{u , -\underline{z}})$ (resp. $H^1_{\overline{D}} ( S ; {\cal N}_{u , -\underline{z}})$) le noyau (resp. conoyau) de $\overline{D}$.
\[propmod\] L’espace ${\cal M}_{r_\C}^{v_\C} (\Gamma , \underline{x})$ est une variété de Banach séparable de classe $C^{l-k}$ La projection $\pi$ est Fredholm, de noyau (resp. conoyau) isomorphe à $H^0_{\overline{D}} ( S ; {\cal N}_{u , -\underline{z}})$ (resp. $H^1_{\overline{D}} ( S ; {\cal N}_{u , -\underline{z}})$). En particulier, l’indice de $\pi$ vaut $2(n-1)\sum_{i=1}^{v_\C} k_i + 2v_\C - 2 - 2(n-1)r_\C - 2(n-1)v_\C^-$ si $L$ est une sphère, $(n-1)\sum_{i=1}^{v_\C} k_i + 2v_\C - 2 - 2(n-1)r_\C - 2(n-1)v_\C^-$ si $L$ est un plan projectif réel et $2v_\C + 2n - 6 - 2(n-1)r_\C - (n-1)v_\C^-$ si $L$ est un tore.
[**Démonstration :**]{}
La première partie de la Proposition \[propmod\] est classique. L’identification des noyau et conoyau de $\pi$ avec ceux de $ \overline{D}$ se démontre comme le Théorème $2$ de [@IShev]. Le caractère Fredhom de $D$ découle de la Proposition $5.2$ de [@Bour] (voir aussi le sixième paragraphe de [@HWZ]). Le calcul des indices découle des Propositions \[propmaslov\] et \[propcotangent\]. $\square$\
Supposons à présent que $v_\C = 2 v_\R$, $v_\C^- = 2 v_\R^-$ et $r_\C = r + 2r_L$. Pour $1 \leq i \leq v_\R$ (resp. $1 \leq j \leq v_\R^-$ ), on note $y_{v_\R + i } = \overline{y}_i$ (resp. $\gamma_{v_\R^- + j } = \overline{\gamma}_j$). De même, pour $1 \leq i \leq r_L$, on note $z_{r + r_L + i} = \overline{z}_{r+i}$ et $x_{r + r_L + i} = \overline{x}_{r+i}$. On suppose cette fois-ci que $x_1 , \dots , x_r \in L \subset T^*L$ et que pour $1 \leq i \leq r_L$ (resp. $1 \leq j \leq v_\R^-$ ), $\overline{x}_{r+i} = c_L (x_{r+i})$ (resp. $ \overline{\gamma}_j = -c_L (\gamma_j )$). On note ${\cal D}iff (S , \underline{z} , \underline{y})$ le groupe des difféomorphismes de classe $C^{l+1}$ de $S$ qui fixent $\underline{z}$ et $\underline{y}$ s’ils préservent l’orientation ou bien fixent $z_1 , \dots z_r$ et échangent $y_i$, $\overline{y}_i$ et $z_{r+j}$, $\overline{z}_{r+j}$ s’ils renversent l’orientation, $1 \leq i \leq v_\R$, $1 \leq j \leq r_L$. Sous les hypothèses que l’on vient de faire, cette extension d’indice deux de ${\cal D}iff^+ (S , \underline{z} , \underline{y})$ agit également sur ${\cal P}^* (\Gamma)$ par $(\phi , (u , J_S , J)) \in {\cal D}iff (S , \underline{z} , \underline{y}) \times {\cal P}^* (\Gamma) \mapsto (c_L \circ u \circ \phi^{-1} ,
\phi^* J_S , \overline{c}_L^* J) \in {\cal P}^* (\Gamma) $ lorsque $\phi \notin {\cal D}iff^+ (S , \underline{z} , \underline{y})$, où $\overline{c}_L^* J = -dc_L \circ J \circ dc_L$. Par suite, le quotient ${\cal M}_{r_\C}^{v_\C} (\Gamma , \underline{x}) = {\cal P}^* (\Gamma) / {\cal D}iff^+ (S , \underline{z} , \underline{y})$ se trouve à présent équipé d’une action de $\Z / 2\Z = {\cal D}iff (S , \underline{z} , \underline{y}) / {\cal D}iff^+ (S , \underline{z} , \underline{y})$, notée $c_{\cal M}$. La projection $\pi : ({\cal M}_{r_\C}^{v_\C} (\Gamma , \underline{x}) , c_{\cal M}) \to ({\cal J}_\lambda , \overline{c}_L^*) $ est alors $\Z / 2\Z $-équivariante. On note $\R {\cal M}_{(r , r_L)}^{v_\R} (\Gamma , \underline{x})$ le lieu fixe de $c_{\cal M}$ et $\pi_\R : \R {\cal M}_{(r , r_L)}^{v_\R} (\Gamma , \underline{x}) \to \R {\cal J}_\lambda$ la projection induite par $\pi$. De la même manière que dans [@Wels1], les seuls éléments de ${\cal D}iff (S , \underline{z} , \underline{y}) $ qui peuvent avoir des points fixes dans ${\cal P}^* (\Gamma) $ sont d’ordre deux et renversent l’orientation de $S$ (voir le Lemme $1.3$ de [@Wels1]) et l’opérateur $D$ est ${\cal D}iff (S , \underline{z} , \underline{y}) $-équivariant (Lemme $1.5$ de [@Wels1]). Si $c_S \in {\cal D}iff (S , \underline{z} , \underline{y}) $ est un tel élément d’ordre deux et $ (u , J_S , J) \in {\cal P}^* (\Gamma) $ un point fixe de $c_S$, on note $L^{k,p}_d (S \setminus \{y_1, \dots , y_{v_\C} \} , u^*TT^*L)_{+1}$, $L^{k,p}_d (S \setminus \{y_1, \dots , y_{v_\C} \} , TS)_{+1}$ et $ L^{k-1,p}_d (S \setminus \{y_1, \dots , y_{v_\C} \} , \Lambda^{0,1} S \otimes u^*TT^*L)_{+1}$, $ L^{k-1,p}_d (S \setminus \{y_1, \dots , y_{v_\C} \} , \Lambda^{0,1} S \otimes TS)_{+1}$ les espaces propres associés aux valeurs propres $+1$ de l’action de $c_S$ sur $L^{k,p}_d (S \setminus \{y_1, \dots , y_{v_\C} \} , u^*TT^*L)$, $L^{k,p}_d (S \setminus \{y_1, \dots , y_{v_\C} \} , TS)$ et $ L^{k-1,p}_d (S \setminus \{y_1, \dots , y_{v_\C} \} , \Lambda^{0,1} S \otimes u^*TT^*L)$, $ L^{k-1,p}_d (S \setminus \{y_1, \dots , y_{v_\C} \} , \Lambda^{0,1} S \otimes TS)$ respectivement. On note alors $\overline{D}_\R $ l’opérateur induit $L^{k,p}_d (S \setminus \{y_1, \dots , y_{v_\C} \} , u^*TT^*L)_{+1} / du(L^{k,p}_d (S \setminus \{y_1, \dots , y_{v_\C} \} , TS)_{+1})
\to L^{k-1,p}_d (S \setminus \{y_1, \dots , y_{v_\C} \} , \Lambda^{0,1} S \otimes u^*TT^*L)_{+1} / du( L^{k-1,p}_d (S \setminus \{y_1, \dots , y_{v_\C} \} , $ $\Lambda^{0,1} S \otimes TS)_{+1})$ et $H^0_{\overline{D}} ( S ; {\cal N}_{u , -\underline{z}})_{+1}$, $H^1_{\overline{D}} ( S ; {\cal N}_{u , -\underline{z}})_{+1}$ ses noyau et conoyau.
\[propmodreel\] L’espace $\R {\cal M}_{(r , r_L)}^{v_\R} (\Gamma , \underline{x})$ est une variété de Banach séparable de classe $C^{l-k}$. La projection $\pi_\R$ est Fredholm, de noyau (resp. conoyau) isomorphe à $H^0_{\overline{D}} ( S ; {\cal N}_{u , -\underline{z}})_{+1}$ (resp. $H^1_{\overline{D}} ( S ; {\cal N}_{u , -\underline{z}})_{+1}$). En particulier, l’indice de $\pi_\R$ vaut $2(n-1)\sum_{i=1}^{v_\R} k_i + 2v_\R - 1 - (n-1)r - 2(n-1)r_L - 2(n-1)v_\R^-$ si $L$ est une sphère, $(n-1)\sum_{i=1}^{v_\R} k_i + 2v_\R - 1 - (n-1)r - 2(n-1)r_L - 2(n-1)v_\R^-$ si $L$ est un plan projectif réel et $2v_\R + n - 3 - (n-1)r - 2(n-1)r_L - (n-1)v_\R^-$ si $L$ est un tore. $\square$
La démonstration de cette proposition est strictement analogue à celle de la Proposition $1.9$ de [@Wels1] et n’est pas reproduite ici.
### Définition des invariants {#subsubinvariants}
Nous allons compter les courbes $J$-holomorphes rationnelles réelles pointées d’énergie de Hofer finie proprement immergées dans $T^* L$ en fonction d’un signe $\pm 1$ de façon à obtenir un invariant associé à $T^* L$. Rappelons que d’après le Théorème $1.2$ de [@HWZ1] et d’après [@Bour], ces courbes rationnelles pointées convergent en leurs pointes vers des orbites de Reeb parcourues un nombre entier de fois, que l’on appelle multiplicité. La dimension de l’espace des modules de telles courbes a été calculée dans la Proposition \[propcotangent\] et dépend du nombre de pointes et des multiplicités associées. Afin d’obtenir un nombre fini de courbes, nous allons soumettre ces courbes à quelques contraintes, soit en les forçant à converger vers des orbites de Reeb prescrites, soit en les forçant à passer par des points de $L$ ou des paires de points complexes conjuguées de $T^* L \setminus L$. Soit $e_i$, $i \geq 1$, la suite d’entiers partout nulle sauf au $i$-ème rang où elle vaut un. Soient $\alpha = \sum_{i \in \Bbb{N}^*} \alpha_i e_i$ et $\beta = \sum_{i \in \Bbb{N}^*} \beta_i e_i$ deux suites d’entiers positifs qui s’annulent à partir d’un certain rang. Ces deux suites codent respectivement le nombre de paires d’orbites de Reeb complexes conjuguées limites prescrites et non prescrites de nos courbes, avec leur multiplicités $i \in \Bbb{N}^*$. Le nombre de pointes de nos courbes vaut donc $2v = 2 \sum_{i \in \Bbb{N}^*} (\alpha_i + \beta_i )$ et nous choisissons un ensemble $\Gamma$ de $\sum_{i \in \Bbb{N}^*} \alpha_i $ géodésiques fermées disjointes de $L$ pour prescrire nos paires d’orbites de Reeb limites. À présent, afin de fixer nos contraintes ponctuelles, soient $r \in \Bbb{N}$ et $x_1 , \dots , x_r$ des points distincts de $L$. De même, soient $r_L \in \Bbb{N}$ et $\xi_1, \overline{\xi}_1 , \dots ,
\xi_{r_L}, \overline{\xi}_{r_L}$ des paires distinctes de points complexes conjugués de $T^* L \setminus L$, c’est-à-dire satisfaisant $c_L (\xi_i) = \overline{\xi}_i$. Nous supposons que $$\label{dimsphere}
(n-1)r + 2(n-1)r_L + 2(n-1) \# \Gamma = 2 v + \epsilon (n-1) \sum_{i \in \Bbb{N}^*} i (\alpha_i + \beta_i ) + n - 3,$$ où $\epsilon = 2$ si $L$ est homéomorphe à une sphère et $\epsilon = 1$ si $L$ est homéomorphe à un espace projectif réel, tandis que nous supposons $$\label{dimtore}
(n-1)r + 2(n-1)r_L = 2 v + n - 3 \text{ et } \alpha = 0$$ si $L$ est homéomorphe à un tore.
Alors, lorsque la structure presque-complexe $J \in \Bbb{R} {\cal J}_\lambda$ est générique, il n’y a qu’un nombre fini de courbes $J$-holomorphes rationnelles réelles d’énergie de Hofer finie, proprement immergées dans $T^* L$ et ayant $2v$ pointes qui passent par $\underline{x}$, par chaque paire $\{ \xi_i ,
\overline{\xi}_i \}$ et qui convergent vers les orbites de Reeb relevant les éléments de $\Gamma$ ainsi que vers $\beta_j$ autres paires d’orbites, $j \in \Bbb{N}^*$, chacune avec multiplicité $j$ ou de classe d’homologie donnée si $L$ est un tore. En effet, si $L$ est un tore, il y a une infinité de géodésiques fermées primitives non homologues et la dimension (\[dimtore\]) ne dépend pas du choix des classes d’homologies de sorte qu’il y a une infinité d’espaces de modules ayant la même dimension. Pour garantir la finitude, nous imposons les classes d’homologies des orbites de Reeb limites. Notons ${\cal R} (\alpha , \beta , \Gamma , \underline{x} , \underline{\xi} , J)$ cet ensemble fini de courbes, la généricité de $J$ garantit qu’elles sont toutes immergées. Si $L$ est de dimension deux, on définit alors comme dans [@Wels1] la masse $m(C)$ d’une telle courbe $C$ comme le nombre fini de ses points doubles réels isolés, c’est-à-dire de ses points doubles situés sur $L$ et qui sont l’intersection transverses de deux branches complexes conjuguées. On pose $$F_{(r, r_L)} (\alpha , \beta , \Gamma , \underline{x} , \underline{\xi} , J) = \sum_{C \in {\cal R} (\alpha , \beta , \Gamma , \underline{x} , \underline{\xi} , J)} (-1)^{m(C)} \in \Bbb{Z}.$$ Si $L$ est de dimension trois, on l’équipe d’une structure spin. Ceci permet d’associer un état spinoriel $\text{sp} (C)$ à chaque courbe $C \in {\cal R} (\alpha , \beta , \Gamma , \underline{x} , \underline{\xi} , J)$ comme expliqué au paragraphe $5.2$ de [@Wels3]. Cet état spinoriel est défini comme suit. La linéarisation de l’équation de Cauchy-Riemann en $C$ fournit un opérateur surjectif de Cauchy-Riemann généralisé défini sur un espace de Banach de sections du fibré normal de $C$ à valeurs dans un espace de Banach de formes de type $(0,1)$ à valeurs dans le fibré normal de $C$. Si cet opérateur est $\C$-linéaire, il induit une structure de fibré vectoriel holomorphe sur le fibré normal de $C$ qui se décompose comme somme équilibrée de fibrés en droites complexes. Cette décomposition fournit un repère mobile le long de la partie réelle de $C$ qui permet de définir l’état spinoriel de $C$ comme l’obstruction à relever ce repère à un repère du fibré des spineurs, voir [@Wels2], [@Wels3]. Si cet opérateur surjectif de Cauchy-Riemann généralisé n’est que $\R$-linéaire, il peut être relié à un opérateur $\C$-linéaire par un chemin transverse à l’espace des opérateurs non-surjectifs. L’état spinoriel de $C$ est alors défini comme état spinoriel de l’opérateur $\C$-linéaire corrigé par la parité du nombre d’intersection du chemin choisi avec le mur des opérateurs non-surjectifs, voir le §$5.2$ de [@Wels3]. On pose alors $$F_{(r, r_L)} (\alpha , \beta , \Gamma , \underline{x} , \underline{\xi} , J) = \sum_{C \in {\cal R} (\alpha , \beta , \Gamma , \underline{x} , \underline{\xi} , J)} \text{sp} (C) \in \Bbb{Z}.$$
\[theoinv\] Soit $L$ une sphère, un tore ou un espace projectif réel de dimension $n=2$ ou $3$ muni d’une métrique à courbure constante. Soient $\alpha , \beta$ deux suites d’entiers positifs qui s’annulent à partir d’un certain rang. On choisit comme ci-dessus un ensemble $\Gamma$ de géodésiques fermées et des ensembles $\underline{x}$, $\underline{\xi}$ de $r$ et $r_L$ points dans $L$ et $T^* L \setminus L$ respectivement de sorte que ces nombres satisfassent (\[dimtore\]) dans le cas du tore et (\[dimsphere\]) sinon. Lorsque $n=3$, on suppose $r \neq 0$ et lorsque de plus $L \in \{ S^3 , \R P^3 \}$, on suppose que $J$ est invariante par le flot de Reeb pour $\rho \gg 1$. Alors, l’entier $F_{(r, r_L)} (\alpha , \beta , \Gamma , \underline{x} , \underline{\xi} , J)$ défini ci-dessus ne dépend ni du choix des contraintes $\Gamma , \underline{x} , \underline{\xi}$, ni du choix générique de la structure presque-complexe $J \in \Bbb{R} {\cal J}_\lambda$.
Nous n’utiliserons ce Théorème \[theoinv\] que dans le cas de la sphère et du plan projectif réel mais incluons toutefois le cas du tore ou de l’espace projectif de dimension trois puisque la démonstration est analogue. Remarquons que dans le cas de la sphère ou de l’espace projectif réel, $F_{(r, r_L)} (\alpha , \beta)$ n’est autre qu’un invariant réel analogue à celui défini dans [@Wels1], [@Wels2] relatif à la quadrique imaginaire pure si $L \in \{ \R P^2 , \R P^3 \}$ ou bien relatif à une section hyperplane réelle de la quadrique disjointe de l’ellipsoïde si $L \in \{ S^2 , S^3 \}$. L’existence d’un tel invariant relatif a été indépendamment observée par Cheol-Hyun Cho dans [@Cho]. Le lien entre la théorie des invariants relatifs et le point de vue de la théorie symplectique des champs est développé dans [@Katz].\
[**Démonstration :**]{}
On considère l’espace des modules $\R {\cal M}^v_{r, r_L} (\Gamma , \underline{x} , \underline{\xi})$ des sphères $J$-holomorphes réelles d’énergie de Hofer finie proprement immergées dans $T^* L$ ayant $2v$ pointes qui passent par $\underline{x}$, par chaque paire $\{ \xi_i ,\overline{\xi}_i \}$ et qui convergent vers les orbites de Reeb relevant les éléments de $\Gamma$ ainsi que vers $\beta_j$ autres paires d’orbites, $j \in \Bbb{N}^*$, chacune avec multiplicité $j$, voir le §\[subsubmodules\]. Soient $J_0$, $J_1 \in \R {\cal J}_\lambda$ deux structures presque complexes génériques de sorte que $F_{(r, r_L)} (\alpha , \beta , \Gamma , \underline{x} , \underline{\xi} , J_0)$ et $F_{(r, r_L)} (\alpha , \beta , \Gamma , \underline{x} , \underline{\xi} , J_1)$ soient bien définis. Soient $\gamma : [0,1] \to \Bbb{R} {\cal J}_\lambda$ une homotopie générique reliant $J_0$ à $J_1$, $\R {\cal M}_\gamma = \R {\cal M}^v_{r, r_L} (\Gamma , \underline{x} , \underline{\xi}) \times_\gamma [0,1] $ et $\pi_\gamma : \R {\cal M}_\gamma \to [0,1] $ la projection associée. Supposons pour commencer que $n=2$. Les seuls points à étudier sont l’absence de compacité de $\R {\cal M}_\gamma$ et les points critiques de $\pi_\gamma$. En effet, en dehors de ce nombre fini de valeurs de $[0,1]$, les seules autres valeurs $t$ où $F_{(r, r_L)} (\alpha , \beta , \Gamma , \underline{x} , \underline{\xi} , J_t)$ n’est pas défini correspondent à des courbes ayant un point triple réel ordinaire ou un point de tangence non-dégénéré, et comme dans [@Wels1], l’invariance de $F_{(r, r_L)} (\alpha , \beta , \Gamma , \underline{x} , \underline{\xi} , J_t)$ au passage de ces valeurs se vérifie facilement. Les points critiques de $\pi_\gamma $ correspondent aux courbes ayant un unique point de rebroussement de première espèce ordinaire. En effet, d’après la Proposition \[propmodreel\], en un tel point critique l’espace $H^0_{\overline{D}} ( S ; {\cal N}_{u , -\underline{z}})_{+1}$ ne s’annule pas. Or par définition, l’indice total des zéros d’une section du fibré normal à une courbe immergée vaut l’indice de Maslov de cette courbe. Une telle section doit par ailleurs s’annuler en $\underline{x} , \underline{\xi}$ et en les $v^-$ pointes prescrites, ce qui ne se peut pas. Par suite, le générateur de $H^0_{\overline{D}} ( S ; {\cal N}_{u , -\underline{z}})_{+1}$ est forcément de torsion, de sorte que la courbe n’est pas immergée. La généricité de $\gamma$ assure l’unicité du point de rebroussement et son caractère ordinaire. Ces courbes sont des points critiques non-dégénérés de $\pi_\gamma$ qui correspondent à l’apparition ou la disparition de deux courbes dont la masse diffère de un. Nous ne reproduisons pas la démonstration de ces deux faits ici puisqu’elle est strictement analogue à celle de [@Wels1]. L’invariance de $F_{(r, r_L)} (\alpha , \beta , \Gamma , \underline{x} , \underline{\xi} , J)$ au passage de telles valeurs critiques $t$ de $\pi_\gamma$ en découle. D’après le théorème de compacité en théorie symplectique des champs [@BEHWZ], l’absence de compacité de $\R {\cal M}_\gamma $ peut provenir de trois phénomènes, à savoir la dégénérescence d’une suite d’éléments de $\R {\cal M}_\gamma $ vers une courbe multiple, réductible ou à plusieurs étages. Supposons pour commencer qu’une telle suite dégénère vers un revêtement $l$-uple d’une courbe $C'$ de $T^*L$, $l \geq 2$. Alors, le nombre de pointes de $C'$ est inférieur à celui de $C$ et la somme des multiplicités associées est inférieure au $l^{\text{\`eme}}$ de celle de $C$. Par suite, dans le cas d’une sphère, l’espace des modules contenant $C'$ vient avec une projection Fredholm sur $\Bbb{R} {\cal J}_\lambda $ dont l’indice est majoré par celui de $C$ qui est nul moins le double de la somme des multiplicité des pointes. De telles courbes multiples ne peuvent apparaître en codimension un. Elles le peuvent dans le cas du plan projectif uniquement lorsque $C$ est un revêtement double d’un cylindre $C'$ sur des orbites de Reeb simples, ramifié en les pointes. Dans ce cas, $r$ vaut un ou trois et la courbe $C'$ a une partie réelle connexe, sans point double et non triviale dans $H_1 (\R P^2 ; \Z/2\Z)$. Le complémentaire de ces parties réelles est donc toujours connexe par arc et par suite le complémentaire dans $\Bbb{R} {\cal J}_\lambda $ des structures presque-complexes pour lesquelles une courbe multiple satisfait nos conditions d’incidence est lui aussi connexe par arc. Dans le cas du tore enfin, l’espace des modules contenant $C'$ vient avec une projection Fredholm sur $\Bbb{R} {\cal J}_\lambda $ dont l’indice est majoré par celui de $C$ qui est nul moins deux sauf si le nombre de pointes de $C'$ est le même que celui de $C$. Notant $v'$ ce nombre de pointes, la formule de Riemann-Hurwitz impose que l’indice total des points de ramification situés au dessus de ces pointes vaille $(l-1)v'$. Cet indice de ramification étant majoré par $2l-2$, cela force $C'$ et $C$ à être des cylindres et $F_{(r, r_L)} (\alpha , \beta , \Gamma , \underline{x} , \underline{\xi} , J) = 1$. Dans tous les cas, l’éventuelle dégénérescence vers des courbes multiples ne fait pas obstacle à l’invariance de $F_{(r, r_L)} (\alpha , \beta , \Gamma , \underline{x} , \underline{\xi} , J)$. Supposons à présent qu’une telle suite d’éléments de $\R {\cal M}_\gamma $ converge vers une courbe $J_{t_0}$-holomorphe réductible $C_{t_0}$. La généricité de $\gamma$ impose alors que cette courbe réductible possède deux composantes irréductibles, toutes deux réelles et que ses points singuliers soient des points doubles ordinaires. De plus, d’après ce que l’on vient de voir, ces composantes doivent être toutes deux simples lorsque $L$ est une sphère ou un plan projectif réel, mais peuvent être des revêtements de cylindres dans le cas du tore. Excluons ce dernier cas pour commencer. Il se produit alors le même phénomène que dans les variétés fermées, voir [@Wels1], à savoir que pour toute structure presque-complexe $J_t$ proche de $J_{t_0}$, à l’exclusion de $J_{t_0}$, et pour chaque point d’intersection réel $p$ entre les deux composantes irréductibles de $C_{t_0}$, il y a exactement une courbe $J_t$-holomorphe dans $T^* L$ satisfaisant nos conditions d’incidence. En effet, s’il y en avait deux, elles s’intersecteraient en chaque point de notre configuration $\underline{x} , \underline{\xi}$, en deux points au voisinage de chaque point double de $C_{t_0}$ autre que $p$ et en $2 i $ (resp. $2i - 2$) points au voisinage de chaque pointe convergeant vers une orbite de Reeb prescrite (resp. non prescrite) de multiplicité $i$, ce qui découle du Théorème $1.5$ de [@HWZ1]. D’après le Lemme \[lemmepointsdoubles\] et (\[dimsphere\]), cela ferait au total $ r + 2r_L + 2k^2 - 4k + 2 + 2k -2v + 2\# \Gamma = r + 2r_L + 2\# \Gamma + 2k^2 - 2k - 2v + 2 = 2k^2 + 1$ (resp. $ r + 2r_L + k^2 - 3k + 2 + 2k -2v + 2\# \Gamma = r + 2r_L + 2\# \Gamma + k^2 - k - 2v + 2 = k^2 + 1$) dans le cas de la sphère (resp. du plan projectif réel), ce qui ne se peut pas d’après la Remarque \[reminters\]. La contradiction à laquelle nous venons d’aboutir provient du fait que le nombre de points de notre configuration est strictement supérieur à l’indice de Maslov de la courbe $C_t$ calculé dans la Proposition \[propcotangent\]. Ceci vaut également lorsque $L$ est un tore, de sorte que nous aboutissons à la même conclusion pour peu qu’aucune des composantes de la courbe réductible ne soit multiple. Ainsi, dans tous ces cas, pour toute structure presque-complexe $J_t$ proche de $J_{t_0}$, à l’exclusion de $J_{t_0}$, et pour chaque point d’intersection réel $p$ entre les deux composantes irréductibles de $C_{t_0}$, il y a au plus une courbe $J_t$-holomorphe dans $T^* L$ satisfaisant nos conditions d’incidence. La démonstration du fait qu’il y a au moins une courbe $J_t$-holomorphe satisfaisant nos conditions d’incidence est la même que celle de la Proposition $2.14$ de [@Wels1] et nous ne reproduisons pas ici cet argument local. Si une des deux composantes de la courbe est multiple, on applique le cas précédent au recollé d’un voisinage de la composante simple avec un revêtement d’un voisinage du cylindre multiple, lequel recollé se projette sur un voisinage $C_{t_0}$ dans $T^* L$, pour aboutir à la même conclusion. On procède de même si les deux composantes sont multiples de sorte que dans tous les cas, l’éventuelle dégénérescence vers des courbes réductibles ne fait pas obstacle à l’invariance de $F_{(r, r_L)} (\alpha , \beta , \Gamma , \underline{x} , \underline{\xi} , J)$. Il reste à étudier la possibilité qu’une suite d’éléments de $\R {\cal M}_\gamma $ dégénère vers une courbe à plusieurs étages. Cela ne se produit pas lorsque le chemin $\gamma $ est choisi de façon suffisamment générale. En effet, soit $D_X$ une composante non-triviale d’un étage $\R \times S^* L$. Le nombre d’orbites de Reeb positives limites de $D_X$ comptées avec multiplicité moins le nombre d’orbite de Reeb négatives comptées avec multiplicité vaut au moins deux (resp. quatre) si $L$ est une sphère (resp. un plan projectif réel), ce qui découle de la positivité de l’aire $\int_{D_X} d\lambda$ et de l’isomorphisme $H_1 (S^* L ; \Z) \cong \Z / 2 \Z$ (resp. $H_1 (S^* L ; \Z) \cong \Z / 4 \Z$). De même, si $L$ est un tore, le nombre de pointes positives de $D_X$ est strictement plus grand que un. La courbe à plusieurs étages que l’on considère possède une composante réelle dans l’étage $T^* L$ que l’on note $D_L$. Soit $m$ le nombre de telles composantes non-triviales $D_X$ de $\R \times S^* L$ adjacentes à $D_L$. D’après les Propositions \[propmaslov\] et \[propcotangent\], le nombre d’asymptotes non-prescrites de $D_L$ est majoré par $v- \# \Gamma + m$. La dimension virtuelle de l’espace des modules contenant $D_L$ est donc majorée par $2 v + \epsilon k - 4m - 1 - r - 2r_L - 2 \# \Gamma + 2m$ (resp. $2(v-m) - 1 - r - 2r_L $) si $L$ est une sphère ou un plan projectif réel (resp. si $L$ est un tore). Dans tous les cas cette dimension est inférieure à $-2$, ce qu’il fallait démontrer. Ceci achève la démonstration du Théorème \[theoinv\] dans le cas où $n=2$. Lorsque $n=3$, le passage par un point critique ou la dégénérescence vers une courbe multiple ou réductible se traite à nouveau de la même manière que dans le cas absolu [@Wels2], [@Wels3]. Le seul phénomène nouveau à exclure est la dégénérescence vers une courbe à plusieurs étages. Lorsque $L$ est un tore, le nombre de pointes positives de chaque composante $D_X$ est à nouveau strictement plus grand que un ce qui force la dimension virtuelle de l’espace des modules contenant $D_L$ à être majorée par $-2$. Lorsque $L \in \{ S^3 , \R P^3 \}$, on a supposé que $J$ est invariante par le flot de Reeb pour $\rho \gg 1$, de sorte que les espaces de modules contenant chaque composante $D_X$ de $S^* L \times \R$ sont munis d’une action de $\C^*$. Ceci force la dimension virtuelle de l’espace des modules contenant $D_L$ à chuter de deux de sorte que cette dégénérescence en une courbe à plusieurs étages ne peut se produire en codimension un. $\square$
Lorsque $L \in \{ S^3 , \R P^3 \}$, le Théorème \[theoinv\] utilise une hypothèse qui n’apparaît pas en dimension deux, à savoir que la structure presque-complexe $J$ est invariante par le flot de Reeb pour $\rho \gg 1$. Cette hypothèse semble nécessaire en général pour la raison suivante. Munissons la symplectisation $\R \times S^* L$ d’une structure presque-complexe asymptotiquement cylindrique générique $J$. Pour tout $k$ strictement positif, cette symplectisation possède un cylindre $J$-holomorphe convergent positivement vers une orbite de Reeb revêtue $k+1$ fois (resp. $k+2$ fois) et convergeant négativement vers une orbite de Reeb revêtue $k$ fois si $L$ est une sphère (resp. un espace projectif réel) de dimension trois. La dimension attendue d’un tel cylindre vaut huit d’après la Proposition \[propcotangent\]. Fixons l’orbite de Reeb positive d’un tel cylindre. Sans l’hypothèse d’invariance de $J$ par le flot de Reeb, l’orbite de Reeb négative appartient à un espace de dimension trois d’orbites tandis qu’avec cette condition, elle n’appartient qu’à un espace de dimension deux, puisque l’espace des modules de tels cylindres est muni d’une action de $\R$ par translation dans le premier cas et de $\C^*$ dans le second, lesquelles actions préservent l’évaluation de l’orbite négative dans son espace de dimension quatre d’orbites de Reeb. Par conséquent, au-dessus d’un chemin générique de structures presque-complexes asymptotiquement cylindriques de $T^* L$, on ne peut éviter que les courbes $J$-holomorphes rigides d’énergie de Hofer finie ayant une orbite de Reeb prescrite de multiplicité plus grande que deux ou trois, selon que $L = S^3$ ou $\R P^3$, se brisent en courbes à deux étages dont l’étage supérieur possède un tel cylindre non-trivial ainsi que des cylindres triviaux. Le nombre de telles courbes n’est donc pas invariant. Il l’est si l’on se restreint aux structures invariantes par le flot de Reeb à l’infini.
### Quelques calculs {#subsubsectcalculs}
L’entier $F_{(r, r_L)} (\alpha , \beta , \Gamma , \underline{x} , \underline{\xi} , J)$ étant indépendant de $\Gamma , \underline{x} , \underline{\xi} , J$ d’après le Théorème \[theoinv\], nous le noterons $F_{(r, r_L)} (\alpha , \beta)$. Afin d’alléger encore cette notation, nous noterons cet entier $F (\alpha , \beta)$ lorsque $r_L = 0$, puisque la valeur de $r$ est alors définie sans ambiguïté par les calculs de dimensions (\[dimsphere\]) et (\[dimtore\]).
\[lemmecalc1\] Si $L$ est homéomorphe à une sphère de dimension deux et $r_L = 0$, on a $F (e_1 , 0) = F (0 , e_1) = 1$, $F (e_2 , 0) = 2$, $F (0 , e_2) = 8$, $F (2e_1 , 0) = 2$, $F (e_1 , e_1) = 4$ et $F (0 , 2e_1) = 6$.
[**Démonstration :**]{}
D’après le Lemme \[lemmepointsdoubles\], les cylindres asymptotes à des orbites de Reeb simples sont plongés et il ressort de la Remarque \[reminters\] que deux tels cylindres s’intersectent en deux points au maximum. Par suite, $F_{(3,0)} (0 , e_1) = 1$ et $F_{(1,0)} (e_1 , 0) = F_{(1,1)} (0 , e_1) = 1$. De même, $F_{(1,1)} (e_2 , 0) = 0$ puisqu’en faisant tendre la paire de points complexes conjugués vers l’infini les courbes devraient converger vers des courbes à deux étages non-triviales et n’ayant qu’une orbite double comme limite positive. De telles courbes à deux étages n’existent pas. Pour comparer $F_{(1,1)} (e_2 , 0) $ à $F_{(3,0)} (e_2 , 0)$, on procède comme au §$3$ de [@Wels1] en faisant tendre la paire de points complexes conjugués vers un point $y$ de $L$. Il n’y a comme précédemment qu’un seul cylindre asymptote à une orbite double, passant par un point $x$ de $L$ et ayant un point double en $y$. On montre de la même manière que le Théorème $3.2$ de [@Wels1] la relation $F_{(3,0)} (e_2 , 0) - F_{(1,1)} (e_2 , 0) = 2$, d’où on déduit $F_{(3,0)} (e_2 , 0) = 2$. On établit de même la relation $F_{(5,0)} (0 , e_2) = 2F_{(3 + \times ,0)} (0 , e_2) + F_{(3,1)} (0 , e_2) $ où $F_{(3 + \times ,0)} (0 , e_2) $ désigne le nombre algébrique de cylindres $J_L$-holomorphes réels de $T^* L$ asymptotes à une orbite de Reeb double non prescrite passant par trois points de $L$ et ayant son point double imposé en un quatrième point, voir le $\S 3.1$ de [@Wels1]. On a $F_{(3,1)} (0 , e_2) = 2 F_{(3,0)} (e_2 , 0) = 4$. De même, $ F_{(3 + \times ,0)} (0 , e_2) = F_{(1 + \times ,1)} (0 , e_2) =
2 F_{(1 + \times ,0)} (e_2, 0) = 2$. D’où finalement $F_{(5,0)} (0 , e_2) = 4 + 4 =8$. Enfin, $F_{(3,0)} (2e_1 , 0) = 2F_{(1 + \times ,0)} (2e_1 , 0) + F_{(1,1)} (2e_1 , 0) = 2 + 0 = 2 $, $F_{(5,0)} (e_1 , e_1) = 2 F_{(3 + \times ,0)} (e_1 , e_1) + F_{(3 , 1)} (e_1 , e_1) = 2 + F_{(3,0)} (2e_1 , 0) = 4$, $F_{(7,0)} (0, 2e_1) = 2F_{(5 + \times ,0)} (0, 2e_1) + F_{(5,1)} (0, 2e_1) =
2 + F_{(5,0)} (e_1 , e_1) = 6$. $\square$
\[lemmecalc2\] Si $L$ est homéomorphe à un plan projectif réel et $r_L = 0$, on a $F (e_1 , 0) = F (0 , e_1) = F (e_2 , 0) = F (2e_1 , 0) = F (e_1 , e_1) = F (0 , 2e_1) = 1$ et $F (0 , e_2 ) = 4$.
[**Démonstration :**]{}
D’après le Lemme \[lemmepointsdoubles\], les cylindres asymptotes à des orbites de Reeb simples sont plongés et d’après la Remarque \[reminters\], deux tels cylindres s’intersectent en un point au maximum. Par suite, $F_{(2,0)} (0 , e_1) = F_{(0,0)} (e_1 , 0) = 1$. De même, des cylindres asymptotes à des orbites de Reeb doubles ou des sphères ayant quatre pointes asymptotes à des orbites de Reeb simples s’intersectent en quatre points au maximum, de sorte que $F_{(1,0)} (e_2 , 0) = F_{(1,0)} (2e_1 , 0) = F_{(3,0)} (e_1 , e_1) = F _{(5,0)} (0 , 2e_1) = 1$. Enfin, $F_{(1,1)} (0 , e_2 ) = 2F_{(1,0)} (e_2 , 0 ) = 2$. Lorsque l’on fait converger la paire de points complexes conjugués vers un point $y$ de $L$, les deux courbes réelles comptées par $F_{(1,1)} (0 , e_2 )$ convergent vers deux courbes réelles ayant une tangente prescrite en $y$. En effet, ces dernières ne peuvent converger vers des courbes ayant un point double en $y$ d’après le Lemme \[lemmepointsdoubles\] et le revêtement double du cylindre sur une orbite simple passant par $y$ ne peut se déformer en un cylindre interpolant une paire de points complexes conjugués. Par contre, ce cylindre double se déforme en au moins un cylindre asymptote à une paire d’orbites doubles et passant par trois points de $L$, de sorte que $F (0 , e_2 ) \geq 4$. Or, le nombre de cylindres complexes de $T^* L$ asymptotes à deux orbites doubles et passant par trois points vaut quatre, ce que l’on obtient en faisant tendre deux points vers l’infini. D’où le résultat. $\square$
\[lemmecalc3\] Si $L$ est homéomorphe à un plan projectif réel et $r_L = 0$, on a $F (e_3 , 0) = 2$, $F (0, e_3) = 12$, $F (e_1 + e_2 , 0) = 2$, $F (e_1 , e_2) = 8$, $F (e_2 , e_1) = 4$, $F (0 , e_1 + e_2) = 24$, $F (3e_1 , 0) = 2$, $F (2e_1 , e_1) = 4$, $F (e_1 , 2e_1) = 6$ et $F (0 , 3e_1) = 8$.
[**Démonstration :**]{}
On procède comme dans la démonstration du Lemme \[lemmecalc1\]. On obtient avec les mêmes notations $F_{(2,0)} (e_3 , 0) = 2 F_{( \times ,0)} (e_3 , 0) +
F_{(0, 1)} (e_3 , 0) = 2 + 0 = 2$. De même, $F_{(4,0)} (0, e_3) = 2 F_{( 2 + \times ,0)} (0 , e_3 ) + F_{(2, 1)} (0 , e_3) = 2 F_{( 2 + \times ,0)} (0 , e_3 ) + 3F_{(2, 0)} ( e_3 , 0) = 2 F_{( 2 + \times ,0)} (0 , e_3 ) + 6$ et $F_{( 2 + \times ,0)} (0 , e_3 ) = F_{( \times ,1)} (0 , e_3 ) = 3 F_{( \times ,0)} (e_3 , 0 ) = 3$, de sorte que $F_{(4,0)} (0, e_3) = 12$. De même, $F_{(2,0)} (e_1 + e_2 , 0) =
2 F_{(\times,0)} (e_1 + e_2 , 0) + F_{(0 , 1)} (e_1 + e_2 , 0) = 2 + 0 = 2$ ; $F_{(4,0)} (e_1, e_2 ) =
2 F_{(2 + \times,0)} (e_1 , e_2) + F_{(2 , 1)} (e_1 , e_2 ) = 2 F_{( \times,1)} (e_1 , e_2) + 2 F_{(2 , 0)} (e_1 + e_2 , 0 ) = 4 F_{( \times,0)} (e_1 + e_2 , 0) + 4 = 8$ ; $F_{(4,0)} (e_2, e_1 ) = 2 F_{(2 + \times,0)} (e_2 , e_1) + F_{(2 , 1)} (e_2 , e_1) = 2 + F_{(2 , 0)} (e_1 + e_2 , 0 ) = 4$ ; $F_{(6,0)} (0 , e_1 + e_2) = 2 F_{(4 + \times,0)} (0 , e_1 + e_2)
+ F_{(4, 1)} (0 , e_1 + e_2) = 2 F_{(2 + \times,1)} (0 , e_1 + e_2) + F_{(4, 0)} (e_1 , e_2) + 2F_{(4, 0)} (e_2 , e_1) = 2 F_{(2 + \times,0)} (e_1 , e_2) + 4 F_{(2 + \times,0)} (e_2 , e_1) +
8 + 8 = 4 + 4 + 16 = 24$. Enfin, $F_{(2,0)} (3e_1 , 0) = 2 F_{(\times,0)} (3e_1 , 0) + F_{(0,1)} (3e_1 , 0) = 2 + 0 = 2$ ; $F_{(4,0)} (2e_1 , e_1) = 2 F_{(2 + \times,0)} (2e_1 , e_1) + F_{(2,1)} (2e_1 , e_1)
= 2 + F_{(2,0)} (3e_1 , 0) = 4$ ; $F_{(6,0)} (e_1 , 2e_1) = 2 F_{(4 + \times,0)} (e_1 , 2e_1) + F_{(4,1)} (e_1 , 2e_1) = 2 + F_{(4,0)} (2e_1 , e_1) = 6$ ; $F_{(8,0)} (0 , 3e_1) = 2 F_{(6 + \times,0)} (0 , 3e_1) + F_{(6,1)} (0 , 3e_1) = 2 + F_{(6,0)} (e_1 , 2e_1) = 8$. $\square$
Calculs dans le plan projectif complexe
---------------------------------------
### Arbres projectifs
Soient $r, r_X$ et $d$ trois entiers naturels satisfaisant la relation $r + 2r_X = 3d - 1$.
Un arbre projectif est un arbre fini connexe dont toutes les arêtes sont étiquetées par des entiers strictement positifs. De plus, un tel arbre possède une racine $s_0$ et tous les sommets à distance paire de $s_0$ sont soit monovalents connectés à une arête double, soit bivalents connectés à deux arêtes simples.
On note ${\cal B}_r$ l’ensemble des arbres projectifs qui satisfont $$\sum_{a \in {\cal A}(s_0)} k(a) - 1 \leq r \leq \sum_{a \in {\cal A}(s_0)} k(a) - 1 + 2v (s_0) \text{ et } r = \sum_{a \in {\cal A}(s_0)} k(a) - 1 \mod (2),$$ où $v(s)$ désigne la valence d’un sommet $s$, $ {\cal A}(s)$ l’ensemble des arêtes adjacentes à $s$ et $k (a)$ désigne la multiplicité de l’arête $a$. Notons également $k_s$ la somme des multiplicités des arêtes adjacentes au sommet $s$ et $k$ la multiplicité totale de toutes les arêtes de l’arbre. On pose $r_L (s_0) = \frac{1}{2} \big( \sum_{a \in {\cal A}(s_0)} k(a) - 1 + 2v (s_0) - r \big)$, de sorte que $0 \leq r_L (s_0) \leq v (s_0)$. Enfin, on note $S_1$ (resp. $S_2$) l’ensemble des sommets à distance impaire (resp. paire) de $s_0$.
Un arbre projectif décoré est un arbre projectif $A \in {\cal B}_r$ équipé d’une partition $S_1^+ \sqcup S_1^-$ de l’ensemble des sommets adjacents à $s_0$ telle que $\# S_1^- = r_L (s_0)$ et $\# S_1^+ = v(s_0) - r_L (s_0)$. Cet arbre est de plus équipé des fonctions :
- $f_A : S_1 \to {\cal P} (\{1, \dots , r_X \})$ satisfaisant $f_A (s) \cap f_A (s') = \emptyset$ dès que $s \neq s'$ et $\cup_{s \in S_1} f_A (s) = \{1, \dots , r_X \}$.
- $g_A : S_1 \to \N$ telle que $k + 4\sum_{s \in S_1} g_A (s) = d$ et pour tout $s \in S_1^+$ (resp. $s \in S_1^-$ ), $6g_A (s) + k_s + v(s) - 1 = \# f_A (s) + 1$ (resp. $6g_A (s) + k_s + v(s) - 1 = \# f_A (s)$).
On note ${\cal B}_r^d$ l’ensemble fini des arbres projectifs décorés. Soit $A \in {\cal B}_r^d$, on pose $m_1^- (A) = \prod_{s \in S_1^-} \#{\{ a \in {\cal A}(s)} \; \vert \; k(a) = k(ss_0)\}$, où $k(ss_0)$ désigne la multiplicité de l’arête reliant $s$ à $s_0$. On note de même $m_1^+ (A)$ le nombre d’injections $\phi : \{s \in S_1^+ \; \vert \; f_A (s) \neq \emptyset \} \to {\cal A}^+ (s_0)$ satisfaisant $k (\phi (s)) = k(ss_0)$ pour tous les sommets $s \in S_1^+$, où ${\cal A}^+ (s_0)$ désigne l’ensemble des arêtes reliant $s_0$ à un sommet de $S_1^+$. Notons enfin $S_2^b$ l’ensemble des sommets bivalents de $S_2 \setminus \{ s_0 \}$. L’arbre $A$ privé de ces sommets bivalents n’est pas connexe. On note $m_2 (A)$ le nombre de façons de reconnecter $A \setminus S_2^b$ de manière à obtenir un arbre isomorphe à $A$. Posons $$\text{mult} (A) = 2^{\sum_{s \in S_1^+} \# f_A (s) + \sum_{s \in S_1 \setminus S_1^+} \max (\# f_A (s) - 1 , 0) + \#S_2^b} m_1^+ (A) m_1^- (A) m_2 (A) \prod_a k(a).$$ C’est la [*multiplicité*]{} de l’arbre $A \in {\cal B}_r^d$.
\[figarbresprojectifs\]
### Relation avec l’invariant relatif
Soit $\Sigma_4$ la surface rationnelle réglée de degré quatre, $e$ la classe d’une section holomorphe d’autointersection quatre et $f$ la classe d’une fibre. Étant donnés $a,b \in \N$ et $\alpha, \beta$ des suites d’entiers positifs, on note $N_4^{ae+bf} (\alpha, \beta)$ le nombre de courbes rationnelles de $\Sigma_4$, homologues à $ae+bf$, ayant $\alpha_i + \beta_i$ points de tangence d’ordre $i$ avec la section exceptionnelle de $\Sigma_4$ parmi lesquels $\alpha_i $ sont prescrits et qui passent par le nombre adéquat de points fixés.
\[theocalcproj\] Soit $(X, \omega , c_X)$ une variété symplectomorphe au plan projectif complexe et $r, r_X, d \in \N$ satisfaisant la relation $r + 2r_X = 3d - 1$. Alors, $$\chi^d_r = \sum_{A \in {\cal B}_r^d} (-1)^{\# S_2 + 1} \text{mult} (A) F_{(r,0)} (\alpha_A^-, \beta_A^+) \prod_{s \in S_1 \setminus S_1^+} N_4^{g_A (s) e + k(s) f} (0, \beta_A)
\prod_{s \in S_1^+} N_4^{g_A (s) e + k(s) f} (e_{k(ss_0)} , \beta_A^0),$$ où $(\alpha_A^-)_i$ (resp. $(\beta_A^+)_i$) vaut le nombre d’arêtes de multiplicité $i$ reliant $S_1^-$ (resp. $S_1^+$) à $s_0$, $( \beta_A)_i$ (resp. $( \beta_A^0)_i$) vaut le nombre d’arêtes de multiplicité $i$ adjacentes à $s$ (resp. moins un si l’arête reliant $s$ à $s_0$ est de multiplicité $i$) et $F_{(r,0)} (\alpha_A^-, \beta_A^+)$ désigne l’invariant défini dans le fibré cotangent du plan projectif réel au §\[subsubinvariants\].
Un algorithme permettant le calcul de $\chi^d_{3d-1}$ a déjà été proposé par G. Mikhalkin dans [@Mikh] et étendu par E. Shustin dans [@Shu] aux autres invariants $\chi^d_r$, $0 \leq r \leq 3d-1$. Par ailleurs, I. Itenberg, V. Kharlamov et E. Shustin [@IKS2] ont adapté la formule de récurrence [@Gath] au cas réel et ainsi déduit une formule calculant l’invariant $\chi^d_{3d-1}$ en fonction d’invariants relatifs tropicaux dans les surfaces de Del Pezzo toriques réelles. J’avais introduit des invariants relatifs réels par rapport à une courbe réelle ayant une partie réelle non-vide dans [@Wels5]. Les formules apparaissant dans les Théorèmes \[theocalcproj\], \[theocal2spher\] et \[theocal3spher\] calculent $\chi^d_r$ en fonction d’invariants relatifs à un diviseur réel ayant une partie réelle vide ; il serait intéressant de calculer ces invariants $F_{(r, r_L)} (\alpha , \beta )$ et de déterminer exactement dans quelles situations l’expression de $\chi^d_r$ en fonction de tels invariants relatifs à un diviseur réel de partie réelle vide permet le calcul effectif de $\chi^d_r$. Lorsque $r \leq 2$, ces invariants $F_{(r, r_L)} (\alpha , \beta )$ sont calculés au §\[subsubsectcalculs\] tandis que $N_4^{ae+bf} (\alpha, \beta)$ est calculé dans [@Vak], de sorte que le Théorème \[theocalcproj\] permet le calcul effectif de $\chi^d_r$, voir le Corollaire \[corcalcproj\] pour les premières valeurs de cet invariant.
[**Démonstration du Théorème \[theocalcproj\] :**]{}
On poursuit la stratégie générale énoncée au $\S$ \[subsubsectstrat\] en allongeant le cou d’une structure presque complexe générique jusqu’à briser la variété en deux morceaux $T^* \R P^2$ et $\C P^2 \setminus \R P^2$. Les courbes $J$-holomorphes rationnelles réelles comptées par $\chi^d_r$ se brisent en courbes à deux étages interpolant $r$ points de $\R P^2$ et $r_X$ paires de points complexes conjugués de $\C P^2 \setminus \R P^2$. Il est apparu au cours de la démonstration du Théorème \[theocong3\] que ces courbes à deux étages sont codées par les arbres projectifs décorés $A \in {\cal B}_r^d$. Il s’agit donc de dénombrer les courbes à deux étages qui sont codées par un arbre donné $A \in {\cal B}_r^d$, puis de dénombrer les courbes $J$-holomorphes rationnelles réelles comptées par $\chi^d_r$ qui dégénèrent sur une courbe à deux étages donnée. Le nombre de façons de répartir les points complexes conjugués parmi les composantes de la courbe à deux étages qui se trouvent dans $\C P^2 \setminus \R P^2$ a été calculé dans la démonstration du Théorème \[theocong3\] et vaut $2^{\sum_{s \in S_1^+} \# f_A (s) + \sum_{s \in S_1 \setminus S_1^+} \max (\# f_A (s) - 1 , 0)}$. Les composantes codées par $S_1 \setminus S_1^+$ sont rigides avec leurs conditions d’incidence, il y en a $\prod_{s \in S_1 \setminus S_1^+} N_4^{g_A (s) e + k(s) f} (0, \beta_A)$. Puis, il y a $m_1^- (A)$ façons de choisir les orbites de Reeb prescrites de la courbe codée par $s_0$. Le nombre de courbes réelles codées par $s_0$ satisfaisant nos conditions d’incidence et comptées avec signe vaut $F_{(r,0)} (\alpha_A^-, \beta_A^+)$. Il y a alors $m_1^+ (A)$ façons de choisir la manière de connecter les courbes codées par $S_1^+$ aux orbites de Reeb restées libres de la courbe codée par $s_0$. Il y a enfin $2^{ \#S_2^b} m_2(A)$ façons de connecter ces composantes entre elles par des paires de cylindres complexes conjugués de $T^* \R P^2$ codés par les sommets bivalents de $S_2^b$. Ceci fournit $2^{\sum_{s \in S_1^+} \# f_A (s) + \sum_{s \in S_1 \setminus S_1^+} \max (\# f_A (s) - 1 , 0) + \#S_2^b} m_1^+ m_1^- m_2 F_{(r,0)} (\alpha_A^-, \beta_A^+)$ $\prod_{s \in S_1
\setminus S_1^+} N_4^{g_A (s) e + k(s) f} (0, \beta_A) \prod_{s \in S_1^+} N_4^{g_A (s) e + k(s) f} (k(ss_0) , \beta_A^0)$ courbes codées par un arbre donné $A \in {\cal B}_r^d$. Or, d’après le théorème de recollement de théorie symplectique des champs [@Bour], il y a $ \prod_a k(a)$ courbes $J$-holomorphes rationnelles réelles comptées par $\chi^d_r$ qui dégénèrent sur une courbe à deux étages donnée. Le résultat découle à présent du fait que chaque courbe codée par $S_2 \setminus \{ s_0 \}$ intersecte $\R P^2$ en un point et contribue donc à la masse des courbes $J$-holomorphes rationnelles réelles en question, d’où le signe $(-1)^{\# S_2 + 1}$. $\square$
\[corcalcproj\] Soit $(X, \omega , c_X)$ une variété symplectomorphe au plan projectif complexe. Alors, $\chi^4 (T) = o(T^2)$, $\chi^5 (T) = 64 + 64T^2 + o(T^3)$, $\chi^6 (T) = 1024T + 1536T^3 + o(T^4)$, $\chi^7 (T) = -14336 + 11776T^2 + o(T^3)$ et $\chi^8 (T) = -280576T + o(T^2)$.
Les valeurs $\chi^4_1 = 0$, $\chi^5_0 = \chi^5_2 = 64$ ont déjà été obtenues dans [@IKS1] à l’aide d’un ordinateur et de l’algorithme [@Shu].
[**Démonstration du Corollaire \[corcalcproj\] :**]{}
L’annulation de $\chi^4_1$ tient au fait que l’ensemble d’arbres $B^4_1$ est vide. Les arbres intervenant dans la démonstration de ce Corollaire \[corcalcproj\] sont représentés dans la Figure \[figarbresprojectifs\]. Lorsque $d=5$ et $r \leq 2$, un seul arbre décoré intervient. Le Théorème \[theocalcproj\] fournit $\chi^5_0 = 2^6 F (e_1 , 0) N^{e+f} (0 , e_1) = 64$ et $\chi^5_2 = 2^6 F (0 , e_1) N^{e+f} (e_1 , 0) = 64$. Lorsque $d=6$ et $r=1$, deux arbres projectifs décorés interviennent qui sont représentés par la Figure \[figarbresprojectifs\]. La contribution du premier vaut $C_8^1 2^6 F (2e_1 , 0) N^{e+f} (0 , e_1) = 512$ et celle du second donné par cette figure vaut $2^7 2 F (e_2 , 0) N^{e+2f} (0 , e_2) = 2^9
N^{e+2f} (e_2 , 0) = 512$, voir le Lemme \[lemmecalc2\], de sorte que $\chi^6_1 = 1024$. Lorsque $d=6$ et $r=3$, les arbres intervenant sont les mêmes. Toutefois, la fonction $f_A$ du premier arbre peut affecter soit six, soit sept paires de points complexes conjugués au sommet $e+f$. Dans le premier cas, l’arbre contribue à hauteur de $C_7^1 2^6 F (e_1 , e_1) N^f (0 , e_1) N^{e+f} (e_1 , 0) = 448$ ; dans le second, il contribue à hauteur de $2^6 F (e_1 , e_1) N^f (e_1 , 0) N^{e+f} (0 , e_1) = 64$, soit une contribution totale de $512$. La contribution du second arbre vaut $2^7 2 F (0 , e_2) N^{e+2f} (e_2 , 0) = 2^{10}$, de sorte que $\chi^6_3 = 1536$. Lorsque $d=7$ et $r=0$, deux arbres contribuent. Le premier donné par la Figure \[figarbresprojectifs\] contribue à hauteur de $- C_{10}^1 2^9 2 F (e_1 , 0) N^{e+2f} (0 , 2e_1) = -10240$ tandis que le second contribue à hauteur de $- 2^9 2 F (e_1 , 0) N^{e+3f} (0 , e_1 + e_2)$. Or $N^{e+3f} (0 , e_1 + e_2) = N^{e+3f} (e_1 , e_2) + 2N^{e+3f} (e_2 , e_1) = 4 N^{e+3f} (e_1 + e_2 , 0) = 4$, de sorte que finalement $\chi^7_0 = -10240 - 4096 = -14336$. Lorsque $d=7$ et $r=2$, cinq arbres contribuent. La contribution du premier arbre donné par la Figure \[figarbresprojectifs\] vaut $C_9^2 2^6 F (3e_1 , 0) N^{e+f} (0 , e_1) = 4608$. La contribution du deuxième arbre vaut $C_9^1 2^7 2 F (e_1 + e_2 , 0) N^{e+2f} (0 , e_2) =
9 2^9 2 N^{e+2f} (e_2 , 0) = 9216$. La contribution du troisième arbre vaut $2^8 3 F (e_3 , 0) N^{e+3f} (0 , e_3) = 9 2^9 N^{e+3f} (e_3 , 0) = 4608$. La contribution du quatrième arbre vaut $-C_9^1 2^9 F (0 , e_1) N^{e+2f} (e_1 , e_1) N^f (0 , e_1) = -4608$. Enfin, la contribution du cinquième arbre vaut $-2^9 2 F (0 ,e_1) N^{e+3f} (e_1 , e_2) = -2^{11} N^{e+3f} (e_1 + e_2 , 0) = -2048$. On en déduit $\chi^7_2 = 4608 + 9216 + 4608 - 4608 - 2048 = 11776$. Lorsque $d=8$ et $r=1$, quatre arbres projectifs décorés contribuent. La contribution du premier arbre donné par la Figure \[figarbresprojectifs\] vaut $- 2 C_{11}^2 2^9 2 F (2e_1 , 0)
N^{e+2f} (0 , 2e_1) N^f (0 , e_1)^2 = -112640$. La contribution du deuxième arbre vaut $-C_{11}^1 2^9 2 F (2e_1 , 0) N^{e+3f} (0 , e_1 + e_2) N^f (0 , e_1) = -45056$. La contribution du troisième arbre vaut $-C_{11}^1 2^{10} 2 F (e_2 , 0) N^{e+3f} (0 , e_1 + e_2) N^f (0 , e_1) = -90112$. Enfin, la contribution du quatrième arbre vaut $-2 2^{10} 2^2 F (e_2 , 0) N^{e+4f} (0 , 2e_2) = -2^{15} N^{e+4f} (2e_2 , 0)
= -32768$. On en conclut $\chi^8_1 = -112640 - 45056 - 90112 - 32768 = -280576$. $\square$
Calculs dans l’ellipsoïde de dimension deux
-------------------------------------------
### Arbres deux-sphériques
Soient $r, r_X$ et $d$ trois entiers naturels satisfaisant la relation $r + 2r_X = 4d - 1$, laquelle impose que $r$ soit impair.
Un arbre deux-sphérique est un arbre fini connexe dont toutes les arêtes sont étiquetées par des entiers strictement positifs. De plus, un tel arbre possède une racine $s_0$ et tous les sommets à distance paire de $s_0$ sont monovalents connectés à une arête simple.
En particulier, la distance d’un sommet à $s_0$ est majorée par deux et seules les arêtes connectées à $s_0$ peuvent avoir une multiplicité non-triviale. On note ${\cal A}_r$ l’ensemble des arbres deux-sphériques qui satisfont $$2\sum_{a \in {\cal A}(s_0)} k(a) - 1 \leq r \leq 2\sum_{a \in {\cal A}(s_0)} k(a) - 1 + 2v (s_0),$$ où $v(s)$ désigne la valence d’un sommet $s$, $ {\cal A}(s)$ l’ensemble des arêtes adjacentes à $s$ et $k (a)$ désigne la multiplicité de l’arête $a$. Notons également $k_s$ la somme des multiplicités des arêtes adjacentes au sommet $s$ et $k$ la multiplicité totale de toutes les arêtes de l’arbre. On pose $r_L (s_0) = \frac{1}{2} \big( 2\sum_{a \in {\cal A}(s_0)} k(a) - 1 + 2v (s_0) - r \big)$, de sorte que $0 \leq r_L (s_0) \leq v (s_0)$. Enfin, on note $S_1$ (resp. $S_2$) l’ensemble des sommets à distance impaire (resp. paire) de $s_0$.
Un arbre deux-sphérique décoré est un arbre deux-sphérique $A \in {\cal A}_r$ équipé d’une partition $S_1^+ \sqcup S_1^-$ de l’ensemble des sommets adjacents à $s_0$ telle que $\# S_1^- = r_L (s_0)$ et $\# S_1^+ = v(s_0) - r_L (s_0)$. Cet arbre est de plus équipé des fonctions :
- $f_A : S_1 \to {\cal P} (\{1, \dots , r_X \})$ satisfaisant $f_A (s) \cap f_A (s') = \emptyset$ dès que $s \neq s'$ et $\cup_{s \in S_1} f_A (s) = \{1, \dots , r_X \}$.
- $g_A : S_1 \to \N$ telle que $k + 2\sum_{s \in S_1} g_A (s) = d$ et pour tout $s \in S_1^+$ (resp. $s \in S_1^-$ ), $4g_A (s) + k_s + v(s) - 1 = \# f_A (s) + 1$ (resp. $4g_A (s) + k_s + v(s) - 1 = \# f_A (s)$).
On note ${\cal A}_r^d$ l’ensemble des arbres deux-sphériques décorés, c’est un ensemble fini. Soit $A \in {\cal A}_r^d$, on pose $m_1^- (A) = \prod_{s \in S_1^-} \#{\{ a \in {\cal A}(s)} \; \vert \; k(a) = k(ss_0)\}$, où $k(ss_0)$ désigne la multiplicité de l’arête reliant $s$ à $s_0$, de sorte que chaque terme du produit vaille un ou $v(s)$. On note de même $m_1^+ (A)$ le nombre d’injections $\phi : \{s \in S_1^+ \; \vert \; f_A (s) \neq \emptyset \} \to {\cal A}^+ (s_0)$ satisfaisant $k (\phi (s)) = k(ss_0)$ pour tous les sommets $s \in S_1^+$. On pose alors $$\text{mult} (A) = 2^{\sum_{s \in S_1^+} \# f_A (s) + \sum_{s \in S_1 \setminus S_1^+} \max (\# f_A (s) - 1 , 0) + \#S_2 -1} m_1^+ (A) m_1^- (A) \prod_a k(a),$$ c’est la [*multiplicité*]{} de l’arbre $A \in {\cal A}_r^d$.
\[figarbres2spheriques\]
### Relation avec l’invariant relatif
Soit $\Sigma_2$ la surface rationnelle réglée de degré deux, $e$ la classe d’une section holomorphe d’autointersection deux et $f$ la classe d’une fibre. Étant donnés $a,b \in \N$ et $\alpha, \beta$ des suites d’entiers positifs, on note $N^{ae+bf}_2 (\alpha, \beta)$ le nombre de courbes rationnelles de $\Sigma_2$, homologues à $ae+bf$, ayant $\alpha_i + \beta_i$ points de tangence d’ordre $i$ avec la section exceptionnelle de $\Sigma_2$ parmi lesquels $\alpha_i $ sont prescrits et qui passent par le nombre adéquat de points fixés.
\[theocal2spher\] Soit $(X, \omega , c_X)$ une variété symplectomorphe à la quadrique ellipsoïde de dimension deux, $r, r_X, d \in \N$ satisfaisant la relation $r + 2r_X = 4d - 1$ et $h$ la classe d’une section plane réelle de bidegré $(1,1)$. Alors, $$\chi^{dh}_r = \sum_{A \in {\cal A}_r^d} (-1)^{\# S_2 + 1} \text{mult} (A) F_{(r,0)} (\alpha_A^-, \beta_A^+) \prod_{s \in S_1 \setminus S_1^+} N_2^{g_A (s) e + k(s) f} (0, \beta_A)
\prod_{s \in S_1^+} N_2^{g_A (s) e + k(s) f} (e_{k(ss_0)} , \beta_A^0),$$ où $(\alpha_A^-)_i$ (resp. $(\beta_A^+)_i$) vaut le nombre d’arêtes de multiplicité $i$ reliant $S_1^-$ (resp. $S_1^+$) à $s_0$, $( \beta_A)_i$ (resp. $( \beta_A^0)_i$) vaut le nombre d’arêtes de multiplicité $i$ adjacentes à $s$ (resp. moins un si l’arête reliant $s$ à $s_0$ est de multiplicité $i$) et $F_{(r,0)} (\alpha_A^-, \beta_A^+)$ désigne l’invariant défini dans le fibré cotangent de la sphère de dimension deux au §\[subsubinvariants\].
Un algorithme permettant le calcul de $\chi^d_r$, $1 \leq r \leq 4d-1$, est proposé par E. Shustin dans [@Shu1]. Remarquons que cet invariant $\chi^d_r$ peut se définir purement en termes de fractions rationnelles complexes. Lorsque $r = 4d-1$ par exemple, il compte algébriquement le nombre de fractions rationnelles $u = P/Q$, $P, Q \in \C [X]$ de degrés $d$, modulo reparamétrage par les homographies réelles de $PGL_2 (\R)$, telles que l’image $u(\R P^1)$ interpole un ensemble donné générique de $4d-1$ points de la sphère de Riemann. Le signe en fonction duquel il convient de compter ces fractions rationnelles $u$ est pair si $u$ possède un nombre pair de points critiques dans chaque hémisphère $\C P^1 \setminus \R P^1$ et impair sinon. Il serait intéressant d’étudier cet invariant à l’aide de la théorie des fractions rationnelles (un problème que j’avais proposé au MSRI au printemps $2004$ lors d’une séance de problèmes ouverts).
[**Démonstration du Théorème \[theocal2spher\] :**]{}
On poursuit la stratégie générale énoncée au $\S$ \[subsubsectstrat\] en allongeant le cou d’une structure presque complexe générique jusqu’à briser la variété en deux morceaux $T^* S^2$ et $X \setminus S^2$. Les courbes $J$-holomorphes rationnelles réelles comptées par $\chi^d_r$ se brisent en courbes à deux étages interpolant $r$ points de $S^2$ et $r_X$ paires de points complexes conjugués de $X \setminus S^2$. Il est apparu au cours de la démonstration du Théorème \[theocong1\] que ces courbes à deux étages sont codées par les arbres deux-sphériques décorés $A \in {\cal A}_r^d$. Il s’agit donc de dénombrer les courbes à deux étages qui sont codées par un arbre donné $A \in {\cal A}_r^d$, puis de dénombrer les courbes $J$-holomorphes rationnelles réelles comptées par $\chi^d_r$ qui dégénèrent sur une courbe à deux étages donnée. Le nombre de façons de répartir les points complexes conjugués parmi les composantes de la courbe à deux étages qui se trouvent dans $X \setminus S^2$ a été calculé dans la démonstration du Théorème \[theocong1\] et vaut $2^{\sum_{s \in S_1^+} \# f_A (s) + \sum_{s \in S_1 \setminus S_1^+} \max (\# f_A (s) - 1 , 0)}$. Les composantes codées par $S_1 \setminus S_1^+$ sont rigides avec leurs conditions d’incidence, il y en a $\prod_{s \in S_1 \setminus S_1^+} N_2^{g_A (s) e + k(s) f} (0, \beta_A)$. Puis, il y a $m_1^- (A)$ façons de choisir les orbites de Reeb prescrites de la courbe codée par $s_0$. Le nombre de courbes réelles codées par $s_0$ satisfaisant nos conditions d’incidence et comptées avec signe vaut $F_{(r,0)} (\alpha_A^-, \beta_A^+)$. Il y a alors $m_1^+ (A)$ façons de choisir la manière de connecter les courbes codées par $S_1^+$ aux orbites de Reeb restées libres de la courbe codée par $s_0$. Enfin, chaque sommet de $S_2$ autre que $s_0$ code un plan $J$-holomorphe de $T^* S^2$ asymptote à une orbite de Reeb simple. Il y a deux tels plans pour une structure presque complexe générique $J$ de $T^* S^2$ qui sont les deux relevés du plan de $T^* \R P^2$ asymptote à une orbite de Reeb double et se compactifient en les deux droites de la quadrique complexe passant par un point donné. Il y a donc $2^{ \#S_2 -1}$ façons de choisir les plans codés par les éléments de $S_2 \setminus \{ s_0 \}$. Ceci fournit $2^{\sum_{s \in S_1^+} \# f_A (s) +
\sum_{s \in S_1 \setminus S_1^+} \max (\# f_A (s) - 1 , 0) + \#S_2 -1} m_1^+ (A) m_1^- (A) F_{(r,0)} (\alpha_A^-, \beta_A^+)$ $\prod_{s \in S_1 \setminus S_1^+} N_2^{g_A (s) e + k(s) f} (0, \beta_A)
\prod_{s \in S_1^+} N_2^{g_A (s) e + k(s) f} (e_{k(ss_0)} , \beta_A^0)$ courbes codées par un arbre donné $A \in {\cal A}_r^d$. Or, d’après le théorème de recollement de théorie symplectique des champs [@Bour], il y a $ \prod_a k(a)$ courbes $J$-holomorphes rationnelles réelles comptées par $\chi^d_r$ qui dégénèrent sur une courbe à deux étages donnée. Le résultat découle à présent du fait que chaque plan codé par $S_2 \setminus \{ s_0 \}$ intersecte $S^2$ en un point et contribue donc à la masse des courbes $J$-holomorphes rationnelles réelles en question, d’où le signe $(-1)^{\# S_2 + 1}$. $\square$
\[corcalc2spher\] Soit $(X, \omega , c_X)$ une variété symplectomorphe à la quadrique ellipsoïde de dimension deux. On note $h$ la classe d’une section plane réelle de bidegré $(1,1)$. Alors, $\chi^{2h} (T) = 2T^3 + 4T^5 + 6T^7$, $\chi^{3h} (T) = 16T + 16T^2 + o(T^3)$, $\chi^{4h} (T) = -256T + 320T^3 + o(T^4)$ et $\chi^{5h} (T) = 26880T + o(T^2)$.
Les valeurs $\chi^{4h}_1$, $\chi^{4h}_3$ et $\chi^{5h}_1$ énoncées dans la Proposition $2.3$ de [@Wels4] sont incorrectes et corrigées ici, note [@Wels4] qui fut d’ailleurs soumise en l’état en janvier et non décembre $2006$.
[**Démonstration du Corollaire \[corcalc2spher\] :**]{}
Le calcul de $\chi^{2h} (T)$ découle immédiatement du Lemme \[lemmecalc1\], puisque les seules composantes de $X \setminus L$ apparaissant sont des fibres. Les arbres intervenant dans la démonstration de ce Corollaire \[corcalc2spher\] sont représentés dans la Figure \[figarbres2spheriques\]. Lorsque $d=3$ et $r \leq 3$, un seul arbre décoré intervient. Le Théorème \[theocalcproj\] fournit $\chi^{3h}_1 = 2^4 F (e_1 , 0) N^{e+f} (0 , e_1) = 16$ et $\chi^{3h}_3 = 2^4 F (0 , e_1) N^{e+f} (e_1 , 0) = 16$. Lorsque $d=4$ et $r=1$, un seul arbre deux-sphérique décoré $A$ intervient dans le calcul de $\chi^{dh}_r$. On obtient $\chi^{4h}_1 = - 2^7 2 F (e_1 , 0) N^{e+2f} (0 , 2e_1) = -256$, puisque $m_1^- (A) = 2$. L’ensemble ${\cal A}_3^{4h}$ contient quant à lui trois arbres deux-sphériques décorés. La contribution du premier arbre donné par la Figure \[figarbres2spheriques\] vaut $C_6^1 2^4 F (2e_1 , 0) N^{f} (0 , e_1) N^{e+f} (0 , e_1) = 192$ puisqu’il y a $C_6^1$ façons de choisir la fonction $f_A$ ; celle du second vaut $2^5 2 F (e_2 , 0) N^{e+2f} (0 , e_2) = 2^8 N^{e+2f} (e_2 , 0) = 256$ et celle du troisième $- 2^7 F (0 , e_1) N^{e+2f} (e_1 , e_1) = -128$, de sorte que $\chi^{4h}_3 = 192 + 256 - 128 = 320$. L’ensemble ${\cal A}_1^{5h}$ contient deux arbres deux-sphériques décorés. La contribution du premier arbre donné par la Figure \[figarbres2spheriques\] vaut $2^8 F (e_1 , 0) N^{2e+f} (0 , e_1) = 2^8 93$, puisque d’après le Théorème $6.8$ de [@Vak], $N^{2e+f} (0 , e_1) = 93$. La contribution du deuxième arbre vaut $2^{10} 3 F (e_1 , 0) N^{e+3f} (0 , 3e_1) = 3072$, de sorte que $\chi^{5h}_1 =
23808 + 3072 = 26880$. $\square$
Calculs dans l’ellipsoïde de dimension trois {#subsec3spher}
--------------------------------------------
### Arbres trois-sphériques
Soient $r, r_X$ et $d$ trois entiers naturels satisfaisant la relation $2r + 4r_X = 3d$, laquelle impose que $d$ soit pair.
Un arbre trois-sphérique est un arbre connexe fini dont toutes les arêtes sont étiquetées par des entiers strictement positifs. De plus, un tel arbre possède une racine $s_0$ et tous les sommets qui lui sont adjacents sont monovalents.
En particulier, la distance maximale d’un sommet à $s_0$ vaut un. On note ${\cal C}_r$ l’ensemble des arbres trois-sphériques qui satisfont $$4\sum_{a \in {\cal A}(s_0)} k(a) - 2v(s_0) \leq 2r \leq 4\sum_{a \in {\cal A}(s_0)} k(a) + 2v (s_0) \text{ et } r = v (s_0) \mod (2),$$ où $v(s)$ désigne la valence d’un sommet $s$, $ {\cal A}(s)$ l’ensemble des arêtes adjacentes à $s$ et $k (a)$ désigne la multiplicité de l’arête $a$. Notons également $k_s$ la somme des multiplicités des arêtes adjacentes au sommet $s$ et $k$ la multiplicité totale de toutes les arêtes de l’arbre. On pose $r_L (s_0) = \frac{1}{4} \big( 4\sum_{a \in {\cal A}(s_0)} k(a) + 2v (s_0) - 2r \big)$, de sorte que $0 \leq r_L (s_0) \leq v (s_0)$. Enfin, on note $S_1$ l’ensemble des sommets adjacents à $s_0$.
Un arbre trois-sphérique décoré est un arbre trois-sphérique $A \in {\cal C}_r$ équipé d’une partition $S_1^+ \sqcup S_1^-$ de l’ensemble des sommets adjacents à $s_0$ telle que $\# S_1^- = r_L (s_0)$ et $\# S_1^+ = v(s_0) - r_L (s_0)$. Cet arbre est de plus équipé des fonctions :
- $f_A : S_1 \to {\cal P} (\{1, \dots , r_X \})$ satisfaisant $f_A (s) \cap f_A (s') = \emptyset$ dès que $s \neq s'$ et $\cup_{s \in S_1} f_A (s) = \{1, \dots , r_X \}$.
- $g_A : S_1 \to \N$ telle que $2k + 2\sum_{s \in S_1} g_A (s) = d$ et pour tout $s \in S_1^+$ (resp. $s \in S_1^-$ ), $3g_A (s) + k_s + 1 = 2\# f_A (s) + 2$ (resp. $3g_A (s) + k_s + 1 = 2\# f_A (s)$).
On note ${\cal C}_r^d$ l’ensemble des arbres trois-sphériques décorés, c’est un ensemble fini. Soit $A \in {\cal C}_r^d$, on pose $m_1^+ (A)$ le nombre d’injections $\phi : \{s \in S_1^+ \; \vert \; f_A (s) \neq \emptyset \} \to {\cal A}^+ (s_0)$ satisfaisant $k (\phi (s)) = k(ss_0)$ pour tous les sommets $s \in S_1^+$. On pose alors $$\text{mult} (A) = 2^{\sum_{s \in S_1^+} \# f_A (s) + \sum_{s \in S_1^-} \max (\# f_A (s) - 1 , 0)} m_1^+ (A) \prod_a k(a),$$ c’est la [*multiplicité*]{} de l’arbre trois-sphérique $A \in {\cal C}_r^d$.
### Relation avec l’invariant relatif
Soit $Y$ la variété réglée $P({\cal O}_Q (1,1) \oplus {\cal O}_Q)$ sur la quadrique de dimension deux $Q$ et $f$ la classe d’une fibre de $Y$. Étant donnés $a,b,c \in \N$ et $\alpha, \beta$ des suites d’entiers positifs, on note $N^{(a,b)+cf}_3 (\alpha, \beta)$ le nombre de courbes rationnelles de $Y$, homologues à $(a,b)+cf$, ayant $\alpha_i + \beta_i$ points de tangence d’ordre $i$ avec la section exceptionnelle $P({\cal O}_Q)$ de $Y$ parmi lesquels $\alpha_i $ sont prescrits et qui passent par le nombre adéquat de points fixés.
\[theocal3spher\] Soient $(X, \omega , c_X)$ une variété symplectomorphe à la quadrique ellipsoïde de dimension trois et $r, r_X, d \in \N$ satisfaisant la relation $2r + 4r_X = 3d$. Alors, $$\chi^{d}_r = \sum_{A \in {\cal C}_r^d} \text{mult} (A) F_{(r,0)} (\alpha_A^-, \beta_A^+) \prod_{s \in S_1^-} \sum_{a+b = g_A (s)} N_3^{(a,b) + k_s f} (0, e_{k_s})
\prod_{s \in S_1^+} \sum_{a+b = g_A (s)} N_3^{(a,b)+ k_s f} (e_{k_s} , 0),$$ où $(\alpha_A^-)_i$ (resp. $(\beta_A^+)_i$) vaut le nombre d’arêtes de multiplicité $i$ reliant $S_1^-$ (resp. $S_1^+$) à $s_0$ et $F_{(r,0)} (\alpha_A^-, \beta_A^+)$ désigne l’invariant défini dans le fibré cotangent de la sphère de dimension trois au §\[subsubinvariants\].
[**Démonstration :**]{}
On poursuit la stratégie générale énoncée au $\S$ \[subsubsectstrat\] en allongeant le cou d’une structure presque complexe générique jusqu’à briser la variété en deux morceaux $T^* S^3$ et $X \setminus S^3$. Les courbes $J$-holomorphes rationnelles réelles comptées par $\chi^d_r$ se brisent en courbes à deux étages interpolant $r$ points de $S^3$ et $r_X$ paires de points complexes conjugués de $X \setminus S^3$. Il est apparu au cours de la démonstration du Théorème \[theocong2\] que ces courbes à deux étages sont codées par les arbres trois-sphériques décorés $A \in {\cal C}_r^d$. Il s’agit donc de dénombrer les courbes à deux étages qui sont codées par un arbre donné $A \in {\cal C}_r^d$, puis de dénombrer les courbes $J$-holomorphes rationnelles réelles comptées par $\chi^d_r$ qui dégénèrent sur une courbe à deux étages donnée. Le nombre de façons de répartir les points complexes conjugués parmi les composantes de la courbe à deux étages qui se trouvent dans $X \setminus S^3$ a été calculé dans la démonstration du Théorème \[theocong2\] et vaut $2^{\sum_{s \in S_1^+} \# f_A (s) + \sum_{s \in S_1^-} \max (\# f_A (s) - 1 , 0)}$. Les composantes codées par $S_1^-$ sont rigides avec leurs conditions d’incidence, il y en a $\prod_{s \in S_1^-} \sum_{a+b = g_A (s)} N_3^{(a,b) + k_s f} (0, k_s) $. Le nombre de courbes réelles codées par $s_0$ satisfaisant nos conditions d’incidence et comptées avec signe vaut $F_{(r,0)} (\alpha_A^-, \beta_A^+)$. Il y a alors $m_1^+ (A)$ façons de choisir la manière de connecter les courbes codées par $S_1^+$ aux orbites de Reeb restées libres de la courbe codée par $s_0$. Ceci fournit $2^{\sum_{s \in S_1^+} \# f_A (s) + \sum_{s \in S_1^-} \max (\# f_A (s) - 1 , 0)} m_1^+ (A) F_{(r,0)} (\alpha_A^-, \beta_A^+) \prod_{s \in S_1^-} \sum_{a+b = g_A (s)} N_3^{(a,b) + k_s f} (0, k_s)$\
$\prod_{s \in S_1^+} \sum_{a+b = g_A (s)} N_3^{(a,b)+ k_s f} (k_s , 0) $ courbes codées par un arbre donné $A \in {\cal C}_r^d$. Or, d’après le théorème de recollement de théorie symplectique des champs [@Bour], il y a $ \prod_a k(a)$ courbes $J$-holomorphes rationnelles réelles comptées par $\chi^d_r$ qui dégénèrent sur une courbe à deux étages donnée. Le résultat en découle. $\square$
\[corcalc3spher\] Soit $(X, \omega , c_X)$ une variété symplectomorphe à la quadrique ellipsoïde de dimension trois. Alors, $\chi^{2}_1 = - 1$, $\chi^{6}_1 = 0$ et $\chi^{10}_1 = -896$.
L’invariant $\chi^{4k}_0$, $k \in \N$, n’est pas défini. Toutefois, d’après le Théorème \[theomin2\], lorsque les points complexes conjugués sont suffisamment proches d’une section hyperplane réelle disjointe du lieu réel de $X$, il n’y a aucune courbe rationnelle réelle ayant une partie réelle non-vide qui satisfait nos conditions d’incidence. Remarquons aussi que l’invariant défini dans [@Wels2], [@Wels3] a été étendu aux variétés symplectiques réelles de dimension six non fortement semi-positives dans [@Sol] (voir aussi [@Cho]) et calculé dans le cas des quintiques réelles de $\C P^4$ dans [@PSW].
[**Démonstration du Corollaire \[corcalc3spher\] :**]{}
Il n’y a qu’une seule section plane de $X$ qui passe par trois points. De plus, l’état spinoriel de cette conique vaut $-1$, de sorte que $\chi^{2}_1 = - 1$. En effet, l’état spinoriel d’une conique dans une quadrique de dimension deux vaut $-1$ et le fibré normal d’une section hyperplane réelle de $X$ est trivial en restriction à sa partie réelle. L’annulation de $\chi^{6}_1$ tient au fait qu’il n’y a pas de courbe rationnelle de bidegré $(a,b)$ qui passe par quatre points dans la quadrique de dimension deux. Enfin, l’ensemble $ {\cal C}_1^{10}$ ne contient qu’un seul arbre qui n’a que deux sommets, est de multiplicité $2^6$, pour lequel $S_1^+$ est vide et $F (\alpha_A^-, \beta_A^+) = -1$ d’après ce qui précède. Par sept point de la quadrique $Q$ de dimension deux, il passe douze courbes rationnelles de bidegré $(2,2)$, une courbe de bidegré $(3,1)$ et une de bidegré $(1,3)$. La restriction du réglage $Y \to Q$ à ces courbes est isomorphe à la surface réglée rationnelle de degré quatre $\Sigma_4$. Par sept points de $\Sigma_4$, il ne passe qu’une seule courbe rationnelle homologue à $e+f$. On en déduit que $N_3^{(3,1) + f} (0, e_1) = N_3^{(1, 3) + f} (0, e_1) = 1$ et $N_3^{(2,2) + f} (0, e_1) = 12$. D’où $\chi^{10}_1 = -896$. $\square$
[10]{}
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Unité de mathématiques pures et appliquées de l’École normale supérieure de Lyon,\
CNRS - Université de Lyon.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We study periodically driven pure Kitaev model and ferromagnetic phase of the Kitaev-Heisenberg model on the honeycomb lattice by off-resonant linearly and circularly-polarized lights at zero magnetic field. Using a combination of linear spin wave and Floquet theories, we show that the effective time-independent Hamiltonians in the off-resonant regime map onto the corresponding anisotropic static spin model, plus a tunable photoinduced magnetic field along the $[111]$ direction, which precipitates Floquet topological magnons and chiral magnon edge modes. They are tunable by the light amplitude and polarization. Similarly, we show that the thermal Hall effect induced by the Berry curvature of the Floquet topological magnons can also be tuned by the laser field. Our results pave the way for ultrafast manipulation of topological magnons in irradiated Kitaev magnets, and could play a pivotal role in the investigation of ultrafast magnon spin current generation in Kitaev materials.'
author:
- 'S. A. Owerre'
- Paula Mellado
- 'G. Baskaran'
title: Photoinduced Floquet topological magnons in Kitaev magnets
---
**Introduction.–** Topological band theory of solid-state materials has dominated many aspects of condensed-matter physics over the past decade [@top3; @top4]. The original concept of topological band theory is rooted in insulating electronic systems possessing a nontrivial gap in their energy band structures. They are characterized by the appearance of gapless chiral edge electron modes traversing the bulk gap, which are topologically protected by the Chern number or the $\mathbb{Z}_2$ index of the bulk bands [@top3; @top4].
Generally, the concept of topological band structure is independent of the statistical nature of the quasiparticle excitations and therefore is not restricted to insulating electronic systems. Recently, there has been a tremendous interest in the topological properties of spin excitations in insulating quantum magnets. In fact, bosonic topological spin excitations (magnons and triplons) have been studied in many different insulating quantum magnets [@rshin; @Zhang; @th6; @owerre; @chern; @romh; @rchi; @cr; @mcC; @Kitaeva; @Kitaevb; @flu], and the appearance of chiral edge modes and bulk Chern number have been demonstrated [@Zhang; @th6; @owerre]. Recently, bosonic topological spin excitations mimicking electronic topological insulators have been experimentally observed in kagome ferromagnet Cu(1,3-bdc) [@rchi], dimerized quantum magnet SrCu$_2$(BO$_3$)$_2$ [@mcC], and honeycomb ferromagnet CrI$_3$ [@cr].
The Mott-insulating honeycomb Kitaev magnets are currently of great interest [@Kitaev1; @Kitaev2a; @Kitaev2; @Kitaev3; @Kitaev4; @Kitaev4a; @Kitaev4b; @Kitaev5; @kit1; @kit2; @kit3; @kit4; @kit5; @kit6; @kit7; @kit8; @kit9; @kit10; @zyou]. Candidate Kitaev materials include Na$_2$IrO$_3$ and $\alpha$-RuCl$_3$ [@Kitaev2; @Kitaev3; @Kitaev2a; @Kitaev4; @Kitaev5]. Recently, topologically protected spin waves have been predicted in the fully-polarized phase of the pure Kitaev model [@Kitaeva] and the Kitaev-Heisenberg model [@Kitaevb] at high magnetic field. In the former, the topological magnons and chiral edge states present in linear spin-wave approximation survive magnon-magnon interactions and therefore are robust [@Kitaeva]. Indeed, the manipulation of topological magnons and magnon spin currents is essential for their practical applications in ultrafast magnetic data storage, magnetic switching, and magnon spintronics [@magn].
[The tremendous interest in topological quantum phases of matter has led to different alternative ways for inducing them in quantum materials]{}. Recently, irradiated solid-state materials have provided an alternative route to extend the search for topological quantum materials in electronic systems [@pho1; @pho2; @pho3; @pho4; @pho5; @pho5a; @pho6]. In this formalism, topologically trivial systems can be periodically driven to nontrivial topological systems termed Floquet topological insulators [@pho6; @pho3]. They have an advantage over their static (equilibrium) topological counterpart, in that their intrinsic properties can be manipulated and different topological phases can be achieved. In irradiated insulating quantum magnets with charge-neutral spin excitations [@sowe; @kar; @ely; @claas], the Floquet physics can emerge from the coupling of the electron spin magnetic dipole moment to the laser electric field through the time-dependent version of the static Aharonov-Casher phase [@aha; @spin3], which acts as a vector potential or gauge field to the spin current [@spin1a]. In this case, the resulting Floquet physics can reshape the underlying Hamiltonian to stabilize magnetic phases and provides a promising avenue for inducing and tuning Floquet topological spin excitations [@sowe; @kar; @ely], with a direct implication of generating and manipulating ultrafast spin current using terahertz ([THz]{}) radiation [@ultra]. Lately, [THz]{} electric field amplitude exceeding $100~{\rm MV/cm}$ between $10~{\rm THz}$ ($1~{\rm THz} \sim 4~{\rm meV}$) and $72~ {\rm THz}$ has been reported [@sell]. In this respect, resonant time-domain [THz]{} spectroscopy has been recently performed in the candidate Kitaev material $\alpha$-RuCl$_3$ [@lwu].
In this paper, we propose a tunable mechanism to induce and manipulate topological magnons in irradiated Kitaev magnets at zero magnetic field. We study the pure Kitaev model [@kita] and the ferromagnetic phase of the Kitaev-Heisenberg model, which are already present in the zero magnetic-field classical phase diagram of the Kitaev-Heisenberg model on the honeycomb lattice [@Kitaev2]. Using linear spin wave and Floquet theories, we show that when the models are periodically driven by off-resonant linearly- and circularly-polarized lights, they effectively map onto the corresponding static spin model plus a tunable photoinduced magnetic field along the $[111]$ direction, which is perpendicular to the honeycomb plane. The photoinduced magnetic field precipitates the existence of Floquet topological magnons and chiral edge modes, in a similar fashion to a homogeneous magnetic field in the undriven systems [@Kitaeva; @Kitaevb]. However, the Floquet topological magnons can be tuned by the amplitude and polarization of the laser field. Likewise, we demonstrate that the resulting Floquet thermal Hall conductivity can be tuned by the laser field. The photoinduced magnetic field required to induce magnetic order and Floquet topological magnons in the pure Kitaev model lies in the interval $0<h(\mathcal E_0,\phi)<2AS$, where $\mathcal E_0,\phi$ are the amplitude and polarization of the laser field, $A>0$ [ is the overall energy scale of the spin exchange interactions]{} and $S$ is the spin value. Therefore, $h(\mathcal E_0,\phi)$ is much smaller than the high magnetic field $h>4AS$ required to induce topological magnons in the undriven pure Kitaev model [@Kitaeva]. Interestingly, the Floquet topological magnons in the irradiated Kitaev magnets do not require an explicit time-reversal symmetry breaking term from the second-order virtual-photon absorption and emission processes [@pho4], which is strictly required in order to induce Floquet topological states in other irradiated quantum systems [@pho5a; @sowe; @kar; @pho4; @pho1].
**Model.–** We study the Kitaev-Heisenberg model on the honeycomb lattice with nearest-neighbour interaction. The spin Hamiltonian reads [@Kitaev1; @Kitaev2a; @Kitaev2; @Kitaev3; @Kitaev4; @Kitaev4a; @Kitaev4b; @Kitaev5] $$\begin{aligned}
\mathcal H&= 2J_K\sum_{ \la ij\ra \gamma} {S}_{i}^{\gamma}{S}_{j}^{\gamma}+J_H\sum_{\la ij\ra}{\vec S}_{i}\cdot{\vec S}_{j},
\label{KH}\end{aligned}$$ where the first term corresponds to the bond-dependent Kitaev interaction and the second term to the isotropic Heisenberg interaction. The bond directions are denoted by $\gamma = \lbrace x,y,z\rbrace$ as shown in Fig. . We parameterize the interactions as $J_H=A\cos\vartheta$ and $J_K= A\sin\vartheta$, where $\vartheta\in [0,2\pi]$ and $A=\sqrt{J_H^2+J_K^2}>0$ is the overall energy scale of the exchange interactions, with $A\sim 8~{\rm meV}$ in some real materials [@Kitaev4]. The classical phase diagram of Eq. has been established in the $\vartheta$ space [@Kitaev5; @Kitaev2]. The zig-zag phase of Eq. is believed to describe the honeycomb magnetic materials Na$_2$IrO$_3$ and $\alpha$-RuCl$_3$ [@Kitaev2; @Kitaev4]. Recent studies have shown that the fully-polarized phase of the pure Kitaev model ($\vartheta = \pi/2$) [@Kitaeva] and the Kitaev-Heisenberg model ($\vartheta = 5\pi/4$) [@Kitaevb] at high magnetic field possess topological magnon modes. The purpose of this paper is to periodically drive the magnon topologically trivial phases of Eq. to Floquet topological magnon insulators for $\vartheta = \pi/2$ and $\vartheta = 5\pi/4$.
![Color online. (a) Honeycomb lattice of the Kitaev model with bond links $\gamma = \lbrace x,y,z\rbrace$ for the Kitaev interaction in Eq. . The primitive lattice vectors are ${\vec{a}_{1,2}}=(\pm \frac{\sqrt{3}}{2}, \frac{3}{2})a$ and the nearest-neighbour vectors are ${\vec{\delta}_{1}}=(\frac{\sqrt{3}}{2}, -\frac{1}{2})a$, ${\vec{\delta}_{2}}=(-\frac{\sqrt{3}}{2}, -\frac{1}{2})a$, ${\vec{\delta}_{3}}=(0, 1)a$. Here, $\alpha_A$ and $\alpha_B$ denote the two sublattices of the honeycomb lattice. (b) Brillouin zone of the honeycomb lattice with high symmetry points. []{data-label="lattice"}](lattice){width="1\linewidth"}
**Irradiated Kitaev magnets.–** In the presence of an intense laser field with a dominant time-dependent electric field component $ {\vec {\mathcal E}}(\tau)$, the spin magnetic dipole moment of an electron ${\vec \mu}_S= -g\mu_B\hat{n}$ hopping along the magnetization direction $\hat{n}$ will accumulate a time-dependent Aharonov-Casher phase [@sowe; @kar; @ely; @claas] $$\begin{aligned}
\Phi_{ij}(\tau) = \mu_m\int_{{\vec r}_i}^{{\vec r}_j}{\vec \Xi}(\tau) \cdot d{\vec \ell},\end{aligned}$$ where $\mu_m = g\mu_B/\hbar c^2$, $g$ is the spin-g factor, $\mu_B$ is the Bohr magneton, $\hbar$ is the reduced Plank’s constant, and $c$ is the speed of light. Here, ${\vec \Xi}(\tau)= {\vec {\mathcal E}}(\tau) \times \hat{n}$ with $ {\vec {\mathcal E}}(\tau)=-\partial_\tau {\vec {\mathcal A}}(\tau)$, where ${\vec {\mathcal A}}(\tau)$ is the time-dependent vector potential of the applied laser field.
It is convenient to introduce orthonormal basis vectors $(\hat{l},\hat{m}, \hat{n})$, where $\hat{n}$ points along the cubic $[111]$ direction, perpendicular to the honeycomb plane [@chal]. We can now write Eq. in the new basis. In this new basis, the spin dipole moment of an electron couples to the laser electric field through the Aharonov-Casher phase, in the same way the electron charge couples through the Peierls phase [@pho2; @pho4]. Therefore, the terms that contribute to linear spin-wave approximation can be written as (see Supplemental material (SM) [@sm]) $$\begin{aligned}
\mathcal H(\tau) &= \big(J_H+\frac{2J_K}{3}\big)\sum_{\la ij\ra}\big [S_i^nS_j^n + \frac{1}{2}\lbrace S_i^+S_j^-e^{i\Phi_{ij}(\tau)}+{\rm H.c.}\rbrace\big] \label{rotH}\\&\nonumber +\frac{2J_K}{3}\sum_{\la ij\ra\gamma}\big[\frac{1}{2}\lbrace e^{i\varphi_\gamma}S_i^+S_j^+e^{i\Phi_{ij}(\tau)}+{\rm H.c.}\rbrace\big],
\end{aligned}$$ where $S_j^\pm = S_j^l\pm i S_j^m$ are the usual raising and lowering spin operators, and the angle $\varphi_\gamma$ comes from the rotation of the bond directions (see SM), with $\varphi_\gamma = 2\pi/3,4\pi/3,0$ for $x,y,z$ bond directions respectively. The Aharonov-Casher phase acts as a vector potential or gauge field to the spin current [@spin1a]. We consider light propagating along the \[111\] direction ([*i.e.*]{} perpendicular to the honeycomb plane), given by $$\begin{aligned}
{\vec \Xi}(\tau)=E_0\big[\sin(\omega \tau),\sin(\omega \tau + \phi), 0\big],\end{aligned}$$ where $E_0$ is the amplitude of the time-dependent electric field, $\omega$ is the angular frequency of light and $\phi$ is the polarization. Linearly and circularly polarized lights correspond to $\phi =0$ and $\phi=\pi/2$ respectively. [We perform linear spin-wave theory in the polarized phase, which is valid in the large $S$ limit and for low-energy excitations]{}. This can be done by writing the spin operators in Eq. in terms of the linearized Holstein-Primakoff bosons [@hp]: $S_i^n = S-a_i^\dg a_i,~S_i^+ \approx \sqrt{2S}a_i$ for $i \in \alpha_A$, and $S_j^n = S-b_j^\dg b_j,~S_j^+ \approx \sqrt{2S}b_j$ for $j \in \alpha_B$. The resulting linear spin-wave bosonic Hamiltonian is time-periodic $\mathcal H_2(\tau+T)=\mathcal H_2(\tau)$, where $T$ is the period of the driving field.
We can now implement the machinery of Floquet theory [@floq], to study the dynamics of irradiated Kitaev magnets. In the off-resonant limit $\hbar\omega\gg A$, light simply modifies the band structures [@pho4]. The effect of such off-resonant light is captured in a static effective Hamiltonian $\mathcal H_{eff}$ [@pho4; @pho2], defined through the evolution Floquet operator $U$ of the system after one period $T= 2\pi/\omega$ as H\_[eff]{}= (U),where $U=\mathcal{T}\exp\big(-i\int_0^T\mathcal{H}_2(\tau) d\tau\big)$ and $\mathcal T$ is the time-ordering operator. The effective Hamiltonian can be written as $\mathcal H_{eff}=\sum_{i\geq 0}\mathcal H_{eff}^{(i)}/(\hbar\omega)^i$. We work in the off-resonant limit where the photon energy is much larger than the energy scale of the static system, i.e. $\hbar\omega\gg A$. That means we focus on the zero-photon sector [@pho2], $\mathcal H_{eff}^{(0)}=\mathcal{H}_2^0$, where $\mathcal H_2^n=\frac{1}{T}\int_0^T d\tau e^{-in\omega \tau} \mathcal H_2(\tau)$ are the discrete Fourier components and $n\in \mathbb{Z}$. Next, we Fourier transform $\mathcal{H}_2^0$ into momentum space and use the basis vector $\big[\psi^{(0)}({ \bo})\big]^\dg=\big(a_{\bo,\alpha_A}^{(0),\dg},b_{\bo,\alpha_B}^{(0),\dg},a_{-\bo,\alpha_A}^{(0)},b_{-\bo,\alpha_B}^{(0)}\big)$. The effective time-independent Hamiltonian is given by
$$\begin{aligned}
\mathcal H_{eff}^{(0)}({\bo})=S
\begin{pmatrix}
\mathcal M^{(0)}(\bo)&\mathcal N^{(0)}(\bo)\\ \big[\mathcal N^{(0)}(\bo)\big]^\dg& \big[\mathcal M^{(0)}(-\bo)\big]^T
\end{pmatrix},
\label{hamp}\end{aligned}$$
$$\begin{aligned}
\mathcal M^{(0)}({\bo})=
\begin{pmatrix}
\rho_0^{(0)}& \rho_1^{(0)}(\bo)\\ \rho_1^{(0)}(\bo)^*& \rho_0^{(0)}
\end{pmatrix},\end{aligned}$$
$$\begin{aligned}
\mathcal N^{(0)}({\bo})=
\begin{pmatrix}
0 & \rho_2^{(0)}(\bo)\\ \rho_2^{(0)}(-\bo)& 0
\end{pmatrix},\end{aligned}$$
where $$\begin{aligned}
&\rho_0^{(0)} = -3J_H -2J_K,
\label{kita9}\end{aligned}$$
$$\begin{aligned}
\rho_1^{(0)}(\bo) &= \big(J_H+2J_K/3\big)\big[\mathcal J_0(\mathcal E_0)\nonumber\\&+\mathcal J_0(\mathcal E_+(\phi))e^{i\bo\cdot {\vec a}_1}+\mathcal J_0(\mathcal E_-(\phi))e^{i \bo\cdot {\vec a}_2}\big],\end{aligned}$$
$$\begin{aligned}
\rho_2^{(0)}(\bo) &=(2J_K/3)\big[\mathcal J_0(\mathcal E_0)+\mathcal J_0(\mathcal E_+(\phi))e^{i (\bo\cdot {\vec a}_1+ 2\pi/3)}\nonumber\\&+\mathcal J_0(\mathcal E_-(\phi))e^{i (\bo\cdot {\vec a}_2-2\pi/3)}\big],\end{aligned}$$
where $\mathcal J_\ell(x)$ is the Bessel function of order $\ell \in\mathbb{Z}$, and $\mathcal E_\pm(\phi) = \frac{\mathcal E_0}{2}\sqrt{4\pm 2\sqrt{3}\cos\phi}$. The dimensionless quantity that characterizes the light intensity is $\mathcal E_0 = g\mu_BE_0a/\hbar c^2$. The static effective Hamiltonian in Eq. can be diagonalized by performing a bosonic Bogoliubov transformation (see SM).
![Color online. Floquet magnon bands for FM Kitaev-Heisenberg model $\vartheta = 5\pi/4$ (top panel) and AFM Kitaev point $\vartheta = \pi/2$ ( bottom panel).[]{data-label="band"}](band){width="1\linewidth"}
![Color online. Chern number of the lowest Floquet topological magnon band as a function of $\mathcal E_0$ in the ferromagnetic Kitaev-Heisenberg model $\vartheta =5\pi/4$ and at the antiferromagnetic Kitaev point $\vartheta =\pi/2$ for $\phi=0$. Inset shows the Chern number for $\phi=\pi/2$. Note that the Chern number of the magnon topology is not well-defined at $\mathcal E_0=0$ (not shown).[]{data-label="ChernN"}](ChernN){width="1\linewidth"}
**Photoinduced topological magnon bands.–** In Fig. , we have shown the Floquet magnon bands for the ferromagnetic (FM) Kitaev-Heisenberg model (top panel) and at the antiferromagnetic (AFM) Kitaev point (bottom panel) for $\mathcal E_0=0$, $(\mathcal E_0=1,~\phi=\pi/2)$, and $(\mathcal E_0=1,~\phi=0)$. In the FM Kitaev-Heisenberg model $\vartheta = 5\pi/4$ (top panel), the magnon bands for the undriven system at $\mathcal E_0 =0$ are already separated by a finite energy gap at the ${\rm K}$ point. In this case, however, the magnon topology of the system is not well-defined and was not discussed in Refs. [@Kitaeva; @Kitaevb]. By applying a laser drive, the gap at ${\rm K}$ point does not close, however the system is now driven to a well-defined topological magnon insulator as we will show below. At the AFM Kitaev point[^1] $\vartheta = \pi/2$ [@kita; @yong] (bottom panel), the lowest magnon band is a zero energy mode in the undriven system for $\mathcal E_0 =0$ [@bask]. The presence of zero energy mode in the spin wave excitations of frustrated magnets is an artifact of an extensive classical degeneracy, and points to the onset of a classical spin liquid [@cla2]. As the laser field is applied, the zero energy mode is lifted for $\phi=\pi/2$ and $\phi=0$, which implies a photoinduced magnetic order without a high applied magnetic field [@Kitaeva].
To investigate the magnon topology of the system, we define the Chern number of the Floquet magnon bands as the flux of the Berry curvature threading the entire Brillouin zone (BZ): $\mathcal C_{eff}^\alpha(\mathcal E_0, \phi) = \frac{1}{2\pi}\int_{BZ} d^2 k~\Omega^z_{\alpha}(\bo)$, where $\Omega^z_{\alpha}(\bo)$ is the Berry curvature of the Floquet magnon bands labeled by $\alpha=1,2$ (see SM). The Chern number has been computed using the discretized BZ method [@fuk]. In the main panel of Fig. , we show the evolution of the lowest Floquet Chern number as a function of $\mathcal E_0$ for $\phi=0$ with $\vartheta= 5\pi/4$ and $\vartheta=\pi/2$. While the inset shows the Chern number for $\phi=\pi/2$. As we mentioned above, the magnon topology of the system is not well-defined at equilibrium $\mathcal E_0=0$, thus we do not consider this case. For $\phi =0$ and $0<\mathcal E_0\lesssim 1.35$, where $h(\mathcal E_0\sim 1.35, \phi=0)\sim AS$ for $\vartheta=\pi/2$ (see Eq. below), the Chern number of the lowest band is $\mathcal C_{eff}=+1$ in the FM Kitaev-Heisenberg model $\vartheta =5\pi/4$ and $\mathcal C_{eff}=-1$ at the AFM Kitaev point $\vartheta =\pi/2$; but the Chern number is zero for $\mathcal E_0> 1.35$. For $\phi = \pi/2$, the Chern number is nonzero provided $\mathcal E_0\neq 0$.
**Effective spin Hamiltonian in real space.–** To understand the origin of the photoinduced topological magnons, we can map the off-resonant effective static Hamiltonian in Eq. back to the real-space spin operators keeping in mind the Holstein-Primakoff bosons. In the original cubic coordinate system, the real-space effective static spin Hamiltonian which reproduces Eq. is given by[^2]
$$\begin{aligned}
\mathcal H_{eff}^{(0)}&= \sum_{ \la ij\ra \gamma}J_\gamma(\mathcal E_0,\phi){S}_{i}^{\gamma}{S}_{j}^{\gamma}+\sum_{ \la ij\ra}J_{ij}(\mathcal E_0,\phi) {\vec S}_i\cdot{\vec S}_j \label{KH0}\\&\nonumber + h(\mathcal E_0,\phi)\sum_{i}\big(S_i^x+S_i^y+S_i^z\big),\end{aligned}$$
which is a renormalized Kitaev-Heisenberg model plus a photoinduced magnetic field along the $[111]$ direction. The anisotropic Kitaev interactions are given by $J_z(\mathcal E_0) = 2J_K\mathcal J_0(\mathcal E_0)$, $J_y(\mathcal E_0,\phi) = 2J_K\mathcal J_0(\mathcal E_-(\phi))$, and $J_x(\mathcal E_0,\phi) = 2J_K\mathcal J_0(\mathcal E_+(\phi))$. The Heisenberg interactions are distorted with $J_{ij}(\mathcal E_0)=J_H\mathcal J_0(\mathcal E_0)$ along the vertical ${\vec{\delta}_3}$ bond, $J_{ij}(\mathcal E_0,\phi)=J_H\mathcal J_0(\mathcal E_+(\phi))$ along the diagonal ${\vec{\delta}_1}$ bond, and $J_{ij}(\mathcal E_0,\phi)=J_H\mathcal J_0(\mathcal E_-(\phi))$ along the diagonal ${\vec{\delta}_2}$ bond (see Fig. ). The photoinduced magnetic field is given by $$\begin{aligned}
h(\mathcal E_0,\phi) &= (2J_K +3J_H)S\Big[1 -\frac{\mathscr J(\mathcal E_0,\phi)}{3}\Big],
\label{Eq13}\end{aligned}$$ where $\mathscr J(\mathcal E_0,\phi) = \mathcal J_0(\mathcal E_0)+\mathcal J_0(\mathcal E_+(\phi))+\mathcal J_0(\mathcal E_-(\phi))$. Eq. stems from the non-renormalized Kitaev-Heisenberg interaction in Eq. . Note that Eq. vanishes at $\mathcal E_0=0$, hence Eq. reduces to Eq. . For $\mathcal E_0\neq 0$, however, Eq. lies in the interval $0<h(\mathcal E_0,\phi)<(2J_K+3J_H)S$. Thus, at the AFM Kitaev point $\vartheta=\pi/2~(J_H=0)$, the photoinduced magnetic field is $0<h(\mathcal E_0,\phi)<2AS$, which is much smaller than the high homogeneous magnetic field $h>4AS$ required to induce topological magnons in the undriven pure Kitaev model [@Kitaeva]. On the contrary, at the FM Heisenberg point $\vartheta =\pi~(J_K=0)$, the effective Hamiltonian is simply a distorted fully-polarized honeycomb ferromagnet, which does not possess any topological magnon modes (see SM).
![Color online. Bulk Floquet magnon bands with tunable zigzag chiral edge states (red curves). []{data-label="Edge"}](Edge){width="1\linewidth"}
![Color online. Tunable photoinduced Floquet thermal Hall conductivity $\kappa_{xy}$ as a function of $\mathcal E_0$ for $\phi =0$, $S=1/2$ in the Kitaev-Heisenberg model $(\vartheta =5\pi/4)$ and at the AFM Kitaev point $\vartheta=\pi/2$ (inset).[]{data-label="THE"}](THE){width="1\linewidth"}
One of the hallmarks of 2D topological systems is the existence of gapless chiral edge modes on the boundary of the system [@top3; @top4]. In insulating topological magnets, the chiral edge modes can play a pivotal role in spin transport [@Zhang]. They are a consequence of the topological properties of the bulk bands. In Fig. , we show the tunable zigzag chiral edge modes (red curves) traversing the bulk gap for $k_x\in \big[\frac{2}{3}\pi,~\frac{4}{3}\pi\big]$ and they cross at the time-reversal invariant momentum $k_x = \pi$ in the topological regime. In the non-topological regime for $\phi=0$ and $\mathcal E_0>1.35$ with $h(\mathcal E_0\sim 1.35, \phi=0)\sim AS$ for $\vartheta=\pi/2$, the chiral edge modes are completely detached from the bulk bands and they are degenerate along a continuous line, which signifies that the system is topologically trivial as the Chern number plot in Fig. shows.
**Photoinduced magnon thermal Hall effect.–** The thermal Hall effect is a consequence of the Berry curvature of topological magnons in magnetically ordered systems [@kasa; @th1; @th2; @th5; @th7; @th4]. In the non-equilibrium Floquet system, we consider the limit where the Bose distribution function of magnon is close to thermal equilibrium. In this limit, the thermal Hall effect mimics that of equilibrium systems where a longitudinal temperature gradient $-{ \partial}_y { T}$ induces a transverse heat current $ J^q_{x}=-\kappa_{xy}\partial_{y} T$, where $\kappa_{xy}$ is the thermal Hall conductivity, derived in Ref. [@th5] (see SM [@sm]). In Fig. , we show the $\mathcal E_0$-dependence of $\kappa_{xy}$ for $\phi=0$ and $T/A=0.3,0.35,0.4$, in the FM Kitaev-Heisenberg model $\vartheta =5\pi/4$ and at the AFM Kitaev point $\vartheta =\pi/2$ (inset). We note that $\kappa_{xy}$ is ill-defined for $\mathcal E_0=0$ at low temperatures (not shown). The thermal Hall conductivity is dominated by the Berry curvature of the lowest magnon band at low temperatures and its sign is consistent with the sign of the Berry curvature (Chern number) of the lowest magnon band. At low temperature $T/A\ll 1$ and for $\mathcal E_0> 1.35$, $\kappa_{xy}$ is very small and approaches zero consistent with the vanishing of the Chern number and the absence of traversing chiral edge modes for $\phi=0$ as shown above. The low-temperature dependence of $\kappa_{xy}$ for $\phi=0$ is shown in SM.
**Conclusion and Outlook.–** We have proposed the existence of Floquet topological magnon insulators in periodically driven pure Kitaev model and ferromagnetic phase of the Kitaev-Heisenberg model at zero magnetic field. The main result of our study can be summarized as follows. In the off-resonant limit, the Floquet physics stabilizes magnetic order and the effective time-independent Hamiltonians map onto the corresponding anisotropic static spin model, plus a tunable photoinduced magnetic field along the $[111]$ direction, which facilitates the existence of Floquet topological magnon modes in a similar fashion to a homogenous magnetic field in the undriven systems [@Kitaeva; @Kitaevb]. One of the advantages of the current results is that the photoinduced topological magnons and the chiral edge modes can be tuned by varying the amplitude and polarization of the laser field. Another interesting feature of irradiated Kitaev magnets is that the existence of the Floquet topological magnon insulators does not require the explicit time-reversal symmetry breaking term from the second-order virtual-photon absorption and emission processes, which is mandatory for the existence of Floquet topological states in irradiated graphene [@pho4; @pho1] and irradiated honeycomb ferromagnets [@sowe; @kar; @ely]. We also showed that irradiated Kitaev magnets exhibit a tunable photoinduced thermal Hall effect. A direct experimental implication of the current proposal is that ultrafast magnon spin currents can be generated in irradiated Kitaev materials using different experimental techniques such as the inverse Faraday effect [@ultra] and [THz]{} spectroscopy [@lwu]. This could pave the way for topological opto-magnonics and opto-spintronics [@magn] using Kitaev materials.
In future work, we plan to address the effect of magnon-magnon interactions and see how they modify Eq. . However, it has been shown that the high magnetic-field-induced undriven topological magnons and chiral edge modes present in linear spin-wave approximation remain intact in the presence of magnon-magnon interactions [@Kitaeva]. We also plan to study the non-equilibrium distribution function [@deh; @gbas] of magnon in this system. Moreover, it would also be interesting to investigate whether tunable topological magnons can be photoinduced in the zigzag phase of the Kitaev-Heisenberg model.
**Acknowledgements.–** Research at Perimeter Institute is supported by the Government of Canada through Industry Canada and by the Province of Ontario through the Ministry of Research and Innovation. PM acknowledges Fondecyt Grant No 1160239.
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[^1]: The antiferromagnetic Kitaev point $\vartheta = \pi/2$ is exactly solvable for spin-$1/2$ in terms of Majorana fermions [@kita] and Jordan-Wigner transformation [@yong].
[^2]: Note that Eq. is valid in linear spin wave approximation for the magnetically-ordered state considered in this paper. Conversely, the effective static spin Hamiltonian that manifests directly from Eq. will be different, because no specific magnetically-ordered state is assumed in Eq. .
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We resolve a 25 year old problem by showing that The Paving Conjecture is equivalent to The Paving Conjecture for Triangular Matrices.'
address: 'Department of Mathematics, University of Missouri, Columbia, MO 65211-4100'
author:
- 'Peter G. Casazza and Janet C. Tremain'
title: the paving conjecture is equivalent to the paving conjecture for triangular matrices
---
[^1]
Introduction {#Intro}
============
The Kadison-Singer Problem [@KS] has been one of the most intractable problems in mathematics for nearly 50 years.
Does every pure state on the (abelian) von Neumann algebra $\D$ of bounded diagonal operators on ${\ell}_2$ have a unique extension to a (pure) state on $B({\ell}_2)$, the von Neumann algebra of all bounded linear operators on the Hilbert space ${\ell}_2$?
A [**state**]{} of a von Neumann algebra ${\cal R}$ is a linear functional $f$ on ${\cal R}$ for which $f(I) = 1$ and $f(T)\ge 0$ whenever $T\ge 0$ (i.e. whenever $T$ is a positive operator). The set of states of ${\cal R}$ is a convex subset of the dual space of ${\cal R}$ which is compact in the $w^{*}$-topology. By the Krein-Milman theorem, this convex set is the closed convex hull of its extreme points. The extremal elements in the space of states are called the [**pure states**]{} (of ${\cal R}$). The Kadison-Singer Problem had been dorment for many years when it was recently brought back to life in [@CT] and [@CFTW] where it was shown that KS is equivalent to fundamental unsolved problems in a dozen different areas of research in pure mathematics, applied mathematics and engineering.
A significant advance on the Kadison-Singer Problem was made by Anderson [@A3] in 1979 when he reformulated KS into what is now known as the [**Paving Conjecture**]{} (Lemma 5 of [@KS] shows a connection between KS and Paving). Before we state this conjecture, let us introduce some notation. For an operator $T$ on $\ell_2^n$, its matrix representation $(\langle Te_i,e_j\rangle )_{i,j\in I}$ is with respect to the natural orthonormal basis. If $A\subset \{1,2,\ldots ,n\}$, the [**diagonal projection**]{} $Q_A$ is the matrix all of whose entries are zero except for the $(i,i)$ entries for $i\in A$ which are all one.
For $\epsilon >0$, there is a natural number $r$ so that for every natural number $n$ and every linear operator $T$ on $l_2^n$ whose matrix has zero diagonal, we can find a partition (i.e. a [*paving*]{}) $\{{A}_j\}_{j=1}^r$ of $\{1, \ldots, n\}$, so that $$\|Q_{{A}_j} T Q_{{A}_j}\| \le \epsilon \|T\|
\ \ \ \text{for all $j=1,2,\ldots ,r$.}$$
It is important that $r$ not depend on $n$ in PC. We will say that an arbitrary operator $T$ satisfies PC if $T-D(T)$ satisfies PC where $D(T)$ is the diagonal of $T$. It is known that the class of operators satisfying PC (the [**pavable operators**]{}) is a closed subspace of $B({\ell}_2)$. Also, to verify PC we only need to verify it for any one the following classes of operators [@AA; @CFTW; @CEKP]: 1. unitary operators, 2. positive operators, 3. orthogonal projections (or just orthogonal projections with $1/2's$ on the diagonal), 4. Gram operators of the form $T^{*}T=(\langle f_i,f_j\rangle )_{i,j\in I}$ where $\|f_i\|=1$ and $Te_i = f_i$ is a bounded operator. The only large classes of operators which have been shown to be pavable are “diagonally dominant” matrices [@BCHL; @BCHL2; @G], matrices with all entries real and positive [@BHKW; @HKW] and matrices with small entries [@BT].
Since the beginnings of the [*paving era*]{}, it has been a natural question whether PC is equivalent to PC for triangular operators This question was formally asked several times at meetings by Gary Weiss and Lior Tzafriri and appeared (for a short time) on the AIM website (http://www.aimath.org/The Kadison-Singer Problem) as an important question for PC. In this paper we will verify this conjecture. Given two conjectures $C_1,\ C_2$ we say that $C_1$ [**implies**]{} $C_2$ if a positive answer to $C_1$ implies a positive answer for $C_2$. They are [**equivalent**]{} if they imply each other.
Preliminaries {#Prelim}
=============
Recall that a family of vectors $\{f_i\}_{i\in I}$ is a [**Riesz basic sequence**]{} in a Hilbert space $\H$ if there are constants $A,B>0$ so that for all scalars $\{a_i\}_{i\in I}$ we have: $$A^2\sum_{i\in I}|a_i |^2 \le \|\sum_{i\in I}a_i f_i \|^2 \le
B^2\sum_{i\in I}|a_i|^2.$$ We call $A,B$ the [**lower and upper Riesz basis bounds**]{} for $\{f_i\}_{i\in I}$. If $\epsilon >0$ and $A = 1-\epsilon, B=1+\epsilon$ we call $\{f_i\}_{i\in I}$ an $\epsilon$-[**Riesz basic sequence**]{}. If $\|f_i\|=1$ for all $i\in I$ this is a [**unit norm**]{} Riesz basic sequence. A natural question is whether we can improve the Riesz basis bounds for a unit norm Riesz basic sequence by partitioning the sequence into subsets.
For every $\epsilon >0$, every unit norm Riesz basic sequence is a finite union of $\epsilon$-Riesz basic sequences.
The $R_{\epsilon}$-Conjecture was posed by Casazza and Vershynin [@CV] where it was shown that KS implies this conjecture. It is now known that the $R_{\epsilon}$-Conjecture is equivalent to KS [@CT]. We will show that PC for triangular operators implies a positive solution to the $R_{\epsilon}$-Conjecture. Actually, we need the finite dimensional quantative version of this conjecture.
Given $0< \epsilon, A,B$, there is a natural number $r=r(\epsilon,
A,B)$ so that for every $n\in \N$ and every unit norm Riesz basic sequence $\{f_i\}_{i=1}^{n}$ for $\ell_2^n$ with Riesz basis bounds $0<A\le B$, there is a partition $\{A_j\}_{j=1}^{r}$ of $\{1,2,\ldots ,n\}$ so that for all $j=1,2,\ldots ,r$ the family $\{f_i\}_{i\in A_j}$ is an $\epsilon$-Riesz basic sequence.
There are standard methods for turning infinite dimensional results into quantative finite dimensional results so we will just outline the proof of their equivalence. We will need a proposition from [@CCLV].
\[Prop1\] Fix a natural number $r$ and assume for every natural number $n$ we have a partition $\{A_{i}^{n}\}_{i=1}^{r}$ of $\{1,2,\ldots , n\}$. Then there are natural numbers $\{n_{1}<n_{2}<\cdots\}$ so that if $j\in A_{i}^{n_{j}}$ for some $i \in \{ 1, \ldots, r\}$, then $j\in A_{i}^{n_{k}}$, for all $k\ge j$. Hence, if $A_{i} = \{j\ |\ j\in A_{i}^{n_{j}}\}$ then
\(1) $\{A_{i}\}_{i=1}^{r}$ is a partition of $\mathbb N$.
\(2) If $A_{i} = \{j_{1}^i<j_{2}^i<\cdots \}$ then for every natural number $k$ we have $\{j_{1}^i,j_{2}^i,\ldots , j_{k}^i\}\subset
A_{i}^{n_{j_{k}}}$.
The $R_{\epsilon}$-Conjecture is equivalent to the Finite $R_{\epsilon}$-Conjecture.
Assume the Finite $R_{\epsilon}$-Conjecture is true. Let $\{f_i\}_{i=1}^{\infty}$ be a unit norm Riesz basic sequence in $\H$ with bounds $0<A,B$ and fix $\epsilon >0$. Then there is a natural number $r\in \N$ so that for all $n\in \N$ there is a partition $\{A_j^n\}_{j=1}^{r}$ of $\{1,2,\ldots ,n\}$ and for every $j=1,2,\ldots ,r$ the family $\{f_i\}_{i\in A_j^n}$ is an $\epsilon$-Riesz basic sequence. Choose a partition $\{A_j\}_{j=1}^r$ of $\N$ satisfying Proposition \[Prop1\]. By (2) of this proposition, for each $j=1,2,\ldots r$, the first $n$-elements of $\{f_i\}_{i\in A_j}$ come from one of the $A_{\ell}^m$ and hence form an $\epsilon$-Riesz basic sequence. So $\{f_i\}_{i\in A_j}$ is an $\epsilon$-Riesz basic sequence.
Now assume the the Finite $R_{\epsilon}$-Conjecture fails. Then there is some $0<\epsilon,A,B$, natural numbers $n_1<n_2< \cdots$ and unit norm Riesz basic sequences $\{f_i^r\}_{i=1}^{n_r}$ for $\ell_2^{n_r}$ so that whenever $\{A_j\}_{j=1}^{r}$ is a partition of $\{1,2,\ldots ,n_r\}$ one of the sets $\{f_i^r\}_{i\in A_j}$ is not an $\epsilon$-Riesz basic sequence. Considering $$\{f_i\}_{i=1}^{\infty} = \{f_i^r\}_{i=1,r=1}^{\ n_r , \ \infty}
\in \left ( \sum_{r=1}^{\infty}\oplus \ell_2^{n_r} \right )^{1/2},$$ we see that this family of vectors forms a unit norm Riesz basic sequence with bounds $0<A,B$ but for any natural number $r$ and any partition $\{A_j\}_{j=1}^{r}$ of $\N$ one of the sets $\{f_i\}_{i\in A_j}$ is not an $\epsilon$-Riesz basic sequence.
The Main Theorem {#Main}
================
Our main theorem is:
\[T\] The Paving Conjecture is equivalent to the Paving Conjecture for Triangular matrices.
Since a paving of $T$ is also a paving of $T^{*}$, we only need to show that The Paving Conjecture for Lower Triangular Operators implies the Finite $R_{\epsilon}$-Conjecture. Fix $0< \epsilon, A,B$, fix $n\in \N$ and let $\{f_i\}_{i=1}^{n}$ be a unit norm Riesz basis for $\ell_2^n$ with bounds $A,B$. We choose a natural number $r\in \N$ satisfying: $$1-\frac{B^4}{A^4r} \ge 1- \frac{\epsilon}{2}.$$ We will do the proof in 5 steps. [**Step 1**]{}: There is a partition $\{A_j\}_{j=1}^{r}$ of $\{1,2,\ldots ,n\}$ so that for every $j=1,2,\ldots ,r$ and every $i\in A_j$ and every $1\le k\not= j \le r$ we have: $$\sum_{i\not= \ell \in A_j}|\langle f_i,f_{\ell}\rangle |^2 \le
\sum_{\ell \in A_k}|\langle f_i,f_{\ell}\rangle |^2.$$ The argument for this is due to Halpern, Kaftal and Weiss ([@HKW], Proposition 3.1) so we will outline it for our case. Out of all ways of partitioning $\{1,2,\ldots ,n\}$ into $r$-sets, choose one, say $\{A_j\}_{j=1}^{r}$, which minimizes $$\label{E1}
\sum_{j=1}^{r}\sum_{i\in A_j}\sum_{i\not= \ell \in A_{j}}
|\langle f_i,f_{\ell}\rangle |^2.$$ We now observe that for each $1\le j\le r$, each $i\in A_j$ and all $1\le k\not= j \le r$ we have $$\sum_{i\not= \ell \in A_j}|\langle f_i,f_{\ell}\rangle |^2
\le \sum_{\ell \in A_k}|\langle f_i,f_{\ell}\rangle |^2.$$ To see this, assume this inequality fails. That is, for some $j_0,
i_0,k_0$ as above we have $$\sum_{i_{0}\not= \ell \in A_{j_0}}|\langle f_{i_0},f_{\ell}\rangle |^2 >
\sum_{\ell \in A_{k_0}}|\langle f_{i_0},f_{\ell}\rangle |^2.$$ We define a new partition $\{B_j\}_{j=1}^{r}$ of $\{1,2,\ldots , n\}$ by: $B_j=A_j$ if $j\not= j_0,k_0$; $B_{j_0} = A_{j_0}-\{i_0\}$; $B_{k_0} = A_{k_0} \cup \{i_0\}$. It easily follows that $$\sum_{j=1}^{r}\sum_{i\in B_j}\sum_{i\not= \ell \in B_j}
|\langle f_i,f_{\ell}\rangle |^2 < \sum_{j=1}^{r} \sum_{i\in A_j}
\sum_{i\not= \ell \in A_j}|\langle f_i,f_{\ell}\rangle |^2,$$ which contradicts the minimality of Equation \[E1\].
[**Step 2**]{}: For every $j=1,2,\ldots ,r$ and every $i\in A_j$ we have $$\sum_{i\not= \ell \in A_j}|\langle f_i,f_{\ell}\rangle |^2 \le
\frac{B^2}{r}.$$ Define an operator $Sf = \sum_{i=1}^{n} \langle f,f_i\rangle f_i$. Then, $$\langle Sf,f\rangle = \sum_{i=1}^{r} |\langle f,f_i\rangle |^2,$$ and since $\{f_i\}_{i=1}^{n}$ is a Riesz basis with bounds $A,B$ we have $$A^2I \le S \le B^2I.$$ Now, by Step 1, $$\begin{aligned}
\sum_{i\not= \ell \in A_j}|\langle f_i,f_{\ell}\rangle |^2 &\le&
\frac{1}{r} \left [ \sum_{i\not= \ell \in A_j}|\langle f_i,f_{\ell}\rangle
|^2 + \sum_{j\not= k=1}^{r} \sum_{\ell \in A_k}|\langle f_i,f_{\ell}
\rangle |^2 \right ]\\
&\le& \frac{1}{r}\sum_{i=1}^{n}|\langle f_i,f_{\ell}\rangle |^2\\
&\le& \frac{1}{r}\|S\|\|f_i\|^2 \le \frac{B^2}{r}.\end{aligned}$$ [**Step 3**]{}: For each $j=1,2,\ldots ,r$ and all $i\in A_j$, if $P_{ij}$ is the orthogonal projection of span $\{f_{\ell}\}_{\ell
\in A_j}$ onto span $\{f_{\ell}\}_{i\not= \ell \in A_j}$ then $$\|P_{ij}f_i\|^2 \le \frac{B^4}{A^4r}.$$ Define the operator $S_{ij}$ on span $\{f_{\ell}\}_{\ell \in A_j}$ by $$S_{ij}(f) = \sum_{i\not= \ell \in A_j}\langle f,f_{\ell}\rangle f_{\ell}.$$ Then $A^2I \le S_{ij}\le B^2I$ and $\{S_{ij}^{-1}f_{\ell}\}_{i\not= \ell \in A_j}$ are the dual functionals for the Riesz basic sequence $\{f_{\ell}\}_{i\not= \ell
\in A_j}$. Also, as in Step 1, $A^2I \le S_{ij}\le B^2I$. So by Step 2, $$\begin{aligned}
\|P_{ij}f_i\|^2 &=& \|\sum_{i\not= \ell \in A_j}\langle f_i,f_{\ell}
\rangle S_{ij}^{-1}f_{\ell}\|^2\\
&\le&\|S_{ij}^{-1}\|^2 \|\sum_{i\not= \ell \in A_j}
\langle f_i,f_{\ell}\rangle f_{\ell}\|^2\\
&\le& \frac{B^2}{A^4}\sum_{i\not= \ell \in A_j}|\langle f_i,f_{\ell}
\rangle |^2| \le \frac{B^4}{A^4 r}.\end{aligned}$$ [**Step 4**]{}: Fix $1\le j\le r$ and let $A_j = \{i_1,i_2,
\ldots ,i_k\}$. If we Gram-Schmidt $\{f_{i_{\ell}}\}_{\ell =1}^{k}$ to produce an orthonormal basis $\{e_{i_{\ell}}\}_{\ell =1}^{k}$ then for all $1\le m\le k$ we have $$|\langle f_{i_m},e_{i_m}\rangle |^2 \ge 1- \frac{\epsilon}{2}.$$ Fix $1\le m\le k$ and let $Q_m$ be the orthogonal projection of span $\{e_{i_{\ell}}\}_{\ell = 1}^{k}$ onto span $\{e_{i_{\ell}}\}_{\ell =1}^{m}$ = span $\{f_{i_{\ell}}\}_{\ell =1}^{m}$. By Step 3, $$\|Q_{m}f_{i_m}\|^2 \le \|P_{mj}f_{i_m}\|^2 \le \frac{B^4}{A^4r}.$$ Since $$f_{i_m} = \sum_{\ell =1}^{m} \langle f_{i_{\ell}},e_{i_{\ell}}\rangle e_{i_{\ell}},$$ we have $$\begin{aligned}
|\langle f_{i_m},e_{i_m}\rangle |^2 &=& \|f_{i_m}\|^2 -
\|Q_{m-1}f_{i_m}\|^2\\
&\ge& 1-\frac{B^4}{A^4r} \ge 1-\frac{\epsilon}{2},\end{aligned}$$ where the last inequality follows from our choice of $r$.
[**Step 5**]{}: We complete the proof. Let $$M = \left ( \langle f_{i_s},e_{i_t}\rangle \right ) _{s\not= t =1}^{k},$$ where by this notation we mean the $k\times k$-matrix with zero diagonal and the given values off the diagonal. By the Gram-Schmidt Process, $M$ is a lower triangular matrix with zero diagonal. Define an operator $T:\ell_2^k \rightarrow span \ \{e_{i_{\ell}}\}_{\ell =1}^{k}$ by $$T\left ( (a_{i_{\ell}})_{\ell = 1}^{k} \right ) =
\sum_{\ell =1}^{k} a_{i_{\ell}}f_{i_{\ell}}.$$ If $K$ is the matrix of $T$ with respect to the orthonormal basis $\{e_{i_{\ell}}\}_{\ell =1}^{k}$ and $D=D(K)$ is the diagonal of $K$ then $M = (K-D)^{*}$ and so $$\|M\| \le \|K\| + \|D\| = \|T\|+1 \le B+1.$$ By The Paving Conjecture for lower triangular matrices, there is a natural number $L_j$ (which is a function of $0 < \epsilon$ and $B$ only) and a partition $\{B_{\ell}^{j}\}_{\ell =1}^{L_j}$ of $\{i_1,i_2 \ldots , i_k\}$ so that $$\|Q_{B_{\ell}^j}MQ_{B_{\ell}^j}\| \le \frac{\epsilon}{2},$$ for all $\ell = 1,2, \ldots ,L_j$ ($Q_{B_{\ell}^j}$ was defined in the introduction). Now, for all scalars $(a_{i_s})_{i_s \in B_{\ell}^j}$, if $$f = \sum_{i_s \in B_{\ell}^j}a_{i_s}f_{i_s},$$ then $$\begin{aligned}
\|\sum_{i_s \in B_{\ell}^j}a_{i_s}f_{i_s}\|
&=& \| D(f) + Q_{B_{\ell}^j}M^{*}Q_{B_{\ell}^j}(f)
\|\\
&\ge& \|Df\| - \|Q_{B_{\ell}^j}M^{*}Q_{B_{\ell}^j}(f)\|\\
&\ge& (1-\frac{\epsilon}{2})\|f\| - \frac{\epsilon}{2}\|f\|\\
&\ge& (1-\epsilon)\|f\|.\end{aligned}$$ Similarly, $$\|\sum_{i_s \in B_{\ell}^j}a_{i_s}f_{i_s} \| \le (1+\epsilon)\|f\|.$$ It follows that $\{f_i\}_{i\in B_{\ell}^j}$ is an $\epsilon$-Riesz basic sequence for all $j=1,2,\ldots,r$ and all $\ell = 1,2,\ldots ,L_j$. Hence, the Finite $R_{\epsilon}$-Conjecture holds which completes the proof of the theorem.
Let us make an observation concerning the proof of the main theorem.
Let $\{f_i\}_{i=1}^{\infty}$ be a sequence of vectors in a Hilbert space $\H$. For each $i=1,2,\ldots $ let $P_i$ be the orthogonal projection of $\H$ onto span $\{f_{\ell}\}_{i\not= \ell \in \N}$. Our sequence is said to be [**$\epsilon$-minimal**]{} if $\|P_i\| \le \epsilon$ for all $i=1,2,\ldots$.
The first three steps of the proof of Theorem \[T\] yields:
If $\{f_i\}_{i=1}^{\infty}$ is a unit norm Riesz basic sequence in a Hilbert space $\H$ then for every $\epsilon >0$ there is a partition $\{A_j\}_{j=1}^{r}$ of $\N$ so that for all $j=1,2,\ldots ,r$, the family $\{f_i\}_{i\in A_j}$ is $\epsilon$-minimal.
[10]{} C.A. Akemann and J. Anderson, [*Lyapunov theorems for operator algebras*]{}, Mem. AMS [**94**]{} (1991).
J. Anderson, [*Extreme points in sets of positive linear maps on $B(\H)$*]{}, Jour. Functional Analysis [**31**]{} (1979) 195–217.
R. Balan, P.G. Casazza, C. Heil and Z. Landau, [*Density, overcompleteness and localization of frames. I. Theory*]{}, Preprint.
R. Balan, P.G. Casazza, C. Heil and Z. Landau, [*Density, overcompleteness and localization of frames. II. Gabor systems*]{}, Preprint.
K. Berman, H. Halpern, V. Kaftal and G. Weiss, [*Matrix norm inequalities and the relative Dixmier property*]{}, Integ. Eqns. and Operator Theory [**11**]{} (1988) 28–48.
J. Bourgain and L. Tzafriri, [*Invertibility of “large” submatrices and applications to the geometry of Banach spaces and Harmonic Analysis*]{}, Israel J. Math. [**57**]{} (1987) 137–224.
P.G. Casazza, O. Christensen, A. Lindner and R. Vershynin, [*Frames and the Feichtinger conjecture*]{}, Proceedings of AMS, [**133**]{} No. 4 (2005) 1025–1033.
P.G. Casazza, D. Edidin, D. Kalra and V. Paulsen, The Kadison-Singer Problem and Projections, Preprint.
P.G. Casazza and J.C. Tremain, [*The Kadison-Singer Problem in Mathematics and Engineering*]{}, Proceedings of the National Academy of Sciences, [**103**]{} No. 7 (2006) 2032-2039.
P.G. Casazza, M. Fickus, J.C. Tremain, and E. Weber, [*The Kadison-Singer Problem in Mathematics and Engineering: Part II: A detailed account*]{}. (Accepted for The Proceedings of The 2005 Great Plains Operator Theory Symposium (GPOTS), Contemp. Math., Amer. Math. Soc., to appear in 2006).
P.G. Casazza and R. Vershynin, [*Kadison-Singer meets Bourgain-Tzafriri*]{}, Preprint.
K.H. Gröchenig, [*Localized frames are finite unions of Riesz sequences*]{}, Adv. Comp. Math. [**18**]{} (2003) 149–157.
H. Halpern, V. Kaftal and G. Weiss, [*Matrix pavings and Laurent operators*]{}, J. Op. Th. [**16**]{} (1986) 121–140.
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[^1]: The first author was supported by NSF DMS 0405376
|
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"pile_set_name": "ArXiv"
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|
---
abstract: 'In [@anzman-det] we gave a variational definition of the nonlinear membrane energy under the constraint “$\det\nabla u\not=0$". In this paper we obtain the nonlinear membrane energy under the more realistic constraint “$\det\nabla u>0$".'
address:
- 'Institut für Mathematik, Universität Zürich, Winterthurerstrasse 190, CH-8057 Zürich, Switzerland.'
- '“Equipe AVA (Analyse Variationnelle et Applications)", Centre Universitaire de Formation et de Recherche de Nîmes, Site des Carmes, Place Gabriel Péri - Cedex 01 - 30021 Nîmes, France.I3M (Institut de Mathématiques et Modélisation de Montpellier) UMR - CNRS 5149, Université Montpellier II, Place Eugène Bataillon, 34090 Montpellier, France.'
author:
- Omar Anza Hafsa
- 'Jean-Philippe Mandallena'
title: 'The nonlinear membrane energy: variational derivation under the constraint $``\det\nabla u>0"$'
---
Introduction
============
Consider an elastic material occupying in a reference configuration the bounded open set $\Sigma_\eps\subset\RR^3$ given by $$\Sigma_\eps:=\Sigma\times\left]-{\eps\over 2},{\eps\over 2}\right[,$$ where $\eps>0$ is very small and $\Sigma\subset\RR^2$ is Lipschitz, open and bounded. A point of $\Sigma_\eps$ is denoted by $(x,x_3)$ with $x\in\Sigma$ and $x_3\in]-{\eps\over 2},{\eps\over 2}[$. Let $
W:\MM^{3\times 3}\to[0,+\infty]
$ be the stored-energy function supposed to be [*continuous*]{} and [*coercive*]{}, i.e., $W(F)\geq C|F|^p$ for all $F\in\MM^{3\times 3}$ and some $C>0$. In order to take into account the important physical properties that the interpenetration of matter does not occur and that an infinite amount of energy is required to compress a finite volume into zero volume, i.e., $$W(F)\to+\infty\ \hbox{ as }\ \det F\to 0,$$ where $\det F$ denotes the determinant of the $3\times 3$ matrix $F$, we assume that: $$\label{Nim}
\hbox{\em $W(F)=+\infty$ if and only if $\det F\leq 0$};$$ $$\label{Icv}
\hbox{\em for every $\delta>0$, there exists $c_\delta>0$ such that for all $F\in\MM^{3\times 3}$},$$ $$\hbox{\em if }\det F\geq\delta\hbox{\em\ then } W(F)\leq c_\delta(1+|F|^p).$$ Our goal is to show that as $\eps\to 0$ the three-dimensional free energy functional ${E}_\eps:W^{1,p}(\Sigma_\eps;\RR^3)\to[0,+\infty]$ (with $p>1$) defined by $$\label{SEF}
{E}_\eps(u):={1\over\eps}\int_{\Sigma_\eps}W\big(\nabla u(x,x_3)\big)dxdx_3$$ converges in a variational sense (see Definition \[variationalconvergence\]) to the two-dimensional free energy functional ${E}_{\rm mem}:W^{1,p}(\Sigma;\RR^3)\to[0,+\infty]$ given by $$\label{MSEF}
{E}_{\rm mem}(v):=\int_\Sigma W_{\rm mem}\big(\nabla v(x)\big)dx$$ with $W_{\rm mem}:\MM^{3\times 2}\to[0,+\infty]$. Usually, $E_{\rm mem}$ is called the nonlinear membrane energy associated with the two-dimensional elastic material with respect to the reference configuration $\Sigma$. Furthermore we wish to give a representation formula for $W_{\rm mem}$.
To our knowledge, the problem of giving a variational definition of the nonlinear membrane energy was studied for the first time by Percivale in [@percivale]. His paper deals with the constraint “$\det\nabla u>0$" but seems to contain some mistakes (it never was published). Nevertheless, Percivale introduced the “good" formula for $W_{\rm mem}$, i.e., $W_{\rm mem}=\mathcal{Q}W_0$ where $W_0$ is given by (\[PercivaleFormula\]) and $\mathcal{Q}W_0$ denotes the quasiconvex envelope of $W_0$. Then, in [@ledretraoult1] Le Dret and Raoult gave a complete proof of percivale’s conjecture in the simpler case where $W$ is of $p$-polynomial growth, i.e., $W(F)\leq c(1+|F|^p)$ for all $F\in\MM^{3\times 3}$ and some $c>0$. Although the $p$-polynomial growth case is not compatible with (\[Nim\]) and (\[Icv\]) their paper established a suitable framework to deal with dimensional reduction problems (it is the point of departure of many works on the subject). After Percivale, Ben Belgacem also considered the constraint “$\det\nabla u>0$". In [@benbelgacem1 Theorem 1] he announced to have succeed to handle (\[Nim\]) and (\[Icv\]). In [@benbelgacem], which is the paper corresponding to the note [@benbelgacem1], the statement [@benbelgacem1 Theorem 1] is partly proved (however, a more detailled proof, but not fully complete, can be found in his thesis [@benbelgacem-thesis]). Moreover, for Ben Belgacem $W_{\rm mem}=\mathcal{Q}\mathcal{R}W_0$ where $\mathcal{R}W_0$ denotes the rank one convex envelope of $W_0$ (in fact, as we proved in [@anzman2; @anzman-det], $\mathcal{Q}\mathcal{R}W_0=\mathcal{Q}W_0$). Nevertheless, Ben Belgacem’s thesis highlighted the role of approximation theorems for Sobolev functions by smooth immersions in the studying of the passage 3D-2D in presence of (\[Nim\]) and (\[Icv\]). Recently, in [@anzman-det] we gave a variational definition of the nonlinear membrane energy under the constraint “$\det\nabla u\not=0$". In the present paper, using the same method as in [@anzman-det] and some results of Ben Belgacem’s thesis (mainly, Theorem \[LemmaBBB\] and Lemma \[LeMMaBiS\]), we obtain the nonlinear membrane energy under the more realistic constraint “$\det\nabla u>0$".
An outline of the paper is as follows. The variational convergence of $E_\eps$ to $E_{\rm mem}$ as $\eps\to 0$ as well as a representation formula for $W_{\rm mem}$ are given by Corollary \[corollary\] in Sect. 2.4. Corollary \[corollary\] is a consequence of Theorems \[first\_main\_result\], \[AdDitioNal\] and \[anzman\]. Roughly, Theorems \[first\_main\_result\] and \[AdDitioNal\] establish the existence of the variational limit of ${E}_\eps$ as $\eps\to 0$ (see Sect. 2.2), and Theorem \[anzman\] gives an integral representation for the corresponding variational limit, and so a representation formula for $W_{\rm mem}$ (see Sect. 2.3). In fact, Theorem \[anzman\] is obtained from Theorem \[AdDitioNal\] which furnishes a “simplified" formula for the variational limit.
Theorem \[first\_main\_result\] is proved in Section 4. The principal ingredients are Theorem \[AdDitioNal\] and Theorem \[basictheorem\] whose proof (given in Section 3) uses an interchange theorem of infimum and integral that we obtained in [@anzman]. (Note that the techniques used to prove Theorems \[first\_main\_result\] and \[basictheorem\] are the same as in [@anzman-det Sections 3 and 4].)
Theorem \[AdDitioNal\] is proved is Section 5. The main arguments are two approximation theorems developed by Ben Belgacem-Bennequin (see [@benbelgacem-thesis]) and Gromov-Eliashberg (see [@gromovEliashberg]). These theorems are stated in Appendix A.
Theorem \[anzman\] is proved in [@anzman-det Appendix A] (see also [@anzman2]).
Results
=======
Variational convergence
-----------------------
To accomplish our asymptotic analysis, we use the notion of convergence introduced by Anzellotti, Baldo and Percivale in [@anzebalper] in order to deal with dimension reduction problems in mechanics. Let $\pi=\{\pi_\eps\}_\eps$ be the family of maps $\pi_\eps:W^{1,p}(\Sigma_\eps;\RR^3)\to W^{1,p}(\Sigma;\RR^3)$ defined by $$\pi_\eps(u):={1\over\eps}\int_{-{\eps\over 2}}^{\eps\over 2}u(\cdot,x_3)dx_3.$$
\[variationalconvergence\] We say that ${E}_\eps$ $\Gamma(\pi)$-converges to ${E}_{\rm mem}$ as $\eps\to 0$, and we write ${E}_{\rm mem}=\Gamma(\pi)\hbox{\rm -}\lim_{\eps\to 0}{E}_\eps$, if the following two assertions hold[:]{}
- for all $v\in W^{1,p}(\Sigma;\RR^3)$ and all $\{u_\eps\}_\eps\subset W^{1,p}(\Sigma_\eps;\RR^3)$, $$\hbox{if }\pi_\eps(u_\eps)\to v\hbox{ in }L^{p}(\Sigma;\RR^3)\hbox{ then }{E}_{\rm mem}(v)\leq\liminf_{\eps\to 0}{E}_\eps(u_\eps);$$
- for all $v\in W^{1,p}(\Sigma;\RR^3)$, there exists $\{u_\eps\}_\eps\subset W^{1,p}(\Sigma_\eps;\RR^3)$ such that[:]{} $$\pi_\eps(u_\eps)\to v\hbox{ in }L^{p}(\Sigma;\RR^3)\hbox{ and }{E}_{\rm mem}(v)\geq\limsup_{\eps\to 0}{E}_\eps(u_\eps).$$
In fact, Definition \[variationalconvergence\] is a variant of De Giorgi’s $\Gamma$-convergence. This is made clear by Lemma \[link\]. Consider $\mathcal{\I}_\eps:W^{1,p}(\Sigma;\RR^3)\to[0,+\infty]$ defined by $$\mathcal{\I}_\eps(v):=\inf\Big\{{E}_\eps(u):\pi_\eps(u)=v\Big\}.$$
\[gammavariationalconvergence\] We say that $\mathcal{\I}_\eps$ $\Gamma$-converges to ${E}_{\rm mem}$ as $\eps\to 0$, and we write ${E}_{\rm mem}=\Gamma\hbox{\rm -}\lim_{\eps\to 0}\mathcal{\I}_\eps$ if for every $v\in W^{1,p}(\Sigma;\RR^3)$, $$\left(\Gamma\hbox{-}\liminf_{\eps\to 0}\mathcal{\I}_\eps\right)(v)=\left(\Gamma\hbox{-}\limsup_{\eps\to 0}\mathcal{\I}_\eps\right)(v)=E_{\rm mem}(v),$$ where $
\left(\Gamma\hbox{-}\liminf_{\eps\to 0}\mathcal{\I}_\eps\right)(v):=\inf\big\{\liminf_{\eps\to 0}\mathcal{\I}_\eps(v_\eps):v_\eps\to v\hbox{ in }L^{p}(\Sigma;\RR^3)\big\}
$ and $
\left(\Gamma\hbox{-}\limsup_{\eps\to 0}\mathcal{\I}_\eps\right)(v):=\inf\big\{\limsup_{\eps\to 0}\mathcal{\I}_\eps(v_\eps):v_\eps\to v\hbox{ in }L^{p}(\Sigma;\RR^3)\big\}.
$
For a deeper discussion of the $\Gamma$-convergence theory we refer to the book [@dalmaso0]. Clearly, Definition \[gammavariationalconvergence\] is equivalent to assertions (i) and (ii) in definition \[variationalconvergence\] with “$\pi(u_\eps)\to v$” replaced by “$v_\eps\to v$”. It is then obvious that
\[link\] ${E}_{\rm mem}=\Gamma(\pi)\hbox{\rm -}\lim_{\eps\to 0}{E}_\eps$ if and only if ${E}_{\rm mem}=\Gamma\hbox{\rm -}\lim_{\eps\to 0}\mathcal{\I}_\eps$.
The $\Gamma(\pi)$-convergence of ${E}_\eps$ in (\[SEF\]) to $E_{\rm mem}$ in (\[MSEF\]) as $\eps\to 0$ as well as a representation formula for $W_{\rm mem}$ are given by Corollary \[corollary\]. It is a consequence of Theorems \[first\_main\_result\], \[AdDitioNal\] and \[anzman\]. Roughly, Theorems \[first\_main\_result\] and \[AdDitioNal\] establish the existence of the $\Gamma(\pi)$-limit of ${E}_\eps$ as $\eps\to 0$ (see Sect. 2.2), and Theorem \[anzman\] gives an integral representation for the corresponding $\Gamma(\pi)$-limit, and so a representation formula for $W_{\rm mem}$ (see Sect. 2.3).
$\Gamma$-convergence of $\mathcal{\I}_\eps$ as $\eps\to 0$
----------------------------------------------------------
Denote by $\C^1(\overline{\Sigma};\RR^3)$ the space of all restrictions to $\overline{\Sigma}$ of $C^1$-differentiable functions from $\RR^2$ to $\RR^3$, and set $$\C^1_*(\overline{\Sigma};\RR^3):=\Big\{v\in\C^1(\overline{\Sigma};\RR^3):\partial_1v(x)\land\partial_2 v(x)\not=0\hbox{ for all }x\in\overline{\Sigma}\Big\},$$ where $\partial_1v(x)$ (resp. $\partial_2 v(x)$) denotes the partial derivative of $v$ at $x=(x_1,x_2)$ with respect to $x_1$ (resp. $x_2$). (In fact, $\C^1_*(\overline{\Sigma};\RR^3)$ is the set of all $C^1$-immersions from $\overline{\Sigma}$ to $\RR^3$.) Let ${\mathcal E}:W^{1,p}(\Sigma;\RR^3)\to[0,+\infty]$ be defined by $${\mathcal E}(v):=\left\{
\begin{array}{cl}
\displaystyle\int_\Sigma W_0\big(\nabla v(x)\big)dx&\hbox{if }v\in\C^1_*(\overline{\Sigma};\RR^3)\\
+\infty&\hbox{otherwise,}
\end{array}
\right.$$ where $W_0:\MM^{3\times 2}\to[0,+\infty]$ is given by $$\label{PercivaleFormula}
W_0(\xi):=\inf_{\zeta\in\RR^3}W(\xi\mid\zeta)$$ with $(\xi\mid\zeta)$ denoting the element of $\MM^{3\times 3}$ corresponding to $(\xi,\zeta)\in\MM^{3\times 2}\times\RR^3$. (As $W$ is coercive, it is easy to see that [*$W_0$ is coercive*]{}, i.e., $W_0(\xi)\geq C|\xi|^p$ for all $\xi\in\MM^{3\times 2}$ and some $C>0$.) The following lemma gives three elementary properties of $W_0$ (the proof is left to the reader). Note that conditions (\[Nim\]) and (\[Icv\]) imply $W_0$ is not of $p$-polynomial growth.
\[propertiesofW\_0\] Denote by $\xi_1\land\xi_2$ the cross product of vectors $\xi_1,\xi_2\in\RR^3$.
- $W_0$ is continuous.
- If [(\[Nim\])]{} holds then
- If [(\[Icv\])]{} holds then[:]{} $$\label{Icvbis}
\hbox{for all $\delta>0$, there exists $c_\delta>0$ such that for all $\xi=(\xi_1\mid\xi_2)\in\MM^{3\times 2}$},$$ $$\hbox{if }|\xi_1\land\xi_2|\geq\delta\hbox{ then } W_0(\xi)\leq c_\delta(1+|\xi|^p).$$
Taking Lemma \[link\] into account, we see that the existence of the $\Gamma(\pi)$-limit of $E_\eps$ as $\eps\to 0$ follows from Theorem \[first\_main\_result\].
\[first\_main\_result\] Let assumptions [(\[Nim\])]{} and [(\[Icv\])]{} hold. Then $\Gamma\hbox{\rm -}\lim_{\eps\to 0}\mathcal{\I}_\eps=\overline{{\mathcal E}}$ with $\overline{{\mathcal E}}:W^{1,p}(\Sigma;\RR^3)\to[0,+\infty]$ given by $$\overline{{\mathcal E}}(v):=\inf\left\{\liminf_{n\to+\infty}{\mathcal E}(v_n):W^{1,p}(\Sigma;\RR^3)\ni v_n\to v\hbox{ in }L^{p}(\Sigma;\RR^3)\right\}.$$
The proof of Theorem \[first\_main\_result\] is established in Section 4. It uses Theorem \[basictheorem\] (see Section 3) and Theorem \[AdDitioNal\].
\[AdDitioNal\] If [(\[Icvbis\])]{} holds then $\overline{{\mathcal E}}(v)={\mathcal I}(v)$ for all $v\in W^{1,p}(\Sigma;\RR^3)$, where ${\mathcal I}:W^{1,p}(\Sigma;\RR^3)\to[0,+\infty]$ is given by $${\mathcal I}(v):=\inf\left\{\liminf_{n\to+\infty}\int_\Sigma W_0\big(\nabla v_n(x)\big)dx:W^{1,p}(\Sigma;\RR^3)\ni v_n\to v\hbox{ in }L^{p}(\Sigma;\RR^3)\right\}.$$
Theorem \[AdDitioNal\] is proved in Section 6 by using two approximation theorems developed by Ben Belgacem-Bennequin (see [@benbelgacem-thesis]) and Gromov-Eliashberg (see [@gromovEliashberg]). These theorems are stated in Appendix A.
Integral representation of $\mathcal{I}$
----------------------------------------
From now on, given a bounded open set $D\subset\RR^2$ with $|\partial D|=0$, we denote by $\Aff(D;\RR^3)$ the space of all continuous piecewise affine functions from $D$ to $\RR^3$, i.e., [*$v\in\Aff(D;\RR^3)$ if and only if $v$ is continuous and there exists a finite family $(D_i)_{i\in I}$ of open disjoint subsets of $D$ such that $|\partial D_i|=0$ for all $i\in I$, $|D\setminus \cup_{i\in I} D_i|=0$ and for every $i\in I$, $\nabla v(x)=\xi_i$ in $D_i$ with $\xi_i\in\MM^{3\times 2}$*]{} (where $|\cdot|$ denotes the Lebesgue measure in $\RR^2$). Define $\Z W_0:\MM^{3\times 2}\to[0,+\infty]$ by $$\label{DefinitionofZW0}
\Z W_0(\xi):=\inf\left\{\int_Y W_0\big(\xi+\nabla\phi(y)\big)dy:\phi\in \Aff_0(Y;\RR^3)\right\}$$ where $Y:=]0,1[^2$ and $\Aff_0(Y;\RR^3):=\{\phi\in \Aff(Y;\RR^3):\phi=0\hbox{ on }\partial Y\}$. (As $W_0$ is coercive, it is easy to see that [*$\Z W_0$ is coercive*]{}.) Recall the definitions of quasiconvexity and quasiconvex envelope:
\[DEFofQUasiRankoNEConvexityandEnvelOPe\] Let $f:\MM^{3\times 2}\to[0,+\infty]$ be a Borel measurable function.
- We say that $f$ is quasiconvex if for every $\xi\in\MM^{3\times 2}$, every bounded open set $D\subset\RR^2$ with $|\partial D|=0$ and every $\phi\in W^{1,\infty}_0(D;\RR^3)$, $$f(\xi)\leq{1\over|D|}\int_D f(\xi+\nabla\phi(x))dx.$$
- By the quasiconvex envelope of $f$, we mean the unique function (when it exists) $\mathcal{Q}f:\MM^{3\times 2}\to[0,+\infty]$ such that:
- $\mathcal{Q}f$ is Borel measurable, quasiconvex and $\mathcal{Q}f\leq f$;
- for all $g:\MM^{3\times 2}\to[0,+\infty]$, if $g$ is Borel measurable, quasiconvex and $g\leq f$, then $g\leq\mathcal{Q}f$.
(Usually, for simplicity, we say that $\mathcal{Q}f$ is the greatest quasiconvex function which less than or equal to $f$.)
Under [(\[Icvbis\])]{}, we proved that $\Z W_0$ is of $p$-polynomial growth and so continuous (see [@anzman-det Propositions A.3 and A.1(iii)]) and that $\Z W_0$ is the quasiconvex envelope of $W_0$, i.e., $\Z W_0=\mathcal{Q}W_0$ (see [@anzman-det Proposition A.5]). Taking Theorems \[first\_main\_result\] and \[AdDitioNal\] together with Lemmas \[link\] and \[propertiesofW\_0\](iii) into account, we see that Theorem \[anzman\] gives an integral representation for the $\Gamma(\pi)$-limit of $E_\eps$ as $\eps\to 0$ as well as a representation formula for $W_{\rm mem}$.
\[anzman\] If [(\[Icvbis\])]{} holds then for every $v\in W^{1,p}(\Sigma;\RR^3)$, $${\mathcal I}(v)=\int_\Sigma \mathcal{Q}W_0\big(\nabla v(x)\big)dx.$$
Theorem \[anzman\] is proved in [@anzman-det Appendix A] (see also [@anzman2]).
$\Gamma(\pi)$-convergence of ${E}_\eps$ to ${E}_{\rm mem}$ as $\eps\to 0$
-------------------------------------------------------------------------
According to Lemmas \[link\] and Lemma \[propertiesofW\_0\](iii), a direct consequence of Theorems \[first\_main\_result\], \[AdDitioNal\] and \[anzman\] is the following.
\[corollary\] Let assumptions [(\[Nim\])]{} and [(\[Icv\])]{} hold. Then as $\eps\to 0$, ${E}_\eps$ in [(\[SEF\])]{} $\Gamma(\pi)$-converge to ${E}_{\rm mem}$ in [(\[MSEF\])]{} with $W_{\rm mem}=\mathcal{Q}W_0$.
Corollary \[corollary\] can be applied when $W:\MM^{3\times 3}\to[0,+\infty]$ is given by $$W(F):=
h(\det F)+|F|^p,$$ where $h:\RR\to[0,+\infty]$ is a continuous function such that:
- [$h(t)=+\infty$ if and only if $t\leq0$]{};
- [for every $\delta>0$, there exists $r_\delta>0$ such that $h(t)\leq r_\delta$ for all $t\geq\delta$]{}.
Representation of ${\mathcal E}$
================================
The goal of this section is to show Theorem \[basictheorem\]. To this end, we begin by proving two lemmas.
For every $v\in\C^1_*(\overline{\Sigma};\RR^3)$ and $j\geq 1$, we define the multifunction $\Lambda^j_v:\overline{\Sigma}\dto\RR^3$ by $$\Lambda_v^j(x):=\left\{\zeta\in\RR^3: \det(\nabla v(x)\mid\zeta)\ge {1\over j}\right\}.$$
\[Lemma2\] Let $v\in\C^1_*(\overline{\Sigma};\RR^3)$. Then[:]{}
- for every $j\geq 1$, $\Lambda_v^j$ is a nonempty convex closed-valued lower semicontinuous[^1] multifunction[;]{}
- for every $x\in\overline{\Sigma}$, $\Lambda_v^1(x)\subset\cdots\subset\Lambda_v^j(x)\subset\cdots\subset\cup_{j\ge 1}\Lambda_v^j(x)=\Lambda_v(x)$, where $\Lambda_v(x):=\{\zeta\in\RR^3: \det(\nabla v(x)\mid\zeta)>0\}$.
\(ii) is obvious. Prove then (i). Let $j\geq 1$. It is easy to see that for every $x\in\overline{\Sigma}$, $\Lambda_v^j(x)$ is nonempty, convex and closed. Let $X$ be a closed subset of $\RR^3$, let $x\in\overline{\Sigma}$, and let $\{x_n\}_{n\geq 1}\subset\overline{\Sigma}$ such that $|x_n-x|\to 0$ as $n\to+\infty$ and $\Lambda^j_{v}(x_n)\subset X$ for all $n\geq 1$. Let $\zeta\in \Lambda_v^j(x)$ and let $\{\zeta_m\}_{m\geq 1}\subset\RR^3$ be given by $\zeta_m:=\zeta+{1\over m}\zeta$. Then, for every $m\geq 1$, $$\label{DetEq}
\det\big(\nabla v(x)\mid\zeta_m\big)=\det\big(\nabla v(x)\mid\zeta\big)+{1\over m}\det\big(\nabla v(x)\mid\zeta\big)\geq{1\over j}+{1\over mj}.$$ Fix any $m\geq 1$. Since $\det(\nabla v(x_n)\mid\zeta_m)\to\det(\nabla v(x)\mid\zeta_m)$ as $n\to +\infty$, using (\[DetEq\]) we see that $\det(\nabla v(x_{n_0})\mid\zeta_m)>{1\over j}$ for some $n_0\geq 1$, so that $\zeta_m\in\Lambda_v^j(x_{n_0})$. Thus $\zeta_m\in X$ for all $m\geq 1$. As $X$ is closed we have $\zeta=\lim_{m\to +\infty}\zeta_m\in X$.
In the sequel, given $\Lambda:\overline{\Sigma}\dto\RR^3$ we set $$C(\overline{\Sigma};\Lambda):=\Big\{\phi\in C\big(\overline{\Sigma};\RR^3\big):\phi(x)\in\Lambda(x)\hbox{ for all }x\in\overline{\Sigma}\Big\},$$ where $C(\overline{\Sigma};\RR^3)$ denotes the space of all continuous functions from $\overline{\Sigma}$ to $\RR^3$.
\[Lemma3\] Given $v\in\C^1_*(\Sigma;\RR^3)$ and $j\geq 1$, if [(\[Icv\])]{} holds, then $$\inf_{\phi\in C(\overline{\Sigma};\Lambda_v^j)}\int_{\Sigma}W\big(\nabla v(x)\mid\phi(x)\big)dx=\int_\Sigma \inf_{\zeta\in\Lambda_v^j(x)}W\big(\nabla v(x)\mid\zeta\big)dx.$$
To prove Lemma \[Lemma3\] we need the following interchange theorem of infimum and integral (that we proved in [@anzman Corollary 5.4]).
\[AHM\] Let $\Gamma:\overline{\Sigma}\dto\RR^3$ and let $f:\overline{\Sigma}\times\RR^3\to[0,+\infty]$. Assume that[:]{}
- $f$ is a Carathéodory integrand[;]{}
- $\Gamma$ is a nonempty convex closed-valued lower semicontinuous multifunction[;]{}
- $C(\overline{\Sigma};\Gamma)\not=\emptyset$ and for every $\phi,\hat\phi\in C(\overline{\Sigma};\Gamma)$, $$\int_\Sigma \max_{\alpha\in[0,1]}f\big(x,\alpha\phi(x)+(1-\alpha)\hat\phi(x)\big)dx<+\infty.$$
Then, $$\inf_{\phi\in C(\overline{\Sigma};\Gamma)}\int_{\Sigma}f\big(x,\phi(x)\big)dx=\int_\Sigma\inf_{\zeta\in\Gamma(x)}f(x,\zeta)dx.$$
[*Proof of Lemma [\[Lemma3\].]{}* ]{}Since $W$ is continuous, (H$_1$) holds with $f(x,\zeta)=W(\nabla v(x)\mid\zeta)$. Lemma \[Lemma2\] shows that (H$_2$) is satisfied with $\Gamma=\Lambda_v^j$, and $C(\overline{\Sigma};\Lambda^j_v)\not=\emptyset$ (for example $\Phi:\overline{\Sigma}\to\RR^3$ defined by (\[ExampleOfContinuousSelection\]) belongs to $C(\overline{\Sigma};\Lambda^j_v)$). Given $\phi,\hat\phi\in C(\overline{\Sigma};\Lambda^j_v)$, it is clear that $
\det(\nabla v(x)\mid\alpha\phi(x)+(1-\alpha)\hat\phi(x))\geq{1/ j}
$ for all $\alpha\in[0,1]$ and all $x\in\overline{\Sigma}$. By (\[Icv\]) there exists $c>0$ depending only on $j$, $v$, $\phi$ and $\hat\phi$ such that $
W(\nabla v(x)\mid\alpha\phi(x)+(1-\alpha)\hat\phi(x))\leq c
$ for all $x\in\overline{\Sigma}$. Thus (H$_3$) is verified with $f(x,\zeta)=W(\nabla v(x)\mid\zeta)$ and $\Gamma=\Lambda_v^j$, and Lemma \[Lemma3\] follows from Lemma \[AHM\].$\square$
Here is our (non integral) representation theorem for ${\mathcal E}$.
\[basictheorem\] If [(\[Nim\])]{} and [(\[Icv\])]{} hold, then for every $v\in C^1_*(\overline{\Sigma};\RR^3)$, $$\label{NIRF}
{\mathcal E}(v)=\inf_{j\geq 1}\inf_{\phi\in C(\overline{\Sigma};\Lambda_v^j)}\int_{\Sigma}W\big(\nabla v(x)\mid\phi(x)\big)dx.$$
Fix $v\in \C^1_*(\overline{\Sigma};\RR^3)$ and denote by $\hat{\mathcal E}(v)$ the right-hand side of (\[NIRF\]). It is easy to verify that ${\mathcal E}(v)\leq\hat{\mathcal E}(v)$. We are thus reduced to prove that $$\label{inequality}
\hat{\mathcal E}(v)\leq{\mathcal E}(v).$$ Using Lemma \[Lemma3\], we obtain $$\label{inequality1}
\hat{\mathcal E}(v)\leq\inf_{j\geq 1}\int_{\Sigma}\inf_{\zeta\in\Lambda^j_v(x)}W\big(\nabla v(x)\mid\zeta\big)dx.$$ Consider the continuous function $\Phi:\overline{\Sigma}\to\RR^3$ defined by $$\label{ExampleOfContinuousSelection}
\Phi(x):={{\partial_1 v(x)\land\partial_2 v(x)}\over \vert\partial_1v(x)\land\partial_2v(x)\vert^2}.$$ Then, $\det(\nabla v(x)\mid\Phi(x))=1$ for all $x\in\overline{\Sigma}$. Using [(\[Icv\])]{} we deduce that there exists $c>0$ depending only on $p$ such that $$\int_\Sigma\inf_{\zeta\in\Lambda^{1}_v(x)}W(\nabla v(x)\mid\zeta)dx\leq c\big(|\Sigma|+\|\nabla v\|^p_{L^p(\Sigma;\MM^{3\times 2})}+\|\Phi\|^p_{L^p(\Sigma;\RR^3)}\big).$$ It follows that $\inf_{\zeta\in\Lambda^{1}_v(\cdot)}W(\nabla v(\cdot)\mid\zeta)\in L^1(\Sigma)$. From Lemma \[Lemma2\](i) and (ii), we see that $\{\inf_{\zeta\in\Lambda^j_v(\cdot)}W(\nabla v(\cdot)\mid\zeta)\}_{j\geq 1}$ is non-increasing, and that for every $x\in \overline{\Sigma}$, $$\label{equality}
\inf_{j\geq 1}\inf_{\zeta\in\Lambda^j_v(x)}W\big(\nabla v(x)\mid\zeta\big)=W_0\big(\nabla v(x)\big),$$ and (\[inequality\]) follows from (\[inequality1\]) and ([\[equality\]]{}) by using Lebesgue’s dominated convergence theorem.
Existence of $\Gamma\hbox{-}\lim\limits_{\eps\to 0}\mathcal{\I}_\eps$
=====================================================================
In this section we prove Theorem \[first\_main\_result\]. Since $\Gamma\hbox{-}\liminf_{\eps\to 0}\mathcal{\I}_\eps\leq\Gamma\hbox{-}\limsup_{\eps\to 0}\mathcal{\I}_\eps$, we only need to show that:
- $\displaystyle\overline{{\mathcal E}}\leq\Gamma\hbox{-}\liminf_{\eps\to 0}\mathcal{\I}_\eps$;
- $\displaystyle\Gamma\hbox{-}\limsup_{\eps\to 0}\mathcal{\I}_\eps\leq\overline{{\mathcal E}}$.
In the sequel, we follow the notation used in Section 3.
Proof of (a)
------------
Let $v\in W^{1,p}(\Sigma;\RR^3)$ and let $\{v_\eps\}_\eps\subset W^{1,p}(\Sigma;\RR^3)$ be such that $v_\eps\to v$ in $L^{p}(\Sigma;\RR^3)$. We have to prove that $$\label{main_ineq1}
\liminf_{\eps\to 0}{\mathcal E}_\eps(v_\eps)\geq \overline{{\mathcal E}}(v).$$ Without loss of generality we can assume that $\sup_{\eps>0}{\mathcal E}_\eps(v_\eps)<+\infty$. To every $\eps>0$ there corresponds $u_\eps\in\pi_{\eps}^{-1}(v_\eps)$ such that $$\label{main_ineq2}
\E_\eps(v_\eps)\geq E_\eps(u_\eps)-\eps.$$ Defining $\hat u_\eps:\Sigma_1\to\RR^3$ by $\hat u_\eps(x,x_3):=u_\eps(x,\eps x_3)$ we have $$\label{main_ineq3}
E_\eps\big(u_\eps\big)=\int_{\Sigma_1}W\Big(\partial_{1} \hat u_\eps(x,x_3)\mid\partial_{2} \hat u_\eps(x,x_3)\mid{1\over\eps}\partial_3 \hat u_\eps(x,x_3)\Big)dxdx_3.$$ Using the coercivity of $W$, we deduce that $\left\|{\partial_3 \hat u_\eps}\right\|_{L^p(\Sigma_1;\RR^3)}\le c\eps^{p}$ for all $\eps>0$ and some $c>0$, and so $
\|\hat u_\eps-v_\eps\|_{L^p(\Sigma_1;\RR^3)}\leq c^\prime\eps^p
$ by Poincaré-Wirtinger’s inequality, where $c^\prime>0$ is a constant which does not depend on $\eps$. It follows that $\hat u_\eps\to v$ in $L^p(\Sigma_1;\RR^3)$. For $x_3\in]-{1\over 2},{1\over 2}[$, let $w_\eps^{x_3}\in W^{1,p}(\Sigma;\RR^3)$ given by $w_\eps^{x_3}(x):=\hat u_\eps(x,x_3)$. Then (up to a subsequence) $w_\eps^{x_3}\to v$ in $L^p(\Sigma;\RR^3)$ for a.e. $x_3\in ]-{1\over 2},{1\over 2}[$. Taking (\[main\_ineq2\]) and (\[main\_ineq3\]) into account and using Fatou’s lemma, we obtain $$\liminf_{\eps\to 0}{{\mathcal E}}_\eps(v_\eps)\geq\int_{-{1\over 2}}^{1\over 2}\left(\liminf_{\eps\to 0}\int_\Sigma W_0\big(\nabla w_\eps^{x_3}(x)\big)dx\right)dx_3,$$ and so $\liminf_{\eps\to 0}{{\mathcal E}}_\eps(v_\eps)\geq{\mathcal I}(v)$, and (\[main\_ineq1\]) follows by using Theorem \[AdDitioNal\].$\square$
Proof of (b)
------------
As $\Gamma\hbox{-}\limsup_{\eps\to 0}\mathcal{\I}_\eps$ is lower semicontinuous with respect to the strong topology of $L^{p}(\Sigma;\RR^3)$ (see [@dalmaso0 Proposition 6.8 p. 57]), it is sufficient to prove that for every $v\in\C^1_*(\overline{\Sigma};\RR^3)$, $$\label{limsupequality}
\limsup_{\eps\to 0}\mathcal{\I}_\eps(v)\leq\mathcal{\I}(v).$$ Given $v\in\C^1_*(\overline{\Sigma};\RR^3)$, fix any $j\geq 1$, and any $n\geq 1$. Using Theorem \[basictheorem\] we obtain the existence of $\phi\in C(\overline{\Sigma};\Lambda^j_v)$ such that $$\label{mediainequality}
\int_\Sigma W\big(\nabla v(x)\mid\phi(x)\big)dx\leq{\mathcal E}(v)+{1\over n}.$$ By Stone-Weierstrass’s approximation theorem, there exists $\{\phi_k\}_{k\geq 1}\subset C^\infty(\overline{\Sigma};\RR^3)$ such that $$\label{uniformityconvergence}
\phi_k\to\phi\hbox{ uniformly as }k\to+\infty.$$ We claim that:
- $\displaystyle\det\big(\nabla v(x)\mid\phi_k(x)\big)\geq{1\over 2j}$ for all $x\in \overline{\Sigma}$, all $k\geq k_v$ and some $k_v\geq 1$;
- $\displaystyle\lim\limits_{k\to+\infty}\int_\Sigma W\big(\nabla v(x)\mid\phi_k(x)\big)dx=\int_\Sigma W\big(\nabla v(x)\mid\phi(x)\big)dx$.
Indeed, setting $\mu_v:=\sup_{x\in\overline{\Sigma}}\vert\partial_1v(x)\land\partial_2v(x)\vert$ ($\mu_v>0$) and using (\[uniformityconvergence\]), we deduce that there exists $k_v\geq 1$ such that for every $k\geq k_v$, $$\label{supequality}
\sup_{x\in\overline{\Sigma}}\big|\phi_k(x)-\phi(x)\big|<{1\over 2j\mu_v}.$$ Let $x\in \overline{\Sigma}$, and let $k\geq k_v$. As $\phi\in C(\overline{\Sigma};\Lambda^j_v)$ we have $$\label{supequality1}
\det\big(\nabla v(x)\mid\phi_k(x)\big)\geq{1\over j}-\det\big(\nabla v(x)\mid\phi_k(x)-\phi(x)\big).$$ Noticing that $\det(\nabla v(x)\mid\phi_k(x)-\phi(x))\leq|\partial_1 v(x)\land\partial_2 v(x)||\phi_k(x)-\phi(x)|$, from (\[supequality\]) and (\[supequality1\]) we deduce that $
\det\big(\nabla v(x)\mid\phi_k(x)\big)\geq{1\over 2j},
$ and (c$_1$) is proved. Combining (c$_1$) with (\[Icv\]) we see that $
\sup_{k\geq k_v}W(\nabla v(\cdot)\mid\phi_k(\cdot))\in L^1(\Sigma).
$ As $W$ is continuous we have $
\lim_{k\to+\infty}W(\nabla v(x)\mid\phi_k(x))=W(\nabla v(x)\mid\phi(x))
$ for all $x\in V$, and (c$_2$) follows by using Lebesgue’s dominated convergence theorem, which completes the claim.
Fix any $k\geq k_v$ and define $\theta:]-{1\over 2},{1\over 2}[\to\RR$ by $
\theta(x_3):=\inf_{x\in \overline{\Sigma}}\det(\nabla v(x)+x_3\nabla\phi_k(x)\mid\phi_k(x)).
$ Clearly $\theta$ is continuous. By (c$_1$) we have $\theta(0)\geq{1\over 2j}$, and so there exists $\eta_v\in]0,{1\over 2}[$ such that $\theta(x_3)\geq{1\over 4j}$ for all $x_3\in]-\eta_v,\eta_v[$. Let $u_k:\Sigma_1\to\RR$ be given by $
u_k(x,x_3):=v(x)+x_3\phi_k(x).
$ From the above it follows that
- $\det\nabla u_k(x,\eps x_3)\geq{1\over 4j}$ for all $\eps\in]0,\eta_v[$ and all $(x,x_3)\in \overline{\Sigma}\times]-{1\over 2},{1\over 2}[$.
As in the proof of (c$_1$), from (c$_3$) together with (\[Icv\]) and the continuity of $W$, we obtain $$\label{finalequality}
\lim_{\eps\to 0}{E}_\eps(u_k)=\lim_{\eps\to 0}\int_{\Sigma_1} W\big(\nabla u_k(x,\eps x_3)\big)dxdx_3=\int_\Sigma W\big(\nabla v(x)\mid\phi_k(x)\big)dx.$$
For every $\eps>0$ and every $k\geq k_v$, since $\pi_\eps(u_k)=v$ we have ${\mathcal E}_\eps(v)\leq {E}_\eps(u_k)$. Using (\[finalequality\]), (c$_2$) and (\[mediainequality\]), we deduce that $$\limsup_{\eps\to 0}{\mathcal E}_\eps(v)\leq{\mathcal E}(v)+{1\over n},$$ and (\[limsupequality\]) follows by letting $n\to+\infty$.$\square$
A simplified formula for $\overline{\I}$
========================================
In this section, we prove of Theorem \[AdDitioNal\]. It is based upon two approximation theorems by Ben Belgacem-Bennequin (see Sect. A.1) and Gromov-Eliasberg (see Sect. A.2).
Recall the definition of rank one convexity and rank one convex envelope:
Let $f:\MM^{3\times 2}\to[0,+\infty]$ be a Borel measurable function.
- We say that $f$ is rank one convex if for every $\alpha\in]0,1[$ and every $\xi,\xi^\prime\in\MM^{3\times 2}$ with rank($\xi-\xi^\prime$)=1, $$f(\alpha\xi+(1-\alpha)\xi^\prime)\leq \alpha f(\xi)+(1-\alpha)f(\xi^\prime).$$
- By the rank one convex envelope of $f$, that we denote by $\mathcal{R}f$, we mean the greatest rank one convex function which less than or equal to $f$.
In [@benbelgacem-thesis Proposition 7 p. 32 and Lemma 8 p. 34] (see also [@benbelgacem Sect. 5.1], [@trabelsi Proposition 3.4.4 p. 112] and [@trabelsi1 Lemma 6.5]) Ben Belgacem proved the following lemma that we will use in the proof of Theorem \[AdDitioNal\]. (As $W_0$ is coercive, it is easy to see that ${\mathcal R}W_0$ is coercive.)
\[RW\_0Properties\] If [(\[Icvbis\])]{} holds then[:]{}
- ${\mathcal R}W_0(\xi)\leq c(1+|\xi|^p)$ for all $\xi\in\MM^{3\times 2}$ and some $c>0;$
- ${\mathcal R}W_0$ is continuous.
Define $I:W^{1,p}(\Sigma;\RR^3)\to[0,+\infty]$ by $$I(v):=\inf\left\{\liminf_{n\to+\infty}\int_\Sigma W_0(\nabla v_n(x))dx:\Aff_{\rm li}(\Sigma;\RR^3)\ni v_n\to v\hbox{ in }L^p(\Sigma;\RR^3)\right\}$$ with $\Aff_{\rm li}(\Sigma;\RR^3):=\{v\in\Aff(\Sigma;\RR^3):v\hbox{ is locally injective}\}$ ($\Aff(\Sigma;\RR^3)$ is defined in Sect. 2.3). To prove Theorem \[AdDitioNal\] we will use Proposition \[I\_li=J\_li\].
\[I\_li=J\_li\] $I=J$ with $J:W^{1,p}(\Sigma;\RR^3)\to[0,+\infty]$ given by $$J(v):=\inf\left\{\liminf_{n\to+\infty}\int_\Sigma{\mathcal R} W_0(\nabla v_n(x))dx:\Aff_{\rm li}(\Sigma;\RR^3)\ni v_n\to v\hbox{ in }L^p(\Sigma;\RR^3)\right\}.$$
To prove Proposition \[I\_li=J\_li\] we need Lemma \[LeMMaBiS\] whose proof is contained in the thesis of Ben Belgacem [@benbelgacem-thesis]. Since it is difficult to lay hands on this thesis (which is written in French), we give the proof of Lemma \[LeMMaBiS\] in appendix B.
\[LeMMaBiS\] $\displaystyle I(v)\leq\int_\Sigma{\mathcal R}W_0(\nabla v(x))dx$ for all $v\in\Aff_{\rm li}(\Sigma;\RR^3)$.
[*Proof of Proposition [\[I\_li=J\_li\]]{}.*]{} Clearly $J\leq I$. We are thus reduced to prove that $$\label{inequality}
I\leq J.$$ Fix any $v\in W^{1,p}(\Sigma;\RR^3)$ and any sequence $v_n\to v$ in $L^p(\Sigma;\RR^3)$ with $v_n\in\Aff_{\rm li}(\Sigma;\RR^3)$. Using Lemma \[LeMMaBiS\] we have $
I(v_n)\leq\int_\Sigma \mathcal{R} W_0(\nabla v_n(x))dx
$ for all $n\geq 1$. Thus, $$I(v)\leq\liminf_{n\to+\infty}I(v_n)\leq\liminf_{n\to+\infty}\int_\Sigma \mathcal{R} W_0(\nabla v_n(x))dx,$$ and (\[inequality\]) follows.$\square$
[*Proof of Theorem [*\[AdDitioNal\]*]{}.* ]{}We first prove that $$\label{AddTheorEq1}
\overline{\mathcal{E}}\leq I.$$ As in the proof of Proposition \[I\_li=J\_li\], it suffices to show that if $v\in \Aff_{\rm li}(\Sigma;\RR^3)$ then $$\label{AddTheorEq2}
\overline{\mathcal{E}}(v)\leq\int_{\Sigma}W_0(\nabla v(x))dx.$$ Let $v\in \Aff_{\rm li}(\Sigma;\RR^3)$. By Theorem \[LemmaBBB\]-bis (and Lemma \[Aff=AffV\]), there exists $\{v_n\}_{n\geq 1}\subset C^1_*(\overline{\Sigma};\RR^3)$ such that (\[BB\_1\]) and (\[BB\_2\]) holds and $\nabla v_n(x)\to\nabla v(x)$ a.e. in $\Sigma$. As $W_0$ is continuous (see Lemma \[propertiesofW\_0\](i)), we have $$\lim_{n\to +\infty}W_0\big(\nabla v_n(x)\big)=W_0\big(\nabla v(x)\big)\;\hbox{ a.e. in }\Sigma.$$ Using (\[Icvbis\]) together with (\[BB\_2\]), we deduce that there exists $c>0$ such that for every $n\geq 1$ and every measurable set $A\subset\Sigma$, $$\int_A W_0\big(\nabla v_n(x)\big)dx\leq c\Big(|A|+\int_A|\nabla v_n(x)-\nabla v(x)|^pdx+\int_A|\nabla v(x)|^pdx\Big).$$ But $\nabla v_n\to\nabla v$ in $L^p(\Sigma;\MM^{3\times 2})$ by (\[BB\_1\]), hence $\{W_0(\nabla v_n(\cdot))\}_{n\geq 1}$ is absolutely uniformly integrable. Using Vitali’s theorem, we obtain $$\lim_{n\to+\infty}\int_{\Sigma}W_0(\nabla v_n(x))dx=\int_{\Sigma}W_0(\nabla v(x))dx,$$ and (\[AddTheorEq2\]) follows.
We now prove that $$\label{AddTheorEq3}
J\leq\overline{J},$$ with $\overline{J}:W^{1,p}(\Sigma;\RR^3)\to[0,+\infty]$ given by $$\overline{J}(v):=\inf\left\{\liminf_{n\to+\infty}\int_\Sigma{\mathcal R} W_0(\nabla v_n(x))dx:W^{1,p}(\Sigma;\RR^3)\ni v_n\to v\hbox{ in }L^p(\Sigma;\RR^3)\right\}.$$ It is sufficient to show that $$\label{AddTheorEq4}
J(v)\leq\int_{\Sigma}\mathcal{R}W_0(\nabla v(x))dx.$$ Let $v\in W^{1,p}(\Sigma;\RR^3)$. By Corollary \[GromovEliasbergConsequence\], there exists $\{v_n\}_{n\geq 1}\subset\Aff_{\rm li}(\Sigma;\RR^3)$ such that $\nabla v_n\to\nabla v$ in $L^p(\Sigma;\RR^3)$ and $\nabla v_n(x)\to\nabla v(x)$ a.e. in $\Sigma$. Taking Lemma \[RW\_0Properties\] into account, from Vitali’s lemma, we see that $$\lim_{n\to+\infty}\int_{\Sigma}\mathcal{R}W_0(\nabla v_n(x))dx=\int_{\Sigma}\mathcal{R}W_0(\nabla v(x))dx,$$ and (\[AddTheorEq4\]) follows.
Noticing that $\mathcal{I}\leq\overline{\mathcal{E}}$ and $\overline{J}\leq\mathcal{I}$, and combining Proposition \[I\_li=J\_li\] with (\[AddTheorEq1\]) and (\[AddTheorEq3\]), we conclude that $\overline{\mathcal{E}}=\mathcal{I}$.$\square$
Approximation theorems
======================
Ben Belgacem-Bennequin’s theorem
--------------------------------
Denote by $\Aff^{ET}(\Sigma;\RR^3)$ the space of Ekeland-Temam continuous piecewise affine functions from $\Sigma$ to $\RR^3$, i.e., [*$u\in\Aff^{ET}(\Sigma;\RR^3)$ if and only if $v$ is continuous and there exists a finite family $(V_i)_{i\in I}$ of open disjoint subsets of $\Sigma$ such that $|\Sigma\setminus \cup_{i\in I} V_i|=0$ and for every $i\in I$, the restriction of $v$ to $V_i$ is affine.*]{} Note that from Ekeland-Temam [@ekeland], we know that $\Aff^{ET}(\Sigma;\RR^3)$ is strongly dense in $W^{1,p}(\Sigma;\RR^3)$. Set $$\Aff^{ET}_{\rm li}(\Sigma;\RR^3):=\Big\{v\in\Aff^{ET}(\Sigma;\RR^3):v\hbox{ is locally injective}\Big\}.$$ In [@benbelgacem-thesis Lemma 8 p. 114] (see also [@trabelsi Proposition C.0.4 p. 127] and [@trabelsi1 Lemma 1.3]) Ben Belgacem and Bennequin proved the following result.
\[LemmaBBB\] For every $v\in\Aff^{ET}_{\rm li}(\Sigma;\RR^3)$, there exists $\{v_n\}_{n\geq 1}\subset C^1_*(\overline{\Sigma};\RR^3)$ such that[:]{} $$\label{BB_1}
\hbox{$v_n\to v$ {\em in } $W^{1,p}(\Sigma;\RR^3)${\rm;}}$$ $$\label{BB_2}
\hbox{$|\partial_1v_n(x)\land\partial_2v_n(x)|\geq\delta$ for all $x\in\overline{\Sigma}$, all $n\geq 1$ and some $\delta>0$.}$$
Denote by $\Aff^V(\Sigma;\RR^3)$ the space of Vitali continuous piecewise affine functions from $\Sigma$ to $\RR^3$ (introduced by Ben Belgacem in [@benbelgacem-thesis; @benbelgacem]), i.e., [*$v\in\Aff^V(\Sigma;\RR^3)$ if and only if $v$ is continuous and there exists a finite or countable family $(O_i)_{i\in I}$ of dsjoint open subsets of $\Sigma$ such that $|\partial O_i|=0$ for all $i\in I$, $|\Sigma\setminus\cup_{i\in I}O_i|=0$, and $v(x)=\xi_i\cdot x+a_i$ if $x\in O_i$, where $a_i\in\RR^3$, $\xi_i\in\MM^{3\times 2}$ and ${\rm Card}\{\xi_i:i\in I\}$ is finite.*]{} In [@trabelsi Lemma 3.1.5 p. 99] Trabelsi remarked that Theorem \[LemmaBBB\] can be generalized replacing the space $\Aff^{ET}_{\rm li}(\Sigma;\RR^{3})$ by $$\Aff^V_{\rm li}(\Sigma;\RR^3):=\Big\{v\in\Aff^V(\Sigma;\RR^3):v\hbox{ is locally injective}\Big\}.$$
[**Theorem \[LemmaBBB\]-bis.**]{} [*For every $v\in\Aff^{V}_{\rm li}(\Sigma;\RR^3)$, there exists $\{v_n\}_{n\geq 1}\subset C^1_*(\overline{\Sigma};\RR^3)$ satisfying*]{} (\[BB\_1\]) and (\[BB\_2\]).
Here we consider the space $\Aff(\Sigma;\RR^3)$ defined in Sect. 2.3. It is clear that $\Aff^{ET}(\Sigma;\RR^3)\subset\Aff(\Sigma;\RR^3)$, and so $\Aff(\Sigma;\RR^3)$ is strongly dense in $W^{1,p}(\Sigma;\RR^3)$. Moreover, we have
\[Aff=AffV\] $\Aff^V(\Sigma;\RR^3)=\Aff(\Sigma;\RR^3)$.
Setting $D_i:=\{x\in\cup_{i\in I}O_i:\nabla v(x)=\xi_i\}$ with $v\in\Aff^V(\Sigma;\RR^3)$, we see that ${\rm Card}\{D_i:i\in I\}$ is finite. Thus $\Aff^V(\Sigma;\RR^3)\subset\Aff(\Sigma;\RR^3)$. Given $v\in\Aff(\Sigma;\RR^3)$, let $(O_j)_{j\in J_i}$ be the connected components of $D_i$ with $i\in I$ (where $I$ is finite). Since $D_i$ is open, $O_j$ is open for all $j\in J_i$, hence $J_i$ is finite or countable because $\QQ^2$ is dense in $\RR^2$. Moreover, for each $j\in J_i$, the restriction of $v$ to $O_j$ is affine. Thus $\Aff(\Sigma;\RR^3)\subset\Aff^V(\Sigma;\RR^3)$.
Gromov-Eliashberg’s theorem
---------------------------
In [@gromovEliashberg Theorem 1.3.4B] (see also [@gromov Theorem B$^\prime_1$ p. 20]) Gromov and Eliashberg proved the following result.
Let $1\leq N<m$ be two integers and let $M$ be a compact $N$-di-mensional manifold which can be immersed in $\RR^m$. Then, for each $C^1$-differentiable function $v$ from $M$ to $\RR^m$ there exists a sequence $\{v_n\}_n$ of $C^1$-immersions from $M$ to $\RR^m$ such that $v_n\to v$ in $W^{1,p}(M;\RR^m)$.
In our context, we have
\[GromovEliasberg\] For every $v\in C^1(\overline{\Sigma};\RR^3)$ there exists $\{v_n\}_{n\geq 1}\subset C^1_*(\overline{\Sigma};\RR^3)$ such that $v_n\to v$ in $W^{1,p}(\Sigma;\RR^3)$.
Moreover, from [@trabelsi Proposition 3.1.7 p. 100], we have
\[TrabelsiThesis\] For every $v\in C^1_*(\overline{\Sigma};\RR^3)$ there exists $\{v_n\}_{n\geq 1}\subset\Aff^{ET}_{\rm li}(\Sigma;\RR^3)$ such that $v_n\to v$ in $W^{1,p}(\Sigma;\RR^3)$.
Thus, as a consequence of Theorem \[GromovEliasberg\] and Proposition \[TrabelsiThesis\], we obtain
\[GromovEliasbergConsequence\] $\Aff^{ET}_{\rm li}(\Sigma;\RR^3)$ is strongly dense in $W^{1,p}(\Sigma;\RR^3)$.
Ben Belgacem’s lemma
====================
In this appendix we prove Ben Belgacem’s lemma, i.e., Lemma \[LeMMaBiS\].
Preliminaries.
--------------
Define the sequence $\{\R_i W_0\}_{i\geq 0}$ by $\R_0 W_0=W_0$ and for every $i\geq 1$ and every $\xi\in\MM^{3\times 2}$, $$\R_{i+1}W_0(\xi):=\infff\limits_{t\in[0,1]}\Big\{(1-t)\R_i W_0(\xi-t a\otimes b)+t\R_i W_0(\xi+(1-t)a\otimes b)\Big\}.$$ Recall that $W_0$ is coercive and continuous (see Lemma \[propertiesofW\_0\](i)). The following lemma is due to Kohn and Strang [@kohnstrang].
\[Khon-Strang\] $\R_{i+1} W_0\leq \R_iW_0$ for all $i\geq 0$ and $\R W_0=\inf_{i\geq 0} \R_i W_0$.
Fix any $i\geq 0$ and any $v\in\Aff_{\rm li}(\Sigma;\RR^3):=\{v\in\Aff(\Sigma;\RR^3):v\hbox{ is locally injective}\}$ (with $\Aff(\Sigma;\RR^3)$ defined in Sect. 2.3). By definition, there exists a finite family $(V_j)_{j\in J}$ of open disjoint subsets of $\Sigma$ such that $|\partial V_j|=0$ for all $j\in J$, $|\Sigma\setminus\cup_{j\in J}V_j|=0$ and, for every $j\in J$, $\nabla v(x)=\xi_j$ in $V_j$ with $\xi_j\in\MM^{3\times 2}$. (As $v$ is locally injective we have ${\rm rank}(\xi_j)=2$ for all $j\in J$.) Fix any $j\in J$. For a proof of Lemmas \[BBLemma2\] and \[BBLemma1\] we refer to [@trabelsi Proposition 3.1.2 p. 96].
\[BBLemma2\] $\R_i W_0$ is continuous.
\[BBLemma1\] There exist $a\in\RR^2$, $b\in\RR^3$ and $t\in[0,1]$ such that $$\R_{i+1}W_0(\xi_j)=(1-t)\R_i W_0(\xi_j-t a\otimes b)+t\R_i W_0(\xi_j+(1-t)a\otimes b).$$
Without loss of generality we can assume that $a=(1,0)$. For each $n\geq 3$ and each $k\in\{0,\cdots,n-1\}$, consider $A^-_{k,n},A^+_{k,n},B_{k,n},B^-_{k,n},B^+_{k,n},C_{k,n},C^-_{k,n},C^+_{k,n}\subset Y$ given by:
$A^-_{k,n}:=\big\{(x_1,x_2)\in Y:{k\over n}\leq x_1\leq{k\over n}+{1-t\over n}\hbox{ and }{1\over n}\leq x_2\leq 1-{1\over n}\big\}$;
$A^+_{k,n}:=\big\{(x_1,x_2)\in Y:{k\over n}+{1-t\over n}\leq x_1\leq {k+1\over n}\hbox{ and }{1\over n}\leq x_2\leq 1-{1\over n}\big\}$;
$B_{k,n}:=\big\{(x_1,x_2)\in Y:{k\over n}\leq x_1\leq{k+1\over n}\hbox{ and }0\leq x_2\leq -x_1+{k+1\over n}\big\}$;
$B^-_{k,n}:=\big\{(x_1,x_2)\in Y:-x_2+{k+1\over n}\leq x_1\leq-tx_2+{k+1\over n}\hbox{ and }0\leq x_2\leq {1\over n}\big\}$;
$B^+_{k,n}:=\big\{(x_1,x_2)\in Y:-tx_2+{k+1\over n}\leq x_1\leq {k+1\over n}\hbox{ and }0\leq x_2\leq {1\over n}\big\}$;
$C_{k,n}:=\big\{(x_1,x_2)\in Y:{k\over n}\leq x_1\leq{k+1\over n}\hbox{ and }x_1+1-{k+1\over n}\leq x_2\leq 1\big\}$;
$C^-_{k,n}:=\big\{(x_1,x_2)\in Y:x_2-1+{k+1\over n}\leq x_1\leq t(x_2-1)+{k+1\over n}\hbox{ and }0\leq x_2\leq {1\over n}\big\}$;
$C^+_{k,n}:=\big\{(x_1,x_2)\in Y:t(x_2-1)+{k+1\over n}\leq x_1\leq {k+1\over n}\hbox{ and }0\leq x_2\leq {1\over n}\big\}$,
and define $\{\sigma_{n}\}_{n\geq 1}\subset\Aff_0(Y;\RR)$ by $$\sigma_{n}(x_1,x_2):=\left\{
\begin{array}{ll}
-t(x_1-{k\over n})&\hbox{if }(x_1,x_2)\in A^-_{k,n}\\
(1-t)(x_1-{k+1\over n})&\hbox{if }(x_1,x_2)\in A^+_{k,n}\cup B^+_{k,n}\cup C^+_{k,n}\\
-t(x_1+x_2-{k+1\over n})&\hbox{if }(x_1,x_2)\in B^-_{k,n}\\
-t(x_1-x_2+1-{k+1\over n})&\hbox{if }(x_1,x_2)\in C^-_{k,n}\\
0&\hbox{if }(x_1,x_2)\in B_{k,n}\cup C_{k,n}
\end{array}
\right.$$ (see Figure B.1).
(300,255) (-5,40)[(1,0)[210]{}]{} (195,34)[$x_1$]{} (-10,240)[$x_2$]{} (0,39.9)[(1,0)[180]{}]{} (0,40.1)[(1,0)[180]{}]{} (0,40) (-5,34)[$0$]{} (20,40) (12,31)[$1-t\over n$]{} (30,40) (26.5,31)[$1\over n$]{} (50,40) (42,31)[${2-t\over n}$]{} (60,40) (56.5,31)[$2\over n$]{} (71,32)[$\cdots$]{} (90,40) (86.5,31)[$k\over n$]{} (120,40) (115,31)[$k+1\over n$]{} (93,31)[$k+1-t\over n$]{} (110,40) (131.5,32)[$\cdots$]{} (150,40) (144,31)[$n-1\over n$]{} (170,40) (180,40) (162,31)[$n-t\over n$]{} (178,32)[$1$]{} (0,35)[(0,1)[210]{}]{} (0,220) (-5,218)[$1$]{} (0,190) (-17,188)[$n-1\over n$]{} (0,70) (-8,68)[$1\over n$]{} (0,220)[(1,0)[180]{}]{} (0,220.1)[(1,0)[180]{}]{} (0,219.9)[(1,0)[180]{}]{} (0,190)[(1,0)[20]{}]{} (30,190)[(1,0)[20]{}]{} (60,190)[(1,0)[20]{}]{} (90,190)[(1,0)[20]{}]{} (120,190)[(1,0)[20]{}]{} (150,190)[(1,0)[20]{}]{} (150,70)[(1,0)[20]{}]{} (120,70)[(1,0)[20]{}]{} (90,70)[(1,0)[20]{}]{} (60,70)[(1,0)[20]{}]{} (30,70)[(1,0)[20]{}]{} (0,70)[(1,0)[20]{}]{} (-0.1,40)[(0,1)[180]{}]{} (0.1,40)[(0,1)[180]{}]{} (180,40)[(0,1)[180]{}]{} (180.1,40)[(0,1)[180]{}]{} (179.9,40)[(0,1)[180]{}]{} (150,40)[(0,1)[180]{}]{} (150.1,40)[(0,1)[180]{}]{} (149.9,40)[(0,1)[180]{}]{} (120,40)[(0,1)[180]{}]{} (120.1,40)[(0,1)[180]{}]{} (119.9,40)[(0,1)[180]{}]{} (90,40)[(0,1)[180]{}]{} (90.1,40)[(0,1)[180]{}]{} (89.9,40)[(0,1)[180]{}]{} (60,40)[(0,1)[180]{}]{} (60.1,40)[(0,1)[180]{}]{} (59.9,40)[(0,1)[180]{}]{} (30,40)[(0,1)[180]{}]{} (30.1,40)[(0,1)[180]{}]{} (29.9,40)[(0,1)[180]{}]{} (170,70)[(0,1)[120]{}]{} (140,70)[(0,1)[120]{}]{} (110,70)[(0,1)[120]{}]{} (80,70)[(0,1)[120]{}]{} (50,70)[(0,1)[120]{}]{} (20,70)[(0,1)[120]{}]{} (0,70)[(1,-1)[30]{}]{} (30,70)[(1,-1)[30]{}]{} (60,70)[(1,-1)[30]{}]{} (90,70)[(1,-1)[30]{}]{} (120,70)[(1,-1)[30]{}]{} (150,70)[(1,-1)[30]{}]{} (0,190)[(1,1)[30]{}]{} (30,190)[(1,1)[30]{}]{} (60,190)[(1,1)[30]{}]{} (90,190)[(1,1)[30]{}]{} (120,190)[(1,1)[30]{}]{} (150,190)[(1,1)[30]{}]{} (20,70)[(1,-3)[10]{}]{} (50,70)[(1,-3)[10]{}]{} (80,70)[(1,-3)[10]{}]{} (110,70)[(1,-3)[10]{}]{} (140,70)[(1,-3)[10]{}]{} (170,70)[(1,-3)[10]{}]{} (20,190)[(1,3)[10]{}]{} (50,190)[(1,3)[10]{}]{} (80,190)[(1,3)[10]{}]{} (110,190)[(1,3)[10]{}]{} (140,190)[(1,3)[10]{}]{} (170,190)[(1,3)[10]{}]{} (260,40)[(1,0)[30]{}]{} (260,40.1)[(1,0)[30]{}]{} (260,39.9)[(1,0)[30]{}]{} (235,52)[$B_{k,n}$]{}(252,52)[(1,0)[17]{}]{} (235,208)[$C_{k,n}$]{}(252,208)[(1,0)[17]{}]{} (235,128)[$A^-_{k,n}$]{}(252,128)[(1,0)[20]{}]{} (307,128)[$A^+_{k,n}$]{}(304,128)[(-1,0)[20]{}]{} (307.5,198)[$C^+_{k,n}$]{}(304.5,198)[(-1,0)[20]{}]{} (307.5,62)[$B^+_{k,n}$]{}(304.5,62)[(-1,0)[20]{}]{} (235,62)[$B^-_{k,n}$]{}(252,62)[(1,0)[25]{}]{} (235,198)[$C^-_{k,n}$]{}(252,198)[(1,0)[25]{}]{} (260,220)[(1,0)[30]{}]{} (260,220.1)[(1,0)[30]{}]{} (260,219.9)[(1,0)[30]{}]{} (260,40) (256.5,31)[$k\over n$]{} (290,40) (285,31)[$k+1\over n$]{} (263,31)[$k+1-t\over n$]{} (280,40) (290,40)[(0,1)[180]{}]{} (290.1,40)[(0,1)[180]{}]{} (289.9,40)[(0,1)[180]{}]{} (260,40)[(0,1)[180]{}]{} (260.1,40)[(0,1)[180]{}]{} (259.9,40)[(0,1)[180]{}]{} (260,70)[(1,-1)[30]{}]{} (280,70)[(1,-3)[10]{}]{} (280,190)[(1,3)[10]{}]{} (260,190)[(1,1)[30]{}]{} (280,70)[(0,1)[120]{}]{} (260,70)[(1,0)[30]{}]{} (260,190)[(1,0)[30]{}]{} (0,10)[Figure B.1. The function $\sigma_n$ and the sets $A^-_{k,n},A^+_{k,n},B_{k,n},B^-_{k,n},B^+_{k,n},C_{k,n},C^-_{k,n},C^+_{k,n}$.]{}
Set $$b_{\ell}:=\left\{
\begin{array}{ll}
b&\hbox{if }b\not\in{\rm Im}\xi_j\\
b+{1\over\ell}\nu&\hbox{if }b\in{\rm Im}\xi_j
\end{array}
\right.$$ (with ${\rm Im}\xi_j:=\{\xi_j\cdot x:x\in\RR^2\}$) where $\ell\geq 1$ and $\nu\in\RR^3$ is a normal vector to ${\rm Im}\xi_j$.
\[LeMMaBBFund2\] Define $\{\theta_{n,\ell}\}_{n,\ell\geq 1}\subset\Aff_0(Y;\RR^3)$ by $$\theta_{n,\ell}(x):=\sigma_{n}(x)b_{\ell}.$$ Then[:]{}
- for every $\ell\geq 1$, $\theta_{n,\ell}\to 0$ in $L^p(Y;\RR^3);$
- $\displaystyle\lim_{\ell\to+\infty}\lim_{n\to+\infty}\int_Y\R_iW_0(\xi_j+\nabla \theta_{n,\ell}(x))dx=\R_{i+1}W_0(\xi_j)$.
\(i) It suffices to prove that $\sigma_n\to 0$ in $L^p(Y;\RR)$. For every $k\in\{0,\cdots,n-1\}$, it is clear that $|\sigma_n(x)|^p\leq {t^p(1-t)^p\over n^p}$ for all $x\in]{k\over n},{k+1\over n}[\times]0,1[$, and so $$\int_{]{k\over n},{k+1\over n}[\times]0,1[}|\sigma_n(x)|^pdx\leq {t^p(1-t)^p\over n^{p+1}}.$$ As $$\int_Y|\sigma_{n}(x)|^pdx=\sum_{k=0}^{n-1}\int_{]{k\over n},{k+1\over n}[\times]0,1[}|\sigma_{n}(x)|^pdx$$ it follows that $$\int_Y|\sigma_{n}(x)|^pdx\leq{t^p(1-t)^p\over n^{p}},$$ which gives the desired conclusion.
\(ii) Recalling that $a=(1,0)$ we see that $$\xi_j+\nabla\theta_{n,\ell}(x):=\left\{
\begin{array}{ll}
\xi_j-ta\otimes b_\ell&\hbox{if }x\in {\rm int}(A^-_{k,n})\\
\xi_j+(1-t)a\otimes b_\ell&\hbox{if }x\in {\rm int}(A^+_{k,n}\cup B^+_{k,n}\cup C^+_{k,n})\\
\xi_j-t(a+a^\perp)\otimes b_\ell&\hbox{if }x\in {\rm int}(B^-_{k,n})\\
\xi_j-t(a-a^\perp)\otimes b_\ell&\hbox{if }x\in {\rm int}(C^-_{k,n})\\
\xi_j&\hbox{if }x\in {\rm int}(B_{k,n})\cup{\rm int}(C_{k,n})
\end{array}
\right.$$ with $a^\perp=(0,1)$ (and ${\rm int}(E)$ denotes the interior of the set $E$). Moreover, we have:
${\displaystyle\int_{\cup_{k=0}^{n-1} A^-_{k,n}}\R_iW_0(\xi_j-ta\otimes b_\ell)dx}=(1-t)(1-{2\over n})\R_iW_0(\xi_j-ta\otimes b_\ell)$;
${\displaystyle\int_{\cup_{k=0}^{n-1} A^+_{k,n}}\R_i W_0(\xi_j+(1-t)a\otimes b_\ell)dx}=t(1-{2\over n})\R_i W_0(\xi_j+(1-t)a\otimes b_\ell)$;
${\displaystyle\int_{\cup_{k=0}^{n-1} (B^+_{k,n}\cup C^+_{k,n})}\R_i W_0(\xi_j+(1-t)a\otimes b_\ell)dx}={t\over n}\R_i W_0(\xi_j+(1-t)a\otimes b_\ell)$;
${\displaystyle\int_{\cup_{k=0}^{n-1} B^-_{k,n}}\R_i W_0(\xi_j-t(a+a^\perp)\otimes b_\ell)dx}={1-t\over 2n}\R_i W_0(\xi_j-t(a+a^\perp)\otimes b_\ell)$;
${\displaystyle\int_{\cup_{k=0}^{n-1} C^-_{k,n}}\R_i W_0(\xi_j-t(a-a^\perp)\otimes b_\ell)dx}={1-t\over 2n}\R_i W_0(\xi_j-t(a-a^\perp)\otimes b_\ell)$;
${\displaystyle\int_{\cup_{k=0}^{n-1} (B_{k,n}\cup C_{k,n})}\R_i W_0(\xi_j)dx}={1\over n}\R_i W_0(\xi_j)$.
Hence $$\begin{aligned}
\int_Y\R_i W_0(\xi_j+\nabla\theta_{n,\ell}(x))dx&=&\Big(1-{2\over n}\Big)\Big[(1-t)\R_iW_0(\xi_j-ta\otimes b_\ell)+t\R_i W_0(\xi_j\\
&&+(1-t)a\otimes b_\ell)\Big]+{1\over n}\Big[t\R_i W_0(\xi_j+(1-t)a\otimes b_\ell)\\
&&+{1-t\over 2}\big(\R_i W_0(\xi_j-t(a+a^\perp)\otimes b_\ell)+\R_i W_0(\xi_j\hskip-0.8mm-\\
&&t(a-a^\perp)\otimes b_\ell)\big)+\R_i W_0(\xi_j)\Big]\end{aligned}$$ for all $n,\ell\geq 1$. It follows that for every $\ell\geq 1$, $$\begin{aligned}
\lim_{n\to+\infty}\int_Y\R_i W_0(\xi_j+\nabla\theta_{n,\ell}(x))dx&=&(1-t)\R_iW_0(\xi_j-ta\otimes b_\ell)\\
&&+t\R_i W_0(\xi_j+(1-t)a\otimes b_\ell).\end{aligned}$$ Taking Lemma \[BBLemma2\] into account and noticing that $b_\ell\to b$, we deduce that $$\begin{aligned}
\lim_{\ell\to+\infty}\lim_{n\to+\infty}\int_Y\R_i W_0(\xi_j+\nabla\theta_{n,\ell}(x))dx&=&(1-t)\R_iW_0(\xi_j-ta\otimes b)\\
&&+t\R_i W_0(\xi_j+(1-t)a\otimes b),\end{aligned}$$ and (ii) follows by using Lemma \[BBLemma1\].
Consider $V^j_{q}\subset V_j$ given by $V^j_{q}:=\{x\in V_j:{\rm dist}(x,\partial V_j)>{1\over q}\}$ with $q\geq 1$ large enough. By Vitali’s covering theorem, there exists a finite or countable family $(r_{m}+\rho_{m}Y)_{m\in M}$ of disjoint subsets of $V^j_{q}$, with $r_{m}\in\RR^2$ and $\rho_{m}\in]0,1[$, such that $
|V^j_{q}\setminus\cup_{m\in M}(r_{m}+\rho_{m}Y)|=0
$ (and so $\sum_{m\in M}\rho_{m}^2=|V^j_{q}|$). Let $\{\phi_{n,\ell,q}\}_{n,\ell,q\geq 1}\subset\Aff_0(V_j;\RR^3)$ be given by $$\phi_{n,\ell,q}(x):=\left\{
\begin{array}{ll}
\displaystyle\rho_{m}\theta_{n,\ell}\left({x-r_{m}\over \rho_{m}}\right)&\hbox{ if }x\in r_{m}+\rho_{m}Y\subset V^j_{q}\\
0&\hbox{if }x\in V_j\setminus V^j_{q}.
\end{array}
\right.$$
\[LeMMaBBFund3\] Define $\{\Phi^j_{n,\ell,q}\}_{n,\ell,q\geq 1}\subset\Aff(V_j;\RR^3)$ by $$\label{BBFunct2}
\Phi^j_{n,\ell,q}(x):=v(x)+\phi_{n,\ell,q}(x).$$ Then[:]{}
- for every $n,\ell,q\geq 1$, $\Phi^j_{n,\ell,q}$ is locally injective[;]{}
- for every $\ell,q\geq 1$, $\Phi^j_{n,\ell,q}\to v$ in $L^p(V_j;\RR^3);$
- $\displaystyle\lim_{q\to+\infty}\lim_{\ell\to+\infty}\lim_{n\to+\infty}\int_{V_j}\R_iW_0(\nabla \Phi^j_{n,\ell,q}(x))dx=|V_j|\R_{i+1}W_0(\xi_j)$.
\(i) Let $x\in V_j$ and let $W\subset V_j$ be the connected component of $V_j$ such that $x\in W$ (as $V_j$ is open, so is $W$). Since $\nabla v=\xi_j$ in $W$, there exists $c\in\RR^3$ such that $v(x^\prime)=\xi_j\cdot x^\prime+c$ for all $x^\prime\in W$. We claim that ${\Phi^j_{n,\ell,q}}{\lfloor_{W}}$ is injective. Indeed, let $x^\prime\in W$ be such that $\Phi^j_{n,\ell,q}(x)=\Phi^j_{n,\ell,q}(x^\prime)$. One the three possibilities holds:
- $\Phi^j_{n,\ell,q}(x)=\xi_j\cdot x+c+\rho_m\sigma_{n}\big({x-r_m\over\rho_m}\big)b_\ell$ and $\Phi^j_{n,\ell,q}(x^\prime)=\xi_j\cdot x^\prime+c+\rho_{m^\prime}\sigma_{n}\big({x^\prime-r_{m^\prime}\over\rho_{m^\prime}}\big)b_\ell$;
- $\Phi^j_{n,\ell,q}(x)=\xi_j\cdot x+c+\rho_m\sigma_{n,\ell}\big({x-r_m\over\rho_m}\big)b_\ell$ and $\Phi^j_{n,\ell,q}(x^\prime)=\xi_j\cdot x^\prime+c$;
- $\Phi^j_{n,\ell,q}(x)=\xi_j\cdot x+c$ and $\Phi^j_{n,\ell,q}(x^\prime)=\xi_j\cdot x^\prime+c$.
Setting $\alpha:=\rho_m\sigma_n({x-r_m\over\rho_m})-\rho_{m^\prime}\sigma_n({x^\prime-r_{m^\prime}\over \rho_{m^\prime}})$ and $\beta:=\rho_m\sigma_n({x-r_m\over\rho_m})$ we have: $$\left\{
\begin{array}{ll}
\xi_j(x^\prime-x)=0&\hbox{if }\alpha=0\\
b_\ell={1\over \alpha}\xi_j(x^\prime-x)&\hbox{if }\alpha\not=0
\end{array}
\right.
\hbox{ when (a) is satisfied;}$$ $$\left\{
\begin{array}{ll}
\xi_j(x^\prime-x)=0&\hbox{if }\beta=0\\
b_\ell={1\over \beta}\xi_j(x^\prime-x)&\hbox{if }\beta\not=0
\end{array}
\right.
\hbox{ when (b) is satisfied;}$$ $$\xi_j(x^\prime-x)=0\hbox{ when (c) is satisfied.}$$ It follows that if $x\not=x^\prime$ then either ${\rm rank}(\xi_j)<2$ or $b_\ell\in{\rm Im}\xi_j$ which is impossible. Hence $x=x^\prime$, and the claim is proved. Thus $\Phi^j_{n,\ell,q}$ is locally injective.
\(ii) As $\rho_m\in]0,1[$ for all $m\in M$ and $\sum_{m\in M}\rho_m^2=|V^j_q|$ we have $$\int_{V^j_q}|\phi_{n,\ell,q}(x)|^pdx\leq|V^j_q|\int_Y|\theta_{n,\ell}(x)|^pdx.$$ Using Lemma \[LeMMaBBFund2\](i) we deduce that for every $\ell,q\geq 1$, $$\lim_{n\to+\infty}\int_{V^j_q}|\phi_{n,\ell,q}(x)|^pdx=0,$$ and (ii) follows.
\(iii) Recalling that $\sum_{m\in M}\rho_m^2=|V^j_q|$ we see that $$\begin{aligned}
\int_{V_j}\R_iW_0(\nabla \Phi^j_{n,\ell,q}(x))dx\hskip-2mm&=&\hskip-2mm\int_{V_j}\R_iW_0(\xi_j+\nabla \phi_{n,\ell,q}(x))dx\\
&=&\hskip-2mm\int_{V^j_q}\R_iW_0(\xi_j+\nabla \phi_{n,\ell,q}(x))dx+|V_j\setminus V^j_q|\R_iW_0(\xi_j)\\
&=&\hskip-2mm|V^j_q|\int_Y\R_iW_0(\xi_j+\nabla\theta_{n,\ell}(x))dx+|V_j\setminus V^j_q|\R_iW_0(\xi_j).\end{aligned}$$ Using Lemma \[LeMMaBBFund2\](ii) we deduce that for every $q\geq 1$, $$\lim_{\ell\to+\infty}\lim_{n\to+\infty}\int_{V_j}\R_iW_0(\nabla \Phi^j_{n,\ell,q}(x))dx=|V^j_q|\R_{i+1}W_0(\xi_j)+|V_j\setminus V^j_q|\R_iW_0(\xi_j),$$ and (iii) follows by noticing that $|V^j_q|\to|V_j|$ and $|V_j\setminus V^j_q|\to 0$.
Proof of Lemma \[LeMMaBiS\]
---------------------------
According to Lemma \[Khon-Strang\], it is sufficient to show that for every $i\geq 0$, $$I(v)\leq \int_\Sigma\R_iW_0(\nabla v(x))dx\hbox{ for all }v\in\Aff_{\rm li}(\Sigma;\RR^3).\leqno (P_i)$$ The proof is by induction on $i$. As $R_0 W_0=W_0$ it is clear that $(P_0)$ is true. Assume that $(P_i)$ is true, and prove that $(P_{i+1})$ is true. Let $v\in\Aff_{\rm li}(\Sigma;\RR^3)$. By definition, there exists a finite family $(V_j)_{j\in J}$ of open disjoint subsets of $\Sigma$ such that $|\partial V_j|=0$ for all $j\in J$, $|\Sigma\setminus\cup_{j\in J}V_j|=0$ and, for every $j\in J$, $\nabla v(x)=\xi_j$ in $V_j$ with $\xi_j\in\MM^{3\times 2}$. Define $\{\Psi_{n,\ell,q}\}_{n,\ell,q\geq 1}\subset\Aff(\Sigma;\RR^3)$ by $$\Psi_{n,\ell,q}(x):=\Phi^j_{n,\ell,q}(x)\hbox{ if }x\in V_j$$ with $\Phi^j_{n,\ell,q}$ given by (\[BBFunct2\]). Taking Lemma \[LeMMaBBFund3\](i) into account (and recalling that $v$ is locally injective), it is easy to see that $\Psi_{n,\ell,q}$ is locally injective. Using $(P_i)$ we can assert that $$I(\Psi_{n,\ell,q})\leq\int_{\Sigma}R_iW_0(\nabla\Psi_{n,\ell,q}(x))dx\hbox{ for all }n,\ell,q\geq 1.%=\sum_{j\in J}\int_{V_j}R_iW_0(\nabla\Phi^j_{n,\ell,q}(x))dx$$ By Lemma \[LeMMaBBFund3\](ii) it is clear that for every $\ell,q\geq 1$, $\Psi_{n,l,q}\to v$ in $L^p(\Sigma;\RR^3)$. It follows that $$I(v)\leq \lim_{n\to+\infty}I(\Psi_{n,\ell,q})\leq\lim_{n\to+\infty}\int_{\Sigma}R_iW_0(\nabla\Psi_{n,\ell,q}(x))dx\hbox{ for all }\ell,q\geq 1.$$ Moreover, from Lemma \[LeMMaBBFund3\](iii) we see that $$\lim_{q\to+\infty}\lim_{\ell\to+\infty}\lim_{n\to+\infty}\int_{\Sigma}R_iW_0(\nabla\Psi_{n,\ell,q}(x))dx=\int_\Sigma\R_{i+1}W_0(\nabla v(x))dx.$$ Hence $$I(v)\leq \int_\Sigma\R_{i+1}W_0(\nabla v(x))dx,$$ and the proof is complete.$\square$
[99]{}
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[^1]: A multifunction $\Lambda:\overline{\Sigma}\to\RR^3$ is said to be lower semicontinuous if for every closed subset $X$ of $\RR^3$, every $x\in\overline{\Sigma}$ and every $\{x_n\}_{n\geq 1}\subset\overline{\Sigma}$ such that $|x_n-x|\to 0$ as $n\to+\infty$ and $\Lambda(x_n)\subset X$ for all $n\geq 1$, we have $\Lambda(x)\subset X$ (see [@aubinfrankowska] for more details).
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'In this short note we consider a variation of the connectivity Waiter-Client game $WC(n,q,\mathcal{A})$ played on an $n$-vertex graph $G$ which consists of $q+1$ disjoint spanning trees. In this game in each round Waiter offers Client $q+1$ edges of $G$ which have not yet been offered. Client chooses one edge and the remaining $q$ edges are discarded. The aim of Waiter is to force Client to build a connected graph. If this happens Waiter wins. Otherwise Client is the winner. We consider the case where $2 < q+1 < \lfloor \frac{n-1}{2}\rfloor$ and show that for each such $q$ there exists a graph $G$ for which Client has a winning strategy. This result stands in opposition to the case where $G$ consists of just 2 spanning trees or where $G$ is a complete graph, since it has been shown that for such graphs Waiter can always force Client to build a connected graph.'
address:
- 'Adam Mickiewicz University, Faculty of Mathematics and Computer Science [(on leave)]{.nodecor}, ul. Umultowska 87, 61-614 Poznań, Poland'
- 'Instytut Matematyczny, Polska Akademia Nauk, ul. Śniadeckich 8, 00-656 Warszawa, Poland'
- 'Freie Universität, Berlin, Germany'
author:
- Sylwia Antoniuk
- Codru Grosu
- Lothar Narins
title: 'On the connectivity Waiter-Client game'
---
Introduction
============
Waiter-Client games were first defined and studied by Beck (see e.g [@B2002]) under the name of Picker-Chooser games. These are positional games closely related to the well-studied Maker-Breaker games and Avoider-Enforcer games. Waiter-Client game $WC(n,q,\mathcal{A})$ is a two player, perfect information game played on the complete graph $K_n$, which proceeds in rounds. In each round, the first player, called Waiter, offers $q+1$ edges of $G$ which have not yet been offered. The second player, called Client, chooses one edge, and the remaining edges are discarded. The aim of Waiter is to force Client to build a graph that satisfies a given monotone property $\mathcal{A}$. If this happens Waiter wins. Otherwise Client wins.
We consider the following version of Waiter-Client game $WC(n,q,\mathcal{A})$ which we call the connectivity Waiter-Client game. This time the game is played on an $n$-vertex graph $G$ which is the union of $q+1$ disjoint spanning trees and the aim of Waiter is to force Client to build a connected graph, i.e. $\mathcal{A}$ is the property of being connected.
Csernenszky et al. [@CMP2009] showed that for $q+1=2$ Waiter always has a winning strategy. Bednarska-Bzdega et al. [@BBHKL2014] showed that the same is true for $G$ being a complete graph, in which case $n$ is necessarily even. This result follows from the more general Theorem 3.3 in [@BBHKL2014] which says that in the Waiter-Client connectivity game $WC(n,q,\mathcal{A})$ played on the complete graph $K_n$, Waiter can always force Client to build a graph of size at least $\min\{n,2(n-q-1)\}$, provided that $n$ is sufficiently large. In particular, if $n$ is even and $q+1=n/2$ then Waiter wins. In [@BBHKL2014], the authors posed a question whether in the connectivity Waiter-Client game $WC(n,q,\mathcal{A})$ played on a graph $G$ which is a disjoint union of $q+1$ spanning trees Waiter always has a winning strategy. We show that the answer to this question is negative.
\[th1\] Let $2 < q+1 < \lfloor \frac{n-1}{2} \rfloor$. Then there exists an $n$-vertex graph $G$ which is a union of $q + 1$ disjoint spanning trees and such that in the connectivity Waiter-Client game $WC(n,q,\mathcal{A})$ played on $G$, Client has a winning strategy.
This leaves open the cases $q+1 = (n-1)/2$ for $n$ odd, and $q+1 = n/2 - 1$ for $n$ even. It may be tempting to believe that in these cases Client wins as well. However, we found that for $n=7$ and $n=9$, Waiter has a winning strategy.
The proof
=========
We first fix some notation. Whenever the game is played on a graph $G$, we proceed in rounds. In each round we delete from $G$ all edges offered by Waiter in this round and we denote by $G_i$ the graph obtained in this way, where $i$ is the number of the round. In particular, $G_0 = G$ and $G_{n-1} = \emptyset$. The edges which have not yet been offered by Waiter are called *free edges*. Moreover, we let $H_i$ denote the graph built on the same vertex set as $G$ and consisting of all the edges chosen by Client up till the $i$-th round. In particular, $H_0=\emptyset$ and $H = H_{n-1}$ is the graph built by Client when the game has finished. The Client wins if $H_{n-1}$ is not connected.
We make the following easy observations which hold for all rounds $i$.
\[obs:obs1\] If in the $i$-th round, there is a connected component $C \neq H_i$ in $H_i$ that has at most $q+2$ outgoing edges in $G_i$, then Waiter has the following options:
- he offers only edges not incident to $C$;
- all the edges he offers are incident to $C$;
- in case the number of edges incident to $C$ in $G_i$ is exactly $q+2$, he offers one of them and $q$ other edges not incident to $C$.
Indeed, if Waiter offers $2 \leq i < q+1$ edges incident to $C$, then Client can refuse all of them. Then at any later moment of time, Waiter can offer at most $q+2 - i < q+1$ edges incident to $C$. Consequently, Client can always refuse all of them, leading to $C$ becoming an isolated component in $H_{n-1}$.
\[obs:obs2\] If $C \neq H_i$ is a connected component in $H_i$ that has at most $q$ outgoing edges in $G_i$ then Client wins the game.
\[obs:obs3\] If $u$ and $v$ are isolated in $H_i$, $u$ and $v$ have degree $q+1$ in $G_i$ and are adjacent, then Client wins the game.
Indeed, suppose without loss of generality that Waiter offers all $q+1$ edges incident to $u$. Then Client chooses the edge $uv$. At any later moment of time, Client can discard any offered edge that is incident to $v$, leading to $uv$ becoming an isolated component in $H_{n-1}$.
Before we proceed to the proof of Theorem \[th1\], we need the following auxiliary lemma.
\[lemma1\] Let $K_{3k}$ be the complete graph on $3k$ vertices, where $k \geq 2$. Then there exist $k+1$ disjoint spanning trees in $K_{3k}$.
It follows from Nash-Williams theorem [@NWilliams] that $K_n$ has $\left\lfloor n/2 \right\rfloor$ edge-disjoint spanning trees. This in particular proves Lemma \[lemma1\].
We now show the following.
\[lemma2\] For any $q\geq 2$ there exists a graph $G=G(q)$ with $3(q+1)$ vertices, which is a disjoint union of $q+1$ spanning trees and such that Client has a winning strategy on $G$.
Note that this proves Theorem \[th1\] in the case $n=3q+3$. We will later extend this construction to all $q$ satisfying $2 < q+1 < \lfloor \frac{n-1}{2} \rfloor$.
The construction goes as follows.
We divide the vertices of $G$ into three sets $U=\{u_0, u_1, \ldots, u_q\}$, $V=\{v_0, v_1, \ldots, v_q\}$ and $W=\{ w_0, w_1, \ldots, w_q\}$. Next, we construct $q+1$ spanning trees $T_i$, $i=0,\ldots,q$. We first put the edges $\{u_0u_1, u_0v_0, v_0w_0\}$ into $T_0$, $\{v_0v_1, u_0w_0, w_0w_1\}$ into $T_1$, and $\{u_0u_i,v_0v_i,w_0w_i\}$ into $T_i$, where $i=2,\ldots, q$ (see Figure \[fig:fig1\]). By Lemma \[lemma1\], we can find $q+1$ disjoint spanning trees $T_0', T_1',\ldots ,T_q'$ on the vertices $(U \cup V \cup W) \setminus \{u_0, v_0, w_0\}$. We put the edges of $T_i'$ into $T_i$, for $i=0,\ldots,q$. Hence, we get $q+1$ disjoint spanning trees on the vertex set $U\cup V\cup W$. This is our graph $G=G(q)$.
\[fig:fig1\]
(9\*10:1.5cm) circle (1.5pt); (21\*10:1.5cm) circle (1.5pt); (33\*10:1.5cm) circle (1.5pt);
(11\*10:3.5cm) circle (1.5pt); (10\*10:3.5cm) circle (1.5pt); (7\*10:3.5cm) circle (1.5pt); (83:3.5cm) circle (0.5pt); (85:3.5cm) circle (0.5pt); (87:3.5cm) circle (0.5pt);
(23\*10:3.5cm) circle (1.5pt); (22\*10:3.5cm) circle (1.5pt); (19\*10:3.5cm) circle (1.5pt); (203:3.5cm) circle (0.5pt); (205:3.5cm) circle (0.5pt); (207:3.5cm) circle (0.5pt);
(35\*10:3.5cm) circle (1.5pt); (34\*10:3.5cm) circle (1.5pt); (31\*10:3.5cm) circle (1.5pt); (323:3.5cm) circle (0.5pt); (325:3.5cm) circle (0.5pt); (327:3.5cm) circle (0.5pt);
(9\*10:1.5cm) – (21\*10:1.5cm) – (33\*10:1.5cm) – (9\*10:1.5cm);
(9\*10:1.5cm) – (11\*10:3.5cm); (9\*10:1.5cm) – (10\*10:3.5cm); (9\*10:1.5cm) – (7\*10:3.5cm);
(21\*10:1.5cm) – (23\*10:3.5cm); (21\*10:1.5cm) – (22\*10:3.5cm); (21\*10:1.5cm) – (19\*10:3.5cm);
(33\*10:1.5cm) – (35\*10:3.5cm); (33\*10:1.5cm) – (34\*10:3.5cm); (33\*10:1.5cm) – (31\*10:3.5cm);
at (9\*10:1.5cm) [$u_0$]{}; at (335:1.7cm) [$v_0$]{}; at (205:1.7cm) [$w_0$]{};
at (115:3.7cm) [$u_1$]{}; at (103:3.7cm) [$u_2$]{}; at (67:3.7cm) [$u_q$]{};
at (350:3.7cm) [$v_1$]{}; at (339:3.7cm) [$v_2$]{}; at (307:3.8cm) [$v_q$]{};
at (235:3.8cm) [$w_1$]{}; at (221:3.8cm) [$w_2$]{}; at (190:3.7cm) [$w_q$]{};
We present a strategy for Client which ensures that he we will win the game. Let $G'$ be the subgraph of $G$ consisting of all edges adjacent to at least one of the vertices $u_0$, $v_0$ or $w_0$ (this is precisely the graph in Figure \[fig:fig1\]). We show that Client can isolate a subgraph of the triangle $u_0v_0w_0$, that is Client can ensure that in $H$ one of the vertices $u_0, v_0, w_0$ or one of the edges $u_0v_0$, $v_0w_0$, $w_0u_0$, or the whole triangle $u_0v_0w_0$ is a connected component.
Now notice that the vertices $u_0$, $v_0$ and $w_0$ have degree $q+2$ in $G$. Therefore, by Observation \[obs:obs1\], the first time Waiter presents an edge incident to any of the above vertices, say to $u_0$, he must present either exactly one edge incident to $u_0$ or $q+1$ edges incident to that vertex. By symmetry the same argument works for $v_0$ and $w_0$.
Moreover, whenever Waiter presents an edge outside of $G'$, Client can choose this edge and thus the number of free edges in $G'$ can only drop down. It is easy to see that if Client has a winning strategy on the graph $G'$ then the same strategy works for the graph $G'$ with some of the edges deleted. Thus we may focus on the game played entirely on the graph $G'$ (and the same parameter $q$).
We shall assume by symmetry that in the first round one of the edges offered is adjacent to $u_0$. We have two subcases.
*Case 1: All of the edges offered are incident to $u_0$.*
Then either Waiter presents both $u_0v_0$ and $u_0w_0$, or just one of them, say $u_0v_0$.
In the first case, Client chooses a third edge (not adjacent to $v_0$ nor $w_0$) and by Observation \[obs:obs3\] he wins the game.
In the second case, Client chooses $u_0v_0$. There are only $q+2$ free edges incident to $u_0v_0$ now. If in the next round, both edges $u_0w_0$ and $v_0w_0$ are offered, then all the other edges offered must be incident to $w_0$ too. Client chooses one of these edges, leaving only $q$ free edges incident to $u_0v_0$, and thus winning the game. If in the second round, exactly one of $u_0w_0$ and $v_0w_0$ is offered, then Client chooses this edge. There are only $q$ free edges left incident to the triangle $u_0v_0w_0$, and so again Client wins. So in the second round, by Observation \[obs:obs1\], Waiter offers $q$ edges incident to $v_0$ and $1$ edge incident to $w_0$. Client then picks the edge incident to $w_0$, leaving only $2 \leq q$ free edges incident to $u_0v_0$. So Client wins in this last situation as well.
*Case 2: Exactly one of the edges offered is incident to $u_0$.*
We may assume that Case 1 does not apply to $v_0$ nor $w_0$, otherwise we are done. Then by Observation \[obs:obs1\], Waiter offers one edge incident to $u_0$, one incident to $v_0$, and one incident to $w_0$. Thus $q=2$. Furthermore the edge incident to $u_0$ is either $u_0u_1$ or $u_0u_2$. Client then chooses this edge.
In the second round Waiter has to offer at least one edge incident to $\{v_0,w_0\}$. Consequently by Observation \[obs:obs1\], Waiter must offer three edges adjacent to one of the vertices $v_0$ or $w_0$. Then Client chooses the edge $v_0w_0$. There are only $2$ edges left incident with $v_0w_0$, and so Client wins by Observation \[obs:obs2\].
The crucial part of the above argument is the existence of a subgraph $G'$ in which we have three vertices spanning a triangle and such that the degree of each of them in $G$ is equal precisely to $q+2$. By repeating the same reasoning, one can ensure oneself that in the connectivity game played on the graph $G$ of this form, that is having such a subgraph $G'$, Client always has a winning strategy. In the remaining part of the proof we show that for any $3 \leq q+1 < \lfloor \frac{n-1}{2}\rfloor$ there exists such a graph $G = G(n,q)$.
Let $n \geq 9$ and $3 \leq q+1 < \left\lfloor \frac{n-1}{2}\right\rfloor$. We use the induction to construct $G(n,q)$ from $G(n-2,q-1)$. To start the induction, we need a graph $G(n - 2(q-2),2)$ on $n - 2(q-2) \geq 9$ vertices, which is the union of three spanning trees and which has three fixed vertices $u_0$, $v_0$, $w_0$ each of degree four and such that they span a triangle. For example, we can take the graph $G(2)$ which has 9 vertices, add to it $n - 2(q-2) - 9$ new vertices and add three edges between each new vertex and three arbitrary vertices from $G(2)$ different from $u_0$, $v_0$, $w_0$.
The construction goes as follows. Assume that we have already constructed the graph $G(n_0,q_0)$ on $n_0$ vertices which is the union of $q_0+1$ spanning trees and in which $u_0$, $v_0$, $w_0$ all have degree $q_0+2$. We now split vertices of $G(n_0,q_0)$ into three sets $V_1$, $V_2$ and $V_3$. The set $V_1$ consists of the vertices $u_0$, $v_0$, $w_0$, and we split the remaining vertices into two sets of size $\lfloor \frac{n_0-3}{2} \rfloor$ and $\lceil \frac{n_0-3}{2} \rceil$ respectively. Let $T_0,T_1,\ldots,T_{q_0}$ denote the spanning trees of $G(n_0,q_0)$. We add two new vertices $u$ and $v$. We then use the edges between $u$ and $V_2$ to extend each spanning tree $T_i$ to a spanning tree $T_i'$ of $G(n_0,q_0) \cup \{u\}$. Note that this is possible, as $\lfloor \frac{n_0-3}{2} \rfloor = \left\lfloor \frac{n_0-1}{2}\right\rfloor - 1 \geq q_0+1$. Similarly, we use the edges between $v$ and $V_3$ to extended each $T_i'$ to a spanning tree of $G(n_0,q_0) \cup \{u, v\}$. Finally, we construct a new spanning tree $T$ of $G(n_0,q_0) \cup \{u, v\}$ from the edges $uv$, $uu_0$, $uv_0$, $uw_0$ and all edges between $u$ and $V_3$ and all edges between $v$ and $V_2$.
*Acknowledgements*. We would like to thank Tibor Szabó for bringing the problem studied in this paper to our attention.
J. Beck, [*Positional games and the second moment method*]{}, Combinatorica 22 (2002), 169–216.
M. Bednarska-Bzdega, D. Hefetz, M. Krivelevich, T. Łuczak, [*Manipulative waiters with probabilistic intuition*]{}, preprint.
A. Csernenszky, C. I. Mándity, A. Pluhár, [*On Chooser-Picker positional games*]{}, Discrete Mathematics 309 (2009), 5141–5146.
C.S.J.A. Nash-Williams, [*Edge-Disjoint Spanning Trees of Finite Graphs*]{}, J. London Math. Soc. (1961) s1-36 (1): 445-450.
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{
"pile_set_name": "ArXiv"
}
|
---
author:
- 'A. Vallenari'
title: Young stellar populations in the Magellanic Clouds
---
Introduction
============
The process of star formation in different environments is far from being understood. In particular, it is difficult to reconcile the prominent influence of the local environment (turbulence, compression, initial trigger) on small scales with the universality of the Schmidt and Kennicut law on Galactic scales which suggests that Galactic-scale gravity is involved in the first stages of star formation. Detailed understanding of how an entire molecular cloud converts into stars is still lacking. To analyze star forming regions can cast light on the formation scenarios, since spatial distributions of different components (molecular clouds, ionized gas), and of stars in different evolutionary stages can help to distinguish between the proposed models. Local Group galaxies, and in particular the Magellanic Clouds (MCs), are very promising regions to study the process of star formation. Their young population of stars and clusters allow to derive hints about the formation process in low metal content environments. This is particularly important since the formation of stars depends on the balance between the gas heating and the cooling, on which the presence of metals has a significant effect. In this paper, in Section \[forma\] we first discuss the formation mechanism of field stars and clusters, in section \[trigger\] the effects on the star formation of the gravitational interaction between the MCs and the Milky Way are discussed, in section \[n11a\] and \[n11b\] the discovery of pre-main sequence candidates in N 11 in the LMC is presented. Final remarks are given in section \[conclu\].
Field star and cluster formation scenario {#forma}
=========================================
Several mechanisms of cluster and field star formation have been proposed in literature. In the turbulence theory, star and cluster formation are regulated by the balance between turbulence and gravitation. The net effect of highly compressible turbulence is to prevent collapse globally, while on local scale it causes local density enhancements that might produce a collapse, under suitable conditions. If the surrounding flow is not strong enough to continue to drive the cloud, the turbulence will quickly dissipate giving rise to an active star formation. The properties of the fragmenting clouds and the mass of the individual proto-stellar cores, giving birth to isolated stars rather then clusters of objects, depend in a complex and still unclear way on the values of the politropic index which, in turn, is related to the metal content of the gas. When turbulence dominates, the star formation is inefficient and slow and stars build up in small groups. Turbulent control of the star formation predicts that star clusters form in regions where the support of the turbulence is insufficient or where only large scale driving is working (McLow & Klessen 2004). Star formation on scales of galaxies as a whole is expected to be controlled by the balance between turbulence and self-gravity just like star formation on scale of individual gas clouds, but might be modulated by additional effects, such as cooling, differential rotation, gravitational interaction (Sasao 1973, Li et al 2004). An alternative mechanism is suggested by Bekki et al (2004): during galaxy interactions and mergers clusters can form as a result of relatively high velocity cloud-cloud collisions.
Young Cluster and field SF in the SMC: star formation by gravitational trigger {#trigger}
==============================================================================
Star formation by gravitational trigger is well known to take place in mergers/ interactions. The Magellanic Clouds represents one of the best studied examples. Formation episodes involving both cluster and field star formation happened at 5, 20, and finally at 100-150 Myr, in coincidence with [SMC]{} perigalactic passage (Chiosi et al 2006). As far as the old population is concerned, the star formation was not continuous but proceeded in a number of bursts taking place at 3 Gyr and 6 Gyr ago. These bursts are temporally coincident with past peri-galactic passages of the SMC about the Milky Way. Only a very low efficiency of the star formation at epochs older than 6-8 Gyr is found (Chiosi & Vallenari 2007, Tosi et al 2007, Noel et al 2007). These results point out that the most recent interactions between [SMC]{} and [LMC]{} triggered cluster and field star formation in the [SMC]{}, in agreement with the expectations from Bekki & Chiba (2005) while at older ages, the tidal interaction between the Magellanic Clouds and the Milky Way was not able to give rise to significant star formation events (see Vallenari 2007 for a wider discussion). However, as expected, the star formation process cannot be explained uniquely as the result of gravitational trigger. Comparing field stars with the cluster age distribution, Chiosi et al (2006) find that there is not a complete coincidence between young cluster and field star formation, suggesting that different modes of formation might be at work.
The stellar population of N 11 in the LMC: a case of triggered star formation? {#n11a}
==============================================================================
Most stars form in groups and clusters. This implies that recently formed stellar objects must interact with their environment. In this respect, one of the most intriguing aspects of the star formation is that of the triggering. Young massive stars are expected to inject energy into the nearby interstellar medium, heating and compressing the surrounding gas. This can have destructive or constructive effects, depending on the balance between heating and gravity but it is still unclear what regulates it. To distinguish between sequential and triggered star formation is indeed very difficult. Star formation is often found to be sequential, i.e. stellar population can independently form in adjacent clouds with little age difference, but no causal relationship is present. A hierarchical system of three or more generations of stars in the same region is often interpreted as more stringent sign of triggers (i.e. causal relationship between episodes of star formation)(Oey et al 2005, Deharveng et al 2005). The stellar population in N 11 is often presented as one of the best example of triggered star formation. N 11 is the second largest nebula of the LMC after the 30 Dor Nebula. It is located in the north-western corner of the LMC. It has a peculiar morphology with a central hole with no emissions and several filaments around. The central cavity has been evacuated by the OB stars in the association LH 9, located near the center of N 11. The whole complex has a diameter of 45’, corresponding to a linear extent of 705 pc, assuming a distance for the LMC of 54 kpc. In addition to LH 9, this complex includes the OB associations LH 10 in N11B, LH 13 in N11C, LH 14 in N11E. Bica & Dutra (2000) find at least 18 objects, including associations and clusters, in the age range 0-30 Myr. Walborn & Parker (1992) suggest a two stage star-burst hypothesis: an initial star burst in LH 9 triggered a secondary burst in LH 10, a few Myr later. This hypothesis was recently confirmed by the results by Hatano et al (2006), Mokiem et al (2007) who detect 127 Herbig Ae/Be star candidates from near-infrared photometry in this region, mainly in the periphery of LH 9. Herbig Ae/Be star are intermediate mass pre-main sequence stars (7-3 M$_\odot$) and have age range 1-3 Myr. Hatano et al (2006) find a spatial correlation of OB stars and Herbig Ae/Be star candidates with the radio continuum, H$_\alpha$, CO and X-ray. Herbig Ae/Be stars are found in all the associations, but inside LH 9. This suggests that in LH 9 such very young objects would have already disappeared, losing their circumstellar envelopes/disks. This leads to the conclusion that LH 9 is slightly older than the other associations and has possibly triggered star formation episodes in the surrounding regions.
Pre-MS candidate stars in N11 {#n11b}
=============================
PMS stars are found in the LMC only in few young associations/clusters: in the region of SN 1987 (Panagia et al 2000), in the young double cluster NGC 1850 (Gilmozzi et al 1994), in 30 Dor region (Brandner et al 2001, Romaniello et al 2006), in LH 95 (Gouliermis et al 2002) and in LH 52(Gouliermis et al 2007).
HST ACS/WFC archive photometry in the region of N 11 shows the presence of pre-main sequence candidates (Vallenari & Chiosi 2007). The studied fields are partially covering the associations LH 9, LH 10, and LH 13. The most conspicuous concentrations of pre-main sequence stars are seen in N11B associated with LH 10, and in LH 9. These concentrations are corresponding to the location of OB stars. A small group of PMS candidates are found in N11C (associated with LH 13). Spectroscopic determination of the age of LH 13 are in the range 3-5 Myr, while LH 10 is 3$\pm 1$ Myr old (Heydari-Malayeri et al 2000). The presence of OB stars in LH 9 set its age at 7.0 $\pm 1 $ Myr (Mokiem et al 2007). Fig.\[li\_vhel\] shows the CMD of the cluster HD 3228 located at the periphery of the association LH 9, where the PMS star candidates are clearly visible as a sequence well separated from the field population. The age of this cluster is set to about 3-4 Myr by Walborn et al (1999) based on the presence of a WC star. Pre-main sequence tracks by Siess et al (2000) suggest an age ranging from 1 to 10 Mys, depending on the adopted extinction (A$_V$=0.2 –0.8). The comparison with the isochrones shows that pre-main sequence stars have masses from 1.3 M$_\odot$ to 2.0 M$_\odot$. Far infrared Spitzer Space Telescope IRAC and MIPS observations of the region are made in the framework of the SAGE project (Meixner et al 2006). These observations are complementary to HST photometry: while HST observations can give information about faint, exposed pre-main sequence candidates, near-IR data allow to detect embedded young stellar objects(YSO). Comparing the magnitudes and colors of the objects with photometric models by Robitaille et al (2006), Vallenari & Chiosi (2007) select young stellar object candidates. They find out that YSO type I (showing large infalling envelopes) and II (characterized by the presence of an optically thick disk) having ages from 0.1 to 1 Myr are found at the same location than the candidate PMS stars. While it cannot be excluded that LH 9 triggered the star formation in the surroundings, however the data seems to suggest that the star formation in the region is a long lasting process where stars from 0.1 to 7 Myr are widely distributed. This seems to be in agreement with a turbulent scenario of star formation.
Conclusions {#conclu}
===========
In this paper we discuss the young population of stars and clusters in the Magellanic Clouds. The last gravitational interaction between LMC and SMC has triggered a star formation episode in both the cluster and field stars. We find that the SMC was relatively quiescent at ages older than 6-8 Gyr. This result suggests that at older ages, the tidal interaction between the Magellanic Clouds and the Milky Way was not able to trigger significant star formation events. Field star and cluster formation are not completely correlated, as expected on the basis of theoretical models and different star formation modes are expected to act in the field and in cluster populations. Young stars are well known to trigger star formation on local scale by injecting energy in the surrounding interstellar medium. N 11 region in the LMC is one of the most promising candidate of triggered star formation. In this paper we discuss the star formation process in this region. We report on the discovery of a PMS candidates associated with N 11 from HST ACS/WFC photometry. The PMS are consistent with an age going from 1 to 10 Myr, although a more precise determination is not possible due to the uncertainties on the interstellar extinction. The comparison with the isochrones shows that pre-main sequence stars having masses from 1.3 M$_\odot$ to 2.0 M$_\odot$ are present. Spitzer IRAC and MIPS observations of the region, made in the framework of the SAGE project reveals the presence of young stellar object (YSO) candidates. YSO type I and II having ages $< 1 Myr$ are found at the same location than the candidate PMS stars. The data seems to suggest that the star formation in the region is a long lasting process where stars from 0.1 to 7 Myr are widely distributed, as expected in a turbulent scenario of star formation.
This work was done in collaboration with E. Chiosi, E. Held, A. Moretti, G. Bertelli, L. Rizzi
Bekki, K. , Couch, W. J., Beasley M. A., et al. 2004, , 610, L93
Bekki, K., Chiba, M. 2005, , 356, 680
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|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'Pseudodifferential operators of several variables are formal Laurent series in the formal inverses of $\partial_1, \ldots, \partial_n$ with $\partial_i = d/dx_i$ for $1 \leq i \leq n$. As in the single variable case, Lax equations can be constructed using such pseudodifferential operators, whose solutions can be provided by Baker functions. We extend the usual notion of tau functions to the case of pseudodifferential operators of several variables so that each Baker function can be expressed in terms of the corresponding tau function.'
---
\[lee-firstpage\]
Introduction
============
One of the most actively studied areas in mathematics for the past few decades is the theory of integrable nonlinear partial differential equations (see e.g. [@Ca91; @Ch96; @Di91; @Ku00]). Such equations are also known as soliton equations because they possess localized nonlinear waves called solitons as solutions. Examples of soliton equations include many well-known equations in mathematical physics such as the nonlinear Schrödinger equation, the Sine-Gordon equation, the Korteweg-de Vries (KdV) equation, and the Katomtsev–Petviashvili (KP) equation.
The main tool used in a systematic study of soliton equations is the notion of Lax equations, which describe certain compatibility conditions for pairs of differential operators. A system of soliton equations called a KP hierarchy is produced by a set of Lax equations, and as a result, solutions of Lax equations can be used to construct solutions of the associated soliton equations. The interpretation of soliton equations in terms of Lax equations leads to the derivation of the integrability as well as other interesting properties of soliton equations.
A few decades ago, Krichever (see e.g. [@Kr77]) introduced the method of constructing an infinite dimensional subspace of ${\mathbb C} (( z))$ associated to some algebro-geometric data, where ${\mathbb C}(( z))$ is the space of Laurent series. This construction is nowadays called the Krichever map, and it has been used successfully in the soliton theory and is closely linked to the theory of moduli of algebraic curves (cf. [@AD88; @Kr77; @SW85]). Thus the Krichever map provides a connection of soliton theory with algebraic geometry, which is one of the most intriguing features of the theory of soliton equations. More specifically, to each subspace of ${\mathbb C}(( z))$ produced by the Krichever map there corresponds a so-called Baker–Akhiezer function, which determines an algebro-geometric solution of a soliton equation (see [@BB94; @Ch96; @Kr77; @SW85]). Baker functions are a generalized version of Baker–Akhiezer functions, and they supply formal solutions of Lax equations. Tau functions also play an important role in algebro-geometric theory of solitons, and in particular, each Baker function can be expressed in terms of the associated tau function. Such an expression of a Baker function in terms of a tau function is an important contribution of the Japanese school (see e.g. [@DK83]). Tau functions can be used to construct soliton solutions of soliton equations, and they are essential in linking soliton theory to quantum field theory as well as to the theory of Virasoro algebras or vertex operators.
Pseudodifferential operators are formal Laurent series in the formal inverse $\partial^{-1}$ of the differentiation operator $\partial = d/dx$ with respect to the single variable $x$, and they are essential ingredients in the construction of Lax equations. For this reason pseudodifferential operators have played a major role in the theory of soliton equations. In a recent paper, Parshin [@Pa99] studied pseudodifferential operators of several variables by considering formal Laurent series in the formal inverses of $\partial_1, \ldots, \partial_n$ with $\partial_i = d/dx_i$ for $1 \leq i \leq
n$. Among other things, he constructed Lax equations associated to such pseudodifferential operators and studied some of their properties. Since then, algebro-geometric connections of those pseudodifferential operators have been studied in [@Pa99a] and [@Os01], where the possibility of extending the Krichever map to the case of higher dimensional varieties was discussed. Baker functions which provide solutions to Lax equations of Parshin type have also been investigated in [@L0f], where some of the properties of the usual Baker functions were extended to the case of pseudodifferential operators of several variables. The goal of this paper is to prove the existence of tau functions associated to Baker functions constructed in [@L0f].
Pseudodifferential operators {#S:ps}
============================
In this section we review pseudodifferential operators of several variables studied by Parshin [@Pa99] as well as the associated Lax equations. We also describe an example of a system of partial differential equations determined by such a Lax equation.
We fix a positive integer $n$ and consider the variables $x_1, \ldots,
x_n$. We denote by $${\mathbb C} ((x_1)) \cdots ((x_n))$$ the associated field of iterated Laurent series over $\mathbb C$, and let $P$ be the space of iterated formal Laurent series of the form $$P= {\mathbb C} ((x_1)) \cdots ((x_n))
\left(\left(\partial^{-1}_1\right)\right) \cdots
\left(\left(\partial^{-1}_n\right)\right)$$ in the formal inverses of the differential operators $$\partial_1 = \frac{\partial} {\partial x_1}, \ \ \ldots, \ \ \partial_n = \frac{\partial} {\partial x_n} .$$ Throughout this paper we shall often use the usual multi-index notation. Thus, given $\alpha = (\alpha_1, \ldots, \alpha_n) \in {\mathbb Z}^n$, we may write $$\partial^\alpha = \partial_1^{\alpha_1} \cdots \partial_n^{\alpha_n}, \qquad |\alpha| = \alpha_1 +
\cdots +\alpha_n ,$$ with $\partial = (\partial_1, \ldots, \partial_n)$. We also write $\alpha \geq \beta$ for $\beta = (\beta_1, \ldots, \beta_n) \in {\mathbb Z}^n$ if $\alpha_i \geq \beta_i$ for each $i$, and use $\mathbf 0$ and $\mathbf 1$ to denote the elements $(0, \ldots, 0)$ and $(1, \ldots, 1)$ in ${\mathbb Z}^n$, respectively. Thus, for example, an element $\psi \in P$ can be written in the form $$\label{E:zx}
\psi = \sum_{\alpha \leq \nu} f_\alpha (x) \partial^\alpha$$ for some $\nu \in {\mathbb Z}^n$. We introduce a multiplication operation on $P$ defined by the Leibniz rule, which means that $$\left( \sum_{\alpha} f_\alpha (x) \partial^\alpha \right) \left( \sum_{\beta} h_\beta (x)
\partial^\beta \right) = \sum_{\alpha, \beta} \sum_{\gamma \geq {\mathbf 0}}
\left(\begin{array}{c} \alpha \\ \gamma\end{array}\right) f_\alpha
(x) (\partial^\gamma h_\beta (x)) \partial^{\alpha+\beta -\gamma} ,$$ where $\left(\begin{array}{c} \alpha \\ \gamma\end{array}\right) =
\left(\begin{array}{c} {\alpha_1}\\ {\gamma_1}\end{array}\right) \cdots
\left(\begin{array}{c} {\alpha_n}\\ {\gamma_n}\end{array}\right)$ for elements $\alpha = (\alpha_1, \ldots, \alpha_n)$ and $\gamma = (\gamma_1, \ldots,
\gamma_n)$ of ${\mathbb Z}^n$ with $\gamma \geq {\mathbf 0}$. We now set $${\mathbb Z}^n_+ = \{ \alpha \in {\mathbb Z}^n \mid \alpha \geq {\mathbf 0}, \, |\alpha| \geq 1 \},$$ and assume that each coefficient $f_\alpha (x)$ in is a function of the infinitely many variables $\{t_\alpha \mid \alpha \in {\mathbb Z}^n_+ \}$. Let ${\mathbf e}_1 = (1,0,\ldots, 0)$, $\ldots$, ${\mathbf e}_n = (0,\ldots, 0,1)$ be the standard basis for the ${\mathbb Z}$-module ${\mathbb Z}^n$, and assume that $$\label{E:sa}
t_{{\mathbf e}_1} = x_1, \ \ \ldots, \ \ t_{{\mathbf e}_n} = x_n .$$ Thus we may write $\psi \in P$ in in the form $$\psi = \sum_{\alpha \leq \nu} f_\alpha (t) \partial^\alpha$$ with $t = (t_\alpha)_{\alpha \in {\mathbb Z}_+^n}$.
If $\psi$ is an element of $P$ which can be written in the form $$\psi= \sum_{i= -\infty}^{\nu_n} a_i \partial_n^i = \sum_{i= -\infty}^{\nu_n}
a_i (t; \partial_1, \ldots, \partial_{n-1}) \partial_n^i$$ with $\nu_n \geq 0$, we set $$\psi_+ = \sum_{i= 0}^{\nu_n} a_i \partial_n^i, \qquad \psi_- = \psi- \psi_+ =
\sum_{i= -\infty}^{-1} a_i \partial_n^i ;$$ if $\nu_n < 0$, we set $\psi_+ = 0$ and $\psi_- = \psi$ . Thus we have $\psi = \psi_+ + \psi_-$ for all $\psi \in P$, and therefore $P$ can be decomposed as $$P = P_+ + P_- ,$$ where $P_+$ is the set of elements of $P$ of the form $\sum\limits_{i= 0}^m a_i
\partial_n^i$ for some nonnegative integer $m$, and $P_-$ is the set of elements of the form $\sum\limits_{j= 0}^k b_j \partial_n^j$ with $k <0$. Let $P^n$ be the Cartesian product of $n$ copies of $P$, and consider an element $L =
(L_1, \ldots, L_n) \in P^n$ which satisfies the generalized Lax equation $$\label{E:4q}
\partial_{t_\alpha} L= [L^\alpha_+,L] = L^\alpha_+ L - L L^\alpha_+$$ for all $\alpha \in {\mathbb Z}^n_+$, where $L^\alpha_+ = (L^\alpha)_+ = (L^{\alpha_1}_1
\cdots L^{\alpha_n}_n)_+ \in P_+$ and $$\partial_{t_\alpha} L= \frac{\partial L} {\partial t_\alpha} = \left( \frac {\partial L_1} {\partial
t_\alpha}, \ldots , \frac {\partial L_n} {\partial t_\alpha} \right) .$$ Thus is equivalent to the system of equations $$\frac {\partial L_i} {\partial t_\alpha} = [L^\alpha_+,L_i]$$ for $1\leq i \leq n$.
We now consider an element $\phi \in 1 + P_-$ satisfying the relation $$\label{E:w4}
\partial_{t_\alpha} \phi = - \left(\phi \partial^\alpha \phi^{-1}\right)_- \phi$$ for each $\alpha \in {\mathbb Z}^n_+$, and set $$\label{E:ak}
L= \phi \partial \phi^{-1} = \left(\phi \partial_1 \phi^{-1}, \ldots, \phi \partial_n
\phi^{-1}\right) \in P^n .$$ Thus, if $L = (L_1, \ldots, L_n)$, then each $L_i$ is of the form $$L_i = \phi \partial_i \phi^{-1} = \partial_i +u_i$$ for some $u_i \in P_-$. Then it can be shown that the pseudodifferential operator $L$ given by satisfies the Lax equation for each $\alpha \in {\mathbb Z}^n_+$. The Lax equation also implies the relation $$\label{E:p9}
\frac {\partial L^\beta_+} {\partial t_\alpha} - \frac {\partial L^\alpha_+} {\partial t_\beta} =
[ L^\alpha_+, L^\beta_+]$$ for all $\alpha, \beta \in {\mathbb Z}^n_+$ (see [@Pa99 Proposition 4]). For each pair $(\alpha, \beta)$ of elements of ${\mathbb Z}^n_+$ the relation determines a system of partial differential equations as can be seen in the following example.
We shall derive partial differential equations which are determined by the Lax equation for $n=2$ associated to the pseudodifferential operators $L_1,
L_2 \in P$ given by $$\begin{gathered}
L_1 = \partial_2 + a \partial_1 \partial_2^{-1} + b \partial_2^{-2} + O\left(\partial_2^{-3}\right),
\label{E:l1}\\
L_2 = \partial_2 + c \partial_2^{-1} + d \partial_1 \partial_2^{-2} + O\left(\partial_2^{-3}\right)
\label{E:l2}\end{gathered}$$ for some functions $a = a (t)$, $b = b (t)$, $c = b (t)$ and $d = d (t)$ with $t = (t_\alpha)_{\alpha \in {\mathbb Z}_+^n}$. We also consider the indices $$\alpha = (1,1), \qquad \beta = (1,2) ,$$ so that $L^\alpha = L_1 L_2$ and $L^\beta = L_1 L_2^2$, where $L= (L_1, L_2) \in
P^2$. Then by the differential operators $L^\alpha_+$ and $L^\beta_+$ satisfy $$\label{E:pf}
\frac {\partial L^\beta_+} {\partial t_\alpha} - \frac {\partial L^\alpha_+} {\partial t_\beta} =
L^\alpha_+ L^\beta_+ - L^\beta_+ L^\alpha_+ .$$ Using and , we obtain $$\begin{gathered}
L_2^2 = \partial_2^2 + c_y \partial_2^{-1} + 2c + 2d \partial_1 \partial_2^{-1} + d_y \partial_1
\partial_2^{-2} + c^2 \partial_2^{-2} + O\left(\partial_2^{-3}\right),\\
L_1 L_2 = \partial_2^2 + a\partial_1 + c + O\left(\partial_2^{-1}\right) ,\\
L_1 L_2^2 = \partial_2^3 +a \partial_1 \partial_2 + 2c \partial_2 + 2d \partial_1 + 3c_y+ b +
O\left(\partial_2^{-1}\right) ,\end{gathered}$$ where the subscripts $x$ and $y$ denote the partial derivatives with respect to $x= x_1$ and $y = x_2$, respectively. Hence we have $$\begin{gathered}
L^\alpha_+ = (L_1 L_2)_+ = \partial_2^2 + a\partial_1 + c,
\label{E:w1}\\
L^\beta_+ = \left(L_1 L_2^2\right)_+ = \partial_2^3 +a \partial_1 \partial_2 + 2c \partial_2 + 2d \partial_1 +
3c_y + b .
\label{E:w2}\end{gathered}$$ Using and , we obtain $$\begin{gathered}
L^\alpha_+ L^\beta_+ = \partial_2^5 + 2a \partial_1 \partial_2^3 + 3c \partial_2^3 + (2a_y +2d) a
\partial_1 \partial_2^2 + (7 c_y +b) \partial_2^2 + a^2 \partial_1^2 \partial_2\\
\phantom{L^\alpha_+ L^\beta_+ =}{} + (a_{yy} + 4d_y + aa_x + 3ac)
\partial_1 \partial_2 + \left(8c_{yy} +2b_y+ 2ac_x + 2c^2\right) \partial_2\\
\phantom{L^\alpha_+ L^\beta_+ =}{}+ 2ad \partial_1^2 + (2d_{yy} +2ad_x + 3ac_y +ab +cd) \partial_1\\
\phantom{L^\alpha_+ L^\beta_+ =}{} + 3c_{yyy} + b_{yy} + 3ac_{yx} + ab_x + 3cc_y +bc,\\
L^\beta_+ L^\alpha_+ = \partial_2^5 + 2a \partial_1 \partial_2^3 + 3c \partial_2^3 + (3a_y +2d) a
\partial_1 \partial_2^2 + (6 c_y +b) \partial_2^2 + a^2 \partial_1^2 \partial_2\\
\phantom{L^\beta_+ L^\alpha_+ =}{} + (3a_{yy} + aa_x + 3ac) \partial_1 \partial_2 +
\left(3c_{yy} + ac_x +2c^2\right) \partial_2 + (aa_y +2ad) \partial_1^2\\
\phantom{L^\beta_+ L^\alpha_+ =}{} + (a_{yyy} +aa_{yx} +4a c_y +2 a_y c +2a_x d +ab +2cd)
\partial_1\\
\phantom{L^\beta_+ L^\alpha_+ =}{} + c_{yyy} + ac_{yx} + 5cc_y +2c_x d +bc.\end{gathered}$$ If we set $t_\alpha =s$ and $t_\beta =t$, then by and the left hand side of becomes $$\frac {\partial L^\beta_+} {\partial s} - \frac {\partial L^\alpha_+} {\partial t} = a_s \partial_1
\partial_2 + 2c_s \partial_2 + (2d_s -a_t) \partial_1 + 3 c_{ys} + b_s - c_t .$$ Thus by comparing the coefficients we see that determines the system of partial differential equations given by $$\begin{gathered}
a_y = c_y =0,\qquad
2c_s = 2b_y + ac_x,\\
a_t + 2ad_x +2d_{yy} = 2a_x d +2d_s + cd,\qquad
b_s +2c_x d = ab_x +b_{yy} + c_t.\end{gathered}$$
Baker functions {#S:bf}
===============
Baker functions associated to pseudodifferential operators of several variables discussed in Section \[S:ps\] were introduced in [@L0f]. As in the single variable case, these Baker functions provide solutions of Lax equations of the from . In this section we review the construction of such Baker functions.
First, we need to introduce an additional set of complex variables $z_1,
\ldots, z_n$. We then consider the formal series given by $$\label{E:7n}
\xi (t, z) = \sum_{\alpha \in {\mathbb Z}^n_+} t_\alpha z^\alpha ,$$ where $z = (z_1, \ldots, z_n)$ so that $z^\alpha = z_1^{\alpha_1} \cdots
z_n^{\alpha_n}$ for $\alpha = (\alpha_1, \ldots, \alpha_n)$. If $\phi \in 1+ P_-$ is as in Section \[S:ps\] satisfying , we define the associated [*Baker function*]{} $w$ by $$\label{E:q0}
w = w(t, z) = \phi e^{\xi (t, z)} .$$ Since $x_i = t_{{\mathbf e}_i}$ for $1 \leq i \leq n$ by , we see that $$\partial_i e^{\xi (t, z)} = \frac {\partial} {\partial x_i} e^{\xi (t, z)} = \frac
{\partial} {\partial t_{\mathbf e_i}} e^{\xi (t, z)} = z^{{\mathbf e}_i} e^{\xi (t, z)} = z_i e^{\xi
(t, z)}.$$ Thus, if $\alpha = (\alpha_1, \ldots, \alpha_n) \in {\mathbb Z}^n_+$, we have $$\begin{gathered}
\partial^\alpha e^{\xi (t, z)} = \partial_1^{\alpha_1} \cdots \partial_n^{\alpha_n} e^{\xi (t, z)}
= \partial_{\mathbf e_1}^{\alpha_1} \cdots \partial_{\mathbf e_n}^{\alpha_n} e^{\xi (t, z)}\\
\phantom{\partial^\alpha e^{\xi (t, z)}}{}=
z_1^{\alpha_1} \cdots z_n^{\alpha_n} e^{\xi (t, z)} = z^\alpha e^{\xi (t, z)}.\end{gathered}$$ Hence, if $\phi = 1 + \sum\limits_\alpha a_\alpha (t) \partial^\alpha\in 1+ P_-$, then the Baker function in can be written in the form $$\label{E:ne}
w (t,z) = \widehat{w} (t,z) e^{\xi (t, z)} ,$$ where $\widehat{w} (t,z)$ is a formal power series in $z_1, \ldots, z_n$ of the form $$\widehat{w} (t,z) = 1 + \sum_\alpha a_\alpha (t) z^\alpha .$$ If $L= (L_1, \ldots, L_n) = \phi \partial \phi^{-1} \in P^n$ is an element associated to $\phi \in 1+ P_-$ satisfying as in , then the Baker function $w$ given by satisfies $Lw = zw$, that is, $L_i w = z_i w$ for each $i$ (see [@L0f Lemma 3.1]). In addition, it can also be shown that $\partial_{t_\alpha} w = L^\alpha_+ w$ for each $\alpha \in
{\mathbb Z}^n_+$ (cf. [@L0f Lemma 3.2]).
Given an element $\psi = \sum\limits_{\alpha \leq \nu} f_\alpha (t) \partial^\alpha \in P$, we define its adjoint $\psi^* \in P$ by $$\label{E:yb}
\psi^* = \sum_{\alpha \leq \nu} (-1)^{|\alpha|} \partial^\alpha f_\alpha (t) ,$$ and its residue with respect to $\partial$ by $${\rm Res}_\partial \psi = f_{-{\mathbf 1}} (t) = f_{(-1, \ldots, -1)} (t) .$$ On the other hand, if $h (z) = h (z_1, \ldots, z_n)$ is a Laurent series in $z_1, \ldots, z_n$ which can be written in the form $h(z) = \sum\limits_\alpha b_\alpha
z^\alpha$, then its residue with respect to $z$ is given by $$\label{E:h6}
{\rm Res}_z h(z) = b_{-\mathbf 1} = b_{(-1, \ldots, -1)} .$$ If $\psi = \sum\limits_\alpha a_\alpha
\partial^\alpha \in P$ and $\eta = \sum\limits_\beta b_\beta \partial^\beta
\in 1 +P_-$, then we have $${\rm Res}_z \left(\psi e^{\xi (t,z)}\right) \left(\eta e^{-\xi (t,z)}\right) = {\rm Res}_\partial \psi \eta^*,$$ where $\eta^*$ is the adjoint of $\eta$ given by (see [@L0f Lemma 3.3]).
We define the adjoint $w^*$ of the Baker function $w$ in by $$\label{E:r4}
w^* (t,z) = (\phi^*)^{-1} e^{-\xi (t,z)} ,$$ where $\phi^*$ is the adjoint of $\phi$ given by . Then it can be shown that the Baker function $w$ in satisfies $$\label{E:w9}
{\rm Res}_z w (t',z) w^\ast (t, z) = 0$$ for all $t$, $t'$ (see [@L0f]).
We now consider the subset $\widehat{P}_-$ of $P_-$ defined by $$\label{E:h5}
\widehat{P}_- = \left\{ \sum_\alpha f_\alpha (t) \partial^\alpha \; \bigg| \; \alpha \leq
-{\mathbf 1} = (-1, \ldots, -1) \ \mbox{whenever} \ f_\alpha (t) \neq 0 \right\} .$$ Then the following theorem extends the result in [@Di91 Proposition 7.3.5] to the case of several variables.
Let $w$ and $w^\#$ be formal power series of the form $$w = \phi e^{\xi (t, z)}, \qquad w^\# = \psi e^{-\xi (t, z)}$$ with $\phi, \psi \in 1+ \widehat{P}_-$ satisfying the condition $${\rm Res}_z \left(\partial^\alpha w w^\#\right) = 0.$$ Then there exists an operator $L= (L_1, \ldots, L_n) \in P^n$ with $L_i =
\partial_i + u_i$ and $u_i \in P_-$ for $1\leq i \leq n$ which satisfies the Lax equation with $w$ and $w^\#$ being the associated Baker function and adjoint Baker function, respectively.
See [@L0f Theorem 3.6].
Tau functions
=============
In this section we extend the notion of tau functions associated to the usual pseudodifferential operators to the case of pseudodifferential operators of several variables. As in the single variable case, a Baker function given by can be expressed in terms of such a tau function.
Let $t= (t_\alpha)_{\alpha \in {\mathbb Z}_+^n}$ and $z = (z_1, \ldots, z_n)$ be the complex variables considered in Section \[S:bf\]. Given a vector $s =
(s_1, \ldots, s_n) \in {\mathbb C}^n$, we define the operator $G(s)$ on functions of the form $f(t,z) = f((t_\alpha)_{\alpha \in {\mathbb Z}_+^n},(z_1,
\ldots, z_n))$ by $$\label{E:yr}
G(s) f(t,z) = f \left(\left(t_\alpha - \alpha^{-1} s^{-\alpha}\right)_{\alpha \in {\mathbb Z}_+^n}, z\right),$$ where $\alpha^{-1} s^{-\alpha} = \alpha_1^{-1} \cdots \alpha_n^{-1} s_1^{-\alpha_1} \cdots
s_n^{-\alpha_n}$ according to the multi-index notation. Thus, if $\xi (t,z)$ is as in , we have $$G(s) \xi (t,z) = \sum_{\alpha \in {\mathbb Z}_+^n} \left(t_\alpha - \alpha^{-1}
s^{-\alpha}\right) z^\alpha = \xi (t,z) - \sum_{\alpha \in {\mathbb Z}_+^n} \alpha^{-1} s^{-\alpha}
z^\alpha .$$ Hence it follows that $G(s)$ operates on the Baker function $w(t,z)$ in associated to an element $\phi \in 1+P_-$ and on the adjoint Baker function $w(t,z)$ in by $$\begin{gathered}
G(s) w(t,z) = w(t,z) \exp \left( - \sum_{\alpha \in {\mathbb Z}_+^n}
\alpha^{-1} s^{-\alpha} z^\alpha \right) ,\\
G(s) w^* (t,z) = w^* (t,z) \exp \left( \sum_{\alpha \in {\mathbb Z}_+^n}
\alpha^{-1} s^{-\alpha} z^\alpha \right) .\end{gathered}$$ Using the relation $$\ln \left( 1- \sum^n_{r=1} \frac {z_r} {s_r} \right) = - \sum_{\alpha \in
{\mathbb Z}_+^n} \frac {z_1^{\alpha_1} \cdots z_n^{\alpha_n}} {\alpha_1 \cdots \alpha_n
s_1^{\alpha_1} \cdots s_n^{\alpha_n}} = - \sum_{\alpha \in {\mathbb Z}_+^n} \alpha^{-1}
s^{-\alpha} z^\alpha,$$ we see that the operation of $G(s)$ on $w^* (t,z)$ can be written in the form $$\label{E:e9}
G(s) w^* (t,z) = w^* (t,z) \left( 1- \sum^n_{r=1} \frac {z_r} {s_r}
\right)^{-1} .$$ We now consider some calculations involving the residue operator ${\rm Res}_z$ given by .
\[L:u1\] Let $s = (s_1, \ldots, s_n) \in {\mathbb C}^n$, and consider a formal power series of the form $\eta (z) = 1+ \sum\limits_{\alpha \leq -{\mathbf 1}} f_\alpha z^\alpha$. Then we have $${\rm Res}_z \eta (z) \left( 1- \sum^n_{r=1} \frac {z_r} {s_r} \right)^{-1} =
s_1 \cdots s_n (\eta (s) -1)$$ for all $z = (z_1, \ldots, z_n) \in {\mathbb C}^n$.
First, we write the formal power series $\eta (z)$ in the form $$\eta (z) = 1 + \sum^{-1}_{\alpha_1 = -\infty} \cdots \sum^{-1}_{\alpha_n =
-\infty} f_{(\alpha_1, \ldots, \alpha_n)} z_1^{\alpha_1} \cdots z_n^{\alpha_n} .$$ Using this and the power series expansion $$\begin{gathered}
\left( 1- \sum^n_{r=1} \frac {z_r} {s_r} \right)^{-1} = \sum^\infty_{r=0}
\left(s_1^{-1} z_1 + \cdots + s_n^{-1} z_n\right)^r
= \sum_{\beta \geq {\mathbf 0}} s^{-\beta} z^\beta = \sum_{\beta \geq {\mathbf 0}} \frac
{z_1^{\beta_1} \cdots z_n^{\beta_n}} {s_1^{\beta_1} \cdots s_n^{\beta_n}}\end{gathered}$$ with $\beta = (\beta_1, \ldots, \beta_n)$, we see that $$\begin{gathered}
{\rm Res}_z \eta (z) \left( 1- \sum^n_{r=1} \frac {z_r} {s_r} \right)^{-1} =
\sum^{-1}_{\alpha_1 = -\infty} \cdots \sum^{-1}_{\alpha_1 = -\infty} \frac
{f_{(\alpha_1, \ldots, \alpha_n)}} {s_1^{-\alpha_1-1} \cdots s_n^{-\alpha_n-1}}\\
\qquad{}= s_1 \cdots s_n \sum^{-1}_{\alpha_1 = -\infty} \cdots \sum^{-1}_{\alpha_n =
-\infty} f_{(\alpha_1, \ldots, \alpha_n)} s_1^{\alpha_1} \cdots s_n^{\alpha_n}\\
\qquad{}= s_1 \cdots s_n \sum_{\alpha \leq -\mathbf 1} f_\alpha s^\alpha
= s_1 \cdots s_n (\eta (s) -1);\end{gathered}$$ hence the lemma follows.
\[L:u2\] Let $s = (s_1, \ldots, s_n)$ and $s' = (s'_1, \ldots, s'_n)$ be elements of ${\mathbb C}^n$, and consider a formal power series of the form $\eta (z) = 1+
\sum\limits_{\alpha \leq - {\mathbf 1}} f_\alpha z^\alpha$. Then we have $$\begin{gathered}
{\rm Res}_z \eta (z) \left( 1- \sum^n_{r=1} \frac {z_r} {s_r} \right)^{-1}
\left( 1- \sum^n_{r=1} \frac {z_r} {s'_r} \right)^{-1} \\
\qquad {}= (\eta (s) - 1) \sum_{\alpha \geq {\mathbf 0}} \frac {s^{\alpha+1}} {{s'}^{\alpha}}
= (\eta (s') - 1) \sum_{\alpha \geq {\mathbf 0}} \frac {{s'}^{\alpha+1}} {s^{\alpha}}\end{gathered}$$ for all $z = (z_1, \ldots, z_n) \in {\mathbb C}^n$.
Using power series expansions and the formal relation $$\sum^\infty_{r=0} (u_1 + \cdots +u_n)^r = \sum_{\alpha \geq {\mathbf 0}} u^\alpha$$ for each $n$-tuple $u = (u_1, \ldots, u_n)$, we see that $$\begin{gathered}
{\rm Res}_z \eta (z) \left( 1 - \sum^n_{r=1} \frac {z_r} {s_r} \right)^{-1}
\left( 1- \sum^n_{r=1} \frac {z_r} {s'_r} \right)^{-1}\\
\qquad {}= {\rm Res}_z \eta (z) \left( \sum^\infty_{r=0} \left( \sum^n_{r=1} \frac
{z_r} {s_r} \right)^r \right) \left( \sum^\infty_{r=0} \left(
\sum^n_{r=1} \frac {z_r} {{s'}_r} \right)^r \right)\\
\qquad {}= {\rm Res}_z \eta (z)
\left( \sum_{\beta \geq {\mathbf 0}} \frac {z^\beta} {s^\beta} \right) \left(
\sum_{\gamma \geq {\mathbf 0}} \frac {z^\gamma} {{s'}^\gamma} \right)
= \sum_{\alpha \leq -{\mathbf 1}} f_\alpha \sum_{\beta +\gamma = -\alpha-{\mathbf 1}}
\frac 1{s^\beta {s'}^\gamma},\end{gathered}$$ where the second summation in the previous line is over multi-indices $\beta,
\gamma \geq {\mathbf 0}$ such that $\beta + \gamma = - \alpha -{\mathbf 1}$. Using $\beta = -\gamma -\alpha-1$, we obtain $$\begin{gathered}
{\rm Res}_z \eta (z) \left( 1- \sum^n_{r=1} \frac {z_r} {s_r} \right)^{-1}
\left( 1- \sum^n_{r=1} \frac {z_r} {s'_r} \right)^{-1} \\
\qquad {}= \sum_{\alpha \leq
-{\mathbf 1}} f_\alpha \sum_{\gamma \geq {\mathbf 0}} \frac 1{s^{-\gamma-\alpha-1} {s'}^\gamma}
= \sum_{\alpha \leq -{\mathbf 1}} f_\alpha s^\alpha \sum_{\gamma \geq {\mathbf 0}} \frac
{s^{\gamma+1}} {{s'}^\gamma}
= (\eta (s) - 1) \sum_{\gamma \geq {\mathbf 0}} \frac {s^{\gamma+1}} {{s'}^{\gamma}}.\end{gathered}$$ Similarly, by using $\gamma = -\beta -\alpha-1$ we have $$\begin{gathered}
{\rm Res}_z \eta (z) \left( 1- \sum^n_{r=1} \frac {z_r} {s_r} \right)^{-1}
\left( 1- \sum^n_{r=1} \frac {z_r} {s'_r} \right)^{-1} \\
\qquad {}= \sum_{\alpha \leq
-{\mathbf 1}} f_\alpha \sum_{\beta \geq {\mathbf 0}} \frac 1{s^\beta {s'}^{-\beta-\alpha -1}}
= \sum_{\alpha \leq -{\mathbf 1}} f_\alpha {s'}^\alpha \sum_{\beta \geq {\mathbf 0}} \frac
{{s'}^{\beta+1}} {s^\beta}
= (\eta (s') - 1) \sum_{\beta \geq {\mathbf 0}} \frac {{s'}^{\beta+1}} {s^{\beta}}.\end{gathered}$$ Hence the lemma follows.
We now state the main theorem in this section, which shows the existence of the tau function $\tau (t)$ corresponding to a Baker function of the type discussed in Section \[S:bf\].
\[T:hw\] Let $w(t,z)$ be the Baker function in corresponding to an element $\phi \in 1+P_-$, and let $\widehat{w} (t,z)$ be the associated formal power series given by . Then there is a function $\tau (t)$ with $t = (t_\alpha)_{\alpha \in {\mathbb Z}_+^n}$ such that $$\widehat{w} (t,z) = G(z) \tau (t) / \tau (t)$$ for $z \in {\mathbb C}^n$ and $t = (t_\alpha)_{\alpha \in {\mathbb Z}_+^n}$, where $G(z)$ is the operator given by .
By we have $${\rm Res}_z w(t,z) G(s) w^* (t,z) = 0$$ for each $s \in {\mathbb C}^n$. Using this and , we have $${\rm Res}_z \widehat{w}(t,z) G(s) \widehat{w}^* (t,z) \left( 1- \sum^n_{r=1} \frac
{z_r} {s_r} \right)^{-1} = 0 .$$ Thus by Lemma \[L:u1\] we see that $$s_1 \cdots s_n (\widehat{w}(t,s) G(s) \widehat{w}^* (t,s) -1) = 0 ;$$ hence we obtain $$\label{E:r1}
\widehat{w}(t,s)^{-1} = G(s) \widehat{w}^* (t,s) .$$ Similarly, we have $${\rm Res}_z w(t,z) G(s) G(s') w^* (t,z) = 0$$ for all $s, s' \in {\mathbb C}^n$, which implies that $${\rm Res}_z \widehat{w}(t,z) G(s) G(s') \widehat{w}^* (t,z) \left( 1- \sum^n_{r=1}
\frac {z_r} {s_r} \right)^{-1} \left( 1- \sum^n_{r=1} \frac {z_r} {s'_r}\right)^{-1} = 0 .$$ Using this and applying Lemma \[L:u2\] to the formal power series $$\phi (z) = \widehat{w}(t,z) G(s) G(s') \widehat{w}^* (t,z) ,$$ we obtain $$\widehat{w}(t,s) G(s) G(s') \widehat{w}^* (t,s) = \widehat{w}(t,s') G(s) G(s') \widehat{w}^*
(t,s') =1 .$$ By combining this with we have $$\label{E:mb}
\widehat{w}(t,s) (G(s') \widehat{w}(t,s))^{-1} = \widehat{w}(t,s) (G(s)
\widehat{w}(t,s'))^{-1} .$$ We now set $$h(t,s) = \ln (\widehat{w}(t,s)) .$$ Then by taking the logarithm of both sides of we obtain $$(1-G(s')) h(t,s) = (1-G(s)) h(t,s') .$$ Replacing $s$ and $s'$ by $z$ and $\zeta$, respectively, gives us $$\label{E:r2}
h(t,z) - G(\zeta) h(t,z) = h(t,\zeta) - G(z) h(t,\zeta) .$$ For each $k \in \{ 1, \ldots, n\}$ we define the differential operator ${\mathcal D}_k (z)$ by $${\mathcal D}_k (z) = \sum_{\alpha \in {\mathbb Z}_+^n} \alpha_k \alpha^{-1} z^{-\alpha- {\mathbf
e}_k} \partial_\alpha - \frac \partial {\partial z_k} ,$$ where $\partial_\alpha
= \partial_{t_\alpha} = \partial/\partial t_\alpha$ with $\alpha = (\alpha_1, \ldots,
\alpha_n)$. For any function $\varphi (t)$, we have $$\begin{gathered}
{\mathcal D}_k (z) G(z) \varphi (t) = {\mathcal D}_k (z) \varphi \left(\left(t_\alpha - \alpha^{-1}
z^{-\alpha}\right)_{\alpha \in {\mathbb Z}_+^n}\right)\\
\qquad {}= \sum_{\alpha \in {\mathbb Z}_+^n}
\alpha_k \alpha^{-1} z^{-\alpha- {\mathbf e}_k} \partial_\alpha \varphi
\left(\left(t_\alpha - \alpha^{-1} z^{-\alpha}\right)_{\alpha \in \mathbb Z_+^n}\right)\\
\qquad {} - \sum_{\alpha \in {\mathbb Z}_+^n} \alpha_k \alpha^{-1} z^{-\alpha- {\mathbf e}_k}
\partial_\alpha \varphi\left(\left(t_\alpha - \alpha^{-1} z^{-\alpha}
\right)_{\alpha \in {\mathbb Z}_+^n}\right) =0 .\end{gathered}$$ Using this and , we see that $${\mathcal D}_k (z) h(t,z) - G(\zeta) {\mathcal D}_k (z) h(t,z) = {\mathcal D}_k (z) h(t,
\zeta) = \sum_{\alpha \in {\mathbb Z}_+^n}
\alpha_k \alpha^{-1} z^{-\alpha- {\mathbf e}_k} \partial_\alpha h(t,\zeta) .$$ Thus for each $\beta \in {\mathbb Z}_+^n$ we obtain $$\begin{gathered}
{\rm Res}_z z^\beta {\mathcal D}_k (z) h(t,z) - G(\zeta) \,{\rm Res}_z z^\beta {\mathcal D}_k (z)
h(t,z)\\
\qquad {}= {\rm Res}_z \sum_{\alpha \in {\mathbb Z}_+^n}
\alpha_k \alpha^{-1} z^{-\alpha+\beta - {\mathbf e}_k}
\partial_\alpha h(t,\zeta)
= \beta_k (\beta+{\mathbf 1}-{\mathbf e}_k)^{-1} \partial_{\beta+{\mathbf 1}-{\mathbf e}_k} h(t,\zeta) ,\end{gathered}$$ where we used the fact that the $k$-component of $\beta+{\mathbf 1}-{\mathbf e}_k$ is $\beta_k$. Thus, if we set $a_{\alpha,k} = {\rm Res}_z z^\alpha \mathcal D_k (z) h(t,z)$ for each $\alpha \in {\mathbb Z}_+^n$, we have $$\label{E:r3}
(1- G(\zeta)) a_{\alpha,k} = \alpha_k (\alpha+{\mathbf 1}-{\mathbf e}_k)^{-1}
\partial_{\alpha+{\mathbf 1}-{\mathbf e}_k} h(t,\zeta)$$ for all $\alpha \in {\mathbb Z}_+^n$ and $k \in \{ 1, \ldots, n\}$. Hence we obtain $$\begin{gathered}
\alpha_k (\alpha+{\mathbf 1}-{\mathbf e}_k)^{-1}
\partial_{\alpha+{\mathbf 1}-{\mathbf e}_k} a_{\beta,k} - \beta_k
(\beta+{\mathbf 1}-{\mathbf e}_k)^{-1} \partial_{\beta+{\mathbf 1}-{\mathbf e}_k} a_{\alpha,k}\\
\qquad {}= G(\zeta) \left(\alpha_k (\alpha+{\mathbf 1}-{\mathbf e}_k)^{-1}
\partial_{\alpha+{\mathbf 1}-{\mathbf e}_k} a_{\beta,k}
- \beta_k (\beta+{\mathbf 1}-{\mathbf e}_k)^{-1} \partial_{\beta+{\mathbf 1}-
{\mathbf e}_k} a_{\alpha,k}\right) ,\end{gathered}$$ which implies that $$\alpha_k (\alpha+{\mathbf 1}-{\mathbf e}_k)^{-1} \partial_{\alpha+{\mathbf 1}-
{\mathbf e}_k} a_{\beta,k} = \beta_k
(\beta +{\mathbf 1}-{\mathbf e}_k)^{-1} \partial_{\beta+{\mathbf 1}-{\mathbf e}_k} a_{\alpha,k}$$ for all $\alpha, \beta \in {\mathbb Z}_+^n$. Therefore there is a function $\tau
(t)$ such that $$a_{\alpha,k} = - \alpha_k (\alpha+{\mathbf 1}-{\mathbf e}_k)^{-1}
\partial_{\alpha+{\mathbf 1}-{\mathbf e}_k}
\ln \tau (t) .$$ By combining this with we obtain $$\begin{gathered}
\partial_{\alpha+{\mathbf 1}-{\mathbf e}_k} h(t,\zeta)
= \alpha_k^{-1} (\alpha+{\mathbf 1}-{\mathbf e}_k)
(1-G(\zeta)) a_{\alpha,k}
= - (1-G(\zeta)) \partial_{\alpha+{\mathbf 1}-{\mathbf e}_k} \ln \tau (t)\end{gathered}$$ for all $\alpha \in {\mathbb Z}_+^n$ and $k \in \{ 1, \ldots, n\}$; hence we see that $$h(t,\zeta) = - (1-G(\zeta)) \ln \tau (t) .$$ Thus it follows that $$\begin{gathered}
\ln (\widehat{w}(t,\zeta)) = h(t,\zeta) = -\ln \tau (t) + G(\zeta) \ln \tau (t)
= \ln (G(\zeta) \tau (t)/\tau (t)) .\end{gathered}$$ Thus we obtain $$\widehat{w}(t,\zeta)= G(\zeta) \tau (t)/\tau (t) ,$$ and therefore the proof of the theorem is complete.
Concluding remarks
==================
As is mentioned in the introduction, Baker functions associated to single-variable pseudodifferential operators provide formal solutions of soliton equations. Baker functions for pseudodifferential operators of several variables also determine solutions of soliton equations, and by Theorem \[T:hw\] we see that the Baker function in can be written in the form $$\begin{gathered}
w (t,z) = \widehat{w} (t,z) e^{\xi (t,z)} = (G(z) \tau (t)/\tau (t)) e^{\xi (t,z)} ,\end{gathered}$$ where $\xi (t,z) = \sum\limits_{\alpha \in {\mathbb Z}_+^n} t_\alpha z^\alpha$. The function $\tau (t)$ with $t = (t_\alpha)_{\alpha \in {\mathbb Z}_+^n}$ is a tau function for pseudodifferential operators of several variables. Thus we have extended the expression of a Baker function in term of the corresponding tau function to the case of pseudodifferential operators of several variables.
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\[lee-lastpage\]
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'This work provides an experimental method for simultaneously measuring finite time Lyapunov exponent fields for multiple particle groups, including non-flow tracers, in three-dimensional multiphase flows. From sequences of particle images, e.g., from experimental fluid imaging techniques, we can directly compute the flow map and coherent structures, with out performing the computationally costly numerical integration. This is particularly useful to find three-dimensional Lagrangian coherent structures for inertial particles, that do not follow the bulk fluid velocity, as we demonstrate for a grid turbulence experiment. The technique described may provide a new means for exploring the physics of experimental multi-phase flows.'
author:
- 'Samuel G. Raben[^1], Shane D. Ross, Pavlos P. Vlachos[^2]'
bibliography:
- 'library.bib'
title: Demonstration of Experimental Three Dimensional Finite Time Lyapunov Exponents with Inertial Particles
---
Finite Time Lyapunov Exponents (FTLE) are a powerful and increasingly popular tool for describing mixing and transport in both turbulent and laminar flow fields [@Haller2001; @Brunton2010]. FTLEs provide a measure of the exponential rate of divergence or convergence of Lagrangian particle trajectories. They can be used both experimentally and numerically to describe a flow field, which may have a high degree of spatiotemporal complexity [@Haller2001; @Shadden2007; @Shadden2006]. While FTLEs are primarily used to describe single-phase flow behavior [@Haller2001; @Haller2005; @Shadden2006] some works have attempted to account for inertial particles by modeling the particles’ motion through simulations [@Haller2008; @Tallapragada2008; @Peng2009]. This procedure can provide insight, but does not provide direct information about the true observable inertial particle trajectories. Often, the equations for inertial particle motion make simplifying assumptions (e.g., the Maxey-Riley equations [@Maxey1983]) that can lead to significant differences between the modeled and true motion. This brief communication describes a method to directly determine FTLEs from experimental data for inertial particles through the use of particle tracking velocimetry (PTV) without any [*a-priori*]{} assumptions about particle motion.
FTLEs are computed via the Cauchy-Green deformation tensor $C_{jk}$, $$C=\left(\nabla \Phi^{t_0+T}_{t_0}\right)^* \cdot \nabla \Phi^{t_0+T}_{t_0}$$ where \* denotes the matrix transpose, and $\Phi^{t_0 + T}_{t_0}$ is the flow map (diffeomorphism) of particle locations from time $t_0$ to $t_0+T$, where $T$ is the time over which the FTLEs will be computed. From the maximum eigenvalue of C, the FTLE field defined in the measurement volume is, $$\sigma^{t_0+T}_{t_0} = \frac{1}{T} \ln \left( \sqrt{\lambda_{max} \left(C\right)}\right)$$ typically when computing FTLEs from experimental data to use a numerical integration routine to numerically advect artificial tracer particles to determine the flow map from the estimated velocity field [@Shadden2006; @Shadden2007; @Lekien2010]. While this can be effective for single-phase flow it neglects the fact that inertial particles, bubbles, or active particles may fail to follow the bulk fluid motion or the fact that tracking individual particles can provide a direct measure of a short duration flow map. Lagrangian tracking can provide a measure of the flow map over longer times but is more prone to experimental errors [@Raben2013]. While numerical routines can be modified to estimate the inertial particle behavior via modeling as mentioned above, this procedure does not directly measure inertial particle trajectories. However, using time resolved PTV to obtain snap shots of the particle motion allows direct measurement of the particle flow map while also allowing for parameterization of the particle flow map based on unique identifying characteristics, such as size, shape or color, providing, e.g., a one-parameter family of particle flow maps with particle size as the parameter. The concept of merging small flow map snap shots to estimate a complete flow map was put forth by Brunton and Rowley [@Brunton2010] for results of fluid computations and later adapted for experimental data as PTV interpolation by Raben et al. [@Raben2013]. Through this method it is possible to simultaneously determine FTLEs for multiple particle groups within the same measurement volume and compare them to the bulk flow field. It has also been shown that this method can provide high accuracy flow map computation results even when the particle concentration drops below what is typically used for PIV/PTV [@Raben2013]. This is an important aspect; when the particles are separated into groups, some groups will have smaller particle population densities requiring a method suitable to provide high resolution FTLE information with low resolution velocity information in order to properly determining the FTLE values.
To study the motion of inertial particles in an experimental environment, data were collected in a vertical water tunnel that was designed to generate homogeneous isotropic grid turbulence, as described in [@Raben2012]. For this experiment a bar thickness of the grid, $b=0.3175$ cm was used with the gap between bars equal to the width of the bar. Overlapping bars created a square lattice, which was located 8 cm upstream from the measurement location. Two different types of particles where added to the flow: $85 \pm 20$ $\mu$m diameter silver coated hollow glass spheres that were tuned to be neutrally buoyant and were used to act as flow tracers; and solid glass particles with diameters ranging from approximately 150 - 200 $\mu$m that were added downstream (top of the tunnel) and had an approximate mass density of 2600 kg/m$^3$. The vertical nature of the tunnel created opposing motion as gravity pulled the negatively buoyant particles down while the bulk flow was moving mostly upward.
Time resolved imaging techniques such as particle image velocimetry (PIV) have made it possible to study the Lagrangian motion of a flow field experimentally [@Mathur2007; @Shadden2007]. With the recent development of volumetric image techniques [@Elsinga:2006wo] it is now possible to investigate particle trajectories in a fully three-dimensional flow field. Because these imaging techniques make no assumptions on particle motion (e.g., must be a tracer following the bulk flow) they can be effective in capturing non-flow tracer particle motion (e.g., inertial particles) as well as bulk flow motion.
Time resolved tomographic imaging was used to collect information on the complete particle field as well as fully resolve the three-dimensional fluid motion. A New Wave Pegasus laser was used to illuminate all the particles in the measurement volume. Three Photron FASTCAM APX-RS high-speed CMOS cameras were used to simultaneously image this light field, recording images at 250 Hz. These images were reconstructed into a three-dimensional light intensity distribution using the Multiplicative Algebraic Reconstruction Technique (MART) employ in the DaVis 8.1 software [@Herman1976; @Elsinga:2006wo].
Once the images had been reconstructed, the particles’ size and motion were determined. Particles were first located in the volume using a simple thresholding method and then sized using an intensity weighted pixel count. In an effort to track the particles, a multi-component particle tracking algorithm developed for single and multiphase flows [@Cardwell2011] was adapted to three-dimensional data. The method worked by comparing unique particle identifiers, such as size, peak intensity, and proximity, to match particles in consecutive images. This method has been shown to work well in turbulent flows even with non-flow tracers [@Cardwell2011].
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 (A) Normalized particle diameter distribution within the measurement volume. Iso-surfaces of the forward (B) and backward (C) FTLE fields based on the different components in the flow. ](particle_hist.png "fig:"){width="30.00000%"}
[A]{}
 (A) Normalized particle diameter distribution within the measurement volume. Iso-surfaces of the forward (B) and backward (C) FTLE fields based on the different components in the flow. ](combo_tecplot_forward.png "fig:"){width="40.00000%"}
[B]{}
 (A) Normalized particle diameter distribution within the measurement volume. Iso-surfaces of the forward (B) and backward (C) FTLE fields based on the different components in the flow. ](combo_tecplot_backward.png "fig:"){width="40.00000%"}
[C]{}
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Figure \[fig:hist\]A shows a histogram of the particle sizes present in the measurement volume. Due to factors such as camera arrangement and the MART reconstruction algorithm [@Herman1976], the particle size may be over-estimated. As these factors should affect all particles equally, and the concern here is not exact particle size but rather relative size, this should not affect the results. For this study, the particle size distribution was divided into only three groups. The first group was composed of the smallest particles, most likely including the tracer particles, which should follow the bulk fluid motion. The second group was composed of the medium particles, which contained a mixture of flow tracers and smaller glass particles. The final group included the largest particles, which were primarily the large glass particles that will tend not to follow the bulk fluid motion. When computing the FTLE field, the complete particle distribution was used as a control, as this total group provides an estimate of the FTLE field that would be found if no particle sizing procedure had been applied to the data and all the particles were (erroneously) treated as flow tracers.
The FTLE field was calculated for each particle group with an integration time of 1s which is equal to 250 frames. For two-dimensional flows, FTLE fields are often characterized by the elevated ridges, or connected lines with high FTLE values, which are referred to as Lagrangian coherent structures (LCS) and reveal hyperbolic or shear-dominated structures. In three-dimensional fields, the locus of elevated values are two-dimensional surfaces. Figure \[fig:hist\]B and C shows iso-surfaces of high FTLE values as proxies for true ridges for both the forward and backward FTLE fields. Ridges in the forward FTLE field reveal repelling surfaces where particles are exponentially diverging away from one another while the backward FTLE shows attracting surfaces which may be related to clustering cores for inertial particles. From Figure \[fig:hist\]B it can be seen that there is a significant difference in the FTLE fields based on the particle size. The iso-surface for the large particle group is dominated by a large structure in the upper left of the domain. It could be seen from the raw data that during this time that there was an influx of larger particles that begin to spread throughout the volume, which would explain the elevated FTLE values in this region. For the small particle group the iso-surface shows a structure that extends from the lower right of the domain up to the top. This structure could indicate that the influx of large particles forced the flow tracers to be redirected around the large particle cluster causing a divergence in the small particle trajectories.
Figure \[fig:hist\]C shows the backward FTLE, which will indicate locations of particle clustering. Previous works that have investigated particle clustering have used the second invariant of the velocity gradient tensor, Q, sometimes referred to as the Okubo-Weiss parameter, as an indicator for where particles are likely to concentrate, [@Squires1991; @Eaton1994; @Guala2008; @Haller2008] where Q is defined as, $$Q=\frac{1}{2}\left( \omega^2 - s^2 \right)$$ with $\omega$ and $s$ representing vorticity and strain rate, respectively. For scaling purposes $Q$ is often normalized by the ensemble average of vorticity squared, $Q^*= Q/\left< \omega \right>$, as was done here. This produced normalized values between -1.5 and 0.5 which is in agreement with the literature for turbulent flow [@Guala2008]. When $Q^*$ is negative this indicates a region of high strain and low vorticity, which, when particles are added to the flow, has been shown to correlate with preferential particle concentration [@Squires1991; @Eaton1994; @Guala2008; @Haller2008]. To illustrate regions where particles should cluster a $Q^*$ iso-surface showing the location of three standard deviations away from the zero in the negative direction based on the mean field, is also shown in Figure \[fig:hist\]. It can be seen from Figure \[fig:hist\]C that while there exist some smaller regions of high backward FTLE throughout the domain, the attracting LCS locations are predominantly located near the location of higher negative $Q^*$. Since the flow is time-dependent, there is no reason to expect perfect agreement between the Eulerian $Q^*$ field and attracting LCSs.
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{width="31.00000%"} {width="31.00000%"}
[A]{} [B]{}
[ ]{} [ ]{}
{width="31.00000%"} {width="31.00000%"}
[C]{} [D]{}
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To further investigate the locations of particle clustering, Figure \[fig:slice\] shows backward FTLE values on the center Z plane for each of the 3 different particle groupings along with the total particle collection, with a thick black line representing the same iso-contour of $Q^*$ is included. In addition an iso-contour -1.5 times the standard deviation and a zero contour are also included. It can be seen from this figure that while there are some similarities in the locations of the elevated backward FTLE values between the different groups, there are also some important differences. Figure \[fig:slice\]A shows the FTLE field for the total particle group, which we note is not a superposition of the FTLE field for the size-based groups. Elevated FTLE values are seen in close proximity to the highly negative $Q^*$ values as this will be a location where particles will cluster [@Guala2008]. For the large particles, Figure \[fig:slice\]D, elevated values are again seen near highly negative $Q^*$ but in a different location from that seen with the total particle group. In this case the large particles appear attracted to a region just above the $Q^*$ iso-contour, on the opposite side from zero $Q^*$ iso-contour (the zero iso-contour would suggest particle repulsion). The large particles also have a lower maximum FTLE value, which may indicate that their attraction to this region is not as strong as some of the other particles groups.
The medium particle group also has elevated FTLE values in close proximity to the $Q^*$ strongly negative iso-contour, as seen in Figure \[fig:slice\]C. As this group is most likely a collection of flow tracers and smaller inertial particles it is interesting to see that very high FTLE values appear to be located inside the $Q^*$ iso-contour mean that particle clustering associated with this group most closely coincides with the $Q^*$ grouping. For the smallest particles, Figure \[fig:slice\]B, it can again be seen that the elevated FTLE values are located near the $Q^*$ iso-contour. This particle group appears to have more scatter than the other groups which is mostly due to the fact that as flow tracers these particles are more susceptible to the turbulent fluctuations in the volume and thus will have a more spatially distributed structure. Again, because $Q^*$ is an Eulerian field and ours is a temporally varying flow there is no expectation of perfect agreement with the LCS but it does help to illustrate the behavior.
To summarize, this work has shown that three-dimensional FTLE fields can be calculated for inertial particles in experiments through the use a non-flow tracer flow map determination technique that uses particle tracking and sizing information to directly measure the size-parameterized families of flow maps. The use of particle tracking for the direct calculation of the FTLEs is an important advancement as it is capable of uniquely determining the flow maps for different groups of particles, e.g., grouped by size in our experiment, but other parameterizations are possible. Using this method it is possible to directly measure inertial particle FTLE fields and Lagrangian coherent structures without making assumptions about the underlying particle equations of motion. This will have relevance for the experimental study of inertial particle motion in fluids and multi-phase flows.
SDR gratefully acknowledges partial support from NSF Grant 1150456.
[^1]: Virginia Tech
[^2]: Purdue University
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We study quantum phase transitions between competing orders in one-dimensional spin systems. We focus on systems that can be mapped to a dual-field double sine-Gordon model as a bosonized effective field theory. This model contains two pinning potential terms of dual fields that stabilize competing orders and allows different types of quantum phase transition to happen between two ordered phases. At the transition point, elementary excitations change from the topological soliton of one of the dual fields to that of the other, thus it can be characterized as a topological transition. We compute the dynamical susceptibilities and the entanglement entropy, which gives us access to the central charge, of the system using a numerical technique of infinite time-evolving block decimation and characterize the universality class of the transition as well as the nature of the order in each phase. The possible realizations of such transitions in experimental systems both for condensed matter and cold atomic gases are also discussed.'
author:
- Shintaro Takayoshi
- 'Shunsuke C. Furuya'
- Thierry Giamarchi
title: Topological transition between competing orders in quantum spin chains
---
Introduction
============
Low dimensional quantum magnets show rich phase diagrams due to the interplay between strong correlations and quantum fluctuations. This competition is at the root of the existence of phases with very different physics, separated by quantum phase transitions when parameters of the system are varied. In one dimensional (1D) quantum magnets, these transitions often have a topological nature. The simplest example of such a transition is the one between a massless phase dominated by XY correlations and the massive Ising phase existing in an anisotropic Heisenberg spin-1/2 chain. The universality class of this transition is the celebrated Berezinskii-Kosterlitz-Thouless (BKT) transition [@Berezinskii1971JETP; @Berezinskii1972JETP; @Kosterlitz1973JPhysC], which is characterized by a set of topological excitations. A field theoretical description is instrumental in understanding the properties of such transitions. In the above mentioned case, the corresponding field theory is the sine-Gordon model [@Giamarchi2004Book] and the low-energy excitations are solitons and carry a topological index. Another example of system described by the sine-Gordon theory is the Heisenberg chain with a staggered magnetic field such as Cu benzoate [@Oshikawa1997PRL; @Affleck1999PRB; @Dender1997PRL]. A field theoretical approach to topological phases has been used with success for more complicated phases, e.g. the Haldane phase in $S=1$ quantum spin chains [@Haldane1983PhysLettA; @Haldane1983PRL].
In this paper, we focus on the phase transitions in quantum magnets which are caused by the competition between two dual fields having a topological nature. Such systems are mapped onto a dual-field double sine-Gordon (DDSG) model [@Giamarchi1988JPhys; @Lecheminant2002NPB; @Delfino2002Inbook; @Sarkar2016SciRep]. This model contains two different potential terms pinning the dual fields. If the strength of these potentials is varied, the stabilized order is changed and a quantum phase transition occurs. In addition to quantum magnets, the DDSG model appears in a broad context such as in XY models with symmetry breaking fields, in mixtures of electric charges and magnetic monopoles [@Jose1977PRB; @Fertig2002PRL], and in quantum ladder systems [@Orignac2017PRB; @Citro2018PRB; @Robinson2018ArXiv]. Experimentally the DDSG model has been realized in the material $\mathrm{BaCo_{2}V_{2}O_{8}}$ [@Faure2018NatPhys]. This compound has a strong Ising anisotropy and when an external uniform magnetic field is applied, an effective staggered field is introduced in the direction perpendicular to both the anisotropy axis and the external magnetic field. Thus the Néel orders along the anisotropy axis and along the effective staggered field are competing in this system. The quantum phase transition between them can be triggered by increasing the strength of the external magnetic field, and it is measured directly in inelastic neutrons scattering (INS) experiments.
In the following, we examine various possible realizations of the DDSG model in quantum magnets, and study quantitatively the resulting transitions. We combine the field theory with a numerical analysis based on the infinite time-evolving block decimation (iTEBD), which utilizes a matrix product state (MPS) such as the density matrix renormalization group [@White1992PRL]. We compute various observables such as the staggered magnetization, the entanglement entropy and the dynamical spin-spin susceptibility. In particular, the dynamical susceptibility not only has a theoretical interest but also is directly related with the experiments such as inelastic neutron scattering (INS), electron spin resonance (ESR), and nuclear magnetic resonance (NMR).
This paper is organized as follows. In Sec. \[sec:bosonization\], we quickly review the bosonization and give some examples of quantum spin systems described by the DDSG model. In Sec. \[sec:QPT\], we study the quantum phase transition between competing orders using the examples given in Sec. \[sec:bosonization\]. Section \[sec:DynSuscep\] discusses how the dynamical susceptibility changes below and above the transition. Section \[sec:application\] is devoted to discussing applications to real materials. We summarize our results and discuss future problems in Sec. \[sec:summary\].
Bosonization and dual-field double sine-Gordon model {#sec:bosonization}
====================================================
In this section, we briefly review the bosonization of 1D spin chains [@Giamarchi2004Book]. We map the spin operators to bosonic scalar fields using the formula, $$\label{eq:bosonspin}
\begin{split}
S_{j}^{z}&=-\frac{a}{\pi}\frac{d\phi(x)}{dx}
+a_{1}(-1)^{j}\cos(2\phi(x))+\cdots,\\
S_{j}^{+}&=e^{-i\theta(x)}
[b_{0}(-1)^{j}+b_{1}\cos(2\phi(x))+\cdots],
\end{split}$$ where $x=ja$ is a spatial coordinate ($a$ is the lattice constant) and $a_{0}$, $b_{0}$ and $b_{1}$ are nonuniversal constants which can be estimated numerically [@Hikihara2004PRB; @Takayoshi2010PRB; @Bouillot2011PRB; @Hikihara2017PRB] $\phi(x)$ and $\theta(x)$ are dual bosonic fields satisfying the commutation relation $[\phi(x),\theta(x')]=-i\pi\vartheta_{\mathrm{step}}(x-x')$ ($\vartheta_{\mathrm{step}}(x-x')$ is the step function). The fields $2\phi(x)$ and $\theta(x)$ can be intuitively interpreted as polar and azimuthal angles of the staggered magnetization.
The Hamiltonian of Heisenberg chains with an Ising anisotropy (XXZ models) $$\begin{aligned}
{\cal H}_{\mathrm{XXZ}}&=J\sum_{j}(S_{j}^{x}S_{j+1}^{x}
+S_{j}^{y}S_{j+1}^{y}+\Delta S_{j}^{z}S_{j+1}^{z})
\label{eq:Hamil_XXZ}\end{aligned}$$ is bosonized as $$\begin{aligned}
\mathcal{H}_{\mathrm{XXZ}}^{\mathrm{eff}}&=\frac{v}{2\pi}\int dx
\Big[\frac{1}{K}\Big(\frac{d\phi(x)}{dx}\Big)^{2}
+K\Big(\frac{d\theta(x)}{dx}\Big)^{2}\Big]\nonumber\\
&\quad-\lambda\int dx\cos(4\phi(x))
+\cdots,\nonumber\end{aligned}$$ where $\lambda$ is some constant, $v$ is spinon velocity, and $K$ is the Luttinger parameter. The $\cos(4\phi(x))$ term has the scaling dimension $4K$, and it is relevant in the Ising anisotropic ($\Delta>1$, $K<1/2$) region. It works as a potential to pin the field $\phi(x)$. When $\phi(x)$ is fixed at $n\pi/2$ ($n$: integer), the system has Néel order along the $z$ axis and the excitations are gapped. If we add a pinning potential for $\theta(x)$, it competes with the $\phi(x)$ pinning potential, since $\phi(x)$ and $\theta(x)$ are conjugate they cannot be simultaneously fixed. The resulting model is the DDSG model, $$\begin{aligned}
&\mathcal{H}_{\mathrm{DDSG}}=
\frac{v}{2\pi}\int dx\Big[
\frac{1}{K}\Big(\frac{d\phi(x)}{dx}\Big)^{2}
+K\Big(\frac{d\theta(x)}{dx}\Big)^{2}\Big]\nonumber\\
&-g_{1}\int dx\cos(m\phi(x))
-g_{2}\int dx\cos(n\theta(x)).
\label{eq:DDSG}\end{aligned}$$ where $m$ and $n$ are integers and $g_{1}$, $g_{2}$ are nonuniversal constants.
In the following, we study several microscopic situations for which the bosonized field theory is a DDSG model.
XXZ model with a staggered magnetic field along the $x$ direction {#sec:XXZhx}
-----------------------------------------------------------------
Let us add a staggered magnetic field along the $x$ axis $-h_{x}\sum_{j}(-1)^{j}S_{j}^{x}$ to the XXZ model . This staggered field is bosonized as $$-h_{x}\sum_{j}(-1)^{j}S_{j}^{x}
=-h_{x}b_{0}\int dx\cos\theta(x)+\cdots.
\nonumber$$ The $\cos\theta(x)$ term has a scaling dimension $1/(4K)$ and is relevant for $K>1/8$. Therefore, the total bosonized Hamiltonian is the DDSG model with $m=4$, $n=1$. For $\Delta>1$ and $h_{x}=0$, the ground state has Néel order (staggered magnetization) along the $z$ axis and the $\phi$ field is pinned. Since $\cos\theta(x)$ dominates over $\cos(4\phi(x))$ with increasing $h_{x}$ and the $\theta$ field is pinned, there is a quantum phase transition. The staggered field $h_{x}$ immediately creates a finite staggered magnetization along the $x$ axis, but the staggered magnetization along the $z$ axis becomes $0$ in the high $h_{x}$ phase and thus works as an order parameter. Note that we could also use $\langle \cos(\nu \theta(x))\rangle$ as an order parameter, where $\nu$ is any noninteger number (for example $\nu=1/2$) since it becomes zero in the $\phi$ pinned phase and nonzero only in the high field phase. Such order parameter is however nonlocal in terms of the spin operators [@Berg2008PRB] and thus its measurement can only be done in particular systems, as is discussed in Sec. \[sec:application\]. Using the spin current operator [@Giamarchi2004Book] $$\mathcal{J}_{j}^{\mathrm{s}}\equiv
\frac{i}{2}(S_{j}^{+}S_{j+1}^{-}-S_{j}^{-}S_{j+1}^{+})
=-vK\frac{a}{\pi}\frac{d\theta(x)}{dx}+\cdots,
\nonumber$$ $\cos(\nu \theta(x))$ is represented as $$\cos\Big(\frac{\nu\pi}{vK}\sum_{l=-\infty}^{j}\mathcal{J}_{l}^{\mathrm{s}}\Big)
=\cos\Big(\nu\int_{-\infty}^{x}dx'\frac{d\theta(x')}{dx'}\Big)
+\cdots.
\nonumber$$ Thus nonlocal measurements are needed for the experimental observation of $\langle \cos(\nu \theta(x))\rangle$. For quantities related to particle density (or $S^{z}$), such nonlocal quantity could be measured in cold atomic systems (see Sec. \[sec:coldatom\]).
Another order parameter which is local and can thus be directly measured in condensed matter experiments is the staggered magnetization $\cos(2\phi(x))$. The lowest energy excitation is a soliton of the $\phi(x)$ ($\theta(x)$) field in the low (high) $h_{x}$ phase. The phase properties are summarized in Table \[tab:PhasePropertiesHx\].
low $h_{x}$ phase high $h_{x}$ phase
------------------------------------------------------------------------------ --------------------- ----------------------
pinned field $\phi(x)$ $\theta(x)$
$\langle\cos(2\phi(x))\rangle\propto\langle\sum_{j}(-1)^{j}S_{j}^{z}\rangle$ $\neq 0$ $0$
$\langle\cos\theta(x)\rangle\propto\langle\sum_{j}(-1)^{j}S_{j}^{x}\rangle$ $\neq 0$ $\neq 0$
$\langle\cos(\nu\theta(x))\rangle$ ($\nu$: noninteger) $ 0$ $\neq 0$
soliton $\phi(x)=0\to\pi/2$ $\theta(x)=0\to2\pi$
: Summary of the phase properties of the XXZ model with a staggered magnetic field in the $x$ direction.[]{data-label="tab:PhasePropertiesHx"}
XXZ model with XY anisotropy
----------------------------
Let us now consider another type of perturbation to the XXZ chain, which is the XY anisotropy. When such a term is bosonized, it has the form of $$D_{xy}\sum_{j}(S_{j}^{x}S_{j+1}^{x}-S_{j}^{y}S_{j+1}^{y})
=-D_{xy} c_{1}\int dx\cos(2\theta(x))+\cdots,
\nonumber$$ where $c_{1}$ is a nonuniversal constant. The $\cos(2\theta(x))$ term has the scaling dimension $1/K$ and it is relevant for $K>1/2$. The total bosonized Hamiltonian is the DDSG model with $m=4$, $n=2$, instead of $m=4$ and $n=1$ of the previous section. In this case, the two cosine potential terms are simultaneously marginal at $K=1/2$, and a controlled perturbative renormalization can be constructed [@Giamarchi1988JPhys] around the marginal point. The properties of such a transition will thus be quite different and are summarized in Table \[tab:PhasePropertiesAnis\].
low $D_{xy}$ phase high $D_{xy}$ phase
------------------------------------------------------------------------------ --------------------- ---------------------
pinned field $\phi(x)$ $\theta(x)$
$\langle\cos(2\phi(x))\rangle\propto\langle\sum_{j}(-1)^{j}S_{j}^{z}\rangle$ $\neq 0$ $0$
$\langle\cos\theta(x)\rangle\propto\langle\sum_{j}(-1)^{j}S_{j}^{x}\rangle$ $0$ $\neq 0$
soliton $\phi(x)=0\to\pi/2$ $\theta(x)=0\to\pi$
: Summary of the phase properties in the XXZ model with XY anisotropy.[]{data-label="tab:PhasePropertiesAnis"}
Other perturbations
-------------------
Although we focus mostly on the two above mentioned models below, it is also possible to consider other perturbations such as a staggered field along $z$ axis $-h_{z}\sum_{j}(-1)^{j}S_{j}^{z}$ and a dimerization $\delta\sum_{j}(-1)^{j}{\boldsymbol{S}}_{j}\cdot{\boldsymbol{S}}_{j+1}$. These perturbations are bosonized as $$\begin{aligned}
-h_{z}\sum_{j}(-1)^{j}S_{j}^{z}
=-h_{z}a_{1}\int dx\cos(2\phi(x))+\cdots,\nonumber\\
\delta\sum_{j}(-1)^{j}{\boldsymbol{S}}_{j}\cdot{\boldsymbol{S}}_{j+1}
=\delta d_{1}\int dx\sin(2\phi(x))+\cdots.\nonumber\end{aligned}$$ These terms give another type of DDSG model, but some of them can be related through a transformation since the fields $\phi$ and $\theta$ can be rescaled by the transformation $$\begin{split}
\phi &\to b \phi \\
\theta &\to \frac1b \theta
\end{split}$$ that preserves the commutation relation. For example, the Heisenberg model with a staggered $z$ field and XY anisotropy is equivalent to the DDSG model with $m=2$, $n=2$. This can be mapped to the $m=4$, $n=1$ case through the transformation $\phi\to2\tilde{\phi}$, $\theta\to\tilde{\theta}/2$, $K/4\to\tilde{K}$. However the operators that correspond to local observable are different since the formula is unchanged.
Quantum phase transition between competing orders {#sec:QPT}
=================================================
In this section, we study the properties of the quantum phase transition between competing orders for the models mentioned in Sec. \[sec:bosonization\].
![Staggered magnetization curves for $m_{\mathrm{N}}^{x}$ and $m_{\mathrm{N}}^{z}$ in the XXZ model with (a) staggered $x$ field ($\Delta=1.9$) and (b) XY anisotropy ($\Delta=1.6$). The saturation value of $m_{\mathrm{N}}^{x(z)}$ is normalized to $1$. []{data-label="fig:magcur_Dxy_hx"}](magcur_Dxy_hx_v03.pdf){width="45.00000%"}
First, we consider the XXZ model with staggered $x$ field, $$\begin{aligned}
\mathcal{H}=\mathcal{H}_{\mathrm{XXZ}}
-h_{x}\sum_{j}(-1)^{j}S_{j}^{x}.
\label{Hamil_XXZhx}\end{aligned}$$ In Fig. \[fig:magcur\_Dxy\_hx\](a), we show the staggered magnetization per site $m_{\mathrm{N}}^{x(z)}$ along $x(z)$ axis calculated by iTEBD. The phase transition is characterized by the disappearance of $m_{\mathrm{N}}^{z}$, and the critical field is $h_{x,\mathrm{c}}/J\simeq0.071$. Let us determine the universality class of this transition. In Fig. \[fig:Ising\](a), we show the log-log plot of the order parameter $m_{\mathrm{N}}^{z}$ as a function of $h_{x,\mathrm{c}}-h_{x}$. The fitting function is given as $m_{\mathrm{N}}^{z}=1.055((h_{x,\mathrm{c}}-h_{x})/J)^{0.129}$, and the critical exponent is $\beta=0.129\simeq 1/8$. We also calculate the entanglement entropy for a finite interval. When the system is bipartitioned into the subsystems $A$ and $B$, where $A$ is an interval consisting of $l$ spins and $B$ is the remainder, the reduced density matrix of the subsystem $A$ is defined as $\rho_{A}=\mathrm{Tr}_{B}|\Psi\rangle\langle\Psi|$ ($|\Psi\rangle$ is the ground state). Then the entanglement entropy is represented as $S_{\mathrm{EE}}=\mathrm{Tr}\rho_{A}\ln\rho_{A}$. In systems described by a conformal field theory, the entanglement scales as [@Calabrese2004Jstat] $$S_{\mathrm{EE}}=\frac{c}{3}\ln l+\mathrm{const},
\label{eq:EntangleEnt}$$ where $c$ is the central charge. The entanglement entropy $S_{\mathrm{EE}}$ as a function of the subsystem size $l$ that is calculated at the transition point $h_{x,\mathrm{c}}$ is shown in Fig. \[fig:Ising\](b). When the data are fitted by , the function is $S_{\mathrm{EE}}=0.157\ln l+0.892$ and the central charge is estimated as $c=0.471\simeq 1/2$. These results $\beta\simeq 1/8$ and $c\simeq 1/2$ indicate that the transition belongs to the Ising universality class. In terms of a field theory, the DDSG model is equivalent to two Majorana fermions [@Lecheminant2002NPB; @Tsvelik2012NJP]. At the transition point, one of the Majorana fermions is gapped out while the other remains gapless, thus the transition is of the Ising type.
![(a) Log-log plot of $m_{\mathrm{N}}^{z}$ as a function of $h_{x,\mathrm{c}}-h_{x}$. (b) Semi-log plot of entanglement entropy for a finite interval $S_{\mathrm{EE}}$ as a function of the size of the interval $l$ at $h_{x}=h_{x,\mathrm{c}}$. $M$ is the bond dimension of MPS (see Appendix \[sec:DetailNumerics\]). []{data-label="fig:Ising"}](Ising_v14.pdf){width="45.00000%"}
In Fig. \[fig:Ising\](a), we see that the data points are deviated from the fitting line in the region of and $(h_{x,\mathrm{c}}-h_{x})/J\gtrsim 0.03$. Let us comment on this point. Figure \[fig:mz8\_EE\](a) shows the plot of $(m_{\mathrm{N}}^{z})^{8}$ as a function of $h_{x}$. The solid line represents a linear fitting, and data points are away from the line in $h_{x}/J\leq 0.04$. This indicates that the deviation in the region of $(h_{x,\mathrm{c}}-h_{x})/J\gtrsim 0.03$ in Fig. \[fig:Ising\](a) is due to getting away from the critical region. From the equation of the fitting line $(m_{\mathrm{N}}^{z})^{8}=-1.45(h_{x}/J-0.0707)$, the critical field is obtained as $h_{x,\mathrm{c}}/J=0.0707$. We can also determine $h_{x,\mathrm{c}}$ from the divergence of half-infinite entanglement entropy $S_{\mathrm{half}}$, which is calculated by the bipartition of the system into two half-infinite chains. In Fig. \[fig:mz8\_EE\](b), we plot the half-infinite entanglement entropy $S_{\mathrm{half}}$ as a function of $h_{x}$, and the critical value is $h_{x,\mathrm{c}}/J=0.0712$. Thus, it is estimated as $h_{x,\mathrm{c}}/J=0.071\pm 0.0003$, which causes the error bars in Fig. \[fig:Ising\](a).
![Plot of (a) $(m_{\mathrm{N}}^{z})^{8}$ (b) half-infinite entanglement entropy $S_{\mathrm{half}}$ as a function of $h_{x}$. []{data-label="fig:mz8_EE"}](mz8_EE_v14.pdf){width="45.00000%"}
Next we consider the XXZ model with XY anisotropy, $$\begin{aligned}
\mathcal{H}=\mathcal{H}_{\mathrm{XXZ}}
+D_{xy}\sum_{j}(S_{j}^{x}S_{j+1}^{x}-S_{j}^{y}S_{j+1}^{y}).
\label{Hamil_XXZDxy}\end{aligned}$$ This Hamiltonian is nothing but the XYZ model, which is exactly solvable [@Baxter2016Book]. Staggered magnetization $m_{\mathrm{N}}^{x}$ and $m_{\mathrm{N}}^{z}$ calculated by iTEBD is shown in Fig. \[fig:magcur\_Dxy\_hx\](b). In contrast to Fig. \[fig:magcur\_Dxy\_hx\](a), the orders $m_{\mathrm{N}}^{x}$ and $m_{\mathrm{N}}^{z}$ are exclusively competing, i.e., if one of the two orders is nonzero, the other is zero. The critical value of $D_{xy}$ is $D_{xy,\mathrm{c}}=(\Delta-1)J$. Since $J-D_{xy,\mathrm{c}}<J+D_{xy,\mathrm{c}}=\Delta J$, the Hamiltonian is the easy-plane XXZ model at the critical point and the ground state is Tomonaga-Luttinger liquid (a conformal field theory with central charge $c=1$). Hence the transition is the BKT type, which is consistent with the renormalization analysis [@Giamarchi1988JPhys].
Dynamical susceptibility {#sec:DynSuscep}
========================
Let us now compute how the critical behavior of the models of Sec. \[sec:QPT\] can be measured experimentally. In addition to the static staggered magnetization, we show that the dynamical susceptibility captures well the properties of the quantum phase transition. This quantity is directly accessible in INS and ESR experiments.
The spin-spin retarded correlation function is defined as $$\chi^{\alpha\beta}({\boldsymbol{r}},t)
=-i\vartheta_{\mathrm{step}}(t)
\langle[S_{{\boldsymbol{r}}}^{\alpha}(t),S_{0}^{\beta}(0)]\rangle,
\label{eq:RetCorr}$$ where $\vartheta_{\mathrm{step}}(t)$ is the Heaviside function. For 1D lattice systems, ${\boldsymbol{r}}$ is replaced with the site index $j$. The dynamical susceptibility is obtained from the Fourier transform of the retarded correlation function , $$\chi^{\alpha\beta}({\boldsymbol{q}},\omega)
=\int_{-\infty}^{\infty}dt\sum_{{\boldsymbol{r}}}
e^{i(\omega t-{\boldsymbol{q}}\cdot{\boldsymbol{r}})}
\chi^{\alpha\beta}({\boldsymbol{r}},t)
\label{eq:SuscepFourier3D}$$ This quantity is related to the differential scattering cross section of INS by $$\begin{aligned}
\frac{d^{2}\sigma}{d\Omega dE}\propto
\frac{|{\boldsymbol{q}}_{\mathrm{out}}|}{|{\boldsymbol{q}}_{\mathrm{in}}|}|F({\boldsymbol{Q}})|^{2}
\sum_{\alpha,\beta=x,y,z}&
\Big(\delta_{\alpha\beta}
-\frac{Q_{\alpha}Q_{\beta}}{|{\boldsymbol{Q}}|^{2}}\Big)\nonumber\\
&\times
\mathrm{Im}\chi^{\alpha\beta}({\boldsymbol{Q}},\omega),
\label{eq:CrossSec}\end{aligned}$$ where $F({\boldsymbol{Q}})$ is the magnetic form factor and ${\boldsymbol{q}}_{\mathrm{in}}$, ${\boldsymbol{q}}_{\mathrm{out}}$ is the direction of incoming and outgoing fluxes, respectively. ${\boldsymbol{Q}}$ is a scattering vector defined as ${\boldsymbol{Q}}={\boldsymbol{q}}_{\mathrm{in}}-{\boldsymbol{q}}_{\mathrm{out}}$. If the system is $U(1)$ symmetric (i.e., $\sum_{j}S_{j}^{z}$ is conserved), Eq. is rewritten as [@Lovesey1986Book] $$\begin{aligned}
\frac{d^{2}\sigma}{d\Omega dE}\propto&
\frac{|{\boldsymbol{q}}_{\mathrm{out}}|}{|{\boldsymbol{q}}_{\mathrm{in}}|}|F({\boldsymbol{Q}})|^{2}
\Big\{\Big(1-\frac{Q_{z}^{2}}{|{\boldsymbol{Q}}|^{2}}\Big)
\mathrm{Im}\chi^{zz}({\boldsymbol{Q}},\omega)\nonumber\\
&\qquad
+\Big(1+\frac{Q_{z}^{2}}{|{\boldsymbol{Q}}|^{2}}\Big)
\mathrm{Im}\chi^{xx}({\boldsymbol{Q}},\omega)\Big\},
\label{eq:CrossSec2}\end{aligned}$$ since $\chi^{xx}=\chi^{yy}$. In ESR experiments, since electromagnetic waves in the GHz frequency region are used, the wavelength is much larger than the lattice constant and only the response at $|{\boldsymbol{q}}|= 0$ is relevant. When such electromagnetic waves are applied to the system, the energy absorption rate is given by $$I(\omega)\propto \omega\mathrm{Im}
\chi^{\alpha\alpha}({\boldsymbol{q}}=0,\omega),$$ where $\alpha$ is the direction of oscillating magnetic field. $I(\omega)$ corresponds with spectrum of ESR.
We compute the dynamical susceptibility numerically. We first obtain the ground state of the system by infinite density matrix renormalization group (iDMRG) [@Mcculloch2008arXiv], then perform the time evolution by iTEBD [@Vidal2007PRL] with the infinite boundary condition [@Phien2012PRB]. In this way, we can calculate space-time correlation function $\langle S_{{\boldsymbol{r}}}^{\alpha}(t)S_{0}^{\beta}(0)\rangle$, and dynamical susceptibility through Fourier transform. The details of numerical calculation are given in Appendix \[sec:DetailNumerics\].
![Dynamical susceptibility (a) $\chi^{xx}(q=\pi)$ and (b) $\chi^{zz}(q=\pi)$ for the XXZ model ($\Delta=1.9$) with staggered $x$ field. The dominant low energy excitation in the low (high) $h_{x}$ phase corresponds to $\chi^{xx}$ ($\chi^{zz}$). We see that $\chi^{zz}$ diverges at the transition point $h_{x}/J\simeq 0.071$ while $\chi^{xx}$ does not. []{data-label="fig:chiqw_hx"}](chiqw_hx_v03.pdf){width="45.00000%"}
In Fig. \[fig:chiqw\_hx\], we show the dynamical susceptibility at $q=\pi$ in the XXZ model with staggered $x$ field . In the low (high) $h_{x}$ phase, the dominant low energy elementary excitation corresponds to $\chi^{xx}$ ($\chi^{zz}$). The order is in the $z$ direction at $h_{x}=0$, and $m_{\mathrm{N}}^{z}$ decreases while $m_{\mathrm{N}}^{x}$ increases as $h_{x}$ becomes larger. Above the critical $h_{x}$, the order is in the $x$ direction. Hence the behavior of $\chi^{xx}$ and $\chi^{zz}$ indicates that the low energy excitation is generated by a spin flip. We can also see that $\chi^{zz}$ diverges at the transition point while $\chi^{xx}$ does not in Fig. \[fig:chiqw\_hx\]. That is because $m_{\mathrm{N}}^{z}$ becomes zero at the transition point while $m_{\mathrm{N}}^{x}$ changes smoothly. \[see Fig. \[fig:magcur\_Dxy\_hx\](a)\].
![Dynamical susceptibility (a) $\chi^{xx}(q=\pi)$ and (b) $\chi^{zz}(q=\pi)$ for the XXZ model ($\Delta=1.6$) with XY anisotropy. Both $\chi^{xx}$ and $\chi^{zz}$ diverge at the transition point $D_{xy}/J=0.6$. []{data-label="fig:chiqw_Dxy"}](chiqw_Dxy_v07.pdf){width="45.00000%"}
Let us now compare with the dynamical susceptibility at $q=\pi$ for the XXZ model with XY anisotropy in Fig. \[fig:chiqw\_Dxy\]. Similarly to the staggered $x$ field case, in the low (high) $D_{xy}$ phase, the dominant elementary excitation corresponds to $\chi^{xx}$ ($\chi^{zz}$). There are however an important difference on the susceptibilities, which stems from the different nature of the transition. It is directly visible that both $\chi^{xx}$ and $\chi^{zz}$ diverge at the transition point in Fig. \[fig:chiqw\_Dxy\]. This is the consequence of the exclusive competition between $m_{\mathrm{N}}^{x}$ and $m_{\mathrm{N}}^{z}$, both of which become zero at the transition point \[see Fig. \[fig:magcur\_Dxy\_hx\](b)\].
![ Dynamical susceptibility (a) $\chi^{xx}(q=0)$ and (b) $\chi^{zz}(q=0)$ for the XXZ model ($\Delta=1.9$) with staggered $x$ field and (c) $\chi^{xx}(q=0)$ and (d) $\chi^{zz}(q=0)$ for the XXZ model ($\Delta=1.6$) with XY anisotropy. []{data-label="fig:chiqw_q0"}](chiqw_q0_v14.pdf){width="45.00000%"}
We also discuss the dynamical susceptibility at $q=0$ which is relevant with ESR experiments. Figure \[fig:chiqw\_q0\] shows $\chi^{xx}(q=0)$ and $\chi^{zz}(q=0)$ for the XXZ model ($\Delta=1.9$) with staggered $x$ field and with XY anisotropy. We first note that the intensity of the dynamical susceptibility is extremely small at $q=0$ compared with $q=\pi$ since antiferromagnetic correlation is dominant in the present system. As seen in Figs. \[fig:chiqw\_q0\](a) and (b), gap does not close at $q=0$ for the XXZ model with staggered $x$ field. Small intensity of the low energy region ($\omega/J\lesssim 0.3$) near the critical field $h_{x}\simeq 0.07$ is numerical artifact. On the contrary, Figs. \[fig:chiqw\_q0\](c) and (d) show that gap closes at $q=0$ for the XXZ model with XY anisotropy. This is natural since the critical point corresponds to an easy plain XXZ model and the gapless des Cloizeaux-Pearson mode exists at $q=0$.
As for the XXZ model with staggered $x$ field, the band at $q=\pi$ is folded to the band at $q=0$ due to the perturbation that breaks one-site translational symmetry. Thus, ESR measurements captures the mixing of $q=0$ and $q=\pi$ components of dynamical susceptibility. This effect is seen in Cu benzoate [@Oshikawa2002PRB], KCuGaF$_{6}$ [@Furuya2012PRL], and $\mathrm{BaCo_{2}V_{2}O_{8}}$ [@Kimura2007PRL]. The similar mixing is also measured in $\mathrm{(C_{7}H_{10}N)_{2}CuBr_{4}}$ [@Ozerov2015PRB].
The above calculations clarifies that the spin-spin susceptibility shows very clear signatures of the nature of these two different topological transitions. Although these measurements do not directly give access to the nonlocal (topological) order, they nevertheless provide clear signatures of the change of the nature of the excitations.
Application to real materials {#sec:application}
=============================
In the above, we discussed the models that can be mapped to DDSG models and their quantum phase transitions. In order to apply the above theoretical analysis to realistic materials, one has to consider several important elements depending on whether the system is condensed matter or cold atomic gas.
Condensed matter systems
------------------------
For the condensed matter realizations, two elements are to be taken into account. First, in the present experiments, one can expect to measure only the local observable (magnetization, spin-spin susceptibility, etc.). Nonlocal order parameters (e.g., $\cos(\theta(x)/2)$ in Sec. \[sec:XXZhx\]) are difficult to measure experimentally in condensed matter systems. Second, in quasi-1D materials, spin chains are coupled and form three dimensional system while the analysis done in the previous parts is strictly 1D.
Recently, the DDSG model discussed above was found to be realized in the compound $\mathrm{BaCo_{2}V_{2}O_{8}}$ [@Faure2018NatPhys]. In this material, Co$^{2+}$ ions effectively form the $S=1/2$ quasi-1D antiferromagnet with Ising anisotropy. When an external magnetic field perpendicular to the anisotropy axis is applied in this system, an effective staggered transverse field arises since nondiagonal components of $g$ tensor are nonzero due to the slight deviation of the magnetic principal axes from the crystallographic axes [@Kimura2013JPSJ]. The model Hamiltonian of this compound is essentially equivalent to the XXZ model with staggered $x$ field , and the quantum phase transition discussed in Sec. \[sec:XXZhx\] happens. Note that an effective staggered field $-h_{\mathrm{eff}}\sum_{j}(-1)^{j}S_{j}^{z}$ along the $z$ axis arises from the interchain interaction, determined self-consistently, with the Néel order along the $z$ axis in the mean field theory has also to be taken into account [@Faure2018NatPhys]. Due to this staggered $z$ field, the critical field is shifted to a higher value than the case without the interchain interaction and the gap opens at the transition point with $h_{\mathrm{eff}}=0$. Thus, the gap is not closed at the quantum phase transition caused by the transverse field in $\mathrm{BaCo_{2}V_{2}O_{8}}$. As discussed in Sec. \[sec:DynSuscep\], the dynamical susceptibility is measured by INS experiments. For a direct comparison with the neutrons, one has to use the actual position of the spin sites (the Co$^{2+}$ ions) in the Fourier transform of retarded correlation function since the neutrons are directly sensitive to the actual position of the spins.
It would be interesting if other examples of the topological transitions discussed in the previous sections also could be realized. The potential of the field $\phi$ is provided by dimerization, Ising anisotropy, and staggered Dzyaloshinskii-Moriya (DM) interaction $\sum_{j}(-1)^{j}{\boldsymbol{D}}\cdot({\boldsymbol{S}}_{j}\times{\boldsymbol{S}}_{j+1})$ with $D\parallel z$ axis. The strategy for material search is to find systems that have these perturbations as well as nondiagonal staggered $g$ tensor. The application of effective staggered field introduces effective staggered field, which gives the potential of the field $\theta$. Then the transition is provoked by increasing the external field. In addition to spin chains, searching for materials which realize the DDSG model in spin ladders with magnetic anisotropy or DM interaction is an interesting future direction.
Cold atomic systems {#sec:coldatom}
-------------------
Another important route to realize the topological transitions described in the previous sections is provided by cold atomic systems [@Bloch2008RMP; @Ritsch2013RMP]. Although initial simulations of quantum magnetism were done in bosonic systems by using the mapping between spin-1/2 and hard core bosons [@Struck2011Science; @Vedmedenko2013NJP] and thus the realization is limited to XX models due to the absence of long range interactions, recent advance allows to probe the quantum magnetism in fermionic systems as well. Short-range quantum magnetism has been observed for ultracold fermions in an optical lattice [@Greif2013Science], and measurements of various physical quantities such as dynamical structure factor [@Landig2015NatComm] and magnetic order and correlations [@Parsons2016Science; @Boll2016Science; @Cheuk2016Science]. In addition to systems with fermions, quantum simulation of spin systems are also realized by using Rydberg atoms [@Browaeys2016JPhysB; @Lienhard2018PRX].
There are several advantages for the cold atomic realization. The first is the controllability of parameters. While the parameters are fixed for each material in condensed matter systems, particle-particle interaction can be varied by using Feshbach resonance in cold atomic systems. Controlling the population of up-spins and down-spins allows the equivalent of a magnetic field along $z$. The second advantage is that cold atomic systems provide the probes complementary to the condensed matter ones, in particular to measure nonlocal order parameters. For example, a string order parameter in the Haldane phase can be observed by repeating snapshot measurements [@Endres2011Science] in cold atomic systems. This technique can be also potentially applicable for measuring nonlocal order parameters such as $\cos(\theta(x)/2)$ discussed in Sec. \[sec:XXZhx\]. Measurements are so far limited to equal time correlations but schemes have been proposed to overcome such limitations [@Knap2013PRL].
One of the challenges in this field is cooling the system enough to simulate the low temperature phenomena of the corresponding condensed matter systems. However, since the experimental technique of cooling has been improving [@Mazurenko2017Nature], we can expect that some of the phases described here could be observed in the near future.
Conclusion {#sec:summary}
==========
We studied quantum phase transitions between competing orders in the models which is mapped to the DDSG field theory. We specifically considered two types of systems: the XXZ chain with staggered $x$ field and with XY anisotropy. The universality class of the transition is of the Ising type in the former case while it is of the BKT type in the latter case. We showed numerically that the difference of the transition properties appears in the dynamical susceptibilities, which can be directly compared with the spectra measured by INS experiments. We discussed the possibility of observation of the phases and the phase transitions studied in the present paper in condensed matter systems and cold atomic ones. For condensed matter realizations, one of the quantum phase transition between competing orders has been seen in a real material $\mathrm{BaCo_{2}V_{2}O_{8}}$, which is a quasi-1D Heisenberg antiferromagnet with Ising anisotropy [@Faure2018NatPhys]. Other quantum spin systems either chains or ladders with anisotropic perturbations could serve as a basis for studying the other universality classes discussed here. In that respect the dynamical susceptibilities, directly measured by INS or ESR experiments, computed in the present paper, provide a clear distinction between the various transitions and can thus be used as an experimental signature.
Another broad class of systems in which the phenomena can be investigated is provided by cold atomic systems of fermions or Rydberg atoms. Such systems have the advantage of a good control of the various parameters in the Hamiltonian as well as the possibility of measure the nonlocal (topological) order parameters which are a direct signature of the various phases. Relatively high temperature as well as the size limitation is the current drawbacks, but the situation is rapidly evolving. These systems also offer the fascinating possibility to study time-dependent Hamiltonians, allowing to investigate the effect of time dependent perturbations in the future, either quenches or periodic perturbations (Floquet systems) on such topological phase transitions.
The authors gratefully thank the many fruitful discussions with Q. Faure, B. Grenier, S. Petit, V. Simonet and Ch. Rüegg on quantum spin chains. S. T. is supported by the Swiss National Science Foundation under Division II and ImPact project (No. 2015-PM12-05-01) from the Japan Science and Technology Agency. S. C. F. is supported by JSPS KAKENHI (No. JP16J04731).
Details of numerical simulations {#sec:DetailNumerics}
================================
In this appendix, we describe the detail of numerical simulations. Time evolution is calculated by iTEBD [@Vidal2007PRL] after the ground state is obtained by iDMRG [@Mcculloch2008arXiv]. The iTEBD uses the MPS representation of quantum states, and the time evolving operator is applied through the second order Trotter decomposition. Time is discretized with the unit $dt/J^{-1}=0.05$ in this study. The initial state (ground state) is represented as infinite MPS, which assumes translational invariance of the system, but in order to calculate the space-time correlation function, we have to break the translational invariance by applying an operator at $t=0,j=0$. Thus, we prepare a finite spatial interval and the matrices at both edges of the interval is determined in the way that they represent a semi-infinite extension of the system, which is called the infinite boundary condition [@Phien2012PRB]. The advantage of this method is that there is no finite-size effect. The space-time correlation function Eq. is calculated for a finite temporal interval $0\leq t \leq T$, and dynamical susceptibility is obtained as the numerical Fourier transform of the space-time correlation function. Gaussian filter is utilized in the Fourier transformation, $$\chi(q,\omega)
=\int_{-T}^{T}dt\sum_{r}e^{i(\omega t-qr)}\chi(r,t)G(t),
\nonumber$$ where $G(t)=e^{-(2t/T)^{2}}$.
![The dependence of iTEBD calculations (a) on the truncation dimension $M$ with fixed $T/J^{-1}=80$ and (b) on the temporal interval $T$ with fixed $M=60$. The results of $\mathrm{Im}\chi^{xx}(q=\pi,\omega)$ for the model are shown with $\Delta=1.9$ and $h_{x}/J=0.02$. []{data-label="fig:M_T_depend"}](M_T_depend_v13.pdf){width="45.00000%"}
In the iTEBD and iDMRG calculations, quantum states are optimally approximated by MPS with finite bond dimension (also called truncation dimension) $M$. As the bond dimension $M$ is larger, the calculation is more precise. In Fig. \[fig:M\_T\_depend\](a), we show $\chi^{xx}(q=\pi,\omega)$ calculated with Eq. for different bond dimensions $M=40,60,80$ while $T/J^{-1}=80$ is fixed. We can see that the dependence of the result on $M$ is small. In the real-time calculation, an error also arises from a finite time effect. Figure \[fig:M\_T\_depend\](b) shows $\chi^{xx}(q=\pi,\omega)$ calculated with Eq. for final time $T/J^{-1}=40,60,80$ while $M=60$ is fixed. The dependence of the result on $T$ is also small.
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---
abstract: 'We consider nonlinear elliptic equations that are naturally obtained from the elliptic Schrödinger equation $-\Delta u +Vu=0$ in the setting of the calculus of variations, and obtain $L^q$-estimates for the gradient of weak solutions. In particular, we generalize a result of Shen in \[Ann. Inst. Fourier 45 (1995), no. 2, 513–546\] in the nonlinear setting by using a different approach. This allows us to consider discontinuous coefficients with a small BMO semi-norm and non-smooth boundaries which might not be Lipschitz continuous.'
address:
- 'Department of Mathematics and Statistics, FI-20014 University of Turku, Finland'
- 'Department of Applied Mathematics and Institute of Natural Science, Kyung Hee University, Yongin 17104, Republic of Korea'
author:
- Mikyoung Lee
- Jihoon Ok
title: 'Interior and boundary $W^{1,q}$-estimates for elliptic quasi-linear equations of Schrödinger type'
---
**Introduction** {#secintro}
================
The present paper is devoted to the study of interior and boundary $L^q$-integrability for the gradient of weak solutions to time independent quasi-linear equations of the $p$-Schrödinger type $$\label{maineqm}
-\mathrm{div}\, (|Du|^{p-2}Du) + V |u|^{p-2}u = 0 \ \ \textrm{ in } \ \Omega,$$ where $1<p<\infty$, $\Omega\subset \mr^n$($n\geq2$) is open and bounded, and the non-negative potential $V$ is taken in an appropriate class. We notice that if $p=2,$ the equation becomes $$\label{propeq}
- \Delta u + Vu=0\ \ \ \textrm{in}\ \ \Omega,$$ which is the classical (elliptic) Schrödinger equation. In the viewpoint of the calculus of variations, the equation is the Euler-Lagrange equation of the following functional $$W^{1,p}(\Omega)\ni u \ \ \mapsto\ \ \int_{\Omega} \left[|Du|^p+V|u|^p\right]\, dx,$$ hence it is one of nonlinear generalizations of the Schrödinger equation in a natural way. Moreover, problems of this type raise in various areas of physics, such as nonlinear quantum field theory, nonlinear optics, plasma physics, condensed matter physics, biophysics, fluid mechanics, etc. We refer to [@APT1; @BS1; @LS1; @MF1; @SS] for the general physical background of this equation.
Research on the Schrödinger type equations which are fundamental ones of quantum mechanics plays a significant role in the fields of mathematical physics. In particular, $L^q$-regularity theory for linear Schrödinger equations was first introduced by Shen [@Sh1]. He obtained $L^q$-estimates by assuming that $V$ belongs to the $\mathcal B_{\gamma}$ class for some $\gamma\geq \frac n2$ which is a certain reverse Hölder class (see below for the definition of $\mathcal B_\gamma$). More precisely, for the Schrödinger equations with non-divergence data of the form $-\Delta u+Vu=f$ in $\mr^n$, he showed $\Vert D^2u\Vert_{L^q(\mr^n)}+\Vert Vu\Vert_{L^q(\mr^n)} \leq c(q)\|f\|_{L^q(\mr^n)}$ for all $1<q\leq \gamma$, and for the equations with divergence data of the form $$\label{divschrodinger}
-\Delta u+Vu=-\textrm{div}\, F\ \ \ \text{in}\ \ \mr^n,$$ he also did $$\Vert Du \Vert_{L^{q}(\mr^n)} +\chi_{\{q\leq 2\gamma\}} \Vert V^{\frac12}u \Vert_{L^{q}(\mr^n)} \leq c(q)\Vert F \Vert_{L^{q}(\mr^n)} , \ \ \textrm{for all}\ \ (\gamma^*)' \leq q \leq \gamma^*,$$ where $\gamma^*=\frac{n\gamma}{n-\gamma}$ when $\gamma<n$ (if $\gamma\geq n$, then $q$ can be any number in $(1,\infty)$). Here, we remark that the range of $q$ is optimal, see [@Sh1 Section 7]. These results have been recently extended to linear elliptic/parabolic Schrödinger equations with discontinuous coefficients on sufficiently smooth domains in several papers for instance [@BHS1; @BBHV1; @PT1], by using the results in [@Sh1] together with the commutator method and the standard flattening and covering arguments. We also refer to [@CFG1; @D1; @FPR; @K1; @PT1; @Sh0; @Sh1] for the regularity theory for (elliptic) Schrödinger equations.
The general aim of this paper is to establish interior and boundary $L^q$-regularity theory for nonlinear Schrödinger equations in non-smooth domains. In particular, as mentioned earlier, we deal with quasi-linear equations of $p$-Laplacian type which are the natural generalizations of the classical Schrödinger equation in the divergence setting. Moreover, the domains we consider here might be non-graph domains which are beyond the class of Lipschitz domains. We point out that the approach used in [@Sh1] cannot be applied to the nonlinear setting. Indeed, Shen in [@Sh1] derived the decay estimates for the fundamental solution by means of the Fefferman-Phong Lemma in [@Fe] by introducing an auxiliary function $m(x,V)$ which is well-defined for $q \geq \frac{n}{2}.$ Furthermore, on the boundary region we cannot make use of the flattening argument since our domain is supposed to be non-smooth. Therefore, an alternative approach must be adopted in order to handle the structures of the nonlinear operators and the non-smooth domains. In our best knowledge, the present paper is a new one treating $L^q$-estimates for Schrödinger equations in a non-linear setting and even for linear Schrödinger equations on non-smooth domains.
Now let us present our main equations. We are concerned with the Dirichlet problem for the quasi-linear Schrödinger equation of the form $$\label{maineq}
\left\{\begin{array}{rclcc}
-\mathrm{div}\, \mathbf{a}(x,Du) + V |u|^{p-2}u& = & -\mathrm{div}\, (|F|^{p-2}F) & \textrm{ in } & \Omega, \\
u & = & 0 & \textrm{ on } & \partial \Omega,
\end{array}\right.$$ where $1<p<\infty$, $\Omega$ is open and bounded in $\mr^{n}$ with $n\geq2$, and $V:\Omega\to \mr$ is non-negative and at least satisfies $V\in L^{n/p}(\Omega)$ if $p<n$ and $V\in L^t(\Omega)$ for some $t>1$ if $p\geq n$. A given vector valued function $ \mathbf{a} : \mr^n \times \mr^n \rightarrow \mr^n $ is a Carathéodory function, that is, $ \mathbf{a}$ is measurable in the $x$-variable and differentiable in the $\xi$-variable. We will always assume that $\mathbf{a}$ satisfies the following growth and ellipticity conditions: $$\label{aas1}
| \mathbf{a}(x,\xi)|+ | D_{\xi} \mathbf{a}(x,\xi)||\xi| \leq L |\xi|^{p-1}$$ and $$\label{aas2}
D_{\xi} \mathbf{a}(x,\xi)\, \eta \cdot \eta \geq \nu |\xi|^{p-2}|\eta|^2$$ for almost all $x \in \mr^n$ and any $\xi, \eta \in \mr^n$ and for some constants $L, \nu$ with $0< \nu \leq 1 \leq L.$ A prime example of the nonlinearlity $\mathbf{a}$ is $$\mathbf{a}(x,\xi) = a(x)|\xi|^{p-2}\xi,\ \ \nu\leq a(\cdot)\leq L,$$ which is the $p$-Laplacian with the coefficient $a(\cdot).$ We also remark that the above condition implies the monotonicity condition: $$\label{mono}
\left( \mathbf{a}(x,\xi) - \mathbf{a}(x,\eta) \right) \cdot (\xi-\eta) \geq c(p,\nu) \left( |\xi|^2 + |\eta|^2 \right)^{\frac{p-2}{2}} |\xi-\eta|^2$$ for any $\xi, \eta \in \mr^n$ and a.e. $x \in \mr^n.$ In particular, if $p \geq 2,$ it can be the following $$\label{mono1}
\left( \mathbf{a}(x,\xi) - \mathbf{a}(x,\eta) \right) \cdot (\xi-\eta) \geq c(p,\nu) |\xi-\eta|^p.$$ Under the above basic setting, we say that $u \in W^{1,p}_0(\Omega)$ is a weak solution to the problem if $$\label{weakform}
\int_{\Omega} \mathbf{a}(x, Du) \cdot D\varphi \, dx + \int_{\Omega} V |u|^{p-2}u \cdot \varphi \, dx =\int_{\Omega} |F|^{p-2} F \cdot D \varphi\, dx$$ holds for any $\varphi \in W_0^{1,p}(\Omega).$ We note that if $u\in W^{1,p}_0(\Omega)$, $\|Du\|_{L^p(\Omega)}$ and $\|Du\|_{L^p(\Omega)}+\|V^{\frac1p}u\|_{L^p(\Omega)}$ are equivalent by the condition of the potential $V$ and Sobolev-Poincaré’s inequality, and that the existence and the uniqueness of the weak solution of (even in the case of a non-zero Dirichlet boundary condition such that $u=g$ on $\Omega$ with $g\in W^{1,p}(\Omega)$) follow from the theory of nonlinear functional analysis, see for instance [@Sho1 Chapter 2].
For the potential $V:\Omega\to \mr$ considered in the problem , we suppose that $V$ belongs to $\mathcal{B}_{\gamma}$ for some $ \gamma \in [ \frac{n}{p}, n)$ when $p<n$ and for some $\gamma \in (1,n)$ when $p\geq n.$ We say that $V:\mr^n\to [0,\infty)$ belongs to $\mathcal{B}_\gamma$ for some $ \gamma >1$ if $V\in L^{\gamma}_{loc}(\mr^n)$ and there exists a constant $b_{\gamma}>0$ such that the *reverse Hölder inequality* $$\label{VBqclass}
\left( \frac{1}{|B|} \int_{B} V^{\gamma} \, dx \right)^{\frac{1}{\gamma}} \leq b_\gamma \left( \frac{1}{|B|} \int_{B} V \, dx \right)$$ holds for every ball $B$ in $\mr^n.$ This $\mathcal{B}_\gamma$ class which is a wide class including all nonnegative polynomials was introduced independently by Muckenhoupt [@Mu] and Gehring [@Ge] in the study of weighted norm inequalities and quasi-conformal mapping, respectively. One notable example of this element is $ V(x) = |x|^{-n/ \gamma} $ which actually belongs to the $\mathcal{B}_{\tilde{\gamma}}$ class for all $\tilde{\gamma} < \gamma.$ Moreover, the $B_\gamma$ class is strongly connected to the Muckenhoupt class, for which we will discuss later in Section \[Preliminaries\].
Our main result is the global integrability of $Du$ and also $V^{\frac1p}u$ for the weak solutions $u$ to the problem with respect to the one of $F$, under a suitable discontinuity condition on the nonlinearity $\mathbf{a}$ and a minimal structure condition on the boundary of the domain $\Omega$ that will be described later in Definition \[smallbmo\] and \[Defreifenberg\], respectively. More precisely, we prove that $$\label{implication1}
F\in L^q\ \Longrightarrow \ Du \in L^q\ \ \text{for each } \left\{\begin{array}{cl}
q\in[p,\gamma^*(p-1))& \text{when }\gamma\in [\frac np,n),\\
q\in[p,\infty)& \text{when } \gamma\in[n,\infty),
\end{array}\right.$$ $$\label{implication2}
F\in L^q \ \ \Longrightarrow \ \ V^{\frac1p}u \in L^q \ \ \ \text{for each } p\leq q \leq p \gamma,$$ by obtaining relevant estimates, see Corollary \[maincor\] and Remark \[mainrmk\] in the next section. We would like to emphasize that for the Schrödinger equation , that is, the equation when $p=2$ and $\ba(x,\xi)\equiv \xi,$ our results cover the ones in [@Sh1 Corollary 0.10] for $q\geq p=2$. Note that, in this linear case, the validity of the implications and for $ \gamma^*<q<2$ can be achieved via the duality argument, see for instance [@Um1].
For the equation with the null potential, i.e., $V\equiv0$, the $L^q$-estimates, which is sometimes called the (nonlinear) Calderón-Zygmund estimates, have been widely studied by many authors. Iwaniec [@Iw1] first obtained the $L^q$-estimates for the $p$-Laplace equations with $p\geq 2$, and then DiBenedetto & Manfredi [@DM1] extended his result to the $p$-Laplace systems with $1<p<\infty$. Later, Caffarelli & Peral [@CP1] considered general equations of the $p$-Laplacian type with discontinuous nonlinearities. Furthermore, Acerbi & Mingione generalized $L^q$-estimates for the parabolic $p$-Laplace systems with discontinuous coefficients [@AM1]. We also refer to [@BR1; @LO1; @MP1; @KZ1; @Mis1] for problems with $p$-Laplacian type and [@AM0; @BO1; @BOR1; @CM1; @Ok1] for problems with nonstandard growth.
We briefly discuss the outline of the proof of the $L^q$-estimates. As mentioned earlier, our approach is different from the one used in [@Sh1] which is based on the linear operator theory. We adopt a perturbation argument which has turned out to be very useful for the study on the regularity theory for linear and nonlinear PDEs. In particular, we employ the method introduced by Acerbi & Mingione in [@AM1], see also [@Min1] for its origin. To be more concrete, we apply an exit time argument to a nonlinear functional of $Du, V^{\frac1p}|u|$ and $F,$ in order to construct a suitable family of balls which covers the level set for $|Du|+V^{\frac1p}|u|$. Then, on each ball, we compare our equation with the homogeneous equation $$-\mathrm{div}\, \ba(x,Dw)+V|w|^{p-2}w=0.$$ The main part at this step is to find the maximal integrability of $Dw$ and $V^{\frac1p}w$ with corresponding estimates. In view of the classical regularity theory we know the $L^\infty$-boundedness of $w$ (see Lemma \[supvlem\]), from which together with the result in our recent paper [@LO1] (see Theorem \[thmDwbdd\]), we see that $Dw \in L^{\gamma^*(p-1)}$ and $V^{\frac1p}w\in L^{p\gamma}$ (see Lemma \[lem42\]). Here, we point out that the corresponding estimates and are derived in a very delicate way. Especially, at this stage, the $\mathcal B_\gamma$ condition of $V$ plays a crucial role, so that we take advantage of the idea of Fefferman & Phong in [@Fe] to obtain the modified version of Fefferman-Phong Lemma (see Lemma \[lemrVbddpq\]). Then from those corresponding estimates, the $L^q$-estimates for $|Du|+V^{\frac1p}|u|$ is derived by the comparison argument when $q\leq p\gamma$. Furthermore, applying the results in [@LO1], we eventually obtain the $L^q$-estimates for $|Du|$ when $p\gamma<q\leq \gamma^*(p-1).$
The remainder of this paper is organized as follows. In the next section, we state our main results with primary assumptions imposed on the nonlinearlity $\mathbf{a}$ and the domain $\Omega.$ Section \[Preliminaries\] deals with the basic properties of $\mathcal{B}_{\gamma}$ class and the auxiliary lemmas to prove the main results. In Section \[sechomo\], we show higher integrability of $Du$ and $V^{\frac1p}u$ for weak solutions $u$ to localized equations of our main problem with $F\equiv 0.$ In Section \[secComestimates\], we obtain the comparison estimates, and finally prove main results, Theorem \[mainthm\] and Corollary \[maincor\], in Section \[sec gradient estimates\].
**Main result** {#secpre}
================
We start this section with standard notation and definitions. We denote the open ball $\mr^n$ with center $y\in \mr^n$ and radius $r>0$ by $B_r(y)= \{ x \in \mr^{n} : |x-y|< r \}.$ We also denote $\Omega_r (y)= B_r(y) \cap \Omega$ and $ \partial_w\Omega_{r}(y) = B_r(y) \cap \partial \Omega.$ For the sake of simplicity, we write $B_r=B_r(0),$ $B_r^+=B_r^+(0)$ and $\Omega_{r} = \Omega_{r}(0).$ We shall use the notation $$\mint_{U} g \; dx := \frac{1}{|U|} \int_{U} g \;dx.$$
The following two definitions are associated with the main assumptions imposed on the nonlinearlity $\mathbf{a}$ and the domain $\Omega.$
\[smallbmo\] We say that $\mathbf{a}=\mathbf{a}(x,\xi)$ is *$(\delta,R)$-vanishing* if $$\sup_{0<\rho\leq R} \ \sup_{y\in\mathbb{R}^n} \mint_{B_{\rho}(y) } \left|\Theta\left( \mathbf{a},B_{\rho}(y) \right)(x) \right| \, dx \leq \delta,$$ where $$\Theta\left( \mathbf{a},B_{\rho}(y) \right)(x):= \sup_{\xi \in \mr^n \setminus \{ 0\} } \frac{ \left|\mathbf{a}(x,\xi)-\overline{\mathbf{a}}_{B_{\rho}(y)}(\xi)\right|}{|\xi|^{p-1} }$$ and $$\overline{\mathbf{a}}_{B_{\rho}(y)}(\xi) := \mint_{B_{\rho}(y)} \mathbf{a}(x,\xi) \;dx.$$
The above definition implies that the map $x\mapsto \ba(x,\xi)/|\xi|^{-p}$ is a (locally) BMO function with the BMO semi-norm less than or equal to $\delta$ for all $\xi\in\mr^n$. Hence we see that the nonlinearity $\ba$ can be discontinuous for the $x$-variable. In particular, if $\ba(x,\xi)=a(x)|\xi|^{p-2}\xi$, then this definition means that $a(\cdot)$ is a BMO function.
\[Defreifenberg\] Given $\delta \in (0,\frac18)$ and $R>0,$ we say that $\Omega$ is a $(\delta, R)$-Reifenberg flat domain if for every $x \in \partial\Omega$ and every $\rho \in (0, R],$ there exists a coordinate system $\{ y_1, y_2, \dots, y_n\}$ which may depend on $\rho$ and $x,$ such that in this coordinate system $x=0$ and that $$B_{\rho}(0) \cap \{ y_n > \delta \rho \} \subset B_{\rho}(0) \cap \Omega \subset B_{\rho}(0) \cap \{ y_n > -\delta \rho \}.$$
In the above definition of the Reifenberg flat domain, $\delta$ is usually supposed to be less than $\frac18$. This number comes from the Sobolev embedding, see for instance [@To1]. However, it is not important since we will consider $\delta$ sufficiently small. We note that the Lipschitz domains with the Lipschitz constant less than or equal to $\delta$ belong to the class of $(\delta,R)$-Reifenberg flat domains for some $R>0$. In addition, we remark that the $(\delta,R)$-Reifenberg flat domain $\Omega$ has the following measure density conditions: $$\label{dencon}
\sup_{0<\rho\leq R} \sup_{y \in \overline{\Omega}} \frac{\left|B_{\rho}(y)\right|}{\left|\Omega \cap B_{\rho}(y)\right|} \leq \left( \frac{2}{1-\delta} \right)^{n} \leq \left( \frac{16}{7} \right)^n,$$ $$\label{dencon1}
\inf_{0<\rho\leq R} \inf_{y \in \partial\Omega} \frac{|\Omega\cap B_{\rho}(y) |}{\left|B_{\rho}(y)\right|} \geq \left( \frac{7}{16} \right)^n.$$ We refer to [@BW1; @PS1; @Re1; @To1] for more details on the Reifenberg flat domains and their applications.
Now let us state the main results in this paper.
\[mainthm\] Let $u\in W^{1,p}_0(\Omega)$ be a weak solution to . Suppose that $V \in \mathcal{B}_{\gamma}$ for some $ \gamma \in [ \frac{n}{p}, n)$ when $p<n$ and for some $\gamma \in (1,n)$ when $p\geq n.$ For $p\leq q <\gamma^*(p-1)$, there exists a small $\delta = \delta(n, p, L, \nu)>0$ so that if $\mathbf{a}$ is $( \delta, R)$-vanishing and $\Omega$ is a $( \delta, R)$-Reifenberg flat domain for some $R\in(0,1),$ then we have for any $x_0\in \overline{\Omega}$ and $r \in (0, \frac{R}{4}]$ satisfying $ (4r)^{p - \frac{n}{\gamma}} \Vert V \Vert_{L^{\gamma}(\Omega_{4r}(x_0))} \leq 1,$ $$\begin{aligned}
\label{mainlocalest}
\nonumber\left( \mint_{\Omega_{ r}(x_0)} |Du|^q +\chi_{\{q<p\gamma\}} \left[V^{\frac1p}|u|\right]^{q}\, dx\right)^{\frac1q} &&\\
&&\hspace{-6cm}\leq c \left( \mint_{\Omega_{4r}(x_0)} |Du|^p + \left[ V^{\frac1p} |u|\right]^p \,dx \right)^{\frac{1}{p}}+ c \left(\mint_{\Omega_{4 r}(x_0)} \left|F \right|^{q} \, dx\right)^{\frac1q}\end{aligned}$$ for some $c=c(n,p,q,\gamma,L,\nu,b_{\gamma})>0,$ where $\chi_{\{q<p\gamma\}}:= 1$ if $q<p\gamma$ and $\chi_{\{q<p\gamma\}}:= 0$ if $q\geq p\gamma.$
Let $\Omega$ be a $(\delta,R)$ Reifenberg flat domain for some small $\delta>0$ and $R>0$ and $V\in \mathcal B_\gamma$ with $\gamma\geq \frac{n}{p}$ and $p>1$. Define $$\rho(y,V):= \sup\left\{r\in(0,R]:r^{p - \frac{n}{\gamma}} \Vert V \Vert_{L^{\gamma}(\Omega_{r}(y))} \leq 1 \right\},\quad y\in \overline{\Omega}.$$ Then by Hölder’s inequality, the $\mathcal B_\gamma$ condition of $V$ and , we see that the function $\rho(y,V)$ is comparable to $$\tilde\rho(y,V):= \sup\left\{r\in(0,R]:\frac{1}{r^{n-p}} \int_{\Omega_{r}(y)} V \,dx \leq 1 \right\},\quad y\in \overline{\Omega},$$ i.e. $\frac{1}{c}\tilde\rho(y,V) \leq \rho(y,V)\leq c \tilde\rho(y,V)$ for all $y\in\overline\Omega$ with constant $c$ independent of $y$. When $p=2$, recalling the function $m(y,V)$ defined in [@Sh1 Definition 1.3], we notice that $\tilde \rho(y,V)$ is a local version of $\frac{1}{m(y,V)}$. In view of this observation, it seems that the restriction $(4r)^{p - \frac{n}{\gamma}} \Vert V \Vert_{L^{\gamma}(\Omega_{4r}(x_0))} \leq 1$ in Theorem \[mainthm\] is reasonable.
In Theorem \[mainthm\], we can obtain the estimate uniformly with respect to $x_0$ by taking $r>0$ such that $$r \leq \frac{1}{4}\min\left\{R,\Vert V \Vert_{L^{\gamma}(\Omega)}^{-\frac{\gamma}{p\gamma-n}}\right\},$$ since this together with the fact that $p\gamma>n$ implies $$(4r)^{p - \frac{n}{\gamma}} \Vert V \Vert_{L^{\gamma}(\Omega_{4r}(x_0))} \leq (4r)^{p-\frac{n}{\gamma}} \Vert V \Vert_{L^{\gamma}(\Omega)} \leq 1.$$
As a consequence of Theorem \[mainthm\] and the preceding remark, we obtain the global gradient estimates for solutions to .
\[maincor\] Let $u\in W^{1,p}_0(\Omega)$ be a weak solution to . Suppose that $V \in \mathcal{B}_{\gamma}$ for some $ \gamma \in [ \frac{n}{p}, n)$ when $p<n$ and for some $\gamma \in (1,n)$ when $p\geq n.$ For $p\leq q <\gamma^*(p-1)$, there exists a small $ \delta = \delta(n, p, L, \nu) \in (0,\frac18)$ so that if $\mathbf{a}$ is $( \delta, R)$-vanishing and $\Omega$ is a $( \delta, R)$-Reifenberg flat domain for some $R\in(0,1),$ then we have $$\label{maingloest}
\Vert Du \Vert_{L^{q}(\Omega)} +\chi_{\{q<p\gamma\}} \Vert V^{\frac1p}u \Vert_{L^{q}(\Omega)} \leq c\left(\frac{\mathrm{diam}(\Omega)}{\tilde{R}}\right)^{n\left(\frac{1}{q}-\frac{1}{p}\right)}\Vert F \Vert_{L^{q}(\Omega)}$$ for some $c=c(n,p,q,\gamma,L,\nu,b_{\gamma})>0,$ where $\tilde{R} := \min\left\{ R, \Vert V \Vert_{L^{\gamma}(\Omega)}^{-\frac{1}{p-\frac{n}{\gamma}}}\right\}.$ Here, $\chi_{\{q<p\gamma\}}:= 1$ if $q<p\gamma$ and $\chi_{\{q<p\gamma\}}:= 0$ if $q\geq p\gamma.$
\[mainrmk\] If $V\in \mathcal B_{\gamma}$, then $V$ belongs to the $\mathcal B_{\gamma+\epsilon}$ class for some small $\epsilon>0$ by virtue of the self improving property of the $\mathcal B_\gamma$ class in Lemma \[lemself\] below. Therefore, by considering $\gamma+\epsilon$ instead of $\gamma$ in Theorem \[mainthm\] and Corollary \[maincor\], the range of $q$ can be extended to $p\leq q\leq \gamma^*(p-1)$, and the estimates and can be replaced by $$\begin{aligned}
\left( \mint_{\Omega_{ r}(x_0)} |Du|^q +\chi_{\{q\leq p\gamma\}} \left[V^{\frac1p}|u|\right]^{q}\, dx\right)^{\frac1q} &&\\
&&\hspace{-6cm}\leq c \left( \mint_{\Omega_{4r}(x_0)} |Du|^p + \left[ V^{\frac1p} |u|\right]^p \,dx \right)^{\frac{1}{p}}+ c \left(\mint_{\Omega_{4 r}(x_0)} \left|F \right|^{q} \, dx\right)^{\frac1q}\end{aligned}$$ and $$\Vert Du \Vert_{L^{q}(\Omega)} +\chi_{\{q\leq p\gamma\}} \Vert V^{\frac1p}u \Vert_{L^{q}(\Omega)} \leq c\,\Vert F \Vert_{L^{q}(\Omega)},$$ respectively.
Finally, if the map $x\mapsto \ba(x,\xi)|\xi|^{-(p-1)}$ is in VMO uniformly for the $\xi$-variable, that is, $$\lim_{\rho\to0}\left(\sup_{y\in\mathbb{R}^n} \mint_{B_{\rho}(y) } \left|\Theta\left( \mathbf{a},B_{\rho}(y) \right)(x) \right| \, dx\right)=0,$$ and the boundary of $\Omega$ is $C^1$, we have the implications and for every $q$ in the ranges stated in there.
Under the assumption that $
V\in \mathcal B_{\gamma}\ \ \text{for some }\gamma\in[n,\infty),
$ in stead of $\gamma\in(\frac{n}{p},n),$ we see that the results of Theorem \[mainthm\] and Corollary \[maincor\] hold for any $q\in[p,\infty).$ Indeed, if $
V\in \mathcal B_{\gamma}$ for some $\gamma\in[n,\infty),
$ it is easily seen that $V\in \mathcal B_{\gamma'}$ for any $\gamma'\in (1,\gamma)$ with the constant $b_{\gamma'}=b_{\gamma}$, by the definition of the $\mathcal B_{\gamma}$ class. Then for any $q\in[p,\infty),$ choosing $\gamma' = \gamma' (n,p,q) \in (\frac{n}{p},n)$ such that $$q<(\gamma')^*(p-1),$$ we consequently obtain the results of Theorem \[mainthm\] and Corollary \[maincor\] for any $q\in[p,\infty).$ Hence, we have the implications for $\gamma\in[n,\infty)$ and .
Preliminaries {#Preliminaries}
=============
$\mathcal{B}_\gamma$ class
--------------------------
\
In order to introduce primary features of the $\mathcal{B}_\gamma$ class, let us first recall the Muckenhoupt $A_p$ and $A_\infty$ classes. We say that nonnegative function $V\in L^1_{loc}(\mr^n)$ is in the $A_p$ class, $V\in A_p$, for some $1\leq p<\infty$ if and only if $$\sup_B \left(\mint_BV\, dx\right)\left(\mint_B V^{-\frac{1}{p-1}}\, dx\right)^{p-1}<\infty$$ and that $V\in L^1_{loc}(\mr^n)$ is in the $A_\infty$ class, $V\in A_\infty$, if and only if $$\sup_B \left(\mint_BV\, dx\right)\exp\left(\mint_B \log V^{-1}\,dx\right)<\infty,$$ where the supremum is taken over all balls $B \subset \mr^n.$ From the definition of $\mathcal{B}_{\gamma}$ in , we notice that $V\in \mathcal{B}_{\gamma}$ for $\gamma\in(1,\infty)$ if and only if $$\sup_B\left(\mint_BV\, dx\right)^{-1} \left(\mint_{B}V^\gamma\,dx\right)^{\frac1\gamma}<\infty,$$ where the supremum is taken over all balls $B \subset \mr^n,$ which is very similar to the condition of $A_p$, or $A_\infty$, class. Indeed, we have the following equivalent condition. For its proof, we refer to [@G1 Theorem 9.3.3].
\[lemequiv\]Let $V\in L^1_{loc}(\mr^n)$ be nonnegative. The following are equivalent
- $V\in A_\infty$.
- There exist $\theta,\sigma\in(0,1)$ such that $$\left|\left\{x\in B: V(x)\leq \theta \mint_{B}V\, dy \right\}\right|\leq \sigma|B|$$ for every ball $B$ in $\mr^n$.
- $V\in B_{\gamma}$ for some $\gamma>1$.
- $V\in A_p$ form some $p>1$.
In particular, if $V\in A_\infty$, then there exists $\theta\in(0,1)$ such that $$\left|\left\{x\in B: V(x)\leq \theta \mint_{B}V\, dy \right\}\right|\leq \frac12|B|$$ for every ball $B$ in $\mr^n$, that is, one can choose that $\sigma=\frac12$.
From the above equivalent conditions and the self improving property of the $A_p$ classes, one can deduce the self improving property of the $\mathcal{B}_\gamma$ classes as follows.
\[lemself\] If $V\in \mathcal{B}_\gamma$ for some $\gamma>1$, then $V\in B_{\gamma+\epsilon}$ for some small $\epsilon>0$.
Gradient estimates for equations with mixed data
------------------------------------------------
\
The next two results are local Calderón-Zygmund estimates for elliptic equations of $p$-Laplace type involving mixed data. Let us consider the following problem $$\label{homoeqfF}
\left\{\begin{array}{rclcc}
-\mathrm{div}\, \mathbf{a}(x,Dw)& = &f - \mathrm{div}\,(|F|^{p-2}F) & \textrm{ in } & \Omega_{2r}(x_0), \\
w & = & 0 & \textrm{ on } & \partial_w\Omega_{2r}(x_0)\ \text{if}\ B_{2r}(x_0)\not\subset\Omega.
\end{array}\right.$$ Here, the ‘mixed data’ means $f-\mathrm{div}\,(|F|^{p-2}F)$. We note that if $f\equiv 0$, the Calderón-Zygmund estimates have been obtained in for instance [@BR1; @MP1], and if $F\equiv 0$ and $2-\frac1n <p<n$, these can be found in for instance [@Ph1]. From those results, we can expect a similar result for the mixed problem , and the authors recently obtained the desired one in [@LO1]. By the Sobolev’s embedding, we consider two cases that $q>\max\{p,\frac{(p-1)n}{n-1}\}$ with $1<p<\infty$ and $p<q\leq \frac{(p-1)n}{n-1}$ with $p>n$.
\[thmDwbdd\] Let $1<p<\infty$ and $q>\max\{p,\frac{(p-1)n}{n-1}\}$. There exists a small $ \delta = \delta(n, L, \nu, p, q) \in (0,\frac18) $ so that if $\mathbf{a}$ is $( \delta, R)$-vanishing and $\Omega$ is a $(\delta,R)$-Reifenberg flat domain for some $R\in(0,1)$, then for any $x_0\in\overline\Omega$, $r\in(0,\frac{R}{2}]$ and weak solution $w\in W^{1,p}(\Omega_{2r}(x_0))$ of with $F\in L^q(\Omega_{2r}(x_0))$ and $f\in L^{(q/(p-1))_*}(\Omega_{2r}(x_0))$, we have
$$\begin{aligned}
\label{homoeqestimate}
\nonumber \left(\mint_{\Omega_r(x_0)} |Du|^{q} \, dx\right)^{\frac{1}{q}}
\nonumber &\leq& c \left( \mint_{\Omega_{2r}(x_0)} |Du|^{p}\, dx\right)^{\frac{1}{p}}+c\left(\mint_{\Omega_{2r}(x_0)} |F|^q \,dx\right)^\frac{1}{q} \\
&&+ c \left( \mint_{\Omega_{2r}(x_0)} |r f |^{\left( \frac{q}{p-1}\right)_*} \, dx \right)^{\frac{1}{\left( \frac{q}{p-1}\right)_*(p-1)}}\end{aligned}$$
for some $c=c(n, L,\nu, p,q)>0.$
\[thmDwbdd1\] Let $n<p<\infty$, $p<q\leq \frac{(p-1)n}{n-1}$ and $1<\tilde q <n$. There exists a small $\delta = \delta(n, L, \nu, p, q)\in (0,\frac18) $ so that if $\mathbf{a}$ is $( \delta, R)$-vanishing, $\Omega$ is a $(\delta,R)$-Reifenberg flat for some $R\in(0,1)$, then for any $x_0\in\overline\Omega$, $r\in(0,\frac{R}{2}]$ and for any weak solution $w\in W^{1,p}(\Omega_{2r}(x_0))$ of with $F\in L^q(\Omega_{2r}(x_0))$ and $f\in L^{\tilde q}(\Omega_{2r}(x_0))$, we have $$\begin{aligned}
\left(\mint_{\Omega_r(x_0)} |Dw|^{q} \, dx\right)^{\frac{1}{q}} &\leq& c \left( \mint_{\Omega_{2r}(x_0)} |Dw|^{p}\, dx\right)^{\frac{1}{p}}+c\left(\mint_{\Omega_{2r}(x_0)} |F|^q \,dx\right)^\frac{1}{q} \\
&&+ c \left( \mint_{\Omega_{2r}(x_0)} |r f |^{\tilde q} \, dx \right)^{\frac{1}{\tilde q(p-1)}}\end{aligned}$$ for some constant $c=c(n, L,\nu, p,q, \tilde q)>0.$
Auxiliary lemmas
----------------
\
We first recall the local boundedness (up to boundaries) for weak solutions to the equation with $F\equiv 0$, which is a classical regularity result and we refer to [@LU Chapter 2.5] and [@Gi Chapter 7]. We point out that Reifenberg flat domains $\Omega$ considered in this paper have the measure density conditions and , which are enough to obtain the boundedness for weak solutions.
\[supvlem\] Let $1<p<\infty$ and suppose that the bounded domain $\Omega\subset\mr^n$ is $(\delta,R)$-Reifenberg flat for some $\delta\in(0,1/2)$ and $R>0$. Assume that $\mathbf{a}$ satisfies $$\label{aAss}
|\ba(x,\xi)|\leq L|\xi|^{p-1} \ \ \text{and}\ \ \ba(x,\xi)\cdot \xi\geq \nu |\xi|^p$$ for any $x,\xi\in \mr^n$ and for some $0<\nu\leq L$, and that the nonnegative function $V$ satisfies $V \in L^{\gamma}(\Omega)$ for some $\gamma \in (\frac{n}{p}, n)$ when $p<n$ and for some $\gamma>1$ when $p\geq n.$ Then for any ball $B_{2r}(x_0)$ with $x_0\in\overline\Omega$ and $r\in(0,\frac{R}2]$ satisfying $ (2r)^{p - \frac{n}{\gamma}} \Vert V \Vert_{L^{\gamma}(\Omega_{2r}(x_0))} \leq 1,$ and for any weak solution $w\in W^{1,p}(\Omega_{2r}(x_0))$ of $$\left\{\begin{array}{rclcc}
-\mathrm{div}\, \mathbf{a}(x,Dw) + V |w|^{p-2}w& = &0 & \textrm{ in } & \Omega_{2r}(x_0), \\
w & = & 0 & \textrm{ on } & \partial_w\Omega_{2r}(x_0)\ \text{if}\ B_{2r}(x_0)\not\subset\Omega,\end{array}\right.$$ we have that $$\|w\|_{L^\infty(\Omega_r (x_0))} \leq c \left( \mint_{\Omega_{2r}(x_0)} |w|^p \, dx \right)^{\frac{1}{p}}$$ for some constant $c = c(n, p, L, \nu,\gamma) >0.$
Let us define the rescaled maps $$\tilde{\mathbf{a}}(x,\xi) = \mathbf{a}(rx, \xi),\ \tilde{w}(x) = \frac{w(rx)}{r}, \ \tilde{V}(x) = r^p V(rx), \text{ and } \tilde{\Omega} = \left\{ \frac{x}{r} : x \in \Omega \right\}.$$ Then one can check that $ \tilde{\mathbf{a}}$ satisfies the assumption with the same constants $L$ and $\nu$, $\tilde{\Omega}$ is $(\delta,\frac{R}{r})$-Reifenberg flat, $\tilde{V} \in L^{\gamma}(\tilde{\Omega})$, and $\tilde{w} \in W^{1,p}(\Omega_{2}(x_0)) $ is a weak solution of $$\left\{\begin{array}{rclcc}
-\mathrm{div}\, \tilde{\mathbf{a}}(x,D\tilde{w}) + \tilde{V} |\tilde{w}|^{p-2}\tilde{w}& = &0 & \textrm{ in } & \tilde{\Omega}_{2}(x_0), \\
\tilde{w} & = & 0 & \textrm{ on } & \partial_w\tilde{\Omega}_{2}(x_0)\ \text{if}\ B_{2}(x_0)\not\subset\tilde{\Omega}. \end{array}\right.$$ By the classical local boundedness result (see, for instance, [@LU Chapter 2.5] and [@Gi Chapter 7]), we see that $$\|\tilde{w}\|_{L^\infty(\tilde{\Omega}_1 (x_0))} \leq c \left( \mint_{\tilde{\Omega}_{2}(x_0)} |\tilde{w}|^p \, dx \right)^{\frac{1}{p}}$$ for some constant $c = c(n, p, L, \nu,\gamma, \Vert \tilde{V} \Vert_{L^{\gamma}(\tilde{\Omega}_{2}(x_0))}) >0.$ Here, since the constant $c$ in the above estimate is increasing as a function of $\Vert \tilde{V} \Vert_{L^{\gamma}(\tilde{\Omega}_{2}(x_0))}$ and $$\Vert \tilde{V} \Vert_{L^{\gamma}(\tilde{\Omega}_{2}(x_0))} \leq (2r)^{p - \frac{n}{\gamma}} \Vert V \Vert_{L^{\gamma}(\Omega_{2r}(x_0))} \leq 1,$$ $c$ can be replaced by a larger constant independent of $\Vert \tilde{V} \Vert_{L^{\gamma}(\tilde{\Omega}_{2}(x_0))}$. Therefore, after scaling back, we can arrive at the desired bound of $w$.
The following is the standard iteration lemma, whose proof can be found in for instance [@HL1].
\[teclem\] Let $ g :[a,b] \to \mr $ be a bounded nonnegative function. Suppose that for any $s_1,s_2$ with $ 0< a \leq s_1 < s_2 \leq b $, $$g(s_1) \leq \tau g(s_2) + \frac{A}{(s_2-s_1)^{\beta}}+B$$ where $A,B \geq 0, \beta >0$ and $0\leq \tau <1$. Then we have $$g(s_1) \leq c\left( \frac{A}{(s_2-s_1)^{\beta}}+ B \right)$$ for some constant $c=c(\beta, \tau) >0$.
We end this section by introducing a basic inequality which will be used later. Although its proof is elementary, we shall give it in detail for the sake of readability.
\[lemineq\] For any function $g\in W^{1,p}(B_r)$ with any $r>0,$ we have $$\frac{1}{r^{n+p}}\int_{B_r} \int_{B_r} \left| g(x)-g(y) \right|^p \,dx dy \leq c \int_{B_r} \left| Dg(x) \right|^p \,dx$$ for some $c=c(n,p)>0$.
Without loss of generality, we shall assume that $g\in C^1(B_r).$ Using Hölder’s inequality, Fubini’s theorem and the fact that $|x-y|\leq 2r,$ we observe that $$\begin{aligned}
\int_{B_r} \int_{B_r} \left| g(x)-g(y) \right|^p \,dx dy &=& \int_{B_r} \int_{B_r} \left| \int_0^1Dg(t(x-y)+y)\cdot (x-y)\,dt \right|^p\, dxdy\\
&\leq & (2r)^p\int_0^1 \int_{B_r} \int_{B_r} |Dg(t(x-y)+y)|^p\, dx dy dt.\end{aligned}$$ Here we point out that $t(x-y)+y\in B_r$ for any $x,y\in B_r$. Then we use a change of variable with $x=\tilde x+y$ and apply Fubini’s theorem to obtain that $$\begin{aligned}
\int_{B_r} \int_{B_r} \left| g(x)-g(y) \right|^p \,dx dy &\leq & (2r)^p\int_0^1 \int_{B_r} \int_{B_r(-y)} |Dg(t\tilde x+y)|^p\, d\tilde x dy dt\\
&\leq & (2r)^p\int_0^1 \int_{B_{2r}} \int_{B_{r-\frac{|\tilde x|}{2}}(-\frac{\tilde x}{2})} |Dg(t\tilde x+y)|^p\, dy d\tilde x dt.\end{aligned}$$ Note that $B_{r-\frac{|\tilde x|}{2}}(t\tilde x-\frac{\tilde x}{2})\subset B_r$ for any $\tilde x\in B_{2r}$ and any $t\in (0,1).$ Hence, by letting $\tilde y=t\tilde x+y$, we have $$\int_{B_{r-\frac{|\tilde x|}{2}}(-\frac{\tilde x}{2})} |Dg(t\tilde x+y)|^p\, dy= \int_{B_{r-\frac{|\tilde x|}{2}}(t\tilde x-\frac{\tilde x}{2})} |Dg(\tilde y)|^p\, d\tilde y \leq \int_{B_{r}} |Dg(\tilde y)|^p\, d\tilde y,$$ which implies that $$\int_{B_r} \int_{B_r} \left| g(x)-g(y) \right|^p \,dx\, dy \leq (2r)^{n+p}|B_1|\int_{B_{r}} |Dg(\tilde y)|^p\, d\tilde y.$$ This completes the proof.
Gradient estimates for homogenous equations {#sechomo}
===========================================
In this section we obtain gradient estimates for weak solutions to localized equations of with $F\equiv 0$. Let us start with the following lemma, which is in fact a key lemma in our proofs.
\[lemrVbddpq\] Let $1 < p < \infty$ and suppose $V \in \mathcal{B}_{\gamma}$ for some $ \gamma \in [ \frac{n}{p}, n)$ when $p<n$ and for some $\gamma \in (1,n)$ when $p\geq n.$ Then for any function $w \in W^{1,p}(B_{r})$ with $0<r<1,$ we have $$\begin{aligned}
\label{prVbddpq}
\nonumber \left( \mint_{B_{r}}\left| \frac{w}{r}\right|^{p} \, dx\right)^{\frac1p} \left( \mint_{B_{r}} [r^pV]^{\gamma}\, dx\right)^{\frac{1}{p\gamma }} &\leq& c\max\left\{\left( \mint_{B_{r}} [r^pV]^{\gamma}\, dx\right)^{\frac{1}{p\gamma }},1\right\} \\
&&\hspace{-1.5cm}\times\, \left[ \left( \mint_{B_{r}} \left| Dw \right|^p \,dx\right)^{\frac1p} + \left(\mint_{B_{r}} V\left| w \right|^p \,dx\right)^{\frac1p} \right]\end{aligned}$$
for some constant $c=c\left(n,p, b_\gamma \right)>0.$
By Lemma \[lemineq\], we have $$\int_{B_r} \left| Dw(x) \right|^p \,dx \geq \frac{c}{r^{n+p}}\int_{B_r} \int_{B_r} \left| w(x)-w(y) \right|^p \,dxdy.$$ Moreover, we also have $$\int_{B_r} V(x) \left| w(x) \right|^p \,dx = \frac{1}{r^{n}|B_1|}\int_{B_r} \int_{B_r} V(y) \left| w(y) \right|^p\,dxdy.$$ Then we have that for any constant $c_0>0$, $$\begin{aligned}
\label{pDwVwc0}
\nonumber&&\int_{B_r} \left| Dw(x) \right|^p \,dx + \int_{B_r} V(x)\left| w(x) \right|^p \,dx \\
\nonumber&&\qquad \geq \frac{c}{\max\{c_0,1\}\,r^{n}} \bigg( \int_{B_r}\int_{B_r} \frac{c_0\left| w(x)-w(y) \right|^p}{r^p}\,dy dx \\
&& \hspace{4.5cm} + \int_{B_r} \int_{B_r} V(y)\left| w(y) \right|^p \,dydx\bigg).\end{aligned}$$ Note that it is easily seen that $$\begin{aligned}
&&\frac{c_0\left| w(x)-w(y) \right|^p}{r^p} + V(y)|w(y)|^p \\
&&\quad \geq \min\left\{\frac{c_0}{r^p},V(y)\right\} \left( \left| w(x)-w(y) \right|^p\ + |w(y)|^p \right) \geq \min\left\{\frac{c_0}{r^p},V(y)\right\} \frac{|w(x)|^p}{2^{p-1}} .\end{aligned}$$ Hence, inserting this into , we obtain $$\begin{aligned}
&&\int_{B_r} \left| Dw(x) \right|^p \,dx + \int_{B_r} \left| w(x) \right|^p V(x) \,dx \\
&&\qquad \geq \frac{c}{\max\{ c_0,1\}\,r^n} \int_{B_r} \left(\int_{B_r} \min_{y \in B_r}\left\{ \frac{c_0}{r^p} , V(y)\right\} \,dy\right)\left| w(x)\right|^p\,dx.\end{aligned}$$
On the other hand, since $V \in \mathcal{B}_\gamma,$ by Lemma \[lemequiv\] there exists $\theta>0$ such that $$\left| \left\{ x \in B : V(x) \geq \theta \mint_{B} V(y)\,dy \right\} \right| \geq \frac12 |B|$$ for every ball $B \subset \mr^n.$ Then we take $$c_0:= \theta\, r^p \mint_{B_r} V(y) dy$$ to discover that $$\int_{B_r} \min_{y \in B_r}\left\{ \frac{c_0}{r^p} , V(y)\right\} dy \geq \frac{c_0}{2r^p} |B_r| = \frac{c_0\, r^{n-p}|B_1|}{2}.$$ Therefore we get $$\begin{aligned}
\frac{c_0r^{-p}}{\max\{ c_0,1\}} \int_{B_r} |w|^p\,dx &\leq& \frac{c}{\max\{ c_0,1\}\,r^n} \int_{B_r} \int_{B_r} \min_{y \in B_r}\left\{ \frac{c_0}{r^p} , V(y)\right\} \left| w(x)\right|^p \,dydx
\\
&\leq& c\left(\int_{B_r} \left| Dw \right|^p \,dx + \int_{B_r}V \left| w \right|^p \,dx\right),\end{aligned}$$ which implies that $$\label{lem41pf1}
\frac{c_0}{\max\{ c_0,1\}} \mint_{B_r} \left|\frac{w}{r} \right|^{p} dx \leq c \left( \mint_{B_r} \left| Dw \right|^p \,dx + \mint_{B_r} V\left| w \right|^p \,dx\right) .$$
At this stage, if $c_0 < 1,$ we see that $$\begin{aligned}
\nonumber \left( \mint_{B_r} r^p\,V \, dx \right)^{\frac1p} \left( \mint_{B_r} \left| \frac{w}{r}\right|^{p} dx\right)^{\frac1p} &= & \left( c_0\mint_{B_r} \left| \frac{w}{r}\right|^{p} dx\right)^{\frac1p} \\
&& \hspace{-2cm}\leq\ c\left[ \left( \mint_{B_r} \left| Dw \right|^p \,dx\right)^{\frac1p} + \left(\mint_{B_r} V \left| w \right|^p \,dx\right)^{\frac1p} \right].\end{aligned}$$ Using this and the fact $V\in \mathcal{B}_\gamma$, we have $$\begin{aligned}
\label{pc0>1}
\nonumber\left( \mint_{B_{r}}\left| \frac{w}{r}\right|^{p} \,dx\right)^{\frac1p} \left( \mint_{B_{r}} [r^pV]^{\gamma}\,dx\right)^{\frac{1}{p\gamma }} &&\\
\nonumber&&\hspace{-6.3cm} \leq b_{\gamma}^{\frac1p} \left( \mint_{B_{r}}\left| \frac{w}{r}\right|^{p} dx \right)^{\frac1p} \left( \mint_{B_{r}}r^p V \, dx \right)^{\frac1p}\\
&&\hspace{-6.3cm} \leq c b_{\gamma}^{\frac1p} \left[ \left( \mint_{B_{r}} \left| Dw \right|^p \,dx\right)^{\frac1p} + \left(\mint_{B_{r}} V \left| w \right|^p \,dx\right)^{\frac1p} \right].
\end{aligned}$$ Otherwise, that is, if $c_0 \geq 1,$ we see from that $$\label{pc0<1}
\left( \mint_{B_r} \left| \frac{w}{r}\right|^{p} dx\right)^{\frac1p} \leq c \left[ \left( \mint_{B_r} \left| Dw \right|^p \,dx\right)^{\frac1p} + \left(\mint_{B_r} V \left| w \right|^p\,dx\right)^{\frac1p}\right].$$
Then combining and , we finally obtain the desired estimate .
Now, let us consider a weak solution $w\in W^{1,p}(\Omega_{4r}(x_0))$ of $$\label{homoeq1}
\left\{\begin{array}{rclcc}
-\mathrm{div}\, \mathbf{a}(x,Dw) + V |w|^{p-2}w& = &0 & \textrm{ in } & \Omega_{4r}(x_0), \\
w & = & 0 & \textrm{ on } & \partial_w\Omega_{4r}(x_0)\ \text{if}\ B_{4r}(x_0)\not\subset\Omega,\end{array}\right.$$ and then we can obtain its gradient estimates as follows.
\[lem42\] Let $1 < p < \infty$, and suppose that $\ba:\mr^n\times\mr^n\to\mr^n$ satisfies and and $V \in \mathcal{B}_{\gamma}$ for some $ \gamma \in [ \frac{n}{p}, n)$ when $p<n$ and for some $\gamma \in (1,n)$ when $p\geq n.$ There exists a small $\delta = \delta(n, p, \gamma, \Lambda, \nu) > 0 $ so that if $\mathbf{a}$ is $( \delta, R)$-vanishing and $\Omega$ is $(\delta,R)$-Reifenberg flat for some $R\in(0,1)$, then for any $x_0\in\overline\Omega$, $r\in(0,\frac{R}{4}]$ satisfying $ (4r)^{p - \frac{n}{\gamma}} \Vert V \Vert_{L^{\gamma}(\Omega_{4r}(x_0))} \leq 1,$ and for any weak solution $w\in W^{1,p}(\Omega_{4r}(x_0))$ of we have $|Dw|\in L^{\gamma^*(p-1)}(\Omega_r(x_0))$ with the estimate $$\begin{aligned}
\label{DwDwVwest}
\nonumber&&\left( \mint_{{\Omega}_r(x_0)} \left| D{w}\right|^{\gamma^* (p-1)} \, dx \right)^{\frac{1}{\gamma^* (p-1)}}\\
&&\qquad \leq c \left( \mint_{\Omega_{4r}(x_0)} \left| Dw \right|^p \,dx\right)^{\frac1p} + c \left(\mint_{\Omega_{4r}(x_0)} V\left| w \right|^p \,dx\right)^{\frac1p}.\end{aligned}$$ Moreover, we have $V^{\frac1p}|w|\in L^{p\gamma }(\Omega_r(x_0))$ with the estimate $$\begin{aligned}
\label{DwDwVwest1}
\nonumber&&\left( \mint_{{\Omega}_r(x_0)} \left[ V^\frac{1}{p}|w|\right]^{p\gamma } \, dx \right)^{\frac{1}{p\gamma }}\\
&&\qquad \leq c\left( \mint_{\Omega_{4r}(x_0)} \left| Dw \right|^p \,dx\right)^{\frac1p} + c \left(\mint_{\Omega_{4r}(x_0)} V\left| w \right|^p \,dx\right)^{\frac1p}.\end{aligned}$$ Here, the constants $c>0$ in the above estimates depend on $n,p,\gamma,\nu,L$ and $b_\gamma$.
For simplicity we shall denote $\Omega_{\rho} :=\Omega_{\rho}(x_0)$ and $B_{\rho} := B_{\rho}(x_0)$ for any $\rho>0$ in this proof. We first observe that, in view of Lemma \[supvlem\] with $r$ replaced by $2r$, $$\label{lem42pf1}
\|w\|_{L^\infty(\Omega_{2r})} \leq c \left( \mint_{\Omega_{4r}} |w|^p \, dx \right)^{\frac{1}{p}}.$$ Then from the fact $V\in L^\gamma(\Omega),$ we see that $V|w|^{p-2}w\in L^\gamma(\Omega_{2r})$. Therefore, applying Theorem \[thmDwbdd\] with $q=\gamma^*(p-1)$, $f=V|w|^{p-2}w$ and $F=0$, we have $$\begin{aligned}
\label{DwDwrVwes}
\nonumber&& \left(\mint_{\Omega_r} |Dw|^{\gamma^*(p-1)} \, dx\right)^{\frac{1}{\gamma^*(p-1)}} \\
&&\qquad \leq c \left( \mint_{\Omega_{2r}} |Dw|^{p}\, dx\right)^{\frac{1}{p}}+ c \left( \mint_{\Omega_{2r}} \left[r V|w|^{p-1} \right]^\gamma \, dx\right)^{\frac{1}{\gamma(p-1)}}.\end{aligned}$$ We now estimate the last term on the right hand side in the previous inequality. Using and with the assumption $ (4r)^{p - \frac{n}{\gamma}} \Vert V \Vert_{L^{\gamma}(\Omega_{4r})} \leq 1,$ we have $$\begin{aligned}
\label{rVwDwVwes}
\nonumber \left( \mint_{\Omega_{2r}} \left[ r V |w|^{p-1}\right]^{\gamma} dx\right)^{\frac{1}{\gamma(p-1)}}
\nonumber& \leq& \frac{ \|w\|_{L^\infty(\Omega_{2r})}}{r} \leq c \left( \mint_{\Omega_{4r}}\left| \frac{w}{r}\right|^{p} \,dx\right)^{\frac1p} \\
& \leq& c \left[ \left( \mint_{\Omega_{4r}} \left| Dw \right|^p \,dx\right)^{\frac1p} + \left(\mint_{\Omega_{4r}} V\left| w \right|^p \,dx\right)^{\frac1p} \right].
\end{aligned}$$ Here, we let $w\equiv 0$ in $B_{4r}\setminus \Omega$ and have used . Hence, inserting into , we obtain . In the same way as , we can derive .
Comparison estimates {#secComestimates}
====================
In this section, we shall derive comparison estimates between the weak solution to and weak solutions to localized equations of with $F\equiv0.$
\[comestlem\] Let $1 < p < \infty$, and suppose that $\mathbf{a}:\mr^n\times\mr^n\to\mr^n$ satisfies -. For any $\epsilon \in (0,1),$ there exists a small $ \delta = \delta( \epsilon, n, p, L, \nu) \in(0,1) $ such that if $u \in W^{1,p}(\Omega)$ is the weak solution to with $$\label{lDuurbdd}
\left( \mint_{\Omega_{4r}} \left[|Du| + V^\frac{1}{p}|u|\right]^p \,dx\right)^{\frac{1}{p}} <\lambda$$ and $$\label{lFbdd}
\left( \mint_{\Omega_{4r}} |F|^p \,dx\right)^{\frac{1}{p}}<\delta\, \lambda$$ for some $r>0$ and $\lambda>0$, then we have $$\label{lDumDvibdd}
\mint_{\Omega_{4r}} |Du-Dw|^{p} + V |u-w|^{p} \, dx \leq \epsilon \lambda^p,$$ where $w \in W^{1,p}(\Omega_{4r})$ is the unique weak solution to $$\label{lhomoeq}
\left\{\begin{array}{rclcc}
-\mathrm{div}\, \mathbf{a}(x,Dw) + V |w|^{p-2}w& = & 0 & \textrm{ in } & \Omega_{4r}, \\
w & = & u & \textrm{ on } & \partial \Omega_{4r}.
\end{array}\right.$$
We first test the equations with the testing function $\varphi=w-u$ in order to discover $$\int_{\Omega_{4r}} \mathbf{a}(x, Dw)\cdot ( Dw-Du)\, dx + \int_{\Omega_{4r}} V |w|^{p-2}w \cdot (w-u)\, dx= 0,$$ and then, in view of and , we obtain $$\int_{\Omega_{4r}} |Dw|^p\, dx + \int_{\Omega_{4r}} V |w|^p\, dx\leq c\int_{\Omega_{4r}} |Dw|^{p-1}|Du|\, dx + c \int_{\Omega_{4r}} V |w|^{p-1}|u|\, dx.$$ Therefore, applying Young’s inequality and we have $$\label{lem51pf1}
\left(\int_{\Omega_{4r}} |Dw|^p\, dx \right)^{\frac1p}+\left( \int_{\Omega_{4r}} V |w|^p\, dx\right)^{\frac1p}\leq c\lambda.$$
We next test the equations and with the testing function $\varphi=u-w$ in order to discover $$\begin{aligned}
\nonumber &&\int_{\Omega_{4r}} \left( \mathbf{a}(x, Du) - \mathbf{a}(x, Dw) \right) \cdot ( Du-Dw)\, dx \\
&& \quad+ \int_{\Omega_{4r}} V \left( |u|^{p-2}u - |w|^{p-2}w \right) \cdot (u-w)\, dx= \int_{\Omega_{4r}} |F|^{p-2}F \cdot (Du-Dw)\, dx.\end{aligned}$$ By virtue of the monotonicity condition , we derive that $$\begin{aligned}
&&\mint_{\Omega_{4r}}\left( |Du|^{2}+ |Dw|^2 \right)^{\frac{p-2}{2}} | Du-Dw|^2\, dx \\
&&\qquad +\mint_{\Omega_{4r}}V \left( |u|^{2}+ |w|^2 \right)^{\frac{p-2}{2}} | u-w|^2\, dx \leq c \mint_{\Omega_{4r}} |F|^{p-1}|Du-Dw|\, dx.\end{aligned}$$ Note that if $p\geq 2$, by we have $$\mint_{\Omega_{4r}}|Du-Dw|^p\, dx +\mint_{\Omega_{4r}}V |u-w|^p\, dx \leq c \mint_{\Omega_{4r}} |F|^{p-1}|Du-Dw|\, dx.$$ On the other hand, if $1<p<2$, then by Young’s inequality we have $$\begin{aligned}
|Du-Dw|^p &=& |Du-Dw|^p \left( |Du|^2 + |Dw|^2 \right)^{\frac{p(p-2)}{4}} \left( |Du|^2 + |Dw|^2 \right)^{\frac{p(2-p)}{4}}\\
&\leq& \kappa \left( |Du|^2 + |Dw|^2 \right)^{\frac{p}{2}} +c(\kappa) \left( |Du|^2 + |Dw|^2 \right)^{\frac{p-2}{2}} |Du-Dw|^2
\end{aligned}$$ and $$V |u-w|^p \leq \kappa V \left( |u|^2 + |w|^2 \right)^{\frac{p}{2}} +c(\kappa) V \left( |u|^2 + |w|^2 \right)^{\frac{p-2}{2}} |u-w|^2$$ for any small $\kappa>0.$ Therefore, combining the above results with , we have that $$\begin{aligned}
\nonumber&&\mint_{\Omega_{4r}} |Du-Dw|^{p} \, dx+\mint_{\Omega_{4r}} V|u-w|^{p} \, dx \\
\nonumber&&\quad \leq \kappa \mint_{\Omega_{4r}} (|Du|^2+|Dw|^2)^\frac{p}{2} \, dx +\kappa \mint_{\Omega_{4r}} V (|u|^2+|w|^2)^\frac{p}{2} \, dx\\\
\nonumber&&\quad \quad+ c(\kappa) \mint_{\Omega_{4r}} \left( |Du|^2 + |Dw|^2 \right)^{\frac{p-2}{2}} |Du-Dw|^{2} \, dx\\
\nonumber&&\quad \quad+ c(\kappa) \mint_{\Omega_{4r}} V\left( |u|^2 + |w|^2 \right)^{\frac{p-2}{2}} |u-w|^{2} \, dx \\
\nonumber&&\quad \leq c_1 \kappa \, \lambda^p+ c(\kappa) \mint_{\Omega_{4 r}} |F|^{p-1}|Du-Dw|\, dx\end{aligned}$$ for some $c_1=c_1(n,p,L,\nu)>0$ and $c(\kappa)= c(\kappa,n,p,L,\nu)\geq1$.
Therefore, in any case, we obtain $$\begin{aligned}
\nonumber&&\mint_{\Omega_{4r}} |Du-Dw|^{p} \, dx+\mint_{\Omega_{4r}} V \left|u-w\right|^{p} \, dx\\
&& \qquad\leq c_1\kappa \, \lambda^p+ c(\kappa,\tau)\mint_{\Omega_{4r}} |F|^{p}\, dx + c(\kappa)\,\tau \mint_{\Omega_{4r}} |Du-Dw|^p\, dx \\
& & \qquad \leq c_1\kappa \, \lambda^p+ c(\kappa,\tau)\delta^p \,\lambda^p + c(\kappa)\,\tau \mint_{\Omega_{4r}}|Du-Dw|^p\, dx\end{aligned}$$ for any small $\kappa,\tau>0$ and for some $c(\kappa,\tau)=c(\kappa,\tau,n,p,L,\nu)\geq 1$. Here, we have used Young’s inequality and . Taking $\kappa$, $\tau$ and $\delta$ sufficiently small such that $$\kappa=\frac{\epsilon}{4c_1},\ \ \ \tau = \frac{1}{2c(\kappa)}\ \ \ \text{and}\ \ \ \delta =\left(\frac{\epsilon}{4c(\kappa,\tau)}\right)^{\frac1p},$$ we finally obtain .
We notice that $\gamma^*(p-1)>\max\{p,\frac{n(p-1)}{n-1}\}$. Therefore, applying the results in Lemma \[lem42\] to the weak solution $w$ of in the previous lemma, we obtain the following gradient estimates.
\[comestlem1\] Let $1 < p < \infty$, and suppose that $\mathbf{a}:\mr^n\times\mr^n\to\mr^n$ satisfies - and $V \in \mathcal{B}_\gamma$ for some $\gamma \in [ \frac{n}{p}, n)$ when $p<n$ and for some $\gamma \in (1,n)$ when $p\geq n.$ There exists a small $ \delta = \delta(n, p, \gamma, L, \nu) > 0 $ such that if $\mathbf{a}$ is $( \delta, R)$-vanishing, $\Omega$ is $( \delta, R)$-Reifenberg flat and $u \in W^{1,p}(\Omega)$ is the weak solution to with and for some $r\in(0,\frac R4]$ satisfying $ (4r)^{p - \frac{n}{\gamma}} \Vert V \Vert_{L^{\gamma}(\Omega_{4r})} \leq 1$ and $\lambda>0$, then we have $$\left( \mint_{\Omega_{r}} \left| Dw\right|^{\gamma^* (p-1)} \, dx \right)^{\frac{1}{\gamma^* (p-1)}} \leq c\, \lambda$$ and $$\left( \mint_{\Omega_{r}} \left[V^{\frac1p} |w|\right]^{p\gamma } \, dx \right)^{\frac{1}{p\gamma }} \leq c\, \lambda$$ for some $c=c(n,p, \gamma, L, \nu, b_\gamma)>0,$ where $w$ is the unique weak solution to .
The estimates above directly follow from , and .
$L^q$-estimates {#sec gradient estimates}
===============
Now we are ready to prove our main results, Theorem \[mainthm\] and Corollary \[maincor\]. As we mentioned in the introduction, we employ so-called an exit-time argument introduced by Mingione in [@AM1; @Min1].
Proof of Theorem \[mainthm\]
----------------------------
\
Assume that $\ba:\mr^n\times \mr^n\to\mr^n$ is $(\delta,R)$-vanishing and $\Omega$ is $(\delta,R)$-Reifenberg flat for some $R\in(0,1),$ where $\delta\in(0,1)$ will be chosen sufficiently small later. Now, we prove the estimate . Fix any $x_0\in\overline{\Omega}$ and $r>0$ satisfying $r \leq \frac{R}{4}$ and $(4r)^{p - \frac{n}{\gamma}} \Vert V \Vert_{L^{\gamma}(\Omega_{4r}(x_0))} \leq 1.$ Note that $$\rho^{p - \frac{n}{\gamma}} \Vert V \Vert_{L^{\gamma}(\Omega_{\rho}(y))}\leq 1$$ for any $B_\rho(y)\subset B_{4r}(x_0)$ with $y\in B_{4r}(x_0)\cap \overline{\Omega}.$ For the sake of simplicity, we shall write $\Omega_{\rho}:=\Omega_{\rho}(x_0)$, $\rho>0$. Also, we define $$\label{Phi}
\Phi(u,V):= |Du|+V^\frac{1}{p}|u|,$$ and for $\lambda,\rho>0$ $$E(\lambda, \rho) := \{x \in \Omega_{\rho} : \Phi(u,V)(x)>\lambda \}.$$
The proof goes in five steps.
*Step 1. Covering argument.* \
Fix any $s_1, s_2$ with $1\leq s_1 < s_2 \leq 2.$ Then we have $ \Omega_{r}\subset \Omega_{s_1 r } \subset \Omega_{s_2 r } \subset \Omega_{2 r } $. We define $$\label{lambda0}
\lambda_0:= \left( \mint_{\Omega_{2r}} \left[\Phi(u,V)\right]^p \,dx\right)^{\frac1p}+\frac{1}{\delta} \left(
\mint_{\Omega_{2r}} |F|^p \,dx \right)^{\frac1p},$$ and consider $\lambda>0$ large enough so that $$\label{lambdarg}
\lambda > A\, \lambda_0,\ \ \textrm{where } A : = \left( \frac{16}{7}\right)^{\frac{n}{p}}\left( \frac{40}{s_2 - s_1}\right)^{\frac{n}{p}}.$$ We note that $ \Omega_{\rho}(y) \subset \Omega_{2r}$ for any $y \in E(\lambda, s_1 r)$ and any $ \rho \in \left( 0, (s_2-s_1)\,r \right].$ By virtue of the measure density condition and the definition of $\lambda_0$ in , we then deduce that $$\begin{aligned}
&&\left(\mint_{\Omega_{\rho}(y)} \left[\Phi(u,V)\right]^p \,dx\right)^\frac1p+
\frac{1}{\delta}\left( \mint_{\Omega_{\rho}(y)} |F|^p \,dx\right)^\frac1p\\
&& \leq \left(\frac{|\Omega_{2r}|}{|\Omega_{\rho}(y)|}\right)^{\frac{1}{p}} \lambda_0 \leq \left( \frac{16}{7}\right)^{\frac np} \left( \frac{2r}{\rho}\right)^{\frac np}\, \lambda_0 \leq A\, \lambda_0 < \lambda,\end{aligned}$$ provided that $$\frac{(s_2 - s_1)\,r}{20} \leq \rho \leq (s_2-s_1)\,r.$$
On the other hand, Lebesgue’s differentiation theorem yields that for almost every $y \in E(\lambda, s_1 r),$ $$\lim_{\rho \rightarrow 0} \left[\left(\mint_{\Omega_{\rho}(y)} \left[\Phi(u,V)\right]^p \,dx \right)^{\frac1p}+\frac{1}{\delta} \left(
\mint_{\Omega_{\rho}(y)} |F|^p \,dx\right)^{\frac1p}\right] > \lambda.$$ Therefore the continuity of the integral implies that for almost every $y \in E(\lambda, s_1 r),$ there exists $$\rho_{y} = \rho(y) \in \left( 0, \frac{(s_2-s_1)\,r}{20}\right)$$ such that $$\left(\mint_{\Omega_{\rho_y}(y)} \left[\Phi(u,V)\right]^p \,dx \right)^{\frac1p}+\frac{1}{\delta} \left(
\mint_{\Omega_{\rho_y}(y)} |F|^p \,dx\right)^{\frac1p} = \lambda$$ and, for any $ \rho \in ( \rho_y, (s_2-s_1)r],$ $$\left(\mint_{\Omega_{\rho}(y)} \left[\Phi(u,V)\right]^p \,dx \right)^{\frac1p}+\frac{1}{\delta} \left(
\mint_{\Omega_{\rho}(y)} |F|^p \,dx\right)^{\frac1p} <\lambda.$$
Applying Vitali’s covering theorem, we have the following:
\[coveringlem\] Given $\lambda > A\, \lambda_0,$ there exists a disjoint family of $\{ \Omega_{\rho_i}(y^i)\}_{i=1}^{\infty}$ with $y^i \in E(\lambda, s_1 r)$ and $\rho_{i} \in \left(0, \frac{(s_2-s_1)\,r}{20} \right)$ such that $$E(\lambda, s_1 r) \subset \bigcup_{i=1}^{\infty} \Omega_{5\rho_i}(y^i),$$ $$\label{covering1}
\left(\mint_{\Omega_{\rho_i}(y^i)} \left[\Phi(u,V)\right]^p \,dx\right)^{\frac1p} +
\frac{1}{\delta}\left( \mint_{\Omega_{\rho_i}(y^i)} |F|^p \,dx\right)^\frac1p = \lambda,$$ and for any $ \rho \in ( \rho_i, (s_2-s_1)\,r]$, $$\label{covering2}
\left(\mint_{\Omega_{\rho_i}(y^i)} \left[\Phi(u,V)\right]^p \,dx\right)^{\frac1p} + \frac{1}{\delta}\left( \mint_{\Omega_{\rho_i}(y^i)} |F|^p \,dx\right)^\frac1p<\lambda.$$
Furthermore, we can deduce from Lemma \[coveringlem\], in particular, , that $$\label{covering3}
\mint_{\Omega_{\rho_i}(y^i)} \left[\Phi(u,V)\right]^p \,dx \geq \left(\frac{\lambda}{2}\right)^p\ \ \ \text{or}\ \ \
\mint_{\Omega_{\rho_i}(y^i)} |F|^p \,dx \geq \left(\frac{\delta\lambda}{2}\right)^p.$$ If the first inequality holds, we have $$\left|\Omega_{\rho_i}(y^i)\right|\leq \frac{2^p}{\lambda^p}\left( \int_{\Omega_{\rho_i}(y^i)\cap\left\{\Phi(u,V)>\frac{\lambda}{2^{(p+1)/p}} \right\}} \left[\Phi(u,V)\right]^p \,dx+\frac{\left|\Omega_{\rho_i}(y^i)\right|\lambda^p}{2^{p+1}} \right)$$ and so $$\left|\Omega_{\rho_i}(y^i)\right|\leq \frac{2^{p+1}}{\lambda^p} \int_{\Omega_{\rho_i}(y^i)\cap\left\{\Phi(u,V)>\frac{\lambda}{2^{(p+1)/p}} \right\}} \left[\Phi(u,V)\right]^p \,dx.$$ Similarly, if the second inequality in holds, we have $$\left|\Omega_{\rho_i}(y^i)\right|\leq \frac{2^{p+1}}{(\delta\lambda)^p} \int_{\Omega_{\rho_i}(y^i)\cap\left\{|F|>\frac{\lambda\delta}{2^{(p+1)/p}} \right\}} |F|^p \,dx.$$ Therefore, in any case, we have $$\begin{aligned}
\label{omegai}
\nonumber \left|\Omega_{\rho_i}(y^i)\right|&\leq & \frac{2^{p+1}}{\lambda^p} \Bigg(\int_{\Omega_{\rho_i}(y^i)\cap\left\{\Phi(u,V)>\frac{\lambda}{2^{(p+1)/p}} \right\}} \left[\Phi(u,V)\right]^p \,dx\\
&&\qquad \qquad \quad+ \int_{\Omega_{\rho_i}(y^i)\cap\left\{\frac{|F|}{\delta}>\frac{\lambda}{2^{(p+1)/p}} \right\}} \left[\frac{|F|}{\delta}\right]^p \,dx\Bigg).\end{aligned}$$
*Step 2. Comparison estimates.* \
From Lemma \[coveringlem\], in particular, , we note that $$\left(\mint_{\Omega_{20\rho_i}(y^i)} \left[|Du|+ V^{\frac1p} |u|\right]^p \,dx\right)^\frac1p <\lambda \ \ \ \text{and}\ \ \
\left(\mint_{\Omega_{20\rho_i}(y^i)} |F|^p \,dx\right)^\frac1p <\delta\lambda.$$ Then applying Lemma \[comestlem\] and Lemma \[comestlem1\], for any $\epsilon \in (0,1),$ there exists a small $\delta= \delta( \epsilon, n, p, \gamma, L, \nu)>0$ such that $$\label{tDumDvibdd}
\left( \mint_{\Omega_{20\rho_i}(y^i)} |Du-Dw_i|^{p} \, dx\right)^{\frac1p} +
\left( \mint_{\Omega_{20\rho_i}(y^i)} \left|V^{\frac1p}u-V^{\frac1p}w_i\right|^{p} \, dx\right)^{\frac1p} \leq \epsilon \lambda,$$ $$\label{tDvihibdd}
\left( \mint_{\Omega_{5\rho_i}(y^i)} \left| Dw_i\right|^{\gamma^* (p-1)} \, dx \right)^{\frac{1}{\gamma^* (p-1)}} \leq c\lambda$$ and $$\label{tDvihibdd1}
\left( \mint_{\Omega_{5\rho_i}(y^i)} \left[V^\frac{1}{p}\left| w_i\right|\right]^{p\gamma } \, dx \right)^{\frac{1}{p\gamma }} \leq c\lambda,$$ where $w_i \in W^{1,p}(\Omega_{20\rho_i}(y^i))$ is the unique weak solution to $$\left\{\begin{array}{rclcc}
-\mathrm{div}\, \mathbf{a}(x,Dw_i) + V |w_i|^{p-2}w_i& = & 0 & \textrm{ in } & \Omega_{20\rho_i}(y^i), \\
w_i & = & u & \textrm{ on } & \partial \Omega_{20\rho_i}(y^i).
\end{array}\right.$$ Furthermore, recalling the definition of $\Phi$ in and the fact $\gamma^* (p-1)>p\gamma $, we have from and that $$\label{Vvibdd}
\mint_{\Omega_{5\rho_i}(y^i)} \left[\Phi(w_i,V)\right]^{p\gamma } \, dx \leq c\lambda^{p\gamma},$$ for some constant $c=c(n,p, \gamma ,L, \nu,b_\gamma)>0.$
*Step 3. Estimates for $\Phi(u,V)$.* \
Let $y \in \Omega_{5\rho_i}(y^i)$ such that $\Phi(u,V)(y)> K\lambda,$ where $K\geq 1$ will be chosen later. We then note that $$\Phi(u,V)(y) \leq \Phi(w_i,V)(y)+|Du(y)-Dw_i(y)|+[V(y)]^{\frac1p}|u(y)-w_i(y)|.$$ Here, we need to consider the two cases: $$\begin{aligned}
\textrm{(i)} && \Phi(w_i,V)(y) \leq|Du(y)-Dw_i(y)|+[V(y)]^{\frac1p}|u(y)-w_i(y)|,\\
\textrm{(ii)} && \Phi(w_i,V)(y) > |Du(y)-Dw_i(y)|+[V(y)]^{\frac1p}|u(y)-w_i(y)|.\end{aligned}$$ For the case (i), it is clear that $$\Phi(u,V)(y) \leq 2\left(|Du(y)-Dw_i(y)|+[V(y)]^{\frac1p}|u(y)-w_i(y)|\right).$$ For the case (ii), we have that $$K\lambda < \Phi(u,V)(y) \leq 2\Phi(w_i,V)(y),$$ from which, it follows that $$\Phi(u,V)(y)\leq 2 \Phi(w_i,V)(y) \left[ \frac{2\Phi(w_i,V)(y)}{K\lambda} \right]^{\gamma -1} = \frac{2^\gamma}{(K \lambda)^{\gamma-1}} [\Phi(w_i,V)(y)]^{\gamma} .$$ In turn, for the both cases (i) and (ii), we have that $$\begin{aligned}
[\Phi(u,V)(y)]^p &\leq& 2^{2p-1} \left( |Du(y)-Dw_i(y)|^p + V(y)|u(y)-w_i(y)|^p \right)\\
&& + \frac{2^{p\gamma}}{(K \lambda)^{p\gamma-p}} [\Phi(w_i,V)(y)]^{p\gamma}\end{aligned}$$ for any $y \in \Omega_{5\rho_i}(y^i)$ such that $\Phi(u,V)(y)> K\lambda.$
Then applying -, we deduce that $$\begin{aligned}
\int_{\Omega_{5\rho_i}(y^i) \cap E(K\lambda , s_2 r)} [\Phi(u,V)]^p\, dx &\leq & c \int_{\Omega_{5\rho_i}(y^i)} \left[ |Du - Dw_i|^p +V |u- w_i|^p \right]\, dx\\
&&\quad + \frac{c}{(K \lambda)^{p\gamma-p}} \int_{\Omega_{5\rho_i}(y^i)} [\Phi(w_i,V)(y)]^{p\gamma} \, dx \\
& \leq & c \left(\epsilon \lambda^p + \frac{\lambda^{p\gamma} }{(K \lambda)^{p\gamma-p}}\right) \left| \Omega_{5\rho_i}(y^i)\right|\\
& \leq & c\, \lambda^p \left( \epsilon + \frac{1}{K^{p\gamma-p}} \right) \left| \Omega_{\rho_i}(y^i)\right|\\
& = & c\,\tilde\epsilon \lambda^p \left| \Omega_{\rho_i}(y^i)\right|\end{aligned}$$ for some constant $c=c(n,p, \gamma, L, \nu, b_\gamma)>0,$ where $$\label{tepsilon}
\tilde \epsilon:= \epsilon + \frac{1}{K^{p\gamma-p}}.$$ Therefore, inserting into the previous estimate, we have that $$\begin{aligned}
\int_{\Omega_{5\rho_i}(y^i) \cap E(K\lambda, s_1 r)} [\Phi(u,V)]^p\, dx & \leq& c\tilde \epsilon \Bigg( \int_{\Omega_{\rho_i}(y^i)\cap\left\{\Phi(u,V)>\frac{\lambda}{2^{(p+1)/p}} \right\}} \left[\Phi(u,V)\right]^p \,dx\\
&&\qquad\quad + \int_{\Omega_{\rho_i}(y^i)\cap\left\{\frac{|F|}{\delta}>\frac{\lambda\delta}{2^{(p+1)/p}} \right\}} \left[\frac{|F|}{\delta}\right]^p \,dx\Bigg).\end{aligned}$$
According to Lemma \[coveringlem\], we note that $\Omega_{\rho_i}(y^i)$ is mutually disjoint and $$E(K\lambda, s_1 r) \subset E(\lambda, s_1 r )\subset \bigcup_{i=1}^{\infty} \Omega_{5\rho_i}(y^i) \subset \Omega_{s_2 r} ,$$ since $K \geq 1.$ Then we have that $$\begin{aligned}
\label{EDubddcal}
\nonumber \int_{ E(K\lambda, s_1 r)} [\Phi(u,V)]^p\, dx &\leq& \sum_{i=1}^{\infty} \int_{\Omega_{5\rho_i}(y^i) \cap E(K\lambda, s_1 r)} [\Phi(u,V)]^p\, dx \\
\nonumber & \leq & c\tilde \epsilon \Bigg( \int_{\Omega_{s_2r}\cap\left\{\Phi(u,V)>\frac{\lambda}{2^{(p+1)/p}} \right\}} \left[\Phi(u,V)\right]^p \,dx\\
& &\qquad \qquad + \int_{\Omega_{s_2r}\cap\left\{\frac{|F|}{\delta}>\frac{\lambda}{2^{(p+1)/p}} \right\}} \left[\frac{|F|}{\delta}\right]^p \,dx\Bigg)\end{aligned}$$ for some constant $c=c(n,p, \gamma, L, \nu, b_\gamma)>0.$
*Step 4. Proof of when $q\in(p,p\gamma)$.* \
We shall use a truncation argument. For $k \geq A\lambda_0,$ let us define $$\Phi(u,V)_k := \min\left\{ \Phi(u,V), k \right\},$$ and denote the upper level sets with respect to $\Phi(u,V)_k $ by $$E_k ( \tilde \lambda, \rho):= \left\{ y \in \Omega_{\rho} : \Phi(u,V)_k> \tilde \lambda\right\}\ \ \text{for }\tilde \lambda,\rho>0.$$ Then since $E_k ( K\lambda, s_1 r) \subset E ( K\lambda, s_1 r)$ and $$\left\{ \Phi(u,V)_k >\frac{\lambda}{2^{(p+1)/p}} \right\} = \left\{ \Phi(u,V)>\frac{\lambda}{2^{(p+1)/p}} \right\},$$ we see from that $$\begin{aligned}
\int_{ E_k(K\lambda, s_1 r)} [\Phi(u,V)]^p\, dx & \leq & c\tilde \epsilon \Bigg( \int_{E_k\left(\frac{\lambda}{2^{(p+1)/p}},s_2r\right)} \left[\Phi(u,V)\right]^p \,dx\\
& &\qquad \quad +\int_{\Omega_{s_2r}\cap\left\{\frac{|F|}{\delta}>\frac{\lambda}{2^{(p+1)/p}} \right\}} \left[\frac{|F|}{\delta}\right]^p \,dx\Bigg).\end{aligned}$$ Then by multiplying both sides by $\lambda^{q-p-1}$ and integrating with respect to $\lambda$ over $(A\lambda_0, \infty)$, we have that $$\begin{aligned}
\label{lamDuEk}
\nonumber I_0 &:=&\int_{A\lambda_0}^\infty \lambda^{q-p-1}\int_{ E_k(K\lambda, s_1 r)} [\Phi(u,V)]^p\, dxd\lambda\\
\nonumber & \leq & c\tilde \epsilon \Bigg( \int_{A\lambda_0}^\infty \lambda^{q-p-1} \int_{E_k\left(\frac{\lambda}{2^{(p+1)/p}},s_2r\right)} \left[\Phi(u,V)\right]^p \,dxd\lambda\\
\nonumber & &\qquad \quad +\int_{A\lambda_0}^\infty \lambda^{q-p-1} \int_{\Omega_{s_2r}\cap\left\{\frac{|F|}{\delta}>\frac{\lambda}{2^{(p+1)/p}} \right\}} \left[\frac{|F|}{\delta}\right]^p \,dxd\lambda\Bigg)\\
&=:& c\tilde\epsilon (I_1+I_2).\end{aligned}$$ Here, Fubini’s theorem allows us to deduce that $$\begin{aligned}
I_0 &=& \int_{E_k(KA\lambda_0, s_1 r)} [\Phi(u,V)]^p \left( \int_{A\lambda_0}^{\Phi(u,V)_k(x)/K} \lambda^{q-p-1} \, d\lambda \right) dx\\
&=& \frac{1}{q-p}\, \Bigg\{ \int_{E_k(KA\lambda_0, s_1 r)} [\Phi(u,V)]^p \left[\frac{\Phi(u,V)_k}{K}\right]^{q-p} \, dx \\
&&\qquad\qquad\qquad\qquad -(A\lambda_0)^{q-p}\int_{E_k(KA\lambda_0, s_1 r)} [\Phi(u,V)]^p \,dx \Bigg\}.\end{aligned}$$ We also employ Fubini’s theorem to discover $$\begin{aligned}
I_1&=& \int_{E_k\left(\frac{A\lambda_0}{2^{(p+1)/p}},s_2r\right)} [\Phi(u,V)]^p \left(\int_{A\lambda_0}^{ 2^{(p+1)/p} \Phi(u,V)_k(x)} \lambda^{q-p-1} \,d\lambda\right) dx \\
&\leq& \frac{1}{q-p} \int_{E_k\left(\frac{A\lambda_0}{2^{(p+1)/p}},s_2r\right)} [\Phi(u,V)]^p \left[2^{(p+1)/p} \Phi(u,V)_k\right]^{q-p}\, dx \\
&\leq & \frac{2^{(p+1)(q-p)/p}}{q-p} \int_{\Omega_{s_2 r}} [\Phi(u,V)]^p \left[\Phi(u,V)_k\right]^{q-p} \, dx.\end{aligned}$$ Similarly, we obtain that $$I_2\leq \frac{2^{(p+1)(q-p)/p}}{q-p} \int_{\Omega_{s_2 r}} \left[\frac{|F|}{\delta}\right]^q \, dx.$$ Therefore, inserting the previous estimates for $I_0$, $I_1$, $I_2$ into , we derive $$\begin{aligned}
&&\int_{E_k(KA\lambda_0, s_1 r)} [\Phi(u,V)]^p [\Phi(u,V)_k]^{q-p} \, dx \\
&& \quad \leq (KA\lambda_0)^{q-p}\int_{\Omega_{s_1 r}} [\Phi(u,V)]^p \,dx\\
&&\quad \quad +c\tilde \epsilon K^{q-p} \left( \int_{\Omega_{s_2 r}} [\Phi(u,V)]^p [\Phi(u,V)_k]^{q-p} \, dx + \int_{\Omega_{s_2 r}} \left[\frac{|F|}{\delta} \right]^{q} \, dx \right).\end{aligned}$$ We also notice that $$\int_{\Omega_{s_1 r} \setminus E_k(KA\lambda_0, s_1 r)} [\Phi(u,V)]^p[\Phi_k(u,V)_k]^{q-p}\, dx \leq (KA\lambda_0)^{q-p} \int_{\Omega_{s_1 r}} [\Phi(u,V)]^p \, dx.$$ Finally, from the last two estimates we have that $$\begin{aligned}
\int_{\Omega_{s_1 r}}[\Phi(u,V)]^p [\Phi(u,V)_k]^{q-p} \, dx & \leq & (KA\lambda_0)^{q-p}\int_{\Omega_{s_1 r}}[\Phi(u,V)]^p\,dx\\
&&\hspace{-4.5cm}+c_2\tilde \epsilon K^{q-p} \left( \int_{\Omega_{s_2 r}} [\Phi(u,V)]^p [\Phi(u,V)_k]^{q-p} \, dx + \int_{\Omega_{s_2 r}} \left[\frac{|F|}{\delta} \right]^{q} \, dx \right)\end{aligned}$$ for some $c_2=c_2(n,p,\gamma,L,\nu,b_\gamma,q)>0$. At this stage, we recall the definition of $\tilde \epsilon$ in , and then take large $K>1$ and small $\epsilon\in(0,1)$ depending on $n,p,\gamma,L,\nu,b_\gamma,q$ such that $$K\geq (4c_2)^{\frac{1}{p\gamma-q}} \ \ \ \text{and}\ \ \ \epsilon\leq\frac{1}{4c_2K^{q-p}},$$ hence $\delta=\delta(n,p,\gamma,L,\nu,b_\gamma,q)\in(0,1)$ is finally determined. Consequently, recalling the definition of $A$ in we have $$\begin{aligned}
\int_{\Omega_{s_1 r}}[\Phi(u,V)]^p [\Phi(u,V)_k]^{q-p} \, dx & \leq & \frac12 \int_{\Omega_{s_2 r}} [\Phi(u,V)]^p [\Phi(u,V)_k]^{q-p} \, dx \\
&& \hspace{-2cm}+\frac{c\lambda_0^{q-p}}{(s_2-s_1)^{\frac np}}\int_{\Omega_{2r}}[\Phi(u,V)]^p\,dx+ c \int_{\Omega_{2 r}} |F|^q \, dx .\end{aligned}$$ Then applying Lemma \[teclem\], we derive that $$\int_{\Omega_{r}}[\Phi(u,V)]^p [\Phi(u,V)_k]^{q-p} \, dx \leq c\lambda_0^{q-p}\int_{\Omega_{2r}}[\Phi(u,V)]^p\,dx+ c \int_{\Omega_{2 r}} |F|^q \, dx$$ for any $k>A\lambda_0$. Finally, by Lebesgue’s monotone convergence theorem, the definition of $\lambda_0$ in , Hölder’s inequality and Young’s inequality, we obtain that $$\begin{aligned}
\mint_{\Omega_{r}}[\Phi(u,V)]^q\,dx &= &\lim_{k\to\infty} \mint_{\Omega_{r}}[\Phi(u,V)]^p [\Phi(u,V)_k]^{q-p} \, dx\\
& \leq& c\lambda_0^{q-p}\mint_{\Omega_{2r}}[\Phi(u,V)]^p\,dx+ c \mint_{\Omega_{2 r}} |F|^q \, dx\\
& \leq& c\left(\mint_{\Omega_{2r}}[\Phi(u,V)]^p\,dx\right)^{\frac{q}{p}}+ c \mint_{\Omega_{2 r}} |F|^q \, dx,\end{aligned}$$ and so, recalling the definition of $\Phi(u,V)$ in , $$\begin{aligned}
\label{mainlocalest1}
\nonumber\left( \mint_{\Omega_{r}} |Du|^q +\left[V^{\frac1p}|u|\right]^{q}\, dx\right)^{\frac1q} &\leq& c \left( \mint_{\Omega_{2r}} |Du|^p + \left[ V^{\frac1p} |u|\right]^p \,dx \right)^{\frac{1}{p}}\\
&&+ c \left(\mint_{\Omega_{2r}} \left|F \right|^{q} \, dx\right)^{\frac1q},\end{aligned}$$ which derives the estimate for $q\in(p,p\gamma)$.
*Step 5. Proof of when $q\in[p\gamma,\gamma^*(p-1))$.* \
Finally, we prove the estimate for the remaining range of $q$. Note that we only consider the gradient of $u$ since $\chi_{\{q<p\gamma\}}=0$.
We first suppose that $q\in[p\gamma,\gamma^*(p-1))$ satisfies $$\label{case1}
\max\left\{p,\frac{n(p-1)}{n-1}\right\}<q.$$ Note that if $p\leq n$ we have $\max\{p,\frac{n(p-1)}{n-1}\}=p$ and so the previous inequality is trivial. Then let us set $\tilde q\in (1,\gamma)$ such that $$\label{tq}
q=(p-1)\tilde q^*.$$ Then we see from Hölder’s inequality that $$\begin{aligned}
\left( \mint_{\Omega_{2r}} \left[ r V |u|^{p-1}\right]^{\tilde{q}}\, dx \right)^{\frac{1}{\tilde{q}}} & = &\left( \mint_{\Omega_{2r}} \left[ r V^{\frac1p} \right]^{\tilde{q}} \left[ V^{\frac1p} |u| \right]^{(p-1)\tilde{q}}\, dx \right)^{\frac{1}{\tilde q}} \\
&\leq& \left( \mint_{\Omega_{2r}} [r^pV]^{\tilde{q}}\, dx\right)^{\frac{1}{p\tilde{q}}} \left( \mint_{\Omega_{2r}} \left[V^{\frac{1}{p}}|u|\right]^{p\tilde{q}} \, dx \right)^{\frac{p-1}{p\tilde{q}}} \\
& \leq & c \left( r^{p\gamma-n} \int_{\Omega_{2r}} V^{\gamma}\, dx\right)^{\frac{1}{p\gamma}} \left( \mint_{\Omega_{2r}} \left[V^{\frac{1}{p}}|u|\right]^{p\tilde{q}} \, dx \right)^{\frac{p-1}{p\tilde{q}}}\\
& \leq & c \left( \mint_{\Omega_{2r}} \left[V^{\frac{1}{p}}|u| \right]^{p\tilde{q}} \, dx \right)^{\frac{p-1}{p\tilde{q}}}.\end{aligned}$$ Here we have used the facts that $\tilde{q} \in (1,\gamma)$ and $(2r)^{p - \frac{n}{\gamma}} \Vert V \Vert_{L^{\gamma}(\Omega_{2r})}\leq 1.$ Therefore, applying the estimate with $q$ and $r$ replaced by $p\tilde q$ and $2r$, respectively, we have that $V |u|^{p-2}u\in L^{\tilde q}(\Omega_{2r})$ with the estimate $$\begin{aligned}
\label{mainpf1}
\nonumber \left( \mint_{\Omega_{2r}} \left[ r V |u|^{p-1}\right]^{\tilde{q}}\, dx \right)^{\frac{1}{\tilde{q}}} &\leq& c \left( \mint_{\Omega_{4r}}\left[ |Du| + V^\frac{1}{p}|u|\right]^p \,dx \right)^{\frac{p-1}{p}}\\
&&+c \left(\mint_{\Omega_{4 r}} |F|^{p\tilde{q}} \, dx\right)^{\frac{p-1}{p\tilde{q}}}.\end{aligned}$$ Finally, by Theorem \[thmDwbdd\] with $f=V |u|^{p-2}u$, the previous estimate and Hölder’s inequality, we have that $$\begin{aligned}
\label{mainpf2}
\nonumber \left(\mint_{\Omega_r} |Du|^{q} \, dx \right)^{\frac{1}{q}} &\leq& c \left( \mint_{\Omega_{2r}} |Du|^{p}\, dx\right)^{\frac{1}{p}} + c \left( \mint_{\Omega_{2r}} \left[ r V |u|^{p-1}\right]^{\tilde{q}} \, dx\right)^{\frac{1}{\tilde{q}(p-1)}} \\
\nonumber && + c \left( \mint_{\Omega_{2r}} |F|^{q} \, dx\right)^{\frac{1}{q}}\\
\nonumber &\leq& c \, \left( \mint_{\Omega_{4r}} \left[ |Du|+ V^{\frac1p} |u|\right]^p \,dx \right)^{\frac{1}{p}} + c \left(\mint_{\Omega_{4 r}} \left|F \right|^{p \tilde{q}} \, dx\right)^{\frac{1}{p \tilde{q}}} \\
\nonumber && + c \left(\mint_{\Omega_{4 r}} \left|F \right|^{q} \, dx\right)^{\frac{1}{q}}\\
&\leq& c \, \left( \mint_{\Omega_{4r}} \left[ |Du|+ V^{\frac1p} |u|\right]^p \,dx \right)^{\frac{1}{p}} + c \left(\mint_{\Omega_{4 r}} \left|F \right|^{q} \, dx\right)^{\frac{1}{q}},\end{aligned}$$ which proves the estimate .
We next assume that $q\in[p\gamma,\gamma^*(p-1))$ does not satisfies , that is, $$p \gamma \leq q\leq \max\left\{p,\frac{n(p-1)}{n-1}\right\}.$$ Note that this happens only for the case that $p>n$ and $1<\gamma\leq \frac{n(p-1)}{p(n-1)}$, and that, in this case, we cannot find $\tilde q\in (1,\gamma)$ satisfying . Instead, let us set $$\tilde q:= \frac{1+\gamma}{2} \in (1,\gamma).$$ Then, in the same argument above, we have the estimate . Using this, Theorem \[thmDwbdd1\] (instead of Theorem \[thmDwbdd\]) and Hölder’s inequality, we obtain the estimate . Hence, holds for the remaining range for $q$. This completes the proof.
Proof of Corollary \[maincor\]
------------------------------
\
We take the test function $\varphi=u$ in the weak formulation , and then use Young’s inequality to arrive at $$\begin{aligned}
&&c(p,\nu)\int_{\Omega} |Du|^{p}\,dx + \int_{\Omega} V|u|^p \, dx \\
&&\quad \leq \int_{\Omega} \mathbf{a}(x, Du) \cdot Du \, dx + \int_{\Omega} V |u|^{p-2}u \cdot u \, dx = \int_{\Omega} |F|^{p-2} F \cdot D u\, dx \\
&&\quad \leq \int_{\Omega} |F|^{p-1} |Du| \,dx \leq c(\tau) \int_{\Omega} |F|^{p} \, dx + \tau \int_{\Omega} |Du|^{p} \,dx\end{aligned}$$ for any small $\tau>0.$ Here we have used the inequality that $\mathbf{a}(x, \xi)\cdot \xi \geq c(p,\nu) |\xi|^p,$ which can be easily obtained from . We choose $\tau>0$ so small that $$\label{energyest}
\int_{\Omega} |Du|^{p}\,dx + \int_{\Omega} V|u|^p \, dx \leq c \int_{\Omega} |F|^{p} \, dx.$$
On the other hand, from the resulting estimates with $r = \frac{\tilde{R}}{4}$ where $\tilde{R} := \min\Big\{ R, \Vert V \Vert_{L^{\gamma}(\Omega)}^{-\frac{1}{p-\frac{n}{\gamma}}}\Big\}$ and $x_0 \in \overline{\Omega},$ we get that $$\begin{aligned}
\label{Duestcov}
\nonumber&&\int_{\Omega_{\tilde{R}/4}(x_0)} |Du|^{q} + \chi_{\{q<p\gamma\}} \left[ V^{\frac1p} |u| \right]^q \, dx \\
&&\qquad \leq \frac{c}{\tilde{R}^{n\left(\frac{q}{p}-1\right)}}\left( \int_{\Omega_{\tilde{R}}(x_0)} |Du|^p + \left[ V^{\frac1p} |u| \right]^p \,dx \right)^{\frac{q}{p}} + c \int_{\Omega_{\tilde{R}}(x_0)} \left|F \right|^{q} \, dx.\end{aligned}$$ Since $\overline{\Omega}$ is compact, by Vitali’s covering lemma, there exist finitely many points $x_0^1, \cdots, x_0^N$ in $\overline{\Omega}$ such that $B_{\tilde{R}/20}(x_0^k)$, $k=1,2,\dots,N$ are mutually disjoint and $\Omega \subseteq \bigcup_{k=1}^{N} B_{\tilde{R}/4}(x_0^k).$ Here we note that $\sum_{k=1}^N\chi_{B_{\tilde{R}}(x_0^k)}\leq c(n)$. Therefore from , we deduce that $$\begin{aligned}
\label{glDuestcov}
\nonumber&&\int_{\Omega} |Du|^{q} + \chi_{\{q<p\gamma\}} \left[ V^{\frac1p} |u| \right]^q \, dx \\
\nonumber&&\quad \leq \sum_{k=1}^{N} \int_{\Omega_{\tilde{R}/4}(x_0^k)} |Du|^{q} + \chi_{\{q<p\gamma\}} \left[ V^{\frac1p} |u| \right]^q \, dx \\
\nonumber&&\quad\leq c\,\sum_{k=1}^{N} \left\{\frac{1}{\tilde{R}^{n\left(\frac{q}{p}-1\right)}}\left( \int_{\Omega_{\tilde{R}}(x_0^k)} |Du|^p + \left[ V^{\frac1p} |u| \right]^p \,dx \right)^{\frac{q}{p}} + \int_{\Omega_{\tilde{R}}(x_0^k)} \left|F \right|^{q} \, dx\right\}\\
&&\quad \leq \frac{c}{\tilde{R}^{n\left(\frac{q}{p}-1\right)}} \left( \int_{\Omega} |Du|^p + \left[ V^{\frac1p} |u| \right]^p \,dx \right)^{\frac{q}{p}} + c \int_{\Omega} \left|F \right|^{q} \, dx.\end{aligned}$$ In turn, inserting into and using Hölder’s inequality together with the fact that $\mathrm{diam}(\Omega)> \tilde{R}$, we obtain $$\begin{aligned}
\nonumber&&\int_{\Omega} |Du|^{q} + \chi_{\{q<p\gamma\}} \left[ V^{\frac1p} |u| \right]^q \, dx \leq c\left(\frac{\mathrm{diam}(\Omega)}{\tilde{R}}\right)^{\frac{n(q-p)}{p}}\int_{\Omega} \left|F \right|^{q} \, dx,\end{aligned}$$ which implies the desired estimates .
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|
{
"pile_set_name": "ArXiv"
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|
---
abstract: 'We give sharp and explicit upper bounds for the first positive eigenvalue $\lambda_1(\Box_b)$ of the Kohn-Laplacian on compact strictly pseudoconvex hypersurfaces in $\mathbb{C}^{n+1}$ in terms of their defining functions. As an application, we show that in the family of real ellipsoids, $\lambda_1(\Box_b)$ has a unique maximum value at the CR sphere.'
address:
- 'Department of Mathematics, University of California, Irvine, CA 92697-3875'
- 'College of Mathematics and Informatics, Fujian Normal University, Fuzhou 350108, Fujian, China'
- 'Texas A&M University at Qatar, Science Program, PO Box 23874, Education City, Doha, Qatar'
author:
- 'Song-Ying Li'
- Guijuan Lin
- Duong Ngoc Son
title: 'The sharp upper bounds for the first positive eigenvalue of the Kohn-Laplacian on compact strictly pseudoconvex hypersurfaces'
---
[^1] [^2] [^3]
Introduction
============
Let $(M^{2n+1},\theta)$ be a compact strictly pseudoconvex pseudohermitian manifold of real dimension $2n+1 \geq 3$. Let $\bar{\partial}_b \colon L^2(M) \to L^2_{0,1}(M)$ be the tangential Cauchy–Riemann operator and $\bar{\partial}_{b}^{\ast}$ the formal adjoint with respect to the volume measure $dv = \theta\wedge (d\theta)^n$. The Kohn-Laplacian acting on functions is given by $ \Box_b = \bar{\partial}_b^{\ast}\bar{\partial}_b$ and the sub-Laplacian is given by $\Delta_b=2\Re \Box_b$. There has been growing interest in the relation between the spectra of the sub-Laplacian and the Kohn-Laplacian and the geometric qualities of the underlying CR manifolds. We mention here, for example, the Lichnerowicz-type estimate for the first positive eigenvalue of the sub-Laplacian on compact manifolds with a lower bound on Ricci and torsion was studied in, e.g., [@ADE; @BD; @Gr; @LL1; @Chiu]. The characterization of extremal case, the Obata-type problem, was studied in, e.g., [@CC; @LW; @IV]. In particular, X. Wang and the first author proved an Obata-type theorem in CR geometry for compact manifolds in [@LW] which characterizes the CR sphere (among compact manifolds) as the only extremal case in the Lichnerowicz-type estimate for the sub-Laplacian. We refer the reader to the aforementioned papers and references therein for more details on these problems.
The eigenvalue problem for $\Box_b$ is more involved. It is well-known that on a *non-embeddable* compact strictly pseudoconvex manifold of three-dimension, $\mathrm{Spec}\, (\Box_b)$ contains a sequence of “small” eigenvalues converging rapidly to zero. In this case, we can *not* define the first positive eigenvalue of $\Box_b$. In fact, by the theorems of Boutet de Monvel, Burns, and Kohn, zero is an isolated eigenvalue of $\Box_b$ if and only if $M$ is embeddable in some complex space $\mathbb{C}^N$ [@Burns; @BdM; @K]; see also [@BE]. Thus, for embedded manifolds, it makes sense to define and study the first positive eigenvalue $\lambda_1$ of $\Box_b$.
In [@CCY], Chanillo, Chiu, and Yang proved a Lichnerowicz-type lower bound for $\lambda_1$ for three-dimensional manifolds (which are not assumed to be embedded *a priori*). Their method also gives the same estimate for five dimensional case. In a preprint [@CW], Chang and Wu gave a lower bound in general dimension and proved some partial results on characterizing the equality case. In [@LSW], X. Wang, the first, and the third author completely analyzed the equality case by establishing an Obata-type theorem for the Kohn-Laplacian; we refer to [@LSW] for more details.
In this paper, we shall give sharp *upper* bounds for $\lambda_1$ on compact strictly pseudoconvex CR manifolds embedded in $\mathbb{C}^{n+1}$. Suppose $\rho$ is a smooth strictly plurisubharmonic function on $\mathbb{C}^{n+1}$ and $\nu$ is a regular value of $\rho$ such that $M:=\rho^{-1}(\nu)$ is compact. On $M$, consider the “usual” pseudohermitian structure $\theta$ “induced” by $\rho$: $$\label{b}
\theta
=
\iota^{\ast} (i/2)(\bar{\partial}\rho - \partial \rho),$$ where $\iota \colon M \to \mathbb{C}^{n+1}$ is the usual embedding. This pseudohermitian structure gives rise to a volume form $dv: = \theta\wedge (d\theta)^n$ on $M$. Furthermore, $\rho$ induces a Kähler metric $\rho_{j{\bar{k}}}dz^jd{\bar{z}}^{k}$ in a neighborhood $U$ of $M$. Let $[\rho^{j\bar{k}}]^t$ be the inverse of $[\rho_{j{\bar{k}}}]$. For a smooth function $u$ on $U$, the length of $\partial u$ in the Kähler metric is given by $$|\partial u|^2_{\rho} = \rho^{j\bar{k}} u_j \bar{u}_{\bar{k}}.$$ Here we use the usual summation convention: repeated Latin indices are summing from $1$ to $n+1$. We also use $\rho^{j\bar{k}}$ and $\rho_{j\bar{k}}$ to raise and lower the indices, e.g., $u^{\bar{k}} = \rho^{ l \bar{k}} u_{l}$, so that $|\partial u|_{\rho}^2 = \bar{u}_{\bar{k}} u^{{\bar{k}}}$. We define the following degenerate differential operator $$\tilde{\Delta}_{\rho} = \left(|\partial\rho|_\rho^{-2}\rho^{j}\rho^{{\bar{k}}} - \rho^{j{\bar{k}}}\right)\partial_{j}\partial_{{\bar{k}}}.$$ Our first result in this paper is the following sharp upper bound for $\lambda_1$.
\[thm:special\] Let $\rho$ be a smooth strictly plurisubharmonic function defined on an open set $U$ of $\mathbb{C}^{n+1}$, $M$ a compact connected regular level set of $\rho$, and $\lambda_1$ the first positive eigenvalue of $\Box_b$ on $M$. Assume that for some $j$, $$\begin{aligned}
\label{special}
\Re \rho_{{\bar{j}}}\tilde{\Delta}_{\rho}\, \rho_j + \tfrac{1}{n}\, |{\partial}\rho|_\rho^2 \, |\tilde{\Delta}_{\rho} \rho_j|^2 \le 0\ \text{on}\ M.\end{aligned}$$ Then $$\begin{aligned}
\label{specialbound}
\lambda_1(M,\theta) \leq n\max_M |{\partial}\rho|_\rho^{-2}\end{aligned}$$ and the equality holds only if $|\partial\rho|_\rho^{2}$ is constant along $M$.
The upper bound in is sharp and the equality occurs on the sphere with the standard pseudohermitian structure. Moreover, in Example \[cex\] below, we shall see that the condition can *not* be relaxed.
Notice that condition is satisfied if there exists $j$ such that $\rho_{j{\bar{k}}l} = 0$ for all $k$ and $l$ and hence we can easily construct examples for which Theorem \[thm:special\] does apply. In particular, if $\rho_{j{\bar{k}}}=\delta_{jk}$, then holds. We shall show that in this case, we can improve the estimate by taking the average value of $|\partial\rho|^{-2}_{\rho}$ instead of its maximum. Thus, we define $v(M) = \int_{M}\theta \wedge (d\theta)^n$ be the volume of $M$.
\[thm:flat\] Let $\rho$ be a smooth strictly plurisubharmonic function defined on an open set $U$ of $\mathbb{C}^{n+1}$, $M$ a compact connected regular level set of $\rho$, and $\lambda_1$ the first positive eigenvalue of $\Box_b$ on $M$. Suppose that $\rho_{j{\bar{k}}} = \delta_{jk}$, then $$\label{e:average}
\lambda_1 \leq \frac{n}{v(M)} \int_{M} |\partial\rho|^{-2}_{\rho} \theta\wedge (d\theta)^n.$$ The equality occurs only if $|\partial\rho|_{\rho}^{2}$ is constant on $M$. If furthermore, $\rho$ is defined in the domain bounded by $M$, then $M$ must be a sphere.
The estimate is a special case of a more general estimate in Theorem \[thm:upperbound\] below which provides a sharp upper bound for $\lambda_1$ in terms of the eigenvalues of the complex Hessian matrix $[\rho_{j{\bar{k}}}]$.
Our main motivation for proving the upper bound in Theorem \[thm:flat\] comes from its application to the eigenvalue problems on the real *ellipsoids*, the compact regular level sets of a real plurisubharmonic quadratic polynomial. The ellipsoids was studied by Webster [@We] who showed that an ellipsoid is not biholomorphic equivalent to the sphere unless it is complex linearly equivalent to the sphere. (It is now well-known that two generic ellipsoids are not biholomorphic equivalent). The eigenvalue problem on ellipsoids was also studied by Tran and the first author [@LiTran]. This paper provides an upper bound for the first positive eigenvalue of $\Delta_b$ on the real ellipsoids in $\mathbb{C}^2$. We shall show that on real ellipsoids, the upper bound in Theorem \[thm:flat\] can be computed explicitly.
\[cor:ellipsoid\] Let $\rho(Z)$ be a real-valued strictly plurisubharmonic homogeneous quadratic polynomial satisfying $\rho_{j{\bar{k}}} = \delta_{jk}$. Suppose that $M = \rho^{-1}(\nu)$ ($\nu>0$) is a compact connected regular level set of $\rho$. Then $$\begin{aligned}
\label{e:ellipsoid}
\lambda_1(M,\theta) \leq \lambda_1(\sqrt{\nu}\,\mathbb{S}^{2n+1},\theta_0) = n/\nu.\end{aligned}$$ The equality occurs if and only if $(M,\theta) = (\sqrt{\nu}\,\mathbb{S}^{2n+1},\theta_0)$.
Here, $\sqrt{\nu}\,\mathbb{S}^{2n+1}$ is the sphere $\|Z\|^2 =\nu$ and $\theta_0 = \iota^{\ast} (i\bar{\partial} \|Z\|^2)$ is the “standard” pseudohermitian structure on the sphere.
The paper is organized as follows. In Section 2, we shall give two simple formulas for the Kohn-Laplacian on compact real hypersurfaces in complex manifolds. These formulas allow us to compute $\Box_b$ explicitly in terms of the defining function $\rho$; see Proposition \[prop:kl\]. These formulas will be crucial for the latter sections. In Section 3, we shall prove a general estimate for $\lambda_1(\Box_b)$ and Theorem \[thm:special\]. In Section 4, we shall give a sharp upper bound for $\lambda_1$ in terms of the eigenvalues of the complex Hessian $[\rho_{j{\bar{k}}}]$, implying the estimate in Theorem \[thm:flat\], and prove the characterization of equality case. We also give a family of examples (beside the ellipsoids) where we can apply this bound. These examples also show that the condition in Theorem \[thm:special\] can not be relaxed. In Section 5, we shall compute the bound in Theorem \[thm:flat\] explicitly in the case of ellipsoids, proving Corollary \[cor:ellipsoid\].
The Kohn-Laplacian on compact real hypersurfaces
================================================
In this section, we shall give two formulas for $\Box_b$ on a compact regular level set of a Kähler potential $\rho$ in terms of $\partial \rho$ and the metric $\rho_{j\bar{k}}dz^jdz^{\bar{k}}$. First, let us start with a compact real hypersurface in $\mathbb{C}^{n+1}$ arising as a regular level set of a strictly plurisubharmonic function $\rho$: $$M = \rho^{-1}(\nu):=\{Z \in U \colon \rho(Z) = \nu\}.$$ Here $\rho$ is smooth on a neighborhood $U$ of $M$ and $d\rho\ne 0$ along $M$. We assume that the complex Hessian $H(\rho): = [\rho_{j{\bar{k}}}]$ is positive definite and thus $\rho$ defines a Kähler metric $\rho_{j\bar{k}}dz^jd\bar{z}^{k}$ on $U$. Let $[\rho^{j\bar{k}}]^t$ be the inverse of $H(\rho)$. For a smooth function $u$ on $U$, the length of $\partial u$ in the Kähler metric is then given by $$|\partial u|^2_{\rho} = \rho^{j\bar{k}} u_j \bar{u}_{\bar{k}}.$$ We shall always equip $M$ with the pseudohermitian structure $\theta$ “induced” by $\rho$: $$\theta = \iota^{\ast} (i/2)(\bar{\partial} \rho - \partial \rho).$$ For local computations, it is convenient to work in the local admissible holomorphic coframe $\{\theta^{\alpha} \colon \alpha = 1,2,\dots, n\}$ on $M$ given by $$\label{e:2.4}
\theta^{\alpha} = dz^{\alpha} - i h^{\alpha} \theta,
\quad
h^{\alpha} = |\partial \rho|_{\rho}^{-2}\rho^{\alpha}
=
|\partial \rho|_{\rho}^{-2}\rho_{{\bar{j}}}\rho^{\alpha{\bar{j}}},
\quad
\alpha = 1,2\dots n.$$ This admissible coframe is valid when $\rho_{n+1} \ne 0$. It is shown by Luk and the first author [@LL p. 679] that at the point $p$ with $\rho_{n+1} \ne 0$, $$d\theta = ih_{\alpha\bar{\beta}} \theta^{\alpha}\wedge \theta^{\bar{\beta}},$$ where the Levi matrix $[h_{\alpha\bar{\beta}}]$ is given explicitly: $$h_{\alpha\betabar}=\rho_{\alpha \betabar}-\rho_\alpha {\partial}_{\betabar}\log \rho_{n+1}-\rho_{\betabar}{\partial}_{\alpha}\log \rho_{\overline{n+1}}+\rho_{n+1\overline{n+1}}
\frac{\rho_\alpha \rho_{\betabar}}{|\rho_{n+1}|^2}.$$ We can check directly that the inverse $[h^{\gamma\bar{\beta}}]$ of the Levi matrix is given by $$h^{\gamma \betabar}
=
\rho^{\gamma\betabar}- \frac{\rho^\gamma \rho^{\betabar}}{|{\partial}\rho|^2_\rho},
\quad
\rho^{\gamma} = \sum_{k=1}^{n+1}\rho_{{\bar{k}}} \rho^{\gamma{\bar{k}}}.$$ We use the Levi matrix and its inverse to lower and raise the Greek indices; repeated Greek indices are summing from $1$ to $n$. The Tanaka-Webster covariant derivatives are given by $$\nabla_{\alpha}\nabla_{\bar{\beta}}f
=
Z_{\alpha} Z_{\betabar} f - \omega_{\bar{\beta}}{}^{\bar{\sigma}}(Z_{\alpha}) Z_{\bar{\sigma}} f$$ where $\{Z_{\alpha}\}$ is the holomorphic frame dual to $\{\theta^{\alpha}\}$ and $\omega_{\bar{\beta}}{}^{\bar{\sigma}}$ are the connection forms. More precisely, $$Z_{\alpha}
=
\frac{{\partial}}{{\partial}z^{\alpha}} - \frac{\rho_{\alpha}}{\rho_{n+1}} \frac{{\partial}}{{\partial}z_{n+1}},$$ and the Tanaka-Webster connection forms are computed in [@LL]; see also [@We]. $$\omega_{\betabar\alpha}
=
(Z_{\bar{\gamma}} h_{\alpha\betabar} - h_{\betabar}h_{\alpha\gamabar}) \theta^{\gamabar} + h_{\alpha}h_{\gamma\betabar}\theta^{\gamma} + ih_{\alpha\bar{\sigma}}Z_{\betabar} h^{\bar{\sigma}} \theta,\quad h_\alpha =h_{\alpha \betabar} h^{\betabar}.$$ Also, the Reeb vector field is given by $$T=i \sum_{j=1}^{n+1}\left(h^j \frac{{\partial}}{ {\partial}z^j}-h^{{\bar{j}}} \frac{{\partial}}{ {\partial}{\bar{z}}^j}\right),\quad h^j= \frac{\rho^j }{|{\partial}\rho|_\rho^2}.$$ The formula below, expressing $\Box_b$ in terms of $\rho$, will be crucial for our analysis.
\[prop:kl\] Let $U$ be an open set in a Kähler manifold $X$ and $\rho$ a Kähler potential on $U$. Let $M$ be a smooth, compact, connected, regular level set of $\rho$, $\theta=\frac{i}{2}(\bar{\partial} \rho - \partial \rho)$, and $\Box_b$ the Kohn-Laplacian defined on $M$ with respect to $dv=\theta\wedge (d\theta)^n$.
(i) If $f$ is a smooth function on $U$, then the following identity holds on $M$. $$\label{e:kohnformula}
\Box_b f
=
- \operatorname{trace}(i\partial \bar{\partial} f)
+ |\partial \rho|_\rho^{-2} \langle \partial \bar{\partial} f, \partial \rho \wedge \bar{\partial} \rho \rangle
+ n|\partial \rho|_\rho^{-2}\langle \partial \rho, \bar{\partial} f \rangle,$$
(ii) Suppose that $(z^1,z^2,\dots, z^{n+1})$ is a local coordinate system on an open set $V$. Define the vector fields $$X_{jk} = \rho_{k}\partial_j - \rho_{j}\partial_k,
\quad
X_{\bar{j}\bar{k}} = \overline{X_{jk}}.$$ Then the following holds on $M \cap V$. $$\label{e:gengel}
\Box_b f
=
-\frac{1}{2}|\partial \rho|_\rho^{-2} \rho^{p\bar{k}} \rho^{q\bar{j}} X_{pq}X_{\bar{j}\bar{k}} f.$$
<!-- -->
(a) The trace operator is taken with respect to the Kähler form and thus $- \operatorname{trace}(i\partial \bar{\partial} f)$ is the Laplace-Beltrami operator acting on $f$. In local coordinates, can be written as $$\Box_b f
=
\left(|\partial \rho|_\rho^{-2} \rho^{k}\rho^{\bar{j}} -\rho^{\bar{j} k}\right) f_{\bar{j} k } + n |\partial \rho|_\rho^{-2} \rho^{\bar{k}} f_{\bar{k}}.$$
(b) Formulas and are generalizations of two formulas for the Kohn-Laplacian on the sphere appeared in [@Geller1980]. This paper also studies the Kohn-Laplacian for forms on the sphere (with volume element induced from $\mathbb{C}^{n+1}$). Notice that the fields $X_{jk}$ are *tangential* Cauchy-Riemann vector fields on $M$ generating $T^{1,0}$ at each point.
We first prove (i). It is well-known [@L] that the Kohn Laplacian acting on function can be given locally by $$\begin{aligned}
-\Box_b f
=
h^{\betabar\alpha} \nabla_{\alpha}\nabla_{\bar{\beta}}f.\end{aligned}$$ Thus, we can work in a local coordinate $(z^1,z^2,\dots, z^{n}, w=z^{n+1})$ on $X$ and assume that $\rho_w = \partial_w \rho \ne 0$. Choose the local frame and coframe as in . Notice that $$\begin{aligned}
Z^{\betabar}
= h^{\alpha\betabar} Z_\alpha
=
h^{\alpha\betabar} {\partial}_{\alpha}- h^{\alpha\betabar} \frac{\rho_\alpha }{ \rho_{n+1}} {\partial}_{n+1}
= \rho^{k\betabar}{\partial}_k-\frac{\rho^{\betabar}}{ |{\partial}\rho|_\rho^2}\rho^k {\partial}_k.
\end{aligned}$$ Therefore, $$\begin{aligned}
-\Box_b f
= &
Z^{\betabar}Z_{\betabar} f -n h^{\bar{\sigma}} f_{\bar{\sigma}}\notag \\
=&
\left[ \rho^{k\betabar}{\partial}_k - |{\partial}\rho|_\rho^{-2} \rho^{\betabar} \rho^k {\partial}_k \right]
\left[f_{\betabar} - \frac{\rho_{\betabar}}{\rho_{{\bar{w}}}}f_{{\bar{w}}}\right] -n h^{\bar{\sigma}} f_{\bar{\sigma}} \notag\\
= &
\rho^{k\betabar}f_{\betabar k} - \frac{\rho_{\betabar}\rho^{k\betabar} f_{{\bar{w}}k}}{\rho_{{\bar{w}}}} -\rho^{k\betabar} f_{{\bar{w}}} \left[\frac{\rho_{{\bar{w}}} \rho_{\betabar k} - \rho_{\betabar} \rho_{{\bar{w}}k}}{\rho_{{\bar{w}}}^2}\right] \notag\\
&
- \frac{\rho^{k}\rho^{\betabar}f_{\betabar k}}{|\partial \rho|_\rho^2}
+ \frac{\rho^{k}\rho^{\betabar}\rho_{\betabar}f_{{\bar{w}}k}}{|\partial \rho|_\rho^2 \rho_{{\bar{w}}}}
+\frac{\rho^{k}\rho^{\betabar}f_{{\bar{w}}}}{|\partial \rho|_\rho^2}\left[\frac{\rho_{{\bar{w}}}\rho_{\betabar k} - \rho_{\betabar}\rho_{{\bar{w}}k}}{\rho_{{\bar{w}}}^2}\right] \notag\\
&
-\frac{n\rho^{\bar{k}}f_{\bar{k}}}{|\partial \rho|_\rho^2} + \frac{nf_{{\bar{w}}}}{\rho_{{\bar{w}}}}.\end{aligned}$$ Here we use summation convention: $k$ runs from $1$ to $n+1$ and $\beta$ runs $1$ to $n$. Simplifying the right hand side, we easily obtain $$-\Box_b f
=
\left( \rho^{\bar{j} k} - |\partial \rho|_\rho^{-2} \rho^{k}\rho^{\bar{j}}\right) f_{\bar{j} k } - n |\partial \rho|_\rho^{-2} \rho^{\bar{k}} f_{\bar{k}},$$ which is clearly equivalent to .
To prove (ii), we notice that $$X_{\bar{j}\bar{k}} f
= \rho_{\bar{k}} f_{\bar{j}} - \rho_{\bar{j}} f_{\bar{k}}.$$ Therefore, $$\begin{aligned}
X_{pq}X_{\bar{j}\bar{k}} f
= &
\rho_q \rho_{\bar{k}} f_{\bar{j}p} + \rho_{q}\rho_{\bar{k}p}f_{\bar{j}} - \rho_{q}\rho_{\bar{j}p}f_{\bar{k}} - \rho_{q}\rho_{\bar{j}} f_{\bar{k} p} \notag \\
&
-\rho_{p}\rho_{\bar{k}} f_{\bar{j} q} - \rho_{p}\rho_{\bar{k}q} f_{\bar{j}}
+\rho_{p}\rho_{\bar{j}} f_{\bar{k}q} + \rho_{p}\rho_{\bar{j}q}f_{\bar{k}}.\end{aligned}$$ Contracting both sides with $\rho^{p\bar{k}}\rho^{q\bar{j}}$, using (i), we easily obtain (ii).
An estimate for eigenvalues and proof of Theorem \[thm:special\]
================================================================
We denote by $S\colon L^2(M) \to \ker \Box_b$ ($=\ker \bar{\partial}_b$) the Szegő orthogonal projection with respect to the volume measure $dv:=\theta\wedge (d\theta)^n$. It is well-known that if $M$ is embeddable, then $\mathrm{Spec}(\Box_b)$ consists of zero and a sequence of point eigenvalues $\{\lambda_k\}$ increasing to infinity. The positive eigenvalues of $\Box_b$ are of finite multiplicity and eigenfunctions are smooth [@BG; @BE]. Furthermore, we have the following orthogonal decomposition: $$\begin{aligned}
L^2(M,dv)
=
\bigoplus_{k=0}^{\infty} E_k, \quad E_0 = \ker \Box_b.\end{aligned}$$ Note that $E_0$ is of infinite dimension.
\[thm:generalestimate\] Let $(M,\theta)$ be an embedded compact strictly pseudoconvex pseudohermitian manifold and $0=\lambda_0<\lambda_1<\lambda_2<\cdots <\lambda_k <\cdots $ the eigenvalues for $\Box_b$. Define $$m(a)=\inf\left\{\left|a - \frac{1}{\lambda_k}\right|^2: k\in {\mathbb{N}}\right\},
\quad
M(a) = \sup\left\{\left|a - \frac{1}{\lambda_k}\right|^2: k\in {\mathbb{N}}\right\}.$$ Then for any $a\in \mathbb{R}$, any function $u \not \in \ker\Box_b$, $$\label{e:mainestimate}
(m(a) - a^2) \|\Box_b u\|^2 \leq \|u - S(u)\|^2 - \int_M |\bar{\partial}_b u|^2 \leq (M(a) - a^2)\|\Box_b u\|^2.$$
Let $E_{k}$ be the eigenspace of $\Box_b$ associated to the eigenvalue $\lambda_k$. Then $m_k:=\hbox{dim}(E_k)<\infty$. Let $\{f_{k, j}\}_{j=1}^{m_k}$ be an orthonormal basis for $E_{k}$. For any $k,\ell$, using integration by parts, we obtain $$\int_M (\Box_b u - \lambda_k u) \bar{f}_{k,\ell}
= \int_M ( u\overline{\Box_b f_{k,\ell}} - \lambda_k u \bar{f}_{k,\ell})
= \int_M u(\overline{\Box_b f_{k,\ell} - \lambda_k f_{k,\ell}}) = 0.$$ This implies that for any real number $a$, $$\langle u-a \Box_b u, f_{k,\ell}\rangle = -(a-1/\lambda_k)\langle \Box_b u,f_{k,\ell}\rangle.$$ Therefore, since $\Box_b u \in (\ker \Box_b)^{\perp}$, $$\begin{aligned}
M(a)\|\Box_b u\|^2
= &
\sum_{k=1}^{\infty}\sum_{\ell=1}^{m_k}M(a)\left|\langle \Box_b u, f_{k,\ell}\rangle\right|^2 \label{e:1}\\
\geq &
\sum_{k=1}^{\infty}\sum_{\ell=1}^{m_k} \left|a-\frac{1}{\lambda_k}\right|^2\left|\langle \Box_b u, f_{k,\ell}\rangle\right|^2 \\
= &
\sum_{k=1}^{\infty}\sum_{\ell=1}^{m_k} \left|\langle u-a\,\Box_b u, f_{k,\ell}\rangle\right|^2 \\
= &
\|u-a\,\Box_b u\|^2 - \|S(u-a\,\Box_b u)\|^2 \\
= &
\|u\|^2 + a^2 \|\Box_b u\|^2 - 2a \int_M \bar{u}\, \Box_b u - \|S(u)\|^2.\end{aligned}$$ Here we have used $\|S(u-a\, \Box_b u)\|^2 = \|S(u)\|^2$. We conclude that $$(M(a)-a^2) \|\Box_b u\|^2 \geq \|u-S(u)\|^2 - 2a \int_M |\bar{\partial}_b u|^2.$$ This proves the second inequality. The first inequality can be proved similarly.
The following two corollaries are undoubtedly known, but we can not find in the literature.
\[cor:32\] Let $(M,\theta)$ be as in Theorem \[thm:generalestimate\], then $$\lambda_1
=
\inf\left\{\bigl\| \Box_b u\bigr\|^2 \colon \int_M |\bar{\partial}_b u|^2 = 1\right\}
=
\inf\left\{\int_M |\bar{\partial}_b u|^2 \colon \|u-S(u)\|^2 =1\right\}.$$
For any $a> \frac{1}{\lambda_1}$, we have $$m(a) = \left|a - \frac{1}{\lambda_1}\right|^2.$$ From Theorem \[thm:generalestimate\], we have for any $u$ with $\int_M |\bar{\partial}_b u|^2 = 1$, $$\left[\left|a - \frac{1}{\lambda_1}\right|^2 - a^2\right]\bigl\| \Box_b u\bigr\|^2
\leq \|u-S(u)\|^2 - 2a.$$ This is equivalent to $$\left(\frac{1}{\lambda_1}\right)^2
-2a\left(\frac{1}{\lambda_1} - \frac{1}{\| \Box_b u\|^2}\right)
\leq
\frac{ \|u-S(u)\|^2}{\| \Box_b u\|^2}.$$ Letting $a\to +\infty$, we easily obtain $$\lambda_1 \leq \| \Box_b u\|^2.$$ Since $u$ is arbitrary, we conclude that $$\lambda_1 \leq \inf\left\{\bigl\| \Box_b u\bigr\|^2 \colon \int_M |\bar{\partial}_b u|^2 = 1\right\}.$$ The reverse inequality is trivial. To prove the second we take $a = \frac{1}{2}\frac{1}{\lambda_1}$ and notice that $M(a) = a^2$. Then from Theorem \[thm:generalestimate\], we deduce that for any $u$ satisfying $\|u-S(u)\|^2 = 1$, $$0=(M(a)-a^2) \|\Box_b u\|^2 \geq \|u-S(u)\|^2 - 2a \int_M |\bar{\partial}_b u|^2
=1- 2a \int_M |\bar{\partial}_b u|^2.$$ Hence, $$\lambda_1
=
\frac{1}{2a}
\leq \int_M |\bar{\partial}_b u|^2.$$ The proof of the reverse inequality is simple and omitted.
Let $(M,\theta)$ be as in Theorem \[thm:generalestimate\], then for any function $u$, $$\|u-S(u)\|\cdot \|\Box_b u\|
\geq
\int_M |\bar{\partial}_b u|^2.$$
Without lost of generality, we may assume that $\int_M |\bar{\partial}_b u|^2 = 1$. For each $k$, we take $a_k = \frac{1}{2}\left(\frac{1}{\lambda_k}+\frac{1}{\lambda_{k+1}}\right)$. Clearly, $$m(a_k) = \left|a_k - \frac{1}{\lambda_k}\right|^2 = \left|a_k - \frac{1}{\lambda_{k+1}}\right|^2$$ By Theorem \[thm:generalestimate\], we have $$\left[\left|a_k - \frac{1}{\lambda_k}\right|^2 - a_k^2\right] \|\Box_b u\|^2
\leq \|u-S(u)\|^2 - 2a_k.$$ By direct calculation, we have that $$\label{e:syli}
\left(\frac{1}{\lambda_{k}} - \frac{1}{\|\Box_b u\|^2}\right)\left(\frac{1}{\lambda_{k+1}} - \frac{1}{\|\Box_b u\|^2}\right)
\geq
\frac{1}{\|\Box_b u\|^4} - \frac{\|u-S(u)\|^2}{\|\Box_b u\|^2}.$$ By Corollary \[cor:32\], $\lambda_1 \leq \|\Box_b u\|^2$. Moreover, $\lambda_k \to \infty$ as $k\to \infty$. We deduce that there exists $k_0$ such that $$\lambda_{k_0} \leq \|\Box_b u\|^2 < \lambda_{k_0+1}.$$ Therefore, with $k=k_0$ implies that $$\frac{1}{\|\Box_b u\|^4} - \frac{\|u-S(u)\|^2}{\|\Box_b u\|^2} \leq 0.$$ This completes the proof.
\[prop:Bz\] Let $(M,\theta)$ be a compact strictly pseudoconvex pseudohermitian manifold. If there is a smooth non-CR function $f$ on $M$ such that $|\Box_b f|^2\le B(z) \Re {\bar{f}}\Box_b f$ for some non-negative function $B$ on $M$, then $$\lambda_1\le \max_M B(z).$$ If the equality holds, then $B$ must be a constant.
Since $|\Box_b f|^2\le B(z) \Re {\bar{f}}\Box_b f$, by Corollary \[cor:32\], $$\label{estimate:3.26}
\lambda_1 \int_M {\bar{f}}\Box_b f
\leq
\int_M |\Box_b f|^2
\leq
\int_M B(z) \Re ({\bar{f}}\Box_b f).$$ By the Mean Value Theorem of the integral, there is $z_0\in M$ such that $$0
\leq
\int_M (B-\lambda_1) \Re ({\bar{f}}\Box_b f)
=
(B(z_0)-\lambda_1)\int_M {\bar{f}}\Box_b f
=
(B(z_0)-\lambda_1)\int_M |\bar{\partial}_b f|^2.$$ This implies $$\lambda_1\le B(z_0)\le \max_M B(z).$$ It is clear that if $\lambda_1=\max_M B$ then $B$ is a constant.
We end this section by proving the Theorem \[thm:special\].
By the condition and the expression for the Kohn-Laplacian given by , we have $$\Box_b \rho_j
=
\tilde{\Delta}_{\rho} \rho_j + n|\partial\rho|_\rho^{-2} \rho^{{\bar{k}}} \rho_{j{\bar{k}}}
=\tilde{\Delta}_{\rho} \rho_j + n|\partial\rho|_\rho^{-2} \rho_{j}.$$ Then $$|\Box_b \rho_j|^2=\frac{n}{|{\partial}\rho|_\rho^2}\Re\left(\rho_{{\bar{j}}} \Box_b \rho_j +\rho_{{\bar{j}}} \tilde{\Delta}_{\rho}\rho_j+\tfrac{1}{n}|{\partial}\rho|_\rho^2\, |\tilde{\Delta}_{\rho} \rho_j|^2\right)
\leq
\frac{n}{|{\partial}\rho|_\rho^2}\Re \left( \rho_{{\bar{j}}} \Box_b \rho_j \right).$$ Applying Proposition \[prop:Bz\] with $B(z)=n |{\partial}\rho|^{-2}_\rho$, we obtain $$\lambda_1
\leq
n \max_M |{\partial}\rho|_\rho^{-2}.$$ The equality holds only if $|{\partial}\rho|_\rho$ is a constant on $M$. The proof of Theorem \[thm:special\] is complete.
Proof of Theorem \[thm:flat\]
=============================
The following theorem gives a sharp upper bound for $\lambda_1(\Box_b)$ in terms of the eigenvalues of the complex Hessian matrix $[\rho_{j{\bar{k}}}]$ and the length of $\partial\rho$. This theorem implies the estimate in Theorem \[thm:flat\].
\[thm:upperbound\] Let $\rho$ be a smooth strictly plurisubharmonic function defined on an open set $U$ of $\mathbb{C}^{n+1}$, $M$ a compact connected regular level set of $\rho$, and $\lambda_1$ the first positive eigenvalue of $\Box_b$ on $M$. Let $r (z)$ be the spectral radius of the matrix $[\rho^{j{\bar{k}}}(z)]$ and $s (z) = \mathrm{trace}\, [\rho^{j{\bar{k}}}] - r (z)$. Then $$\label{e:upperbound}
\lambda_1 \leq \frac{n^2\int_{M} r (z)|\partial\rho|^{-2}_{\rho}}{\int_M s (z)}.$$
Here the spectral radius of a square matrix is the maximum of the moduli of its eigenvalues.
First, we define $$C_j
=
\int_{M} \frac{|\rho^{{\bar{j}}}|^2}{|\partial\rho|_{\rho}^4},
\quad
D_{j}
=
\int_{M} \left(\rho^{j{\bar{j}}} - \frac{|\rho^{{\bar{j}}}|^2}{|\partial\rho|_{\rho}^2}\right).$$ From Proposition \[prop:kl\], we can compute $$\Box_b {\bar{z}}^j
=
n|\partial\rho|^{-2}_{\rho}\rho^{{\bar{j}}}.$$ Therefore, $$\|\Box_b {\bar{z}}^j \|^2
=
n^2\int_{M} \frac{|\rho^{{\bar{j}}}|^2}{|\partial\rho|_{\rho}^4}
=
n^2C_j.$$ We can also compute $$|\bar{\partial}_b \bar{z}^j|^2
=
\delta_{j\alpha}\delta_{j\beta}\left(\rho^{\alpha\bar{\beta}} - \frac{\rho^{\alpha}\rho^{\bar{\beta}}}{|\partial\rho|_{\rho}^2}\right)
=
\rho^{j{\bar{j}}} - \frac{|\rho^{{\bar{j}}}|^2}{|\partial\rho|_{\rho}^2}.$$ Here without lost of generality, we assume $j\ne n+1$. Therefore, $$\int_M |\bar{\partial}_b \bar{z}^j|^2
=
D_j.$$ Thus, from Corollary \[cor:32\] above, we obtain for all $j$, $$\label{e:lambdaest}
\lambda_1
\leq
n^2C_j/D_j.$$ Next, observe that $1/r (z)$ is the smallest eigenvalue of the Hermitian matrix $[\rho_{j{\bar{k}}}(z)]$, and thus, for all $(n+1)$-vector $v^j$, $$\frac{1}{r (z)} \sum_{j=1}^{n+1} |v^j|^2 \leq v^j\rho_{j{\bar{k}}} v^{{\bar{k}}}.$$ Plugging $v^j = \rho^j$ into the inequality, we easily obtain $\sum\limits_{j=1}^{n+1}|\rho^j|^2\leq r(z){|\partial\rho|^2_{\rho}} $. Consequently $$\label{e:cest}
\sum_{j} C_j
=
\sum_{j=1}^{n+1} \int_M \frac{|\rho^j|^2}{|\partial\rho|^4_{\rho}}
\leq
\int_M r (z)|\partial\rho|^{-2}_{\rho},$$ and therefore, $$\label{e:dest}
\sum_{j} D_j
=
\sum_{j=1}^{n+1} \int_{M} \left(\rho^{j{\bar{j}}} - \frac{|\rho^j|^2}{|\partial\rho|^2_{\rho}}\right)
\geq
\int_{M} \left[\operatorname{trace}[\rho^{j{\bar{k}}}] - r (z)\right]
=\int_M s (z).$$ Thus, from , , and , we obtain $$\lambda_1 \leq n^2\min_j (C_j/D_j) \leq \frac{n^2\sum_j C_j}{\sum_j D_j}
=
\frac{n^2\int_{M} r (z)|\partial\rho|^{-2}_{\rho}}{\int_M s (z)}.$$ The proof is complete.
Since $\rho_{j{\bar{k}}} = \delta_{jk}$, we have $r (z) = 1$ and $s (z) = n$. Therefore, by Theorem \[thm:upperbound\], $$\lambda_1
\leq
\frac{n^2\int_{M} r (z)|\partial\rho|^{-2}_{\rho}}{\int_M s (z)}
=\frac{n}{v(M)} \int_M |\partial\rho|^{-2}_{\rho}.$$ which proves the inequality.
Next we suppose that $\lambda_1 = \frac{n}{v(M)} \int_M |\partial\rho|_{\rho}^{-2}$. We shall show that $|\partial\rho|^{2}_{\rho}$ is constant along $M$. Put $$b_j = n^{-1}\Box_b {\bar{z}}^j = |\partial\rho|_{\rho}^{-2}\rho_j.$$ Then by inspecting the proof of Theorem \[thm:generalestimate\] above, in particular, the estimate , we have for all $j$, $$\label{e:c}
\langle b_j, f_{k,\ell}\rangle = 0,\quad \text{for all}\ \ell, \ \text{for all} \ k\ne 1.$$ Thus, $b_j \perp \ker\Box_b$ and imply that $b_j\in E_1$ (the eigenspace corresponding to $\lambda_1$). Therefore, $$\begin{aligned}
\Box_b b_j = \lambda_1 b_j.\end{aligned}$$ Recall that $\Box_b {\bar{z}}^j = nb_j$. We then deduce that $$\begin{aligned}
\Box_b\left[{\bar{z}}^j - \frac{n}{\lambda_1}\frac{\rho_{j}}{|\partial\rho|^2_{\rho}} \right]=0.\end{aligned}$$ Hence, ${\bar{z}}^j -n\rho^{{\bar{j}}}/(\lambda_1 |\partial \rho|^2_{\rho})$ restricted to $M$ is a CR function. Since $X_{\bar{l}\bar{k}}$ is a tangential CR vector fields on $M$, we have $$\begin{aligned}
X_{\bar{l}\bar{k}}\left[{\bar{z}}^j - \frac{n}{\lambda_1}\frac{\rho_j}{|\partial\rho|_{\rho}^2} \right]=0.\end{aligned}$$ By direct calculation, this is equivalent to $$\frac{n}{\lambda_1} \rho_j X_{\bar{l}\bar{k}} (|\partial\rho|_{\rho}^2)/ |\partial \rho|_{\rho}^4
=
\left(1- \frac{n}{\lambda_1|\partial \rho|_{\rho}^2}\right)\left(\rho_{\bar{l}}\delta_{jk} - \rho_{\bar{k}} \delta_{jl}\right).$$ Since $M$ is compact, there exists point $x\in M$ such that $$|\partial\rho(x)|_{\rho}^2 = \max_{M}|\partial\rho|_{\rho}^2.$$ At the maximum point $x$, we also have $X_{\bar{l}\bar{k}} |\partial\rho|_{\rho}^4= 0$. Thus, $$\left[1 - \frac{n}{\lambda_1 |\partial\rho|_{\rho}^2}\right] \left(\rho_{\bar{l}}\delta_{jk} - \rho_{\bar{k}} \delta_{jl}\right)
=0 \quad \text{at} \ x.$$ Since $\partial \rho(x) \ne 0$, we can assume that $\rho_{\bar{1}}(x) \ne 0$. Taking $j=k=2$, we have at $x$, $$1 - \frac{n}{\lambda_1 |\partial\rho|_{\rho}^2}
=
0.$$ Therefore, $$\min |\partial\rho|_{\rho}^{-2}
=
|\partial \rho(x)|_{\rho}^{-2}
=
\frac{\lambda_1}{n}
=
\frac{1}{v(M)}\int_{M} |\partial\rho|_{\rho}^{-2}.$$ As the right most term is the average of $|\partial \rho|_{\rho}^{-2}$ on $M$, we deduce from above that $|\partial \rho|_{\rho}^{-2}$ must be constant on $M$.
Finally, suppose that $|\partial\rho|_{\rho}^2$ is constant along $M$ and $\rho$ extends to the domain bounded by $M$ and satisfies $\rho_{j{\bar{k}}} = \delta_{jk}$ on the domain. We shall show in the lemma below that $M$ must be a sphere and complete the proof of Theorem \[thm:flat\].
\[lem:c\] Let $M$ be a compact connected regular level set of $\rho$ which bounds a domain $D$. Suppose that $\rho_{j{\bar{k}}} = \delta_{jk}$ on $D$. If $|\partial \rho|_{\rho}^2$ is constant on $M$, then $M$ must be a sphere.
The proof is an application of Serrin’s theorem [@Ser Theorem 1]. Let $D$ be the domain with $M$ is its boundary. Define $u = \rho - \nu$ on a neighborhood of $\overline{D}$. Since $M$ is smooth and the function $u$ satisfies $\Delta u=-4(n+1)$ in $D$, $u=0$ on ${\partial}D$, and the normal derivative ${\partial}u/{\partial}\mathbf{n} = 2|{\partial}\rho|_{\rho}$ is a constant on ${\partial}D$ by assumption, we can apply the Serrin’s theorem to conclude that $M$ is a standard sphere.
We end this section by the following example which gives a sharp upper bound on the family of compact level sets of Kähler potentials of Fubini-Study metric. This example also shows that the condition in Theorem \[thm:special\] can not be relaxed.
\[cex\] Let $\rho$ be a strictly plurisubharmonic function of the form $$\label{e:fsp}
\rho(Z) = \log (1+\|Z\|^2) + \psi(Z,\bar{Z}),$$ where $\psi$ is a real-valued pluriharmonic function. We suppose that $\rho$ is defined and proper in some domain $U \subset \mathbb{C}^{n+1}$ (e.g., if $\psi = -\log |z_1|$, then $\rho$ is defined and proper on $(\mathbb{C}\setminus\{0\}) \times \mathbb{C}^n$).
Observe that $$\rho_{j{\bar{k}}}
=
\frac{1}{1+\|Z\|^2}\left(\delta_{jk} - \frac{{\bar{z}}^{j} z^k}{1+\|Z\|^2}\right), \quad
\rho^{j{\bar{k}}}
=
(1+\|Z\|^2)\left(\delta_{jk} + {\bar{z}}^{k} z^j\right),$$ By a routine calculation, we see that the characteristic polynomial of $[\rho^{j{\bar{k}}}]$ is $$P_{[\rho^{j{\bar{k}}}]}(\lambda) = (1+\|Z\|^2-\lambda)^n\bigl[(1+\|Z\|^2)^2-\lambda\bigr].$$ Thus, the spectral radius of $[\rho^{j{\bar{k}}}]$ is $r (Z) = (1+\|Z\|^2)^2$ and $s (Z) = \operatorname{trace}[\rho^{j{\bar{k}}}] - r (Z) = n(1+\|Z\|^2)$. By Theorem \[thm:upperbound\], if $M$ is a compact, connected, regular level set of $\rho$, then $$\lambda_1
\leq
\frac{n\int_M (1+\|Z\|^2)^2 |\partial\rho|_{\rho}^{-2}}{\int_M (1+\|Z\|^2)} \leq n\max_{M} (1+\|Z\|^2)|\partial\rho|_{\rho}^{-2}.$$ Notice that if $\psi=0$ and then $M_{\nu}:=\rho^{-1}(\nu)$ is the sphere $\|Z\|^2 = e^{\nu}-1$ with $$\theta
=
ie^{-\nu}\sum_{j=1}^{n+1} \left(z^j d{\bar{z}}^j - {\bar{z}}^j dz^j\right).$$ Moreover, $|\partial\rho|^2_{\rho} = e^{\nu}-1$ on $M_{\nu}$ and $\lambda_1 = n e^{\nu}/(e^{\nu}-1)$. Therefore, the condition in Theorem \[thm:special\] can not be relaxed.
The real ellipsoids: Proof of Corollary \[cor:ellipsoid\]
=========================================================
The proof of Theorem \[cor:ellipsoid\] follows from Theorem \[thm:upperbound\] and the proposition below.
\[prop:a\] Let $Q(Z)$ be a quadratic polynomial and let $M_{\nu} = \rho^{-1}(\nu)$ be a compact regular level set of $\rho$, where $\rho$ is given by $$\label{e:elip}
\rho(Z) = \sum_{k=1}^{n+1} |z^k|^2 + 2\Re Q(Z)$$ Then $$C_{\nu}: = \frac{1}{v(M_{\nu})} \int_{M_{\nu}} |\partial \rho|^{-2} = \frac{1}{\nu}.$$
We observe that $$\begin{aligned}
\Re \sum_{j=1}^{n+1} z^j \rho^{{\bar{j}}}
=
\sum_{j=1}^{n+1} |z^j|^2 + \Re \sum_{j=1}^{n+1} z^jQ_j
=
\nu - 2\Re Q + \Re \sum_{j=1}^{n+1} z^jQ_j.\end{aligned}$$ As $Q$ is a quadratic polynomial, we can check that $
\sum_{j=1}^{n+1} z ^j Q_j = 2Q.
$ Hence $$\Re \sum_{j=1}^{n+1} z^j \rho^{{\bar{j}}} = \nu
\quad
\text{on} \ M_{\nu}.$$ Therefore, $$\begin{aligned}
\int_{M_{\nu}} |\partial \rho|_{\rho}^{-2}
=
\Re \frac{1}{\nu} \sum_{j=1}^{n+1}\int_{M_{\nu}} \frac{z^j\rho_j}{|\partial \rho|_{\rho}^2}
=
\Re \frac{1}{\nu} \sum_{j=1}^{n+1} \int_{M_{\nu}} \frac{(z^j+\overline{Q_j})\rho_j}{|\partial \rho|_{\rho}^2}
=
\frac{v(M)}{\nu},\end{aligned}$$ Here, we use $$\int_{M_{\nu}} \frac{\overline{Q_j} \rho_j}{|\partial \rho|_{\rho}^2}
=
\frac{1}{n}\int_{M_{\nu}} \overline{Q_j} \Box_b \bar{z}^j
=
\frac{1}{n}\int_{M_{\nu}} {\bar{z}}^j \overline{\Box_b Q_j}
=
0.$$ Hence, $C_{\nu} = \frac{1}{\nu}$.
From Theorem \[thm:flat\] and Proposition \[prop:a\], we have $$\lambda_1
\leq nC_{\nu}
=
\lambda_1(\sqrt{\nu}\,\mathbb{S}^{2n+1}).$$ Also from Theorem \[thm:flat\], we see that the equality occurs if and only if $M$ is the sphere and hence the proof is complete. However, we provide an elementary proof of this last step below. Notice that $$Q(Z)=\sum\limits_{k,j=1}^n q_{jk} z_k z_j$$ and $Q=\big[q_{jk}\big]$ is $n\times n$ symmetric matrix. By an well-known factorization theorem (see [@Hua Section 3.5]), we can write $Q=U^t \Lambda U$, where $U$ is a unitary matrix and $\Lambda=\hbox{Diag}(A_1,\cdots, A_{n+1})$ is a diagonal matrix with $A_j\ge 0$. We make a holomorphic unitary change of variables $W=U Z$, then $$\rho(Z)=\|W\|^2+\Re \sum_{j=1}^{n+1} A_j w_j^2.$$ Without loss of generality, one may assume that $\rho(Z)=\|Z\|^2+\Re \sum\limits_{j=1}^{n+1}|z_j|^2$. Since $M_{\nu}$ is bounded, it is easy to see that $A_j<1$ for $1\le j\le n+1$. Notice that on $M_{\nu}$, $$c=|{\partial}\rho|_{\rho}^2=\nu+\sum_{j=1}^{n+1} A_j^2 |z_j|^2+\Re\sum_{j=1}^{n+1} A_j z_j^2=2\nu -|Z|^2 +\sum_{j=1}^{n+1} A_j |z_j|^2.$$ If the equality occurs, then $|\partial\rho|_{\rho}^2$ is a constant along $M_{\nu}$. Restricting $z=\lambda {\bf e}_j\in M_{\nu} $, one has $|z_j|^2 $ must be a constant. This can not be true unless $A_j=0$. This proves $\rho(Z)=\|Z\|^2$ and $M_{\nu}$ is a sphere centered at $0$ with radius $\sqrt{\nu}$.
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[^1]: 2000 [*Mathematics Subject Classification*]{}. 32V20, 32W10
[^2]: *Key words and phrases:* eigenvalue, Kohn-Laplacian
[^3]: The second author was partially supported by the Hu Guozan Study-Abroad Grant for graduates (China) for her visit to UC Irvine in 2015–2016 when part of this work was done. The third author was partially supported by the Qatar National Research Fund, NPRP project 7-511-1-098. Part of this work was done while the third author visited Fujian Normal University at Fuzhou, China in July 2016 which he thanks for supports and hospitality.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'Analytical approaches to model the structure of complex networks can be distinguished into two groups according to whether they consider an intensive (e.g., fixed degree sequence and random otherwise) or an extensive (e.g., adjacency matrix) description of the network structure. While extensive approaches—such as the state-of-the-art Message Passing Approach—typically yield more accurate predictions, intensive approaches provide crucial insights on the role played by any given structural property in the outcome of dynamical processes. Here we introduce an intensive description that yields almost identical predictions to the ones obtained with MPA for bond percolation. Our approach distinguishes nodes according to two simple statistics: their degree and their position in the core-periphery organization of the network. Our near-exact predictions highlight how accurately capturing the long-range correlations in network structures allows to easily and effectively compress real complex network data.'
author:
- Antoine
- Laurent
title: Percolation and the effective structure of complex networks
---
The structure of real complex networks lies somewhere in-between order and randomness [@Watts1998; @Goldenfeld1999; @Strogatz2005], with the consequence that it cannot typically be fully characterized by a concise set of synthesizing observables. This *irreductibility* explains why most theoretical approaches to model complex networks are inspired by statistical physics in that they consider ensembles of networks constrained by the values of observables (e.g. density of links, degree-degree correlations, clustering coefficient, degree/motif distribution) and otherwise organized randomly. These approaches have three notable advantages. First, they usually yield analytical treatment. Second, they are *intensive* in network size, meaning that their complexity scales with the support of the observables (i.e., sub-linearly with the numbers of nodes and links). Third, they provide null models, of which many have led to the identification of fundamental properties characterizing the structure of real complex networks [@Newman2010; @Barabasi2016].
Despite important leaps forward in recent years, these approaches still fail to capture enough information to systematically provide accurate quantitative predictions of most dynamical processes on real complex networks. The reason for this shortcoming is that the properties from which the ensembles are constructed are not constraining enough; the ensembles are “too large” such that the original real networks are exceptions, rather than typical instances, in the ensembles. As a result, the current state-of-the-art approach—the so-called *message passing approach* (MPA) [@Karrer2014]—requires the whole structure to be specified as an input (i.e., the adjacency matrix, or a transformation thereof). This method is interesting because it is mathematically principled, meaning that it yields *exact* results on trees, and offers inexact, albeit generally good, predictions on networks containing loops (i.e., most real complex networks) [@Radicchi2015].
However, by considering the whole structure of networks and thereby considering every link on equal footing, the accuracy of the MPA comes at a significant computational and conceptual cost. First, its time and space complexity are *extensive* in the number of links and therefore in the size of the network. Second, and most importantly, it does not provide any insight on the role played by any given structural property in the outcome of a dynamical process. With the MPA, getting good predictions comes at the expense of understanding what led to that outcome.
In this paper, we bridge the gap between intensive and extensive approaches to the mathematical modeling of bond percolation on networks. We introduce a random network ensemble that relies solely on an *intensive* description of the network structure that, nevertheless, yields predictions that are comparable to the ones from the MPA for most of the 111 real complex networks considered in this study. This ensemble is based on the *onion decomposition* (OD), a refined $k$-core decomposition [@Hebert-Dufresne2016a]. Critically, the OD can be translated into local connection rules allowing an exact mathematical treatment using probability generating functions (pgf) in the limit of large network size. This approach leads to exact predictions on trees like the MPA, and highlights the critical contribution of the OD to an accurate effective mathematical description of real complex networks.
![Illustration of the Onion Decomposition (OD) of a simple network. The number of the layer to which each node belongs is indicated and the different $k$-cores are shown using increasingly darker background shades. The color of each stub according the LCCM is also shown.[]{data-label="fig:illutrationOD"}](Fig1){width="\linewidth"}
Results and discussions {#results-and-discussions .unnumbered}
=======================
Most analytical models of complex networks rely on some variation of the tree-like approximation which assumes that complex networks have essentially no loops beyond some local structure of interest [@Karrer2010; @Allard2015]. While this approximation is inaccurate for the vast majority of real complex networks, it nevertheless allows an elegant mathematical treatment which typically works surprisingly well [@Melnik2011]. In the case of the MPA, the tree-like approximation implies that a lot of information given to the model is thrown away due to loops being included in the input information (i.e., the adjacency matrix) to then be mathematically ignored. We here propose to limit the information we give to our model by compressing complex networks following their tree-like decomposition. We therefore rely on a known peeling process, which iteratively removes leaves (i.e., the peripheral nodes of the network) to calculate the depth of every node in the *effective tree*.
Taking this information into account, we then focus on predicting the outcome of bond percolation on complex networks: a canonical problem of network science analogous to many applied problems such as disease propagation or network resilience [@Latora2017]. Given a network structure, this simple stochastic process consists in the occupation of each original link with probability $p$. We aim to predict the size of the largest connected component composed of occupied links, $S$, as well as the percolation threshold, $p_\mathrm{c}$, above which that component corresponds to a macroscopic fraction of the network. The outcome of percolation depends on structural properties at all scales, thus making it a good benchmark for theoretical network models.
Onion decomposition {#onion-decomposition .unnumbered}
-------------------
The $k$-core decomposition is a well-known network metric that identifies a set of nested maximal sub-networks—the $k$-cores—in which each node shares at least $k$ links with the other nodes [@Seidman1983; @Dorogovtsev2006]. A node belonging to the $k$-core but not to the $(k+1)$-core is said to be of *coreness* $k$ and to be part of the $k$-*shell*. Nodes with a high coreness are generally seen as more central whereas nodes with low corenesses are seen as being part of the periphery of the network. The onion decomposition (OD) refines the $k$-core decomposition by assigning a layer $l$ to each node to further indicate its *position* within its shell (e.g., in the middle of the layer or at its boundary). The OD therefore unveils the internal organization of each centrality shell and, unlike the original $k$-core decomposition, can be used to assess whether the structure of a core is more similar to a tree or to a lattice, among other things [@Hebert-Dufresne2016a].
The OD of a given network structure is obtained via the following pruning process (see Fig. \[fig:illutrationOD\]). First we remove every nodes with the smallest degree, $k_\mathrm{min}$; the coreness of these nodes is equal to $k_\mathrm{min}$ and they are part of the first layer ($l=1$). Removing these nodes may yield nodes whose *remaining* degree is now equal to or smaller than $k_\mathrm{min}$; these nodes must also be removed, have a coreness of $k_\mathrm{min}$ as well, but are part of the second layer ($l=2$). If removing nodes of the second layer yields new nodes with a remaining degree equal to or lower than $k_\mathrm{min}$, they will be part of the third layer ($l=3$), will have a coreness of $k_\mathrm{min}$ and will also be removed. This process is repeated until no new nodes with a remaining degree equal to or lower than $k_\mathrm{min}$ are left. We then update the value of $k_\mathrm{min}$ to reflect the lowest remaining degree and repeat this whole process until every node has been assigned a coreness and a layer (the layer number keeps increasing such that each layer corresponds to a unique coreness).
An efficient implementation of this procedure has a run-time complexity of $\mathcal{O}(L \log N)$, where $L$ and $N$ are respectively the number of links and nodes, which implies that the OD can be quickly obtained for virtually any real complex network [@Hebert-Dufresne2016a]. Most importantly, nodes belonging to a same layer are *topologically similar* with regard to the mesoscale centrality organization of the network. Because the layer of a node is only weakly related to its degree (i.e., the coreness of a node provides a lower bound to its degree), the pair layer-degree can therefore be used to indicate how well a node is connected, but also to indicate its “topological position” in the network. It therefore allows us to discriminate central nodes from peripheral ones which, based on their degree alone, would have otherwise been deemed identical.
Effective random network ensemble: the LCCM {#effective-random-network-ensemble-the-lccm .unnumbered}
-------------------------------------------
From the pruning process described above, it can be concluded that a node of coreness $c$ belonging to the $l$-th layer is in one of two scenarios. 1) It must have *exactly* $c$ links to nodes in layers $l^\prime \geq l$ if layer $l$ is the first layer of the $c$-shell (i.e., nodes in layer $l-1$ belong to the $c^\prime$-shell with $c^\prime < c$). 2) Otherwise, if it is not in the first layer of its $c$-shell, it must have *at least* $c+1$ links to nodes of layers $l^\prime \geq l-1$ and *at most* $c$ links to nodes of layers $l^\prime \geq l$. The distinction between the two scenarios is that nodes not in the first layer of their shell require at least one link to the previous layer to *anchor* them to their own layer. Also, the common feature of these scenarios is that a node of coreness $c$ needs at least $c$ links with nodes of equal or greater coreness.
By rewiring the links of a given network using a degree-preserving procedure [@Coolen2009; @Fosdick2016] while ensuring that the aforementioned rules are respected at all time, it is possible to explore the ensemble of all possible single networks with the same fixed layer-degree sequence (i.e., the sequence of every pairs $(l, k)$ in the original network). Exactly preserving the layers—and thus the coreness of every nodes—is of critical significance since previous rewiring approaches could only approximately preserve the $k$-core decomposition [@Hebert-Dufresne2013].
Additionally, the pair layer-degree assigned to each node can be used to enforce two-point correlations (i.e., the (layer-degree)–(layer-degree) correlations), thus reducing the size of a random network ensemble. This correlated ensemble can be explored via a double link swap Markov chain method preserving both the layer-degree sequence and the number of links within and between every node classes (i.e., nodes with the same layer-degree). One way to implement this method is by first choosing one link at random (e.g., joining nodes A and B) and then choosing another link at random (e.g., joining nodes C and D) among the links that are attached to at least one node whose layer-degree pair is the same at one of the two nodes connected by the first link (e.g., A and C have the same layer-degree) [@Colomer-de-Simon2013]. The two links are then swapped (e.g., A becomes connected to D and B to C) if no self-link or multi-link would be created. Doing so ensures that that both the degree sequence and the two-point correlations are preserved at all time. We call *layered and correlated configuration model* (LCCM) the ensemble of maximally random networks with a given joint layer-degree sequence and (layer-degree)–(layer-degree) correlations.
Since it preserves both the degree sequence and the degree-degree correlations, the LCCM is a subset of two commonly used random network ensembles defined by the *configuration model* (CM) [@Newman2002] and the *correlated configuration model* (CMM) [@Vazquez2003]; the latter being known for its fair accuracy in many applications [@Melnik2011]. The LCCM, however, distinguishes itself from these models (and other variants) by enforcing a mesoscopic organization via the layers of the OD. This feature has the critical advantage of making the LCCM a mathematically principled approached in the sense that it exactly preserves the structure of a wide variety of trees (see Fig. \[fig:trees\]). As we show below, this mesoscopic information accounts for a significant portion of the missing gap between the predictions of the intensive configuration models and the extensive, current state-of-the-art MPA.
Percolation on the LCCM {#percolation-on-the-lccm .unnumbered}
-----------------------
We adapt the approach of Ref. [@Allard2015] to solve bond/site percolation on the LCCM in the limit of large network size. This approach requires to specify 1) the classes of nodes, which here correspond to the distinct pairs layer-degree noted $(l,k)$, and 2) the colors of stubs (i.e., half-links), which in the LCCM are identified based on the layer $l^\prime$ of the neighboring node. More precisely, from the connection rules stated in the previous section, the LCCM requires to keep track of the number of links that each node in each layer $l$ shares with nodes i) in layers $l^\prime \geq l$, ii) in layer $l^\prime = l - 1$ and iii) in layers $l^\prime < l - 1$. We identify the corresponding half-links as red, black and green stubs, respectively. For instance, a link between nodes in layers 3 and 5 consists in a red stub stemming out of the node in layer 3 paired with a green stub belonging to the node in layer 5. Note that a link between two given layers can only consist in a unique pair of stub colors, and the only allowed combinations are red-red, red-black and red-green.
From the link correlation matrix $\mathbf{L}$, whose entries specify the fraction of links within and between every classes of nodes, we can derive the function (see Methods) $$\begin{aligned}
\label{eq:varphi_definition}
\varphi_{lk}(\bm{x}) = \sum_{k^\mathrm{r} k^\mathrm{b} k^\mathrm{g}} P_{lk}(k^\mathrm{r},k^\mathrm{b},k^\mathrm{g})
[x_{lk}^\mathrm{r}]^{k^\mathrm{r}} [x_{lk}^\mathrm{b}]^{k^\mathrm{b}} [x_{lk}^\mathrm{g}]^{k^\mathrm{g}}\end{aligned}$$ generating the probability $P_{lk}(k^\mathrm{r},k^\mathrm{b},k^\mathrm{g})$ that a node in class $(l,k)$ has $k^\mathrm{r}$ red stubs, $k^\mathrm{b}$ black stubs and $k^\mathrm{g}$ green stubs, given the connection rules of the LCCM. From the same link correlation matrix, we can also derive the functions (see Methods) $$\begin{aligned}
\label{eq:gamma}
\gamma_{lk}^\mathrm{\alpha}(\bm{x}) & = \sum_{l^\prime k^\prime} \sum_{\alpha^\prime\in\{\mathrm{r},\mathrm{b},\mathrm{g}\}} Q_{lk}^\alpha(l^\prime, k^\prime, \alpha^\prime) x_{l^\prime k^\prime}^{\alpha^\prime} \ ,\end{aligned}$$ for every $\alpha\in\{\mathrm{r},\mathrm{b},\mathrm{g}\}$, generating the probability $Q_{lk}^\alpha(l^\prime, k^\prime, \alpha^\prime)$ that a stub of color $\alpha$ stemming of a node of class $(l,k)$ is attached to a stub of color $\alpha^\prime$ belonging to a node in class $(l^\prime,k^\prime)$. Combining these two functions yields the pgf generating the distribution of the number of nodes of each class that are neighbors of a randomly chosen node of class $(l,k)$ $$\begin{aligned}
\label{eq:g}
g_{lk}(\bm{x}) = \varphi_{lk}(\bm{\gamma(\bm{x})}) \ .\end{aligned}$$ Note that this pgf also includes the colors of the stub through which these neighors are connected to the node of class $(l,k)$. Similarly, the number of such nodes that can be reached from a node of class $(l,k)$ that has itself been reached by one of its stubs of color $\alpha$ is $$\begin{aligned}
\label{eq:f}
f_{lk}^\alpha(\bm{x}) = \frac{1}{\langle k^\alpha \rangle_{lk}} \left. \frac{\partial \varphi_{lk}(\bm{x^\prime})}{\partial x_{lk}^{\prime\alpha}} \right|_{\bm{x^\prime}=\bm{\gamma(\bm{x})}} \ ,\end{aligned}$$ where $\langle k^\alpha \rangle_{lk} = \frac{\partial \varphi_{lk}(\bm{1})}{\partial x_{lk}^\alpha}$ is the average number of stubs of color $\alpha$ nodes of class $(l,k)$ have.
To compute the size of the extensive component, we assume that the networks in the ensemble are locally tree-like, which occurs in the limit of large network size or when the detailed structure of matrix $\mathbf{L}$ only permits exact trees (i.e., when loops are structurally impossible). We define $a_{lk}^\alpha$ as the probability that attempting to reach a node in class $(l,k)$ by one of its stubs of color $\alpha$ does not eventually lead to the extensive component. Noting $p$ the probability that links are occupied, the probabilities $\{a_{lk}^\alpha\}$ are the solution of $$\begin{aligned}
\label{eq:a_lk_self_consistency}
a_{lk}^\alpha = 1 - p + p f_{lk}^\alpha(\bm{a}) \ ,\end{aligned}$$ for all $l$, $k$ and $\alpha$. This last expression encodes the simple self-consistent argument that attempting to reach the node will not lead to the extensive component if 1) the link is unoccupied, which occurs with probability $1-p$, or if 2) the link is occupied, with probability $p$, but the attempts to reach the other neighbors of the node that has just been reached will all fail, which occurs with probability $f_{lk}^\alpha(\bm{a})$. Note that this argument relies on the assumption that the state of these neighbors are independent, which is true for a tree-like structure. Having solved Eq. , the relative size of the extensive component, $S$, is then given by the probability that a randomly chosen node is found in $S$ $$\begin{aligned}
S = 1 - \sum_{lk} P(l,k) g_{lk}(\bm{a}) \ ,\end{aligned}$$ where $P(l,k)$ is the fraction of nodes in class $(l,k)$ which can be extracted from the link correlation matrix $\mathbf{L}$ (see Methods). Notice that since we assume the networks of the ensemble to be tree-like, the relative size of the extensive component if nodes (instead of links) were occupied with probability $p$ is simply $S^\mathrm{site} = pS$ to account for the probability that the initial randomly chosen node is occupied. Note also that the percolation threshold, $p_\mathrm{c}$, is the value of $p$ at which $\bm{a}=\bm{1}$ becomes an unstable solution of Eq. (see Methods), which corresponds to the emergence of the extensive component.
Effective tree-like structure {#effective-tree-like-structure .unnumbered}
-----------------------------
Because it is a subset of both the CM and the CCM, the cardinality of the ensemble defined by the LCCM should, in principle, be smaller than the ensembles considered by the formers. Consequently, if the mesoscale structural information provided by the layers $l$ is of any significance, we expect the predictions of the LCCM to be the closest to the ones obtain with the MPA. Figures \[fig:bifurcation\] and \[fig:thresholds\] confirm this observation. In fact, our results demonstrate that identifying nodes using the layer in the OD alongside their degree does not merely improve the predictions, it drastically changes their nature, making them qualitatively very similar to the ones of the MPA when not strikingly quantitatively identical. As shown on Fig. \[fig:bifurcation\], the LCCM reproduces the general shape of the curves, has the same number of inflection points, and always predict a connected network when all links are occupied (i.e., $S$ must be 1 at $p=1$ since we considered the largest connected components of every datasets). Interestingly, only the LCCM and the MPA are able to capture the mesoscopic core-periphery and/or modular structures that were numerically shown to lead to smeared (or double) phase transitions [@Colomer-de-Simon2014] such as the one observed on the protein-protein interaction network.
Perhaps most importantly, the LCCM approximates to high accuracy the percolation threshold predicted by the MPA, as seen in Fig. \[fig:thresholds\](left), with an relative error of less than 1.5% for 75% of the 111 network datasets considered. Additionally, Fig. \[fig:thresholds\](right) shows the expected error on the size of the extensive component averaged over the entire range of occupation probability $p$. When using the LCCM to compress the network structure, we find that the error, relative to the MPA, to be of the order of $10^{-3}$ for 75% of the datasets considered; an improvement of at least one order of magnitude from existing approaches. Altogether, these results indicate that categorizing nodes with the classes $(l,k)$ captures critical features of the local and mesoscopic tree-like organization of many real complex networks, thus offering an intensive effective description of their structure.
Conclusion {#conclusion .unnumbered}
==========
We introduced a random network ensemble that relies solely on an *intensive* description of the network structure that, nevertheless, yields predictions for percolation that are either essentially quantitatively identical—or at least strikingly qualitatively similar—to the ones obtained with the state-of-the-art MPA. This ensemble assigns two structural features to each node—its degree $k$ (local) and its position $l$ in the Onion Decomposition of the network (mesoscale)—and creates links according to simple connection rules that exactly preserve these two features. This ensemble lends itself to exact analytical calculations using probability generating functions in the limit of large network size, and is mathematically principled, meaning that it leads to exact predictions on trees, like the MPA, but unlike other intensive approaches such as the configuration model and its variants. The accuracy of the predictions of the LCCM shows that the OD easily captures important features of the mesoscale structural organization of many real complex networks, and that this information should be leveraged by the future generations of models of complex networks.
For instance, Eq. , which provides the distribution of different link types (e.g., the number of links leading to lower or higher layers) for any node, could be straightforwardly included in equations for other problems such as the Susceptible-Infectious-Susceptible dynamics. It would thus be possible to track the fraction of infected nodes with a given pair $(l,k)$ whose time evolution would be driven by the transmission events along the connections prescribed by Eqs. –. In a purely numerical context, and using a simpler, less accurate version of the LCCM, this approach was already shown to lead to predictions of SIS dynamics that are an order of magnitude more precise than other network models [@Hebert-Dufresne2016a]. More generally, the pair $(l,k)$ consists in a straightforward and computationally inexpensive observable to characterize and rank nodes based on their local connectivity (through $k$) and global centrality (through $l$).
Finally, the accuracy of the LCCM strongly suggests that the long-range correlations induced by the OD effectively emulate the correlations considered in the MPA, and, consequently, that a large chunk of the structural properties behind the accuracy of the MPA now lend themselves to intensive analytical treatment. This opens the way for future work to focus on bringing the analytical modeling of complex networks beyond the ubiquitous tree-like approximation. Doing so should provide a unified framework for random graphs, regular structures like lattices, and the complex networks that lie in-between.
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Methods {#methods .unnumbered}
=======
Link correlation matrix {#link-correlation-matrix .unnumbered}
-----------------------
We define the symmetrical link correlation matrix $\mathbf{L}$ whose elements, $L_{lk, l^\prime k^\prime}$, correspond to the fraction of links between nodes of class $(l,k)$ and $(l^\prime,k^\prime)$. It has the following properties $$\begin{aligned}
\frac{1}{2} \sum_{lk} \sum_{l^\prime k^\prime} (1 + \delta_{ll^\prime}\delta_{kk^\prime}) L_{lk, l^\prime k^\prime} = 1 \ ,\end{aligned}$$ since each type of links appears twice in the matrix except for the links connecting nodes of the same class (i.e., diagonal elements), and $$\begin{aligned}
\label{eq:links_vs_stubs}
\frac{1}{2} \sum_{l^\prime k^\prime} (1 + \delta_{ll^\prime}\delta_{kk^\prime}) L_{lk, l^\prime k^\prime} = \frac{k P(l,k)}{\langle k \rangle} \ ,\end{aligned}$$ where $P(l,k)$ is the fraction of nodes belonging to the class $(l,k)$ and $\langle k \rangle = \sum_{lk}kP(l,k)$ is the average degree.
Distribution of the number and of the color of stubs {#distribution-of-the-number-and-of-the-color-of-stubs .unnumbered}
----------------------------------------------------
The connection rules of the LCCM indicate that a node of degree $k$ in layer $l$ and coreness $c_l$ have at most $c_l$ red stubs. Since red stubs are defined as half-links toward nodes in layers $l^\prime \geq l$, they represent a fraction $$\begin{aligned}
\frac{1}{2}\sum_{l^\prime \geq l} \sum_{k^\prime} (1 + \delta_{ll^\prime}\delta_{kk^\prime}) L_{lk, l^\prime k^\prime}\end{aligned}$$ of all stubs in the network ensemble, where $\delta_{ll^\prime}\delta_{kk^\prime}$ accounts for the fact that a link connecting two nodes of class $(l,k)$ contribute to two red stubs. This last quantity would be equal to $$\begin{aligned}
\frac{c_l P(l,k)}{\langle k \rangle} \end{aligned}$$ if every of these nodes had exactly $c_l$ red stubs. Consequently, since the LCCM only dictates bounds on the number of each color, the probability that a node of degree $k$ in layer $l$ has exactly $k^\mathrm{r}$ red stubs is simply $$\begin{aligned}
\label{eq:prob_number_red_stubs}
\binom{c_l}{k^\mathrm{r}}& \big[ p_{lk}^{\mathrm{r}} \big]^{k^\mathrm{r}} \big[ 1 - p_{lk}^{\mathrm{r}} \big]^{c_l-k^\mathrm{r}}\end{aligned}$$ where $$\begin{aligned}
\label{eq:prob_red}
p_{lk}^\mathrm{r} & = \frac{\sum_{l^\prime \geq l} \sum_{k^\prime} (1 + \delta_{ll^\prime}\delta_{kk^\prime}) L_{lk, l^\prime k^\prime}}{2 c_l P(l,k) / \langle k \rangle} \ .\end{aligned}$$ Note that whenever layer $l$ is the first layer of its core—when $c_{l}>c_{l-1}$—Eq. reduces to $p_{lk}^{\mathrm{r}}=1$ meaning that each node has exactly $c_l$ red stubs, as prescribed by the connection rules of the LCCM.
Similarly, the fraction of half-links shared with nodes in layers $l^\prime<l-1$ (i.e., green stubs) is $$\begin{aligned}
\label{eq:fraction_green_stubs}
\frac{1}{2} \sum_{l^\prime < l-1} \sum_{k^\prime} L_{lk, l^\prime k^\prime} \ .\end{aligned}$$ The maximal value of this quantity, however, varies in function of $l$. If the layer is the first layer of its shell (i.e., if $c_{l}>c_{l-1}$), then each node has $c_l$ red stubs and up to $k-c_{l}$ green stubs according to the connections rules. If $c_l=c_{l-1}$, nodes that have exactly $c_l$ red stubs can have up to $k-c_l-1$ green stubs since they must have at least one black stubs, and can have up to $k-c_l$ otherwise. The maximal value of Eq. can therefore be summarized as $$\begin{aligned}
\frac{(k - c_l - \delta_{k^\mathrm{r},c_l}\delta_{c_l,c_{l-1}}) P(l,k)}{\langle k \rangle} \ ,\end{aligned}$$ such that the probability that a node of degree $k$ in layer $l$ has exactly $k^\mathrm{g}$ green stubs is $$\begin{aligned}
\label{eq:prob_number_green_stubs}
\binom{k-c_l-\delta_{k^\mathrm{r},c_l}\delta_{c_l,c_{l-1}}}{k^\mathrm{g}} \left[ \frac{k-c_l}{k-c_l-\delta_{k^\mathrm{r},c_l}\delta_{c_l,c_{l-1}}} p_{lk}^{\mathrm{g}} \right]^{k^\mathrm{g}} \left[ 1 - \frac{k-c_l}{k-c_l-\delta_{k^\mathrm{r},c_l}\delta_{c_l,c_{l-1}}} p_{lk}^{\mathrm{g}} \right]^{k-c_l-k^\mathrm{g}-\delta_{k^\mathrm{r},c_l}\delta_{c_l,c_{l-1}}}\end{aligned}$$ with $$\begin{aligned}
p_{lk}^\mathrm{g} = \frac{\sum_{l^\prime < l-1} \sum_{k^\prime} L_{lk, l^\prime k^\prime}}{2(k - c_l) P(l,k)/\langle k \rangle} \ .\end{aligned}$$ Combining Eqs. and yields the probability that a node in layer $l$ and of degree $k$ has $k^\mathrm{r}$, $k^\mathrm{g}$ and $k^\mathrm{b}$ red, green and blacks stubs, respectively $$\begin{aligned}
\label{eq:stub_number_and_color_distribution}
&P_{lk}(k^\mathrm{r},k^\mathrm{g},k^\mathrm{b}) = \displaystyle \delta_{k,k^\mathrm{r}+k^\mathrm{b}+k^\mathrm{g}} \binom{c_l}{k^\mathrm{r}} \big[ p_{lk}^{\mathrm{r}} \big]^{k^\mathrm{r}} \big[ 1 - p_{lk}^{\mathrm{r}} \big]^{c_l-k^\mathrm{r}} \nonumber \\
& \times \binom{k-c_l-\delta_{k^\mathrm{r},c_l}\delta_{c_l,c_{l-1}}}{k^\mathrm{g}} \left[ \frac{k-c_l}{k-c_l-\delta_{k^\mathrm{r},c_l}\delta_{c_l,c_{l-1}}} p_{lk}^{\mathrm{g}} \right]^{k^\mathrm{g}} \left[ 1 - \frac{k-c_l}{k-c_l-\delta_{k^\mathrm{r},c_l}\delta_{c_l,c_{l-1}}} p_{lk}^{\mathrm{g}} \right]^{k-c_l-k^\mathrm{g}-\delta_{k^\mathrm{r},c_l}\delta_{c_l,c_{l-1}}} \ .\end{aligned}$$ Finally, after some elementary algebra, it can be shown that the generating function $\varphi_{lk}(\bm{x})$ associated with this distribution is $$\begin{aligned}
\label{eq:varphi}
\varphi_{lk}(\bm{x}) & = \sum_{k^\mathrm{r},k^\mathrm{b},k^\mathrm{g}} P(k^\mathrm{r},k^\mathrm{g},k^\mathrm{b}|l,k)
[x_{lk}^{\mathrm{r}}]^{k^\mathrm{r}} [x_{lk}^{\mathrm{g}}]^{k^\mathrm{g}} [x_{lk}^{\mathrm{b}}]^{k^\mathrm{b}} \nonumber \\
& = \delta_{c_l, c_{l-1}}
x_{lk}^\mathrm{b} \left[ p_{lk}^{\mathrm{r}} x_{lk}^{\mathrm{r}} \right]^{c_l}
\left[ \left(1 - \frac{k-c_l}{k-c_l-1} p_{lk}^{\mathrm{g}}\right) x_{lk}^{\mathrm{b}} + \frac{k-c_l}{k-c_l-1} p_{lk}^{\mathrm{g}} x_{lk}^{\mathrm{g}} \right]^{k-c_l-1} \nonumber \\
& \qquad\qquad\qquad\qquad\qquad
- \delta_{c_l,c_{l-1}}
\left[ p_{lk}^{\mathrm{r}} x_{lk}^{\mathrm{r}} \right]^{c_l}
\left[ \left(1 - p_{lk}^{\mathrm{g}}\right) x_{lk}^{\mathrm{b}} + p_{lk}^{\mathrm{g}} x_{lk}^{\mathrm{g}} \right]^{k-c_l} \nonumber \\
& \qquad\qquad\qquad\qquad\qquad
+ \left[ (1 - p_{lk}^{\mathrm{r}}) x_{lk}^{\mathrm{b}} + p_{lk}^{\mathrm{r}} x_{lk}^{\mathrm{r}} \right]^{c_l}
\left[ (1 - p_{lk}^{\mathrm{g}}) x_{lk}^{\mathrm{b}} + p_{lk}^{\mathrm{g}} x_{lk}^{\mathrm{g}} \right]^{k-c_l} \ .\end{aligned}$$
Transition probabilities {#transition-probabilities .unnumbered}
------------------------
With the distribution of the number of stubs of each color that nodes have being provided by Eq. , the only missing quantities are the transition probabilities: the probability $Q_{lk}^\alpha(l^\prime,k^\prime,\alpha^\prime)$ that a stub of color $\alpha$ stemming from a node of class $(l,k)$ leads to a stub of color $\alpha^\prime$ attached to a node of class $(l^\prime,k^\prime)$. Once more, this information can be extracted from the link correlation matrix $\mathbf{L}$.
Let us recall that black stubs stemming from nodes of class $(l,k)$ can only lead to red stubs attached to nodes in the previous layer (i.e., $l^\prime=l-1$), which can be summarized by $$\begin{aligned}
\label{eq:black_to_red_stubs}
Q_{lk}^\mathrm{b}(l^\prime,k^\prime,\alpha^\prime) = \frac{\delta_{\alpha^\prime,\mathrm{r}} \delta_{l^\prime,l-1} L_{l^{\prime}k^{\prime},lk}}{\sum_{l^{\prime\prime}}\sum_{k^{\prime\prime}}\delta_{l^{\prime\prime},l-1}L_{l^{\prime\prime}k^{\prime\prime},lk}} \ ,\end{aligned}$$ where the denominator is proportional to the fraction of all stubs that are black and that are stemming from nodes of class $(l,k)$. Similarly, since green stubs can only lead to red stubs attached nodes in layer $l^\prime<l-1$, we have $$\begin{aligned}
\label{eq:green_to_red_stubs}
Q_{lk}^\mathrm{g}(l^\prime,k^\prime,\alpha^\prime) =
\left\{
\begin{array}{cc}
\displaystyle\frac{\delta_{\alpha^\prime,\mathrm{r}} L_{l^\prime k^\prime, lk}}{\sum_{l^{\prime\prime}<l-1} \sum_{k^{\prime\prime}} L_{l^{\prime\prime}k^{\prime\prime},lk}} & \text{if } l^\prime < l-1 \\
&\\
0 & \text{otherwise}
\end{array}\right. \ .\end{aligned}$$ Because red stubs can lead to all three colors of stubs, we first consider the case where a red stubs leads to a black stubs (i.e., to a node in layer $l^\prime=l+1$), which corresponds to $$\begin{aligned}
Q_{lk}^\mathrm{r}(l^\prime,k^\prime,\mathrm{b}) = \frac{\delta_{l^\prime,l+1}L_{lk,l^\prime k^\prime}}{\sum_{l^{\prime\prime} \geq l} \sum_{k^{\prime\prime}} (1 + \delta_{ll^{\prime\prime}}\delta_{kk^{\prime\prime}}) L_{lk,l^{\prime\prime}k^{\prime\prime}}} \ ,\end{aligned}$$ where the denominator is proportional to the fraction of all stubs that corresponds to red stubs stemming from nodes of class $(l,k)$. In the case of red stubs leading to red stubs—i.e., links between nodes in the same layer—, we need to double the contribution of $L_{lk,lk}$ since each link between nodes of the same class contributes to two red stubs, which yields
$$\begin{aligned}
Q_{lk}^\mathrm{r}(l^\prime,k^\prime,\mathrm{r}) = \frac{\delta_{ll^\prime}(1 + \delta_{kk^{\prime\prime}}) L_{lk,l^{\prime\prime}k^{\prime\prime}}}{\sum_{l^{\prime\prime} \geq l} \sum_{k^{\prime\prime}} (1 + \delta_{ll^{\prime\prime}}\delta_{kk^{\prime\prime}}) L_{lk,l^{\prime\prime}k^{\prime\prime}}} \ .\end{aligned}$$
The case of red stubs leading to green stubs is similar to Eq. and is straightforward to obtain $$\begin{aligned}
\label{eq:red_to_green_stubs}
Q_{lk}^\mathrm{r}(l^\prime,k^\prime,\mathrm{g}) =
\left\{
\begin{array}{cc}
\displaystyle\frac{L_{lk, l^\prime k^\prime}}{\sum_{l^{\prime\prime} \geq l} \sum_{k^{\prime\prime}} (1 + \delta_{ll^{\prime\prime}}\delta_{kk^{\prime\prime}}) L_{lk,l^{\prime\prime}k^{\prime\prime}}} & \text{if } l^\prime > l+1 \\
&\\
0 & \text{otherwise}
\end{array}
\right. \ .\end{aligned}$$ Finally, by injecting Eqs. – in Eq. , we obtain
$$\begin{aligned}
\gamma_{lk}^\mathrm{r}(\bm{x}) & = \frac{\sum_{l^{\prime} \geq l} \sum_{k^{\prime}} L_{lk,l^{\prime}k^{\prime}}
[\delta_{ll^\prime} (1 + \delta_{kk^{\prime}}) x_{l^\prime k^\prime}^\mathrm{r} + \delta_{ll^\prime-1}x_{l^\prime k^\prime}^\mathrm{b} + (1 - \delta_{ll^\prime})(1 - \delta_{ll^\prime-1})x_{l^\prime k^\prime}^\mathrm{g}]}{\sum_{l^{\prime\prime} \geq l} \sum_{k^{\prime\prime}} (1 + \delta_{ll^{\prime\prime}}\delta_{kk^{\prime\prime}}) L_{lk,l^{\prime\prime}k^{\prime\prime}}} \label{eq:gammas_r} \\
%
\gamma_{lk}^\mathrm{b}(\bm{x}) & = \frac{\sum_{k^\prime}L_{l-1k^\prime,lk}x_{l-1k^\prime}^\mathrm{r}}{\sum_{k^{\prime\prime}}L_{l-1k^{\prime\prime},lk}} \\
%
\gamma_{lk}^\mathrm{g}(\bm{x}) & = \frac{\sum_{l^\prime<l-1} \sum_{k^\prime} L_{l^\prime k^\prime,lk}x_{l^\prime\mathrm{r}}^\mathrm{r}}{\sum_{l^{\prime\prime}<l-1} \sum_{k^{\prime\prime}} L_{l^{\prime\prime}k^{\prime\prime},lk}} \label{eq:gammas_g} \ .\end{aligned}$$
Percolation threshold {#percolation-threshold .unnumbered}
---------------------
The value of the percolation threshold, $p_\mathrm{c}$, can be computed analytically by a linear stability analysis of the solution $\bm{a}=\bm{1}$ of Eq. . Substituing $a_{lk}^\alpha = 1 - \varepsilon_{lk}^\alpha$, where $\varepsilon_{lk}^\alpha \ll 1$, yields $$\begin{aligned}
\varepsilon_{lk}^{\alpha} = p \sum_{l^\prime k^\prime \alpha^\prime} \left. \frac{\partial f_{lk}^{\alpha}(\bm{x})}{\partial x_{l^\prime k^\prime}^{\alpha^\prime}} \right|_{\bm{x}=\bm{1}} \varepsilon_{l^\prime k^\prime}^{\alpha^\prime} \ ,\end{aligned}$$ when limiting the expansion of $ f_{lk}^{\alpha}(\bm{1} - \bm{\varepsilon})$ to the first order. The last equation can be rewritten as an eigenvalue problem $$\begin{aligned}
\bm{\varepsilon} = p \mathbf{M} \bm{\varepsilon} \ ,\end{aligned}$$ thus indicating that the fixed point $\bm{a}=\bm{1}$ looses its stability—i.e., the extensive component emerges—when the largest eigenvalue of $p\mathbf{M}$ exceeds 1. The percolation threshold, $p_\mathrm{c}$, therefore equals the reciprocal of the largest eigenvalue of $\mathbf{M}$ which, by virtue of the Perron-Frobenius theorem, is real and positive.
The elements of $\mathbf{M}$ can be written as $$\begin{aligned}
\frac{\partial f_{lk}^{\alpha}(\bm{1})}{\partial x_{l^\prime k^\prime}^{\alpha^\prime}}
= \frac{1}{\langle k^\alpha \rangle_{lk}} \sum_{\alpha^{\prime\prime}} \frac{\partial^2 \varphi_{lk}(\bm{1})}{\partial x_{lk}^{\alpha} \partial x_{lk}^{\alpha^{\prime\prime}}} \frac{\partial \gamma_{lk}^{\alpha^{\prime\prime}}(\bm{1})}{\partial x_{l^\prime k^\prime}^{\alpha^\prime}} \ ,\end{aligned}$$ where the derivatives are calculated directly from Eq. and Eqs. –. While the derivatives of $\gamma_{lk}^\alpha\bm{x}$ are straightforward, the derivatives of $\varphi_{lk}(\bm{x})$ require special care with respect to the value of $k-c_l$. To facilitate the numerical implementation of the formalism, we provide the explicit expression of the derivatives of $\varphi_{lk}(\bm{x})$.
$$\begin{aligned}
\langle k^\mathrm{r} \rangle_{lk} & = \frac{\partial \varphi_{lk}(\bm{1})}{\partial x_{lk}^\mathrm{r}} = c_l p_{lk}^\mathrm{r} \\
\langle k^\mathrm{g} \rangle_{lk} & = \frac{\partial \varphi_{lk}(\bm{1})}{\partial x_{lk}^\mathrm{g}} =
\left\{
\begin{array}{lc}
(k - c_l) p_{lk}^\mathrm{g} - \delta_{c_l, c_{l-1}} [p_{lk}^\mathrm{r}]^{c_l} (k - c_l) p_{lk}^\mathrm{g} & \text{if } k - c_l \leq 1 \\
&\\
(k - c_l) p_{lk}^\mathrm{g} & \text{otherwise}
\end{array}\right. \\
\langle k^\mathrm{b} \rangle_{lk} & = \frac{\partial \varphi_{lk}(\bm{1})}{\partial x_{lk}^\mathrm{b}} =
\left\{
\begin{array}{lc}
c_l (1 - p_{lk}^\mathrm{g}) + (k - c_l) (1 - p_{lk}^\mathrm{g}) \\
% \qquad\qquad -\ \delta_{c_l, c_{l-1}} [p_{lk}^\mathrm{r}]^{c_l} (k - c_l -1) \\
\qquad\qquad +\ \delta_{c_l, c_{l-1}} [p_{lk}^\mathrm{r}]^{c_l} (k - c_l) p_{lk}^\mathrm{g} & \text{if } k - c_l \leq 1 \\
&\\
c_l (1 - p_{lk}^\mathrm{r}) + (k - c_l) (1 - p_{lk}^\mathrm{g}) & \text{otherwise}
\end{array}\right.\end{aligned}$$
$$\begin{aligned}
\frac{\partial^2 \varphi_{lk}(\bm{1})}{\partial x_{lk}^{\mathrm{r}\,2}} & = c_l (c_l - 1) [p_{lk}^\mathrm{r}]^2 \\
%
\nonumber \\
%
\frac{\partial^2 \varphi_{lk}(\bm{1})}{\partial x_{lk}^{\mathrm{r}}\partial x_{lk}^{\mathrm{g}}} & =
\left\{
\begin{array}{lc}
c_l (k - c_l) p_{lk}^\mathrm{r} p_{lk}^\mathrm{g} \\
\qquad\qquad -\ \delta_{c_l,c_{l-1}} [p_{lk}^\mathrm{r}]^{c_l} c_l (k - c_l) p_{lk}^\mathrm{g} & \text{if } k - c_l \leq 1 \\
& \\
c_l (k - c_l) p_{lk}^\mathrm{r} p_{lk}^\mathrm{g} & \text{otherwise}
\end{array} \right. \\
%
\nonumber \\
%
\frac{\partial^2 \varphi_{lk}(\bm{1})}{\partial x_{lk}^{\mathrm{r}}\partial x_{lk}^{\mathrm{b}}} & =
\left\{
\begin{array}{lc}
c_l (c_l - 1) p_{lk}^\mathrm{r} (1 - p_{lk}^\mathrm{r}) + c_l (k - c_l) p_{lk}^\mathrm{r} (1 - p_{lk}^\mathrm{g}) \\
% \qquad\qquad +\ \delta_{c_l,c_{l-1}} [p_{lk}^\mathrm{r}]^{c_l} c_l \Big[1 - (k - c_l) (1 - p_{lk}^\mathrm{g}) \Big] & \text{if } k - c_l \leq 1 \\
\qquad\qquad +\ \delta_{c_l,c_{l-1}} [p_{lk}^\mathrm{r}]^{c_l} c_l (k - c_l) p_{lk}^\mathrm{g} & \text{if } k - c_l \leq 1 \\
& \\
c_l (c_l - 1) p_{lk}^\mathrm{r} (1 - p_{lk}^\mathrm{r}) + c_l (k - c_l) p_{lk}^\mathrm{r} (1 - p_{lk}^\mathrm{g}) & \text{otherwise}
\end{array} \right.\\
%
\nonumber \\
%
\frac{\partial^2 \varphi_{lk}(\bm{1})}{\partial x_{lk}^{\mathrm{g}\,2}} & =
\left\{
\begin{array}{lc}
(k - c_l) (k - c_l - 1) [p_{lk}^\mathrm{g}]^2 \\
\qquad\qquad -\ \delta_{c_l, c_{l-1}} [p_{lk}^\mathrm{r}]^{c_l} (k - c_l) (k - c_l - 1) [p_{lk}^\mathrm{g}]^2 & \text{if } k - c_l \leq 2 \\
& \\
(k - c_l) (k - c_l - 1) [p_{lk}^\mathrm{g}]^2 \\
\qquad\qquad -\ \delta_{c_l, c_{l-1}} [p_{lk}^\mathrm{r}]^{c_l} (k - c_l) (k - c_l - 1) [p_{lk}^\mathrm{g}]^2 \\
\qquad\qquad +\ \delta_{c_l, c_{l-1}} [p_{lk}^\mathrm{r}]^{c_l} (k - c_l)^2 [p_{lk}^\mathrm{g}]^2 \displaystyle \frac{k - c_l - 2}{k - c_l -1}& \text{otherwise}
\end{array} \right. \\
%\\
\nonumber \\
%
\frac{\partial^2 \varphi_{lk}(\bm{1})}{\partial x_{lk}^{\mathrm{g}}\partial x_{lk}^{\mathrm{b}}} & =
\left\{
\begin{array}{lc}
c_l (k-c_l) p_{lk}^\mathrm{g} (1 - p_{lk}^\mathrm{r}) & \text{if } k - c_l \leq 1 \\
& \\
c_l (k-c_l) p_{lk}^\mathrm{g} (1 - p_{lk}^\mathrm{r}) + (k - c_l) (k - c_l - 1) p_{lk}^\mathrm{g} (1 - p_{lk}^\mathrm{g}) \\
\qquad\qquad +\ \delta_{c_l, c_{l-1}} [p_{lk}^\mathrm{r}]^{c_l} (k - c_l) (k - c_l - 1) [p_{lk}^\mathrm{g}]^2 & \text{if } k - c_l = 2 \\
& \\
c_l (k-c_l) p_{lk}^\mathrm{g} (1 - p_{lk}^\mathrm{r}) + (k - c_l) (k - c_l - 1) p_{lk}^\mathrm{g} (1 - p_{lk}^\mathrm{g}) \\
\qquad\qquad +\ \delta_{c_l, c_{l-1}} [p_{lk}^\mathrm{r}]^{c_l} (k - c_l) (k - c_l - 1) [p_{lk}^\mathrm{g}]^2 \\
\qquad\qquad -\ \delta_{c_l, c_{l-1}} [p_{lk}^\mathrm{r}]^{c_l} (k - c_l)^2 [p_{lk}^\mathrm{g}]^2 \displaystyle\frac{k - c_l - 2}{k - c_l - 1} & \text{otherwise}
\end{array} \right. \\
%
\nonumber \\
%
\frac{\partial^2 \varphi_{lk}(\bm{1})}{\partial x_{lk}^{\mathrm{b}\,2}} & =
\left\{
\begin{array}{lc}
c_l (c_l - 1) (1 - p_{lk}^\mathrm{r})^2 + 2 c_l (k - c_l) (1 - p_{lk}^\mathrm{r}) (1 - p_{lk}^\mathrm{g}) & \text{if } k - c_l \leq 1 \\
& \\
c_l (c_l - 1) (1 - p_{lk}^\mathrm{r})^2 + 2 c_l (k - c_l) (1 - p_{lk}^\mathrm{r}) (1 - p_{lk}^\mathrm{g}) \\
\qquad\qquad +\ (k - c_l) (k - c_l - 1) (1 - p_{lk}^\mathrm{g})^2 \\
\qquad\qquad -\ \delta_{c_l, c_{l-1}} [p_{lk}^\mathrm{r}]^{c_l} (k - c_l) (k - c_l - 1) [p_{lk}^\mathrm{g}]^2 & \text{if } k - c_l = 2 \\
& \\
c_l (c_l - 1) (1 - p_{lk}^\mathrm{r})^2 + 2 c_l (k - c_l) (1 - p_{lk}^\mathrm{r}) (1 - p_{lk}^\mathrm{g}) \\
\qquad\qquad +\ (k - c_l) (k - c_l - 1) (1 - p_{lk}^\mathrm{g})^2 \\
\qquad\qquad -\ \delta_{c_l, c_{l-1}} [p_{lk}^\mathrm{r}]^{c_l} (k - c_l) (k - c_l - 1) [p_{lk}^\mathrm{g}]^2 \\
\qquad\qquad +\ \delta_{c_l, c_{l-1}} [p_{lk}^\mathrm{r}]^{c_l} (k - c_l)^2 [p_{lk}^\mathrm{g}]^2 \displaystyle\frac{k - c_l - 2}{k - c_l - 1} & \text{otherwise}
\end{array} \right.\end{aligned}$$
Let us recall that $c_l \neq c_{l-1}$ and $p_{lk}^\mathrm{r} = 1$ whenever $k = c_l$ since these nodes are in the first layer of their core by definition, and that we set $c_1 \neq c_0$ to simplify the notation. Note also that $$\begin{aligned}
\sum_{\alpha} \frac{\partial \varphi_{lk}(\bm{1})}{\partial x_{lk}^\alpha} =
\sum_{\alpha} \langle k^\alpha \rangle_{lk} = k
\qquad \text{ and } \qquad
\sum_{\alpha, \alpha^\prime} \frac{\partial^2 \varphi_{lk}(\bm{1})}{\partial x_{lk}^\alpha \partial x_{lk}^{\alpha^\prime}} =
\sum_{\alpha, \alpha^\prime} \langle k^\alpha (k^{\alpha^\prime} - \delta_{\alpha\alpha^\prime}) \rangle_{lk} = k(k - 1)\end{aligned}$$ for $\alpha, \alpha^\prime \in \{\mathrm{r}, \mathrm{g}, \mathrm{b}\}$ and regardless of the value of $k-c_l$, as expected.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We prove that every continuous function on a separable infinite-dimensional Hilbert space $X$ can be uniformly approximated by $C^\infty$ smooth functions [*with no critical points*]{}. This kind of result can be regarded as a sort of very strong approximate version of the Morse-Sard theorem. Some consequences of the main theorem are as follows. Every two disjoint closed subsets of $X$ can be separated by a one-codimensional smooth manifold which is a level set of a smooth function with no critical points; this fact may be viewed as a nonlinear analogue of the geometrical version of the Hahn-Banach theorem. In particular, every closed set in $X$ can be uniformly approximated by open sets whose boundaries are $C^\infty$ smooth one-codimensional submanifolds of $X$. Finally, since every Hilbert manifold is diffeomorphic to an open subset of the Hilbert space, all of these results still hold if one replaces the Hilbert space $X$ with any smooth manifold $M$ modelled on $X$.'
author:
- Daniel Azagra and Manuel Cepedello Boiso
date: 'December 2, 2001'
title: Uniform approximation of continuous functions by smooth functions with no critical points on Hilbert manifolds
---
Introduction and main results
=============================
A fundamental result in differential topology and analysis is the Morse-Sard theorem [@Sard1; @Sard2], which states that if $f:\mathbb{R}^{n}\longrightarrow \mathbb{R}^{m}$ is a $C^r$ smooth function, with $r>\max\{n-m, 0\}$, and $C_{f}$ stands for the set of critical points of $f$ (that is, the points of $X$ at which the differential of $f$ is not surjective), then the set of critical values, $f(C_{f})$, is of (Lebesgue) measure zero in $\mathbb{R}^{m}$. This result also holds true for smooth functions $f:X\longrightarrow Y$ between two smooth manifolds of dimensions $n$ and $m$ respectively.
Several authors have dealt with the question as to what extent one can obtain a similar result for infinite-dimensional spaces or manifolds modelled on such spaces. Let us recall some of their results.
Smale [@Smale] proved that if $X$ and $Y$ are separable connected smooth manifolds modelled on Banach spaces and $f:X\longrightarrow Y$ is a $C^r$ Fredholm map (that is, every differential $df(x)$ is a Fredholm operator between the corresponding tangent spaces) then $f(C_{f})$ is meager, and in particular $f(C_{f})$ has no interior points, provided that $r>\max\{\textrm{index}(df(x)), 0\}$ for all $x\in X$; here index($df(x)$) stands for the index of the Fredholm operator $df(x)$, that is, the difference between the dimension of the kernel of $df(x)$ and the codimension of the image of $df(x)$, which are both finite. However, these assumptions are quite restrictive: for instance, if $X$ is infinite-dimensional then there is no Fredholm map $f:X\longrightarrow\mathbb{R}$. In general, the existence of a Fredholm map $f$ from a manifold $X$ into another manifold $Y$ implies that $Y$ is infinite-dimensional whenever $X$ is. On the other hand, one cannot dream of extending the Morse-Sard theorem to infinite dimensions without imposing strong restrictions. Indeed, as shown by Kupka’s counterexample [@Kupka], there are $C^\infty$ smooth functions $f:X\longrightarrow\mathbb{R}$, where $X$ is a Hilbert space, so that their sets of critical values $f(C_{f})$ contain intervals and in particular have non-empty interior.
More recently, S. M. Bates has carried out a deep study concerning the sharpness of the hypothesis of the Morse-Sard theorem and the geometry of the sets of critical values of smooth functions. In particular he has shown that the above $C^r$ smoothness hypothesis in the statement of the Morse-Sard theorem can be weakened to $C^{r-1,1}$. See [@Bates1; @Bates2; @Bates3; @Bates4; @Bates5].
Nevertheless, for many applications of the Morse-Sard theorem, it is often enough to know that any given function can be uniformly approximated by a map whose set of critical values has empty interior. In this direction, Eells and McAlpin established the following theorem [@EellsMcAlpin]: if $X$ is a separable Hilbert space, then every continuous function from $X$ into $\mathbb{R}$ can be uniformly approximated by a smooth function $f$ whose set of critical values $f(C_{f})$ is of measure zero. This allowed them to deduce a version of this theorem for mappings between smooth manifolds $M$ and $N$ modelled on $X$ and a Banach space $F$ respectively, which they called an [*approximate Morse-Sard theorem*]{}: every continuous mapping from $M$ into $N$ can be uniformly approximated by a smooth function $f:X\longrightarrow Y$ so that $f(C_{f})$ has empty interior. However, this seemingly much more general version of the result is a bit tricky: indeed, as they already observed ([@EellsMcAlpin], Remark 3A), when $F$ is infinite-dimensional, the function $f$ they obtain satisfies that $C_{f}=X$, although $f(X)$ has empty interior in $Y$. Unfortunately, even though all the results of that paper seem to be true, some of the proofs are not correct.
In this paper we will prove a much stronger result: if $M$ is a $C^\infty$ smooth manifold modelled on a separable infinite-dimensional Hilbert space $X$ (in the sequel such a manifold will be called a Hilbert manifold), then every continuous function on $M$ can be uniformly approximated by $C^\infty$ smooth functions [*with no critical points*]{}. This kind of result might be regarded as the strongest possible one of any class of approximate Morse-Sard theorems, when the target space is $\mathbb{R}$.
As a by-product we also obtain the following: for every open set $U$ in a separable Hilbert space $X$ there is a $C^{\infty}$ smooth function $f$ whose support is the closure of $U$ and so that $f'(x)\neq 0$ for every $x\in U$. This result could be summed up by saying that for every open subset $U$ of $X$ there is a function $f$ whose open support is $U$ and which does not satisfy Rolle’s theorem; one should compare this result with the main theorem from [@AJ] (see also the references therein).
Either of these results has in turn interesting consequences related to smooth approximation and separation of closed sets. For instance, every closed set in a separable Hilbert manifold $M$ can be uniformly approximated by open sets whose boundaries are smooth one-codimensional submanifolds of $M$. Moreover, every two disjoint closed subsets in $M$ can be separated by a smooth one-codimensional submanifold of $M$ which is a level set of a smooth function with no critical points. The latter may in turn be regarded as a nonlinear analogue of the geometrical version of the Hahn-Banach theorem.
Let us now formally state our main results.
\[main theorem\] Let $U$ be an open subset of a separable infinite-dimensional Hilbert space $X$. Then, for all continuous functions $f:U\longrightarrow\mathbb{R}$ and $\varepsilon:U\longrightarrow
(0,+\infty)$, there are $C^\infty$ smooth functions $\psi$ on $U$ such that $|f(x)-\psi(x)|\leq\varepsilon(x)$ and $\psi'(x)\neq 0$ whenever $x\in X$.
We will prove this result in the following section. Let us now establish the announced consequences of Theorem \[main theorem\].
One could probably adapt the ideas in our proof to extend Theorem \[main theorem\] to the setting of Hilbert manifolds but, for simplicity, we will instead use another approach. Indeed, bearing in mind a fundamental result on Hilbert manifolds due to Eells and Elworthy [@EE] that every separable Hilbert manifold can be $C^\infty$ embedded as an open subset of the Hilbert space, it is a triviality to observe that Theorem \[main theorem\] still holds if we replace $U$ with a a separable Hilbert manifold.
\[main theorem for manifolds\] Let $M$ be a separable Hilbert manifold. Then, for all continuous functions $f:M\longrightarrow\mathbb{R}$ and $\varepsilon:M\longrightarrow (0,+\infty)$, there are $C^\infty$ smooth functions $\psi:M\longrightarrow\mathbb{R}$ so that $|f(x)-\psi(x)|\leq\varepsilon(x)$, and $d\psi(x)\neq 0$, for all $x\in X$.
According to the main theorem of [@EE], there is a $C^\infty$ embedding of $M$ onto an open subset of the Hilbert space $X$. Therefore $M$ is $C^\infty$ diffeomorphic to an open subset $U$ of $X$; let $h:U\longrightarrow M$ be such a $C^\infty$ diffeomorphism. Consider the continuous functions $g=f\circ
h:U\longrightarrow\mathbb{R}$ and $\delta=\varepsilon\circ
h:U\longrightarrow (0,+\infty)$. By Theorem \[main theorem\] there is a $C^\infty$ smooth function $\varphi:U\longrightarrow\mathbb{R}$ so that $\varphi$ has no critical points, and $$|g(y)-\varphi(y)|\leq\delta(y)$$ for all $y\in U$. Now define $\psi=\varphi\circ
h^{-1}:M\longrightarrow\mathbb{R}$. Since $h$ is a diffeomorphism it is clear that $h$ takes the critical set of $\psi$ onto the critical set of $\varphi=\psi\circ h$. But, as the latter is empty, so is the former; that is, $\psi$ has no critical points either. On the other hand, it is clear that $$|f(x)-\psi(x)|=|g(h^{-1}(x))-\varphi(h^{-1}(x))|\leq
\delta(h^{-1}(x))=\varepsilon(x)$$ for all $x\in M$.
As an easy corollary we can deduce our promised nonlinear version of the geometrical Hahn-Banach theorem.
We will say that an open subset $U$ of a Hilbert manifold $M$ is [*smooth*]{} provided that its boundary $\partial U$ is a smooth one-codimensional submanifold of $M$.
Let $M$ be a separable Hilbert manifold. Then, for every two disjoint closed subsets $C_1$, $C_2$ of $M$, there exists a $C^\infty$ smooth function $\varphi:X\longrightarrow\mathbb{R}$ with no critical points, such that the level set $N=\varphi^{-1}(0)$ is a $1$-codimensional $C^\infty$ smooth submanifold of $M$ that separates $C_{1}$ and $C_{2}$, in the following sense. Define $U_{1}=\{x\in M : \varphi(x)<0\}$ and $U_{2}=\{x\in M : \varphi(x)>0\}$; then $U_1$ and $U_2$ are disjoint $C^\infty$ smooth open sets of $M$ so that $C_{i}\subset
U_{i}$ for $i=1, 2$, and $\partial U_{1}=\partial U_{2}=N$.
By Urysohn’s lemma there exists a continuous function $f:M\longrightarrow [0,1]$ so that $C_{1}\subset f^{-1}(0)$ and $C_{2}\subset f^{-1}(1)$. Taking $\varepsilon=1/3$ and applying Theorem \[main theorem for manifolds\] we get a $C^\infty$ smooth function $\psi:M\longrightarrow\mathbb{R}$ which has no critical points and is so that $$|f(x)-\psi(x)|\leq 1/3$$ for all $x\in M$; in particular $$C_{1}\subseteq f^{-1}(0)\subseteq \psi^{-1}(-\infty,1/2):=U_{1},$$ and $$C_{2}\subseteq f^{-1}(1)\subseteq \psi^{-1}(1/2, +\infty):=U_{2}.$$ The open sets $U_1$ and $U_2$ are smooth because their common boundary $N=\psi^{-1}(1/2)$ is a smooth one-codimensional submanifold of $M$ (thanks to the implicit function theorem and the fact that $d\psi(x)\neq 0$ for all $x\in N$). In order to obtain the result in the above form it is enough to set $\varphi=\psi-1/2$.
A trivial consequence of this result is that every closed subset of $X$ can be uniformly approximated by smooth open subsets of $X$. In fact,
Every closed subset of a separable Hilbert manifold $M$ can be approximated by smooth open subsets of $M$, in the following sense: for every closed set $C\subset M$ and every open set $W$ containing $C$ there is a $C^\infty$ smooth open set $U$ so that $C\subset U\subseteq W$.
Finally, the following result, which also implies the above corollary, tells us that for every open set $U$ in $X$ there always exists a function whose open support is $U$ and which does not satisfy Rolle’s theorem.
\[second main theorem\] For every open subset $U$ of a Hilbert manifold $M$ there is a continuous function $f$ on $M$ whose support is the closure of $U$, so that $f$ is $C^\infty$ smooth on $U$ and yet $f$ has no critical point in $U$.
For the same reasons as in the proof of Theorem \[main theorem for manifolds\] we may assume that $U$ is an open subset of the Hilbert space $X=\ell_2$. Let $\varepsilon:X\longrightarrow
[0,+\infty)$ be the distance function to $X\setminus U$, that is, $$\varepsilon(x)=\textrm{dist}(x, X\setminus U)=\inf\{\|x-y\|: y\in X\setminus U\}.$$ The function $\varepsilon$ is continuous on $X$ and satisfies that $\varepsilon(x)>0$ if and only if $x\in U$. According to Theorem \[main theorem\], and setting $f(x)=2\varepsilon(x)$, there exists a $C^\infty$ smooth function $\psi:U\longrightarrow\mathbb{R}$ which has no critical points on $U$, and such that $\varepsilon$-approximates $f$ on $U$, that is, $$|2\varepsilon(x)-\psi(x)|\leq\varepsilon(x)$$ for all $x\in U$. This inequality implies that $$\lim_{x\to z}\psi(x)=0$$ for every $z\in\partial U$. Therefore, if we set $\psi=0$ on $X\setminus U$, the extended function $\psi:X\longrightarrow[0,+\infty)$ is continuous on the whole of $X$, is $C^\infty$ smooth on $U$ and has no critical points on $U$. On the other hand, $\psi(x)\geq\varepsilon(x)>0$ for all $x\in U$, hence the support of $\psi$ is $\overline{U}$.
Proof of the main result
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The main ideas behind the proof of Theorem \[main theorem\] are as follows. First we use a perturbed smooth partition of unity to approximate the given continuous function $f$. The summands of this perturbed partition of unity are functions supported on scalloped balls and carefully constructed in such a way that the critical points of the approximating sum $\varphi$ are kept under control. More precisely, those critical points consist of a sequence of compact sets $K_n$ that are suitably isolated in pairwise disjoint open sets $U_n$ of small diameter so that the oscillation of both $f$ and $\varphi$ on $U_n$ is small as well.
Then we have to eliminate all of those critical points without losing much of the approximation. To this end we compose the approximating function $\varphi$ with a sequence of deleting diffeomorphisms $h_{n}:X\longrightarrow X\setminus K_{n}$ which extract each of the compact sets of critical points $K_n$ and restrict to the identity outside each of the open sets $U_{n}$. The infinite composition of deleting diffeomorphisms with our function, $\psi=\varphi\circ \bigcirc_{n=1}^{\infty} h_{n}$, is locally finite, in the sense that only a finite number of diffeomorphisms are acting on some neighborhood of each point, while all the rest restrict to the identity on that neighborhood. In this way we obtain a smooth function $\psi$ which has no critical points, and which happens to approximate the function $\varphi$ (which in turn approximates the original $f$) because the perturbation brought on $\varphi$ by that infinite composition is not very important: indeed, recall that each $h_n$ restricts to the identity outside the set $U_n$ (on which $\varphi$ has a small oscillation), and the $U_n$ are pairwise disjoint.
We will make the proof of Theorem \[main theorem\] in the case of a constant $\varepsilon>0$ so as to avoid bearing an unnecessary burden of notation. Later on we will briefly explain what additional technical precautions must be taken in order to deduce the general form of this result (see Remark \[remark for e(x)\]).
The following proposition shows the existence of a function $\varphi$ with the above properties. Recall that $C_{\varphi}$ stands for the set of critical points of $\varphi$.
\[existence of varphi\] Let $U$ be an open subset of the separable Hilbert space $X$. Let $f:U\longrightarrow\mathbb{R}$ be a continuous function on $X$, and $\varepsilon>0$. Then there exist a $C^\infty$ smooth function $\varphi:U\longrightarrow\mathbb{R}$, a sequence $(K_{n})$ of compact sets, a sequence $(U_{n})$ of open sets, and a sequence $(B(y_{n}, r_{n}))$ of open balls which are contained in $U$ and whose union covers $U$, such that:
- $C_{\varphi}\subseteq\bigcup_{n=1}^{\infty} K_{n}$;
- $K_{n}\subset U_{n}\subseteq B(y_{n}, r_{n})$ for all $n\in\mathbb{N}$, and $U_{n}\cap U_{m}=\emptyset$ whenever $n\neq
m$;
- $|\varphi(x)-f(y)|\leq 2\varepsilon$ for all $x,y\in B(y_{n},r_{n})$, $n\in\mathbb{N}$;
- for every $x\in U$ there exist an open neighborhood $V_{x}$ of $x$ and some $n_{x}\in\mathbb{N}$ such that $V_{x}\cap
U_{m}=\emptyset$ for all $m>n_{x}$.
The following theorem ensures the existence of the diffeomorphisms $h_{n}$.
\[removing compact sets\] Let $X$ be an infinite-dimensional Hilbert space. Then, for every compact set $K$ and every open subset $U$ of $X$ with $K\subset
U$, there exists a $C^\infty$ smooth diffeomorphism $h:X\longrightarrow X\setminus K$ so that $h$ restricts to the identity outside $U$.
This result may be regarded, in the Hilbert case, as a (rather technical, but crucial to our purposes) improvement of some known results on smooth negligibility of compact sets (see [@ADo; @Do]; there $h$ is known to be the identity only outside a ball containing $K$).
Assume for a while that Proposition \[existence of varphi\] and Theorem \[removing compact sets\] are already established, and let us see how we can deduce Theorem \[main theorem\].
[**Proof of Theorem \[main theorem\]**]{}
For a given continuous function $f$ and a number $\varepsilon>0$, take a function $\varphi$ and sequences $(K_{n})$ and $(U_{n})$ with the properties of Proposition \[existence of varphi\]. For each compact set $K_{n}$ and each open set $U_{n}$, use Theorem \[removing compact sets\] to find a $C^{\infty}$ diffeomorphism $h_{n}:X\longrightarrow X\setminus K_{n}$ so that $h_{n}(x)=x$ if $x\notin U_{n}$. Note that, since the $U_j$ contain the $K_j$ and are pairwise disjoint, $$h_{n}(x)\notin\bigcup_{j=1}^{\infty}K_{j} \supseteq C_{\varphi} \eqno (1)$$ for all $x\in U$, $n\in\mathbb{N}$. Define then $\psi:U\longrightarrow\mathbb{R}$ by $$\psi=\varphi\circ\bigcirc_{n=1}^{\infty} h_{n}.$$ This formula makes sense and the function $\psi$ is $C^\infty$ smooth because the infinite composition is in fact locally finite. Indeed, for a given $x\in U$, according to Proposition \[existence of varphi\](d), we can find an open neighborhood $V_{x}$ of $x$ and some $n_{x}\in\mathbb{N}$ so that $V_{x}\cap
U_{m}=\emptyset$ for all $m>n_{x}$; hence $h_{m}(y)=y$ for all $y\in V_{x}$ and $m>n_{x}$, and therefore $$\psi(y)=\varphi\circ h_{n_{x}}\circ h_{n_{x}-1}\circ ...\circ
h_{2}\circ h_{1}(y) \eqno(2)$$ for all $y\in V_{x}$. The derivative $\psi'(y)$ is given by $$\psi'(y)=\varphi'\bigl(\bigcirc_{j=1}^{n_{x}}h_{j}(y)\bigr)\circ Dh_{n_{x}}
\bigl(\bigcirc_{j=1}^{n_{x}-1}h_{j}(y)\bigr)\circ ...\circ
Dh_{2}(h_{1}(y))\circ Dh_{1}(y) \eqno(3)$$ for all $y\in V_{x}$. Since $U_{n}\subseteq X\setminus U_{m}$ for $n\neq m$, we have that $h_{m}$ is the identity on $U_{n}$, and therefore $Dh_{m}(x)=I$ (the identity isomorphism of $\ell_{2}$) for all $x\in U_{n}$. By the continuity of $Dh_{n}$ it follows that $Dh_{m}(x)=I$ for all $x\in\overline{U}_{n}$, if $m\neq n$. This implies that, for $y\in\overline{U_{n}}\cap V_{x}$, all the differentials $Dh_{j}(z)$ in $(3)$ are the identity, except perhaps for $j=n$. Hence we get that either $$\psi'(y)=\varphi'(h_{n}(y))\circ Dh_{n}(y), \, \textrm{ and }\,
\psi(y)=\varphi(h_{n}(y)), \eqno(4)$$ if $y$ belongs to some $\overline{U}_{n}$, or else $$\psi'(y)=\varphi'(y), \, \textrm{ and }\,
\psi(y)=\varphi(y), \eqno(5)$$ when $y\notin\bigcup_{n=1}^{\infty}\overline{U}_{n}$.
Now we can easily check that $C_{\psi}=\emptyset$. Take $x\in U$. If we are in the case that $x\in\overline{U}_{n}$ for some $n$ then $
\psi'(x)=\varphi'(h_{n}(x))\circ Dh_{n}(x)\neq 0,
$ because $D h_{n}(x)$ is a linear isomorphism and, according to $(1)$ above, $\varphi'(h_{n}(x))\neq 0$. Otherwise we have that $x\notin\bigcup_{n=1}^{\infty}\overline{U}_{n}\supseteq
C_{\varphi}$, so $\psi'(x)=\varphi'(x)\neq 0$ trivially.
It only remains to check that $\psi$ still approximates $f$. As before, for a given $x\in U$, either $\psi(x)=\varphi(x)$ or $\psi(x)=\varphi(h_{n}(x))$ for some $n$ (with $x\in U_{n}$). In the first case, from Proposition \[existence of varphi\](c) we get that $|\psi(x)-f(x)|\leq 2\varepsilon$. In the second case, bearing in mind that $h_{n}(x)\in U_{n}\subseteq B(y_{n},r_{n})$, and for the same reason, we have that $$|\psi(x)-f(x)|=|\varphi(h_{n}(x))-f(x)|\leq 2\varepsilon;$$ in either case we obtain that $|\psi(x)-f(x)|\leq 2\varepsilon$.
[**Proof of Proposition \[existence of varphi\]**]{}
We will assume that $U=X$, since the proof is completely analogous in the case of a general open set. One only has to take some (easy but rather rambling) technical precautions in order to make sure that the different balls considered in the argument are in $U$.
Let $B(x, r)$ and $\overline{B}(x,r)$ stand for the open ball and closed ball, respectively, of center $x$ and radius $r$, with respect to the usual hilbertian norm $\|\cdot\|$ of $X$.
Let $f:X\longrightarrow\mathbb{R}$ be a continuous function, and let $\varepsilon>0$. By continuity, for every $x\in X$ there exists $\delta_{x}>0$ so that $|f(y)-f(x)|\leq \varepsilon/4$ whenever $y\in B(x, 2\delta_{x})$. Since $X=\bigcup_{x\in X}
B(x,\delta_{x}/2)$ is separable, there exists a countable subcovering, $$X=\bigcup_{n=1}^{\infty}B(x_{n}, s_{n}/2),$$ where $s_{n}=\delta_{x_{n}}$, for some sequence of centers $(x_{n})$. By induction we can choose a sequence of [*linearly independent*]{} vectors $(y_{n})$, with $y_{n}\in B(x_{n}, s_{n}/2)$, so that $$X=\bigcup_{n=1}^{\infty}B(y_{n}, s_{n}).$$ Moreover, we have that $$|f(y)-f(y_{n})|\leq\varepsilon/2$$ provided $\|y-y_{n}\|\leq \frac{3}{2} s_{n}$, as is immediately checked.
[*In the sequel $\mathcal{A}[z_{1},...,z_{k}]$ stands for the affine subspace spanned by a finite sequence of points $z_{1},
..., z_{k}\in X$.*]{}
The following lemma shows that we can slightly move the radii $s_{n}$ so that, for any finite selection of centers $y_{n}$, the spheres that are the boundaries of the balls $B(y_{n}, s_{n})$ have empty intersection with the affine subspace spanned by those centers.
\[controlled intersection of spheres\] We can find a sequence of positive numbers $(r_{n})$ with $s_{n}\leq r_{n}\leq \frac{3}{2} s_{n}$ so that, if we denote $S_{n}=\partial B(y_{n}, r_{n})$ then,
- for each finite sequence of positive integers $k_{1}<k_{2}<...<k_{m}$, $$\mathcal{A}[y_{k_{1}}, ..., y_{k_{m}}]\cap S_{k_{1}}\cap ...\cap
S_{k_{m}}=\emptyset.$$
- for any $n,k\in\mathbb{N}$, $y_{n}\notin S_{k}$.
We will define the $r_{n}$ inductively.
For $n=1$ we may take $r_{1}\in [s_{1}, \frac{3}{2}s_{1}]$ so that $r_{1}$ does not belong to the countable set $\{\|y_{1}-y_{k}\| : k\in\mathbb{N}\}$; this means that $y_{k}\notin S_{1}$ for any $k\in\mathbb{N}$. On the other hand, it is obvious that $\{y_{1}\}\cap S_{1}=\emptyset$.
Assume now that $r_{1}, ..., r_{n}$ have already been chosen in such a way that the spheres $S_{1}$, ..., $S_{n}$ satisfy $(i)$ and $(ii)$, and let us see how we can find $r_{n+1}$. For any finite sequence of integers $0<k_{1}< ... <k_{j}\leq n+1$, let us denote $$\mathcal{A}_{k_{1},...,k_{j}}=\mathcal{A}[y_{k_{1}}, ...,
y_{k_{j}}].$$ For simplicity, and up to a suitable translation (which obviously does not affect our problem), we may assume that $y_{n+1}=0$, so that $\mathcal{A}_{k_{1},...,k_{m},n+1}$ is the $m$-dimensional vector subspace of $X$ spanned by $y_{k_{1}}$, ..., $y_{k_{m}}$. Now, for each finite sequence of integers $0<k_{1}<...<k_{m}\leq
n$, consider the map $F_{k_{1}, ...,
k_{m}}:\mathcal{A}_{k_{1},...,k_{m}, n+1}\longrightarrow
\mathbb{R}^{m}$ defined by $$F_{k_{1}, ..., k_{m}}(x)=\big(\|x-y_{k_{1}}\|^{2}- {r_{k_{1}}}^{2}, ...,
\|x-y_{k_{m}}\|^{2}- {r_{k_{m}}}^{2}\big).$$ Note that $$D F_{k_{1}, ..., k_{m}}(x)=\big(2(x-y_{k_{1}}), ..., 2(x-y_{k_{m}})\big)$$ and therefore $\textrm{rank}\big(D F_{k_{1}, ...,
k_{m}}(x)\big)<m$ if and only if $x\in
\mathcal{A}_{k_{1},...,k_{m}}$. By the induction assumption we know that $$S_{k_{1}}\cap ... \cap S_{k_{m}}\cap \mathcal{A}_{k_{1},...,k_{m}}=\emptyset,$$ hence it is clear that $\textrm{rank}\big(D F_{k_{1}, ...,
k_{m}}(x)\big)=m$ for all $x\in S_{k_{1}}\cap ... \cap
S_{k_{m}}\cap \mathcal{A}_{k_{1},...,k_{m}, n+1}$. This implies that $$M_{k_{1}, ..., k_{m}}:=S_{k_{1}}\cap ... \cap
S_{k_{m}}\cap \mathcal{A}_{k_{1},...,k_{m}, n+1}$$ is a compact $m-m=0$-dimensional submanifold of $\mathcal{A}_{k_{1},...,k_{m}, n+1}$, and in particular $M_{k_{1}, ..., k_{m}}$ consists of a finite number of points (in fact two points, but we do not need to know this). Therefore $$M=\bigcup M_{k_{1}, ..., k_{m}}$$ (where the union is taken over all the finite sequences of integers $0<k_{1}<...<k_{n}\leq n$) is a finite set as well. Now we have that $$I:=\big[s_{n+1}, \frac{3}{2}s_{n+1}\big]\setminus
\Big(\{\|z\| : z\in M\}\cup\{\|y_{j}\| :
j\in\mathbb{N}\}\Big)$$ is an uncountable subset of the real line, so we can find a number $r_{n+1}\in I$. With this choice it is clear that $$S_{k_{1}}\cap ... \cap
S_{k_{m}}\cap S_{n+1}\cap \mathcal{A}_{k_{1},...,k_{m}, n+1}=
M_{k_{1},...,k_{m}}\cap S_{n+1}=\emptyset$$ for all finite sequences of integers $0<k_{1}<...<k_{m}<n+1$, and also $y_{j}\notin S_{n+1}=\partial B(0, r_{n+1})$ for all $j\in\mathbb{N}$. Therefore the spheres $S_{1}$, ..., $S_{n}$, $S_{n+1}$ satisfy $(i)$ and $(ii)$ as well. By induction the sequence $(r_{n})$ is thus well defined.
Since $s_{n}\leq r_{n}\leq \frac{3}{2} s_{n}$ for all $n$, it is clear that the new balls $B(y_{n},r_{n})$ keep the two important properties of the old balls $B(y_{n}, s_{n})$, namely, $$X=\bigcup_{n=1}^{\infty}B(y_{n}, r_{n}), \eqno (6)$$ and $$|f(y)-f(y_{n})|\leq\varepsilon/2 \text{ whenever $\|y-y_{n}\|\leq r_{n}$.} \eqno (7)$$
Now we define the scalloped balls $B_{n}$ that are the basis for our perturbed partition of unity: set $B_{1}=B(y_{1},r_{1})$, and for $n\geq 2$ define $$B_{n}=B(y_{n}, r_{n})\setminus \Bigl(\bigcup_{j=1}^{n-1}
\overline{B}(y_{j}, \lambda_{n} r_{j})\Bigr);$$ where $1/2<\lambda_{2}<\lambda_{3}< ...
<\lambda_{n}<\lambda_{n+1}< ... <1$, with $\lim_{n\to\infty}\lambda_{n}=1$. The $\lambda_{n}$ are to be fixed later on.
Taking into account that $\lim_{n\to\infty}\lambda_{n}=1$, it is easily checked that the $B_{n}$ form a locally finite open covering of $X$, with the nice property that $$|f(y)-f(y_{n})|\leq\varepsilon/2
\text{ whenever $y\in B_{n}$.}$$
Next, pick a $C^{\infty}$ smooth function $g_{1}:\mathbb{R}\longrightarrow [0,1]$ so that:
- $g_{1}(t)=1$ for $t\leq 0$,
- $g_{1}(t)=0$ for $t\geq {r_{1}}^{2}$,
- $g_{1}'(t)<0$ if $0<t< {r_{1}}^{2}$;
and define then $\varphi_{1}:X\longrightarrow\mathbb{R}$ by $$\varphi_{1}(x)=g_{1}(\|x-y_{1}\|^{2})$$ for all $x\in X$. Note that $\varphi_{1}$ is a $C^{\infty}$ smooth function whose open support is $B_{1}$, and $B_{1}\cap
C_{\varphi_{1}}=\{y_{1}\}$, that is, $y_{1}$ is the only critical point of $\varphi_{1}$ that lies inside $B_{1}$.
Now, for $n\geq 2$, pick $C^\infty$ smooth functions $\theta_{(n,j)}:\mathbb{R}\longrightarrow [0,1]$, $j=1, ..., n$, with the following properties. For $j=1, ..., n-1$, $\theta_{(n,j)}$ satisfies that
- $\theta_{(n,j)}(t)=0$ for $t\leq(\lambda_{n}r_{j})^{2}$,
- $\theta_{(n,j)}(t)=1$ for $t\geq {r_{j}}^{2}$,
- $\theta_{(n,j)}'(t)>0$ if $(\lambda_{n}r_{j})^{2}<t<{r_{j}}^{2}$;
while for $j=n$ the function $\theta_{(n,n)}$ is such that
- $\theta_{(n,n)}(t)=1$ for $t\leq 0$,
- $\theta_{(n,n)}(t)=0$ for $t\geq {r_{n}}^{2}$,
- $\theta_{(n,n)}'(t)<0$ if $0<t< {r_{n}}^{2}$.
Then define the function $g_{n}:\mathbb{R}^{n}\longrightarrow
[0,1]$ as $$g_{n}(t_{1}, ..., t_{n})=\prod_{i=1}^{n}\theta_{(n,i)}(t_{i})$$ for all $t=(t_{1}, ..., t_{n})\in\mathbb{R}^{n}$. This function is clearly $C^\infty$ smooth on $\mathbb{R}^{n}$ and satisfies the following properties:
- $g_{n}(t_{1},...,t_{n})>0$ if and only if $t_{j}>(\lambda_{n}r_{j})^{2}$ for all $j=1, ..., n-1$, and $t_{n}< {r_{n}}^{2}$; and $g_{n}$ vanishes elsewhere;
- $g_{n}(t_{1},...,t_{n})=\theta_{(n,n)}(t_{n})$ whenever $t_{j}\geq
{r_{j}}^{2}$ for all $j=1, ..., n-1$;
- $\nabla g_{n}(t_{1},...,t_{n})\neq 0$ provided $(\lambda_{n}r_{j})^{2}<t_{j}$ for all $j=1, ..., n-1$, and $0<t_{n}<{r_{n}}^{2}$.
Moreover, under the same conditions as in (iii) just above we have that $$\frac{\partial g_{n}}{\partial t_{n}}(t_{1},...,t_{n})=
\frac{\partial \theta_{(n,n)}}{\partial t_{n}}(t_{n})
\prod_{i=1}^{n-1}\theta_{(n,i)}(t_{i})<0, \eqno(8)$$ since no function in this product vanishes on the specified set, while for $j<n$, according to the corresponding properties of the functions $\theta_{(n,j)}$ we have that $$\frac{\partial g_{n}}{\partial t_{j}}(t_{1},...,t_{n})=
\frac{\partial \theta_{(n,j)}}{\partial t_{j}}(t_{j})
\prod_{i=1, i\neq j}^{n}\theta_{(n,i)}(t_{i})>0. \eqno(9)$$ If we are not in the conditions of (iii) then the corresponding inequalities do still hold but are not strict.
Let us now define $\varphi_{n}:X\longrightarrow [0,1]$ by $$\varphi_{n}(x)=g_{n}(\|x-y_{1}\|^{2}, ... , \|x-y_{n}\|^{2}).$$ It is clear that $\varphi_{n}$ is a $C^{\infty}$ smooth function whose open support is precisely the scalloped ball $B_{n}$.
As above, let us denote by $C_{\varphi_{n}}$ the critical set of $\varphi_{n}$, that is, $$C_{\varphi_{n}}=\{x\in X :
\varphi_{n}'(x)=0\}.$$ Since our norm $\|\cdot\|$ is hilbertian we have that, if $x\in C_{\varphi_{n}}\cap B_{n}$, then $x$ belongs to the affine span of $y_{1}, ..., y_{n}$. Indeed, if $x\in
B_{n}$, $$\varphi_{n}'(x)=\sum_{j=1}^{n}\frac{\partial g_{n}}{\partial t_{j}}
(\|x-y_{1}\|^{2}, ... , \|x-y_{n}\|^{2})\,2(x-y_{j})=0, \eqno(10)$$ which (taking into account (8) and the fact that the $y_{j}$ are all linearly independent) means that $x$ is in the affine span of $y_{1}, ..., y_{n}$. Here, as is usual, we identify the Hilbert space $X$ with its dual $X^*$, and we make use of the fact that the derivative of the function $x\mapsto \|x\|^{2}$ is the mapping $x\mapsto 2x$.
Similarly, by using (8) it can be shown that $x\in
C_{\varphi_{1}+\cdots+\varphi_{m}}\cap (B_{1}\cup ...\cup B_{m})$ implies that $x$ belongs to the affine span of $y_{1}, ...,
y_{m}$.
In order that our approximating function has a small critical set we cannot use the standard approximation provided by the partition of unity associated with the functions $(\varphi_{j})_{i\in\mathbb{N}}$, namely $$x\mapsto \frac{\sum_{n=1}^{\infty}\alpha_{n}\varphi_{n}(x)}
{\sum_{n=1}^{\infty}\varphi_{n}(x)},$$ where $\alpha_{n}=f(y_{n})$. Indeed, such a function would have a huge set of critical points since it would be constant (equal to $\alpha_{n}$) on a lot of large places (at least on each $B_{n}$ minus the union of the rest of the $B_j$). Instead, we will modify this standard approximation by letting the $\alpha_{n}$ be functions (and not mere numbers) of very small oscillation and with only one critical point (namely $y_{n}$). So, for every $n\in\mathbb{N}$ let us pick a $C^{\infty}$ smooth real function $a_{n}:[0, +\infty)\longrightarrow\mathbb{R}$ with the following properties:
- $a_{n}(0)=f(y_{n})$;
- $a_{n}'(t)<0$ whenever $t>0$;
- $|a_{n}(t)-a_{n}(0)|\leq\varepsilon/2$ for all $t\geq 0$;
and define $\alpha_{n}:X\longrightarrow\mathbb{R}$ by $$\alpha_{n}(x)=a_{n}(\|x-y_{n}\|^{2})$$ for every $x\in X$. It is clear that $\alpha_{n}$ is a $C^\infty$ smooth function on $X$ whose only critical point is $y_{n}$. Besides, $$|\alpha_{n}(x)-f(y_{n})|\leq\varepsilon/2 \text{ for all $x\in X$}.$$
Now we can define our approximating function $\varphi:X\longrightarrow\mathbb{R}$ by $$\varphi(x)=\frac{\sum_{n=1}^{\infty}\alpha_{n}(x)\varphi_{n}(x)}
{\sum_{n=1}^{\infty}\varphi_{n}(x)}$$ for every $x\in X$. Since the sums are locally finite, it is clear that $\varphi$ is a well-defined $C^\infty$ smooth function.
\[good approximation\] The function $\varphi$ approximates $f$ nicely. Namely, we have that
- $|\varphi(x)-f(x)|\leq\varepsilon$ for all $x\in X$, and
- $|\varphi(y)-f(x)|\leq 2\varepsilon$ for all $x, y\in
B(y_{n}, r_{n})$ and each $n\in\mathbb{N}$.
Indeed, for every $n$ we have that $|\alpha_{n}(x)-f(y_{n})|\leq\varepsilon/2$ for all $x\in X$. On the other hand, by $(7)$ above we know that $|f(x)-f(y_{n})|\leq
\varepsilon/2$ whenever $x\in B(y_{n}, r_{n})$. Then, by the triangle inequality, it follows that $$|\alpha_{n}(x)-f(x)|\leq\varepsilon \eqno(11)$$ whenever $x\in B(y_{n}, r_{n})$. In the same way we deduce that $$|\alpha_{m}(x)-f(y_{n})|\leq\varepsilon \eqno(12)$$ whenever $x\in B(y_{n}, r_{n})\cap B(y_{m}, r_{m})$. Since $\varphi_{m}(y)=0$ when $y\notin B(y_{m}, r_{m})$, from $(11)$ we get that $$|\varphi(x)-f(x)|=
\bigg|\frac{\sum_{m=1}^{\infty}(\alpha_{m}(x)-f(x))\varphi_{m}(x)}
{\sum_{m=1}^{\infty}\varphi_{m}(x)}\bigg|\leq
\frac{\sum_{m=1}^{\infty}\varepsilon \varphi_{m}(x)}
{\sum_{m=1}^{\infty}\varphi_{m}(x)}=\varepsilon$$ for all $x\in X$, which shows $(i)$. Similarly, we deduce from $(12)$ that $$|\varphi(y)-f(y_{n})|=
\bigg|\frac{\sum_{m=1}^{\infty}(\alpha_{m}(y)-f(y_{n}))\varphi_{m}(y)}
{\sum_{m=1}^{\infty}\varphi_{m}(y)}\bigg|\leq
\frac{\sum_{m=1}^{\infty}\varepsilon \varphi_{m}(y)}
{\sum_{m=1}^{\infty}\varphi_{m}(y)}=\varepsilon$$ for every $y\in B(y_{n}, r_{n})$, which, combined with the fact that $|f(x)-f(y_{n})|\leq\varepsilon/2$ for $x\in B(y_{n},
r_{n})$, yields that $$|\varphi(y)-f(x)|\leq \varepsilon+\varepsilon/2,$$ for every $x, y\in B(y_{n}, r_{n})$, so $(ii)$ is satisfied as well.
Now let us have a look at the derivative of $\varphi$. To this end let us introduce the auxiliary functions $f_{n}$ defined by $$f_{n}(x)=\frac{\sum_{k=1}^{n}\alpha_{k}(x)\varphi_{k}(x)}
{\sum_{k=1}^{n}\varphi_{k}(x)}, \, \textrm{ for all }\,
x\in \bigcup_{i=1}^{n}B_{i}.$$ Notice that $\varphi$ can be expressed as $$\varphi(x)=\lim_{n\to\infty}f_{n}(x),$$ that the domains of the $f_{n}$ form an increasing tower of open sets whose union is $X$, and that each $f_{n}$ restricts to $f_{n-1}$ on $\bigcup_{i=1}^{n-1}B_{i}\setminus B_{n}$. Moreover, for each $x\in X$ there is some open neighborhood $V_{x}$ of $x$ and some $n_{x}\in\mathbb{N}$ so that $\varphi(y)=f_{n_{x}}(y)$ for all $y\in V_{x}$. In fact we have that $$\varphi(x)=f_{n}(x) \hspace{0.3cm} \textrm{for all} \hspace{0.3cm}
x\in V_{n}:=\big(\bigcup_{j=1}^{n}B_{j}\big)\setminus
\big(\bigcup_{i=n+1}^{\infty}\overline{B}_{i}\big),$$ for every $n$, the $V_{n}$ are open, $V_{n}\subseteq V_{n+1}$, and $\bigcup_{i=1}^{\infty}V_{i}=X$, because the covering of $X$ formed by the $B_j$ is locally finite.
Hence, by looking at the derivatives of the functions $f_{n}$ we will get enough information about the derivative of $\varphi$.
If $x\in\bigcup_{j=1}^{n}B_{j}$ then the expression for the derivative of $f_{n}$ is given by $$f_{n}'(x)=\frac{\sum_{j=1}^{n}[\alpha_{j}'(x)\varphi_{j}(x)+
\alpha_{j}(x)\varphi_{j}'(x)]\sum_{i=1}^{n}\varphi_{i}(x)-
\sum_{j=1}^{n}\varphi_{j}'(x)\sum_{i=1}^{n}\alpha_{i}(x)\varphi_{i}(x)}
{(\sum_{j=1}^{n}\varphi_{j}(x))^{2}}.$$ Therefore, for $x\in\bigcup_{j=1}^{n}B_{j}$ we have that $f_{n}'(x)=0$ if and only if $$\sum_{j=1}^{n}\sum_{i=1}^{n}\varphi_{i}(x)
\Bigl[\alpha_{j}'(x)\varphi_{j}(x)+\bigl(\alpha_{j}(x)-\alpha_{i}(x)\bigr)\varphi_{j}'(x)\Bigr]
=0. \eqno(13)$$ By inserting the expressions for the derivatives of $\varphi_{j}$ and $\alpha_{j}$ in equation $(13)$, we can express the condition $f_{n}'(x)=0$ as a nontrivial linear dependence link on the vectors $(x-y_{j})$, which yields that $x$ is in the affine span of the points $y_{1}, ..., y_{n}$. Indeed, we are going to prove the following.
\[the critical set is locally finite dimensional\] If $x\in C_{f_{n}}\cap B_{n}$, then $x\in
\mathcal{A}_{n}:=\mathcal{A}[y_{1}, ..., y_{n}]$. Moreover, for each $n\in\mathbb{N}$ and for every finite sequence of positive integers $k_{1}<k_{2}<...<k_{m}<n$ we have that $$C_{f_{n}}\cap\big(B_{n}\setminus\bigcup_{j=1}^{m}B_{k_{j}}\big)
\subseteq\mathcal{A}\big[\{y_{1}, ..., y_{n}\}
\setminus\{y_{k_{1}}, ..., y_{k_{m}}\}\big].$$
As above, in all the subsequent calculations, we will identify the Hilbert space $X$ with its dual $X^*$, and the derivative of $\|\cdot\|^{2}$ with the mapping $x\mapsto 2x$. To save notation, let us simply write $$\frac{\partial g_{n}}{\partial
t_{j}}(\|x-y_{1}\|^{2},...,\|x-y_{n}\|^{2})=\mu_{(n,j)},$$ and $$a_{j}'(\|x-y_{j}\|^{2})=\eta_{j}.$$
Notice that, according to $(8)$ and $(9)$ above, $\mu_{(n,j)}\geq
0$ for $j=1, ..., n-1$, while $\mu_{(n,n)}\leq 0$; and $\mu_{(n,n)}\neq 0$ provided $x\in B_{n}$ and $x\neq y_{n}$; on the other hand it is clear that $\eta_{j}<0$ for all $j$ unless $x=y_{j}$ (in which case $\eta_{j}=0$).
Assuming $x\in C_{f_{n}}\cap B_{n}$, and taking into account the expression $(10)$ for $\varphi_{j}'(x)$ and the fact that $\alpha_{j}'(x)=2\eta_{j}(x-y_{j})$, we can write condition $(13)$ above in the form $$2\sum_{j=1}^{n}\sum_{i=1}^{n}\varphi_{i}(x)
\Bigl[\eta_{j}\varphi_{j}(x)\,(x-y_{j})+\bigl(\alpha_{j}(x)-\alpha_{i}(x)\bigr)
\sum_{\ell=1}^{j}\mu_{(j,\ell)}\,(x-y_{\ell})\Bigr]
=0,$$ which in turn is equivalent (taking the common factors of each $(x-y_{j})$ together) to the following one $$\sum_{j=1}^{n}\Biggl[
\eta_{j}\varphi_{j}(x)\sum_{i=1}^{n}\varphi_{i}(x) +
\sum_{k=j}^{n}\Bigl(\sum_{i=1}^{n}\bigl(\alpha_{k}(x)-
\alpha_{i}(x)\bigr)\varphi_{i}(x)\Bigr)
\mu_{(k,j)}\Biggr]\,(x-y_{j})=0. \eqno(14)$$ Now notice that, if we can prove that at least one of the expressions multiplying the $(x-y_{j})$ does not vanish then we are done; indeed, we will have that the vectors $x-y_{1}$, ..., $x-y_{n}$ are linearly dependent, which means that $x$ belongs to the affine span of the points $y_{1}, ..., y_{n}$.
So let us check that not all of those expressions in $(14)$ vanish. In fact we are going to see that at least one of the terms is strictly negative. We can obviously assume that $x$ is not any of the points $y_{1}, ..., y_{n}$ (which are already in $\mathcal{A}_{n}$). In this case we have that $\mu_{(n,n)}<0$ and $\eta_{j}<0$ for all $j=1, ..., n$. For simplicity, we will only make the argument in the case $n=3$; giving a proof in a more general case would be as little instructive as tedious to read.
Let us first assume that $\varphi_{j}(x)\neq 0$ for $j=1,2,3$. We begin by looking at the term that multiplies $(x-y_{3})$ in $(14)$, that is $$\beta_{3}:=\eta_{3}\varphi_{3}(x)\sum_{i=1}^{3}\varphi_{i}(x) +
\sum_{i=1}^{3}\bigl(\alpha_{3}(x)-
\alpha_{i}(x)\bigr)\varphi_{i}(x)
\mu_{(3,3)}.$$ If $\sum_{i=1}^{3}\bigl(\alpha_{3}(x)-\alpha_{i}(x)\bigl) \varphi_{i}(x)\geq
0$ we are done, since in this case we easily see that $\beta_{3}<0$ (remember that $\mu_{(3,3)}\leq 0$, $\eta_{3}<0$, and $\varphi_{3}(x)>0$). Otherwise we have that $$\sum_{i=1}^{3}\Bigl(\alpha_{3}(x)-\alpha_{i}(x)\Bigr) \varphi_{i}(x)<0,$$ and then we look at the term $\beta_{2}$ multiplying $(x-y_{2})$ in $(14)$, namely, $$\beta_{2}:=\eta_{2}\varphi_{2}(x)\sum_{i=1}^{3}\varphi_{i}(x) +
\sum_{k=2}^{3}\Biggl(\sum_{i=1}^{3}\Bigl(\alpha_{k}(x)-
\alpha_{i}(x)\Bigr)\varphi_{i}(x)\Biggr)
\mu_{(k,2)}.$$ Now, since $\mu_{(3,2)}\geq 0$, we have $\sum_{i=1}^{3}\bigl(\alpha_{3}(x)-\alpha_{i}(x)\bigr)
\varphi_{i}(x)\mu_{(3,2)}\leq 0$, and on the other hand $\eta_{2}\varphi_{2}(x)\sum_{i=1}^{3}\varphi_{i}(x)<0$ so that, if $\sum_{i=1}^{3}\bigl(\alpha_{2}(x)-\alpha_{i}(x)\bigr) \varphi_{i}(x)$ happens to be nonnegative, then we also have $\sum_{i=1}^{3}\bigl(\alpha_{2}(x)-\alpha_{i}(x)\bigr)
\varphi_{i}(x)\mu_{(2,2)}\leq 0$, and then we are done since $\beta_{2}$, being a sum of negative terms (one of them strictly negative) must be negative as well. Otherwise, $$\sum_{i=1}^{3}\Bigl(\alpha_{2}(x)-\alpha_{i}(x)\Bigr)
\varphi_{i}(x)$$ is negative, and then we finally pass to the term $\beta_{1}$ multiplying $(x-y_{1})$ in $(14)$, that is, $$\beta_{1}:=\eta_{1}\varphi_{1}(x)\sum_{i=1}^{3}\varphi_{i}(x) +
\sum_{k=1}^{3}\Biggl(\sum_{i=1}^{3}\Bigl(\alpha_{k}(x)-
\alpha_{i}(x)\Bigr)\varphi_{i}(x)\Biggr)
\mu_{(k,1)}.$$ Here, by the assumptions we have made so far and taking into account the signs of $\mu_{(k,j)}$ and $\eta_{j}$, we see that $\sum_{i=1}^{3}\bigl(\alpha_{k}(x)- \alpha_{i}(x)\bigr)\varphi_{i}(x)
\mu_{(k,1)}\leq 0$ for $k=2, 3$. Having arrived at this point, it is sure that $\sum_{i=1}^{3}\bigl(\alpha_{1}(x)-\alpha_{i}(x)\bigr)\varphi_{i}(x)$ must be nonnegative (otherwise the numbers $\sum_{i=1}^{3}\bigl(\alpha_{k}(x)-\alpha_{i}(x)\bigr)\varphi_{i}(x)$ should be strictly negative for all $k=1,2,3$, which is impossible if one takes $\alpha_{k}(x)$ to be the maximum of the $\alpha_{i}(x)$), and now we can deduce as before that $\beta_{1}<0$.
Finally let us consider the case when some of the $\varphi_{i}(x)$ vanish, for $i=1,2$ (remember that $\varphi_{3}(x)\neq 0$ since $x\in B_{3}$, the open support of $\varphi_{3}$). From the definitions of $\mu_{(k,j)}$, $g_{n}$ and $\varphi_{n}$, it is clear that $\mu_{(k,j)}=0$ whenever $\varphi_{j}(x)=0$ or $\varphi_{k}(x)=0$, and bearing this fact in mind we can simplify equality $(14)$ to a great extent by dropping all the terms that now vanish.
If $\varphi_{1}(x)=\varphi_{2}(x)=0$ then $(14)$ reads $$\varphi_{3}(x)^{2}\eta_{3}\,(x-y_{3})=0,$$ which cannot happen since we assumed $x\neq y_{j}$ (this means that the only critical point that $f_{n}$ can have in $B_{3}\setminus(B_{1}\cup B_{2})$ is $y_{3}$).
If $\varphi_{1}(x)=0$ and $\varphi_{2}(x)\neq 0$ then the term $\beta_{1}$ accompanying $(x-y_{1})$ in $(14)$ vanishes, and hence $(14)$ is reduced to $$\sum_{j=2}^{3}\Biggl[
\eta_{j}\varphi_{j}(x)\sum_{i=2}^{3}\varphi_{i}(x) +
\sum_{k=j}^{3}\Bigl(\sum_{i=2}^{3}(\alpha_{k}(x)-
\alpha_{i}(x))\varphi_{i}(x)\Bigr)
\mu_{(k,j)}\Biggr]\,(x-y_{j})=0.$$ Since at least one of the numbers $\sum_{i=2}^{3}(\alpha_{k}(x)-
\alpha_{i}(x))\varphi_{i}(x)$, $k=2,3$, is nonnegative, the same reasoning as in the first case allows us to conclude that either $\beta_{3}$ or $\beta_{2}$ is strictly negative. Finally, in the case $\varphi_{1}(x)\neq 0$ and $\varphi_{2}(x)=0$, it is $\beta_{2}$ that vanishes, and $(14)$ reads $\beta_{1}\,(x-y_{1})+
\beta_{3}\,(x-y_{3})=0$, where $$\beta_{3}=\eta_{3}\varphi_{3}(x)\sum_{i=1, i\neq 2}^{3}\varphi_{i}(x) +
\sum_{i=1, i\neq 2}^{3}\Bigl(\alpha_{3}(x)-
\alpha_{i}(x)\Bigr)\varphi_{i}(x)
\mu_{(3,3)},$$ and $$\beta_{1}=\eta_{1}\varphi_{1}(x)\sum_{i=1, i\neq 2}^{3}\varphi_{i}(x) +
\sum_{k=1, i\neq 2}^{3}\sum_{i=1, i\neq 2}^{3}\Bigl(\alpha_{k}(x)-
\alpha_{i}(x)\Bigr)\varphi_{i}(x)
\mu_{(k,1)}.$$ Again, at least one of the numbers $\sum_{i=1, i\neq
2}^{3}\bigl(\alpha_{k}(x)- \alpha_{i}(x)\bigr)\varphi_{i}(x)$, $k=1,3$, is nonnegative, and the same argument as above applies.
To finish the proof of the proposition we will need even more accurate information about the location of the critical points of $f_{n}$. Bearing in mind the definition of the functions $\varphi_{j}$, whose open support are the $B_{j}$, it is clear that the above discussion shows, in fact, the following inclusions:
- $C_{f_{3}}\cap B_{3}\subseteq\mathcal{A}[y_{1}, y_{2},
y_{3}]$;
- $C_{f_{3}}\cap(B_{3}\setminus B_{1})\subseteq\mathcal{A}[y_{2},
y_{3}]$, and $C_{f_{3}}\cap(B_{3}\setminus
B_{2})\subseteq\mathcal{A}[y_{1}, y_{3}]$ ;
- $C_{f_{3}}\cap(B_{3}\setminus (B_{1}\cup
B_{2}))\subseteq \mathcal{A}[y_{3}]$.
An analogous argument in the case $n\geq 4$ proves the second part of the statement of Fact \[the critical set is locally finite dimensional\].
[*Note that from Fact \[the critical set is locally finite dimensional\] it follows that the set of critical points $C_{\varphi}$ is locally compact, since it is closed and it is locally a bounded set of a finite-dimensional affine subspace.*]{}
So far, all the properties we have shown about our functions $f_{n}$ are independent of the way we may choose the numbers $\lambda_{j}$ in the definitions of $B_j$ and $\varphi_{j}$. Now we are going to be more accurate and see how we can select those numbers $\lambda_{j}$ so as to have more control over the set $C_{\varphi}$ of critical points of $\varphi$. Indeed, we want $C_{\varphi}$ not only to be locally compact, but to consist of a sequence of suitably isolated small compact sets $K_{n}$. That is, we want to write $
C_{\varphi}\subseteq\bigcup_{n=1}^{\infty}K_{n},
$ where the $K_n$ are compact sets which are associated with open sets $U_{n}$ so that $K_{n}\subset U_{n}\subset B(y_{n}, r_{n})$, and $U_{n}\cap U_{m}=\emptyset$ whenever $n\neq m$.
We will choose the numbers $\lambda_{n}$ and the open sets $U_{n}$ inductively.
[**First step.**]{} Define $\varphi_{1}$ as above and put $f_{1}(x)=\alpha_{1}(x)$ for all $x\in B_{1}=B(y_{1}, r_{1})$. Set $\mu_{2}=1/2$, $K_{1}=C_{f_{1}}\cap B_{1}=\{y_{1}\}$, and $U_{1}=B(y_{1},\mu_{2}r_{1})$.
[**Second step.**]{} Fix $\lambda_{2}\in (\mu_{2}, 1)$, and define $B_{2}$, $\varphi_{2}$, and $f_{2}$ as above. According to Fact \[the critical set is locally finite dimensional\], we have that
- $C_{f_{2}}\cap B_{2}\subset\mathcal{A}[y_{1}, y_{2}]$, and
- $C_{f_{2}}\cap (B_{2}\setminus B_{1})
\subseteq\mathcal{A}[y_{2}]$.
We claim that there must exist some $\mu_{3}\in (\lambda_{2}, 1)$ so that $\overline{C_{f_{2}}\cap B_{2}\cap B_{1}}\subset B(y_{1},
\mu_{3}r_{1})$. Otherwise there would exist a sequence $(x_{j})$ in $C_{f_{2}}\cap B_{2}\cap B_{1}$ so that $\|x_{j}-y_{1}\|$ goes to $r_{1}$ as $j$ goes to $\infty$. Since $C_{f_{2}}\cap
B_{2}\subset\mathcal{A}[y_{1}, y_{2}]$, we may assume, by compactness, that $x_{j}$ converges to some point $x_{0}\in\partial B(y_{1}, r_{1})=S_{1}$. If $x_{0}\in B(y_{2},
r_{2})$ then $f_{2}'(x_{0})=0$ (by continuity of $f_{2}'$), and $x_{0}\neq y_{2}$ (because $y_{2}\notin S_{1}$ by ii) of Lemma \[controlled intersection of spheres\]), so $$f_{2}'(x_{0})=\alpha_{2}'(x_{0})\neq 0,$$ a contradiction. Therefore it must be the case that $x_{0}\in\partial B(y_{2}, r_{2})=S_{2}$. But then $$x_{0}\in S_{1}\cap S_{2}\cap\mathcal{A}[y_{1}, y_{2}],$$ and this contradicts Lemma \[controlled intersection of spheres\].
So let us take $\mu_{3}\in (\lambda_{2}, 1)$ so that $\overline{C_{f_{2}}\cap B_{2}\cap B_{1}}\subset B(y_{1},
\mu_{3}r_{1})$. In the case that $y_{2}\in B_{1}$, let us simply set
- $U_{2}=B(y_{2}, r_{2})\cap B(y_{1}, \mu_{3}r_{1})\setminus
\overline{B}(y_{1}, \mu_{2}r_{1})$, and
- $K_{2}=\overline{C_{f_{2}}\cap B_{2}\cap B_{1}}\subset
U_{2}$.
In the case that $y_{2}\notin B_{1}$, find $\delta_{2}\in (0,
\mu_{3} r_{2})$ so that $B(y_{2}, \delta_{2})\subset
B_{2}\setminus\overline{B_{1}}$, and set
- $U_{2}=\big[B(y_{2}, r_{2})\cap B(y_{1}, \mu_{3}r_{1})\setminus
\overline{B}(y_{1}, \mu_{2}r_{1})\big]\cup B(y_{2},\delta_{2})$, and
- $K_{2}=\overline{C_{f_{2}}\cap B_{2}\cap B_{1}}\cup\{y_{2}\}\subset
U_{2}$.
Clearly, we have that $C_{f_{2}}\subseteq K_{1}\cup K_{2}$, and $U_{1}\cap U_{2}=\emptyset$.
[**Third step.**]{} Now choose $\lambda_{3}\in (\mu_{3},
1)$ with $\lambda_{3}>1-1/3$, and define $B_{3}$, $\varphi_{3}$, and $f_{3}$ as above. We have that $f_{3}$ and $f_{2}$ coincide on $(B_{1}\cup B_{2})\setminus B_{3}$. On $B_{3}$, according to Fact \[the critical set is locally finite dimensional\], we know that
- $C_{f_{3}}\cap B_{3}\cap B_{2}\cap B_{1}\subseteq\mathcal{A}[y_{1}, y_{2},
y_{3}]$;
- $C_{f_{3}}\cap(B_{3}\cap B_{2}\setminus B_{1})\subseteq\mathcal{A}[y_{2},
y_{3}]$, and $C_{f_{3}}\cap(B_{3}\cap B_{1}\setminus
B_{2})\subseteq\mathcal{A}[y_{1}, y_{3}]$ ;$(15)$
- $C_{f_{3}}\cap(B_{3}\setminus (B_{1}\cup
B_{2}))\subseteq \mathcal{A}[y_{3}]$.
Again, there must be some $\mu_{4}\in (\lambda_{3}, 1)$ so that $$\overline{C_{f_{3}}\cap B_{3}\cap(B_{1}\cup B_{2})}
\subset B(y_{1}, \mu_{4}r_{1})\cup
B(y_{2}, \mu_{4}r_{2}).$$ Otherwise (bearing in mind the local compactness of $\mathcal{A}[y_{1}, y_{2}, y_{3}]$), there would exist a sequence $(x_{j})$ in $C_{f_{3}}\cap B_{3}\cap (B_{1}\cup B_{2})$ so that $(x_{j})$ converges to some point $x_{0}$ and $(x_{j})$ is not contained in $B(y_{1}, \mu_{4}r_{1})\cup
B(y_{2}, \mu_{4}r_{2})$ for any $\mu_{4}<1$. Since a subsequence of $(x_{j})$ must be contained in one of the sets listed in $(15)$, we deduce that the limit point $x_{0}$ must belong to one of the following sets:
- $S_{2}\cap S_{1}\cap\mathcal{A}[y_{1}, y_{2},
y_{3}]$;
- $S_{2}\cap\mathcal{A}[y_{2},
y_{3}]\setminus B_{1}$;
- $S_{1}\cap\mathcal{A}[y_{1},
y_{3}]\setminus B_{2}$,
Now we have two cases: either $x_{0}\in B_{3}$, or $x\in\partial
B_{3}$. If $x_{0}\in B_{3}$ then $f_{3}'(x_{0})=0$ (by continuity of $f_{3}'$), and $x_{0}\neq y_{3}$ (because $y_{3}\notin
S_{1}\cup S_{2}$ by (ii) of Lemma \[controlled intersection of spheres\]), so it follows that $$f_{3}'(x_{0})=\alpha_{3}'(x_{0})\neq 0,$$ a contradiction. On the other hand, if $x_{0}\in\partial B_{3}$ then $x_{0}\in S_{3}$ as well, and now one of the following must hold:
- $x_{0}\in S_{3}\cap S_{2}\cap S_{1}\cap\mathcal{A}[y_{1}, y_{2},
y_{3}]$;
- $x_{0}\in S_{3}\cap S_{2}\cap\mathcal{A}[y_{2},
y_{3}]$;
- $x_{0}\in S_{3}\cap S_{1}\cap\mathcal{A}[y_{1},
y_{3}]$,
but in any case this contradicts Lemma \[controlled intersection of spheres\].
Hence we can take $\mu_{4}\in (\lambda_{3}, 1)$ so that $$\overline{C_{f_{3}}\cap B_{3}\cap(B_{1}\cup B_{2})}
\subset B(y_{1}, \mu_{4}r_{1})\cup
B(y_{2}, \mu_{4}r_{2}).$$ Now two possibilities arise. If $y_{3}\in B_{1}\cup B_{2}$, let us define $$U_{3}=\Biggl[B(y_{3}, r_{3})\setminus\bigcup_{j=1}^{2}
\overline{B}(y_{j}, \mu_{3}r_{j})\Biggr]\bigcap
\Biggl[\bigcup_{j=1}^{2}B(y_{j}, \mu_{4}r_{j})\Biggr],$$ and $$K_{3}=\overline{C_{f_{3}}\cap B_{3}\cap(B_{1}\cup B_{2})}\subset U_{3}.$$ If $y_{3}\notin B_{1}\cup B_{2}$, since $y_{3}\notin S_{1}\cup
S_{2}$ we can find $\delta_{3}\in (0, \mu_{4}r_{3})$ so that $B(y_{3},\delta_{3})\subseteq B_{3}\setminus (B_{1}\cup B_{2})$, and then we can set $$U_{3}=\Biggl[\Bigl(B(y_{3}, r_{3})\setminus\bigcup_{j=1}^{2}
\overline{B}(y_{j}, \mu_{3}r_{j})\Bigr)\bigcap
\Bigl(\bigcup_{j=1}^{2}B(y_{j}, \mu_{4}r_{j})\Bigr)\Biggr]\bigcup
B(y_{3}, \delta_{3}),$$ and $$K_{3}=\overline{[C_{f_{3}}\cap B_{3}\cap(B_{1}\cup B_{2})]\cup
\{y_{3}\}}\subset U_{3}.$$ Notice that $U_{3}$ does not meet $U_{1}$ or $U_{2}$, and $C_{f_{3}}\subseteq K_{1}\cup K_{2}\cup K_{3}$.
[**N-th step.**]{} Suppose now that $\mu_{j}$, $\lambda_{j}$, $\varphi_{j}$, $B_{j}$, $f_{j}$, $K_{j}$, $U_{j}$ have already been fixed for $j=1, ..., n$ (and also $\mu_{n+1}$ has been chosen) in such a manner that $f_{j}$ agrees with $f_{j-1}$ on $(B_{1}\cup ...\cup B_{j-1})\setminus B_{j}$, and $K_{j}$ and $U_{j}$ are of the form $$K_{j}=\overline{C_{f_{j}}\cap B_{j}\cap(B_{1}\cup... \cup B_{j-1})}
\eqno(16)$$ and $$U_{j}=\biggl[B(y_{j}, r_{j})\setminus\Bigl(\bigcup_{i=1}^{j-1}
\overline{B}(y_{i}, \mu_{j}r_{i})\Bigr)\biggr]\bigcap
\Bigl[\bigcup_{i=1}^{j-1}B(y_{i}, \mu_{j+1}r_{i})\Bigr] \eqno(17)$$ in the case that $y_{j}\in B_{1}\cup ...\cup B_{j-1}$, and are of this form plus $\{y_{j}\}$ and $B(y_{j},\delta_{j})$ respectively when $y_{j}\notin B_{1}\cup ...\cup B_{j-1}$; assume additionally that $U_{j}\cap U_{k}=\emptyset$ whenever $j\neq k$, that $C_{f_{j}}\subseteq \bigcup_{i=1}^{j}K_{i}$, and that $\lambda_{j}>1-1/j$. Let us see how we can choose $\lambda_{n+1}$, $\mu_{n+2}$, $K_{n+1}$ and $U_{n+1}$ so that the extended bunch keeps the required properties.
Pick any $\lambda_{n+1}\in (\mu_{n+1}, 1)$ so that $\lambda_{n+1}>1-1/(n+1)$, and define $\varphi_{n+1}$, $B_{n+1}$ and $f_{n+1}$ as above. We know that $f_{n+1}$ agrees with $f_{n}$ on the set $(B_{1}\cup ... \cup B_{n})\setminus B_{n+1}$. On $B_{n+1}$, according to Fact \[the critical set is locally finite dimensional\], we have that $$C_{f_{n+1}}\cap\big(B_{n+1}\setminus\bigcup_{j=1}^{m}B_{k_{j}}\big)
\subseteq\mathcal{A}\big[\{y_{1}, ..., y_{n+1}\}
\setminus\{y_{k_{1}}, ..., y_{k_{m}}\}\big]$$ for every finite sequence of integers $0<k_{1}<k_{2}< ...
<k_{m}<n+1$.
We claim that there exists some $\mu_{n+2}\in (\lambda_{n+1}, 1)$ so that $$\overline{C_{f_{n+1}}\cap B_{n+1}\cap (B_{1}\cup ... \cup B_{n})}
\subseteq \bigcup_{i=1}^{n}B(y_{i},\mu_{n+2}r_{i}).$$ Otherwise there would exist a finite (possibly empty!) sequence of integers $0<k_{1}<k_{2}< ... <k_{m}<n+1$, and a sequence $(x_{j})_{j=1}^{\infty}$ contained in $$\Biggl[C_{f_{n+1}}\cap B_{n+1}\cap \Bigl(\bigcap_{j=1}^{\ell}B_{i_{j}}\Bigr)\Biggr]
\setminus\Bigl(\bigcup_{j=1}^{m}B_{k_{j}}\Bigr)
\subseteq\mathcal{A}[y_{i_{1}}, ..., y_{i_{\ell}}, y_{n+1}]$$ (where $i_{1}, ...,i_{\ell}$ are the positive integers less than or equal to $n$ that are left when we remove $k_{1}, ...,
k_{m}$), such that $(x_{j})$ converges to some point $x_{0}\in
S_{i_{1}}\cap ...\cap S_{i_{\ell}}$ with $x_{0}\notin\bigcup_{j=1}^{m}B_{k_{j}}$.
If $x_{0}\in B_{n+1}$ then $f_{n+1}'(x_{0})=0$ (by continuity of $f_{n+1}'$), and $x_{0}\neq y_{n+1}$, so we easily see that $$f_{n+1}'(x_{0})=\alpha_{n+1}'(x_{0})\neq 0,$$ a contradiction.
If $x_{0}\in \partial B_{n+1}$ then $x_{0}\in S_{n+1}$ as well, and in this case we have $$x_{0}\in S_{i_{1}}\cap ...\cap S_{i_{\ell}}\cap S_{n+1}\cap
\mathcal{A}[y_{i_{1}}, ..., y_{i_{\ell}}, y_{n+1}],$$ but this contradicts Lemma \[controlled intersection of spheres\].
Therefore we may take $\mu_{n+2}\in (\lambda_{n+1}, 1)$ so that $$\overline{C_{f_{n+1}}\cap B_{n+1}\cap (B_{1}\cup... \cup
B_{n})}\subseteq
\bigcup_{i=1}^{n}B(y_{i},\mu_{n+2}r_{i}).$$ As before, now we face two possibilities. If $y_{n+1}\in\bigcup_{i=1}^{n}B_{i}$, let us define $$U_{n+1}=\Biggl[B(y_{n+1}, r_{n+1})\setminus\bigcup_{i=1}^{n}
\overline{B}(y_{i}, \mu_{n+1}r_{i})\Biggr]\bigcap
\Biggl[\bigcup_{i=1}^{n}B(y_{i}, \mu_{n+2}r_{i})\Biggr],$$ and $$K_{n+1}=\overline{C_{f_{n+1}}\cap B_{n+1}\cap(B_{1}\cup... \cup
B_{n})}.$$ If $y_{n+1}\notin\bigcup_{i=1}^{n}B_{i}$, since $y_{n+1}\notin
S_{i}$ we may find $\delta_{n+1}\in (0, \mu_{n+2}r_{n+1})$ so that $B(y_{n+1},\delta_{n+1})\subseteq B_{n+1}\setminus
\bigcup_{i=1}^{n}B_{i}$, and then we can add this ball to the above $U_{n+1}$, and the point $\{y_{n+1}\}$ to that $K_{n+1}$, in order to obtain sets $U_{n+1}$, $K_{n+1}$ with the required properties.
By induction, the sequences $(\varphi_{n})$, $(f_{n})$, $(U_{n})$, $(K_{n})$ are well defined and satisfy the above properties.
From the construction it is clear that $U_{n}\cap U_{m}=\emptyset$ whenever $n\neq m$, and $$C_{f_{n}}\subseteq\bigcup_{j=1}^{n}K_{j}$$ for all $n$. Note also that $U_{n}\subseteq B(y_{n}, r_{n})$ for all $n$, and $\lim_{n\to\infty}\lambda_{n}=1$.
As observed before, $$\varphi(x)=\frac{\sum_{n=1}^{\infty}\alpha_{n}(x)\varphi_{n}(x)}
{\sum_{n=1}^{\infty}\varphi_{n}(x)}=\lim_{n\to\infty}f_{n}(x)$$ and, moreover, for each $x\in X$ there exists an open neighborhood $V_{x}$ of $x$ and some $n_{x}\in\mathbb{N}$ so that $\varphi(y)=f_{n_{x}}(y)$ for all $y\in V_{x}$. Bearing these facts in mind, it is immediately checked that $C_{\varphi}\subseteq\bigcup_{n=1}^{\infty}K_{n}$. Now it is clear that $\varphi$ satisfies (a), (b) and (d) in the statement of Proposition \[existence of varphi\]. On the other hand, remember that (c) is a consequence of fact \[good approximation\].
\[remark for e(x)\] [*Let us say a few words as to the way one has to modify the above proofs in order to establish Theorem \[main theorem\] when $\varepsilon$ is a positive continuous function. At the beginning of the proof of Proposition \[existence of varphi\], before choosing the $\delta_{x}$, we have to take some number $\alpha_{x}>0$ so that $|\varepsilon(y)-\varepsilon(x)|\leq\varepsilon(x)/2$ whenever $\|y-x\|\leq 2\alpha_{x}$ and then we can find some $\delta_{x}\leq\alpha_{x}$ so that $|f(y)-f(x)|\leq\varepsilon(x)/4$ whenever $y\in B(x,
2\delta_{x})$. Equation $(7)$ above reads now $$|f(y)-f(y_{n})|\leq\varepsilon(y_{n})/2$$ for all $y\in B(y_{n}, r_{n})$. Some obvious changes must be made in the definition of the functions $a_{n}$ and $\alpha_{n}$. Fact \[good approximation\] and Proposition \[existence of varphi\](c) can be reduced to saying that $$|\varphi(y)-f(x)|\leq
2\varepsilon(y_{n})$$ for all $x, y\in B(y_{n}, r_{n})$ and each $n\in\mathbb{N}$. Finally, at the end of the proof of Theorem \[main theorem\] we get that $$|\psi(x)-f(x)|\leq 2\varepsilon(y_{n})$$ whenever $x, y\in B(y_{n}, r_{n})$; now, taking into account that $r_{n}\leq\alpha_{y_{n}}$, we have that $\varepsilon(y_{n})\leq
2\varepsilon(x)$ for all $x\in B(y_{n}, r_{n})$. Hence, by combining these inequalities, we obtain that $
|\psi(x)-f(x)|\leq 4\varepsilon(x)
$ for all $x\in X$.*]{}
[**Proof of Theorem \[removing compact sets\]**]{}
The proof of Theorem \[removing compact sets\] is done in two steps. The first one uses the noncomplete norm technique of deleting compact sets introduced in [@ADo; @Do]. We only sketch the guidelines of this part, referring to the proof of Theorem 2.1 in [@ADo] for the details. We will show that a mapping of the form $G(x)=x+p(f(x))$, $x\in X\setminus K$, for a certain function $f:X\to[0,+\infty)$ with $f^{-1}(K)=0$ and a path $p:(0,+\infty)\to X$, establishes a $C^\infty$ diffeomorphism between $X\setminus K$ and $X$. The map $G$ can be viewed as a [*small*]{} perturbation of the identity. In order that the perturbation $p\circ f$ be small, $p$ and $f$ must satisfy some Lipschitzian-type conditions with respect to a certain distance induced by a smooth [*noncomplete norm*]{} $\omega$. Lemma \[Whitney function for compacta\] provides us with a required function $f(x)$ which can be viewed as a smooth substitute for the $\omega$-distance function from $x$ to the set $K$. Lemma \[deleting path\] gives us a required path $p(t)$ which avoids compact sets and gets lost in the infinitely many dimensions of $X$ as $t$ goes to $0$; by pushing away $\omega$-neighborhoods of $K$ along the path $p$, the mapping $G^{-1}$ will make $K$ disappear. By combining all these tools, the $C^\infty$ diffeomorphism $G$ can be constructed in such a way that $G$ restricts to the identity outside a given $\omega$-neighborhood of $K$.
So far this is the same negligibility scheme as in [@ADo; @Do]. The second step of the proof is to construct a self-diffeomorphism $F$ of $X$ that fixes the compact set $K$ and takes the open set $U$ (which in general is [*not*]{} a $\omega$-neighborhood of $K$) onto a $\omega$-neighborhood of $K$, and then to adjust the definition of $G$ so that it restricts to the identity outside $F(U)$. If we succeed in doing so then the composition $h=F^{-1}\circ G^{-1}\circ F$ will define a diffeomorphism from $X$ onto $X\setminus K$ with the property that $h$ is the identity outside $U$.
\[omega neighbourhoods\] [*Let $(X, \|\cdot\|)$ be a Banach space. We say that a norm $\omega:X\longrightarrow[0,+\infty)$ is a $C^p$ smooth noncomplete norm on $X$ provided $\omega$ is $C^p$ smooth (with respect to $\|\cdot\|$) away from the origin, but the norm $\omega$ is not equivalent to $\|\cdot\|$. Geometrically speaking, this means that the unit ball of $\omega$ is a symmetric $C^p$ smooth convex body that contains no rays and yet is unbounded. We define the (open) $\omega$-ball of center $x$ and radius $r$ as $$B_{\omega}(x,r)=\{y\in X : \omega(y-x)<r\},$$ and the $\omega$-distance from $x$ to $A$ as $$d_{\omega}(x,A)=\inf\{\omega(x-z) : z\in A\}.$$ We say that a set $V$ is an $\omega$-neighborhood of a subset $A$ of $X$ provided that for every $x\in A$ there exists some $r>0$ so that $B_{\omega}(x,r)\subseteq V$.*]{}
We next state the two facts we need for the first part of the proof. All the omitted proofs can be found in [@ADo; @Do].
\[Whitney function for compacta\] Let $\omega:X\longrightarrow [0,+\infty)$ be a $C^\infty$ smooth noncomplete norm in the Hilbert space $X$, and let $K$ be a compact subset of $X$. Then, for each $\varepsilon>0$ there exists a continuous function $f=f_\varepsilon:X\longrightarrow
[0,+\infty)$ such that
1. $f$ is $C^{\infty}$ smooth on $X\setminus K$;
2. $f(x)-f(y)\leq\omega(x-y)$ for every $x,
y\in X$;
3. $f^{-1}(0)=K$;
4. $\inf\{f(x)\mid d_{\omega}(x,K)\geq\eta\}>0$ for every $\eta>0$;
5. $f$ is constant on the set $\{x\in X\mid d_{\omega}(x,K)\geq\varepsilon\}$.
\[deleting path\] Let $\omega$ be a continuous noncomplete norm in the Hilbert space $X$. Then, for every $\delta>0$, there exists a $C^{\infty}$ path $p=p_{\delta}:(0,+\infty)\longrightarrow X$ such that
1. $\omega(p(\alpha)-p(\beta))\leq
\frac{1}{2}(\beta-\alpha)$ if $\beta\ge\alpha>0$;
2. For every compact set $A\subset
X$ there exists $t_{0}>0$ such that $$\inf\{\omega(z-p(t))\mid
0<t\leq t_{0}, z\in A\} >0;$$
3. $p(t)=0$ if and only if $t\geq\delta$.
The following lemma is the key to the second step of the proof, allowing us to improve, at least for the Hilbert case, the negligibility scheme introduced in [@ADo; @Do]. Note also that the norm $\omega$ that we will use in the first step is in fact the one provided by this lemma.
\[radial push\] Let $(X, \|\cdot\|)$ be an infinite-dimensional Hilbert space (with its usual hilbertian norm). Then, for every compact set $K$ and every open set $U$ so that $K\subset U$, there exist a $C^\infty$ diffeomorphism $F:X\longrightarrow X$ and a $C^\infty$ smooth noncomplete norm $\omega$ on $X$ such that $F(K)=K$ and $F(U)$ is an $\omega$-neighborhood of $K$.
Since $K$ is compact and $U$ is an open neighborhood of $K$ we can find points $x_{1}, ..., x_{n}\in K$ and positive numbers $r_{1}, ..., r_{n}$ so that $$K\subset \bigcup_{i=1}^{n} B_{\|.\|}(x_{i}, r_{i}) \subset
\bigcup_{i=1}^{n} B_{\|.\|}(x_{i}, 2r_{i})\subseteq U. \eqno
(18)$$ We may assume that $0\in K$. Let $Y=\textrm{span}\{x_{1}, ...,
x_{n}\}$, and write $X=Y\oplus Z$, where $Z$ is an infinite-dimensional space of finite codimension that is orthogonal to $Y$. Since the norm $\|\cdot\|$ is hilbertian we have that $$\|x\|=\|(y,z)\|=\big( \|y\|^{2}+\|z\|^{2}\big)^{1/2}$$ for every $x=(y,z)\in X=Y\oplus Z$.
Take a normalized basic sequence $(z_{i})$ in $Z$ so that the vectors $z_{i}$ are pairwise orthogonal, and let $W$ be the closed linear subspace spanned by $(z_i)$. Let us write $Z=W\oplus V$, where $V$ is the orthogonal complement of $W$ in $Z$. Define $\omega_{Z}:Z\longrightarrow [0,+\infty)$ by $$\omega_{Z}(w,v)=\biggl[\sum_{j=1}^{\infty}\Bigl(\frac{<w,z_{j}>}{2^{j}}\Bigr)^{2}
+\|v\|^{2}\biggr]^{1/2},$$ where $<,>$ denotes the inner product on $X$. Then $\omega_{Z}$ is a $C^\infty$ smooth noncomplete norm on $Z$, as it is easily checked. We also have that $\omega_{Z}(z)\leq \|z\|$ for every $z\in Z$. If we define now $\omega:X=Y\oplus Z\longrightarrow
[0,+\infty)$ by $$\omega(x)=\omega(y,z)=\big(\|y\|^{2}+\omega_{Z}(z)^{2}\big)^{1/2},$$ it is clear that $\omega$ is a $C^\infty$ smooth noncomplete norm on $X$ (note that in fact both $\omega_{Z}$ and $\omega$ are real-analytic, as they are prehilbertian).
For each $i=1, ..., n$, let us now pick $C^\infty$ smooth functions $\theta_{i}:\mathbb{R}\longrightarrow [0,1]$ so that $\theta_i$ is nondecreasing and $\theta_{i}^{-1}(0)=(-\infty,
r_{i}]$, while $\theta_{i}^{-1}(1)=[2r_{i}, +\infty)$. Define then $g:X=Y\oplus Z \longrightarrow [0, 1]$ by $$g(y,z)=g(x)=\prod_{i=1}^{n}\theta_{i}(\|x-x_{i}\|)$$ for all $x\in X$. Note that the function $g$ is $C^\infty$ smooth on $X$ and has the following properties:
- the function $t\mapsto g(y, tz)$, $t\geq 0$, is nondecreasing, for all $(y,z)\in X=Y\oplus Z$;
- $g(x)=0$ if $x\in\bigcup_{i=1}^{n} B_{\|.\|}(x_{i}, r_{i})$;
- $g(x)=1$ whenever $x\notin\bigcup_{i=1}^{n} B_{\|.\|}(x_{i},
2r_{i})$.
The first property is merely a consequence of the definition of $g$ and the fact that the function $t\mapsto \|((y-x_{i}, tz)\|$, $t\geq 0$, is increasing for every $(y,z)\in Y\oplus Z$ and every $i=1, ..., n$. Note that here we are using that $\|\cdot\|$ is a hilbertian norm; this property is not necessarily true for other norms.
Let us define our mapping $F:X=Y\oplus Z\longrightarrow X$ by $$F(x)=F(y,z)=\biggl( y, \, \Bigl(g(x)\frac{\|z\|}{\omega_{Z}(z)}
+1-g(x)\Bigr)\, z \, \biggr).$$ Clearly, $F$ is $C^\infty$ smooth. By using the facts that the functions $t\mapsto g(y, tz)$, $t\geq 0$, are nondecreasing, and that $\omega_{Z}(z)\leq \|z\|$ for every $z\in Z$, $y\in Y$, it is not difficult to see that $F$ is a bijection from every ray $\{(y,
tz) : t\geq 0\}$ onto itself, and therefore $F$ is one-to-one from $X$ onto $X$. Moreover, a standard application of the implicit function theorem allows to show that $F^{-1}$ is $C^\infty$ smooth as well, and hence $F$ is a diffeomorphism.
Finally, by the definitions of $g$ and $F$, it is clear that $
F(K)=K
$. In fact, $F$ restricts to the identity on the set $\bigcup_{i=1}^{n}
B_{\|.\|}(x_{i}, r_{i})$, which contains $K$, because $g$ takes the value $0$ on this set.
On the other hand, if $x\notin\bigcup_{i=1}^{n} B_{\|.\|}(x_{i}, 2r_{i})$ we have $g(x)=1$, so $F(y,z)=(y, \frac{\|z\|}{\omega(z)} z)$, and therefore $$\begin{aligned}
& &\omega(F(x)-x_{j})=\omega\Bigl(y-x_{j}, \frac{\|z\|}{\omega(z)}z\Bigr)=
\biggl(\|y-x_{j}\|^{2}+\omega\Bigl(\frac{\|z\|}{\omega(z)}z\Bigr)^{2}\biggr)^{1/2}\\
& &=\bigl(\|y-x_{j}\|^{2}+\|z\|^{2}\bigr)^{1/2}=\|x-x_{j}\|\geq 2r_{j}\end{aligned}$$ for each $j=1, ..., n$, which means that $F(x)\notin\cup_{i=1}^{n}
B_{\omega}(x_{i}, 2r_{i})$. Therefore, considering $(18)$, and bearing in mind that, since $\omega(x)\leq\|x\|$, the $\omega$-balls are larger than the $\|\cdot\|$-balls, we deduce that $$K\subset \bigcup_{i=1}^{n} B_{\omega}(x_{i}, r_{i})
\subset \bigcup_{i=1}^{n} B_{\omega}(x_{i}, 2r_{i})\subseteq
F\big(\bigcup_{i=1}^{n} B_{\|.\|}(x_{i}, 2r_{i})\big)\subseteq F(U);$$ in particular we see that $F(U)$ includes a finite union of $\omega$-balls which in turn includes K, and this shows that $F(U)$ is a $\omega$-neighborhood of $K$.
Let us now see how we can finish the proof of Theorem \[removing compact sets\]. First, for the given sets $K\subset
U$, take a non-complete norm $\omega$ and a diffeomorphism $F:X\longrightarrow X$ with the properties of Lemma \[radial push\]. Since $F(U)$ is a $\omega$-neighborhood of $K$ and $K$ is also compact in $(X,\omega)$, we can write $$K\subset \bigcup_{i=1}^{n}B_{\omega}(x_{i}, r_{i})\subseteq
\bigcup_{i=1}^{n}B_{\omega}(x_{i}, 2r_{i})\subseteq F(U)$$ for some points $x_{1}, ..., x_{n}\in K$ and positive numbers $r_{1}, ..., r_{n}$ (in fact such an expression appears in the proof of \[radial push\]). This in turn implies that $d_{\omega}(x, K)\geq \min\{r_{1}, ..., r_{n}\}>0$ whenever $x\in
X\setminus F(U)$, as it is easily seen.
Now, for $\varepsilon=\min\{r_{1}, ..., r_{n}\}$, we can choose a function $f=f_{\varepsilon}$ satisfying the properties of Lemma \[Whitney function for compacta\] (for the already selected $\omega$). Assuming $f(x)=\delta>0$ whenever $d_{\omega}(x,K)\geq\varepsilon$, select a path $p=p_{\delta}$ from Lemma \[deleting path\]. With these choices, for every $x\in X\setminus K$, define $$G(x)=x+p(f(x)).$$ Exactly as in the proof of Theorem 2.1 in [@ADo], it can be checked that $G$ is a $C^\infty$ diffeomorphism from $X\setminus
K$ onto $X$, with the property that $G(x)=x$ whenever $d_{\omega}(x, K)\geq\varepsilon$. In particular, since $d_{\omega}(x, K)\geq \varepsilon=\min\{r_{1}, ..., r_{n}\}$ whenever $x\in X\setminus F(U)$, we have that $G$ restricts to the identity outside $F(U)$.
Finally, let us define $h=F^{-1}\circ G^{-1}\circ F$. Taking into account the properties of the diffeomorphisms $F:X\longrightarrow
X$ and $G:X\setminus K\longrightarrow X$, it is clear that $h$ is a $C^\infty$ diffeomorphism from $X$ onto $X\setminus K$ so that $h$ is the identity outside $U$.
[**Acknowledgements**]{}
We wish to thank Pilar Cembranos, José Mendoza, Tijani Pakhrou and Raúl Romero, who helped us to realize that Fact \[the critical set is locally finite dimensional\] fails when the norm is not hilbertian.
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Departamento de Análisis Matemático. Facultad de Ciencias Matemáticas. Universidad Complutense. 28040 Madrid, SPAIN\
Departamento de Análisis Matemático. Universidad de Sevilla. Sevilla, SPAIN. [*E-mail addresses:*]{} daniel\[email protected], [email protected]
|
{
"pile_set_name": "ArXiv"
}
|
---
author:
- 'Pablo Cerdá-Durán'
- 'José A. Font'
- Harald Dimmelmeier
bibliography:
- '7432.bib'
date: 'Received date / Accepted date'
title: |
General relativistic simulations of passive-magneto-rotational\
core collapse with microphysics
---
Introduction
============
Understanding the dynamics of the gravitational collapse of the core of massive stars leading to supernova explosions still remains one of the primary problems in general relativistic astrophysics, despite the continuous theoretical efforts during the last few decades. This problem stands as a distinctive example of a research field where essential progress has been accomplished through numerical modelling with increasing levels of complexity in the input physics: hydrodynamics, gravity, magnetic fields, nuclear physics, equation of state (EOS), neutrino transport, etc. While studies based upon Newtonian physics are highly developed nowadays, state-of-the-art simulations still fail, broadly speaking, to generate successful supernova explosions under generic conditions (see e.g. @buras [-@buras]; @kifonidis06 [-@kifonidis06] for details on the degree of sophistication achieved in present-day supernova modelling, and @woosley05 [-@woosley05] and references therein for a review on the mechanism of core collapse supernovae). The reasons behind those apparent failures are diverse, all having to do with the limited knowledge of some of the underlying key issues such as realistic precollapse stellar models (including rotation, or the strength and distribution of magnetic fields), the appropriate EOS, as well as numerical limitations due to the need for Boltzmann neutrino transport, multi-dimensional hydrodynamics, and relativistic gravity.
Aside from their assistance to understand the supernova mechanism, numerical simulations of stellar core collapse are highly motivated nowadays by the prospects of a direct detection of the gravitational waves emitted in this scenario. In core collapse events where rotation plays a role, one of the emission mechanism for gravitational waves is the hydrodynamic core bounce, which generates a burst signal. The post-bounce wave signal also exhibits large amplitude oscillations associated with pulsations in the collapsed core [@zwerger_97_a; @rampp_98_a], neutrino-driven convection behind the supernova shock [@mueller_04_a] and (possibly) rotational dynamical instabilities [@ott05; @ott06a; @ott06b]. However, a successful future detection of gravitational radiation from stellar core collapse faces the combined problem of the smallness of the signal strength and of the complexity of the burst signal from bounce. On the other hand, the energy released in gravitational waves is so small that its backreaction to the collapse dynamics is negligible, which can significantly simplify the numerical simulation of this scenario. To pave the road for a successful detection through waveform templates for data analysis, such simulations are essential.
At birth neutron stars have intense magnetic fields ($ \sim 10^{12} \mbox{\,--\,} 10^{13} \mathrm{\ G} $) or in extreme cases even larger ones ($ \sim 10^{14} \mbox{\,--\,} 10^{15} \mathrm{\ G} $), as inferred from studies of anomalous X-ray pulsars and soft gamma-ray repeaters [@kouveliotou98]. For magnetars, the magnetic field can be so strong as to alter the internal structure of the neutron star [@bocquet95]. The emergence of such strong magnetic fields in neutron stars from the initial field configuration in the pre-collapse stellar cores is an active and important field of research. Similarly, the rotation state of the nascent proto-neutron star (PNS) is determined by the amount and distribution of angular momentum in the core of the progenitor, which is still rather unconstrained, being only currently incorporated into stellar evolution codes [@heger05]. Observations of surface velocities imply that a large fraction of progenitor cores is rapidly rotating. The presence of intense magnetic fields, on the other hand, may also affect rotation in the core, as it can be spun down in the red giant phase by magnetic torques via dynamo action which couples to the outer layers of the star [@meier76; @spruit98; @spruit02; @heger05]. The latest numerical calculations of stellar evolution thus predict low pre-collapse core rotation rates, which are in agreement with observed periods of young neutron stars in the range of $ \sim 10 \mbox{\,--\,} 15 \mathrm{\ ms} $. Nevertheless, a recent estimate by @woosley06 indicates that $ \sim 1\% $ of all stars with $ M \ge 10 M_\odot $ could still have rapidly rotating cores, which could also be relevant for the collapsar-type gamma-ray burst scenario.
The presence of intense magnetic fields in a PNS makes magneto-rotational core collapse simulations mandatory. The weakest point of all existing simulations to date is the fact that both the strength and distribution of the initial magnetic field in the core are basically unknown. If the magnetic field is initially weak, there exist several mechanisms which may amplify it to values which can have an impact on the dynamics, among them differential rotation ($ \Omega
$-dynamo[^1], the magneto-rotational instability (MRI hereafter), as well as dynamo mechanisms related to convection or turbulence. The first of these mechanisms transforms rotational energy into magnetic energy, winding up any seed poloidal field into a toroidal field. The MRI leads to a sustained turbulent dynamo which is able to transport angular momentum outwards, although it remains unclear how large the actual amplification by this process can be (see below). The latter mechanisms, which are generically called $\alpha\mbox{-}\Omega$-dynamo and will be discussed below, include a number of processes which can also produce an amplification of the magnetic field.
Magneto-rotational core collapse simulations were first performed as early as in the 1970s [@leblanc70; @bisnovatyi76; @meier76; @mueller79; @ohnishi83; @symbalisty84], in which magneto-rotational core collapse was already proposed as a plausible supernova explosion mechanism. In recent years, an increasing number of authors have performed axisymmetric magneto-hydrodynamic (MHD) simulations of stellar core collapse (within the so-called ideal MHD limit) employing a Newtonian treatment of MHD and gravity, and either a simplified equation of state [@yamada04; @ardeljan05; @sawai05] or a microphysical description of matter [@kotake04_a; @kotake04_b; @kotake05]. The main implications of the presence of strong magnetic fields in the collapse are the redistribution of the angular momentum and the appearance of jet-like explosions. Specific magneto-rotational effects on the gravitational wave signature were first studied in detail by @kotake04_a and @yamada04, who found differences with purely hydrodynamic models only for very strong initial fields ($ \geq 10^{12} \mathrm{\ G} $). The most exhaustive parameter study of magneto-rotational core collapse to date has been carried out very recently by @obergaulinger_06_b [@obergaulinger_06_a]. Their axisymmetric simulations employed rotating polytropes, Newtonian hydrodynamics and gravity (approximating general relativistic effects via an effective relativistic gravitational potential in their latter work), and ad-hoc initial poloidal magnetic field distributions. These authors found that for weak initial fields ($ \leq 10^{11} \mathrm{\ G} $, which is the astrophysically most motivated case) there are no differences compared to purely hydrodynamic simulations, neither in the collapse dynamics nor in the resulting gravitational wave signal. However, strong initial fields ($ \geq 10^{12} \mathrm{\ G} $) manage to slow down the core efficiently (leading even to retrograde rotation in the PNS) which causes qualitatively different dynamics and gravitational wave signals. For the most strongly magnetized models @obergaulinger_06_a found highly bipolar, jet-like outflows.
Nowadays, there exists sophisticated numerical technology to perform general relativistic hydrodynamics simulations [see e.g. @font_03_a]. In recent years this technology has been extended to general relativistic magneto-hydrodynamics (GRMHD) [@Koide99; @DeVilliers03; @delzanna03; @Gammie03; @Duez05; @anton06]. General relativistic simulations involve the challenging computational task of solving Einstein’s field equations coupled to the fluid (and magneto-fluid) evolution. The first general relativistic axisymmetric simulations of rotational stellar core collapse to neutron stars were performed by @dimmelmeier_01_a [@dimmelmeier_02_a; @dimmelmeier_02_b]. These simulations employed simplified models to describe the thermodynamics of the process, in the form of a polytropic EOS modified such that it accounts both for the stiffening of the matter above nuclear density as well as thermal heating by the passing shock front [the so-called hybrid EOS; see @janka_93_a]. The inclusion of relativistic effects within the so-called CFC approximation results primarily in a stronger gravitational pull during the contraction of the core. Thus, higher densities than in Newtonian models are reached during bounce, and the nascent PNS is more compact. Relativistic simulations with improved dynamics and gravitational waveforms were reported by @cerda05, who used the CFC+ framework, which includes second post-Newtonian corrections to CFC. Comparisons of the CFC approach with fully general relativistic simulations (employing also stable reformulations of the Einstein equations in $ 3 + 1 $ form) have been reported by @shibata_04_a, @ott06a, and @ott06b in the context of axisymmetric core collapse simulations. As in the case of CFC+, the differences found in both the collapse dynamics and the gravitational waveforms are minute, which highlights the suitability of CFC for performing accurate simulations of such scenarios without the need for solving the full system of Einstein’s equations. Owing to the excellent approximation of full general relativity offered by CFC in the context of stellar core collapse, extensions to improve the microphysics through the incorporation of a tabulated non-zero temperature EOS and a simplified neutrino treatment have been recently reported by @ott06a and @dimmelmeier_07_a. These simulations allow a direct comparison to the models presented in @dimmelmeier_02_b, @cerda05, and @shibata_04_a, which use the same parameterization of rotation but a simple hybrid EOS. This comparison shows that with a microphysical treatment the influence of rotation on the collapse dynamics and waveforms is significantly reduced. In particular, the most important result of [@dimmelmeier_07_a] is the suppression of core collapse with multiple centrifugal bounces and its associated Type II gravitational waveforms [see @dimmelmeier_02_b].
On the other hand, to further improve the realism of core collapse simulations in general relativity, the incorporation of magnetic fields in numerical codes via solving the MHD equations is also currently being undertaken [@shibata_06_a; @nfnr]. The work of @shibata_06_a is focused on the collapse of initially strongly magnetized cores ($ \sim 10^{12} \mbox{\,--\,} 10^{13} \mathrm{\ G} $). Although these values are probably astrophysically not relevant [as the stellar evolution models of @heger05 predict a poloidal field strength of $ \sim 10^6\mathrm{\ G} $ in the progenitor], they enable them to resolve the scales needed for the MRI to develop, since the MRI length scale grows with the magnetic field. The results of @shibata_06_a show that the magnetic field is mainly amplified by the wind-up of the magnetic field lines by differential rotation. Consequently, the magnetic field is accumulated around the inner region of the PNS, and a MHD outflow forms along the rotation axis removing angular momentum from the PNS. A different approach is followed by @nfnr. Their progenitors are chosen to be weakly magnetized ($ \le 10^{10} \mathrm{\ G} $) which is in much better agreement with predictions from stellar evolution. Under these conditions the so-called “passive” magnetic field approximation (see Sect. \[sec:passive\] below) is appropriate. In addition, the numerical code used in that work, which utilizes spherical coordinates, is more suitable for core collapse simulations than codes based on Cartesian/cylindrical coordinates, as used e.g. by @shibata_06_a.
In this paper we continue the program initiated in @nfnr to build a numerical code which includes all relevant ingredients to study relativistic magneto-rotational stellar core collapse. To this aim we present here the first relativistic simulations of magneto-rotational core collapse which take into account the effects of a microphysical EOS and a simplified neutrino treatment. Those effects have been incorporated in the code following the approach recently presented by @ott06a and @dimmelmeier_07_a. As in @nfnr we employ the passive magnetic field approximation in the treatment of the magnetic field.
The paper is organized as follows: Sect. \[theory\] presents a brief overview of the theoretical framework we use to perform relativistic simulations of core collapse. Sect. \[sec:mag\_ini\_models\] describes how the magnetized initial models for core collapse are built and presents our sample of models. In Sect. \[eos\] we discuss aspects related to incorporating microphysics in the core collapse models and their implementation in the numerical code. A brief outline of our numerical approach is discussed in Sect. \[numerics\]. The evolution of the core collapse initial models is discussed in Sect. \[results\]. The main paper closes with a summary in Sect. \[sec:conclusions\]. Relevant tests of the code are analyzed in Appendix \[app:tests\], while Appendix \[app:odynamo\] provides an estimate for the growth rate of the $ \Omega $-dynamo.
Throughout the paper we use a spacelike signature $ (-, +, +, +) $ and units in which $ c = G = 1 $. Greek indices run from 0 to 3, Latin indices from 1 to 3, and we adopt the standard Einstein summation convention.
Theoretical framework {#theory}
=====================
We adopt the $ 3 + 1 $ formalism of general relativity [@lichnerowicz44] to foliate the spacetime into spacelike hypersurfaces. In this approach the line element reads $$\mathrm{d}s^2 = - \alpha^2 \, \mathrm{d}t^2 +
\gamma_{ij} (\mathrm{d}x^i + \beta^i \,\mathrm{d}t)
(\mathrm{d}x^j + \beta^j \, \mathrm{d}t),$$ where $ \alpha $ is the lapse function, $ \beta^i $ is the shift vector, and $ \gamma_{ij} $ is the spatial three-metric induced in each hypersurface. Using the projection operator $ \perp^\mu_\nu $ and the unit four-vector $ n^\mu $ normal to each hypersurface, it is possible to build the quantities $$\begin{aligned}
E & = & n^{\mu} n^{\nu} T_{\mu\nu} = \alpha^2 T^{00},
\label{eq:tmn_projection_1}
\\
S_i & = & - \perp^{\mu}_{i} n^{\nu} T_{\mu\nu} =
- \frac{1}{\alpha} (T_{0i} - T_{ij} \beta^j),
\label{eq:tmn_projection_2}
\\
S_{ij} & = & \perp^{\mu}_i \perp^{\nu}_j T_{\mu\nu} = T_{ij},
\label{eq:tmn_projection_3}\end{aligned}$$which represent the total energy, the momenta, and the spatial components of the stress-energy tensor, respectively.
To solve the gravitational field equations we choose the ADM gauge in which the three-metric can be decomposed as $ \gamma_{ij} = \phi^4 \hat{\gamma}_{ij} + h^\mathrm{TT}_{ij}$, where $ \phi $ is the conformal factor, $\hat{\gamma}_{ij}$ is the flat three-metric, and $ h^\mathrm{TT}_{ij} $ is the transverse and traceless part of the three-metric. Note that this gauge choice implies the maximal slicing condition in which the trace $ K $ of the extrinsic curvature tensor $ K_{ij} $ vanishes.
The CFC approximation
---------------------
In our work Einstein’s field equations are formulated and solved using the conformally flat condition (CFC hereafter), introduced by @isenberg_78_a and first used in a dynamical context by @wilson_96_a. In this approximation, the three-metric in the ADM gauge is assumed to be conformally flat, $ \gamma_{ij} = \phi^4 \hat\gamma_{ij} $. Note that the same aproximation can be realized for other gauge choices such as the quasi-isotropic gauge or the Dirac gauge, both supplemented by the maximal slicing condition. Under the CFC assumption the gravitational field equations can be written as a system of five nonlinear elliptic equations, $$\begin{aligned}
\hat{\Delta} \phi & = & - 2 \pi \phi^5
\left( E + \frac{K_{ij}K^{ij}}{16 \pi} \right),
\label{eq:cfc1}
\\
\hat{\Delta} (\alpha \phi) & = & 2 \pi \alpha \phi^5
\left( E + 2 S + \frac{7 K_{ij}K^{ij}}{16 \pi} \right),
\label{eq:cfc2}
\\
\hat{\Delta} \beta^i & = & 16 \pi \alpha \phi^4 S^i +
2 \phi^{10} K^{ij} \hat{\nabla}_j
\left(\! \frac{\alpha}{\phi^6} \!\right) -
\frac{1}{3} \hat{\nabla}^i \hat{\nabla}_k \beta^k,
\label{eq:cfc3}\end{aligned}$$ where $ \hat{\Delta} $ and $ \hat{\nabla} $ are the Laplace and nabla operators associated with the flat three-metric, and $ S = \gamma^{ij} S_{ij} $.
General relativistic magnetohydrodynamics
-----------------------------------------
The energy-momentum tensor of a magnetized perfect fluid can be written as the sum of the fluid part and the electromagnetic field part. In the so-called ideal MHD limit (where the fluid is a perfect conductor of infinite conductivity), the latter can be expressed solely in terms of the magnetic field $ b^\mu $ measured by a comoving observer. In this case the total energy-momentum tensor is given by $$T^{\mu \nu} = (\rho h + b^2) \, u^\mu u^\nu +
\left( P + \frac{b^2}{2} \right) g^{\mu \nu} - b^\mu b^\nu,
\label{eq:tmunu_grmhd}$$ where $ \rho $ is the rest-mass density, $ h = 1 + \epsilon + P / \rho $ the relativistic enthalpy, $ \epsilon $ the specific internal energy, $ P $ the pressure, and $ u^\mu $ the four-velocity of the fluid, while $ b^2 = b^\mu b_\mu $. We define the magnetic pressure $ P_\mathrm{mag} = b^2 / 2 $ and the specific magnetic energy $ \epsilon_\mathrm{mag} = b^2 / (2 \rho) $, whose effect on the dynamics is similar to that played by the pressure and specific internal energy of the fluid, respectively. In the ideal MHD limit, the electric field measured by a comoving observer vanishes, and Maxwell’s equations simplify. Under this assumption the electric field four-vector $ E^\mu $ can be expressed in terms of the magnetic field four-vector $ B^\mu $, and only equations for $ B^i $ are needed. For an Eulerian observer, $ U^\mu = n^\mu $, the temporal component of the electric field vanishes, $ E^\mu = (0, - \varepsilon_{ijk} v^j B^k) $. In this case Maxwell’s equations reduce to the divergence-free condition and the induction equation for the magnetic field, $$\hat{\nabla}_i B^{*\,i} = 0,
\qquad
\frac{\partial B^{*\,i}}{\partial t} =
\hat{\nabla}_j (v^{*\,i} B^{*\,j} - v^{*\,j} B^{*\,i}),$$ where $ B^{*\,i} = \sqrt{\bar{\gamma}} B^i $ and $ v^{*\,i} = \alpha v^i - \beta^i $, with $ v^i $ being the fluid’s three-velocity as measured by the Eulerian observer. The ratio of the determinants of the three-metric and the flat three-metric is given by $ \bar{\gamma} = \gamma / \hat{\gamma} $. In the Newtonian limit $ v^{*\,i} \to v^i $ and $ B^{*\,i} \to B^i $, and the Newtonian induction equation and divergence constraint are recovered.
The evolution of a magnetized fluid is determined by the conservation law of the energy-momentum, $ \nabla_\mu T^{\mu \nu} = 0 $, and by the continuity equation, $ \nabla_\mu J^\mu = 0 $, for the rest-mass current $ J^\mu = \rho u^\mu $. Following the procedure laid out in @anton06, in order to cast the GRMHD equations as a hyperbolic system of conservation laws well adapted to numerical work, the conserved quantities are chosen in a way similar to the purely hydrodynamic case presented by @banyuls_97_a: $$\begin{aligned}
D & = & \rho W,
\\
S_i & = & (\rho h + b^2) W^2 v_i - \alpha b_i b^0,
\\
\tau & = & (\rho h + b^2) W^2 - \left( P + \frac{b^2}{2} \right) -
\alpha^2 (b^0)^2 - D,\end{aligned}$$where $ W = \alpha u^0 $ is the Lorentz factor. With this choice the system of conservation equations for the fluid and the induction equation for the magnetic field can be cast as a first-order, flux-conservative, hyperbolic system, $$\frac{\partial \sqrt{\gamma} {\mbox{\boldmath $U$}}}{\partial t} +
\frac{\partial \sqrt{- g} {\mbox{\boldmath $F$}}^i}{\partial x^i} =
\sqrt{- g} {\mbox{\boldmath $S$}},
\label{eq:hydro_conservation_equation}$$ with the state vector, flux vector, and source vector given by $$\begin{aligned}
{\mbox{\boldmath $U$}} & = & [D, S_j, \tau, B^k],
\label{eq:state_vector}
\\
{\mbox{\boldmath $F$}}^i & = & \left[
D \hat{v}^i, S_j \hat{v}^i + \delta^i_j \left( P + \frac{b^2}{2} \right) -
\frac{b_j B^i}{W}, \right.
\nonumber
\\
& & \left. \:\: \tau \hat{v}^i + \left( P + \frac{b^2}{2} \right) v^i -
\alpha \frac{b^0 B^i}{W}, \hat{v}^i B^k - \hat{v}^k B^i \right],
\label{eq:flux_vector}
\\
{\mbox{\boldmath $S$}} & = & \left[ 0, \frac{1}{2} T^{\mu \nu}
\frac{\partial g_{\mu \nu}}{\partial x^j},
\alpha \! \left( \! T^{\mu 0} \frac{\partial \ln \alpha}{\partial x^\mu} -
T^{\mu \nu} {\it \Gamma}^0_{\mu \nu} \! \right), 0^k \! \right],
\label{eq:source_vector}\end{aligned}$$where $ \delta^i_j $ is the Kronecker delta and $ \Gamma^\mu_{\mu \lambda}$ are the Christoffel symbols associated with the four-metric. We note that the above definitions contain components of the magnetic field measured by both a comoving observer and an Eulerian observer. The two are related by $$b^0 = \frac{W B^i v_i}{\alpha},
\qquad
b^i = \frac{B^i + \alpha b^0 u^i}{W}.$$
The hyperbolic structure of Eq. (\[eq:hydro\_conservation\_equation\]) and the associated spectral decomposition (into eigenvalues and eigenvectors) of the flux-vector Jacobians is given in @anton06. This information is needed for numerically solving the system of equations using the class of high-resolution shock-capturing schemes that we have implemented in our code.
The passive field approximation {#sec:passive}
-------------------------------
In the collapse of *weakly magnetized* stellar cores, it is a fair approximation to assume that the magnetic field entering in the energy-momentum tensor of Eq. (\[eq:tmunu\_grmhd\]) is negligible when compared with the fluid part, i.e. $ P_\mathrm{mag} \ll P $, $ \epsilon_\mathrm{mag} \ll \epsilon $, and that the components of the anisotropic term of $ T^{\mu \nu} $ satisfy $ b^{\mu}b^{\nu} \ll \rho h u^{\mu} u^{\nu} + P g^{\mu \nu} $. With these simplifications the evolution of the magnetic field, governed by the induction equation, does not affect the dynamics of the fluid, which is governed solely by the hydrodynamics equations. However, the magnetic field evolution does depend on the fluid evolution, due to the presence of the velocity components in the induction equation. This “test magnetic field” (or passive field) approximation is employed in the core collapse simulations reported in this work.
Within this approach the seven eigenvalues of the GRMHD Riemann problem (entropy, Alfvén, and fast and slow magnetosonic waves) reduce to three [@nfnr], $$\begin{aligned}
\lambda^{i}_{0 \, \mathrm{hydro}} & = &
\lambda^i_\mathrm{e} = \lambda^i_{\mathrm{A} \, \pm} =
\lambda^i_{\mathrm{s} \, \pm},
\\
\lambda^i_{\pm \, \mathrm{hydro}} & = &
\lambda^i_{\mathrm{f} \, \pm},\end{aligned}$$ where $ \lambda^{i}_{0 \, \mathrm{hydro}} $ and $ \lambda^i_{\pm \, \mathrm{hydro}} $ are the eigenvalues of the Jacobian matrices of the purely hydrodynamics equations, as reported by @banyuls_97_a.
This approximation has several interesting properties. First, if we perform a simulation for a given initial magnetic field, we can compute the result for a simulation with the same initial magnetic field scaled by some factor. To do this it is sufficient to increase or reduce the strength of the magnetic field at any given time during the simulation by the same factor. The second property is what we call the “composition rule”. If we perform two simulations with the same hydrodynamics but different initial magnetic fields, $ {\mbox{\boldmath $B$}}^{*\,0}_1 $ and $ {\mbox{\boldmath $B$}}^{*\,0}_2 $, any linear combination $ {\mbox{\boldmath $B$}}^* (t)= a \, {\mbox{\boldmath $B$}}^*_1 (t) + b \, {\mbox{\boldmath $B$}}^*_2 (t) $ of the magnetic field at any time, with $ a $ and $ b $ being constants, will be the solution for the evolution of a model whose initial magnetic field is the same linear combination, i.e.$ {\mbox{\boldmath $B$}}^{*\,0}= a \, {\mbox{\boldmath $B$}}^{*\,0}_1 + b \, {\mbox{\boldmath $B$}}^{*\,0}_2 $. Hence, we can make use of these properties to cover a wide range of magnetic field strengths and structures by performing just a few simulations, and then constructing additional ones by means of the “composition rule”. Needless to say, these two properties are valid only if the magnetic field resulting from the scaling or composition satisfies itself the passive field approximation at all times.
Gravitational waves
-------------------
The Newtonian standard quadrupole formula has been extensively used in numerical simulations of astrophysical systems to compute the gravitational radiation and waveforms without having to consider the full evolution of the spacetime and solving Einstein’s equations. This formula computes the radiative part of the spatial metric as $$h^\mathrm{quad}_{ij} =
P^{\mathrm{TT} kl}_{ij} \frac{2}{R} \ddot{Q}_{ij},
\label{eq:quad_formula}$$ where $ P^{\mathrm{TT} kl}_{ij} $ is the transverse traceless projector operator [@thorne80], $ R $ is the distance to the source, $ Q_{ij} $ is the mass quadrupole moment, and a dot denotes a time derivative. In spite of its simplicity, the particular form in which Eq. (\[eq:quad\_formula\]) is expressed leads to numerical difficulties due to the presence of second time derivatives. A way to circumvent this problem is to eliminate all time derivatives using the equations of motion. Following @finn_89_a and @blanchet_90_a one can arrive to an expression for $ \ddot{Q}_{kl} $ with no explicit appearance of time derivatives. This is the so-called stress formula, $$\ddot{Q}_{ij} \approx \operatorname{STF}\left\{ 2 \int \mathrm{d}^3 {\mbox{\boldmath $x$}} \,
\sqrt{\hat{\gamma}} D^* \left( \hat{\gamma}_{ik} \hat{\gamma}_{jl}
\, v^k v^l + x^k \hat{\gamma}_{ki} \, \hat{\nabla}_j U \right) \right\},
\label{eq:stress_formula}$$ where $ \operatorname{STF}$ means the symmetric and traceless part, and $ D^* = \sqrt{\bar{\gamma}} D $. This formula has proved to be numerically much more accurate than the original formula and we use it in this paper to extract gravitational waveforms.
In the case of a magnetized fluid in the ideal MHD case, the gravitational radiation is also affected by the energetic content of the magnetic field. @kotake04_b have derived an extension of the quadrupole formula for such a case. In a similar way, it is possible to calculate the corresponding stress formula [@obergaulinger_06_a], which reads $$\begin{aligned}
\ddot{Q}_{ij} & \approx & \operatorname{STF}\biggl\{ 2 \int \mathrm{d}^3 {\mbox{\boldmath $x$}} \, \sqrt{\hat{\gamma}} \biggl[ D^*
\left( \hat{\gamma}_{ik} \hat{\gamma}_{jl} \, v^k v^l +
x^k \hat{\gamma}_{ki} \, \hat{\nabla}_j U \right)
\nonumber
\\
& & \qquad \qquad \qquad \qquad \,\;
- \hat{\gamma}_{ik} \hat{\gamma}_{jl} \, b^k b^l \biggr] \biggr\}.
\label{eq:mag_stress_formula}\end{aligned}$$ Note that in the limit of weak magnetic fields the original stress formula is recovered. We use this formula in the magnetized core collapse simulations to calculate the contribution of the magnetic field to the waveforms in the passive field approximation.
Initial data {#sec:mag_ini_models}
============
The structure and strength of the magnetic field in the stellar core collapse progenitors, needed as initial conditions of our numerical simulations, is still an open question in astrophysics. State-of-the-art models from stellar evolution including a description for the influence of the magnetic field [@heger05], predict that the distribution of the magnetic field in the iron core has probably a dominant toroidal component, with a strength of about $ 10^9 \mbox{\,--\,} 10^{10} \mathrm{\ G} $, and a poloidal component of only about $ 10^5 \mbox{\,--\,} 10^6 \mathrm{\ G} $. For such weak fields ($ P_\mathrm{mag} \ll P $), the passive field approximation adopted here is likely to be sufficient to describe the initial models considered in this work. A second consideration is whether or not the initial model should be an equilibrium model. In general, if one tries to construct a stationary model without meridional currents and assuming an isentropic flow, the only possible magnetic field configuration is poloidal [see @bekenstein79]. Stationary models of magnetized stars have been computed under these assumptions by @bocquet95. In the general (but still isentropic) case in which meridional circulation is allowed, a toroidal component of the magnetic field may exist, but the method to calculate stationary models is far more complicated [@gourgulhon93; @ioka03; @ioka04]. When one considers ideal MHD, also purely toroidal magnetic fields exist which maintain the circularity condition [@oron02], and therefore it is possible to generate stationary models without meridional components. Finally, in the case that magnetic pressure does not exceed the hydrostatic pressure, @oron02 has shown that stationary models with mixed toroidal and poloidal component approximately accomplish the circularity condition.
Therefore, it makes sense to construct initial models for magnetized stellar cores by simply adding an ad-hoc magnetic field to a purely hydrodynamic equilibrium configuration. If the condition $ \vec{B}^* \cdot \vec{\hat{\nabla}} \Omega^* = 0 $ is satisfied, where $ \Omega^* = v^{*\,\varphi} / (r \sin{\theta}) $ is the angular velocity of the fluid, then the initial magnetic field does not evolve in time either. Note that in this work we use as initial models both equilibrium and non-equilibrium configurations for the magnetic field, as specified in Table \[tab:MCC\_models\].
Magnetic field configurations
-----------------------------
Since the numerical scheme we use to evolve the MHD equations only preserves the value of $ \vec{\hat{\nabla}} \cdot \vec{B}^* $ but does not impose the divergence constraint of the magnetic field itself, it is necessary to build initial configurations that also satisfy this condition. To do this we calculate the initial magnetic field from a vector potential $ \vec{A}^* $, such that $ \vec{B}^* = \vec{\hat{\nabla}} \times \vec{A}^* $, which is discretized as explained in @nfnr.
For our code tests and core collapse simulations we use three possible magnetic field configurations as initial conditions (or any possible combination of them):
– the homogeneous “starred” magnetic field, in which $ \vec{B}^* $ is constant and parallel to the symmetry axis,
– the poloidal magnetic field generated by a circular current loop of radius $ r_\mathrm{mag} $ [@jackson62], that can be calculated from the only non-vanishing component of the vector potential $ A^*_\varphi $ as $$A^*_\varphi = \frac{r^2_\mathrm{mag} B^*_0}{2}
\sum_{n=0}^{\infty} \frac{(-1)^n (2n-1)!!}{2^n (n+1)!}
\frac{r_<^{2n+1}}{r_>^{2n+2}} P^1_{2n+1} (\cos{\theta}),
\label{eq:circ_loop}$$ where $ r_< = \min(r, r_\mathrm{mag}) $, $ r_> = \max(r, r_\mathrm{mag}) $, and $ B^*_0 $ is the magnetic field at the center, and
– a toroidal magnetic field of the form $$B^{*\,\varphi} =
B^*_0 \frac{r_\mathrm{mag}^2}{r_\mathrm{mag}^2 - \varpi^2},
\label{eq:pure_tor}$$ whose maximum value is reached at $ \varpi = r_\mathrm{mag} $, where $ B^*_0 $ is the initial central magnetic field and $ \varpi $ is the distance to the axis.
Note that in all three cases, we employ the “starred” magnetic field, since the divergence constraint is valid for this quantity when computed with respect to the flat divergence operator. In this way we can extend any analytic prescription for the magnetic field given in flat spacetime in an easy way. Also note that in the presence of strong gravitational fields the magnetic field $ \vec{B} $ is deformed with respect to $ \vec{B}^* $ due to the curvature of the spacetime, although the divergence constraint is automatically fulfilled. Fig. \[fig:homo\_field\] shows examples of the magnetic field structure for the poloidal configurations of the initial magnetic field.
The initial magnetic field configuration is denoted in the names of the models in our sample by adding a label to the purely hydrodynamic model. For the latter we follow the notation of @dimmelmeier_02_a and @ott06a. These models are listed in Table \[tab:CC\_models\]. The label for the magnetic field is constructed following the notation of @obergaulinger_06_a. We add the suffix D3M0 to denote those models with purely poloidal magnetic field generated by a circular loop and $ r_\mathrm{mag} = 400 \mathrm{\ km} $ (M0 denotes the passive field approximation) and we use the suffix T3M0 for models with purely toroidal magnetic field and $ r_\mathrm{mag} = 400 \mathrm{\ km} $. We have also built the DT3M0 model, whose magnetic field distribution is a combination of D3M0 and T3M0 with equal magnetic field strengths at the center. This model allows us to check the validity of the “composition rule” (see Sect. \[sec:passive\]).
Hydrodynamic configurations
---------------------------
### Polytropes in rotational equilibrium
For the simulation with a simplified description of matter using the hybrid EOS (see Sect. \[hybrid\_eos\]), we construct $ \gamma_\mathrm{ini} = 4/3 $ polytropes in rotational equilibrium which we obtain by using the relativistic generalization of Hachisu’s self-consistent field method by @komatsu_89_a[^2]. Their rotation law for the specific angular momentum $ j $ is given by $$j = A^2 (\Omega_\mathrm{c} - \Omega),
\label{rotation_law}$$ where $ A $ parametrizes the degree of differential rotation (stronger differentiality with decreasing $ A $) and $ \Omega_\mathrm{c} $ is the value of the angular velocity $ \Omega $ at the center. In the Newtonian limit, this reduces to $$\Omega = \frac{A^2 \Omega_\mathrm{c}}{A^2 + \varpi^2}.
\label{newtonian_rotation_law}$$ The parameters of the selected models, which are chosen to be identical to some of the models considered by @dimmelmeier_02_a, are described in Table \[tab:CC\_models\]. In addition, as we aim at comparing our results with the recent numerical simulations performed by @obergaulinger_06_a in Newtonian gravity, a subset of our models (those with purely poloidal magnetic field) have been selected as general relativistic counterparts of their models. In Table \[tab:CC\_models\] we also give the values for the gravitational mass $ M_\mathrm{g} $ (which is identical to the ADM mass $ M_\mathrm{ADM} $) and for the initial rotation rate $ \beta = E_\mathrm{rot} / |E_\mathrm{b}| $. In the definition of $ \beta $ we use the following expressions for the rotational kinetic energy $ E_\mathrm{rot} $, the gravitational binding energy $ E_\mathrm{b} $, and the magnetic energy $ E_\mathrm{mag} $: $$\begin{aligned}
E_\mathrm{rot} & = & \frac{1}{2} \int \mathrm{d}^3 {\mbox{\boldmath $x$}} \,
\sqrt{\gamma} \, \alpha \hat{v}^\varphi S_\varphi,
\\
E_\mathrm{b} & = & M_\mathrm{g} - M_\mathrm{p} -
E_\mathrm{rot} - E_\mathrm{mag},
\\
E_\mathrm{mag} & = & \frac{1}{2} \int \mathrm{d}^3 {\mbox{\boldmath $x$}} \,
\sqrt{\gamma} \, W b^2,\end{aligned}$$where $ M_\mathrm{p} $ is the proper mass.
--------- ----------------------------------------- ------ ----------- ------------------ --------------
Model $ \rho_\mathrm{c} $ $ \beta $ $ M_\mathrm{g} $ $ \gamma_1 $
\[$ 10^{10} \mathrm{\, g\, cm}^{-3} $\] \[%\] \[$ M_\odot $\]
A1B3G3 1.00 50.0 0.90 1.46 1.300
A1B3G5 1.00 50.0 0.90 1.46 1.280
A3B3G5 1.00 0.5 0.90 1.46 1.280
A2B4G1 1.00 1.0 1.80 1.50 1.325
A4B5G5 1.00 0.5 4.00 1.61 1.280
s20A1B1 0.88 50.0 0.25 1.58 —
s20A1B5 0.88 50.0 4.00 1.58 —
s20A2B2 0.88 1.0 0.50 1.58 —
s20A2B4 0.88 1.0 1.80 1.58 —
s20A3B3 0.88 0.5 0.90 1.58 —
E20A 0.42 — 0.37 2.00 —
--------- ----------------------------------------- ------ ----------- ------------------ --------------
: Purely hydrodynamic initial models used in the magnetized core collapse simulations.
\[tab:CC\_models\]
For these simplified initial models the gravitational collapse is initiated by slightly decreasing the adiabatic index from its initial value to $ \gamma_1 < \gamma_\mathrm{ini} = 4 / 3 $, which results in a loss of pressure support. If no pressure reduction were imposed, the purely hydrodynamic initial models would remain stationary. However, even in that case the associated initial magnetic field may not remain stationary (see also Table \[tab:MCC\_models\] below). Only a purely toroidal initial magnetic field would not evolve in time, while any magnetic field configuration of initial models labeled A1 would still stay approximately stationary, since these models rotate almost rigidly, and thus the initial magnetic field cannot wind up itself strongly.
### Presupernova models from stellar evolution calculations
As initial models for the simulations where we use a microphysical EOS and deleptonization, we employ the solar-metallicity $ 20 \, M_\odot $ (zero-age main sequence) model of @woosley_02_a (labeled as model s20 in Table \[tab:CC\_models\]). On this spherically symmetric model, which is initially not in equilibrium as it has a non-zero radial velocity profile, we impose the rotation law (\[rotation\_law\]), using the same rotation nomenclature as for the previously described polytropes in equilibrium.
In addition, we perform calculations with the “rotating” presupernova model E20A of @heger00_a, which we map onto our computational grids under the assumption of constant rotation on cylindrical shells of constant $ \varpi $.
Treatment of matter during the evolution {#eos}
========================================
In this work we improve upon preceding relativistic stellar core collapse simulations by using an advanced description of microphysics as presented in @ott06a and @dimmelmeier_07_a. For comparison to previous results, we also perform simulations with the simplified hybrid EOS [@janka_93_a]. In the following, we describe both approaches for the treatment of matter.
Hybrid EOS {#hybrid_eos}
----------
For calculations employing polytropes in rotational equilibrium as initial models, we utilize the hybrid EOS. Here the pressure consists of a polytropic part, $ P_\mathrm{p} = K \rho^\gamma $, with $ K = 4.897 \times 10^{14} $ (in cgs units), plus a thermal part, $ P_\mathrm{th} = \rho \epsilon_\mathrm{th} (\gamma_\mathrm{th} - 1) $, where $ \epsilon_\mathrm{th} = \epsilon - \epsilon_\mathrm{p} $ and where we set $ \gamma_\mathrm{th} = 1.5 $. The thermal contribution is chosen to take into account the rise of thermal energy due to shock heating. As $ \rho $ reaches nuclear density at $ \rho_\mathrm{nuc} = 2.0 \times 10^{14} \mathrm{\ g\ cm}^{-3} $, $ \gamma $ is raised to $ \gamma_2 = 2.5 $ and $ K $ adjusted accordingly to maintain monotonicity of $ P $ and $ \epsilon $. Due to this stiffening of the EOS the core undergoes a so-called pressure-supported bounce. More details of the hybrid EOS can be found e.g. in @dimmelmeier_02_a.
Microphysical EOS, deleptonization scheme, and neutrino pressure
----------------------------------------------------------------
In our more realistic calculations, for which the models s20 and E20A from stellar evolution are taken as initial models, we employ the tabulated non-zero temperature nuclear EOS by @shen_98_a in the variant of @marek_05_a which includes baryonic, electronic, and photonic pressure components. This gives the fluid pressure $ P $ (and additional thermodynamic quantities) as a function of $ \rho $, the temperature $ T $, and the electron fraction $ Y_e $. Since the code operates with the specific internal energy $ \epsilon $ instead of the temperature $ T $, we determine the corresponding value for $ T $ iteratively with a Newton–Raphson scheme.
To determine the evolution of $ Y_e $, the state vector, flux vector, and source vector for the conservation equations (\[eq:hydro\_conservation\_equation\]), as given in Eqs. (\[eq:state\_vector\]–\[eq:source\_vector\]) have to be augmented by the components $$D Y_e,
\qquad
D Y_e \hat{v}^i,
\qquad
S_{Y_e},$$ respectively. The source term $ S_{Y_e} $ is a consequence of the electron captures during collapse, which reduces $ Y_e $. This deleptonization also effectively decreases the size of the homologously collapsing inner core, and has thus a direct influence on the collapse dynamics and the gravitational wave signal. Hence, it is essential to include (at least an approximate scheme for) deleptonization during collapse.
Since multi-dimensional radiation hydrodynamics calculations in general relativity are not yet computationally feasible, in the simulations using the microphysical EOS we make use of a a recently proposed scheme [@liebendoerfer_05_a] where deleptonization is parametrized based on data from detailed spherically symmetric calculations with Boltzmann neutrino transport. As in @dimmelmeier_07_a we take the latest available electron capture rates [@langanke_00_a], which result in lower values for $ Y_e $ in the inner core at bounce compared to recent results [@ott06a] where standard capture rates were used [@rampp_00_a]. Following @liebendoerfer_05_a, deleptonization is stopped at core bounce (i.e. as soon as the specific entropy $ s $ per baryon exceeds $ 3 k_\mathrm{B} $). After core bounce $ Y_e $ is only passively advected, neglecting any further deleptonization in the nascent PNS.
Neutrino pressure is included only in the regime which is optically thick to neutrinos, which we define for $ \rho $ being above the trapping density $ \rho_\mathrm{t} = 2 \times 10^{12} \mathrm{\ g\ cm}^{-3} $. Following @liebendoerfer_05_a, here we approximate the contribution of the neutrino pressure $ P_\nu $ as an ideal Fermi gas and include the radiation stress via additional source terms in the momentum and energy equations for the fluid.
Outline of the numerical approach {#numerics}
=================================
The GRMHD numerical code we use in our simulations is based on the purely hydrodynamic code described in @dimmelmeier_02_a [@dimmelmeier_02_b], and on its extension discussed in @cerda05. It has been described in detail in a previous paper [@nfnr], which allows us to provide here only succint information. The code performs the coupled time evolution of the equations governing the dynamics of the spacetime, the fluid, and the magnetic field in general relativity. The equations are implemented in the code using spherical polar coordinates $ \{ t, r, \theta, \varphi \} $, assuming axisymmetry with respect to the rotation axis and equatorial plane symmetry at $ \theta = \pi / 2 $.
The hydrodynamics solver {#sec:hydro_solver}
------------------------
For the evolution of the matter fields we utilize a high-resolution shock-capturing (HRSC) scheme, which numerically integrates the subset of equations in system (\[eq:hydro\_conservation\_equation\]) that corresponds to the purely hydrodynamic variables ($ D $, $ S_i $, $ \tau $). HRSC methods ensure the numerical conservation of physically conserved quantities and a correct treatment of discontinuities such as shocks [see e.g. @font_03_a for a review and references therein]. We have implemented in the code various cell-reconstruction procedures, either second-order or third-order accurate in space, namely minmod, MC, and PHM [see @toro99 for definitions]. The time update of the state vector $ {\mbox{\boldmath $U$}} $ is done using the method of lines in combination with a second-order accurate Runge–Kutta scheme. The numerical fluxes at the cell interfaces are obtained using either the HLL single-state solver of @harten83 or the symmetric scheme of @kt00 (KT hereafter). Both solvers yield results with an accuracy comparable to complete Riemann solvers (with the full characteristic information), as shown in simulations involving purely hydrodynamic special relativistic flows [@lucas04] and general relativistic flows in dynamical spacetimes [@shibata_font05]. Tests of both solvers in GRMHD have been reported recently by @anton06.
Evolution of the magnetic field {#b_field_evolution}
-------------------------------
The evolution of the magnetic field needs to be performed in a way that is different from the rest of the conservation equations, since the physical meaning of the corresponding conservation equation is different. Although the induction equation can be written in a flux conservative way, a supplementary condition for the magnetic field has to be given (the divergence constraint), which has to be fulfilled at each time iteration. The physical meaning of these two equations is the conservation of the magnetic flux in a close volume, in our case each numerical cell. Therefore, an appropriate numerical scheme has to be used which takes full profit of such a conservation law. Among the numerical schemes that satisfy this property [see @toth00 for a review], the constrained transport (CT) scheme [@evans88] has proved to be adequate to perform accurate simulations of magnetized flows. Our particular implementation of this scheme [see @nfnr for details] has been adapted to the spherical polar coordinates used in the code. The discretized evolution equations for the poloidal components of the magnetic field read $$\begin{aligned}
\partial_t B^{*\,r} _{i+{{1\over2}}\, j} & = &
\frac{[\sin \theta \, E^*_\varphi]_{i+{{1\over2}}\, j+{{1\over2}}} -
[\sin \theta \, E^*_\varphi]_{i+{{1\over2}}\, j-{{1\over2}}}}
{r_{i+{{1\over2}}\, j} \, \Delta (\cos \theta)_j},
\label{eq:ct_r}
\\
\partial_t B^{*\,\theta}_{i \, j+{{1\over2}}} & = &
2 \, \frac{[r \, E^*_\varphi]_{i+{{1\over2}}\, j+{{1\over2}}} -
[r \, E^*_{\varphi}]_{i-{{1\over2}}\, j+{{1\over2}}}}{\Delta r^2_i},
\label{eq:ct_theta}\end{aligned}$$ where (in vectorial form) $ {\mbox{\boldmath $E$}}^* = {\mbox{\boldmath $v$}}^* \times {\mbox{\boldmath $B$}}^* $, and where cell centers are located at $ (i\,j) $, radial interfaces at $ (i+{{1\over2}}\,j) $, angular interfaces at $ (i\,j+{{1\over2}}) $, and cell corners (cell edges along the $ \varphi $-direction) at $ (i+{{1\over2}}\,j+{{1\over2}}) $. We note that the evolution equation for the toroidal magnetic field is analog to that used for the hydrodynamics, since in axisymmetry this component does not play any role in the CT scheme. The previous expressions are used in the numerical code to update the magnetic field. The only remaining aspect is to give an explicit expression for the value of $ E^*_i $. A practical way to calculate $ E^*_\varphi $ from the numerical fluxes in the adjacent interfaces [@balsara99] is $$\begin{aligned}
E^*_{\varphi \, i+{{1\over2}}\, j+{{1\over2}}} & = &
- \frac{1}{4} \left[ (F^r)^\theta_{i \, j+{{1\over2}}} +
(F^r)^\theta_{i+1 \, j+{{1\over2}}} \right.
\nonumber
\\ & & \left. \qquad
- (F^\theta)^r_{i+{{1\over2}}\, j} -
(F^\theta)^r_{i+{{1\over2}}\, j+1} \right],\end{aligned}$$ where the fluxes (\[eq:flux\_vector\]) are obtained in the usual way by solving the Riemann problem at the interfaces. The combination of the CT scheme and this way of computing the electric field is called the flux-CT scheme. It is used in all numerical simulations reported in this paper. Finally, the time discretization of Eqs. (\[eq:ct\_r\]) and (\[eq:ct\_theta\]) is performed in the same way as for the fluid evolution equations.
The metric solver
-----------------
The CFC metric equations are five nonlinear elliptic coupled Poisson-like equations which can be written in compact form as $ \hat{\Delta} {\mbox{\boldmath $u$}} ({\mbox{\boldmath $x$}}) = {\mbox{\boldmath $f$}} ({\mbox{\boldmath $x$}}; {\mbox{\boldmath $u$}} ({\mbox{\boldmath $x$}}))$, where $ {\mbox{\boldmath $u$}} = u^k = (\phi, \alpha \phi, \beta^j) $, and $ {\mbox{\boldmath $f$}} = f^k $ is the vector of the respective sources. These five scalar equations for each component of $ {\mbox{\boldmath $u$}} $ couple to each other via the source terms that in general depend on the various components of $ {\mbox{\boldmath $u$}} $. We use a fix-point iteration scheme in combination with a linear Poisson solver to solve these equations. Further details on this type of metric solver can be found in @cerda05 and @dimmelmeier_02_a.
Setup of the numerical grid
---------------------------
------------------------ ------------ ----------------------------------------------------------------------------- ------------------------------------ --------------------------------- ------------------------ ------------------------ ------------------------- -------------------------- ---- ------
Model stationary $ \rho_\mathrm{max} $ $ |B_\mathrm{polo}|_\mathrm{max} $ $ |B_{\varphi}|_\mathrm{ max} $ $ \beta_\mathrm{rot} $ $ \beta_\mathrm{mag} $ $ \beta_\mathrm{polo} $ $ \Omega_\mathrm{c} $
$ \displaystyle \left[ 10^{14} \frac{\mathrm{g}}{\mathrm{\ cm}^3} \right] $ \[$ 10^{10} \mathrm{\ G} $\] \[$ 10^{10} \mathrm{\ G} $\] \[$ 10^{-2} $\] \[$ 10^{-8} $\] \[$ 10^{-8} $\] \[$ \mathrm{ms}^{-1} $\]
\[0.5 em\] A2B4G1-D3M0 no 0.47 400 1467 15.6 8.3 0.9 0.36 85 10.3
A1B3G3-D3M0 approx. 4.22 1719 2522 2.3 1.2 0.8 3.96 7 0.3
A1B3G3-T3M0 yes 4.22 0 1714 2.3 0.2 0.0 3.96 — —
A1B3G5-D3M0 approx. 4.57 1146 1275 0.9 0.5 0.4 3.91 21 0.7
A1B3G5-T3M0 yes 4.57 0 1542 0.9 0.2 0.0 3.91 — —
A1B3G5-DT3M0 approx. 4.57 1146 1537 0.9 0.6 0.4 3.91 21 0.6
A3B3G5-D3M0 no 3.73 984 1672 2.3 0.6 0.4 3.75 24 1.1
A4B5G5-D3M0 no 1.74 1094 2716 8.5 4.4 1.2 1.18 24 2.1
A4B5G5-T3M0 yes 1.74 0 1626 8.5 0.4 0.0 1.18 — —
s20A1B1-D3M0 approx. 2.69 1221 162 0.6 2.9 2.9 1.34 22 0.5
s20A2B2-D3M0 no 2.75 1849 3574 5.8 7.6 3.2 3.55 7 0.5
s20A2B2-T3M0 yes 2.75 0 1365 5.8 0.9 0.0 3.55 — —
s20A1B5-D3M0 approx. 2.69 1100 1011 7.8 3.2 2.4 3.89 10 0.8
s20A1B5-T3M0 yes 2.69 0 1447 7.8 1.2 0.0 3.89 — —
E20A-D3M0 no 2.29 2343 7503 7.7 23.4 1.8 4.37 6 0.5
E20A-T3M0 yes 2.29 0 2739 7.7 1.9 0.0 4.37 — —
------------------------ ------------ ----------------------------------------------------------------------------- ------------------------------------ --------------------------------- ------------------------ ------------------------ ------------------------- -------------------------- ---- ------
\[tab:MCC\_models\]
We perform all axisymmetric simulations with a resolution ($ n_r \times n_\theta $) of $ 300 \times 30 $ zones, except for models labeled A4B5G5 in which a resolution of $ 300 \times 40 $ is used due to the more complex angular structure. In both cases the radial grid is equally spaced for the first 100 points, with a grid spacing of $ 100 \mathrm{\ m} $. The remaining radial zones are logarithmically distributed to cover the outer parts of the star and the exterior artificial low-density atmosphere. The angular grid is equally spaced and we assume equatorial symmetry. We have performed resolution tests and we have found that such a resolution is adequate for our simulations [see @cerda_phd; @nfnr for details]. As a consequence of our various code tests (see Appendix \[app:tests\]) all results discussed in Sect. \[results\] correspond to simulations performed using PHM reconstruction and the HLL solver for the hydrodynamics.
Results
=======
We now present the main results from our numerical simulations of rotational magnetized core collapse to neutron stars. First, we note that a quantitative summary of our findings is reported in Table \[tab:MCC\_models\], to which we will refer repeteadly. The dynamics of the models we have selected is identical to the dynamics of the unmagnetized ones, since the passive field approximation is used. Therefore, we will not describe here all the morphological features of the hydrodynamics of both models with the hybrid EOS (simplified models hereafter) and models with the microphysical EOS and the deleptonization scheme (microphysical models hereafter), as they have been discussed in detail in @dimmelmeier_02_a [@dimmelmeier_02_b], and @ott06b as well as @dimmelmeier_07_a [@dimmelmeier_07_b], respectively. (It is worth to emphasize, however, the excellent agreement found in the hydrodynamical simulations performed with three independent numerical codes.) We pay more attention instead to the magnetic field evolution. In all our simulations an initial magnetic field strength of $ B^*_0 = 10^{10} \mathrm{\ G} $ is considered. This value is an upper limit for the T3M0 models, since the expected initial toroidal magnetic field is of this order [@heger05]. However, for the D3M0 models, this field strength is already at (or above) the upper end of the astrophysically expected values.
For all models we first present results for identical values of $ B^*_0 $, in a way that we can study the different effects and compare them properly. Afterwards we present the results scaled to lower, astrophysically expected values. We anticipate that our results can change if some of the several assumptions made in our simulations (axisymmetry and passive field approximation) are relaxed. An estimation and discussion of these effects can be found in Sect. \[sub:ampli\].
Evolution of the magnetic energy parameter {#sec:beta_evol}
------------------------------------------
\
\[0.5 em\]
\
\[0.5 em\]
The evolution of the energy parameter for the magnetic field $ \beta_\mathrm{mag} = E_\mathrm{mag} / |E_\mathrm{b}| $ can be seen in Fig. \[fig:beta\_mcollapse\] for model A1B3G5 of our sample. In order to analyze the amplification of the magnetic field, we separate the effects of the different components of the magnetic field into $ \beta_\varphi $ for the toroidal component and $ \beta_\mathrm{polo} =
\beta_\mathrm{mag} - \beta_\varphi $ for the poloidal component, which are also plotted in the figure. As the collapse proceeds the magnetic field grows by at least two reasons: First, the radial flow compresses the magnetic field lines, amplifying the existing poloidal and toroidal magnetic field components. Second, during the collapse of a rotating star differential rotation is produced and increased, even for rigidly rotating initial models (see e.g. @dimmelmeier_02_b). Hence, if a seed poloidal field exists, the $ \Omega $-dynamo mechanism winds up the poloidal field lines into a toroidal component. This (linear) amplification process generates a toroidal magnetic field component, even from purely poloidal initial configurations. The toroidal component of the magnetic field is affected by the two effects, while the poloidal field is only amplified by radial compression of the field lines. Thus, even if the initial magnetic field configuration is purely poloidal, the toroidal component dominates after some dynamical time. To study the differences in the evolution of the magnetic field depending on the initial magnetic field we now describe in detail the features of model A1B3G5 with different initial magnetic field configurations.
In model A1B3G5-D3M0 the initial magnetic field is entirely poloidal. The top panel of Fig. \[fig:beta\_mcollapse\] shows that $ \beta_\varphi $ (dashed line) grows much faster than $ \beta_\mathrm{polo} $, particularly after bounce ($ t \sim 30 \mathrm{\ ms} $) when the radial compression mechanism stops. We note that the magnetic field considered is weak enough not to affect the dynamics, with the final $ \beta_\mathrm{mag} \ll 1 $.
If we consider a purely toroidal magnetic field initially, as model A1B3G5-T3M0, the only amplification mechanism present in our simulations is the radial compression, since no poloidal field can be wound up. The bottom panel of Fig. \[fig:beta\_mcollapse\] shows the behaviour of $ \beta_\mathrm{mag} $ for model A1B3G3-T3M0. It is important to notice that during the collapse $ \beta_\mathrm{mag} $ hardly grows (for other models of the T3M0 series it even decreases) since the radial compression is a very inefficient mechanism to amplify the magnetic field. As a result, for some models the final PNS is “less magnetized” than the progenitor core in the sense that $ \beta_\mathrm{mag} $ at bounce is smaller than it is before the collapse. We note that the evolution of this kind of purely toroidal models could change completely if the axisymmetry condition were removed, since in three dimensions there are mechanisms that can transform a toroidal magnetic field into a poloidal one. Some of these mechanisms are discussed in Sect. \[sub:ampli\] below.
To check whether the “composition rule” (see Sect. \[sec:passive\]) is valid we consider next a mixed configuration of poloidal and toroidal magnetic fields at the beginning of the simulation (model A1B3G5-DT3M0). The top panel of Fig. \[fig:beta\_mcollapse2\] shows with a solid line the time evolution of $ \beta_\mathrm{mag} $ for model A1B3G5-DT3M0 and with a dot-dashed line the composition of the individual values for $ \beta_\mathrm{mag} $ in models A1B3G5-D3M0 and A1B3G5-T3M0 with identical initial field strengths. (The separate evolutions for the latter are also included in the plot as dashed and dotted lines, respectively.) The agreement of the two evolution paths is remarkable, which shows that the “composition rule” works properly for our simulations. Therefore, we can use it to obtain any desirable composition of magnetic fields with a single hydrodynamic evolution of the two models D3M0 and T3M0. For the particular composition showed in this model, the final value of $\beta_\mathrm{mag}$ depends very weakly on the initial toroidal magnetic field component. In other words, the structure of the magnetic field of the PNS will depend almost exclusively on the radial compression of the initial poloidal component of the magnetic field.
Next, we consider a “composition” of these models with different initial magnetic field strength. We keep the initial toroidal component fixed at a realistic value, $ B^{*\,0}_\varphi = 10^{10} \mathrm{\ G} $, and change the initial poloidal component in a range that spans from $ B^{*\,0}_\mathrm{polo} = 10^{10} \mathrm{\ G} $ down to the astrophysically more realistic value of $ 10^6 \mathrm{\ G} $. The bottom panel of Fig. \[fig:beta\_mcollapse2\] shows the time evolution of $ \beta_\mathrm{mag} $ for these different configurations. For lower values of $ B^{*\,0}_\mathrm{polo} $, the $ \Omega $-dynamo mechanism becomes increasingly slower and the initial toroidal component becomes important for the magnetic field configuration of the PNS. In the lowest initial poloidal field case analyzed, the magnetic field of the PNS is completely toroidal and depends exclusively on the initial magnetic field configuration.
The remaining computed models of our sample, including those with microphysics, behave qualitatively in a very similar manner, although quantitative differences can be found in the amplification of the magnetic field during the collapse, and the amplification rates after bounce due to the $ \Omega $-dynamo. Fig. \[fig:beta\_mcollapse\_comp\] shows the evolution of the magnetic energy parameter $ \beta_\mathrm{mag} $ for all the simulated models with initial purely poloidal magnetic field (label D3M0). For all models we find the following relation between the collapse time and the amplification rate of the magnetic field after bounce [which, however, does not hold for model A2B4G1-D3M0, as this is the only model of our sample for which the collapse is halted not by the stiffening of the EOS, but rather by centrifugal forces at subnuclear densities; cf. @dimmelmeier_02_b]: Models with large collapse times, such as all microphysical models as well as the simplified model A1B3G3-D3M0, exhibit a more efficient amplification of the magnetic field as compared to the rapid collapse models (G5 series). To quantify the differences between the models we estimate the time scale $ \tau_\Omega $ for the amplification of the magnetic field by fitting the post-bounce evolution of $ \beta_\mathrm{mag} $ to $$\beta_\mathrm{mag} = \left( \frac{t}{\tau_\Omega} \right)^2.$$ The resulting values can be found in Table \[tab:MCC\_models\]. The time scale should depend on the central angular velocity $ \Omega_\mathrm{c} $ of the PNS, and on the strength of the poloidal magnetic field that can be wound up, which can be estimated from $ \beta_\mathrm{polo} $. Hence, the following expression should be valid in the most efficient scenario (see Appendix \[app:odynamo\] for details): $$\tau_\Omega = \frac{2}{\Omega_\mathrm{c} \sqrt{\beta_\mathrm{polo}}}.
\label{upper_limit}$$ To check this relation we plot in the top panel of Fig. \[fig:beta\_mcollapse\_comp2\] the value of the fit for $ \tau_\Omega $ versus the value from the previous analytic expression. Apparently for all models the growth time of the $ \Omega $-dynamo is always larger than that of the most efficient situation (solid line in the figure), and corresponds to a fraction (30%–90%) of the upper limit (\[upper\_limit\]). This relation shows that in order to obtain higher amplification rates of the magnetic field not only strong rotation is needed, but also a sufficient compression of the poloidal magnetic field during the collapse.
\
\[0.5 em\]
Furthermore, we also find a relation between the value of $
\beta_\mathrm{polo} $ and the mass enclosed in the neutrino sphere[^3], $ M_\mathrm{PNS} $ hereafter (see bottom panel of Fig. \[fig:beta\_mcollapse\_comp2\]). Since most of the magnetic field lines compressed by the collapse are located inside the neutrino sphere, it is easy to understand that more massive PNS have higher magnetic energies. The fit to a power law of the data shown in Fig. \[fig:beta\_mcollapse\_comp2\] yields $$\begin{aligned}
\beta_\mathrm{polo} & = & (3.2 \pm 0.5) \times
\nonumber
\\
& & 10^{-8}
\left( \frac{M_\mathrm{PNS}}{M_\odot} \right)^{(1.6 \pm 0.2)}
\left( \frac{B^*_0}{10^{10} \mathrm{\ G}} \right)^2.
\label{eq:beta_mcollapse_mass}\end{aligned}$$ As discussed in detail by @dimmelmeier_07_a [@dimmelmeier_07_b], in the microphysical models the mass of the homologously collapsing inner core at bounce has a value of $ \sim 0.5 \, M_\odot $ (for the rotation rates considered here). This is also consistent with the high mass $ M_\mathrm{PNS} \sim 0.8 \, M_\odot $ of the PNS in these models, as shown in Fig. \[fig:beta\_mcollapse\_comp2\]. To obtain masses in this range in models with a simple matter treatment, the adiabatic index would require a value $ \gamma_1 \gtrsim 1.32 $, which is close to $ 4 / 3 $. Already for moderate rotation, this choice would cause the core to undergo multiple centrifugal bounces at densities lower than nuclear density (as exemplified here in model A2B4G1), which is a dynamical behavior that does not occur at all in microphysical models [@dimmelmeier_07_a; @dimmelmeier_07_b see also the related discussion in Sect. \[gravitational\_waves\]]. Therefore, only the microphysical models feature a collapse to a PNS that has both high densities and is in addition comparably heavy. This combination, which cannot be realized with the simplified models, explains the higher growth rates of the magnetic field due to the $ \Omega $-dynamo observed if improved microphysics is taken into account.
Combining Eqs. (\[upper\_limit\]) and (\[eq:beta\_mcollapse\_mass\]) we can establish an upper limit to the growth rate of the magnetic field due to the $ \Omega $-dynamo using only hydrodynamic quantities, namely $ \Omega_\mathrm{c} $ and $ M_\mathrm{PNS} $, and the strength of the magnetic field in the progenitor, $ B^*_0 $. This limit is given by $$\begin{aligned}
\tau_{\Omega} & = & (11.18 \pm 0.9) \times
\nonumber
\\
& & \left( \frac{1 \mathrm{\ ms}^{-1}}{\Omega_\mathrm{c}} \right)
\left( \frac{M_\odot}{M_\mathrm{PNS}} \right)^{(0.8 \pm 0.1)}
\left( \frac{10^{10} \mathrm{\ G}}{B^*_0} \right) \mathrm{\ s}.\end{aligned}$$ This relation can be very useful to estimate how fast the magnetic field grows in a collapsed star, under the assumption of a weak magnetic field and with a similar poloidal configuration in the progenitor, using data from purely hydrodynamical simulations (with no magnetic fields). As a proof of consistency and in order to assess the quality of this estimate we have computed $ \tau_\Omega $ with this method. We find that in all cases the estimate is a lower limit for the actual value of $ \tau_\Omega $ obtained from the numerical simulations and deviates by at most 30%.
Convection {#subsec:convection}
----------
One of the most important features that can affect the evolution of the magnetic field in stellar core collapse to a PNS is the presence of convection. We present here a detailed analysis of this effect in our simulations. Since in all of our models the magnetic field is weak, the discussion can be performed without considering its influence. We also note that due to the approximations made in our simulations, specifically the lack of a consistent neutrino transport scheme, our findings regarding convection should not be considered as definite.
The stability conditions for a rotating star are given by the so-called Solberg–H[ø]{}iland criteria [@tassoul78], $$\begin{array}{rcl}
\mathcal{C}_\mathrm{SH1} & = & {\mbox{\boldmath $g$}} \cdot {\bf \mathcal{B}} +
\mathcal{J} \cdot \nabla\varpi > 0,
\\ [0.2 em]
\mathcal{C}_\mathrm{SH2} & = & ({\mbox{\boldmath $g$}} \times \nabla \varpi)
(\mathcal{B} \times \mathcal{J}) > 0,
\end{array}
\label{eq:shcriteria}$$ where $ {\mbox{\boldmath $g$}} $ is the gravitational acceleration, and the buoyancy and rotational terms are respectively given by $$\mathcal{B} = \frac{\nabla \rho}{\rho} - \frac{\nabla P}{P \Gamma_1},
\qquad
\mathcal{J} = \frac{1}{\varpi^3} \nabla (\Omega^2 \varpi^4),$$ with $ \Gamma_1 = (\partial \ln P / \partial \ln \rho)_{s, Y_e} $. Note that in the first condition of Eq. (\[eq:shcriteria\]), $ N^2 = {\mbox{\boldmath $g$}} \cdot \mathcal{B} $ is the Brunt–Väisälä frequency and $ \kappa^2 = \mathcal{J} \cdot \nabla \varpi $ is the epicyclic frequency. In the case of either no rotation or uniform rotation the Solberg–H[ø]{}iland criteria reduce to the well known Schwarzschild criterion, $ N^2 > 0 $. If one of the two conditions is not satisfied, convective instability develops. Following [@miralles04], the time scale of the fastest growing mode can be computed as $$\tau_\mathrm{\,SH} = \operatorname{Im}\left[ \left(
\frac{\mathcal{C}_\mathrm{SH1}}{2} -
\frac{1}{2} \sqrt{\mathcal{C}^2_\mathrm{SH1} -
4 \, \mathcal{C}_\mathrm{SH2}} \right)^{-1/2} \right].
\label{eq:shtimescale}$$ It is very useful to express the buoyancy terms in the conditions (\[eq:shcriteria\]) in terms of the contributions of the entropy and electron fraction gradients, $$\mathcal{B} = \xi \, \nabla s + \delta \, \nabla Y_\mathrm{e},
\label{eq:buoyancy_reformulation}$$ with $ \xi = - \partial \ln P / \partial s|_{\rho, Y_e} / \Gamma_1 $ and $ \delta = - \partial \ln P / \partial Y_e|_{\rho, s} / \Gamma_1 $. We point out that the Solberg–H[ø]{}iland criteria are valid exactly only in Newtonian gravity, and thus we use them here only as estimates. In order to assess the influence of general relativistic corrections, we also evaluate Eq. (\[eq:shcriteria\]) using covariant derivatives with respect to the CFC metric, which yields very similar results. Note also that the Solberg–H[ø]{}iland criteria are based on a local instability analysis, while the convection observed in our simulations covers extended regions.
In Fig. \[fig:convection\] we show the extent of the convectively unstable regions according to the Solberg-H[ø]{}iland criteria (\[eq:shcriteria\]) after core bounce for models of the series s20A1B5, by plotting the time evolution of angle-averaged values for the convective growth time scale $ \tau_\mathrm{\,SH} $. From this figure it becomes apparent that two regions are susceptible to developing instabilities: the region just below the neutrino sphere (between about $ 20 \mathrm{\ km} $ and $ 40 \mathrm{\ km} $) and extended regions behind the shock. The innermost $ 2 \mathrm{\ km} $ of the star are also convectively unstable, but we suspect that the small negative entropy gradient responsible of this unstable region is a numerical artifact of the inner boundary, related to the so-called wall heating effect commonly appearing in shock reflection experiments [@donat_96_a]. In our simulations of models of the series s20A1B5, convective motions indeed occur in those unstable regions as predicted by the instability criteria, as well as in the surrounding regions due to overshooting. We also find that the time scale of the onset of the observed instability is correctly estimated by Eq. (\[eq:shtimescale\]).
Below the neutrino sphere ($ 20\mbox{\--\,}40 \mathrm{\ km} $), convection sets in inmediately after bounce, with typical maximum velocities of about $ 2 \times 10^4 \mathrm{\ km\ s}^{-1} $. The velocities progressively decrease until the end of the simulation (at about $ 65 \mathrm{\ ms} $ after bounce) with average values around $ 100 \mathrm{\ km\ s}^{-1} $, although convection does not disappear completely. Behind the shock ($ 100\mbox{\,--\,}200 \mathrm{\ km} $), the typical convective velocities are of the order of $ 1000 \mathrm{\ km\ s}^{-1} $, with maximum values in some regions of $ 10^4 \mathrm{\ km\ s}^{-1} $. This magnitude remains until the end of the simulation.
For a more detailed analysis we separately evaluate the different contributions in the Solberg–H[ø]{}iland criteria (\[eq:shcriteria\]) with $ \mathcal{B} $ in the form of Eq. (\[eq:buoyancy\_reformulation\]). Since the radial gradient of $ Y_e $ is positive (as deleptonization is stronger towards the center during the collapse), this has an stabilizing effect against convection. Similarly, rotation also suppresses convection, since the epicyclic frequency $ \kappa^2 $ is positive everywhere. Convective instability can thus only appear in regions with a sufficiently large negative radial entropy gradient. Such a gradient occurs in the region already swept by the shock front. Shock heating creates entropy most strongly close to the neutrino sphere at a radius of about $ 30 \mathrm{\ km} $ (see Fig. \[fig:convection\]), producing a steep gradient there $ 1 \mathrm{\ ms} $ after core bounce. Behind the shock front, which then propagates to larger radii at lower densities and decelerates, another region with a negative gradient also appears. All our microphysical models show very similar qualitative behavior with some variations due to different angular momentum distribution and the description of matter.
In models with slower rotation (i.e. the s20A1B1 series), strong convection sets in immediately after the occurence of the negative entropy gradient close to the neutrino sphere. For models with very little rotation (which have not been considered in this work), such convective overturn is strong enough to be clearly visible in the post-bounce gravitational wave signal [@dimmelmeier_07_a; @dimmelmeier_07_b]. Within about $ 20 \mathrm{\ ms} $ after core bounce, convection has managed to smooth out the entropy gradient around the neutrino sphere, thus removing the condition for sustained convection. Accordingly, convection is strongly damped, the vortices disappear quickly, and the low-frequency contribution to the gravitational wave signal is no longer visible. This fast convective transient near the neutrino sphere has been observed in numerical simulations without any neutrino treatment [see e.g. @Burrows1992; @mueller1997], and also in simulations using a neutrino diffusion scheme [@Swesty2005], although in the latter case the time scale for damping of convection is shorter ($ \sim 10 \mathrm{\ ms} $) than in our case. However, in simulations including state-of-the-art Boltzmann neutrino transport [@mueller_04_a], a few ms after core bounce no significant convection remains in this region, and no traces in the gravitational wave signal can be found. We attribute this disagreement with our results to the simplified neutrino treatment in our models, which cannot properly take into account the deleptonization of the PNS after core bounce. As the deleptonization of the PNS is initially strongest when the shock travels through the neutrino sphere, we expect the most significant inaccuracies of our formulation there. We therefore conclude that the convection over $ \sim 20 \mathrm{\ ms} $, which we observe in the neutrino sphere region, is an artifact that should disappear once a more realistic neutrino description is included.
In more rapidly rotating models, the stabilizing effect of rotation in the Solberg–H[ø]{}iland criteria prevents the strong transient we find in the slowly rotating models from developing, and significantly weaker convection is present in this region. However, irrespective of rotation, convection vortices are formed behind the decelerating shock front. On post-bounce evolution times of several $ 10 \mathrm{\ ms} $, the weak but persistent convection is unable to remove the entropy gradient behind the shock, except near the rotation axis, where the specific angular momentum is smaller, and convection is stronger.
Rotation also influences the shape of the convective cells. If the buoyancy terms in the Solberg–H[ø]{}iland criteria (\[eq:shcriteria\]) are much larger than the rotation terms, the convective cells show no preferred direction. We observe this feature particularly in models with slower rotation (the s20A1B1 series), and to a lesser degree also in other convectively unstable models in the first few ms after bounce. If the buoyancy terms are comparable in magnitude to the rotation terms, convection develops preferredly parallel to the rotation axis [see e.g. @miralles04]. This effect is present in our microphysical models at later phases, as the entropy gradient has already been partially smoothed out and the buoyancy terms have become smaller.
In contrast to the microphysical models, which show remarkable convection in the PNS and behind the shock front, models with a simplified matter treatment exhibit either no convection at all, or only close to the neutrino sphere (in the case of models of the A1B3G5 series). This is a consequence of using the hybrid EOS in the latter models, which is unable to properly decelerate the shock after core bounce and turn it into an accretion shock. Hence in these models the entropy gradient is mostly positive behind the shock.
Structure of the magnetic field {#subsec:morpho}
-------------------------------
\
\[0.5 em\]
The main qualitative differences between the various models become apparent when we study the detailed structure of the magnetic field of the resulting PNS. In Fig. \[fig:morphology\] we show two-dimensional snapshots of selected hydrodynamic and magnetic field variables at the final time of the simulations for two representative models of our sample, namely model A1B3G5-D3M0 (top panels) and model s20A1B1-D3M0 (bottom panels). For typical simulations with initial poloidal magnetic fields (D3M0 models) the resulting PNS has two clearly distinct parts (see left panels of Fig. \[fig:morphology\]): an inner region with a size of $ \sim 10 \mathrm{\ km} $, where nuclear density is exceeded and which is almost rigidly rotating, and a surrounding shell extending to the neutrino sphere at $ \sim 30 \mathrm{\ km} $, with subnuclear densities and which is strongly differentially rotating. These two parts are also visible in the distribution of the magnetic field (see center and right panels of Fig. \[fig:morphology\]). The inner region has a mixed toroidal and poloidal magnetic field configuration, with both components having similar strength, which results in a helicoidal structure aligned with the rotation axis. As this part of the PNS is almost rigidly rotating and practically in equilibrium, the magnetic field hardly evolves in time. On the other hand, the outer shell is differentially rotating; thus the toroidal magnetic field component dominates and grows linearly with time due to the $ \Omega $-dynamo mechanism.
If we compare the microphysical with the simplified simulations, we find that some significant morphological differences arise due to the stronger convection in the microphysical models just below the neutrino sphere. These motions affect the magnetic field, since they twist the poloidal magnetic field lines, generating a much more complicated structure of the poloidal field for those models. In particular those strong meridional currents distort the magnetic field in such a way that in some regions the poloidal component changes direction with respect to the rotation axis (see e.g. bottom-right panel of Fig. \[fig:morphology\]). This produces a negative effect in the $ \Omega $-dynamo as in these regions the magnetic field is wound up in the opposite direction. However, the overall $ \Omega $-dynamo mechanism seems not to be affected in a significant way by these local effects.
Model A4B5G5-D3M0 has to be discussed separately, since it has initially significantly stronger differential rotation and more angular momentum than the other models. As a result this model undergoes a core bounce due to centrifugal hang-up before reaching nuclear density. Its structure is toroidal with an off-center maximum density. Although it exhibits stronger differential rotation at the beginning compared to the other models, and the amplification process during collapse is thus more efficient, after bounce its angular velocity $ \Omega $ is smaller (as the PNS is less compact) and therefore the linear amplification due to $ \Omega $-dynamo is less pronounced. The main differences in the magnetic field structure of its PNS with respect to the other models are that, first, the $ \Omega $-dynamo is active not only in the high-density torus, but also in the central lower-density region, and, second, the strong meridional currents twist the magnetic field lines around the torus. However, we point out that in the investigated range of initial rotation configurations all microphysical models are significantly less influenced by rotation than the simplified models (like A4B5G5), and that even for rather extreme rotation such collapse dynamics, leading to a toroidal structure, is strongly suppressed if an advanced description of microphysics is used [which is in accordance with the comprehensive parameter study by @dimmelmeier_07_a].
In the models with initially purely toroidal field at the beginning (T3M0 series), a poloidal field cannot emerge in axisymmetry. Hence, the final magnetic field structure of the PNS consists of a stationary and entirely toroidal magnetic configuration with the highest field strengths found in the high density regions. As the rotational profile does not affect the distribution of the magnetic field, the different regions of the PNS are not visible in the structure of the magnetic field.
Comparison with Newtonian results
---------------------------------
In order to study the general relativistic effects in the evolution of the magnetic field, we choose a subset of our simulations with the hybrid EOS to represent the relativistic version of some of the models of @obergaulinger_06_b [@obergaulinger_06_a]. Their first paper is devoted to Newtonian simulations of magneto-rotational core collapse, while in their second paper an effective relativistic gravitational potential was used to mimic general relativistic effects (while still keeping a Newtonian framework for the hydrodynamics; TOV models in their notation). Since in contrast to their work we use the passive field approximation, the comparison can only be made with the low magnetic field models presented in that work, namely the “M10” models. In these models the magnetic field does not affect the collapse dynamics and our approximation is valid. Although there are no qualitative differences between Newtonian and general relativistic models [aside from those coming purely from the hydrodynamics as described in @dimmelmeier_02_a; @dimmelmeier_02_b], some dissimilarities can be found in the magnetic field strength and amplification rates after core bounce.
We have studied the evolution of the magnetic energy parameter $ \beta_\mathrm{mag} $ for the various models, and plot the results in Fig. \[fig:betamag\_comp\]. Note that for the same initial magnetic field, the magnetic field contribution to the magnetic energy parameter differs between a purely Newtonian treatment, a Newtonian formulation with the effective relativistic TOV potential, and general relativity. As a consequence, the initial value of $ \beta_\mathrm{mag} $ is not the same in these three cases. In order to be able to make an unambiguous comparison, we scale the magnetic fields such that $ \beta_\mathrm{mag} $ in the initial model is equal to the value in general relativity. In general, for a similar hydrodynamic behavior (models A1B3G3-D3M0 and A3B3G5-D3M0) the magnetic energy attained during the evolution is smaller in the general relativistic case than in the Newtonian case (with either regular or effective relativistic TOV potential).
The winding up of magnetic field lines is the main mechanism responsible for the increase of the magnetic field during the collapse. Therefore the amplification rate for $ \beta_\mathrm{mag} $ is determined by what rotation rate is reached and also by how strongly the poloidal component of the magnetic field is compressed. In the general relativistic case both higher densities and also stronger rotation are achieved [@dimmelmeier_02_b]. To investigate the impact of general relativistic gravity on the magnetic field compression, we consider $ \beta_\mathrm{polo} $ as this quantity is the seed for the $ \Omega $-dynamo. In general relativity the PNS has in general a smaller mass $ M_\mathrm{PNS} $ than in the corresponding Newtonian simulation of the same model. Following the relation established in Sect. \[sec:beta\_evol\] (see the bottom panel of Fig. \[fig:beta\_mcollapse\_comp2\]), the smaller PNS mass in the general relativistic simulation leads to a lower value of $ \beta_\mathrm{polo} $. Therefore in that case, despite the larger $ \Omega_\mathrm{c} $ the much smaller magnitude of $ \beta_\mathrm{polo} $ results in a longer time scale for the $ \Omega $-dynamo via Eq. (\[upper\_limit\]), and hence a smaller growth rate of the magnetic field.
In the multiple centrifugal bounce model A2B4G1-D3M0, general relativistic effects lead to a bounce at significantly higher maximum densities than in Newtonian gravity. Therefore, this is the only investigated model where $ M_\mathrm{PNS} $, and consequently $ \beta_\mathrm{polo} $ as well as $ \beta_\mathrm{mag} $ are larger in the general relativistic simulation.
Gravitational waves {#gravitational_waves}
-------------------
We calculate the gravitational wave output from all of our simulations using the Newtonian quadrupole formula given in Eq. (\[eq:mag\_stress\_formula\]), which includes the magnetic terms. Thus, the quadrupole wave amplitude $ A^\mathrm{E2}_{20} $, which is related to the dimensionless quadrupolar strain amplitude $ h^\mathrm{quad} $ as $$h^\mathrm{quad} = \frac{1}{8} \sqrt{\frac{15}{\pi}}
\sin^2 \theta \frac{A^\mathrm{E2}_{20}}{R},$$ contains the contribution $ A^\mathrm{E2}_\mathrm{20\,mag} $ corresponding to the magnetic field. Here $ h^\mathrm{quad} $ is the only independent component of the radiative part $ h^\mathrm{quad}_{ij} $ of the spatial metric as given by Eq. (\[eq:quad\_formula\]). In order to understand how the magnetic field affects the waveforms, we also separately compute $ A^\mathrm{E2}_\mathrm{20\,mag} $. The resulting waveforms for some representative models are shown in Fig. \[fig:mcollapse\_gw\_1\]. As the magnetic field is very low at all times, $ b^2 \ll \rho $, the component of the gravitational wave due to the magnetic field is several orders of magnitude smaller than $ A^\mathrm{E2}_{20} $.
Therefore, during the core bounce and the immediate post-bounce phase, the waveforms we obtain are practically identical to the ones presented for the same model setup in @dimmelmeier_02_a (for the simplified models) and @dimmelmeier_07_a (for the microphysical models), which can also be downloaded from a freely accessible waveform catalog at `www.mpa-garching.mpg.de/rel_hydro/``wave_catalog.shtml`. The values for $ A^\mathrm{E2}_{20} $ lie in the range between about $ 30 \mathrm{\ cm} $ and $ 3000 \mathrm{\ cm} $, which translates to a $ h^\mathrm{quad} $ of roughly $ 3 \times 10^{-22} $ to $ 3 \times 10^{-20} $ (assuming a distance $ R = 10 \mathrm{\ kpc} $ to the source and optimal orientation between the source and the detector).
We also emphasize that all investigated microphysical models yield gravitational wave signals known as Type I in the literature, i.e.the waveform exhibits a positive pre-bounce rise and then a large negative peak, followed by a ring-down. This is to be expected, as recent studies using the same hydrodynamical model setup [@ott06a; @dimmelmeier_07_a] have shown that the inclusion of microphysics in stellar core collapse simulations suppresses the other signal types, which were associated to multiple centrifugal bounce (Type II signals) or rapid collapse with a very small mass of the inner core (Type III signals).
\
\[0.5 em\]
After bounce, the star reaches a quasi-equilibrium state, and thus, the hydrodynamic component of the waveform decreases. At the same time, for models D3M0, the magnetic field grows linearly with time. Such a behavior in the magnetic field produces an increasing gravitational wave signal, which grows quadratically with time due to the dependence on the magnetic field in Eq. (\[eq:mag\_stress\_formula\]). However, at the end of the simulation, the magnetic field component of the waveform is still negligible in comparison with the hydrodynamic component. It is expected that at later times, as the amplification of the magnetic field reaches saturation, the influence of the magnetic field on the waveform becomes significant, both due to its effect on the dynamics and also due to the contribution of the magnetic field to the gravitational radiation itself. We note, however, that the effect of the MRI could additionally lead to noticeable changes in the waveforms, provided it were able to efficiently amplify the magnetic field (see discussion in Sect. \[subsec:mri\]).
For models T3M0, on the other hand, the component of the waveform due to the magnetic field is even smaller than for the D3M0 models. This is a consequence of the inefficient amplification of the magnetic field via the radial compression. After bounce, the magnetic component of the waveforms in models T3M0 does not grow, and hence it is not expected to dominate the waveform later in the evolution, unless other processes amplifying the magnetic field were present.
Amplification of the magnetic field {#sub:ampli}
-----------------------------------
Different mechanisms that amplify the magnetic field can act during a core collapse or the subsequent evolution of the newly formed PNS. This issue is of great importance, since the evolution of the PNS during its first minute of life until a cold NS forms can change drastically depending on the initial conditions at formation. One of the most important aspects is the distribution of angular momentum. A highly differentially rotating PNS can be subject to various types of instabilities, such as the dynamical low-$ \beta $ instability, the classical bar-mode instability (which is unlikely to occur in a PNS on dynamical time scales as it requires very high values of $ \beta $), or the secular CFS instability. Such instabilities are potential sources of detectable gravitational radiation. Therefore, a natural question that arises is whether the magnetic field is going to grow sufficiently fast to act on the PNS dynamics by flattening the rotation profiles (and therefore preventing the instabilities to develop), or whether, instead, the growth process may take a few seconds, allowing the instabilities to grow and the accompanying gravitational waves to become detectable. A number of effects can amplify the magnetic field shortly after PNS formation. In the following, we discuss these effects and estimate their importance for our models of core collapse[^4].
### $ \Omega $-dynamo
Within our passive field approximation we can only compute the amplification rates for the $ \Omega $-dynamo, for which the magnetic field grows linearly with time; therefore $ \beta_\mathrm{mag} $ grows quadratically with time (see Appendix \[app:odynamo\]). The time scale $ \tau_\Omega $ of this amplification process and the estimated time $ t_\mathrm{sat} $ at which the field saturation begins are given in Table \[tab:MCC\_models\]. In the fastest case of our model sample, which occurs for model A1B3G3-D3M0, saturation is reached at about $ 300 \mathrm{\ ms} $, and in most other cases, the $ \Omega $-dynamo saturates at times larger than $ 0.5 \mathrm{\ s} $. Note, however, that these estimates depend on the initial magnetic field strength, which is chosen to be $ B^*_0 = 10^{10} \mathrm{\ G} $. For lower values of the magnetic field these time scales can be scaled as (see Eq. \[eq:tau\_omega\]) $$\tau_{\Omega} \approx
\tau_{\Omega \, 10}
\left( \frac{10^{10} \mathrm{\ G}}{B^*_0} \right),
\qquad
t_\mathrm{sat} \approx t_\mathrm{sat \, 10}
\left( \frac{10^{10} \mathrm{\ G}}{B^*_0} \right).$$ We recall that stellar evolution calculations predict that in a progenitor core the poloidal component of the magnetic field can initially have a strength of about $ 10^6 \mathrm{\ G} $ [@heger05]. For such an initial magnetic field the saturation time scale becomes several hours. This makes the $ \Omega $-dynamo a very inefficient mechanism to amplify the magnetic field during core collapse and bounce, unless the progenitors are highly magnetized ($ B > 10^{10} \mathrm{\ G} $) for which the saturation could be reached within a few dynamical time scales. The magnetic field at the saturation is independent of the initial magnetic field strength and of the order of $ \sim 10^{16} \mathrm{\ G} $.
### Magneto-rotational instability {#subsec:mri}
There are other magnetic field amplification processes that our simulations cannot account for, but for which it is nevertheless possible to estimate the growth rates. It has been suggested that the magneto-rotational instability could amplify the magnetic field from arbitrary weak fields up to values where equipartition between the magnetic field energy and the rotational kinetic energy is reached [@akiyama03]. However, our analysis shows that in the context of core collapse such an amplification is still an open issue. We proceed next to describe the MRI and the uncertainties related to its effect on the amplification of the magnetic field in core collapse.
#### Linear regime:
The MRI is a shear instability that generates turbulence and results in an amplification of the magnetic field in a differentially rotating magnetized plasma [@balbus91; @balbus92], redistributing angular momentum in the plasma. Linear analysis shows that if the magnetic field strength is very low, as in our case, the stability criteria for the MRI in the Newtonian limit [@balbus95] are $$\begin{array}{rcl}
\mathcal{C}_\mathrm{MRI1} & = & {\mbox{\boldmath $g$}} \cdot {\bf \mathcal{B}} +
\mathcal{R} \cdot \nabla\varpi > 0,
\\ [0.2 em]
\mathcal{C}_\mathrm{MRI2} & = & ({\mbox{\boldmath $g$}} \times \nabla \varpi)
(\mathcal{B} \times \mathcal{R}) > 0,
\end{array}
\label{eq:mricriteria}$$ where $ \mathcal{R}= \varpi \nabla (\Omega^2) $. Note that these criteria are very similar to the Solberg–H[ø]{}iland criteria (\[eq:shcriteria\]) for convection, but use an angular velocity gradient $ \mathcal{R} $ instead of an angular momentum gradient $ \mathcal{J} $. Since in the core collapse scenario $ \mathcal{R} \le 0 $ is satisfied almost everywhere, it is important to compute the buoyancy terms given by $ \mathcal{B} $ to estimate the onset of the MRI. For regions with $ \mathcal{B} > 0 $ (i.e with a negative entropy gradient that is strong enough to compensate the positive electron fraction gradient term in Eq. (\[eq:buoyancy\_reformulation\])), the first criterion is not fulfilled. Furthermore, for regions with $ \mathcal{B} < 0 $ (i.e. a positive or sufficiently small negative entropy gradient), the second criterion is neither satisfied. This means that the presence of a adequately strong negative entropy gradient (which also leads to convective instability) enhances the MRI, although a positive entropy gradient does not affect the condition for MRI instability. Note that this peculiarity is caused by the negative value of $ \mathcal{R} $, and does not happen in the Solberg–H[ø]{}iland criteria (\[eq:shcriteria\]) for convection, as $ \mathcal{J} > 0 $ in that case. If at least one of the criteria (\[eq:mricriteria\]) is not satisfied *and* a magnetic field is present, then fluid and magnetic field perturbations grow exponentially in time. Neglecting buoyancy terms, the time scale for the fastest growing unstable mode can be roughly estimated as[^5] $$\tau_\mathrm{MRI} = 4 \pi \,
\left| \frac{\partial\Omega}{\partial\ln{\varpi}} \right|^{-1},$$ which is independent of the magnetic field configuration and strength. Only those modes with a length scale larger than a critical wavelength will grow [@balbus91]. This length scale can roughly be estimated as $ \lambda_\mathrm{MRI} \sim 2 \pi c_\mathrm{A} / \Omega $, where $ c_\mathrm{A} = \sqrt{B^2 / \rho} $ is the Alfvén speed. For the typical values attained in the nascent PNS, in which the dominant magnetic field is toroidal, the critical length scale is $$\lambda_\mathrm{MRI} \approx
62 \left( \frac{B^{*\,0}}{10^{10} \mathrm{\ G}} \right)
\left( \frac{1 \mathrm{\ ms}^{-1}}{\Omega} \right)
\left( \frac{10^{14} \mathrm{\ g\ cm}^{-3}}{\rho} \right)^{1/2}
\mathrm{\ m}.$$ Note that we have scaled the length scale with the typical magnetic field strength $ B^{*\,0} $ of the progenitor, and not with that of the PNS itself. For the poloidal component and realistic values of the initial magnetic field ($ B^{*\,0} \sim 10^6 \mathrm{\ G} $) this length scale is reduced by several orders of magnitude ($ \lambda_\mathrm{MRI\,polo} \sim 0.6 \mathrm{\ cm} $). In any case, resolving the scales needed to simulate the MRI is a challenging problem as, in the case of weak magnetic fields, the wavelength of the fastest growing mode (which is close to the critical length scale) is typically much smaller than the available grid resolution.
#### Non-linear regime:
Linear analysis provides tools to determine the onset of the instability and the typical time and length scales. However, once the perturbations of the magnetic field reach values comparable to the magnetic field itself, linear analysis is no longer valid (although in the weak field case the perturbations of the fluid variables are still small). The amplification of the magnetic field due to the MRI is therefore a nonlinear effect, and can only be studied by means of numerical simulations. The appropriate numerical approach, due to the smallness of the length scales necessary to be resolved, are local simulations of the MRI in a shearing box. Numerical simulations of this kind in three dimensions have been performed by @Hawley1995 in the context of accretion discs. They show that if the instability condition of linear analysis is fulfilled, then the amplification of the magnetic field proceeds by the formation of an axisymmetric channel flow. This is well understood, since the linear MRI solution is also a solution of the nonlinear axisymmetric MHD equations [@Goodman1994]. In the ideal MHD limit, the amplification saturates when nonaxisymmetric perturbations destroy the channel flow. It is important to emphasize the necessity of performing three-dimensional simulations in the shearing box since, in axisymmetry, the channel flow is not destroyed and any magnetic field is able to grow continuously, reaching saturation only when the MRI length scale is of the order of the region in which the MRI is present [@Hawley1992].
For a magnetic field distribution with zero mean at large scales, the amplification proceeds from arbitrarily weak fields [@Hawley1996] and saturates irrespective of the initial magnetic field at average values of $ P_\mathrm{mag} / P_0 \sim 0.01 $, where $ P_0 $ is the initial gas pressure. If a mean magnetic field is present (as in our case), the saturated magnetic field depends on the initial magnetic field strength. In the most favorable case of a vertical magnetic field, the total amplification by the MRI is only about a factor 20 of the original field, this amplification being even smaller in the case of a purely toroidal field [@Hawley1995]. On the other hand, @Sano2004 have suggested that for sufficiently weak magnetic fields the saturation level could be independent of the initial field and equal to that in the zero-mean case. If this were confirmed it would mean that, for the weak magnetic field strength present in our magneto-rotational core collapse models ($ P_\mathrm{mag} / P_0 \sim 10^{-8} $ in the PNS for progenitors with $ B^{*\,0} = 10^{10} \mathrm{\ G} $), a magnetic field of $ \sim 10^{16} \mathrm{\ G} $ could be reached on time scales of $ \tau_\mathrm{MRI} $. Such a strong magnetic field would have a significant effect on the dynamics, similar to that observed in numerical simulations with highly magnetized progenitors [@obergaulinger_06_a; @obergaulinger_06_b; @shibata_06_a]. In the opposite case, the MRI would fail to considerably amplify the magnetic field, and for a purely toroidal field the magnetic field should grow only by a factor of about 3 according to @Hawley1995.
The inclusion of more complex physics relevant for the core collapse scenario (like radiation, diffusion, or resistivity) can significantly change the amplification process, since in the nonideal case the reconnection of magnetic field lines seems to be the dominant effect in the saturation process of the MRI. In general, these effects act towards lowering the values of the saturation; for reasons of simplicity we do not consider them in this discussion [see @Hawley2005 and references therein for a detailed review on this topic].
Furthermore, it has to be noted that all local simulations of the MRI have been performed in the context of Keplerian accretion discs, and, hence, some of the underlying physical conditions are not valid in the case of a PNS. For example, the typical sound speed $ c_\mathrm{s} $ in those simulations is of the order of $ 10^{-3} $. Only the parametric study performed by @Sano2004 covers a wider range of values of $ c_\mathrm{s} \sim 10^{-8} \mbox{\,--\,} 10^{-2} $ in units of $ c $. However, the sound speed in a PNS is higher, $ c_\mathrm{s} \sim 10^{-1} $. Rotational velocities and profiles are also very different in a disc and a PNS. Therefore, appropriate local simulations of the PNS scenario should eventually be performed in order to confirm the growth of the MRI for a weakly magnetized PNS, and to infer the magnetic field at which the instability saturates.
#### Our results:
As the MRI involves a backreaction of the magnetic field onto the dynamics, we cannot study this effect in our simulations, as we assume the passive field approximation. Furthermore, even with “active” magnetic fields, both the resolution needed to resolve the MRI length scale ($ \sim 10 \mathrm{\ m} $) and the requirement for three-dimensional simulations are not affordable with present computers. Therefore, we are limited to analyzing whether our magnetized collapse models are susceptible to developing such an instability according to linear analysis estimates, leaving aside the issue of saturation, whose uncertainties need a deeper analysis which is beyond the scope of this work. In order to estimate how MRI could change our results if it were taken into account properly, we determine the regions where the MRI instability criteria (\[eq:mricriteria\]) are not satisfied. Inside these regions we calculate the time scale for the fastest growing mode. In Fig. \[fig:MRI\] we show the results for the models A1B3G5-D3M0 (left panel) and s20A1B5-D3M0 (right panel). We note that since the onset of the MRI is independent of the magnetic field strength, provided that a poloidal component exists, any composition of D3M0 and T3M0 models has the same instability properties as the D3M0 models.
Our analysis of all computed models shows that during the infall phase the MRI is either not possible or the typical time scales involved are much larger (i.e. $ > 10 \mathrm{\ s} $) than the duration of the collapse itself. Therefore the instability can affect neither the dynamics nor the magnetic field strength in that phase. Around the time of core bounce, the angular velocity gradient is larger and the MRI time scale becomes dynamical. Almost the entire region between the shock formation radius (at $ \sim 10 \mathrm{\ km} $) and the shock itself is MRI unstable with time scales of the order of $ \sim 1\mbox{\,--\,}10 \mathrm{\ ms} $. Note that the innermost part of the PNS rotates rigidly, and therefore the MRI unstable region that appears in the inner $ 2 \mathrm{\ km} $ is possibly a numerical artifact caused by the probably unphysical negative entropy gradient mentioned in Sect. \[subsec:convection\]. Some differences appear when comparing microphysical and simplified models.
A general feature of the microphysical models is the post-bounce appearance of a negative entropy gradient (regions with $ N^2 < 0 $, see Sect. \[subsec:convection\]). This property is much less prominent in the simplified models (except in model A1B3G5). Thus, the presence of a such gradient in the microphysical models enhances the occurence of the MRI behind the shock as compared to the simplified models (see Fig. \[fig:MRI\]), since in these regions the cause for the instability is mainly the presence of a negative entropy gradient. Around the neutrino sphere the presence or absence of a negative entropy gradient does not affect the onset of the instability since it is caused by the strong negative angular velocity gradient. Therefore, only small differences can be found in the latter region between the simplified and the microphysical models.
As a result of this analysis, for collapse progenitors with a magnetic field smaller than $ 10^{10} \mathrm{\ G} $ (hence including astrophysically more relevant initial values of $ 10^6 \mathrm{\ G} $), we infer that perturbations of the magnetic field are going to grow exponentially on dynamical time scales and will reach saturation in the unstable regions mentioned above. However, the value of the magnetic field at which saturation appears is still unknown, which is a key issue in order to establish the effects of the MRI, if any, on the dynamics. Nevertheless, even if the MRI were unable to considerably amplify the magnetic field, it could still play a major role at late times during the evolution of the PNS, provided other amplification mechanisms were capable to increase the magnetic field to significant larger values (see below). In such a situation the MRI could have an impact on the dynamics by transporting angular momentum outwards and driving the PNS towards rigid rotation.
### Dynamo mechanisms
The wind-up process of the magnetic field ($ \Omega $-dynamo) discussed before is a mechanism that works by transforming the poloidal magnetic field into a toroidal field and extracting energy from differential rotation. In axisymmetry this process amplifies the magnetic field linearly with time as long as differential rotation exists. If the axisymmetry condition is relaxed, however, a number of instabilities of the toroidal field can transform the toroidal magnetic field back into a poloidal magnetic field. This feedback then “closes” the dynamo process.
The first group of instabilities are those related to convective unstable regions, neutron-finger instabilities (due to a negative composition gradient) and, in general, turbulence. In these cases the $ \alpha $-effect is the one which closes the dynamo in the $ \alpha\mbox{-}\Omega $-dynamo [@Thompson1993]. Computations of this effect [@bonanno05] suggest that even for a rapidly rotating PNS with a period around $ 1 \mathrm{\ ms} $ (i.e. comparable to the models presented here), the time scale for the growth of the magnetic field is $ \sim 1 \mathrm{\ s} $. Therefore, this effect is probably not important after core bounce on dynamical time scales. However, for larger time scales (i.e. several seconds), and if the MRI is not efficient enough, this mechanism will most likely amplify the magnetic field, leading to magnetic braking of the PNS within a few seconds.
There are also types of instabilities that can act in stably stratified regions, i.e. regions which are convectively stable. @spruit99 has proposed the Tayler instability [@tayler73] as a mechanism to close the dynamo. This dynamo has been confirmed in numerical simulations by @braithwaite06b [@braithwaite06a]. The condition for this kink-type instability to grow in the rotating case ($ m = 1 $ mode) is [@spruit99] $$\partial_\theta \ln B_\varphi^2 \, \sin \theta \, \cos \theta > 0,$$ which is satisfied almost everywhere inside the star in our simulations. The growth rate of the instability is of the order of the Alfvén time scale, $$\tau_\mathrm{T} = \frac{2 \pi}{\Omega_\mathrm{A}}
\quad
(\Omega \ll \Omega_\mathrm{A}),
\quad \quad
\tau_\mathrm{T} = \frac{2 \pi \Omega}{\Omega_\mathrm{A}^2}
\quad
(\Omega \gg \Omega_\mathrm{A}),
\label{eq:tayler_growth_time}$$ where $ \Omega_\mathrm{A} = c_\mathrm{A} / R $ and $ R $ is the typical size of the region considered. In case this instability appears, it destroys the toroidal magnetic field by transforming it into a poloidal field which feeds back the amplification of the toroidal magnetic field via the $ \Omega $-dynamo. Therefore, the dynamo is only effective if the $ \Omega $-dynamo is able to generate a toroidal magnetic field faster than the Tayler instability destroys that field, i.e. $ \tau_\mathrm{T} \gg \tau_\Omega $. Saturation is then reached as $ \tau_\mathrm{T} \approx \tau_\Omega $. Note that depending on the system, the saturated magnetic field can be weak enough not to affect the dynamics.
If we consider the typical toroidal magnetic field at bounce to be $ 10^{13} \mathrm{\ G} $ (as in the T3M0 models) with a typical density in the PNS of $ \rho \sim 2 \times 10^{14} \mathrm{\ g\ cm}^{-3} $ and a typical size of the inner region of $ R \sim 10 \mathrm{\ km} $, then the time scale for the growth of the Tayler instability is strongly increased by rotation, $$\tau_\mathrm{T} \approx
3 \left( \frac{10^{10} \mathrm{\ G}}{B^{*\,0}} \right)^2
\left( \frac{R}{10 \mathrm{\ km}} \right)^2
\left( \frac{\Omega_\mathrm{c}}{1 \mathrm{\ ms}^{-1}} \right)
\mathrm{\ hr},$$ which we obtain from the $ \Omega \gg \Omega_\mathrm{A} $ limit of Eq. (\[eq:tayler\_growth\_time\]).
This means that for a typical progenitor with a toroidal magnetic field of $ B^{*\,0}_\varphi \sim 10^{10} \mathrm{\ G} $, the instability proposed by [@spruit99] is going to be very inefficient in amplifying the magnetic field. However, on a longer time scale, when other mechanisms could amplify the magnetic field (e.g. the $ \alpha\mbox{-}\Omega $-dynamo), the Tayler instability could also become important.
Conclusions {#sec:conclusions}
===========
In this paper we have presented numerical simulations of the collapse of rotating magnetized stellar cores in the CFC approximation of general relativity, as well as tests assessing our numerical approach for solving the ideal general relativistic magneto-hydrodynamics (GRMHD) equations.
As initial models we have set up (either fully or nearly) stationary configurations of weakly magnetized stars in general relativity, with either toroidal or poloidal (or both) magnetic field components. We have used the “test” passive field approximation for evolving these initial models, for which the magnetic pressure in all cases considered is several orders of magnitude smaller than the fluid pressure.
We have performed tests to check the accuracy and convergence properties for the GRMHD extension of our code. For magnetic field quantities we have found an order of convergence above 1 in all of the performed tests. These results are consistent with the second-order accuracy (in space and time) of our numerical scheme, reduced to first order only at shocks and local extrema. The errors in all of the cases in which the theoretical solution is known are below 0.1%, except at shocks, which are correctly captured within only few numerical cells thanks to the use of high-resolution shock-capturing schemes.
For the simulations of magnetized core collapse, we have considered cases with magnetic fields which are initially either purely poloidal (series D3M0), purely toroidal (series T3M0), or a combination of both. The D3M0 models are a general relativistic extension of a subset of the cases evolved in fully coupled MHD by @obergaulinger_06_b [@obergaulinger_06_a], who used a Newtonian formulation (approximating general relativistic effects to some extent in the latter work). One of our aims has been to compare the dynamics and gravitational waveforms with their results. No qualitative differences have been found in the models studied, while quantitatively the strength of the magnetic field at bounce and after the collapse are consistently smaller in general relativity.
We have also compared simulations of models with improved microphysics (employing a tabulated non-zero temperature equation of state (EOS) and an approximate but effective deleptonization scheme) with the simple (though still widely used) analytic hybrid EOS. The results show that the microphysical models (i) lead to a more complex structure of the poloidal magnetic field due to convective motions surrounding the inner region of the PNS, and (ii) exhibit a wind-up of the magnetic field ($ \Omega $-dynamo) that is more efficient than in the simplified models for comparable rotation rates, which is due to the larger compression of the poloidal component during the collapse.
We have found a unified explanation for the magnetic energy of all models, independent of the description of gravity (general relativity or Newtonian) or the EOS, which relates the angular velocity and mass of the PNS with its magnetic energy and the growth rate of the magnetic field due to the $ \Omega $-dynamo. This relation shows that higher rotation rates and masses of the PNS lead to stronger magnetic fields. We have shown that it is not possible to mimic the conditions of the microphysical simulations using a simplified EOS. Simplified models with a mass of the homologously collapsing inner core during contraction and a mass of the PNS after bounce similar to the respective masses of the microphysical models (and identical initial rotation profiles) will undergo multiple centrifugal bounces, a behavior that has recently shown to be an artifact of the neglect of microphysics [@dimmelmeier_07_a].
Further differences appear in the appearance of convective motion in the PNS and behind the shock. This convection is more active in microphysical models than in simplified ones. In models with slow rotation, strong convection in the PNS occurs as a transient and disappears within a few ten ms after bounce. Evidently, this transient is an artifact as it does not appear in simulations of similar models with comparable microphysics but using Boltzmann neutrino transport [@mueller_04_a] instead of our simple advection scheme for the electron fraction after core bounce. In rapidly rotating models convection is not entirely suppressed by rotation but develops and persists on longer time scales, albeit at a lower intensity.
As we have adopted the passive field approximation and the investigated magnetic fields are weak in all phases of the collapse, the waveforms of the gravitational radiation emitted by all our models are practically identical to the corresponding ones in a purely hydrodynamical simulation [@ott06a; @dimmelmeier_07_a], with the contribution due to magnetic fields being several orders of magnitude smaller than the total signal amplitude. However, if the MRI could become dominant for the dynamics in the post-bounce phase, in a fully coupled GRMHD simulation we would expect a clear imprint of such an instability on the signal waveform. As expected, for the microphysical models we obtain gravitational wave signals exclusively of Type I, as all other waveform types (in particular the Type II signals associated with multiple centrifugal bounces) are suppressed if more realistic microphysics is taken into account.
For an astrophysically expected strength of the magnetic field [@heger05], where the initial toroidal component is much larger than the poloidal one, we have obtained a topology of the magnetic field in the PNS that is purely toroidal due to the radial compression of the initial toroidal component. In this case the time scale for the $ \Omega $-dynamo is very long (several hours). For progenitors with stronger poloidal magnetic fields, we have found that a core/shell structure is formed. Inside the core, where nuclear density is exceeded, a mixed configuration of a poloidal and a toroidal magnetic field yields a helicoidal configuration of the field lines. In the surrounding shell (which extends several $ 10 \mathrm{\ km} $) the poloidal magnetic field lines are wound up due to differential rotation ($ \Omega $-dynamo), and shortly after core bounce the magnetic field is dominated by the toroidal component. The growth time scale for the toroidal component due to this process is, in the best case scenario, several $ 100 \mathrm{\ ms} $.
We have also estimated the growth times for several other instabilities that could appear if the passive field approximation or the restriction to axisymmetry are removed. Among these the MRI is apparently the fastest growing instability, although it remains unclear if it is going to amplify the magnetic field sufficiently (from the initially weak field values) to affect the dynamics at all. In addition, we have found that the inclusion of microphysics could enhance the MRI, since the regions behind the shock exhibit a negative entropy gradient, resulting in a growth time of $ \sim 10 \mathrm{\ ms} $ for the MRI. However, the influence of our simplified neutrino treatment or the effects of an alternative microphysical EOS must still be investigated in detail.
In the event that the MRI were unable to sufficiently amplify the magnetic field in the PNS (which is still an open issue), the main amplification mechanism would probably be the $ \alpha\mbox{-}\Omega $-dynamo, which can amplify the magnetic field to values where the magnetic energy is in equipartition with the rotational kinetic energy on a time scale of, at least, several seconds. The study of this effect is well beyond the goals of the work presented in this paper, since the required time scales are much longer than those affordable with current numerical magneto-hydrodynamical (MHD) codes. Moreover, the underlying physics necessary to be included (like neutrino transport, diffusion, radiation, and cooling) is far more complex. However, in the light of the results presented here, in which astrophysically expected values for the magnetic field have been adopted, we can speculate about the following scenario. If the MRI is ineffective, after core bounce the magnetic field does not grow significantly strong during one (or maybe several) seconds, and therefore differential rotation generated in the collapse could persist. This “one-second-window” would provide sufficient time for several instabilities to develop in the PNS. Such instabilities are promising sources of gravitational waves.
The restriction to the passive magnetic field approximation in studying magneto-rotational core collapse of weakly magnetized progenitors can be justified if the MRI is indeed inefficient, since none of the other estimated mechanisms seem to be able to amplify the magnetic field significantly on dynamical time scales. Otherwise, an “active” magnetic field approach becomes necessary. However, it has to be stressed that the use of active magnetic fields alone for core collapse simulations will probably not be sufficient to model all the effects amplifying the magnetic field, since the numerical resolution needed to correctly describe them (probably less than $ 10 \mathrm{\ m} $) is not affordable in current numerical simulations. In addition most of the prospectively relevant effects have to be investigated in three dimensions, which makes the computational task even more challenging.
This research has been supported by the Spanish Ministerio de Educación y Ciencia (grant AYA2004-08067-C03-01), by the DFG (SFB/Transregio 7 and SFB 375), by the DAAD and IKY (IKYDA German–Greek research travel grant), and by a Marie Curie Intra-European Fellowship within the 6th European Community Framework Programme (IEF 040464). It is a pleasure to thank C. D. Ott, A. Marek, and H.-T. Janka for their contributions related to the improved microphysics, L. Antón for many useful discussions in setting up the numerical scheme for the magnetic field evolution, M. Obergaulinger for his help on understanding the MRI and the Newtonian simulations data, and E. Müller as well as N. Stergioulas for useful comments.
Code tests {#app:tests}
==========
Here we discuss several tests we have designed in order to check the accuracy of our numerical code when solving the induction equation with the numerical methods described in this paper [see also @nfnr]. The “toroidal test” is set up for assessing the ability of the code to maintain various magnetic field configurations in equilibrium (labelled TTA and TTB) and to correctly compute the amplification of the toroidal magnetic field as it is wound up by a rotating fluid (TTC). On the other hand, the “poloidal test” (PT) is designed to check whether the code can correctly compute the compression of the poloidal magnetic field in a spherical collapse. Finally, the strong spherical explosion test checks that the code is able to handle the presence of radial shocks. We refer the interested reader to @nfnr for details on the setup of the toroidal and poloidal tests as well as the diagnostics we use to compute the errors and order of convergence of our numerical schemes.
Toroidal tests
--------------
Fig. \[fig:TT\] shows the global error $ \sigma $ in the toroidal magnetic field $ B^\varphi $ for the three tests TTA, TTB, and TTC against $ 1 / f $ and the corresponding fits to a power law. Here $ f $ denotes the factor which specifies the increase in resolution from a coarse reference grid [see @nfnr for details]. The resulting convergence order of each numerical scheme (minmod, MC, and PHM) as well as the errors for the highest resolution grid can be found in Table \[tab:TT\]. Our results show that (i) the order of convergence and the error is almost independent of the cell reconstruction scheme employed, (ii) the order of convergence for the TTC test is smaller than for the TTA and TTB tests, and (iii) the order of convergence for the tests TTA and TTB is $ N > 2 $, and hence higher than the theoretical expectation (which is second order, since it is limited by the order of the time discretization, for which we use a conservative, second order Runge–Kutta scheme).
Test Reconstruction scheme $ N $ $\sigma_{320 \times 40} $
------ ----------------------- ------- --------------------------- -- --
TTA minmod 2.45 $ 1.2 \times 10^{-6} $
TTA MC 2.16 $ 2.4 \times 10^{-6} $
TTA PHM 2.46 $ 2.1 \times 10^{-6} $
TTB minmod 2.64 $ 7.7 \times 10^{-6} $
TTB MC 2.53 $ 1.2 \times 10^{-5} $
TTC PHM 2.85 $ 0.5 \times 10^{-6} $
TTC minmod 1.38 $ 4.0 \times 10^{-5} $
TTC MC 1.48 $ 7.0 \times 10^{-5} $
TTC PHM 1.39 $ 3.5 \times 10^{-5} $
PT minmod 1.41 $ 8.3 \times 10^{-4} $
PT MC 1.11 $ 8.6 \times 10^{-4} $
PT PHM 1.17 $ 8.6 \times 10^{-4} $
: Convergence order $ N $ for the tests performed (TTA, TTB, TTC, and PT) and for different reconstruction procedures (minmod, MC, and PHM). The error $ \sigma_{320 \times 40} $ for the higher resolution grid is also given.
\[tab:TT\]
The numbers reported in Table \[tab:TT\] demonstate that we obtain similar results in all three tests for linear reconstruction schemes (minmod and MC) and for the third order reconstruction scheme (PHM), as the order of the scheme is limited by the second order discretization in time and by the linear interpolation of the cell-centered magnetic fluxes (which is a consequence of using a staggered grid in the flux-CT scheme for the magnetic field). To understand these results we note that in test TTC there is a component of the magnetic field, $ B^{*\,\varphi} $, which grows linearly in time, while in tests TTA and TTB no components evolve. Hence, the order of convergence for the latter is higher than for test TTC. This can be explained by investigating the *local* order of convergence, i.e. the order obtained when computing the error $ \sigma_{ij} $ in each numerical cell instead of the global error $ \sigma $. The results for test TTA are displayed in Fig. \[fig:TTA\_nij\] (similar plots can be obtained for the other two cases). At some particular grid zones the order of convergence is larger than two, while at most locations it remains around two.
Poloidal test
-------------
As mentioned before the setup and specifications of the poloidal test are described in detail in @nfnr. Here we simply focus on showing the comparison and performance of the various numerical schemes employed in our simulations. [Note that in @nfnr only the minmod scheme was assessed.] Fig. \[fig:PTA\] shows the evolution of the error in the quantity $ r\, D^* / B^{*\,\theta} $ at the equatorial plane (which is a quantity that should not change with time with respect to a Lagrangian coordinate system) during the spherical collapse of a 4/3-polytrope for different $ \{r, \theta\} $ grid resolutions ($ 80 \times 10 $, $ 160 \times 20 $, and $ 320 \times 40 $), equally-spaced in the angular direction and logarithmically spaced in the radial direction. Table \[tab:TT\] gives again numbers for the error and the order of convergence of the various schemes computed at the end of the simulation ($ t = 20 \mathrm{\ ms} $). In all cases the errors are below 1%, even for the coarsest grid, and the order of convergence is higher than 1 (the presence of local extrema in the radial profiles of some hydrodynamical variables explains the reduction of the theoretical order as a built-in feature of total-variation diminishing numerical schemes). Comparisons between the HLL approximate Riemann solver and the KT symmetric scheme yield almost identical results [in agreement with @lucas04; @shibata_font05; @anton06].
Strong spherical explosion
--------------------------
\
\[0.5 em\]
Explosions are among the most demanding tests for multi-dimensional codes as they show the ability of numerical schemes to handle shocks. Since the majority of existing MHD codes are written in Cartesian coordinates, the most common test is the cylindrical explosion. For relativistic MHD codes the setup of @komissarov99 for this test has been used by other authors [@delzanna03; @leismann05] to compare different codes. However, in spherical coordinates it is not possible to impose the symmetries needed for this test. The most natural choice is thus the spherical explosion. @koessl90 performed this test in the case of Newtonian MHD. To our knowledge, no spherical explosions tests have been performed in relativistic MHD. Therefore, we have designed such a spherical explosion test in which the initial jump conditions in the variables are the same as for the test by @komissarov99. In this way a relativistic shock is formed which does not occur in the Newtonian case of @koessl90.
Our test setup consists of an initial explosion zone with $ P = 1.0 $ and $ \rho = 10^{-2} $ for $ r < 0.8 $, surrounded by an ambient gas with $ P = 3 \times 10^{-5} $ and $ \rho = 10^{-4} $. The explosion region joins the ambient medium by matching an exponential decline in a transition region region $ 0.8 < r < 1.0 $. The velocities are initially zero, and the magnetic field is homogeneous and parallel to the symmetry axis. The background spacetime is considered to be flat. The inital data are evolved using an ideal gas EOS with adiabatic index $ \gamma = 4 / 3 $. We use an evenly spaced grid with a maximum radius of $ r = 6.0 $. We perform the test for three resolutions ($ 80 \times 10 $, $ 160 \times 20 $, and $ 320 \times 40 $) for all reconstruction schemes.
Fig. \[fig:SEcolor\] shows the Lorentz factor $ W $ at $ t = 4.0 $. A strong spherical shock has formed, propagating close to the speed of light, and as a consequence the magnetic field lines are compressed in the direction perpendicular to the axis. The results for this test are qualitatively comparable to the weakly magnetized case in @komissarov99. Fig. \[fig:SE\] shows radial profiles for $ P $ and $ B^\theta $ along the equatorial plane at the end of the simulation, using various reconstruction schemes. These plots are qualitatively similar to those of the cylindrical explossion [see e.g. Fig. B.4 in @leismann05]. All numerical schemes exhibit first order convergence with increasing resolution, as is expected to happen at shocks. The MC and PHM schemes yield very similar results, while minmod shows slightly lower values.
Estimation of the growth rates of the $ \Omega $-dynamo {#app:odynamo}
=======================================================
To compute the characteristic time scales on which the $ \Omega $-dynamo mechanism amplifies the magnetic field one has to study how the wind-up proceeds. Let us consider a stationary rotating configuration with no meridional flows, $ v^{*\,r} = v^{*\,\theta} = 0 $ and $ v^{*\,\varphi} = \Omega^* (r, \theta) \, r \sin \theta $, where $ \Omega^* (r, \theta) $ stands for the rotation law. Under these conditions and in the passive field approximation, the induction equation can be integrated analytically. The solution shows that the poloidal component of the magnetic field remains constant and the toroidal component grows linearly with time as $$B^{*\,\varphi} (t) =
B^{*\,\varphi} (t = 0) + t \, \varpi \, \vec{B}^* \cdot
\vec{\hat{\nabla}} \Omega^*.
\label{eq:bphi_evolution}$$ This equation specifies the toroidal magnetic field at any given time, provided that the poloidal component is constant and the angular velocity profile is fixed. For a time $ t \gg B^{*\,0}_\varphi /
(\varpi \vec{B}^* \cdot \vec{\hat{\nabla}} \Omega^*) $, which is $ \sim 1 \mathrm{\ ms} $ in our simulations, we can use this expression to compute the magnetic energy $$E_{\mathrm{mag} \, \varphi} \approx
\int \mathrm{d}^3 {\mbox{\boldmath $x$}} \frac{{B^*_{\Omega}}^2}{2}
\left( \varpi |\vec{\hat{\nabla}} \Omega^*| \right)^2
\, t^2,$$ where $ B^*_\Omega $ is the component of $ \vec{B}^* $ parallel to $ \vec{\hat{\nabla}} \Omega^* $. The rotation profiles of the final PNS can be approximated in all our models by the rotation law (\[rotation\_law\]) [@villain04]. In the Newtonian limit (\[newtonian\_rotation\_law\]), which is good enough for this estimate, we can compute an upper limit to the magnetic energy considering the maximum value of $ | \varpi \vec{\hat{\nabla}}
\Omega^* |_\mathrm{max} = \Omega^*_\mathrm{c} / 2 $, which yields $$E_{\mathrm{mag} \, \varphi} \le
E_{\mathrm{mag} \, \Omega} \,
\frac{{\Omega^*_\mathrm{c}}^2}{4} \, t^2.$$ Therefore, an estimate for the upper limit of the amplification of the magnetic energy parameter is $$\beta_\mathrm{mag}\approx\beta_{\varphi} \leq
\beta_{\Omega} \frac{{\Omega^*_\mathrm{c}}^2}{4} \, t^2 \leq
\beta_\mathrm{polo} \frac{{\Omega^*_\mathrm{c}}^2}{4} \, t^2 =
\left( \frac{t}{\tau_{\Omega}} \right)^2,
\label{eq:windup}$$ where we have defined the time scale for amplification of the magnetic field by the $ \Omega $-dynamo as $$\tau_{\Omega} = \frac{2}{\Omega^*_\mathrm{c} \sqrt{\beta_\mathrm{polo}}}.
\label{eq:tau_omega}$$ This gives us the characteristic time scale in which $ \beta_\mathrm{mag} $ reaches a value of 1; therefore, the saturation time $ t_\mathrm{sat} $ should be a fraction of this time. As the $ \Omega $-dynamo operates by transforming rotational energy into magnetic energy, the maximum energy can be extracted by the magnetic field is the one that is contained in the differential rotation of the core. In accordance with numerical simulations using strong magnetic fields [@obergaulinger_06_a] we estimate this amount to be 10% of the total rotational energy, i.e. $ \beta_\mathrm{mag}
(t_\mathrm{sat}) = 0.1 \, \beta_\mathrm{rot} (t_\mathrm{sat}) $. We also assume that the evolution of the magnetic energy parameter is given by Eq. (\[eq:windup\]) and that the energy is conserved, i.e.$ \beta_\mathrm{rot} (t) = \beta_\mathrm{rot} (t_0) - \beta_\mathrm{mag} (t) $.
[^1]: Note that the “$ \Omega $-dynamo” is also referred to in the literature as “wind-up” or “field-wrapping”. We follow in this paper the notation used by @obergaulinger_06_b [@obergaulinger_06_a]
[^2]: The adiabatic index should not be confused with the determinant of the spacetime three-metric, although we use the same symbol $ \gamma $ (following usual practice).
[^3]: In all models, we define the neutrino sphere as the surface inside the core where the density equals the trapping density $ \rho_\mathrm{trap} = 2 \times 10^{12} \mathrm{\ g\ cm}^{-3} $. In the microphysical models, above this density neutrinos are assumed to be trapped in the medium [@liebendoerfer_05_a].
[^4]: For these estimates we utilize the Newtonian limit, since most of the work on linear analysis of instabilities has not yet been extended to general relativity. Furthermore, for an approximate assessment, the restriction to a Newtonian treatment appears sufficiently accurate.
[^5]: We note that @balbus91 derived a complicated expression including bouyancy terms which, however, is only valid in the equatorial plane. To the best of our knowledge the timescale for the fastest growing mode in the general case has not been computed so far. It would require solving the dispersion relation, a task out of the scope of this paper.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We report the first observation of ‘orbital truncation rods’ — the scattering arising from the termination of bulk orbital order at the surface of a crystal. The x-ray measurements, performed on a cleaved, single-layered perovskite, La$_{0.5}$Sr$_{1.5}$MnO$_4$, reveal that while the crystallographic surface is atomically smooth, the orbital ‘surface’ is much rougher, with an r.m.s. deviation from the average ‘surface’ of $\sim$7Å. The temperature dependence of this scattering shows evidence of a surface-induced second order transition.'
address: |
$^1$Photon Factory, Institute of Materials Structure Science, High Energy Accelerator Research Organization, Tsukuba 305-0801, Japan\
$^2$CMPMS, Brookhaven National Laboratory, Upton 11973, NY, USA\
$^3$Institut Néel, CNRS & Université Joseph Fourier, BP 166, F-38042 Grenoble Cedex 9, France\
$^4$National Synchrotron Light Source, Brookhaven National Laboratory, Upton 11973, NY, USA\
$^5$Ames Laboratory and Department of Physics and Astronomy, Iowa State Univ., Ames, 50011, IA, USA\
$^6$Argonne National Laboratory, Argonne, 60439, IL, USA\
author:
- 'Y. Wakabayashi$^{1,2}$, M.H. Upton$^2$, S. Grenier$^{2,3}$, J.P. Hill$^2$, C.S. Nelson$^4$, J.-W. Kim$^{5}$, P.J. Ryan$^{5}$, A.I. Goldman$^{5}$, H. Zheng$^{6}$, and J.F. Mitchell$^{6}$'
title: 'Surface effects on the orbital order in the single layered manganite La$_{0.5}$Sr$_{1.5}$MnO$_4$'
---
An issue destined to be of increasing importance in the science of correlated-electrons systems in the coming years is the question of whether the electronic behaviour, such as charge, orbital or spin order, is different at a surface or interface compared to the bulk of the material. This question is of central importance from an applied perspective — any device one fabricates from these materials will require electron transport across an interface; from a nanoscience perspective — nanoscaled objects are defined by their very large surface-to-volume ratio; and from a fundamental perspective — understanding the role of reduced dimensionality in determining the electronic behaviour of such systems is a key component in understanding the general problem of strongly correlated electron systems.
In many ways manganites represent an ideal system to address the role of the surface, because all the relevant degrees of freedom — charge, spin and orbital — play an active role in determining the ground state. As a result, they are exquisitely sensitive to perturbations, and hence, might be expected to exhibit relatively large surface effects. In fact, while the vast majority of experimental work has focused on bulk properties[@review1; @review2], the very few surface sensitive studies do suggest that the surface of a manganite may be quite different from the bulk. For example, the measurements of Freeland [*et al.*]{}[@Freeland05NatMat] showed that the top bilayer of a cleaved surface of a ferromagnetic bilayered manganite is magnetically dead, though the second bilayer has full bulk magnetic order. Similarly, the half-doped single-layer manganite, which is known to exhibit bulk orbital order below $T_{OO}=231$ K, showed no superlattice reflection corresponding to such ordering down to 80 K, when probed with surface-sensitive LEED [@Plummer01ProgSurfSci]. Thus, there is evidence that the surface of manganite crystals may exhibit different electronic phases than the bulk — a phenomena that has been coined ‘electronic reconstruction’[@Ohtomo02Nature; @Okamoto04Nature] in analogy with more traditional surface science studies of atomic reconstruction driven by surface energetics.
In this paper, we present the first study of orbital order at a surface, utilizing surface x-ray scattering. The advantage of using x-ray scattering in this endeavour is that it provides information on both the electronic surface (specifically, atomic displacements associated with orbital order) and the chemical surface, separately. It is well known that the abrupt truncation of a sample by its surface gives rise to rod-shaped x-ray scattering around the Bragg reflections, the so-called crystal truncation rods (CTR). The intensity profiles of such rods provide detailed, quantitative information about the chemical surface[@Andrews85JPhysC]. Similarly then, the abrupt termination of orbital order by a surface should give rise to orbital truncation rods (OTR) running through the orbital superlattice peaks. These would allow one to quantify the surface orbital order. Here, we present the first observation of such orbital truncation rod scattering. This allows us to quantitatively probe the orbital order in the vicinity of the surface. Working with a freshly cleaved La$_{0.5}$Sr$_{1.5}$MnO$_4$, we show that the orbital order has a significantly rougher interface than the chemical surface and a shorter correlation length than in the bulk. These results imply that the surface order parameter is reduced relative to the bulk. Further, the temperature dependence of the surface orbital order suggests that the surface exhibits a more continuous transition than the bulk ordering transition. Taken together, these results demonstrate that the surface is a significant perturbation in regard to the electronic phenomena of orbital ordering and point to the need for an improved theoretical understanding of how the surface couples to such electronic orders.
We first discuss surface preparation and characterization. Single crystal samples of La$_{0.5}$Sr$_{1.5}$MnO$_4$, which has the layered perovskite structure shown in Fig. \[fig:chem\_sur\] (a), were grown by the floating zone method. The samples were then either cleaved perpendicular to the $c$-axis in He and placed in a sealed He-filled can, or cleaved in air, and put into a closed-cycle refrigerator. Both treatments gave the same results, indicating that the surface is stable to brief exposure to air, a result also found elsewhere[@Freeland06APS]. Figure \[fig:chem\_sur\](b) shows an atomic force microscope (AFM) picture of a freshly cleaved sample. The terraces seen here are on the order of 1 $\mu$m in size, much larger than the coherence length of the x-rays. The step height observed here was 6.2Å$\pm$ 0.2 Å, corresponding to $c/2$. We note that similar AFM pictures were obtained at the end of a week long x-ray experiment in vacuum. Analysis of the CTR intensity allows a determination of the precise termination of the cleaved sample[@You92PRB]. Figure \[fig:chem\_sur\] (c) shows the CTR scattering intensity distribution along the (00$L$) and (11$L$) directions together with the calculated intensity distribution for a La-terminated surface and a Mn-terminated surface (Fig.\[fig:chem\_sur\] (a)). The experimental data are well reproduced by assuming a predominantly La-terminated surface with very little surface relaxation. For comparison, there are conflicting reports in the literature for the termination of thin films, with both La and Mn termination reported[@Choi99PRB].
![(a) The crystal structure of La$_{0.5}$Sr$_{1.5}$MnO$_4$. Green, red and white spheres denote La/Sr, Mn, and O atoms, respectively. (b) AFM image of a cleaved sample used in this study. The step height was 6.2 $\pm$ 0.2 Å. (c) (00$L$) and (11$L$) CTR scattering intensity distribution together with the calculated intensity for La and Mn terminated surfaces. The result of the curve fitting shows the chemical surface is predominantly La terminated.[]{data-label="fig:chem_sur"}](chem_sur.eps){width="10cm"}
As discussed above, the structure of the chemical surface is reflected in the scattering along the ($hkL$) direction, where $h$, $k$ are integer multiples of reciprocal lattice vectors, that is rods along the surface normal $c$\*-direction through the allowed, bulk Bragg reflections. Similarly, scattering originating from the surface of the orbital ordering must be present in ‘orbital truncation rods’ — rods of scattering along the same direction but through the superlattice reflections related to the orbital order, i.e. ($h+\frac14, k+\frac14, L$) where ($\frac14, \frac14, 0$) is the orbital ordering wave-vector[@Dhesi04PRL]. One of the primary goals of this work was to observe such scattering. To this end, we measured the (20$L$)$\pm$($\frac 14 \frac 14 0$) intensity distribution below $T_{OO}=$231 K, the bulk orbital order transition temperature. We discuss first transverse scans taken across such ‘orbital rods’ (figure \[fig:OO-CTR\](a)). For reference a transverse (2 $\delta k$ 0.2) scan profile across the truncation rod arising from the crystallographic surface is also shown (blue data). It is as sharp as the instrumental resolution. As discussed below, this is indicative of a smooth atomic surface. The open circles show the profile of the ($\frac 94 \frac 14 2$) bulk orbital order Bragg peak. The finite width observed here shows that the correlation length of the bulk orbital order is finite, and in fact, we find $\xi_{OO}^{\scriptsize{\rm bulk}}=$ 410Å$\pm$ 8Å, in agreement with earlier work[@Dhesi04PRL]. The transverse scan taken at ($\frac 94 \frac 14 0.2$), that is across the expected OTR is shown in red. A peak is clearly observed at this position. However, this peak is not, by itself, evidence of surface scattering. Such a peak could arise from stacking faults or other defects, in the bulk orbital order which would result in a finite correlation length for the orbital order along the $c$-direction and hence a tail along the $c$\*-direction for superlattice reflections.
![(a) Transverse ($k$) scans through (2 0 0.2), ($\frac 94 \frac 14 2$), and ($\frac 94 \frac 14 0.2$) reciprocal lattice positions shown with blue circles, open circles, and red circles, respectively. The red curve shows the result of a two-Gaussian fitting for ($\frac 94 \frac 14 0.2$) profile. (inset) The incidence angle dependence of the scattered intensity at ($\frac 74 \frac {\bar{1}}{4} 2$) and ($\frac 74 \frac {\bar{1}}{4} 0.3$). The enhancement of the latter at $\alpha$ around $\theta_C=0.4^\circ$ shows that it results from surface scattering. (b) (20$L$) and ($\frac 94 \frac 14 L$) truncation rod intensity for samples A and B at 100K, 170K and 200K. The solid line shows the calculated intensity distribution given by the measured I(20$L$) for sample A multiplied by $\exp(-\sigma^2q_z^2)$ with surface roughness parameter $\sigma=6.7$ Å (see text).[]{data-label="fig:OO-CTR"}](OO-CTR.eps){width="7cm"}
We can, however, distinguish surface scattering from such bulk scattering by measuring the scattering intensity as a function of incidence angle $\alpha$. Surface scattered intensity will be enhanced at $\alpha = \theta_C$, the critical angle for total reflection, as a result of the interference between the incident and the scattered waves of the photon[@Feidenhansl89SurSciRep]. The inset to Fig. \[fig:OO-CTR\](a) shows the scattered intensity as a function of $\alpha$ at the bulk orbital superlattice reflection ($\frac 74 \frac {\bar{1}}{4} 2$) and on the nominal OTR at ($\frac 74 \frac {\bar{1}}{4} 0.3$). The intensity at ($\frac 74 \frac {\bar{1}}{4} 0.3$) has a maximum around $\theta_C=0.4^\circ$ while a smooth increase is seen at ($\frac 74 \frac {\bar{1}}{4} 2$) with increasing $\alpha$, a result of the fact that more of the beam is intercepted by the sample at higher angles. Since the exit angle $\beta$ for ($\frac 74 \frac {\bar{1}}{4} 0.3$) is 3 to 1.5 degrees in this scan region, the observed intensity maximum cannot be due to a maximum in the sample illumination and detector acceptance, which happens at $\alpha = \beta$ in our experimental configuration. Therefore, we conclude that the scattering observed at (2 0 0.2)$\pm$($\frac 14 \frac 14 0$), shown in Fig. \[fig:OO-CTR\] (a) does indeed come from surface scattering and not from stacking faults or other bulk effects. This is the first such observation of the orbital truncation rod scattering.
Having established that such scattering originates from the truncation of the orbital order, we next discuss what can be learned from this new scattering. As shown in Fig. \[fig:OO-CTR\] (a), the transverse scan profile of the rod intensity for the chemical surface and for the orbital surface are quite different. This difference in the scattering profiles directly reflects a difference in the surface roughness.[@Andrews85JPhysC] The sharp single peak observed for the chemical CTR scattering implies a very flat surface, and is consistent with the atomically smooth chemical surface observed in the AFM picture (Fig. \[fig:chem\_sur\] (b)). In contrast, the transverse profile of the orbital truncation rod exhibits two components; a ‘sharp’ one and a ‘broad’ one. Such a line shape is expected when a surface — in this case the orbital surface — has a well-defined average position (a flat plane) but local, finite deviations from this average[@Andrews85JPhysC]. Thus, without recourse to any model-dependent fitting we can immediately conclude from the data of Fig.\[fig:OO-CTR\] (a) that there exists a flat chemical surface and a rougher orbital order surface.
A rougher surface also has the consequence that the intensity along the truncation rod must fall more sharply as a function of $L$ than for a less rough surface. Figure \[fig:OO-CTR\] (b) shows the intensity of the (20$L$) and ($\frac 94 \frac 14 L$) rods as a function of $L$ measured on two samples (cleaved in He, sample A, and in air, sample B) at 100K, 170K, and 200K. For the orbital rod, the intensity of the sharp component integrated in the transverse direction is shown. Note, the $L$-dependence of the (20$L$) intensity of sample A is slightly different from that of sample B, suggesting a slightly different degree of flatness of the chemical surface for the two samples. On the other hand, the ($\frac 94 \frac 14 L$) intensity distribution for both samples is very similar and in both cases, the slope is steeper than that along (20$L$). This is consistent with the result in Fig. \[fig:OO-CTR\] (a): the orbital order surface is rougher than the chemical surface.
We next quantify this result by use of a model for a nearly flat surface due to Andrews and Cowley[@Andrews85JPhysC]. In this model, the surface is characterized by two parameters: $\sigma$, the standard deviation of the surface height, and $\xi^{\scriptsize {\rm sur}}$, the correlation length of the surface (Fig. \[fig:analysis\]). The truncation rod intensity distribution $I(\vec Q)$ from such a surface is given by: $$\begin{aligned}
I(\vec Q)&=&I_s(\vec Q) + I_b(\vec Q),\label{eq:I}\\
{\rm with \hspace{1cm}} I_s(\vec Q)&=&\frac{4\pi^2|F(\vec Q)|^2}{q_z^2}\exp(-\sigma^2 q_z^2) \delta(h,k),
\label{eq:Is}\\
{\rm and \hspace{1cm}} I_b(\vec Q)&=&\pi|F(\vec Q)|^2\frac{(\xi^{\scriptsize {\rm sur}})^2 [1-\exp(-\sigma^2q_z^2)]^2}{\sigma^2q_z^4}
\nonumber\\&&\times
\exp\left[\frac{-(\xi^{\scriptsize {\rm sur}})^2[1-\exp(-\sigma^2q_z^2)]}{4\sigma^2q_z^2}(h^2+k^2)\right],\label{eq:Ib}\end{aligned}$$ where $I_s$ and $I_b$ are the intensities for the sharp component and the broad component respectively, $F(\vec Q)$ is the structure factor at scattering vector $\vec Q$, and $q_z$ is the $c$\*-component of the reduced wavevector (e.g. this is $2\pi L/c$ for $|L|<0.5$). In order to compare our data to the predictions of this model, we replace the delta-function in equation (\[eq:Is\]) with a Gaussian, to account for the finite bulk orbital correlation length, and fit the transverse scans to a two-Gaussian lineshape as shown by the red solid curve in Fig. \[fig:OO-CTR\](a). The $L$-dependence of the parameters derived from these fits are shown in Figure \[fig:analysis\](a), which plots the ratio of the intensity of the broad component and the sharp component, and in Figure \[fig:analysis\](b), which shows the widths of the two components, for the 170 K data set.
![(a) The intensity ratio of the broad component observed in the transverse orbital truncation rod scans to the sharp component, as a function of $L$. (inset) Schematic view of the orbital order region around the surface. (b) Peak widths of broad- and sharp- component. Calculated values for the model discussed in the text are shown by solid lines in both figures.[]{data-label="fig:analysis"}](analysis.eps){width="7cm"}
The intensity ratio $I_b/I_s$ increases rapidly with $L$. This ratio depends mainly on $\sigma$, and hence $\sigma$ may be obtained from the plot. The solid line in Fig. \[fig:analysis\](a) shows the expected behaviour for $\sigma=6.7$Å, which reproduces the data quite well. Note, this same value of $\sigma$ is also consistent with the results shown in Fig. \[fig:OO-CTR\] (b); The solid line in this latter figure is the calculated intensity distribution given by taking the product of the measured $I(20L)$ and $\exp (-\sigma^2q_z^2)$, with $\sigma$=6.7Å; here, we assumed the chemical surface is perfectly flat. The in-plane correlation length $\xi^{\scriptsize {\rm sur}}$ is given by the transverse peak width and is shown in Fig. \[fig:analysis\] (b) as a function of $L$, together with the $L$-dependence predicted by eq. (\[eq:Ib\]) for a particular value of $\xi^{\scriptsize {\rm sur}}$. Using the value of $\sigma=6.7$Å, $\xi^{\scriptsize {\rm sur}}$ is found to be 120Å. That is, the surface correlation length is much shorter than that of the bulk orbital order $\xi_{OO}^{\scriptsize{\rm bulk}}=410$Å. This suggests that some aspect of the surface disrupts the orbital order. This may be due to differing surface energetics resulting from the reduced coordination, or perhaps varying Sr concentrations at the site of the cleave.
We next compare the present results to that of previous reports. LEED measurements on the same material [@Plummer01ProgSurfSci] showed no superlattice intensity down to 80K, implying little orbital order is established in the topmost layer. Our result is consistent with this study. The surface roughness in the orbital order observed here necessarily requires a small order parameter or a small ordered volume fraction at the surface. The same authors also performed STM measurements on the same system and found unidentified ‘electronic roughness at the surface’ having an in-plane correlation length of several nanometres and surface roughness of 6Å. Comparing to the present results, one notes that the length scale of the roughness observed in the STM study is very similar to that of the orbital order interface observed in this study, and it is tempting to surmise that this ‘electronic roughness’ may be related in some way to the rough orbital interface observed here. However, it is difficult to draw any firm conclusions on this point without further study.
Finally, we briefly discuss the temperature dependence of the orbital truncation rod scattering. Figure \[fig:Tdep\] shows the temperature dependence of the ($\frac 94 \frac 14 L$)-sharp component intensity for $L=$2, 0.2, 0.3, and 0.4 r.l.u.
![Temperature dependence of the ($\frac 94 \frac 14 L$)-sharp component intensity for $L=$2, 0.2, 0.3, and 0.4.[]{data-label="fig:Tdep"}](Tdep.eps){width="7cm"}
The temperature dependence of the surface-sensitive OTR scattering intensity ($L$=0.2, 0.3 and 0.4 data) is different from that of the bulk-sensitive superlattice reflection. The bulk OO transition was reported to be ‘probably second order’ with a relatively discontinuous transition[@Larochelle05PRB]. In contrast, the temperature dependence of the OTR scattering intensity is seen to evolve much more continuously through the transition. The bulk superlattice reflection intensity is proportional to the volume of the orbitally ordered phase, and to the square of the order parameter, or the Jahn-Teller distortion caused by the orbital ordering. In contrast, the OTR intensity is proportional to the area of the orbital ordered flat surface and to the square of the order parameter. Since the surface roughness of the OO phase at 100K is the same as that at 200 K, then the change in intensity must be attributed to a change in the magnitude of the atomic displacement, i.e., in the orbital order parameter at the surface. Thus, these data suggest that the surface orbital order parameter decays more continuously than the bulk. Such phenomena may be related to surface-induced second order phase transitions that have been observed in other materials[@Watson96PRL; @Dosch88PRL]. Future work will investigate this possibility in detail.
In conclusion, we have performed a surface x-ray scattering study of a cleaved, single-layered, half-doped manganite. The chemical surface was found to be extremely flat with a La/Sr layer termination. Importantly, we have observed the scattering arising from truncation of the orbital order at a surface for the first time. We find that the orbital surface is rougher than the chemical surface, and that its temperature dependence is different from the bulk orbital order, indicating a surface-induced second order transition. These results are further evidence of the importance of surface effects in determining electronic ground states in strongly correlated systems and provide a direct route to probe such effects quantitatively. Detailed understanding will require theoretical models capable of dealing with such surface effects explicitly.
Methods {#methods .unnumbered}
=======
X-ray diffraction measurements were performed at the beamlines X22C and X21 at the National Synchrotron Light Source, and at 6ID at the Advanced Photon Source. The measurements at X22C and 6ID were performed with 6.5 keV x-rays using a six-circle diffractometer for surface scattering experiments and those at X21 were performed with 8.8 keV x-rays and a standard four-circle diffractometer. Most of the experiments using the six-circle diffractometers were conducted with a fixed-$\alpha$ mode in order to keep the illuminated sample volume constant, where $\alpha$ denotes the incidence angle, i.e., the angle between the crystal surface and the incident x-ray. The angle $\alpha$ was fixed to 0.6$^\circ$, which corresponds to a penetration depth of 600 Å, unless otherwise noted. This angle is higher than the critical angle for total reflection ($\theta_C=0.4^\circ$). The experiments at X21 were performed in a $\alpha =\beta$ mode, where $\beta$ denotes the exit angle. Both modes give very similar results. The temperature was controlled by closed-cycle He refrigerators. The sample was mounted either in thermally-insulating vacuum or in a He exchange gas. The curve fitting for CTR profile was made with the ROD program[@Vlieg00JApplCryst] with Mo for La/Sr site because the form factor of Mo approximates that of the La$_{0.25}$Sr$_{0.75}$ mixture well.
Acknowledgement {#acknowledgement .unnumbered}
===============
We thank Dr. A. Checco for helping with AFM measurements. Y.W. wishes to acknowledge the Yamada Science Foundation, support for long-term visit. US DOE, Basic Energy Sciences supported work at Brookhaven under contract No. DE-AC02-98CH10886, at Argonne under contract No. DE-AC02-06CH11357, and at the MUCAT Sector at the Adcanced Photon Source and Ames Laboratory under contract No. DE-AC02-07CH11358.
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|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: |
The problem of classifying modules over a tame algebra $A$ reduces to a block matrix problem of tame type whose indecomposable canonical matrices are zero- or one-parameter. Respectively, the set of nonisomorphic indecomposable modules of dimension at most $d$ divides into a finite number $f(d,A)$ of modules and one-parameter series of modules.
We prove that the number of canonical parametric block matrices of size $m\times
n$ and a given partition into blocks is bounded by $4^s$, where $s$ is the number of free entries, $s{\leqslant}mn$. Basing on this estimate, we prove that $$f(d,A){\leqslant}{\binom {d+r} r}
4^{d^2(\delta_1^2+\dots +\delta_r^2)}{\leqslant}(d+1)^r4^{d^2(\dim A)^2},$$ where $r$ is the number of nonisomorphic indecomposable projective left $A$-modules and $\delta_1,\dots,\delta_r$ are their dimensions.
[*AMS classification:*]{} 15A21; 16G60.
[*Keywords:*]{} Canonical matrices; Classification; Tame algebras.
author:
- |
Thomas Brüstle\
Fakultät für Mathematik, Universität Bielefeld\
Postfach 100 131 D-33501 Bielefeld, Germany\
[email protected]\
and\
Vladimir V. Sergeichuk[^1]\
Institute of Mathematics\
Tereshchenkivska 3, Kiev, Ukraine\
[email protected]
title: 'Estimate of the number of one-parameter families of modules over a tame algebra'
---
\[section\] \[section\]
Introduction {#s1}
============
Matrices and finite dimensional algebras are considered over an algebraically closed field $k$.
Gabriel, Nazarova, Roiter, Sergeichuk, and Vossieck [@gab_vos] studied matrix problems, in which the row-transformations are given by a category and the column transformations are arbitrary. They interpreted ${m\times n}$ matrices as points of the affine space $k^{m\times n}$ of all ${m\times n}$ matrices and proved that for a tame matrix problem and every ${m\times n}$ there exists a full system of nonisomorphic indecomposable $m\times n$ matrices that consists of a finite number of points and punched straight lines. This result was extended to modules over a tame finite dimensional algebra $A$: for every $d\in\mathbb{N}$ there exists an almost full (except for a finite number of modules) system of nonisomorphic indecomposable $d$-dimensional modules that consists of a finite number $\rho_A(d)$ of punched lines (an $A$-module of dimension $d$ was considered as a point of the affine space $k^{d\times d}\oplus\dots \oplus k^{d\times
d}$; the number of summands $k^{d\times d}$ is a number of generators of $A$).
Brüstle [@bru] proved, that $$\label{0.1}
\rho_A(d){\leqslant}\dim ({\mathop{\rm Rad}\nolimits}A)\cdot e^{2^6
3^{d-1} (d-1)^{2d-1}}.$$
Sergeichuk [@ser] extended the results of [@gab_vos] to block matrix problems in which rows and columns transformations are given by triangular matrix algebras: If the matrix problem is of tame type, then for every $m\times n$ there exists a finite set of zero- and one-parameter matrices $$\label{0.2}
M_1,\dots,M_{t_1},\, N_1(\lambda_1),
\dots, N_{t_2}(\lambda_{t_2})$$ such that the set of indecomposable canonical $m\times n$ matrices is $$\{M_1,\dots,M_{t_1}\}\cup \{N_1(a)\,|\, a\in
k\}\cup \dots \cup \{N_{t_2}(a)\,|\, a\in
k\};$$ it may be interpreted as a set of points and straight lines in the affine space $k^{m\times n}$. The proof was based on Belitskiĭ’s algorithm [@bel] (see also [@bel1]) for reducing a matrix to canonical form; two matrices may be reduced one to the other if and only if they have the same canonical form.
Drozd [@dro1] proposed the following reduction of the problem of classifying modules over an algebra $A$ to a matrix problem. Let $P_1,\dots, P_r$ be all nonisomorphic indecomposable projective right $A$-modules. For every right module $M$ over $A$, there exists an exact sequence $$\label{0.3}
P_1^{p_1}\oplus\dots\oplus P_r^{p_r}
\stackrel{\varphi}{\longrightarrow}
P_1^{q_1}\oplus\dots\oplus P_r^{q_r}
\stackrel{\psi}{\longrightarrow}
M\longrightarrow 0,$$ where $X^{l}:=X\oplus\dots\oplus X$ ($l$ times). The homomorphism $\varphi$ is determined up to transformations $\varphi\mapsto g\varphi f$, where $f$ and $g$ are automorphisms of $\oplus_iP_i^{p_i}$ and $\oplus_iP_i^{q_i}$. The $\varphi$, $f$, and $g$ can be given by their matrices in bases of the spaces $\oplus_iP_i^{p_i}$ and $\oplus_iP_i^{q_i}$ over $k$. This reduces the problem of classifying modules over algebras to block matrix problems, which were studied in [@ser]. The modules that correspond to the canonical matrices form a full system of nonisomorphic modules; indecomposable modules correspond to indecomposable matrices.
In this article, we obtain the following estimates:
- If a block matrix problem is of tame type, then the number of canonical parametric block matrices of size $m\times n$ and a given partition into blocks is bounded by $4^s$, where $s$ is the number of free entries, $s{\leqslant}mn$.
- If an algebra $A$ is of tame type, then the number of zero- and one-parameter matrices that give a full system of nonisomorphic indecomposable modules of dimension at most $d$ is bounded by $${\binom {d+r} r} 4^{d^2(\delta_1^2+\dots
+\delta_r^2)},$$ where $r$ is the number of nonisomorphic indecomposable projective left $A$-modules and $\delta_1,\dots,\delta_r$ are their dimensions.
Here the first estimate is optimal and the second one improves significantly the estimate from \[3\]. The paper is organized as follows: in Section 2, we introduce the concept of standard linear matrix problems and recall Belitskii’s algorithm. Section 3 is devoted to the proof of the estimate (i), Section 4 is concerned with the corresponding estimate (ii) for modules over a tame algebra.
Belitskiĭ’s algorithm for linear matrix problems {#s2}
================================================
A block matrix $M=[M_{ij}]$, $M_{ij}\in
k^{m_i\times n_j}$, will be called an $\underline{m}\times\underline{n}$ [*matrix*]{}, where $\underline{m}=(m_1,m_2,\ldots)$ and $\underline{n}=(n_1,n_2,\ldots)$.
A linear matrix problem is the canonical form problem for $\underline{n}\times\underline{n}$ matrices whose blocks satisfy a certain system of linear homogeneous equations. Solving this system, we select [*free blocks*]{} that are arbitrary; the other blocks are their linear combinations. The set of admissible transformations consists of elementary transformations within strips, additions of linear combinations of rows of the $i$th strip to rows of the $j$th strip for certain $i>j$, and additions of linear combinations of columns of the $i$th strip to columns of the $j$th strip for certain $i<j$. Elementary transformations and additions may be linked: making elementary transformations within a horizontal strip, we must produce the same elementary transformations within all horizontal strips linked with it and inverse elementary transformations within all vertical strips linked with it. Making an addition between strips, we must produce all linked with it additions.
Applying Belitskiĭ’s algorithm ([@bel],[@ser]), we can reduce a block matrix by these transformations to canonical form; two block matrices may be reduced one to the other if and only if they have the same canonical form.
If the matrix problem is of tame type (that is, it does not contain the problem of classifying pairs of matrices up to simultaneous similarity, then the set of direct-sum-indecomposable canonical $\underline{n}\times\underline{n}$ matrices forms a finite number of points and straight lines in the affine space of $\underline{n}\times\underline{n}$ matrices (see [@ser Theorem 3]). In the article, we prove that this number is bounded by $4^s$, where $s$ is the number of entries in free blocks.
Let us sketch a more formal definition of a linear matrix problem (see [@ser Sect. 2.2]).
An algebra $\varGamma \subset k^{t\times t}$ of upper triangular matrices is a [*basic matrix algebra*]{} if $$\label{3.00}
\begin{bmatrix}
a_{11}&\cdots&a_{1t} \\
&\ddots&\vdots \\
\text{\Large 0} & & a_{tt}
\end{bmatrix}\in\varGamma
\quad {\rm implies} \quad
\begin{bmatrix}
a_{11} & & \text{\Large 0} \\
&\ddots& \\
\text{\Large 0} & & a_{tt}
\end{bmatrix}\in\varGamma.$$ The diagonals $(a_{11},a_{22},\dots,a_{tt})$ of the matrices from $\varGamma$ form a subspace in $k^t =k\oplus\dots\oplus k$, which may be given by a system of equations of the form $a_{ii}=a_{jj}$. Define an equivalence relation in $T=\{1,\dots,t\}$ putting $$\label{4.0a}
\text{$i\sim j$ if and only if
${\mathop{\rm diag}\nolimits}(a_1,\dots,a_t)\in \varGamma$ implies
$a_i=a_j$.}$$ We say that a sequence of nonnegative integers $\underline{n}=(n_1,n_2,\dots,n_t)$ is a [*step-sequence*]{} if $i\sim j$ implies $n_i=n_j$.
A [*linear matrix problem given by a pair*]{} $$\label{3.4aa}
(\varGamma,\cal M), \quad \varGamma {\cal
M}\subset {\cal M},\ {\cal M}\varGamma
\subset {\cal M},$$ consisting of a basic $t\times t$ algebra $\varGamma$ and a vector space ${\cal
M}\subset k^{t\times t}$, is the canonical form problem for matrices $M\in {\cal
M}_{\underline{n}\times\underline{n}}$ with respect to transformations $$\label{3.01}
M\longmapsto S^{-1}MS,\qquad S\in
\varGamma_{\underline{n}\times
\underline{n}}^{*},$$ where $\underline{n}=(n_1,\dots,n_t)$ is a step-sequence, $\varGamma_{\underline{n}\times
\underline{n}}$ and ${\cal
M}_{\underline{n}\times\underline{n}}$ consist of $\underline{n}\times\underline{n}$ matrices whose blocks satisfy the same systems of linear homogeneous equations as the entries of $t\times t$ matrices from $\varGamma$ and $\cal M$, respectively, and $\varGamma_{\underline{n}\times
\underline{n}}^{*}$ denotes the set of nonsingular matrices from $\varGamma_{\underline{n}\times
\underline{n}}$. ($\varGamma$ and ${\cal M}$ are subspaces of $k^{t\times t}$; they may be given by systems of linear homogeneous equations of the form $$\sum_{(i,j)\in{\cal
I} \times{\cal J}}d_{ij}x_{ij} = 0,$$ where ${\cal I},{\cal J}\in
\{1,\dots,t\}/\!\sim$ are equivalence classes.)
Let us outline Belitskiĭ’s algorithm (it has been detailed in [@ser]) for reducing a matrix $$M=\begin{bmatrix}
M_{11} & \cdots & M_{1t} \\
\hdotsfor{3} \\
M_{t1} & \cdots & M_{tt}
\end{bmatrix}
\in{\cal
M}_{\underline{n}\times\underline{n}}$$ to canonical form by transformations . We assume that the blocks of $M$ (and of every block matrix) are ordered starting from the lower strip: $$\label{4.1}
M_{t1}<M_{t2}<\dots<M_{tt}<M_{t-1,1}<
M_{t-1,2}<\dots<M_{t-1,t}<\cdots$$ In the set $\{M_{ij}\}$ of blocks of $M$, we select the set of free blocks such that every unfree block is a linear combination of free blocks that preceding it with respect to the ordering . The entries of free blocks will be called the [*free entries*]{}.
On the first step, we reduce the block $M_{t1}$. It is reduced by transformations $$\label{5.1}
M_{t1}\longmapsto
S^{-1}_{tt}M_{t1}S_{11},\qquad S\in
\varGamma_{\underline{n}\times
\underline{n}}^*.$$
If $1\nsim t$, then $M_{t1}$ is reduced by arbitrary equivalence transformations. We reduce it to the form $$\label{5.2}
\left[ \begin{array}{cc}
0 & I \\
0 & 0
\end{array} \right]$$ and extend its division into substrips onto the first vertical and the first horizontal strips of $M$.
If $1\sim t$, then $M_{t1}$ is reduced by arbitrary similarity transformations. We reduce it to a [*Weyr matrix*]{} (which is obtained from a Jordan matrix by simultaneous permutations of rows and columns, see [@ser Sect. 1.3]): $$\label{5.3}
W=W_{\alpha_1}\oplus\dots\oplus
W_{\alpha_r},\quad \alpha_1\prec\dots\prec
\alpha_r,$$ where $\prec$ is a linear order in $k$ (if $k$ is the field of complex numbers, we use the lexicographic ordering), and $$\label{5.4}
W_{\alpha_i}= \left[\begin{tabular}{cccc}
$\alpha_iI_{m_{i1}}$&$W_{i1}$&&{\Large 0}\\
&$\alpha_iI_{m_{i2}}$&$\ddots$&\\
&&$\ddots$&$W_{i,q_i-1}$\\ {\Large
0}&&&$\alpha_iI_{m_{iq_i}}$
\end{tabular}\right],\quad
W_{ij}=
\begin{bmatrix}
I\\0
\end{bmatrix},$$ $m_{i1}{\geqslant}\dots{\geqslant}m_{iq_i}$. We make the most coarse partition of $W$ into substrips for which all diagonal subblocks have the form $\alpha_i I$ and all off-diagonal subblocks are $0$ and $I$ (all matrices commuting with $W$ are upper block triangular with respect to this partition). We extend this division of $M_{t1}=W$ into substrips onto the first vertical and the first horizontal strips of $M$.
Then we restrict the set of admissible transformations with $M$ to those transformations that preserve $M_{t1}$ (that is, $S^{-1}_{tt}M_{t1}S_{11}=
M_{t1}$). It may be proved that the algebra of matrices $$\Lambda_1= \{S=[S_{ij}]\in
\varGamma_{\underline{n}\times
\underline{n}}\, |\, M_{t1}S_{11}=
S_{tt}M_{t1}\}$$ also has the form $\varGamma'_{\underline{n'}\times
\underline{n'}}$, where $\varGamma'$ is a basic matrix algebra. The entries of $M_{t1}$ are the [*reduced entries*]{} of $M$.
On the second step, we take the first unreduced (that is, does not contained in $M_{t1}$) block with respect to the new partition and reduce it.
On each step, we take the first unreduced block $M_{pq}$ (with respect to a new subdivision) and reduce it by those admissible transformations that preserve all reduced entries. If $M_{pq}$ is not free, then it is the linear combination of preceding free blocks that have been reduced, and hence $M_{pq}$ is not changed at this step. If $M_{pq}$ is free, then the following three cases are possible:
\(i) There exists a nonzero admissible addition to $M_{pq}$ from other blocks. Since admissible transformations are given by upper block triangular matrices and we use the ordering , all nonzero additions to $M_{pq}$ are from preceding (reduced) blocks. We make $M_{pq}=0$ by these additions.
\(ii) There exist no nonzero admissible additions to $M_{pq}$ and it is reduced by equivalence transformations. Then we reduce $M_{pq}$ to the form .
\(iii) There exist no nonzero admissible additions to $M_{pq}$ and it is reduced by similarity transformations. Then we reduce $M_{pq}$ to a Weyr matrix.
At the end of this step, we make an additional subdivision of $M$ into strips in accordance with the block form of the reduced $M_{pq}$ and restrict the set of admissible transformations to those that preserve $M_{pq}$.
The process stops after reducing the last unreduced entry of $M$. The obtained canonical matrix will be partitioned into $$\label{9}
M_1,M_2,\dots,M_{l(M)},$$ where $M_i$ is the block that reduces at the $i$th step. Each $M_i$ has the form $0$, , or is a Weyr matrix. We will call the [*boxes*]{} of $M$.
For instance, $$M= \left[ \begin{tabular}{c|c}
$M_3$ &\!\!\!\!\! \begin{tabular}{c|c}
$M_6$ & $M_7$ \\ \hline
$M_4$ &$ M_5$
\end{tabular}\!\!\!
\\ \hline
$ M_1$ & $M_2$
\end{tabular} \right]
=\left[ \begin{tabular}{c|c}
$ \!\!\!\! \begin{array}{cc}-1&1\\
0&-1\end{array}$\!\!\!\! &
\!\!\!\begin{tabular}{c|c}2&$0$ \\
\hline 0&1\end{tabular}\!\!\! \\ \hline
$ 3I_2$ & $0$
\end{tabular} \right],
\qquad l(M)=7,$$ is a canonical $(2,2)\times(2,2)$ matrix for the linear matrix problem given by the pair $(\varGamma, k^{2\times 2})$, where $$\varGamma=
\left\{\left.\begin{bmatrix} a&b\\ 0&a
\end{bmatrix}\,\right|\, a,b\in k\right\}.$$
Let $M$ be a canonical matrix. Replacing all diagonal entries of its free boxes that are Weyr matrices by parameters, we obtain a parametric matrix $M(\lambda_1,\dots,
\lambda_p)$. Its [*domain of parameters*]{} $\cal{D}$ is the set of all $(a_1,\dots,a_p)\in k^p$ for which $M(a_1,\dots,a_p)$ is a canonical matrix. If a parameter $\lambda_i$ is finite (that is, the number of vectors of $\cal{D}$ with distinct $a_i$ is finite), we replace $\lambda_i$ by its values and obtain several parametric matrices with a smaller number of parameters. Repeating this process, we obtain parametric matrices having only infinite parameters. The obtained matrices will be called [*canonical parametric matrices*]{}.
Hence, the canonical form problem for $\underline{n}\times\underline{n}$ matrices with the same $\underline{n}$ reduces to the problem of finding a finite number of canonical parametric matrices and their domains of parameters.
Estimate of the number of canonical parametric matrices {#s3}
=======================================================
In this section, we study a linear matrix problem of tame type. As was proved in [@ser], each of its canonical parametric matrices, up to simultaneous permutations of rows and columns, has the form $$\label{12}
N_1(\lambda_1)\oplus\dots\oplus
N_p(\lambda_p) \oplus R_1\oplus\dots\oplus
R_q,\qquad p{\geqslant}0,\quad q{\geqslant}0,$$ where $N_i(\lambda_i)$ and $R_j$ are indecomposable canonical one- and zero-parameter canonical matrices. The purpose of the section is to prove the following theorem.
\[t1\] If a linear matrix problem is of tame type, then the number of its canonical parametric matrices of size $\underline{n}\times\underline{n}$ is bounded by $4^{s(\underline{n})}$, where $s(\underline{n})$ is the number of free entries in an $\underline{n}\times\underline{n}$ matrix.
We first prove a technical lemma.
\[l1\] Let $$\label{14.0}
A(x,y)=\begin{bmatrix}
a_{11}(x,y) &\dots & a_{1n}(x,y) \\
\hdotsfor{3} \\
a_{m1}(x,y) & \dots & a_{mn}(x,y)
\end{bmatrix}$$ be a matrix whose entries are linear polynomials in $x$ and $y$, and let the rows of $A(\alpha,\beta)$ be linearly independent for all $(\alpha,\beta)\in k^2$ except for $$(\alpha_1,\beta_1),\
(\alpha_2,\beta_2),\dots,
(\alpha_s,\beta_s).$$ Then $s{\leqslant}m^2$; moreover, $s{\leqslant}3$ if $m=2$.
[*Part 1: $s{\leqslant}m^2$*]{}. Clearly, $m{\leqslant}n$. The rows of $A(\alpha,\beta)$ are linearly dependent if and only if $(\alpha,\beta)\in
k^2$ is a common root of all determinants formed by columns of $A(x,y)$. The determinants are polynomials in $x$ and $y$ of degree at most $m$; they are relatively prime (otherwise, they have infinitely many common roots $(\alpha,\beta)\in k^2$). The inequality $s{\leqslant}m^2$ follows from the following statement: $$\label{14.1}
\parbox{25em}
{If $h_1,\dots,h_t\in k[x,y]$ are
polynomials of degree at most $m$ and their
greatest common divisor $(h_1,\dots,h_t)$ is
1, then they have at most $m^2$ common
roots.}$$ For $m=2$, this statement is a partial case of the Bezout theorem [@gri Sect. 1.3]: if $h_1,h_2\in k[x,y]$ and $(h_1,h_2)=1$, then they have at most $\deg(h_1)\cdot\deg(h_2)$ common roots.
Let $m{\geqslant}3$. Applying induction in $t$, we may assume that $d:=(h_1,\dots,h_{t-1})\ne
1$. If $(\alpha,\beta)$ is a common root of $h_1,\dots,h_{t}$, then $(\alpha,\beta)$ is a root of $h_t$ and also a root of $d$ or a common root of $g_1=h_1/d,\dots,g_{t-1}=h_{t-1}/d$. By the Bezout theorem, the number of common roots of $d$ and $h_t$ is at most $\deg(d)m$. By induction, the number of common roots of $g_1,\dots,g_{t-1}$ is at most $(m-\deg(d))^2$. Hence, the number of common roots of $h_1,\dots,h_{t}$ is at most $\deg(d)m+(m-\deg(d))^2{\leqslant}\deg(d)m+
(m-\deg(d))m=m^2$. This proves .
[*Part 2: $s{\leqslant}3$ if $m=2$*]{}. Let $m=2$; assume to the contrary that $s>3$. We will reduce $A(x,y)$ by elementary transformations over $k$ and by substitutions $$
---------------------------------
$x_{\text{new}}=ax+by+c,$
$y_{\text{new}}=a_1x+b_1y+c_1,$
---------------------------------
a&b\
a\_1&b\_1
0; $$ the obtained matrices $A'(x,y)$ will have the same number $s$, and their entries are linear polynomials too. We suppose that each of the matrices $A'(x,y)$ does not contain a zero column; otherwise we can remove it and take the obtained matrix instead of $A(x,y)$.
Let $n=2$. The rows of $A(\alpha,\beta)$ are linearly independent only if $\det
A(\alpha,\beta)\ne 0$. Under the conditions of the lemma, the rows of $A(\alpha,\beta)$ are linearly independent for almost all $(\alpha,\beta)\in k^2$, and so $\det
A(x,y)$ is a nonzero scalar and the rows of $A(\alpha,\beta)$ are linearly independent for all $(\alpha,\beta)\in k^2$.
Hence, $n{\geqslant}3$. By elementary transformations of rows of $A(x,y)$, we make $a_{11}(x,y)=a_{11}\in \{0,1\}$.
If $a_{21}(x,y)=a_{21}\in k$, we make $(a_{11},a_{21})=(1,0)$ by elementary transformations of rows. The rows of $A(\alpha,\beta)$ are linearly dependent only if $$a_{22}(\alpha,\beta)=
a_{23}(\alpha,\beta)= \dots =
a_{2n}(\alpha,\beta)=0.$$ Since $a_{22}(x,y), a_{23}(x,y), \ldots$ are linear polynomial, $s{\leqslant}1$.
Hence $a_{21}(x,y)\notin k$. We make $a_{21}(x,y)=x$ by the substitution $$x_{\text{new}}= a_{21}(x,y),\quad
y_{\text{new}}=
\begin{cases}
y & \text{if $a_{21}(x,y)\notin k[y]$},\\
x & \text{otherwise}.
\end{cases}$$
If there exist distinct $l,r>1$ such that $$\label{15.0}
\begin{tabular}{l}
$a_{1l}(x,y)=ax+by+c,$ \\
$a_{1r}(x,y)=a_1x+b_1y+c_1,$
\end{tabular}\quad
\begin{vmatrix}
a&b \\
a_1&b_1
\end{vmatrix}\ne 0,$$ then we make $a_{12}(x,y)=x+a$ by elementary transformations of columns except for the first column. The rows of $A(\alpha,\beta)$ are linearly dependent if and only if $(\alpha,\beta)$ is a solution of the system $$\label{15.1}
\begin{vmatrix}
a_{11}(x,y)&a_{1j}(x,y) \\
a_{21}(x,y)&a_{2j}(x,y)
\end{vmatrix}=\begin{vmatrix}
a_{11}&a_{1j}(x,y) \\
x&a_{2j}(x,y)
\end{vmatrix}= 0,
\quad j=2,\dots,m.$$ The first equation has the form $$\label{15.2}
\begin{vmatrix}
a_{11}&x+a \\
x&bx+cy+d
\end{vmatrix}= 0.$$
Let $a_{11}c\ne 0$. We present in the form $y=a_1x^2+b_1x+c_1$, substitute it into the other equations of the system , and obtain a system of polynomial equations in $x$ of degree at most 3. This system has at most three solutions, and so $s{\leqslant}3$.
Let $a_{11}c= 0$. Since is a quadratic equation in x, $x=\alpha_1$ or $x=\alpha_2$ for certain $\alpha_1,\alpha_2\in k$. Substituting $x=\alpha_i$ into the other equations of the system gives a system of linear equations with respect to $y$, which has at most one solution, and so $s{\leqslant}2$.
Hence, does not hold for all $l,r>1$. If there exists $j>1$ such that $a_{1j}(x,y)=bx+a,\ b\ne 0$, then we make $b=1$ and reason as in the previous case. The case $a_{1j}(x,y)=a_j\in k$ for all $j>1$ is trivial. Let us consider the remaining case $a_{1j}(x,y)=ax+by+c,\ b\ne
0$, for a certain $j>1$. We make $$A(x,y)=\begin{bmatrix}
a_{11}&y&0&\dots&0 \\
x&a_{22}(x,y)&a_{23}(x,y)&\dots
&a_{2n}(x,y)
\end{bmatrix}$$ by the substitution $y_{\text{new}}=ax+by+c$ and by elementary transformations of columns starting with the second. If $a_{11}=0$, then the rows of $A(\alpha,0)$ are linearly dependent for all $\alpha\in k$. Hence $a_{11}=1$.
If the system $$a_{2j}(x,y)=0,\quad
j=3,\dots,n,$$ has at most one solution, then $s{\leqslant}1$. So this system is equivalent to one equation of the form $y=ax+b$ or $x=a$. Substituting it into $$\begin{vmatrix}
a_{11}&y\\
x&a_{22}(x,y)
\end{vmatrix}=0,$$ we obtain a quadratic equation with respect to $x$ or $y$. Hence $s{\leqslant}2$, a contradiction.
Let a linear matrix problem of tame type be given by a pair $(\varGamma,\cal M)$ and let $M\in {\cal
M}_{\underline{n}\times\underline{n}}$. We sequentially reduce $M$ to the canonical parametric form. If a block is reduced to a Weyr matrix, we replace its diagonal entries by parameters; but as soon as it becomes clear from the form of subsequent boxes in the process of reduction that a parameter may possess only a finite number of values, we replace it by these values.
The matrix that is obtained after reduction of the first $r$ boxes will be called an $r$-[*matrix*]{}; its partition into strips (which refines the ${\underline{n}\times\underline{n}}$ partition) will be called the $r$-[*partition*]{}, its strips and blocks will be called $r$-[*strips*]{} and $r$-[*blocks*]{}. Two $r$-matrices are [*equivalent*]{} if their reduced boxes coincide.
Let $M$ be an $r$-matrix. Denote by $\bar M$ the matrix obtained from it by replacement of all unreduced free entries with zeros. Since the matrix problem is of tame type, $\bar M$ is canonical for all values of parameters, and it is reduced by simultaneous permutations of horizontal and vertical $r$-strips to the form $$\label{14}
\bar{M}^{\vee}=
N_1(\lambda_1I)\oplus\dots\oplus
N_p(\lambda_pI) \oplus (R_1\otimes
I)\oplus\dots\oplus (R_q\otimes I),$$ where $N_i(\lambda_iI)$ and $R_j\otimes I$ are indecomposable canonical one- and zero-parameter canonical matrices ($R_j\otimes I$ is obtained from $R_j$ by replacement of all its entries $a$ with $aI$).
By the same permutation of $r$-strips, we reduce $M$ to $M^{\vee}$ and break up it into $(p+q)\times (p+q)$ strips conformally to . The obtained strips and blocks will be called the [*big strips*]{} and [*big blocks*]{} of $M^{\vee}$. (In the terminology of [@ser], the $r$-strips of $M$ that are contained in the same big strip are [*linked*]{}.)
Define the [*weight*]{} $$t_M=3^{w(M)}$$ of an $r$-matrix $M$, where ${w(M)}$ is the number of entries in all free boxes $M_i$, $i{\leqslant}r$, with the following property: $M_i$ disposes in the same big strip with a free box $M_L$, $L<i$, containing a parameter (that is, $M_i$ is linked with a box having a parameter and reduces after it). Denote by $s(M)$ the number of free entries in the first unreduced $r$-block of $M$.
We say that an $(r+1)$-canonical matrix $B$ is an [*extension*]{} of an $r$-canonical matrix $M$ and write $B\supset M$ if the boxes $B_{1}, B_{2},\dots, B_r$ coincide with the boxes $M_{1}, M_{2},\dots, M_r$ or are obtained from them by replacement of some of their parameters by scalars.
The proof of Theorem \[t1\] bases on the following lemma.
\[l2\] Let $M$ be an $r$-matrix having unreduced entries. Then the number of its nonequivalent extensions $B\supset M$ taken $t_B/t_M$ times is at most $4^{s(M)}$: $$\label{16}
\sum_{\text{nonequiv.\,}B\supset M} t_B/t_M
{\leqslant}4^{s(M)}.$$
Let $M_{r+1}$ be the first unreduced $r$-block of $M$ and let $M^{\vee}_{xy}$ be the big block containing $M_{r+1}$. The following three cases are possible.
[*Case 1: $x>p$ and $y>p$*]{} (see ). Then the horizontal and the vertical big strips of $M^{\vee}_{xy}$ do not contain parameters, and $t_B=t_M$ for all $B\supset M$.
\(i) Let there exist a nonzero addition to $M_{r+1}$. We make $M_{r+1}=0$, then all $B\supset M$ are equivalent and the inequality takes the form $1{\leqslant}4^{s(M)}$.
\(ii) Let there exist no nonzero addition to $M_{r+1}$ and $M_{r+1}$ is reduced by elementary transformations. Then each $B\supset M$ has $B_{r+1}$ of the form , the number of such $z_1\times
z_2$ matrices $B_{r+1}$ is $\min\{z_1,z_2\}+1$. The inequality takes the form $\min\{z_1,z_2\}+1
{\leqslant}4^{z_1z_2}$.
\(iii) Let there exist no nonzero addition to $M_{r+1}$ and $M_{r+1}$ is reduced by similarity transformations. Then the box $B_{r+1}$ of each $B\supset M$ is a parametric Weyr matrix. The number of parametric $z\times z$ Weyr matrices is bounded by $3^{z-1}$ since the structure of a matrix $W$ of the form is determined by the sequence $(n_2,\dots,n_z)\in\{1,2,3\}^{z-1}$, where $n_l=1$ if the $(l,l)$ entry of $W$ is the first entry of $W_{\alpha_i}$, $n_l=2$ if the $(l,l)$ entry is not the first entry of $W_{\alpha_i}$ but the first entry of $\alpha_iI_{m_ij}$ (see ), and $n_l=3$ if the $(l,l)$ entry is not the first entry of $\alpha_iI_{m_ij}$. Hence, the number of nonequivalent extensions $B$ of $M$ is bounded by $3^{z-1}$. This proves since $t_B=t_M$ and $s(M)=z^2$.
[*Case 2: $x{\leqslant}p<y$ or $y{\leqslant}p<x$*]{}. Then a horizontal or vertical big strip of $M^{\vee}_{xy}$ contains a parameter $\lambda_l,\
l\in\{1,\dots,p\}$.
Let the parameters of $M$ take on values from the domain of parameters. There exists no nonzero addition to $M_{r+1}$ if and only if $$\label{21}
M'=SMS^{-1}$$ implies $M'_{r+1}=M_{r+1}$ for all $r$-matrices $M'$ that are equivalent to $M$ and all $S\in \varGamma_{\underline{n}
\times\underline{n}}$ whose main diagonal with respect to $r$-partition consists of the identity $r$-blocks.[^2]
Let us partition $S$ and $M$ into $r$-blocks: $S=[S_{\alpha\beta}]_{\alpha,\beta=1}^e$ and $M=[M_{\alpha\beta}]_{\alpha,\beta=1}^e$. Since $M_{r+1}$ is an $r$-block, $M_{r+1}=M_{\zeta\eta}$ for certain $\zeta$ and $\eta$. Presenting in the form $M'S=SM$ and equating the $(\zeta
,\eta)$ $r$-blocks, we obtain $$\label{21'}
M_{\zeta 1}'S_{1\eta}+\dots+ M_{\zeta
,\eta-1}'S_{\eta-1,\eta}+ M_{\zeta \eta}' =
M_{\zeta\eta}+ S_{\zeta,
\zeta+1}M_{\zeta+1,\eta}+\dots +S_{\zeta
e}M_{e\eta}$$ since $S$ is upper triangular with identity diagonal $r$-blocks.
The blocks $M_{\zeta 1}',\dots, M_{\zeta
,\eta-1}'$ precede $M_{\zeta \eta}'$ so they have been reduced and $M_{\zeta 1}'=M_{\zeta
1}, \dots,M_{\zeta ,\eta-1}'=M_{\zeta
,\eta-1}$. Moreover, each of them is nonzero only when it is contained in the big block $M_{xx}^{\vee}$ (they are contained in the $x$ big horizontal strip of $M^{\vee}$ since $M_{\zeta \eta}$ is contained in $M_{xy}^{\vee}$, but $M^{\vee}$ is big-block-diagonal, see ). Analogously, each of $M_{\zeta+1,\eta},\dots
,M_{e\eta}$ is nonzero only when it is contained in $M_{yy}^{\vee}$. Hence, each $r$-block $S_{\alpha\beta}$ in may have a nonzero factor only when it is contained in $S_{xy}^{\vee}$. This factor has the form $(a\lambda_l+b)I$, $a,b\in k$, since all reduced free $r$-blocks from $M_{xx}^{\vee}$ and $M_{yy}^{\vee}$ are zero matrices, scalar matrices, and $\lambda_lI$.
Therefore, there exists no nonzero addition to $M_{r+1}$ for $\lambda_l=a\in k$ if and only if the following property holds for each $S\in \varGamma_{\underline{n}\times
\underline{n}}$ whose main diagonal with respect to $r$-partition consists of the identity $r$-blocks: if the transformation given by $S$ preserves all boxes preceding $M_{r+1}$, then $$\label{23}
M_{\zeta 1}S_{1\eta}+\dots+ M_{\zeta
,\eta-1}S_{\eta-1,\eta}-
S_{\zeta, \zeta+1}
M_{\zeta+1,\eta}-\dots
-S_{\zeta e}M_{e\eta}=0.$$ The equality is a linear combination of $r$-blocks from $S_{xy}^{\vee}$; its coefficients are linear polynomials in $\lambda_l$.
The conditions on $r$-blocks of $S_{xy}^{\vee}$ that ensure the preservation of all boxes preceding $M_{r+1}$ can be formulated in the form of a system of linear homogeneous equations with respect to $r$-blocks of $S$ that consists of:
\(a) Linear equations with coefficients from $k$ that give the algebra $\varGamma_{\underline{n}
\times\underline{n}}$ as a vector space. We restrict ourselves to those equations that contain $r$-blocks from $S_{xy}^{\vee}$, then they do not contain $r$-blocks outside $S_{xy}^{\vee}$ (see [@ser p. 87]).
\(b) Linear equations with coefficients from $k$ that ensure the preservation of those free $r$-blocks $M_{\alpha\beta}$ that are contained in the intersection of $M_{xy}^{\vee}$ with the boxes $M_1,\dots,M_L$, where $M_L$ is the free box containing the parameter $\lambda_l$. These equations have the form with the indices $(\alpha,\beta)$ instead of $(\zeta
,\eta)$.
\(c) Linear equations, whose coefficients are linear polynomials in $\lambda_l$, that ensure the preservation of free $r$-blocks $M_{\alpha\beta}$ contained in the intersection of $M_{xy}^{\vee}$ with the boxes $M_{L+1},\dots,M_r$; the number of entries in the boxes $M_{\alpha\beta}$ will be denoted by $h$. They also have the form with $(\alpha,\beta)$ instead of $(\zeta ,\eta)$.
Solving the system (a)$\cup$(b), we choose $r$-blocks $S_1,\dots,S_n$ from $S_{xy}^{\vee}$ such that they are arbitrary and the other $r$-blocks from $S_{xy}^{\vee}$ are their linear combinations. Substituting the solution into the system (c) and the equation , we obtain a system of the form $$\label{25}
\begin{matrix}
\qquad\ \ a_{11}(\lambda_l)S_1+
\dots+a_{1n}(\lambda_l)S_n=0
\\ \hdotsfor{1}\\
a_{m-1,1}(\lambda_l)S_1+
\dots+a_{m-1,n}(\lambda_l)S_n=0
\end{matrix}$$ and, respectively, an equation $$\label{25'}
a_{m1}(\lambda_l)S_1+
\dots+a_{mn}(\lambda_l)S_n=0,$$ where $a_{ij}(\lambda_l)$ are linear polynomials in $\lambda_l$. We take the equations – such that the $m\times n$ matrix $A(\lambda_l)=
[a_{ij}(\lambda_l)]$ has linearly independent rows for almost all values of $\lambda_l$; it is possible by [@ser Sect. 3.3.2] since the matrix problem is of tame type. Then $m{\leqslant}n$.
Let there exist no nonzero addition to $M_{r+1}$ for $\lambda_l=\alpha\in k$. Then the equation follows from the system . Therefore, all determinants formed by columns of the matrix $A(\lambda_l)$ become zero for $\lambda_l=\alpha$. These determinants are polynomials in $\lambda_l$ of degree at most $m$. If all the polynomials are identically equal to 0, then the rows of $A(\lambda_l)$ are linearly dependent for all values of $\lambda_l$ and the problem is of wild type. Therefore, they have at most $m$ common roots, and hence there are at most $m$ values $\alpha\in k$ of $\lambda_l$ for which we cannot make $M_{r+1}=0$.
Let $\lambda_l$ be equal to one of these values. The matrix $M_{r+1}$ is transformed by equivalence transformations since $M_{r+1}$ is not contained in a diagonal big block. Hence each extension $B\supset M$ has $B_{r+1}$ in the form ; the number of nonequivalent extensions $B$ with nonzero $B_{r+1}$ and the same value of $\lambda_l$ is $\min\{z_1,z_2\}$, where $z_1\times z_2$ is the size of $M_{r+1}$; their weight $t_B{\leqslant}t_M/3^{m-1}$ (since $\lambda_l$ no longer is a parameter and $m-1{\leqslant}h$, where $h$ is defined in paragraph (c)).
There is also one (up to equivalence) extension $B\supset M$ with $B_{r+1}=0$ and the parameter $\lambda_l$. Its weight $t_B=t_M\cdot 3^{z_1z_2}$.
We have $$\sum_{\text{nonequiv.\,}B\supset
M} t_B/t_M {\leqslant}3^{z_1z_2}+m\cdot\min\{z_1,z_2\}\cdot
3^{-m+1}{\leqslant}4^{z_1z_2}=4^{s(M)}$$ since $m\cdot 3^{-m+1}{\leqslant}1$ and $3^{z_1z_2}+\min\{z_1,z_2\}{\leqslant}4^{z_1z_2}$ for all natural numbers $m$, $z_1$ and $z_2$. This proves .
[*Case 3: $x{\leqslant}p$ and $y{\leqslant}p$*]{}. Then the horizontal and vertical big strips of $M^{\vee}_{xy}$ contain parameters $\lambda_l$ and $\lambda_r$ from free boxes $M_L$ and $M_R$, respectively. We will assume $L{\leqslant}R$.
Let $l=r$. Then $M_L=M_R$ is a Weyr matrix, $\lambda_l=\lambda_r$ is the parameter of its block , and $x=y$. This case is similar to Case 2, but the matrix $M_{r+1}$ is reduced by similarity transformations since $M_{r+1}$ is contained in the diagonal big block $M^{\vee}_{xx}$. In each extension $B\supset M$, the box $B_{r+1}$ is a Weyr matrix. The number of parametric $z\times z$ Weyr matrices is bounded by $3^{z-1}$ (see Case 1(iii)), so we have $$\sum_{\text{nonequiv.\,}B\supset
M} t_B/t_M {\leqslant}3^{z^2}+m\cdot 3^{z-1}\cdot
3^{-m+1}{\leqslant}4^{z^2}=4^{s(M)}$$ since $m\cdot 3^{-m+1}{\leqslant}1$ and $3^{z^2}+3^{z-1}{\leqslant}4^{z^2}$ for all natural numbers $m$ and $z$.
Let $l\ne r$. Then $x\ne y$; in distinction to Case 2, the system (c) consists of linear equations whose coefficients are linear polynomials in $\lambda_l$ and $\lambda_r$. Correspondingly, the system and the equation take the form $$\label{25a}
\begin{matrix}
\qquad\ \ a_{11}(\lambda_l,\lambda_r)S_1+
\dots+a_{1n}(\lambda_l,\lambda_r)S_n=0
\\ \hdotsfor{1}\\
a_{m-1,1}(\lambda_l,\lambda_r)S_1+
\dots+a_{m-1,n}(\lambda_l,\lambda_r)S_n=0
\end{matrix}$$ and $$\label{25'a}
a_{m1}(\lambda_l,\lambda_r)S_1+
\dots+a_{mn}(\lambda_l,\lambda_r)S_n=0,$$ respectively, where $a_{ij}(\lambda_l,\lambda_r)$ are linear polynomials in $\lambda_l$ and $\lambda_r$.
Let there exist no nonzero addition to $M_{r+1}$ for $(\lambda_l,\lambda_r)
=(\alpha,\beta)\in k^2$. Then the equation follows from the system and hence the matrix $A(\alpha,\beta)$ (see ) has linearly dependent rows. The set of values of $(\lambda_l,\lambda_r)$ for which the rows of $A(\lambda_l,\lambda_r)$ are linearly dependent is finite (otherwise the matrix problem is of wild type, see [@ser Sect. 3.3.1]); assume that this set consists of pairs $
(\alpha_1,\beta_1),\
(\alpha_2,\beta_2),\dots,
(\alpha_s,\beta_s)\in k^2.
$
By analogy with Case 2, there are at most $s\cdot\min\{z_1,z_2\}$ nonequivalent extensions $B\supset M$ with nonzero $B_{r+1}$ of size $z_1\times z_2$, their weight $t_B{\leqslant}t_M/3^{m-1}$ (since $\lambda_l$ and $\lambda_r$ no longer are parameters). There is also one extension $B\supset M$ with $B_{r+1}=0$ and the parameters $\lambda_l$ and $\lambda_r$; its weight $t_B=t_M\cdot 3^{z_1z_2}$. We have $$\sum_{\text{nonequiv.\,}B\supset M} t_B/t_M
{\leqslant}3^{z_1z_2}+s\cdot\min\{z_1,z_2\}\cdot
3^{-m+1}{\leqslant}4^{z_1z_2}=4^{s(M)}$$ since $s\cdot 3^{-m+1}{\leqslant}1$ by Lemma \[l1\] and $3^{z_1z_2}+\min\{z_1,z_2\}{\leqslant}4^{z_1z_2}$ for all natural numbers $m$, $z_1$ and $z_2$. This proves .
Let $M$ be an $r$-matrix of size $\underline{n} \times\underline{n}$. We will write $M\Subset C$ if $C$ is a canonical parametric matrix whose boxes $C_{1},
C_{2},\dots, C_r$ coincide with the boxes $M_{1}, M_{2},\dots, M_r$ or are obtained from them by replacement of some of their parameters by scalars. We may add sequentially the boxes of $C$ to the boxes of $M$ and obtain a sequence of extensions $$\label{26}
M\subset B_1 \subset B_2 \subset
\dots \subset B_{l-1}\subset B_l=C,$$ where $B_i$ is an $(r+i)$-matrix and $l+r$ is the number of boxes of $C$. The length $l$ of this sequence may be changed if we change $C$; the greatest length $l$ will be called the [*dept*]{} of $M$ and will be denoted by $l(M)$.
We prove by induction in $l(M)$ that $$\label{27}
\sum_{C\Supset M} t_C/t_M {\leqslant}4^{\bar{s}(M)},$$ where $\bar{s}(M)$ is the number of unreduced free entries in $M$.
If $l(M)=1$, this inequality follows from Lemma \[l2\]. Let $l(M){\geqslant}2$ and holds for all $r'$-matrices whose dept is less than $l(M)$. Then $$\begin{aligned}
{2}
\sum_{C\Supset M} t_C/t_M
&=\sum_{\text{nonequiv.\,}B\supset M}
\sum_{C\Supset B} t_C/t_B \cdot t_B/t_M &&\\
&= \sum_{\text{nonequiv.\,}B\supset M}
t_B/t_M \sum_{C\Supset B} t_C/t_B &&\\
&{\leqslant}\sum_{\text{nonequiv.\,}B\supset M}
t_B/t_M \cdot 4^{\bar{s}(B)} &&
\quad \text{by the induction
hypothesis}\\
&=4^{\bar{s}(M)-s(M)}
\sum_{\text{nonequiv.\,}B\supset M} t_B/t_M
&&\\
&=4^{\bar{s}(M)-s(M)}\cdot 4^{s(M)} &&
\quad \text{by Lemma \ref{l2}}\\
&=4^{\bar{s}(M)}; &&
\end{aligned}$$ that proves . The substitution of the 0-canonical matrix $0$ for $M$ in gives $$\sum_{C\Supset 0} t_C {\leqslant}4^{s(\underline{n})}.$$ This proves Theorem \[t1\] since the sum is taken over all canonical parametric matrices and $t_C{\geqslant}1$ by the definition of weight.
Now we extend Theorem \[t1\] to matrix problems, in which row- and column-transformations are separated.
Let $\varGamma\subset k^{t\times t}$ and $\Delta\subset k^{l\times l}$ be two basic matrix algebras and let ${\cal N} \subset
k^{t\times l}$ be a vector space such that $$\varGamma{\cal N} \subset {\cal N}\quad
\text{and} \quad {\cal N} \Delta \subset
{\cal N}.$$ By a [*separated matrix problem given by*]{} $(\varGamma, \Delta,{\cal N})$, we mean the canonical form problem for matrices $N\in
{\cal N}_{\underline{m}\times\underline{n}}$ in which the row transformations are given by $\varGamma$ and the column transformations are given by $\Delta $: $$N\longmapsto CNS,\quad C\in
\varGamma_{\underline{m}\times
\underline{m}}^*,\ S\in \Delta
_{\underline{n}\times\underline{n}}^*.$$ Following [@ser Lemma 2.3], we may consider this matrix problem as the linear matrix problem given by the pair $(\varGamma\times \Delta ,\ 0\diagdown {\cal
N})$ (see ), where $0\diagdown
{\cal N}$ denotes the vector space of ${(t+l)\times (t+l)}$ matrices of the form $$\begin{bmatrix}
0 & X\\ 0&0
\end{bmatrix},\qquad X\in{\cal N}.$$ This permits to extend Theorem \[t1\] to separated matrix problems.
\[t1’\] If a separated matrix problem is of tame type, then the number of its canonical parametric matrices of size $\underline{m}\times\underline{n}$ is bounded by $4^{s(\underline{m},
\underline{n})}$, where $s(\underline{m},
\underline{n})$ is the number of free entries in an $\underline{m}\times\underline{n}$ matrix.
Number of modules {#s4}
=================
The problem of classifying modules over finite dimensional algebra $A$ reduces to a linear matrix problem; its canonical matrices determine a full system of nonisomorphic modules over $A$ (see [@ser Sect. 2.5]), which will be called [*canonical*]{}. If $A$ is of tame type, then the set of canonical right modules of a fixed dimension partitions into a finite number of series that are determined by canonical parametric matrices of the form . In this section, we prove the following estimate.
\[t2\] If $A$ is an algebra of tame type and $f(d,A)$ is the number of series of canonical right $A$-modules of dimension at most $d$, then $$\label{3.1}
f(d,A){\leqslant}{\binom {d+r} r}
4^{d^2(\delta_1^2+\dots +\delta_r^2)}{\leqslant}(d+1)^r4^{d^2(\dim A)^2},$$ where $r$ is the number of nonisomorphic indecomposable projective left $A$-modules, and $\delta_1,\dots,\delta_r$ are their dimensions.
Without loss of generality, we will prove Theorem \[t2\] for basic matrix algebras (see ). Indeed, $A$ is isomorphic to the subalgebra $B\subset
{\mathop{\rm End}\nolimits}_k A$ consisting of all linear operators $$\label{3.0}
\hat{a}: x\mapsto ax,\qquad a\in A,$$ on the space $_kA$. There exists a basis of $_kA$ in which the matrices of $B$ form an algebra $\varGamma_{\underline{n}\times
\underline{n}}$, where $\varGamma\subset
k^{t\times t}$ is a basic matrix algebra and $\underline{n}=(n_1,\dots,n_t)\in {\mathbb
N}^t$, see [@ser Theorem 1.1]. By the Morita theorem [@dr_ki], the categories of representations of $\varGamma_{\underline{n}\times
\underline{n}}$ and its basic algebra $\varGamma$ are equivalent, hence $$f(d,A)=f(d,\varGamma_{\underline{n}\times
\underline{n}})= f(d,\varGamma).$$ Furthermore, the replacement of $\varGamma_{\underline{n}\times
\underline{n}}$ with $\varGamma$ preserves the number $r$ of nonisomorphic indecomposable projective left modules and reduces their dimensions.
The algebra $\varGamma$ determines the equivalence relation in the set of indices $T=\{1,\dots,t\}$. Let ${\cal
I}_1,\dots, {\cal I}_r$ be the equivalence classes, put $$\label{3.1'}
e_{\alpha}=\sum_{i\in {\cal
I}_{\alpha}} e_{ii},$$ where $e_{ij}$ are the matrix units of $k^{t\times t}$. Define the matrix $$\label{3.1''}
L=[l_{\alpha\beta}]_{\alpha,\beta=1}^r,
\qquad l_{\alpha\beta}=\dim
e_{\alpha}Re_{\beta},$$ where $R={\mathop{\rm Rad}\nolimits}\varGamma$ is the radical of $\varGamma$ consisting of all its matrices with zero diagonal.
\[l3\] If $\varGamma\in k^{t\times t}$ is a basic matrix algebra of tame type, then $$\label{3.1a}
f(d,\varGamma){\leqslant}\sum_{q_1+\dots+q_r{\leqslant}d}
4^{[q_1,\dots,q_r]L\cdot
([q_1,\dots,q_r]L)^T},$$ where $q_1,\dots,q_r$ are nonnegative integers.
Let us show that implies Theorem \[t2\]. By , $$I=e_1+\dots+e_r$$ is a decomposition of the identity of $\varGamma$ into a sum of minimal orthogonal idempotents, and so $\varGamma e_1,\dots,
\varGamma e_r$ are all nonisomorphic indecomposable projective left modules over $\varGamma$. The number of summands in is equal to the number of solutions of the inequality $$\label{3.1aaa}
x_1+\dots+x_r{\leqslant}d$$ in nonnegative integers; it equals ${\binom
{d+r} r}$ by [@sta Sect. 1.2]. Since $q_{\alpha} {\leqslant}d$, $[q_1,\dots,q_r]L\cdot
([q_1,\dots,q_r]L)^T{\leqslant}d^2[1,\dots,1]L\cdot
([1,\dots,1]L)^T= d^2(\delta_1^2+\dots+
\delta_r^2)$, where $\delta_{\beta} =
[1,\dots,1]\cdot [l_{1\beta},
\dots,l_{r\beta}]^T =
l_{1\beta}+\dots+ l_{r\beta}=
\dim e_{1}Re_{\beta}+\dots+
\dim e_{r}Re_{\beta}=
\dim (e_{1}+\dots+ e_{r})Re_{\beta}=
\dim Re_{\beta}=
\dim \varGamma e_{\beta}-1$. This proves the first inequality in . We have $${\binom {d+r} r}{\leqslant}(d+1)^2$$ since each $x_i$ in possesses at most $d+1$ values $0,1,\dots,d$. We also have $\delta_1^2+\dots +\delta_r^2{\leqslant}(\delta_1+\dots +\delta_r)^2= (\dim
\varGamma e_1+\dots+\dim \varGamma e_r)^2=
(\dim \varGamma (e_1+\dots+ e_r))^2=(\dim
\varGamma)^2{\leqslant}(\dim A)^2$. This proves the second inequality in .
[*Step 1: reduction to a matrix problem.*]{} The reduction to a linear matrix problem given in [@ser] is a light modification of Drozd’s reduction [@dro1] (see also [@dro2] and [@cra]). It bases on the construction, for every right module $M$ over $\varGamma$, an exact sequence $$\begin{gathered}
P\stackrel{\varphi}{\longrightarrow} Q
\stackrel{\psi}{\longrightarrow}
M\longrightarrow 0, \label{3.2}\\
{\mathop{\rm Ker}\nolimits}\varphi\subset {\mathop{\rm Rad}\nolimits}P, \quad
{\mathop{\rm Im}\nolimits}\varphi\subset {\mathop{\rm Rad}\nolimits}Q, \label{3.2a}\end{gathered}$$ where $P$ and $Q$ are projective right modules. The homomorphism $\varphi$ is defined by $P$, $Q$, and $M$ up to transformations $$\label{3.3}
\varphi\longmapsto g\varphi f,
\qquad f\in {\mathop{\rm Aut}\nolimits}_{\varGamma} P,\quad g\in
{\mathop{\rm Aut}\nolimits}_{\varGamma} Q.$$
Let us show briefly (details in [@ser]) that the problem of classifying $\varphi$ up to these transformations reduces to a separated matrix problem given by the triple $(\varGamma,\varGamma,{\mathop{\rm Rad}\nolimits}\varGamma)$.
Decompose $P$ and $Q$ from into direct sums of indecomposable projective modules: $$\label{3.3'}
P=(e_1\varGamma)^{p_1}\oplus\dots\oplus
(e_r\varGamma)^{p_r},\quad
Q=(e_1\varGamma)^{q_1}\oplus\dots\oplus
(e_r\varGamma)^{q_r},$$ where $X^{l}:=X\oplus\dots\oplus X$ ($l$ times) and $e_i$ are defined by . Then the homomorphism $\varphi$ becomes the $q\times p=
(q_1+\dots+q_r)\times (p_1+\dots+p_r)$ matrix $\varphi=
[\varphi_{xy}]_{x=1,}^q{}_{y=1}^p$, which we partition into $r$ horizontal and $r$ vertical strips of sizes $q_1,\dots, q_r$ and $p_1,\dots, p_r$. Denote by $$\alpha=\alpha(x)\quad\text{and}\quad
\beta=\beta(y)$$ the indices of the vertical and the horizontal strips containing $\varphi_{xy}$. Then $\varphi_{xy}: e_{\beta}\varGamma \to
e_{\alpha}\varGamma$ and is determined by $\varphi_{xy} (e_{\beta})=
e_{\alpha}\varphi_{xy} (e_{\beta})\in
e_{\alpha}\varGamma$. Since $\varphi_{xy}$ is a homomorphism and $e_{\beta}$ is an idempotent, $\varphi_{xy}
(e_{\beta})=\varphi_{xy}
(e_{\beta}^2)=\varphi_{xy}
(e_{\beta})e_{\beta}$. Hence, $\varphi_{xy}
(e_{\beta})=e_{\alpha}\varphi_{xy}
(e_{\beta}) e_{\beta}\in e_{\alpha}\varGamma
e_{\beta}$. By , $${\mathop{\rm Im}\nolimits}\varphi\subset {\mathop{\rm Rad}\nolimits}Q
=(e_1R)^{q_1}\oplus\dots\oplus (e_rR)^{q_r},$$ where $R={\mathop{\rm Rad}\nolimits}\varGamma$. We have $\varphi_{xy} (e_{\beta(y)})\in
e_{\alpha(x)}R e_{\beta(y)}$.
If a matrix $a=[a_{ij}]_{i,j=1}^t\in
\varGamma$ belongs to $e_{\alpha}R
e_{\beta}$, then it is determined by its submatrix $\bar{a}=[a_{ij}]_{(i,j)\in {\cal
I}_{\alpha}\times{\cal I}_{\beta}}$ since all entries outside of $\bar{a}$ are zero by . The size of $\bar{a}$ is $h(\alpha)\times h(\beta)$, where $
h(\alpha)$ is the number of elements in ${\cal I}_{\alpha}$. Therefore, the homomorphism $\varphi=
[\varphi_{xy}]_{x=1,}^q{}_{y=1}^p$ is determined by the block matrix $$\label{3.4}
[\overline{\varphi_{xy}
(e_{\beta(y)})}]_{x=1,}^q{}_{y=1}^p$$ of size $$(q_1h(1)+\dots +q_rh(r))\times
(p_1h(1)+\dots +p_rh(r)).$$ Permuting rows and columns of this matrix to order them in accordance with their position in $\varGamma$, we obtain a block matrix ${\Phi}\in R_{\underline{m}\times
\underline{n}}$, where $m_i:=q_{\alpha}$ if $i\in{\cal I}_{\alpha}$ and $n_j:=p_{\beta}$ if $j\in{\cal I}_{\beta}$. In the same way, the automorphisms $f\in {\mathop{\rm Aut}\nolimits}_{\varGamma} P$ and $g\in {\mathop{\rm Aut}\nolimits}_{\varGamma} Q$ are determined by nonsingular matrices from $\varGamma_{\underline{m}\times
\underline{m}}$ and $\varGamma_{\underline{n}\times
\underline{n}}$.
Hence, the problem of classifying modules over $\varGamma$ reduces to the canonical form problem for matrices ${\Phi}\in
R_{\underline{m}\times \underline{n}}$ up to transformations $$\label{3.5}
\Phi\longmapsto F\Phi G,
\qquad F\in \varGamma_{\underline{m}\times
\underline{m}}^{*},\quad G\in
\varGamma_{\underline{n}\times
\underline{n}}^{*}.$$ Let $$\label{3.5aa}
H_1,\dots,H_t$$ be the vertical strips of $\Phi$ with respect to $\underline{m}\times
\underline{n}$ partition. The condition ${\mathop{\rm Ker}\nolimits}\varphi\subset {\mathop{\rm Rad}\nolimits}P$ from means that $$\label{3.5a}
\parbox{25em}
{there are not an equivalence class ${\cal
I}_{\alpha}=\{j_1,\dots,j_{h(\alpha)}\}$ and
a transformation \eqref{3.5} making zero the
last column in each of
$H_{j_1},\dots,H_{j_{h(\alpha)}}$
simultaneously.}$$
[*Step 2: an estimate.*]{} Let the module $M$ in has dimension at most $d$. By , , and the condition ${\mathop{\rm Im}\nolimits}\varphi\subset {\mathop{\rm Rad}\nolimits}Q$ from , $$\label{3.6}
q_1+\dots+ q_r= \dim Q/ {\mathop{\rm Rad}\nolimits}Q
{\leqslant}\dim Q/{\mathop{\rm Im}\nolimits}\varphi =
\dim M{\leqslant}d.$$ Each summand $(e_{\alpha}\varGamma)^{p_{\alpha}}$ in the decomposition of $P$ determines the equivalence class ${\cal I}_{\alpha}=
\{j_1,\dots,j_{h(\alpha)}\}$ and corresponds to the strips $H_{j_1},\dots,H_{j_{h(\alpha)}}$ of $\Phi$ (see ); these strips are reduced by simultaneous elementary transformations and each of them has $p_{\alpha}$ columns.
Let us prove that $$\label{3.7}
p_{\alpha}{\leqslant}[q_1,\dots,q_r]\cdot
[l_{1\alpha},\dots,l_{r\alpha}]^T,$$ where $[l_{1\alpha},\dots,l_{r\alpha}]^T$ is a column of the matrix . Put $$n_{\iota}=[q_1,\dots,q_r]\cdot [\dim
e_1\varGamma
e_{j_{\iota}j_{\iota}},\dots,\dim
e_r\varGamma e_{j_{\iota}j_{\iota}}]^T,
\qquad 1{\leqslant}\iota{\leqslant}h(\alpha),$$ where $e_{jj}$ are matrix units. By , $$[q_1,\dots,q_r]\cdot
[l_{1\alpha},\dots,l_{r\alpha}]^T= n_1+\dots
+n_{h(\alpha)}.$$
Suppose that does not hold, i.e. $$p_{\alpha}{\geqslant}n_1+\dots +n_{h(\alpha)}+1,$$ and show that there is a transformation making zero the $(n_1+\dots
+n_{h(\alpha)}+1)$st column in each of $H_{j_1},\dots,H_{j_{h(\alpha)}}$ simultaneously, to the contrary with . It suffices to show that there is a transformation making zero the $(n_1+\dots +n_{h(\alpha)}+1)$st column in all free blocks from $H_{j_1},\dots,H_{j_{h(\alpha)}}$ since the other blocks are their linear combinations.
The number of rows in free blocks of $H_{j_1}$ is equal to $ n_1$; by elementary transformations of columns, we maximize the rank of the first $n_1$ columns of these blocks, and then make zero the other their columns (by the definition of admissible transformations, the same transformations are produced within the strips $H_{j_2},\dots,H_{j_{h(\alpha)}}$). The number of rows in free blocks of $H_{j_2}$ is $n_2$; by elementary transformations with the $n_1+1,n_1+2,\dots$ columns, we maximize the rank of the $n_1+1,n_1+2,\dots$ $n_1+n_2$ columns of these blocks, and then make zero the $n_1+n_2+1,n_1+n_2+2,\dots$ columns in free blocks of $H_{j_2}$ (the same transformations are produced within the strips $H_{j_1}, H_{j_3} \dots,
H_{j_{h(\alpha)}}$; they do not spoil the made zeros in $H_{j_1}$), and so on. At last, we reduce $H_{j_{h(\alpha)}}$ and obtain $\Phi$ in which the $(n_1+\dots+
n_{h(\alpha)}+1)$st column is zero in all free boxes of $H_{j_1},\dots,H_{j_{h(\alpha)}}$. This proves .
Therefore, each module $M$ of dimension at most $d$ may be given by a sequence , in which $P$ and $Q$ are of the form with $p_i$ and $q_j$ satisfying and . To make $$p_{\alpha}= [q_1,\dots,q_r]\cdot
[l_{1\alpha},\dots,l_{r\alpha}]^T,$$ we add, if necessary, additional summands to the decomposition of $P$ and put $\varphi$ equaling 0 on the new summands. Correspondingly, we omit the first condition in and the condition on the matrix $\Phi$. The number of free entries in $\Phi$ becomes equal to $$[q_1,\dots,q_r]L[p_1,\dots,p_r]^T=
{[q_1,\dots,q_r]L\cdot
([q_1,\dots,q_r]L)^T};$$ this proves in view of and Theorem \[t1’\].
[99]{}
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[^1]: The research was done while this author was visiting the University of Bielefeld and the University of Utah supported by Sonderforschungsbereich 343 and NSF grant DMS-0070503.
[^2]: In [@ser Theorem 1.4(b)], the condition “but $M_q'\ne M_q$” must be replaced with “and $M_q'=0$”.
|
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abstract: 'Hawkes processes are a class of simple point processes that are self-exciting and have clustering effect, with wide applications in finance, social networks and many other fields. This paper considers a self-exciting Hawkes process where the baseline intensity is time-dependent, the exciting function is a general function and the jump sizes of the intensity process are independent and identically distributed non-negative random variables. This Hawkes model is non-Markovian in general. We obtain closed-form formulas for the Laplace transform, moments and the distribution of the Hawkes process. To illustrate the applications of our results, we use the Hawkes process to model the clustered arrival of trades in a dark pool and analyze various performance metrics including time-to-first-fill, time-to-complete-fill and the expected fill rate of a resting dark order.'
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**Transform Analysis for Hawkes Processes with Applications in Dark Pool Trading**
[Xuefeng Gao]{}[^1], [Xiang Zhou]{}[^2], Lingjiong Zhu[^3]
Introduction
============
Consider a positive sequence of event arrival times $\tau_{1}<\tau_{2}<\cdots$, that are defined on a complete probability space $(\Omega,\mathcal{F},\mathbb{P})$ with right-continuous and complete information filtration $(\mathcal{F}_{t})_{t\geq 0}$. We define a counting process $N$ and an associated point process $L$ as $$N_{t}=\sum_{n=1}^{\infty}1_{\tau_{n}\leq t} \quad \text{and} \quad L_{t}=\sum_{n=1}^{\infty}\ell_{n}\cdot 1_{\tau_{n}\leq t},$$ where $\{ \ell_{n}: n \ge 1 \}$ is a sequence of independent and identically distributed (i.i.d.) non-negative random variables, and $\ell_{n}$ is $\mathcal{F}_{\tau_{n}}$-measurable for each $n\in\mathbb{N}$.
We consider $\{N_{t}: t \ge 0\}$ to be a Hawkes process with random jump sizes in the intensity, that is a simple point process $N$ with a stochastic intensity given by $$\lambda_{t}=\mu(t)+\int_{0}^{t-}h(t-s)dL_{s} = \mu(t) + \sum_{0<\tau_i<t} h(t-\tau_i) \cdot \ell_i , \label{eq:dynamics}$$ where $\mu(\cdot)\geq 0$ is the time-dependent baseline intensity and $h(\cdot):\mathbb{R}_{\geq 0}\rightarrow\mathbb{R}_{\geq 0}$ is the exciting function encoding the influence of past events on the intensity and we always assume that $\Vert h\Vert_{L^{1}}=\int_{0}^{\infty}h(t)dt<\infty$ and $h$ is locally bounded. In the theory of point processes [@Daley], the random jump sizes $\ell_{i}$ are sometimes referred to as the random marks associated with the point process, and the point process with intensity is a marked Hawkes process.
Two special cases of this Hawkes model have been well studied in the literature. First, when $\ell_i \equiv 1$ for each $i$, the counting process $N$ is the classical linear Hawkes process introduced by A.G. Hawkes in 1971 [@Hawkes; @Hawkes71II]. Hawkes process exhibits both self–exciting (i.e., the occurrence of an event increases the probabilities of future events) and clustering properties. It generalizes the standard Poisson process. Hence Hawkes process is very appealing in point process modeling and it has wide applications in finance. This includes modeling of clustering behavior in stock trade arrivals, default clustering in portfolio credit risk and financial contagion, high-frequency stock prices, etc. See, e.g., @ZhuThesis [@Bacry2015; @Jaisson] and references therein for details.
Second, when the exciting function $h$ is exponential, i.e., $h(t)= \delta e^{-\kappa t}$ for $t \ge 0$, where $\delta, \kappa >0$, @Errais studied the transforms and distributions of this Hawkes process with i.i.d. jumps $\{\ell_{i}\}$ and a special time-dependent baseline intensity in the form of $\mu(t) = \mu+e^{-\kappa t} (\lambda_0 -\mu)$. In this case, the two-dimensional process $(\lambda, N)$ is Markovian. @Errais used this Markovian Hawkes process to model the clustering of corporate defaults, where the random jump times $\tau_i$ represent default times, and the intensity jump magnitudes $\ell_i$ represent the random losses at default. In particular, the intensity model captures the empirical feature that the larger the financial loss of a defaulted firm, the larger the impact of such a event on the other firms, and the bigger the increase of the default intensity at an event. Relying on the Dynkin formula, the authors of @Errais characterized the Fourier transform and the distribution of the Hawkes process using ODEs, and they apply these results in a range of applications in portfolio credit risk, including the valuation, hedging and calibration of portfolio credit derivatives.
This paper considers a Hawkes process with intensity in where the exciting function $h$ is a general function, the baseline intensity is time-dependent, and the random jump sizes $\{ \ell_{n}: n \ge 1 \}$ are i.i.d. nonnegative random variables. We pursue this extended Hawkes model for two reasons: first, we would like to extend the transform analysis of Markovian Hawkes processes in @Errais to the general setting which allows a general time–dependent baseline intensity to account for non–stationarity such as intraday seasonalities in trading activities and non-exponential exciting functions to account for possibly non-Markovian dynamics; second, our motivating application in dark pool trading, which will be illustrated later, naturally fits this general Hawkes model.
In our setting, the Hawkes process can be non-Markovian as a result of the general exciting function $h(\cdot)$. Relying on the immigration-birth representation of linear Hawkes processes given in @HawkesII, and in particular @Karabash for marked linear Hawkes processes, we obtain closed-form formulas for the Laplace transform, moments and the distribution of the Hawkes process $(N, L)$ via integral equations. In the special case of an exponential exciting function, we recover the results obtained in @Errais.
The closed-form formulas of transforms and the probability distribution of Hawkes processes generate computational tractability, and they provide insights into the behavior of Hawkes processes. They could be useful in applications in finance and other fields where event occurrences exhibit self-exciting and clustering. In this paper, we apply our theoretical results to analyze the performance of dark pools.
Dark pools are automated trading facilities which do not display bid and ask quotes to the public, hence they can be used to reduce the market impact of trading big orders. There are around 40 active dark pools in the U.S. for equity trading. Dark pools now account for about 15% of the trading volume in the U.S. equity market and about $7\%$ in Europe. See, e.g., @Mittal2008 [@ZhuHX2014] for an overview. We focus on a typical “midpoint" dark pool using a continuous matching mechanism, where participants submit buy or sell orders with specified quantities for a particular security. Trades can occur at any time if there is liquidity on both sides of the market, and the matching price is the midpoint of the best bid and offer on transparent exchanges. If an investor rests an order in a pool for some time and the order is not completely filled, then the remaining quantity may be cancelled and submitted to a different dark pool or an exchange to seek liquidity.
Several theoretical and empirical studies have suggested that the liquidity in dark pool is clustered, i.e., “liquidity begets liquidity". This means that once a trade has occurred in a dark pool, the probability of observing another one increases. See, e.g., @Buti2011, Chapter 3 in @Lehalle2013 and @Markov2013 for details. Various market events can lead to trade clustering in dark pools. For example, an institutional investor who trades and gets a fill from a particular pool can re-route his orders from another venue back to this pool. In addition, high frequency traders in the market who are fishing in the dark pool may also notice the existence of a big order from a trade occurrence and they may also come to trade in this pool [@Mittal2008]. The clustering of liquidity suggests that strategic traders form liquidity expectations from either their own trades or post-trade information even in the absence of pre-trade market transparency, and this allows them to design liquidity seeking algorithms that exploit the clustered arrivals of liquidity to maximize the fill rate of their orders. It also suggests that in fragmented markets, orders can migrate quickly from one venue to another. A natural model to capture the clustering behavior of trade arrivals in dark pools is the Hawkes process. Indeed, the classical Hawkes process with $\ell_n \equiv 1$ has been widely used to model clustering of trade arrivals on transparent exchanges in the literature. See, e.g., @Bowsher2007 [@BacryMuzy2014; @Cartea2014; @AJ; @Bacry2015] and references therein.
We consider an investor who rests a large midpoint peg (buy) order in a given dark venue, where the execution price of the order floats with the market at the mid-quote derived from transparent exchanges. As in previous studies [@Afeche2014; @Kratz2015], we consider a time-priority rule where orders from counterparties are matched on a first-come-first-served basis[^4]. We model the execution process of the investor’s resting midpoint order by a Hawkes process $(N, L)$ where $\{\tau_i\}$ represent the arrival times of the consolidated trades (eligible-to-match sell orders) from other players in the pool and the random variables $\{\ell_i\}$ represent the sizes of arriving trades which may not be a constant. Empirically, it has been observed that the distribution of resting liquidity in dark pools has fatter tails than exponential distributions. See, e.g., @Ganchev2010. This implies that the larger the size of a trade, the more likely it is that there is more quantity remaining in the pool. Hence, liquidity seekers or high frequency traders may be attracted to put more dark orders to the pool after a trade’s occurrence, leading to a bigger increase of the trading intensity at a trade’s occurrence. Such a feature of positive liquidity feedback could be captured by the self-exciting intensity model . In the special case when $h \equiv 0,$ the self-exciting behavior disappears and the point process $L$ modeling the cumulative arriving volume of dark trades reduces to a compound Poisson process. For tractability purposes, in this paper we do not consider other order attributes such as limit price or minimum execution size which can be attached to a midpoint order as anti-gaming and risk management tools.
Using the transform formulas we obtain for the Hawkes model $(N, L)$, we can efficiently compute performance quantities including time-to-first-fill, time-to-complete-fill, and the expected fill rate in a given time window for a midpoint peg order placed at an empty dark pool. We also analyze the probability of obtaining another fill and the expected fill size conditioned on there is an initial fill of the midpoint order, to understand liquidity expectations after an occurrence of a trade. Furthermore, we extend our analysis to study non-empty dark pools. The performance quantities we compute represent major performance characteristics of dark pools around liquidity [@Mittal2007; @Afeche2014]. They could help give investors a guide to maximize fills and liquidity opportunities from dark pools, and indicate whether and where to trade in a fragmented financial market with multiple dark pools. Hence, such performance quantities are important for smart order routing and allocation of liquidity among different pools to reduce market impact and execution costs in portfolio trading. See, e.g., @Mittal2007 [@Ganchev2010; @Laruelle2011] for detailed discussions.
**Related literature.** Two streams of research that are closely related to our work are Hawkes processes and dark pools. We now explain the difference between our study and the existing literature in these two areas.
*Hawkes processes.* The majority of the works on Hawkes processes in the literature assume a constant baseline intensity $\mu(\cdot)\equiv\mu$. The case when the baseline intensity and/or the exciting function are time-dependent is much less studied. In a recent work, @Euch2016 obtained the characteristic function of a multivariate Hawkes process $N$ with a time-dependent baseline intensity. They did not consider random jump sizes in the intensity. @RSS studied the properties of a locally stationary Hawkes process with both the baseline intensity and exciting function being time-dependent. See also @Toke for the estimations of Hawkes processes with time-dependent baseline intensities and @KL with time-dependent exciting function and zero baseline intensity for various applications. Both @Toke and @RSS also used constant jump sizes, while @KL considered random jump sizes.
Several papers have considered the Hawkes process where the intensity process has random jump sizes as our paper. Almost all of them remain in the Markovian framework. In @Dassios2011, the authors studied a dynamic contagion process by combining the Markovian Hawkes model with i.i.d. intensity jump sizes with externally-excited jumps. They characterized distributional properties of this new process. @Errais and @Zhang2015 studied generalized Markovian Hawkes processes, or affine point processes, where the intensity is an affine function of an affine jump-diffusion. These models belong to the class of affine processes studied in @Duffie. In all these works, the (generalized) Hawkes models are still Markovian. One work that deviates from the Markovian framework, with time-dependent baseline intensity and random jump sizes, similar as this paper, is @Lee2016, where the jump size of the intensity is modulated by a stochastic process described by a stochastic differential equation. They proposed new simulation and model fitting algorithms for the Hawkes model, but they did not obtain distributional properties. The special case $\mu(t)\equiv\mu$ of our model also belongs to the class of the Hawkes process with random marks, see. e.g. @Bremaud2002 who studied the power spectrum, and @Karabash who studied the limit theorems and we refer to Section 2.1.1 of @Bacry2015 for more references.
*Dark pools.* In the dark pool literature, our work is closely related to studies including @Markov2013 and @Afeche2014. The paper @Markov2013 from the industry explicitly modeled the clustering of trade arrivals in a dark pool using the classical Hawkes process with $\ell_n \equiv 1$. They discussed estimation of this classical Hawkes model using exponential exciting functions. @Afeche2014 used a double–sided queueing model to study the operational characteristics of dark pools. They considered Poisson order arrivals and obtained closed-form results for system-level and order-level performances such as fill rates and system times. Our work focuses on the order-level performance, i.e., the experience of a single resting midpoint order placed at a dark venue. We consider more general Hawkes arrival process to capture the clustering behavior of order arrivals. Incorporating Hawkes processes to study system-level performance of dark pools is left for future work. Our work also complements other studies on dark pools, see, e.g. @Ganchev2010 [@Laruelle2011; @Almgren] for order routing algorithms among multiple pools, @Klock2011 [@gatheral2013; @Kratz2014; @Kratz2015] for optimal portfolio trading strategies and price manipulation issues in the presence of a dark pool and a lit exchange, @Hendershott2000 for the conditions under which investors should use a dark pool versus a traditional trading venue, and @Buti2011 [@ZhuHX2014; @Iyer2015] for effects of dark pool trading on the market quality and welfare analysis.
**Organization of this paper.** The rest of the paper is organized as follows. In Section \[sec:2\], we state the main result on the joint Laplace transform of the Hawkes model $(N_T, L_T)$ for a fixed $T>0$. Relying on this result, we obtain explicit formulas for the first two moments of $N_{T}$ and $L_{T}$. We also compute analytically the probability mass function of $N_{T}$ and also that of $L_{T}$ when the jump sizes $\{\ell_i\}$ are lattice distributed. In Section \[sec:application\], we apply the main results to analyze performance problems arising from trading in dark pools. Section \[sec:conclusion\] concludes. Some technical proofs are collected in the Appendix.
Main results {#sec:2}
============
In this section we present the main results. Throughout this section, we use $\mathbb{C}$ to denote the set of complex numbers, $\mathcal{R}(\theta)$ to denote the real part of a complex number $\theta \in \mathbb{C}$, and $|\theta|$ to denote its modulus.
The key mathematical result is the following joint Laplace transform of the Hawkes process $(N_T, L_T)$ for fixed $T>0.$
\[MainThm\] For any $\theta_{1},\theta_{2}\in\mathbb{C}$ with $\mathcal{R}(\theta_{1})\geq 0$, $\mathcal{R}(\theta_{2})\geq 0$, $$\label{eq:laplace}
\mathbb{E}[e^{-\theta_{1}N_{T}-\theta_{2}L_{T}}]
=e^{\int_{0}^{T}\mu(T-s)(F(s)-1)ds},$$ where the function $F$ is the unique solution to the integral equation $$\label{eq:F}
F(t)=e^{-\theta_{1}}\mathbb{E}\left[e^{-\theta_{2}\ell_{1}+\int_{0}^{t}\ell_{1}h(s)(F(t-s)-1)ds}\right],$$ with $|F(t)| \le 1$ for $t \in [0, T]$.
The Equation is a Hammerstein–type Volterra integral equation, and it can be quickly solved numerically using, for example, piecewise polynomial collocation methods. See e.g. Chapter 2 in @Brunner2004 for further details.
We show in this remark that we recover the transform of Hawkes processes in @Errais for an exponential exciting function. Note when $h(x)= \delta e^{-\kappa x}$ with $\delta, \kappa>0$, @Errais derived that (Proposition 2.2 in their paper) with a baseline intensity $\mu(t) = \mu + e^{- \kappa t} (\lambda_{0}-\mu)$ where $\lambda_0 \ge \mu >0$, $$\mathbb{E}[e^{-\theta_{1}N_{T}-\theta_{2}L_{T}}]
=\exp \left( B(T) + \lambda_0 \cdot A(T) \right),$$ where the functions $A(\cdot)$ and $B(\cdot)$ satisfy the ODEs $$\begin{aligned}
A'(t) &= - \kappa A(t) -1 + f(\delta A(t) - \theta_2) e^{-\theta_1},\label{AtEqn}\\
B'(t) &= \kappa \mu A(t), \label{eq:Bt}\end{aligned}$$ with $A(0)=B(0)=0$, and $f$ is defined by $f(\omega) := \mathbb{E} [e^{\omega \cdot \ell_1}]$ for $\omega \in \mathbb{C}$. Thus using we obtain $$\label{eq:Gie}
\mathbb{E}[e^{-\theta_{1}N_{T}-\theta_{2}L_{T}}]
=\exp \left(\mu \int_0^T \kappa A(t)dt + \lambda_0 \cdot A(T) \right).$$ We prove that our result is consistent with the result from @Errais. To see this, we first note that when $h(x)= \delta e^{-\kappa x}$, we obtain from Theorem \[MainThm\] that the function $F$ is given by $$\begin{aligned}
F(t)&=e^{-\theta_{1}}\mathbb{E}\left[e^{-\theta_{2}\ell_{1}+\int_{0}^{t}\ell_{1} \delta e^{-\kappa (t-s)} (F(s)-1)ds}\right] \nonumber\\
&=e^{-\theta_{1}} f\left( -\theta_{2} + \int_{0}^{t} \delta e^{-\kappa (t-s)} (F(s)-1)ds\right). \label{eq:F1}\end{aligned}$$ In view of , and the expression of the baseline intensity $\mu(t)$, it suffices to show that $$\label{eq-siam}
\mu \int_0^T \kappa A(t)dt + \lambda_0 \cdot A(T) = \int_{0}^{T}(\mu + e^{- \kappa (T-s)} (\lambda_{0}-\mu))(F(s)-1)ds.$$ To this end, we first prove that for $t \in [0, T],$ $$F(t) -1
= \kappa A(t) + A'(t)\label{FA}
= -1 + f(\delta A(t) - \theta_2) e^{-\theta_1},$$ where the second equality above is due to . That is, we need to show for $t \in [0, T],$ $$\label{eq:F2}
F(t) = f(\delta A(t) - \theta_2) e^{-\theta_1}.$$ In view of and the fact that $A(0)=0$, this equation clearly holds when $t=0$. Let us verify that is indeed the unique solution for . We write for $t \in [0, T],$ $$\label{eq:xt}
x(t) := \int_{0}^{t} e^{-\kappa (t-s)} (F(s)-1)ds.$$ Taking derivative at both sides, we find that $$x'(t)= -\kappa x(t) + F(t) -1.$$ Now Equation implies that $$\label{eq:F-inter}
F(t)= e^{-\theta_{1}} f( -\theta_{2} + \delta x(t) ).$$ Hence $x$ solves the ODE $$x'(t)= -\kappa x(t) -1 + e^{-\theta_{1}} f( -\theta_{2} + \delta x(t) ).$$ As one can see from , this is exactly the ODE that $A$ satisfies. Since $A(0)=x(0)=0$, we obtain $$\begin{aligned}
\label{eq:XA}
A(t) \equiv x(t), \quad \text{for $t \in [0,T]$.}\end{aligned}$$ Then readily follows from . In addition, we infer from and that $$\label{part1}
(\lambda_{0}-\mu) \cdot A(T) = (\lambda_{0}-\mu) \cdot \int_0^T e^{- \kappa (T-s)}(F(s)-1)ds.$$ Furthermore, Equation implies that $$\label{part2}
\mu \int_0^T \kappa A(t)dt + \mu \cdot A(T) = \int_{0}^{T}\mu(F(s)-1)ds.$$ On combining and , we obtain . Therefore, we have recovered the result in @Errais.
@HawkesII first discovered that a linear Hawkes process has an immigration-birth representation. The immigrants (roots) arrive according to a time-inhomogeneous Poisson process $\bar{N}$ with intensity $\mu(t)\geq 0$ at time $t$. Each immigrant generates children according to a Galton-Watson tree, that is, the number of children of each immigrant follows a Poisson distribution with parameter $\Vert h\Vert_{L^{1}}$, and each child will independently generate children according to the same Poisson distribution, and so on and so forth. In addition, when the children are born, they are born at independent random times with the probability density function $\frac{h(t)}{\Vert h\Vert_{L^{1}}}$ for being born at time $t$. In other words, they are born according to an inhomogeneous Poisson process with intensity $h(t)$. Consider an immigrant arrive at time $0$. Note that in the later computations, we will consider an immigrant that arrives at a positive time $t$, but the computation is the same as shifting the time backwards by $t$ to consider the immigrant that arrives at time $0$. Let $K$ be the number of children of this immigrant and $\ell_{1}$ be the associated jump size. Let $S_{t, j}$ be the number of the total descendants of the $j$-th child of the immigrant that were born before time $t$, including the $j$-th child, and $L_{t, j}$ be the sum of all of jump sizes associated with all the descendants of the $j$-th child, including $j$-th child, where $1\leq j\leq K$. Let $S_{t}$ be the total number of all the descendants of this immigrant that arrives at time $0$ including the immigrant, and let $L_{t}^{S}$ be the associated sum of jump sizes, that is $S_{t}=1+\sum_{j=1}^{K}S_{t, j}$ and $L_{t}^{S}=\ell_{1}+\sum_{j=1}^{K}L_{t, j}$. Then, we have $$\begin{aligned}
F(t)&:=\mathbb{E}\left[e^{-\theta_{1}S_{t}-\theta_{2}L_{t}^{S}}\right] \label{eq:F-def}
\\
&=\sum_{k=0}^{\infty}\mathbb{E}\left[e^{-\theta_{1}S_{t}-\theta_{2}L_{t}^{S}}|K=k\right]\mathbb{P}(K=k)
\nonumber
\\
&=\mathbb{E}\left[e^{-\theta_{1}-\theta_{2}\ell_{1}}
\sum_{k=0}^{\infty}\prod_{i=1}^{k}\mathbb{E}\left[e^{-\theta_{1}S_{t, i}-\theta_{2}L_{t, i}}\right]\mathbb{P}(K=k|\ell_{1})\right]
\nonumber
\\
&=\mathbb{E}\left[e^{-\theta_{1}-\theta_{2}\ell_{1}}
\sum_{k=0}^{\infty}\left(\mathbb{E}\left[e^{-\theta_{1}S_{t, 1}-\theta_{2}L_{t,1 } }\right]\right)^{k}\mathbb{P}(K=k|\ell_{1})\right]
\nonumber
\\
&=\mathbb{E}\left[e^{-\theta_{1}-\theta_{2}\ell_{1}}
\sum_{k=0}^{\infty}\left(\int_{0}^{t}\frac{h(s)}{\Vert h\Vert_{L^{1}}}F(t-s)ds\right)^{k}
e^{-\ell_{1}\Vert h\Vert_{L^{1}}}\frac{\ell_{1}^{k}\Vert h\Vert_{L^{1}}^{k}}{k!}\right]
\nonumber
\\
&=e^{-\theta_{1}}\mathbb{E}\left[e^{-\theta_{2}\ell_{1}+\int_{0}^{t}\ell_{1}h(s)(F(t-s)-1)ds}\right],
\nonumber\end{aligned}$$ where the third and fourth equalities above use the tower property of the conditional expectation, and the fact that $(S_{t, i},L_{t, i})$ are i.i.d. independent of $K$, and the fifth equality above uses the fact that conditional on $\ell_{1}$, $K$ is Poisson distributed with parameter $\ell_{1}\Vert h\Vert_{L^{1}}$ and conditional on the children being born at time $s$, $e^{-\theta_{1}S_{t, 1}-\theta_{2}L_{t , 1}}$ has the expectation $F(t-s)$ by the definition of $F(\cdot)$, and the timing of the children being born at time $s$ has the probability density function $\frac{h(s)}{\Vert h\Vert_{L^{1}}}$.
Next, by the immigration-birth representation, we have $N_{T}=\sum_{i:0<\bar{\tau}_{i} \le T} S_{T-\bar \tau_{i}}(i)$, and $L_{T}=\sum_{i:0<\bar{\tau}_{i} \le T}L_{T- \bar \tau_{i}}^{S}(i)$, where $\bar{\tau}_{i}$ are the arrival times of the time-inhomogeneous Poisson process $\bar{N}$ and $S_{T-t}(i)$ are i.i.d. copies of $S_{T-t}$, and $L_{T-t}^{S}(i)$ are i.i.d. copies of $L_{T-t}^{S}$, where $S_{T-t}$ and $L_{T-t}^{S}$ are defined as before. Thus, we have $$\mathbb{E}\left[e^{-\theta_{1}N_{T}-\theta_{2}L_{T}}\right]
=\mathbb{E}\left[e^{\sum_{i:0<\bar{\tau}_{i} \le T}(-\theta_{1}S_{T-\bar \tau_{i}}(i)-\theta_{2}L_{T- \bar \tau_{i}}^{S}(i))}\right].$$ Hence, we have $$\begin{aligned}
\mathbb{E}\left[e^{-\theta_{1}N_{T}-\theta_{2}L_{T}}\right]
&=\mathbb{E}\left[\prod_{i:0<\bar{\tau}_{i}\leq T}e^{-\theta_{1}S_{T-\bar{\tau}_{i}}(i)-\theta_{2}L_{T-\bar{\tau}_{i}}^{S}(i)}
\right]
\\
&=\mathbb{E}\left[\mathbb{E}\left[\prod_{i:0<\bar{\tau}_{i}\leq T}e^{-\theta_{1}S_{T-\bar{\tau}_{i}}(i)-\theta_{2}L_{T-\bar{\tau}_{i}}^{S}(i)}\bigg|\mathcal{F}_{T}^{\bar{N}}\right]\right],\end{aligned}$$ where we used the tower property and $\mathcal{F}_{T}^{\bar{N}}$ is the natural filtration generated by $\bar{N}$ process on the time interval $[0,T]$. Conditional on $\mathcal{F}_{T}^{\bar{N}}$, $(S_{T-\bar{\tau}_{i}},L_{T-\bar{\tau}_{i}})$ are independent. Thus, we have $$\begin{aligned}
\mathbb{E}\left[e^{-\theta_{1}N_{T}-\theta_{2}L_{T}}\right]
&=\mathbb{E}\left[\prod_{i:0<\bar{\tau}_{i}\leq T}
\mathbb{E}\left[e^{-\theta_{1}S_{T-\bar{\tau}_{i}}(i)-\theta_{2}L_{T-\bar{\tau}_{i}}^{S}(i)}
\bigg|\mathcal{F}_{T}^{\bar{N}}\right]\right]
\\
&=\mathbb{E}\left[e^{\sum_{i:0<\bar{\tau}_{i} \le T}
\log\mathbb{E}\left[e^{-\theta_{1}S_{T-\bar{\tau}_{i}}(i)-\theta_{2}L_{T-\bar{\tau}_{i}}^{S}(i)}\big|\mathcal{F}_{T}^{\bar{N}}\right]}\right]
\\
&=\mathbb{E}\left[e^{\sum_{i:0<\bar{\tau}_{i} \le T}
\log\mathbb{E}\left[e^{-\theta_{1}S_{T-\bar{\tau}_{i}}(1)-\theta_{2}L_{T-\bar{\tau}_{i}}^{S}(1)}\big|\mathcal{F}_{T}^{\bar{N}}\right]}\right]
\nonumber
\\
&=e^{\int_{0}^{T}\mu(s)\left(\mathbb{E}\left[e^{-\theta_{1}S_{T-s}(1)-\theta_{2}L_{T-s}^{S}(1)}\right]-1\right)ds}
\nonumber
\\
&=e^{\int_{0}^{T}\mu(s)(F(T-s)-1)ds},
\nonumber\end{aligned}$$ where the second last equality follows from the fact that for any deterministic and bounded function $g(\cdot)$, and the inhomogeneous Poisson process $\bar{N}$ with intensity $\mu(\cdot)$, we have $\mathbb{E}[e^{\int_{0}^{T}g(s)d\bar{N}_{s}}]=e^{\int_{0}^{T}\mu(s)(e^{g(s)}-1)ds}$, and the last equality follows from the definition of $F$ in .
Finally, we show that $F$ is the unique solution to the integral equation satisfying $|F(t)| \le 1$ for all $t \in [0, T].$ The fact that $|F(t)| \le 1$ is clear from the definition . To show uniqueness, let $F_{1}(t)$ and $F_{2}(t)$ be two solutions of so that $|F_{1}(t)|,|F_{2}(t)|\leq 1$ for $t \in [0,T].$ Then we have $$\begin{aligned}
|F_{1}(t)-F_{2}(t)|
&\leq\mathbb{E}\left[\left|e^{-\theta_{1}}e^{-\theta_{2}\ell_{1}+\int_{0}^{t}\ell_{1}h(s)(F_{1}(t-s)-1)ds}
-e^{-\theta_{1}}e^{-\theta_{2}\ell_{1}+\int_{0}^{t}\ell_{1}h(s)(F_{2}(t-s)-1)ds}\right|\right]
\\
&=\mathbb{E}\left[\left|e^{-\theta_{1}}e^{-\theta_{2}\ell_{1}}\right| \cdot
\left|e^{\int_{0}^{t}\ell_{1}h(s)(F_{1}(t-s)-1)ds}
-e^{\int_{0}^{t}\ell_{1}h(s)(F_{2}(t-s)-1)ds}\right|\right]
\\
&\leq\mathbb{E}\left[\left|e^{\int_{0}^{t}\ell_{1}h(s)(F_{1}(t-s)-1)ds}
-e^{\int_{0}^{t}\ell_{1}h(s)(F_{2}(t-s)-1)ds}\right|\right].\end{aligned}$$ Let $F_{1}(t)=R_{1}(t)+iI_{1}(t)$ and $F_{2}(t)=R_{2}(t)+iI_{2}(t)$. Then, we have $$\begin{aligned}
\label{FEqn}
&|F_{1}(t)-F_{2}(t)|
\\
&\leq\mathbb{E}\left[\left|e^{\int_{0}^{t}\ell_{1}h(s)(R_{1}(t-s)-1)ds+i\int_{0}^{t}\ell_{1}h(s)I_{1}(t-s)ds}
-e^{\int_{0}^{t}\ell_{1}h(s)(R_{2}(t-s)-1)ds+i\int_{0}^{t}\ell_{1}h(s)I_{1}(t-s)ds}\right|\right]
\nonumber
\\
&\qquad
+\mathbb{E}\left[\left|e^{\int_{0}^{t}\ell_{1}h(s)(R_{2}(t-s)-1)ds+i\int_{0}^{t}\ell_{1}h(s)I_{1}(t-s)ds}
-e^{\int_{0}^{t}\ell_{1}h(s)(R_{2}(t-s)-1)ds+i\int_{0}^{t}\ell_{1}h(s)I_{2}(t-s)ds}\right|\right]
\nonumber
\\
&=\mathbb{E}\left[\left|e^{\int_{0}^{t}\ell_{1}h(s)(R_{1}(t-s)-1)ds}
-e^{\int_{0}^{t}\ell_{1}h(s)(R_{2}(t-s)-1)ds}\right|\right]
\nonumber
\\
&\qquad
+\mathbb{E}\left[e^{\int_{0}^{t}\ell_{1}h(s)(R_{2}(t-s)-1)ds}\left|e^{i\int_{0}^{t}\ell_{1}h(s)I_{1}(t-s)ds}
-e^{i\int_{0}^{t}\ell_{1}h(s)I_{2}(t-s)ds}\right|\right].
\nonumber\end{aligned}$$ Notice that $|R_{1}(t)|\leq |F_{1}(t)|\leq 1$ and $|R_{2}(t)|\leq |F_{2}(t)|\leq 1$, thus, $\int_{0}^{t}\ell_{1}h(s)(R_{j}(t-s)-1)ds\leq 0$ for $j=1,2$. The map $x\mapsto e^{x}$ is Lipschitz with constant $1$ for $x\leq 0$. Thus, for any $0\leq t\leq T$. $$\begin{aligned}
\mathbb{E}\left[\left|e^{\int_{0}^{t}\ell_{1}h(s)(R_{1}(t-s)-1)ds}
-e^{\int_{0}^{t}\ell_{1}h(s)(R_{2}(t-s)-1)ds}\right|\right]
&\leq\mathbb{E}\left[\int_{0}^{t}\ell_{1}h(s)|R_{1}(t-s)-R_{2}(t-s)|ds\right]
\nonumber
\\
&\leq
\Vert h\Vert_{L^{\infty}[0,T]}\mathbb{E}[\ell_{1}]\int_{0}^{t}|R_{1}(s)-R_{2}(s)|ds
\nonumber
\\
&\leq
\Vert h\Vert_{L^{\infty}[0,T]}\mathbb{E}[\ell_{1}]\int_{0}^{t}|F_{1}(s)-F_{2}(s)|ds,
\label{REqn}\end{aligned}$$ where $\Vert h\Vert_{L^{\infty}[0,T]}=\sup_{0\leq s\leq T}h(s)$.
Next, let us notice that for any $x,y\in\mathbb{R}$, $$|e^{ix}-e^{iy}|
\leq|\cos(x)-\cos(y)|+|\sin(x)-\sin(y)|
\leq 2|x-y|.$$ Therefore, $$\begin{aligned}
&\mathbb{E}\left[e^{\int_{0}^{t}\ell_{1}h(s)(R_{2}(t-s)-1)ds}\left|e^{i\int_{0}^{t}\ell_{1}h(s)I_{1}(t-s)ds}
-e^{i\int_{0}^{t}\ell_{1}h(s)I_{2}(t-s)ds}\right|\right]
\nonumber
\\
&\leq
\mathbb{E}\left[\left|e^{i\int_{0}^{t}\ell_{1}h(s)I_{1}(t-s)ds}
-e^{i\int_{0}^{t}\ell_{1}h(s)I_{2}(t-s)ds}\right|\right]
\nonumber
\\
&\leq 2\mathbb{E}\left[\int_{0}^{t}\ell_{1}h(s)|I_{1}(t-s)-I_{2}(t-s)|ds\right]
\nonumber
\\
&\leq
2\Vert h\Vert_{L^{\infty}[0,T]}\mathbb{E}[\ell_{1}]\int_{0}^{t}|I_{1}(s)-I_{2}(s)|ds
\nonumber
\\
&\leq
2\Vert h\Vert_{L^{\infty}[0,T]}\mathbb{E}[\ell_{1}]\int_{0}^{t}|F_{1}(s)-F_{2}(s)|ds.
\label{IEqn}\end{aligned}$$ Hence, by applying and to , we get $$|F_{1}(t)-F_{2}(t)|
\leq
3\Vert h\Vert_{L^{\infty}[0,T]}\mathbb{E}[\ell_{1}]\int_{0}^{t}|F_{1}(s)-F_{2}(s)|ds.$$ By Gronwall’s inequality, we conclude that $F_{1}\equiv F_{2}$ on $[0,T]$. The proof is complete.
By letting $\theta_{1}=0$ or $\theta_{2}=0$ in Theorem \[MainThm\], we get the single Laplace transforms of the counting process $N_{T}$ and the point process $L_{T}$.
\[SingleTransform\] (i) For any $\theta\in\mathbb{C}$ with $\mathcal{R}(\theta)\geq 0$, $$\mathbb{E}\left[e^{-\theta N_{T}}\right]
=e^{\int_{0}^{T}\mu(T-s)(F_{N}(s)-1)ds},$$ where the function $F_{N}$ is the unique solution to the integral equation $$\label{FN}
F_{N}(t)=e^{-\theta}\mathbb{E}\left[e^{\int_{0}^{t}\ell_{1}h(s)(F_{N}(t-s)-1)ds}\right],$$ with $|F_{N}(t)|\leq 1$ for $0\leq t\leq T$.
\(ii) For any $\theta\in\mathbb{C}$ with $\mathcal{R}(\theta)\geq 0$, $$\label{eq:L-lap}
\mathbb{E}\left[e^{-\theta L_{T}}\right]
=e^{\int_{0}^{T}\mu(T-s)(F_{L}(s)-1)ds},$$ where the function $F_{L}$ is the unique solution to the integral equation $$\label{FL}
F_{L}(t)=\mathbb{E}\left[e^{-\theta\ell_{1}+\int_{0}^{t}\ell_{1}h(s)(F_{L}(t-s)-1)ds}\right],$$ with $|F_{L}(t)|\leq 1$ for $0\leq t\leq T$.
The result of the single Laplace transform of $N_{T}$ has been obtained in @Karabash by using the immigration-birth representation as a special case of the linear marked Hawkes process.
The Laplace transforms obtained allow us to explicitly compute the moments of the counting process $N_{T}$ and the point process $L_{T}$. We derive the first and second moments in the following result and present the proof in the Appendix. Higher order moments can be derived similarly.
\[prop:moments\] (i) The first moment of the counting process $N$ is given by $$\label{eq:E-NT}
\mathbb{E}[N_{T}]=\int_{0}^{T}\mu(T-t)\psi_{1}(t)dt,$$ where $\psi_{1}$ is the unique solution to the equation: $$\label{psi1}
\psi_{1}(t)
=1+\int_{0}^{t}\mathbb{E}[\ell_{1}]h(t-s)\psi_{1}(s)ds,
\qquad
0\leq t\leq T.$$
\(ii) The first moment of the process $L$ is given by $$\label{eq:ELT}
\mathbb{E}[L_{T}]=\mathbb{E}[\ell_{1}]\int_{0}^{T}\mu(T-t)\psi_{1}(t)dt.$$
\(iii) The second moment of the counting process $N$ is given by $$\mathbb{E}[N_{T}^{2}]=\int_{0}^{T}\mu(T-t)\psi_{2}(t)dt+\left(\int_{0}^{T}\mu(T-s)\psi_{1}(t)dt\right)^{2},$$ where $\psi_{2}$ is the unique solution to the equation: $$\label{psi2}
\psi_{2}(t)
=(\psi_{1}(t))^{2}
+\int_{0}^{t}\mathbb{E}[\ell_{1}]h(s)\psi_{2}(t-s)ds,
\qquad
0\leq t\leq T.$$
\(iv) The second moment of the process $L$ is given by $$\mathbb{E}[L_{T}^{2}]
=\int_{0}^{T}\mu(T-t)\psi_{3}(t)dt+(\mathbb{E}[\ell_{1}])^{2}\left(\int_{0}^{T}\mu(T-t)\psi_{1}(t)dt\right)^{2},$$ where $\psi_{1}$ is defined in and $\psi_{3}$ is the unique solution to the equation: $$\label{psi3}
\psi_{3}(t)
=\mathbb{E}[\ell_{1}^{2}](\psi_{1}(t))^{2}
+\int_{0}^{t}\mathbb{E}[\ell_{1}]h(s)\psi_{3}(t-s)ds,
\qquad
0\leq t\leq T.$$
It follows from Proposition \[prop:moments\] that the first moments $\mathbb{E}[N_{T}]$ and $\mathbb{E}[L_{T}]$ and also the second moment $\mathbb{E}[N_{T}^{2}]$ depend on the distribution of $\ell_{1}$ only via the mean $\mathbb{E}[\ell_{1}]$, and the second moment $\mathbb{E}[L_{T}^{2}]$ depends on the distribution of $\ell_{1}$ only via the $\mathbb{E}[\ell_{1}]$, $\mathbb{E}[\ell_{1}^{2}]$, the first two moments of $\ell_{1}$.
Using the Laplace transform of $N_{T}$, one can also compute the probability mass function of $N_{T}$ analytically, as shown in the following result. The proof relies on the celebrated Faà di Bruno’s formula, and it is given in the appendix.
\[prop:mass\] We have $\mathbb{P}(N_{T}=0)=e^{-\int_{0}^{T}\mu(T-s)ds}$, and for any $k\geq 1$, $$\begin{aligned}
\mathbb{P}(N_{T}=k)
&=
e^{-\int_{0}^{T}\mu(T-s)ds}\sum\frac{1}{m_{1}!1!^{m_{1}}m_{2}!2!^{m_{2}}\cdots m_{k}!k!^{m_{k}}}
\\
&\qquad\qquad\qquad
\cdot\prod_{j=1}^{k}\left(\int_{0}^{T}\mu(T-s)F_{N,j}(s)ds\right)^{m_{j}},\end{aligned}$$ where the summation is over all $k$-tuples of nonnegative integers $(m_1, \ldots, m_k)$ satisfying the constraint $1\cdot m_{1}+2\cdot m_{2}+3\cdot m_{3}+\cdots+k\cdot m_{k}=k$, and $F_{N,0}(t)=0$, $$F_{N,1}(t)=\mathbb{E}\left[e^{-\int_{0}^{t}\ell_{1}h(s)ds}\right],$$ and for every $j\geq 2$, $$\begin{aligned}
F_{N,j}(t)
&=\sum\frac{j!}{m_{1}!1!^{m_{1}}m_{2}!2!^{m_{2}}\cdots m_{j-1}!(j-1)!^{m_{j-1}}}
\\
&\qquad\qquad
\cdot
\mathbb{E}\left[e^{-\int_{0}^{T}\ell_{1}h(s)ds}\prod_{i=1}^{j-1}\left(\int_{0}^{t}\ell_{1}h(s)F_{N,i}(t-s)ds\right)^{m_{i}}\right],\end{aligned}$$ where the summation is over all $(j-1)$-tuples of nonnegative integers $(m_1, \ldots, m_{j-1})$ satisfying the constraint $1\cdot m_{1}+2\cdot m_{2}+3\cdot m_{3}+\cdots+(j-1)\cdot m_{j-1}=j-1$.
Next, let us discuss the distribution of $L_{T}$. First note that by assuming that $\mathbb{P}(\ell_{1}=0)=0$, we always have $\mathbb{P}(L_{T}=0)=\mathbb{P}(N_{T}=0)=e^{-\int_{0}^{T}\mu(s)ds}$ for any nonnegative jump size distribution of $\ell_1$. Next, we assume that the random jump size $\ell_{1}$ has a lattice distribution and takes discrete values $k\delta$, $k\in\mathbb{N}$ with $\mathbb{P}(\ell_{1}=k\delta)=p_{k}$ where $0\leq p_{k}\leq 1$ and $\sum_{k=1}^{\infty}p_{k}=1$ for some $\delta>0.$ Note that this includes the case for geometrically distributed $\ell_{1}$, Poisson distributed $\ell_{1}$ etc. for fixed $\delta=1$. Under this assumption, $L_{T}$ also takes values $k\delta$, for $k\in\mathbb{N}\cup\{0\}$.
Then we have the following result on the distribution of $L_{T}$ when the jump size $\{\ell_i\}$ is lattice distributed. The proof is deferred to the Appendix.
\[prop:LT-distri\] We have $\mathbb{P}(L_{T}=0)=e^{-\int_{0}^{T}\mu(T-s)ds}$, and for any $k\geq 1$, $$\begin{aligned}
\mathbb{P}(L_{T}=k\delta)
&=
e^{-\int_{0}^{T}\mu(T-s)ds}\sum\frac{1}{m_{1}!1!^{m_{1}}m_{2}!2!^{m_{2}}\cdots m_{k}!k!^{m_{k}}}
\\
&\qquad\qquad\qquad
\cdot\prod_{j=1}^{k}\left(\int_{0}^{T}\mu(T-s)F_{L,j}(s)ds\right)^{m_{j}},\end{aligned}$$ where the summation is over all $k$-tuples of nonnegative integers $(m_1, \ldots, m_k)$ satisfying the constraint $1\cdot m_{1}+2\cdot m_{2}+3\cdot m_{3}+\cdots+k\cdot m_{k}=k$, and $F_{L,0}(t)=0$, $$F_{L,1}(t)=p_{1}e^{-\int_{0}^{t}\delta h(s)ds},$$ and for every $j\geq 2$, $$\begin{aligned}
F_{L,j}(t)&=\sum_{k=0}^{j}\binom{j}{k}k!
e^{-\int_{0}^{t}k\delta h(s)ds}p_{k}
\sum\frac{(j-k)!}{m_{1}!1!^{m_{1}}m_{2}!2|^{m_{2}}\cdots
m_{j-k}!(j-k)!^{m_{j-k}}}
\\
&\qquad\qquad\qquad\cdot
\prod_{i=1}^{j-k}\left(\int_{0}^{t}k\delta h(s)F_{L,i}(t-s)ds\right)^{m_{i}},\end{aligned}$$ where the summation is over all $(j-k)$-tuples of nonnegative integers $(m_1, \ldots, m_{j-k})$ satisfying the constraint $1\cdot m_{1}+2\cdot m_{2}+\cdots+(j-k)\cdot m_{j-k}=j-k$.
Numerical methods to calculate the summands in Faà di Bruno’s formula in Propositions \[prop:mass\] and \[prop:LT-distri\] can be found in, e.g. @Kilmko73. In general, when the jump size $\ell_{1}$ is not lattice distributed, one can still efficiently calculate the distributions of $N_T$ and $L_T$ by numerically inverting the Laplace transforms in Corollary \[SingleTransform\]. See e.g. @AbateWhitt95 for numerical Laplace transform inversion methods.
Applications in Dark Pool Trading {#sec:application}
=================================
In this section, we apply the main results to analyze performance problems arising from trading in dark pools. We use the Hawkes process to model executions of a large midpoint peg order placed at an empty dark pool and compute various performances in Section \[sec:performance\]. Non-empty dark pools are discussed in Sections \[sec:nonempty\]. In computing some performance metrics (e.g. the probability of obtaining another fill conditioned on a past fill) and studying non-empty pools, we will see that it is natural to study a Hawkes process with a time-dependent baseline intensity.
Model description and performance analysis {#sec:performance}
------------------------------------------
Suppose an investor rests a large midpoint buy order of size $x>0$ in a dark pool with a continuous first-come-first-served matching mechanism. This order is “pegged” at the mid-quote of transparent exchanges, i.e., the execution price of the order automatically adjusts as the market moves. Considering a sell order is similar. As liquidity in dark pools could be sparse and there could be a high probability of no volume in pools (see, e.g., @Ganchev2010 [@Markov2013]), we assume in this section that when the investor’s order reaches the dark pool there are no other orders sitting in the pool.
We model the successive executions of this midpoint peg order using a Hawkes process. More specifically, we model the consolidated sell trades from other players in the dark pool as a Hawkes process $(N, L)$ with the intensity , where $N_t$ counts the number of eligible-to-trade sell orders (or trades with the investor’s resting buy order) by time $t$ and the i.i.d. sequence $\{\ell_i: i =1, 2, \ldots\}$ models the volumes of arriving sell orders. Such a self-exciting Hawkes process based model of executions of a large order could capture the clustering of trade arrivals and positive liquidity feedback in dark pools.
Since the pool is assumed to be initially empty, there will be no trade occurring at time zero when the investor puts the buy order in the pool. A sample path of the trading intensity $\lambda_t$ and the remaining quantity of the dark order is given in Figure \[fig1\].
For this particular path, we observe from Figure \[fig:mini:subfig:1\] that the resting order matches with incoming sell trades with variable sizes, and it will be completely filled at time $t=7.2$ if the investor leaves the order in the pool for a sufficiently long time. On the other hand, if the investor decides to cancel the order before time $t=7.2$, then this resting order will be partially executed and the remaining quantity could be routed to another dark pool or a lit exchange for liquidity-seeking purposes.
We want to compute various performance quantities of interest such as time-to-first-fill and fill rate which indicate the liquidity of a dark venue. Their mathematical expressions and economic interpretations are summarized below. In a fragmented financial market with multiple dark pools and exchanges, these performance metrics could be useful for smart order routing and allocation of liquidity among different pools to maximize fills and liquidity opportunities from dark pools, which in turn help investors reduce market impact or opportunity cost in trading big orders. In terms of the notations, we differentiate between $\ell_{i}$ and $l_{i}$ by having $\ell_{i}$ being random and $l_{i}$ being deterministic and given.
**Performance quantities we consider**:
1. Time-to-first-fill $\tau(1)$ of the order is defined by $$\tau(1):=\inf\{t \ge 0: N_t =1\}.$$ That is, $\tau(1)$ measures the time between order placement at a given dark venue and the first execution (possibly a partial fill) of that order. Thus we obtain that the probability of a fill within $[0,t]$ is given by $$\mathbb{P}(\tau({1}) \le t)=1 - \mathbb{P}(N_t=0)=1 - e^{-\int_{0}^{t}\mu(s)ds}, \quad \text{for $t \ge 0$}.$$ >From this expression, it is clear that the baseline intensity $\mu(\cdot)$ completely determines the distribution of time-to-first-fill. In particular, when $\mu(\cdot)\equiv\mu$ is constant, $\tau(1)$ is an exponential random variable with mean $1/\mu$.
2. Time-to-complete-fill $\sigma_x$ of a resting order with size $x>0$ is defined by the time it takes for the order to be completely executed. That is, $\sigma_x$ measures the time it takes for the aggregated volumes of matching trades exceed the resting order’s size $x$: $$\label{eq:TTF}
\sigma_x:=\inf\{t \ge 0: L_t\ge x \}.$$ Hence its distribution is given by $$\mathbb{P}(\sigma_{x}\leq t)=\mathbb{P}(L_t\geq x), \quad \text{for $t \ge 0$}.$$ Since we have obtained the Laplace transform of $L_t$ in , we can then use the inverse Laplace transform to obtain the distribution of $L_t$ and that of $\sigma_x$ numerically. In addition, we can also compute the expected time-to-complete-fill $\mathbb{E}[\sigma_x]$ numerically, where $$\label{eq:E-TTF}
\mathbb{E}[\sigma_x] = \int_{0}^{\infty} \mathbb{P}(\sigma_{x}> t) dt = \int_{0}^{\infty} \mathbb{P}(L_t < x) dt.$$
3. The expected fill rate of the resting dark order with size $x>0$ in the time interval $[0, t]$ is defined by $$\label{eq:fillrate}
\frac{1}{x} \cdot \mathbb{E} [\min\{L_{t},x\}],$$ which is equal to $\frac{1}{x}\int_{0}^x \mathbb{P}(L_t >y)dy$. In practice, the deadline $t$ can be deterministic or random. For example, the investor may rest the order in a particular dark pool for one minute which is predetermined at the time of order placement. It is also possible that the investor may cancel a resting order due to exogenous market events such as a significant price move, in which case $t$ is random. We can numerically evaluate the expected fill rate efficiently if $t$ is independent of the execution process $(N, L)$ by first inverting the Laplace transform of $L_t$ in and getting its distribution, and then calculate the expectation in . Alternatively, one can also use Fast Fourier Transform (FFT) methods where the expected fill rates across the whole spectrum of order sizes $x$ can be obtained in one set of FFT calculations. See e.g. [@CarrMadan].
4. The probability of obtaining one fill (or at least one fill) in the next $T$ units of time, given that there is an initial fill of size $l_1<x$ in $(0,t]$. Mathematically we are interested in computing $$\label{eq:cond-hit}
\mathbb{P}(N_{t+T}-N_t=1|N_t=1,\ell_{1}=l_{1}),$$ and $$\label{eq:cond-hit2}
\mathbb{P}(N_{t+T}-N_t\ge 1|N_t=1,\ell_{1}=l_{1}) = 1 - \mathbb{P}(N_{t+T}-N_t=0|N_t=1,\ell_{1}=l_{1}).$$ As argued in the industry paper @Mittal2007, these conditional fill probabilities are particularly interesting in practice. Liquidity in a dark pool is sticky, and the expectation of liquidity changes when a trade occurs. The conditional fill probabilities in and give investors a quantitative view of the liquidity expectation in the future given a prior fill of the resting order.
To compute these conditional fill probabilities, we can use the Laplace transform of $N_T$ in Corollary \[SingleTransform\] and the intensity dynamics to obtain that $$\begin{aligned}
\label{eq:cond-prob}
&\mathbb{P}(N_{t+T} - N_t=0|N_t=1,\ell_{1}=l_{1}) \nonumber
\\
&=\frac{\int_{0}^{t}\mu(\tau_{1})e^{-\int_{0}^{\tau_{1}}\mu(s)ds}e^{-\int_{\tau_{1}}^{t}(\mu(s)+h(s-\tau_{1})l_{1})ds}
e^{-\int_{t}^{t+T}(\mu(s)+h(s-\tau_{1})l_{1})ds}d\tau_{1}}
{\int_{0}^{t}\mu(\tau_{1})e^{-\int_{0}^{\tau_{1}}\mu(s)ds}e^{-\int_{\tau_{1}}^{t}(\mu(s)+h(s-\tau_{1})l_{1})ds}d\tau_{1}},\end{aligned}$$ and $$\begin{aligned}
\label{eq:cond-prob2}
&\mathbb{P}( N_{t+T} - N_t=1|N_t=1,\ell_{1}=l_{1}) \nonumber
\\
&=\left(\int_{0}^{t}\mu(\tau_{1})e^{-\int_{0}^{\tau_{1}}\mu(s)ds}e^{-\int_{\tau_{1}}^{t}(\mu(s)+h(s-\tau_{1})l_{1})ds}d\tau_{1}\right)^{-1}
\nonumber \\
&\quad \cdot\int_{0}^{t}\bigg[\mu(\tau_{1})e^{-\int_{0}^{\tau_{1}}\mu(s)ds}e^{-\int_{\tau_{1}}^{t+T}(\mu(s)+h(s-\tau_{1})l_{1})ds}
\nonumber
\\
&\qquad\qquad\qquad
\cdot
\int_{t}^{t+T}(\mu(s)+h(s-\tau_{1})l_{1})\mathbb{E} \left[e^{-\int_{0}^{T+t-s} \ell_1 h(u)du}\right]ds\bigg]d\tau_{1}.\end{aligned}$$
A detailed derivation of and relies on the distributional properties of a Hawkes process with a time-dependent baseline intensity (due to the conditioning), and it is given in the Appendix.
Two interesting observations are in order. First, we can infer from and that the conditional probability of at least one fill given there is a past fill of size $l_1$ in the last $t$ units of time, is independent of the distribution of the trade size $\ell_1.$ Second, we note from that the conditional probability of exactly one fill in the next $T$ units of time depends on the distribution of $\ell_1$ only through its Laplace transform.
5. The expected fill size of the resting dark order in the next $T$ units of time conditioned on there is an initial fill of size $l_1<x$ in $(0,t]$, is given by $$\label{eq:conditional-expec}
\mathbb{E}\left[(L_{t+T} - L_t) \wedge (x-l_1) |N_t = 1, \ell_1 =l_1\right]
=\mathbb{E}\left[\min\{L_{t+T} , x\} |N_t = 1, \ell_1 =l_1\right] - l_1.$$ Similar as the conditional fill probabilities, such a conditional expectation provides the investor with an indication of the liquidity size in the dark pool based on a prior execution of the dark order.
To compute this conditional expected fill size, we can first infer from Corollary \[SingleTransform\] and the intensity dynamics to obtain the following Laplace transform: $$\begin{aligned}
&\mathbb{E}\left[e^{-\theta (L_{t+T} - L_t) }|N_t=1,\ell_{1}=l_{1}\right] \nonumber
\\
&=\frac{\int_{0}^{t}\mu(\tau_{1})e^{-\int_{0}^{\tau_{1}}\mu(s)ds}e^{-\int_{\tau_{1}}^{t}(\mu(s)+h(s-\tau_{1})l_{1})ds}
e^{\int_{t}^{t+T}(\mu(s)+h(s-\tau_{1})l_{1})(F_{L}(T+t-s)-1)ds}d\tau_{1}}
{\int_{0}^{t}\mu(\tau_{1})e^{-\int_{0}^{\tau_{1}}\mu(s)ds}e^{-\int_{\tau_{1}}^{t}(\mu(s)+h(s-\tau_{1})l_{1})ds}d\tau_{1}},
\label{pgfL}\end{aligned}$$ where for any $0\leq t\leq T$, the function $F_L$ satisfies the integral equation: $$F_{L}(t)=\mathbb{E}\left[e^{-\theta\ell_{1}+\int_{0}^{t}\ell_{1}h(s)(F_{L}(t-s)-1)ds}\right].$$ Then, we can numerical invert this Laplace transform to obtain the conditional distribution of $L_{t+T} - L_t$ and hence the conditional expectation in . The derivation of is similar as the derivation of in the Appendix, where we use the properties of a Hawkes process with a time-dependent baseline intensity. We omit the derivation here.
Two remarks are in order. First, the estimations of the performance metrics of a resting order are relatively straightforward if the investor has his/her own execution data from trading in dark pools. For example, the expected fill rate of an order of size $x$ placed at a dark pool for a given time horizon can be estimated as the arithmetic average of the fill rate of many orders with the same sizes $x$ placed at this pool, assuming that the market conditions and pool characteristics remain stationary. The estimation procedure is similar for other performance quantities. Second, given the investor has the execution data from trading in a dark pool, it is also possible to estimate the Hawkes model by first estimating the trade size distribution $\ell_i$, and then estimate the baseline intensity and the exciting function using parametric or non–parametric methods. See e.g. [@Bacry2015; @Errais] and references therein for details on estimations of Hawkes models.
### Numerical examples {#numerical-examples .unnumbered}
We now present numerical examples to illustrate the computations of the various quantities derived in Section \[sec:performance\]. We also consider different order size distributions for $\ell_i$, different exciting functions $h(\cdot)$, and different baseline intensities $\mu(\cdot)$ to investigate their impact on the performance metrics. Our numerical experiments are implemented in MATLAB on a PC with a 3.30 GHz Intel Processor and 8 GB of RAM.
To compute the performance quantities, we need to get the distribution of $L_t$ numerically. This requires us first to solve the integral equation to obtain the point process transform $F_L(t) $ for a given $\theta \in\mathbb{C}$ with a nonnegative real part, and then use Laplace inversion methods to obtain the distribution of $L_t$ for fixed $t$. The Laplace inversion method we use is a Fourier series method which employs Bromwich contour inversion integral and Euler summation. See @AbateWhitt95 for a detailed description of this Laplace inversion algorithm (called EULER in the paper).
To numerically solve $F_L(t)$ which satisfies a Hammerstein–type Volterra integral equation as in , we apply the collocation method, see e.g. Chapter 2.3.3 in @Brunner2004. The main idea of this method is to select a number of points (collocation points) on $[0, t]$, and use piecewise polynomial functions to approximate the true solution where the piecewise polynomial functions solve the given integral equation at the collocation points. Table \[table:time\] reports the computation time for representative examples where we solve $F_L(t)$ for $ t \in [0,6]$ using piecewise linear approximation on $[0, 6]$ with a uniform mesh consisting of 150 subintervals, a number which balances the speed and accuracy of the algorithm. The computation time is generally around 25 seconds for various different specifications of the mark size $\ell_i$ and the exciting function $h(\cdot)$.
$\ell_i \equiv 1$ $\ell_i \sim$ exponential $\ell_i \sim$ hyper exponential
---------- ------------------- --------------------------- ---------------------------------
$h_1(t)$ 25.411 24.795 25.343
$h_2(t)$ 25.349 25.048 25.472
$h_3(t)$ 25.433 25.778 25.801
: For a given $\theta \in\mathbb{C}$ ($\theta =1+ i$ in this example), this table records the CPU time (in seconds) for using piecewise linear collocation method to solve $F_L(t)$ on the time interval $[0,6]$ with a uniform mesh consisting of 150 subintervals for different combinations of the mark size $\ell_i$ and the exciting function $h(t)$. Here, we have considered three distributions for $\ell_i$: (a) $\ell_i \equiv 1$; (b) $\ell_i$ follows an exponential distribution with mean 1; and (c) $\ell_i$ has mean one and it follows a mixture of an exponential distribution with mean 5 and an exponential distribution with mean 1/5. Three exciting functions $h(t)$ considered are: (a) $h_1(t)=\frac{9}{10}(1+t)^{-2}$; (b) $h_2(t)=\frac{9}{10}(1+t)^{-3}$; and (c) $h_3(t)=\frac{9}{10}e^{-t}$.
\[table:time\]
We now report numerical results for the performance quantities (b)–(e) in Section \[sec:performance\], since the time-to-first-fill is completely determined by the baseline intensity function. Unless otherwise stated, we fix a constant baseline intensity $\mu(t)\equiv \mu=1$.
**Varying the trade size distribution $\ell_i$ while fixing an exciting function $h(t)= \frac{9}{10} \frac{1}{(1+t)^2}$.** Without loss of generality, we consider here three different distributions for $\ell_i$, all with a unit mean: (a) $\ell_i \equiv 1$; (b) $\ell_i$ has an exponential distribution; and (c) $\ell_i$ has a hyper-exponential distribution: here, we consider a concrete example where $\ell_i$ follows a mixture of an exponential distribution with mean 5 and an exponential distribution with mean 1/5. This choice of trade size distributions is motivated by @Afeche2014. In particular, a mixture of exponential distributions with different means can capture the feature that in dark pools, impatient high frequency traders submit small “pinging” orders and liquidity traders may submit relatively larger orders. In addition, the class of hyper-exponential distributions is very rich that it can approximate many heavy-tailed distributions for trade sizes, while maintaining analytical tractability, see e.g. @Cai2009 [@CaiKou2011].
Figure \[fig2\] summarizes the results on the expected time-to-complete-fill as a function of the resting order size $x$. Two key observations stand out from our results in Figure \[fig2\].
![*Expected time-to-complete-fill $\mathbb{E}[\sigma_x]$ in , as a function of the resting order size $x$. Here, $\mu(t) \equiv 1$ and the exciting function $h(t)= \frac{9}{10} \frac{1}{(1+t)^2}$ are fixed. The three curves correspond to three different distributions with a unit mean: (a) $\ell_i \equiv 1$ (red); (b) $\ell_i$ follows an exponential distribution (black); and (c) $\ell_i$ follows a mixture of an exponential distribution with mean 5 and an exponential distribution with mean 1/5 (blue).* []{data-label="fig2"}](time_to_fill_l.png){width="60.00000%"}
First, the expected time-to-complete-fill of the dark order increases in the size $x$ of the dark order and changes significantly when the distribution of the incoming trade size varies. It can be seen from the figure that given the size of the investor’s resting order $x$, when the incoming trade size follows a hyper-exponential distribution (a mixture of exponential distributions), it takes longer on average to completely fill this resting order than the cases of an exponential and a constant order size with the same mean. This observation is similar to the special case when $h \equiv 0$ where the point process $L$ becomes a compound Poisson process, and one can show that (see the Appendix for a proof) $$\label{321}
\mathbb{E}[\sigma^{(3)}_x] > \mathbb{E}[\sigma^{(2)}_x] = x +1 > \mathbb{E}[\sigma^{(1)}_x] = \lceil x \rceil, \quad \text{for $x > 0.$}$$ Here $\mathbb{E}[\sigma^{(i)}_x]$, $i=1,2,3$, are the expected time-to-complete-fill for the compound Poisson arrival with trade sizes $\ell_{i}^{(1)}$ being constant, $\ell_{i}^{(2)}$ being exponential and $\ell_{i}^{(3)}$ being hyper-exponential (all with mean one) respectively. We also remark that the expected time-to-complete-fill with Hawkes trades arrivals depends on the distribution of $\ell_i$, not just its coefficient of variation.
Second, the expected time-to-complete-fill of the first unit of a resting dark order is greater than the second and subsequent units. This reflects the self-exciting modeling of the order execution process which captures the trade clustering behavior. In other words, after a partial execution of the resting dark order, the expectation of another trade and the future trading intensity will increase, which leads to a continuing reduction of the marginal time-to-complete-fill of the resting dark order.
Next in Figure \[fig3\], we plot the expected fill rate of a resting order of size $x=10$, as a function of rest time $t$ for different trade size distributions. We observe that for a hyper-exponential trade size distribution, the expected fill rate of the resting order is much smaller than the case of a constant order size with the same mean. This is consistent with the observations from Figure \[fig2\] which suggest that it is harder to fill an order with “more variable" trade sizes. We provide an informal explanation on this relative order of expected fill rate for different trade size distributions in Figure \[fig3\]. Let $L_{t}^{(1)}$, $L_{t}^{(2)}$, and $L_{t}^{(3)}$ denote the associated $L_{t}$ when $\ell_{1}^{(1)}$ is a constant $1$, $\ell_{1}^{(2)}$ is exponentially distributed and $\ell_{1}^{(3)}$ is hyper-exponentially distributed, all with the same mean. First, notice that $\mathbb{E}[L_{t}^{(1)}]=
\mathbb{E}[L_{t}^{(2)}]=\mathbb{E}[L_{t}^{(3)}]$ from Proposition \[prop:moments\] as $\mathbb{E}[\ell_{1}^{(1)}]=\mathbb{E}[\ell_{1}^{(2)}]=\mathbb{E}[\ell_{1}^{(3)}]$. Next, observe that the hyper-exponential distribution is more “spread out" than the exponential distribution which is more “spread out" than a constant. Now when the jump size increases, the intensity of future arrivals also increase, and as a result $L_{t}$ increases. Similar argument holds when the jump size decreases. This suggests $L^{(3)}_{t}$ with a hyper-exponential jump size is “more variable" than $L^{(2)}_{t}$ with an exponential jump size in the sense that $L^{(3)}_{t}$ is more likely to take on “extreme" values. So intuitively, the expected fill rates satisfy $\frac{1}{x}\mathbb{E}[\min\{L_{t}^{(1)},x\}]
\geq \frac{1}{x}\mathbb{E}[\min\{L_{t}^{(2)},x\}]\geq \frac{1}{x}\mathbb{E}[\min\{L_{t}^{(3)},x\}]$.
![*Expected fill rate $\mathbb{E} [\min\{L_{t},x\}]/x$ in vs rest time $t$, for a resting order with size $x=10$. Here, $\mu(t)\equiv 1$ and the exciting function $h(t)= \frac{9}{10} \frac{1}{(1+t)^2}$ are fixed. The three curves correspond to three different distributions with a unit mean: (a) $\ell_i \equiv 1$ (red); (b) $\ell_i$ follows an exponential distribution (black); and (c) $\ell_i$ follows a mixture of an exponential distribution with mean 5 and an exponential distribution with mean 1/5 (blue).* []{data-label="fig3"}](fillrate_l.png){width="60.00000%"}
Furthermore, we plot in Figure \[fig4\] the conditional probability of one fill and the conditional expected fill size for the resting order as a function of the future $T$ units of time, given that there is a fill of size one in the past two units of time. Mathematically, the event conditioned on is $\{N_2=1,\ell_1=1\}$.
For the conditional probability of one fill in Figure \[fig4\](a), we note that it is biggest when the trade size $\ell_i$ follows a hyper-exponential distribution with mean one, and it is smallest when the trade size is constantly one. Let us explain. It is clear from that this conditional probability of one fill depends monotonically on the following Laplace transform of the random variable $\ell_1$: $$\begin{aligned}
\label{eq:lap-h}
\mathbb{E} \left[e^{-\int_{0}^{T+t-s} h(u)du \cdot \ell_1}\right].\end{aligned}$$ If we denote $\alpha:=\int_{0}^{T+t-s} h(u)du \ge 0$, then computing the Laplace transform in for the three distributions of $\ell_1$ (hyper-exponential, exponential and constant one) yields $$\frac{1}{6} \cdot \frac{0.2}{0.2 +\alpha } + \frac{5}{6} \cdot \frac{5}{5 +\alpha } \ge \frac{1}{1+\alpha} \ge e^{-\alpha}.$$ Now the observation in Figure \[fig4\](a) follows from and the above inequalities. **Varying the exciting function while fixing an exponential trade size distribution.** We next investigate how the exciting function $h$ impacts the performance quantities. For illustration purposes, we consider a family of power-law exciting functions with different tail behaviors: $$\label{eq:h-gamma}
h^{\gamma}(t) = \frac{C}{(1+t)^{\gamma}}, \quad \text{for $C>0, \quad \gamma>1$}.$$ In particular, $\Vert h^{\gamma}\Vert_{L^{1}}=\int_{0}^{\infty}h^{\gamma}(t)dt = \frac{C}{\gamma -1}$. In the literature, this quantity $\Vert h^{\gamma}\Vert_{L^{1}}$ is usually interpreted as a branching ratio, i.e., the expected number of events generated by any parent event. In the following, we will fix $C=0.9$, and $\ell_i$ follows an exponential distribution with mean 1 and the baseline intensity $\mu=1$. We consider three different exciting functions $h^{\gamma}(t)= \frac{9}{10} \frac{1}{(1+t)^\gamma}$ corresponding to $\gamma=2,2.5$ and 3. This allows us to better understand how the exciting functions impact the performance quantities.
We first plot in Figure \[fig5\] the expected time-to-complete-fill of an order as a function of the order size $x$, for different exciting functions $h^{\gamma}$ in . As one can observe from Figure \[fig5\], the larger the $\gamma$, the longer it takes on average to fill a given order completely.
![*Expected time-to-complete-fill $\mathbb{E}[\sigma_x]$ in vs order size $x$ for different exciting functions $h^{\gamma}$ defined in . Here, $\ell_i$ follows an exponential distribution with mean 1 and the baseline intensity $\mu(t) \equiv 1$ for the Hawkes model.*[]{data-label="fig5"}](time_to_fill_h.png){width="60.00000%"}
Next, we plot in Figure \[fig6\] the expected fill rate for a given order with size $x=10$, as a function of the resting time of the order. Consistent with Figure \[fig5\], the larger the $\gamma,$ the harder to fill an order and hence the smaller the expected fill rate for a given resting time.
![*Expected fill rate $\mathbb{E} [\min\{L_{t},x\}]/x$ in vs rest time $t$ for a given order with size $x=10$. Here, $\ell_i$ follows an exponential distribution with mean 1 and the baseline intensity $\mu=1$ for the Hawkes model. The three curves correspond to three different exciting functions $h^{\gamma}$ defined in .*[]{data-label="fig6"}](fillrate_h.pdf){width="60.00000%"}
Let us explain the phenomenon observed in Figures \[fig5\] and \[fig6\]. For $\gamma_1>\gamma_2>1$, we find from that $h^{\gamma_1}(t)<h^{\gamma_2}(t)$ for all $t\ge 0$. This implies that one can find a common probability space such that the associated point processes satisfy $L_t^{\gamma_1}\leq L_t^{\gamma_2}$ for all $t$ almost surely. Then the observations in Figures \[fig5\] and \[fig6\] readily follow from the formulas for the expected time-to-complete-fill in and the expected fill rate in .
We further investigate the conditional probability of another fill and the conditional expected fill size for different exciting functions, for a given resting order of size 10. Again, the event conditioned on is $\{N_2=1,\ell_1=1\}$, i.e., there is a fill of size one in the past two units of time. These two performance quantities are plotted in Figure \[fig7\]. We find from Figure \[fig7\](a) that, unlike in Figure \[fig4\](a), there is no monotonicity for the conditional probability of one fill when we vary $\gamma.$ This is not surprising as we can see from the formula that this conditional probability depends on the exciting function in a delicate way.
**The effect of the baseline intensity $\mu(t)$ on performance metrics.** So far, all the numerical examples on performance metrics are presented assuming a constant baseline intensity $\mu(t) \equiv 1$. We now briefly discuss the effect of a non-constant baseline intensity of the Hawkes process on performance metrics. Such a time-dependent baseline intensity $\mu(t)$ could represent, for example, the intraday pattern of dark pool liquidity.
For illustration purposes, we focus on the representative performance metric: the expected fill rate of a resting dark order given in . We compare in Figure \[fig-mut\] the case of a constant baseline intensity $\mu(t) \equiv 1$ with the following two cases where $\mu(t)$ is piecewise constant: $$\begin{aligned}
\label{eq:mu-t}
\mu_1(t) =
\begin{cases}
2, & \text{for } 0 \leq t \leq 4, \\
0.5 , & \text{for } 4 < t \leq 8, \\
1, & \text{for } t >8.
\end{cases} \quad \text{and} \quad
\mu_2(t) =
\begin{cases}
0.5, & \text{for } 0 \leq t \leq 4, \\
2 , & \text{for } 4 < t \leq 8, \\
1, & \text{for } t > 8.
\end{cases}\end{aligned}$$ We can observe from Figure \[fig-mut\] that the expected fill rate of a resting dark order depends on the baseline intensity of the Hawkes execution process in a delicate way. For the initial time period $[0, 4]$, as $\mu_1(t)>1> \mu_2(t)$, it follows that a higher baseline intensity of the Hawkes execution process leads to a higher expected fill rate of the resting order. On the other hand, on the time interval $(4,8]$, we have $\mu_2(t)>1>\mu_1(t)$. Compared with a Hawkes execution process with a constant one baseline intensity, the expected fill rate of a resting dark order during the time interval $(4, 8]$ is still higher when the trades follow a Hawkes arrival process with a baseline intensity $\mu_1(t)<1$. In addition, the expected fill rate of an order may also become higher when the Hawkes process has a baseline intensity $\mu_2(t)>1$. These two observations are due to the fact the intensity of a Hawkes process depends on both the baseline intensity and its own entire history (i.e. the past occurrence of trades).
![*Expected fill rate $\mathbb{E} [\min\{L_{t},x\}]/x$ in vs rest time $t$ for a given order with size $x=10$. Here, $\ell_i$ follows an exponential distribution with mean 1 and the exciting function $h(t)=\frac{9}{10} \frac{1}{(1+t)^2}$ for the Hawkes model. The three curves correspond to three different baseline intensity functions $\mu(t)$: $\mu(t) \equiv 1$ and $\mu_1(t), \mu_2(t)$ given in .*[]{data-label="fig-mut"}](fillrate_mu.pdf){width="60.00000%"}
Non-empty dark pools {#sec:nonempty}
--------------------
The performance formulas derived in Section \[sec:performance\] can be generalized to non-empty dark pools. To illustrate, we consider computing time-to-first-fill, time-to-complete-fill and expected fill rate of a posted dark order.
Suppose at time zero when the investor’s midpoint peg buy order of size $x>0$ reaches the dark pool, the liquidity size $Y$ in the pool is a random variable with a known or estimated cumulative distribution function $F_Y(y)$ which could possibly have a mass at zero (see e.g. @Ganchev2010). Here $Y>0$ represents the size of existing buy orders at the midpoint, and $Y<0$ represents the size of sell orders resting at the midpoint in the pool. In particular, there will be an immediate execution of the investor’s buy order at time zero if $Y<0$. In this case, when $Y\le -x$, then the investor’s dark order is completely filled at time zero. Otherwise for $Y \in (-x, 0)$, the dark buy order will get partially filled. The trader on the other side of the completed trade then realizes there could potentially be more liquidity on the opposite side of his trade, and then re-routes his other orders to this pool. Other information-seekers may also notice the trade and submit orders to this pool. Hence, this trade against resting sell orders at time zero may also incur a jump of the intensity of the arriving sell trades.
Mathematically, with a random liquidity size $Y$ in the dark venue, the intensity of the Hawkes process $N$ modeling the executions of the dark buy order will be modified as follows (defined till the time the dark order is completely filled): $$\label{eq:modifydynamics}
\lambda_{t}=\mu+ \min\{x, |Y|\}\cdot 1_{Y<0} \cdot h(t) + \sum_{0<\tau_i<t} h(t-\tau_i) \cdot \ell_i ,$$ where $\min\{x, |Y|\}\cdot 1_{Y<0}$ represents the size of a fill at time zero, and the impact on the trading intensity also decays according to the exciting function $h$. Hence, conditioned on $Y=y>0$, suppose these existing buy orders with total size $y$ also rest in the pool until full execution, then the Hawkes process $(N,L)$ will be essentially the same as in the case of an empty dark pool. On the other hand, conditioned on $Y=y<0$, the intensity $\lambda$ follows a different dynamics where now the baseline intensity is time-varying as given in the first two parts of the expression .
We now derive the performance quantities and still use the same notations as in Section \[sec:performance\] for simplicity. First, with a random liquidity size $Y$ in the dark pool, the time-to-first-fill of the investor’s dark buy order is given by $$\tau(1)=\inf\{t \ge 0:L_t >Y\}.$$ Hence we obtain $$\begin{aligned}
\mathbb{P}(\tau(1)=0) &=& \mathbb{P}(Y<0)= F_Y(0-),\\
\mathbb{P}(\tau(1)>t)& =& \mathbb{P}(L_t \le Y) = \int_0^{\infty}\mathbb{P}(L_t \le y) dF_{Y}(y), \quad \text{for $t \ge 0.$}\end{aligned}$$ Since we have derived the transform of $L_t$, we can then use inverse Laplace transform to get $\mathbb{P}(L_t \le y)$ for $y>0$, and hence compute the distribution and the expectation of $\tau(1)$.
Second, the time-to-complete-fill of the investor’s dark buy order with size $x>0$ is given by $$\sigma_x=\inf\{t \ge 0: L_t\ge x +Y \},$$ hence we have $$\begin{aligned}
\mathbb{P}(\sigma_x=0) &=& \mathbb{P}(Y \le -x) =F_Y(-x), \label{eq:tau-x-1}\\
\mathbb{P}(\sigma_x>t) & =& \mathbb{P}(L_t < x+ Y) , \quad \text{for $t \ge 0.$} \label{eq:tau-x-2}\end{aligned}$$ To obtain the distribution of $\sigma_x$, it suffices to compute $\mathbb{P}(L_t < x+ Y)$. Note that $$\begin{aligned}
\mathbb{P}(L_t < x+ Y) & =& \mathbb{P}(L_t -Y < x) \nonumber \\ &=& \int_{-x}^{0}\mathbb{P}(L_t < x + y| Y=y) dF_{Y}(y) \nonumber \\
&&\qquad\qquad\qquad+\int_{0}^{\infty}\mathbb{P}(L_t < x + y) dF_{Y}(y). \label{eq:Lt-Y}\end{aligned}$$ So it only remains to compute $\mathbb{P}(L_t < x + y| Y=y)$ for $-x < y <0.$ Conditioned on $Y=y \in (-x, 0)$, i.e., there is a partial execution of the dark buy order at time zero (which matches with resting sell orders in the pool), then from we infer that the intensity of the Hawkes process becomes $$\lambda_{t}=\mu + |y| \cdot h(t) + \sum_{0<\tau_i<t} h(t-\tau_i) \cdot \ell_i ,$$ where the baseline intensity $\mu(t)=\mu + |y| \cdot h(t)$ is deterministic and time-dependent. Since we have computed the Laplace transform of $(N_t, L_t)$ where the baseline intensity of the Hawkes process can be time-varying, we can then use inverse Laplace transform to compute $\mathbb{P}(L_t < x + y| Y=y)$ for $-x < y <0$. Now the distribution of time-to-complete-fill of the midpoint dark order can be computed using , and . The expected time-to-complete-fill $\mathbb{E}[\sigma_x]$ also readily follows.
Finally, the expected fill rate of the investor’s resting midpoint dark order of size $x$, in a given time interval $[0, t]$, is given by $$\label{eq:fillrate-nonempty}
\frac{1}{x} \cdot \mathbb{E} [\min\{(L_{t}-Y)^+,x\}] = \frac{1}{x} \cdot {\int_{0}^x \mathbb{P}((L_{t}-Y)^+ >z)dz},$$ where $a^+:=\max\{a,0\}$ for a real number $a$. This expected fill rate can hence also be readily computed as we have derived the distribution of $L_{t}-Y$ in .
Therefore, if the dark pool is non-empty at the time of the dark order placement by the investor, these performance quantities can still be similarly derived and efficiently numerically computed.
### Numerical examples {#numerical-examples-1 .unnumbered}
For illustration purposes, we plot in Figure \[fig8\] the expected fill rate of a given midpoint peg order when the initial liquidity size $Y$ in the dark pool has the following distribution: $\mathbb{P}(Y=0)=0.3$ and when $Y \ne 0$, it has a density function $$f_Y(y)=0.35\cdot k |y|^{k-1} e^{-|y|^k}, \quad \text{for $y \ne 0.$}$$ That is, $Y$ has a mass at zero, and it follows a two-sided Weibull distribution with scale parameter 1 and shape parameter $k$. Note that when $k \in (0,1),$ the tail of $Y$ is heavier than that of a two-sided exponential distribution. In addition, a smaller shape parameter $k$ implies a heavier tail of the liquidity size $Y$. Such a choice of $Y$ is motivated by @Ganchev2010 which empirically finds that the distribution of volume in dark pools is heavy-tailed: often, there is no volume available, but sometimes very large volume is present.
![*Expected fill rate in vs rest time $t$ for a given resting order with size $x=10$. Here, $\mu=1$, the trade size $\ell_i$ follows an exponential distribution with mean 1 and the exciting function $h(t)= \frac{9}{10} \frac{1}{(1+t)^2}$. The initial liquidity in the pool is modeled by a random variable $Y$ which has a mass 0.3 at zero, and when $Y \ne 0$, it follows a two-sided Weibull distribution with scale parameter 1 and shape parameter $k.$* []{data-label="fig8"}](nonempty_wbl.png){width="60.00000%"}
We observe from Figure \[fig8\] that the expected fill rate of a posted order at $t=0$ in a non-empty pool is greater than zero, which is different from that in an empty pool. Intuitively, this is clear as there could be contra-side sell orders resting at the midpoint in a non-empty pool which triggers trades at time zero when the investor posts a buy order at the midpoint. In fact, mathematically we can deduce from that the expected fill rate at time zero is simply $$\label{eq:fill-0}
\frac{1}{x} \cdot {\int_{0}^x \mathbb{P}((-Y)^+ >z)dz},$$ since $L_0=0.$ Hence, the initial percentage of fill when the order is posted critically depends on both the posted order size $x$, and the distribution of the volume of contra-side resting orders $(-Y)^+$. While we also observe from Figure \[fig8\] that a smaller shape parameter $k$ of the liquidity $Y$ leads to a larger expected fill rate at time zero, we remark that our extensive numerical experiments show this is not generally true for any order size $x$.
Conclusions {#sec:conclusion}
===========
We study the Hawkes process, a self-exciting point process, where the baseline intensity is time-dependent, the exciting function is a general function and the jump sizes of the intensity process are i.i.d. non-negative random variables. We obtain closed-form formulas for the Laplace transform, moments and the distribution of the Hawkes process. We apply these results to dark pool trading and analyze various performance metrics of a resting dark order which trades against contra-side marketable orders arriving according to a Hawkes process. These performance quantities can be useful for strategic allocation of liquidity among different pools to reduce market impact and execution costs in portfolio trading.
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Proof of Proposition \[prop:moments\]
=====================================
We first compute the first two moments of the counting process $N$. By differentiating the Laplace transform of the counting process $N$ with respect to (w.r.t.) $\theta$ in Corollary \[SingleTransform\], we get $$\frac{\partial}{\partial\theta}\mathbb{E}[e^{-\theta N_{T}}]
=\int_{0}^{T}\mu(T-s)\frac{\partial}{\partial\theta}F_{N}(s)ds
e^{\int_{0}^{T}\mu(T-s)(F_{N}(s)-1)ds},$$ and by differentiating w.r.t. $\theta$ again, we get $$\begin{aligned}
\frac{\partial^{2}}{\partial\theta^{2}}\mathbb{E}[e^{-\theta N_{T}}]
&=\int_{0}^{T}\mu(T-s)\frac{\partial^{2}}{\partial\theta^{2}}F_{N}(s)ds
e^{\int_{0}^{T}\mu(T-s)(F_{N}(s)-1)ds}
\\
&\qquad\qquad
+\left(\int_{0}^{T}\mu(T-s)\frac{\partial}{\partial\theta}F_{N}(s)ds\right)^{2}
e^{\int_{0}^{T}\mu(T-s)(F_{N}(s)-1)ds}.
\nonumber\end{aligned}$$ By differentiating both sides of w.r.t. $\theta$, we get $$\label{FNDiff1}
\frac{\partial}{\partial\theta}F_{N}(t)
=\mathbb{E}\left[\left(-1+\int_{0}^{t}\ell_{1}h(s)\frac{\partial}{\partial\theta}
F_{N}(t-s)ds\right)e^{-\theta+\int_{0}^{t}\ell_{1}h(s)(F_{N}(t-s)-1)ds}\right].$$ By differentiating w.r.t. $\theta$ again, we get $$\begin{aligned}
\label{FNDiff2}
\frac{\partial^{2}}{\partial\theta^{2}}F_{N}(t)
&=\mathbb{E}\left[\left(-1+\int_{0}^{t}\ell_{1}h(s)\frac{\partial}{\partial\theta}
F_{N}(t-s)ds\right)^{2}e^{-\theta+\int_{0}^{t}\ell_{1}h(s)(F_{N}(t-s)-1)ds}\right]
\\
&\qquad\qquad
+\mathbb{E}\left[\int_{0}^{t}\ell_{1}h(s)\frac{\partial^{2}}{\partial\theta^{2}}
F_{N}(t-s)ds
e^{-\theta+\int_{0}^{t}\ell_{1}h(s)(F_{N}(t-s)-1)ds}\right].
\nonumber\end{aligned}$$ By letting $\theta=0$ in , we get $$\frac{\partial}{\partial\theta}F_{N}(t)\bigg|_{\theta=0}
=-1+\int_{0}^{t}\mathbb{E}[\ell_{1}]h(s)\frac{\partial}{\partial\theta}
F_{N}(t-s)\bigg|_{\theta=0}ds.$$ This implies that $$\frac{\partial}{\partial\theta}F_{N}(t)\bigg|_{\theta=0}=-\psi_{1}(t),$$ where $\psi_{1}(\cdot)$ is defined in and thus $$\mathbb{E}[N_{T}]=-\frac{\partial}{\partial\theta}\mathbb{E}[e^{-\theta N_{T}}]\bigg|_{\theta=0}
=-\int_{0}^{T}\mu(T-s)\frac{\partial}{\partial\theta}F_{N}(s)\bigg|_{\theta=0}ds
=\int_{0}^{T}\mu(T-s)\psi_{1}(s)ds.$$ By letting $\theta=0$ in , we get $$\frac{\partial^{2}}{\partial\theta^{2}}F_{N}(t)\bigg|_{\theta=0}
=(\psi_{1}(t))^{2}
+\int_{0}^{t}\mathbb{E}[\ell_{1}]h(s)\frac{\partial^{2}}{\partial\theta^{2}}F_{N}(t-1)\bigg|_{\theta=0}ds.$$ By the definition of $\psi_{2}(\cdot)$ in , we have $\frac{\partial^{2}}{\partial\theta^{2}}F_{N}(t)\big|_{\theta=0}=\psi_{2}(t)$. Finally, we conclude that $$\begin{aligned}
\mathbb{E}[N_{T}^{2}]
&=\frac{\partial^{2}}{\partial\theta^{2}}\mathbb{E}[e^{-\theta N_{T}}]\bigg|_{\theta=0}
\\
&=\int_{0}^{T}\mu(T-t)\frac{\partial^{2}}{\partial\theta^{2}}F_{N}(s)\bigg|_{\theta=0}ds
+\left(\int_{0}^{T}\mu(T-s)\frac{\partial}{\partial\theta}F_{N}(s)\bigg|_{\theta=0}ds\right)^{2}
\nonumber
\\
&=\int_{0}^{T}\mu(T-t)\psi_{2}(t)dt+\left(\int_{0}^{T}\mu(T-s)\psi_{1}(t)dt\right)^{2}.
\nonumber\end{aligned}$$
We next compute the first two moments of the process $L$. We can compute from that $$\frac{\partial}{\partial\theta}
F_{L}(t)\bigg|_{\theta=0}
=-\mathbb{E}[\ell_{1}]+\int_{0}^{t}\mathbb{E}[\ell_{1}]h(s)\frac{\partial}{\partial\theta}
F_{L}(t-s)\bigg|_{\theta=0}ds,$$ which implies that $$\frac{\partial}{\partial\theta}
F_{L}(t)\bigg|_{\theta=0}
=\mathbb{E}[\ell_{1}]\psi_{1}(t).$$ Hence, $$\mathbb{E}[L_{T}]=-\frac{\partial}{\partial\theta}\mathbb{E}[e^{-\theta L_{T}}]\bigg|_{\theta=0}
=-\int_{0}^{T}\mu(T-s)\frac{\partial}{\partial\theta}F_{L}(s)\bigg|_{\theta=0}ds
=\mathbb{E}[\ell_{1}]\int_{0}^{T}\mu(T-s)\psi_{1}(s)ds.$$ We can also compute from that $$\begin{aligned}
\frac{\partial^{2}}{\partial\theta^{2}}F_{L}(t)
&=\mathbb{E}\left[\left(-\ell_{1}+\int_{0}^{t}\ell_{1}h(s)\frac{\partial}{\partial\theta}
F_{L}(t-s)ds\right)^{2}e^{-\theta\ell_{1}+\int_{0}^{t}\ell_{1}h(s)(F_{L}(t-s)-1)ds}\right]
\\
&\qquad\qquad
+\mathbb{E}\left[\int_{0}^{t}\ell_{1}h(s)\frac{\partial^{2}}{\partial\theta^{2}}
F_{L}(t-s)ds
e^{-\theta\ell_{1}+\int_{0}^{t}\ell_{1}h(s)(F_{L}(t-s)-1)ds}\right].
\nonumber\end{aligned}$$ Therefore, by the definition of $\psi_{1}$, $$\begin{aligned}
\frac{\partial^{2}}{\partial\theta^{2}}F_{L}(t)\bigg|_{\theta=0}
&=\mathbb{E}[\ell_{1}^{2}]
\left(-1+\int_{0}^{t}h(s)\frac{\partial}{\partial\theta}
F_{L}(t-s)\bigg|_{\theta=0}ds\right)^{2}
+\int_{0}^{t}\mathbb{E}[\ell_{1}]h(s)\frac{\partial^{2}}{\partial\theta^{2}}
F_{L}(t-s)\bigg|_{\theta=0}ds
\\
&=\mathbb{E}[\ell_{1}^{2}](\psi_{1}(t))^{2}
+\int_{0}^{t}\mathbb{E}[\ell_{1}]h(s)\frac{\partial^{2}}{\partial\theta^{2}}
F_{L}(t-s)\bigg|_{\theta=0}ds.\end{aligned}$$ Now, by recalling that $\psi_{3}(t)= \frac{\partial^{2}}{\partial\theta^{2}}F_{L}(t)\bigg|_{\theta=0}$, we conclude that $$\begin{aligned}
\mathbb{E}[L_{T}^{2}]
&=\frac{\partial^{2}}{\partial\theta^{2}}\mathbb{E}[e^{-\theta L_{T}}]\bigg|_{\theta=0}
\\
&=\int_{0}^{T}\mu(T-s)\frac{\partial^{2}}{\partial\theta^{2}}F_{L}(s)\bigg|_{\theta=0}ds
+\left(\int_{0}^{T}\mu(T-s)\frac{\partial}{\partial\theta}F_{L}(s)\bigg|_{\theta=0}ds\right)^{2}
\nonumber
\\
&=\int_{0}^{T}\mu(T-t)\psi_{3}(t)dt+(\mathbb{E}[\ell_{1}])^{2}\left(\int_{0}^{T}\mu(T-t)\psi_{1}(t)dt\right)^{2}.
\nonumber\end{aligned}$$
Finally, let us show that , and have unique solutions. We will only show uniqueness for the solution of here, and the argument is the same for and . Assume that has two solutions, say $\psi_{1}^{(1)}$ and $\psi_{1}^{(2)}$. Then, for any $0\leq t\leq T$, we have $$\begin{aligned}
\left|\psi_{1}^{(1)}(t)-\psi_{1}^{(2)}(t)\right|
&\leq\int_{0}^{t}h(t-s)\mathbb{E}[\ell_{1}]\left|\psi_{1}^{(1)}(s)-\psi_{1}^{(2)}(s)\right|ds
\\
&\leq\Vert h\Vert_{L^{\infty}[0,T]}\mathbb{E}[\ell_{1}]\int_{0}^{t}\left|\psi_{1}^{(1)}(s)-\psi_{1}^{(2)}(s)\right|ds.\end{aligned}$$ By Gronwall’s inequality, we conclude that $\psi_{1}^{(1)}=\psi_{1}^{(2)}$ on $[0,T]$.
Proof of Proposition \[prop:mass\]
==================================
Note that for any $z\in\mathbb{C}$ with $|z|\leq 1$, by considering $z=e^{-\theta}$, we obtain from the Laplace transform of $N_T$ that $$\label{z1}
\mathbb{E}[z^{N_{T}}]=e^{\int_{0}^{T}\mu(T-s)(F_{N}(s)-1)ds},$$ where (with slight abuse of notations) $F_N(t)$ depends on $z$ and it is given by $$\label{z2}
F_{N}(t)=z\cdot \mathbb{E}\left[e^{\int_{0}^{t}\ell_{1}h(s)(F_{N}(t-s)-1)ds}\right], \quad \text{for any $0\leq t\leq T$.}$$ It is easy to see that $$\mathbb{E}[z^{N_{T}}]
=\sum_{k=0}^{\infty}z^{k}\mathbb{P}(N_{T}=k),$$ and hence $$\mathbb{P}(N_{T}=k)=\frac{1}{k!}\frac{\partial^{k}}{\partial z^{k}}\mathbb{E}[z^{N_{T}}]\bigg|_{z=0}.$$ Let us recall the celebrated Faà di Bruno’s formula, for any smooth functions $f$ and $g$: $$\label{BrunoFormula}
\frac{d^{n}}{dx^{n}}f(g(x))
=\sum\frac{n!}{m_{1}!1!^{m_{1}}m_{2}!2!^{m_{2}}\cdots m_{n}!n!^{m_{n}}}
f^{(m_{1}+\cdots+m_{n})}(g(x))\prod_{j=1}^{n}(g^{(j)}(x))^{m_{j}},$$ where the summation is over all $n$-tuples of nonnegative integers $(m_1, \ldots, m_n)$ satisfying the constraint $1\cdot m_{1}+2\cdot m_{2}+3\cdot m_{3}+\cdots+n\cdot m_{n}=n$. Notice that $$\mathbb{E}[z^{N_{T}}]=e^{-\int_{0}^{T}\mu(T-s)ds}e^{\int_{0}^{T}\mu(T-s)F_{N}(s)ds},$$ and $F_{N}=0$ for $z=0$. By applying Faà di Bruno’s formula (with $f(z)=e^{z}$, $g(z)=\int_{0}^{T}\mu(T-s)F_{N}(s)ds$ and $n=k$), we get: $$\begin{aligned}
\frac{\partial^{k}}{\partial z^{k}}\mathbb{E}[z^{N_{T}}]\bigg|_{z=0}
&=e^{-\int_{0}^{T}\mu(T-s)ds}\sum\frac{k!}{m_{1}!1!^{m_{1}}m_{2}!2!^{m_{2}}\cdots m_{k}!k!^{m_{k}}}
\\
&\qquad\qquad\qquad
\cdot\prod_{j=1}^{k}\left(\int_{0}^{T}\mu(T-s)F_{N,j}(s)ds\right)^{m_{j}},\end{aligned}$$ where $$F_{N,j}(t):=\frac{\partial^{j}}{\partial z^{j}}F_{N}(t)\bigg|_{z=0}.$$ >From it is clear that $F_{N,0}(t)=0$, and $$F_{N,1}(t)=\mathbb{E}\left[e^{-\int_{0}^{t}\ell_{1}h(s)ds}\right].$$ By applying Faà di Bruno’s formula again, we get for any $j\geq 2$ the following recursive equation: $$\begin{aligned}
F_{N,j}(t)&=j\frac{\partial^{j-1}}{\partial z^{j-1}}\mathbb{E}\left[e^{\int_{0}^{t}\ell_{1}h(s)(F_{N}(t-s)-1)ds}\right]\bigg|_{z=0}
\\
&=j\sum\frac{(j-1)!}{m_{1}!1!^{m_{1}}m_{2}!2!^{m_{2}}\cdots m_{j-1}!(j-1)!^{m_{j-1}}}
\\
&\qquad\qquad
\cdot
\mathbb{E}\left[e^{-\int_{0}^{T}\ell_{1}h(s)ds}\prod_{i=1}^{j-1}\left(\int_{0}^{t}\ell_{1}h(s)F_{N,i}(t-s)ds\right)^{m_{i}}\right].\end{aligned}$$ The proof is therefore completed.
Proof of Proposition \[prop:LT-distri\]
=======================================
The proof also relies on Faà di Bruno’s formula. Note that for any $|z|<1$, $$\mathbb{E}\left[z^{\frac{1}{\delta}L_{T}}\right]
=\sum_{k=0}^{\infty}z^{k}\mathbb{P}(L_{T}=k\delta),$$ and thus, $$\mathbb{P}(L_{T}=k\delta)
=\frac{1}{k!}\frac{d^{k}}{dz^{k}}\mathbb{E}\left[z^{\frac{1}{\delta}L_{T}}\right]\bigg|_{z=0}.$$ For any $z\in\mathbb{C}$ with $|z|<1$, $$\mathbb{E}\left[z^{\frac{1}{\delta}L_{T}}\right]
=e^{\int_{0}^{T}\mu(T-s)(F_{L}(s)-1)ds},$$ where for any $0\leq t\leq T$, $$F_{L}(t)=\mathbb{E}\left[z^{\frac{1}{\delta}\ell_{1}}e^{\int_{0}^{t}\ell_{1}h(s)(F_{L}(t-s)-1)ds}\right]
=\sum_{k=1}^{\infty}z^{k}e^{\int_{0}^{t}k\delta h(s)(F_{L}(t-s)-1)ds}p_{k}.$$
By applying Faà di Bruno’s formula, we get: $$\begin{aligned}
\mathbb{E}\left[z^{\frac{1}{\delta}L_{T}}\right]\bigg|_{z=0}
&=e^{-\int_{0}^{T}\mu(T-s)ds}\sum\frac{k!}{m_{1}!1!^{m_{1}}m_{2}!2!^{m_{2}}\cdots m_{k}!k!^{m_{k}}}
\\
&\qquad\qquad\qquad
\cdot\prod_{j=1}^{k}\left(\int_{0}^{T}\mu(T-s)F_{L,j}(s)ds\right)^{m_{j}},\end{aligned}$$ where $$F_{L,j}(t):=\frac{\partial^{j}}{\partial z^{j}}F_{L}(t)\bigg|_{z=0}.$$ It is clear that $F_{L,0}(t)=0$, and $$F_{L,1}(t)=p_{1}e^{-\int_{0}^{t}\delta h(s)ds}.$$ By applying Faà di Bruno’s formula again, we get for any $j\geq 2$ the following recursive equation: $$\begin{aligned}
F_{L,j}(t)&=\frac{\partial^{j}}{\partial z^{j}}\sum_{k=0}^{\infty}z^{k}e^{\int_{0}^{t}k\delta h(s)(F_{L}(t-s)-1)ds}p_{k}\bigg|_{z=0}
\\
&=\sum_{k=0}^{\infty}\sum_{i=0}^{j}\binom{j}{i}\frac{d^{i}}{dz^{i}}z^{k}\bigg|_{z=0}
\frac{d^{j-i}}{dz^{j-i}}
e^{\int_{0}^{t}k\delta h(s)(F_{L}(t-s)-1)ds}p_{k}\bigg|_{z=0}
\\
&=\sum_{k=0}^{j}\binom{j}{k}k!
\frac{d^{j-k}}{dz^{j-k}}
e^{\int_{0}^{t}k\delta h(s)(F_{L}(t-s)-1)ds}p_{k}\bigg|_{z=0}
\\
&=\sum_{k=0}^{j}\binom{j}{k}k!
e^{-\int_{0}^{t}k\delta h(s)ds}p_{k}
\sum\frac{(j-k)!}{m_{1}!1!^{m_{1}}m_{2}!2|^{m_{2}}\cdots
m_{j-k}!(j-k)!^{m_{j-k}}}
\\
&\qquad\qquad\qquad\cdot
\prod_{i=1}^{j-k}\left(\int_{0}^{t}k\delta h(s)F_{L,i}(t-s)ds\right)^{m_{i}},\end{aligned}$$ where the summation is over $1\cdot m_{1}+2\cdot m_{2}+\cdots+(j-k)\cdot m_{j-k}=j-k$.
Derivations of Equations and
==============================
Let us compute for a non-negative integer $k,$ $$\mathbb{P}(N(t,t+T]=k|N_t=1,\ell_{1}=l_{1}),$$ where $N(t,t+T] = N_{t+T}- N_t.$ Our strategy is to first compute the probability generating function of $N(t,t+T]$ conditional on $N_t=1$ and $\ell_{1}=l_{1}$.
Note that on $[0,t]$, the first jump $\tau_{1}$ has the probability density function $\mu(\tau_{1})e^{-\int_{0}^{\tau_{1}}\mu(s)ds}$. Conditional on the time of the first jump $\tau_{1}$, $N_t=1$ if and only if $N(\tau_{1},t]=0$, which occurs with probability $e^{-\int_{\tau_{1}}^{t}(\mu(s)+h(s-\tau_{1})l_{1})ds}$ conditional on $\ell_{1}=l_{1}$. Next, notice that conditional on there is only one jump on $[0,t]$ and the time of the first jump being $\tau_{1}$ and conditional on the first jump size being $l_{1}$, the stochastic process $N(t,t+s]$ as a function of $s\in[0,T]$, is a Hawkes process with an exciting function $h(\cdot)$, i.i.d. jump sizes $\ell_{i}$ and the time-dependent baseline intensity $\mu(s)+h(s-\tau_{1})l_{1}$ at time $t<s<t+T$.
Hence, from the discussions above and the probability generating functions we derived in and in the proof of Proposition \[prop:mass\], we conclude that, for any $z\in\mathbb{C}$ with $0\leq |z|\leq 1$, the probability generating function is given by $$\begin{aligned}
H(z)&:=\mathbb{E}\left[z^{N(t,t+T]}|N_t=1,\ell_{1}=l_{1}\right]
\nonumber
\\
&=\frac{\int_{0}^{t}\mu(\tau_{1})e^{-\int_{0}^{\tau_{1}}\mu(s)ds}e^{-\int_{\tau_{1}}^{t}(\mu(s)+h(s-\tau_{1})l_{1})ds}
e^{\int_{t}^{t+T}(\mu(s)+h(s-\tau_{1})l_{1})(F_{N}(T+t-s)-1)ds}d\tau_{1}}
{\int_{0}^{t}\mu(\tau_{1})e^{-\int_{0}^{\tau_{1}}\mu(s)ds}e^{-\int_{\tau_{1}}^{t}(\mu(s)+h(s-\tau_{1})l_{1})ds}d\tau_{1}},
\label{pgfN}\end{aligned}$$ where for any $0\leq t\leq T$, $$F_{N}(t)=z\mathbb{E}\left[e^{\int_{0}^{t}\ell_{1}h(s)(F_{N}(t-s)-1)ds}\right].$$ The probability generating function yields $$\mathbb{E}\left[z^{N(t,t+T]}|N_t=1,\ell_{1}=l_{1}\right]
=\sum_{k=0}^{\infty}z^{k}\mathbb{P}(N(t,t+T]=k|N_t=1,\ell_{1}=l_{1}),$$ and hence the Taylor expansion coefficient of this generating function gives the probability mass function we need. Hence, we can compute that $$\begin{aligned}
\mathbb{P}(N(t,t+T]=0|N(t)=1,\ell_{1}=l_{1})
&=H(0)=\frac{\int_{0}^{t}\mu(\tau_{1})e^{-\int_{0}^{\tau_{1}}\mu(s)ds}e^{-\int_{\tau_{1}}^{t+T}(\mu(s)+h(s-\tau_{1})l_{1})ds}
d\tau_{1}}
{\int_{0}^{t}\mu(\tau_{1})e^{-\int_{0}^{\tau_{1}}\mu(s)ds}e^{-\int_{\tau_{1}}^{t}(\mu(s)+h(s-\tau_{1})l_{1})ds}d\tau_{1}},\end{aligned}$$ and $$\begin{aligned}
&\mathbb{P}(N(t,t+T]=1|N(t)=1,\ell_{1}=l_{1})
\\
&=H'(0)=\left(\int_{0}^{t}\mu(\tau_{1})e^{-\int_{0}^{\tau_{1}}\mu(s)ds}
e^{-\int_{\tau_{1}}^{t}(\mu(s)+h(s-\tau_{1})l_{1})ds}d\tau_{1}\right)^{-1}
\\
&\qquad
\cdot\int_{0}^{t}\mu(\tau_{1})e^{-\int_{0}^{\tau_{1}}\mu(s)ds}e^{-\int_{\tau_{1}}^{t}(\mu(s)+h(s-\tau_{1})l_{1})ds}
e^{-\int_{t}^{t+T}(\mu(s)+h(s-\tau_{1})l_{1})ds}
\\
&\qquad\cdot
\int_{t}^{t+T}(\mu(s)+h(s-\tau_{1})l_{1})\frac{\partial}{\partial z}F_{N}(T+t-s)\bigg|_{z=0}dsd\tau_{1}
\\
&=\left(\int_{0}^{t}\mu(\tau_{1})e^{-\int_{0}^{\tau_{1}}\mu(s)ds}e^{-\int_{\tau_{1}}^{t}(\mu(s)+h(s-\tau_{1})l_{1})ds}d\tau_{1}\right)^{-1}
\\
&\qquad
\cdot\int_{0}^{t}\bigg[\mu(\tau_{1})e^{-\int_{0}^{\tau_{1}}\mu(s)ds}e^{-\int_{\tau_{1}}^{t+T}(\mu(s)+h(s-\tau_{1})l_{1})ds}
\\
&\qquad\qquad\qquad\qquad
\cdot
\int_{t}^{t+T}(\mu(s)+h(s-\tau_{1})l_{1})\mathbb{E} \left[e^{-\int_{0}^{T+t-s} \ell_1 h(u)du}\right]ds\bigg]d\tau_{1}.\end{aligned}$$
Derivations of
===============
We provide a direct proof for . It is obvious that $\mathbb{E}[\sigma^{(1)}_x] = \lceil x \rceil$, since $\sigma^{(1)}_x$ is the hitting time to level $x>0$ for a Poisson process with rate one. Hence, it suffices to show $$\label{32}
\mathbb{E}[\sigma^{(3)}_x] > \mathbb{E}[\sigma^{(2)}_x] = x +1 \quad \text{for $x > 0,$}$$ where $\mathbb{E}[\sigma^{(i)}_x]$, $i=2,3$, are the expected hitting time to level $x>0$ for the compound Poisson arrival with jump sizes $\ell_{i}^{(2)}$ being exponential and $\ell_{i}^{(3)}$ being hyper-exponential respectively. Let us write the density function of $\ell_{i}^{(3)}$ as $$c\lambda_{1}e^{-\lambda_{1}x}+(1-c)\lambda_{2}e^{-\lambda_{2}x},
\qquad
x>0,$$ where $0<c<1$ and $0 <\lambda_{1} <1< \lambda_{2}$ so that $$\mathbb{E}[\ell_{i}^{(3)}]
=\frac{c}{\lambda_{1}}+\frac{1-c}{\lambda_{2}}=1= \mathbb{E}[\ell_{i}^{(2)}].$$ Since the baseline intensity is one, we have $\{L_{t}^{(j)}- t: t \ge 0\}$ is a martingale for $j=2,3$, where $L_{t}^{(j)}$ is the point process with jump sizes $\ell_{i}^{(j)}$. Now we infer from optional stopping theorem that $$\label{MEqn}
\mathbb{E}\left[\sigma_{x}^{(j)}\wedge M\right]=\mathbb{E}\left[L_{\sigma_{x}^{(j)}\wedge M}^{(j)}\right],
\qquad j=2,3,$$ for any $M>0$. Note that $0\leq L_{\sigma_{x}^{(j)}\wedge M}^{(j)}\leq L_{\sigma_{x}^{(j)}}^{(j)}$. By letting $M\rightarrow\infty$, we apply monotone convergence of the left hand side of and dominated convergence theorem on the right hand side of and we get: $$\label{eq:E-sigma-x}
\mathbb{E}[\sigma_{x}^{(j)}]=x+\mathbb{E}\left[L_{\sigma_{x}^{(j)}}^{(j)}-x\right],
\qquad j=2,3,$$ provided that $\mathbb{E}[L_{\sigma_{x}^{(j)}}^{(j)}]$ is finite for $j=2,3$.
Next, let us compute and estimate the expected overshoot $\mathbb{E}[L_{\sigma_{x}^{(j)}}^{(j)}-x]$. For $j=2,$ it is well known that for exponentially distributed $\ell_{i}^{(2)}$ with mean $1$, the overshoot is also exponentially distributed with mean $1$ and thus $$\label{eq:E-sigma2-x}
\mathbb{E}[\sigma_{x}^{(2)}]=x+\mathbb{E}\left[L_{\sigma_{x}^{(2)}}^{(2)}-x\right] = x+1.$$ For $j=3,$ we note that for a hyper-exponentially distributed $\ell_{i}^{(3)}$ with mean $1$, we can compute that for any $0<z<x$, $$\begin{aligned}
\label{eq:overshott}
\mathbb{E}\left[L_{\sigma_{x}^{(3)} }^{(3)}-x\big|L_{\sigma_{x}^{(3)}- }^{(3)}=x-z\right]
&=\mathbb{E}\left[\ell_{1}^{(3)}\big|\ell_{1}^{(3)}>z\right] \nonumber
\\
&=\frac{\int_{z}^{\infty}c\lambda_{1}(y-z)e^{-\lambda_{1}y}dy+\int_{z}^{\infty}(1-c)(y-z)\lambda_{2}e^{-\lambda_{2}y}dy
}{\int_{z}^{\infty}c\lambda_{1}e^{-\lambda_{1}y}dy+\int_{z}^{\infty}(1-c)\lambda_{2}e^{-\lambda_{2}y}dy}
\nonumber
\\
&=\frac{c\frac{1}{\lambda_{1}}e^{-\lambda_{1}z}+(1-c)\frac{1}{\lambda_{2}}e^{-\lambda_{2}z}}
{ce^{-\lambda_{1}z}+(1-c)e^{-\lambda_{2}z}}.\end{aligned}$$ Notice that $$\frac{c\frac{1}{\lambda_{1}}e^{-\lambda_{1}z}+(1-c)\frac{1}{\lambda_{2}}e^{-\lambda_{2}z}}
{ce^{-\lambda_{1}z}+(1-c)e^{-\lambda_{2}z}}
\leq\frac{1}{\lambda_{1}}+\frac{1}{\lambda_{2}},$$ uniformly in $0<z<x$ and thus $\mathbb{E}[L_{\sigma_{x}^{(3)}}^{(3)}]\leq\frac{1}{\lambda_{1}}+\frac{1}{\lambda_{2}}+x<\infty$ is finite. Moreover, $$\frac{c\frac{1}{\lambda_{1}}e^{-\lambda_{1}z}+(1-c)\frac{1}{\lambda_{2}}e^{-\lambda_{2}z}}
{ce^{-\lambda_{1}z}+(1-c)e^{-\lambda_{2}z}}
> 1,$$ if and only if $$\label{IneqHold}
c\left(\frac{1}{\lambda_{1}}-1\right)e^{-\lambda_{1}z}
>(1-c)\left(1-\frac{1}{\lambda_{2}}\right)e^{-\lambda_{2}z}.$$ Since $\lambda_{2}>\lambda_{1}$ and $z>0$, the strict inequality holds if we can show that $$\label{IneqHold2}
\frac{c}{\lambda_{1}}-c\geq 1-c-\frac{1-c}{\lambda_{2}},$$ This holds and indeed we get the equality in due to $\mathbb{E}[\ell_{i}^{(3)}]=1$. Hence, we can infer from that $\mathbb{E}[L_{\sigma_{x}^{(3)} }^{(3)}-x] >1$ when the jump size is hyper-exponentially distributed. On combining with and , we obtain .
[^1]: Corresponding author. Department of Systems Engineering and Engineering Management, The Chinese University of Hong Kong, Shatin, N.T. Hong Kong; [email protected]
[^2]: Department of Systems Engineering and Engineering Management, The Chinese University of Hong Kong, Shatin, N.T. Hong Kong; [email protected]
[^3]: Department of Mathematics, Florida State University, 1017 Academic Way, Tallahassee, FL-32306, United States of America; [email protected].
[^4]: Matching rules or allocation mechanisms of dark pools are typically complex, partly confidential and frequently updated [@Ye2011]. Time-priority matching rule is used by, e.g., BATS Europe Dark Book, see @Liquidmetrix. Besides time-priority matching, many dark pools use some form of pro-rata matching [@ZhuHX2014]. This matching rule is different from the model we consider here and we leave the study of it for future research.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'Recently, there have been found new relations between the zero forcing number and the minimum rank of a graph with the algebraic co-rank. We continue on this direction by giving a characterization of the graphs with real algebraic co-rank at most 2. This implies that for any graph with at most minimum rank at most 3, its minimum rank is bounded from above by its real algebraic co-rank.'
author:
- 'Carlos A. Alfaro [^1]'
title: 'Graph classes for critical ideals, minimum rank and zero forcing number'
---
**Keywords:** critical ideals, algebraic co-rank, forbidden induced subgraph, minimum rank, Laplacian matrix, zero forcing number.
**MSC:** 05C25, 05C50, 05E99, 13P15, 15A03, 68W30.
Introduction
============
Given a graph $G$ and a set of indeterminates $X_G=\{x_u \, : \, u\in V(G)\}$, the [*generalized Laplacian matrix*]{} $L(G,X_G)$ of $G$ is the matrix whose $uv$-entry is given by $$L(G,X_G)_{uv}=\begin{cases}
x_u& \text{ if } u=v,\\
-m_{uv}& \text{ otherwise},
\end{cases}$$ where $m_{uv}$ is the number of the edges between vertices $u$ and $v$. Moreover, if $\mathcal{R}[X_G]$ is the polynomial ring over a commutative ring $\mathcal{R}$ with unity in the variables $X_G$, then the [*critical ideals*]{} of $G$ are the determinantal ideals given by $$I^{\mathcal{R}}_i(G,X_G)=\langle {\rm minors}_i(L(G,X_G))\rangle\subseteq \mathcal{R}[X_G] \text{ for all } 1\leq i\leq n,$$ where $n$ is the number of vertices of $G$ and ${\rm minors}_i(L(G,X_G))$ is the set of the determinants of the $i\times i$ submatrices of $L(G,X_G)$.
An ideal is said to be [*trivial*]{} if it is equal to $\langle1\rangle$ ($=\mathcal{R}[X]$). The [*algebraic co-rank*]{} $\gamma_\mathcal{R}(G)$ of $G$ is the maximum integer $i$ for which $I^{\mathcal{R}}_i(G,X_G)$ is trivial. For simplicity, we might refer to the [*real algebraic co-rank*]{} to $\gamma_\mathbb{R}(G)$. Note that $I^{\mathcal{R}}_n(G,X_G)=\langle \det L(G,X_G)\rangle$ is always non-trivial, and if $d_G$ denote the degree vector, then $I^{\mathcal{R}}_n(G,d_G)=\langle 0\rangle$.
Critical ideals were defined in [@corrval] and some interesting properties were pointed out there. For instance, it was proven that if $H$ is an induced subgraph of $G$, then $I^{\mathcal{R}}_i(H,X_H)\subseteq I^{\mathcal{R}}_i(G,X_G)$ for all $i\leq |V(H)|$. Thus $\gamma_\mathcal{R}(H)\leq \gamma_\mathcal{R}(G)$. Initinally, critical ideals were defined as a generalization of the critical group, [*a.k.a.*]{} sandpile group, see [@alfacorrval; @alfaval; @corrval]. In [@alfaval2; @merino] can be found an account of the main results on sandpile group. Further, it is also a generalization of several other algebraic objects like Smith group or characteristic polynomials of the adjacency and Laplacian matrices, see [@alfavalvaz Section 4] and [@corrval Section 3.3]. In [@alflin], there were explered its relation with the zero forcing number and the minimum rank. We continue on this direction. For this, we recall these well-known concepts.
The *zero forcing game* is a color-change game where vertices can be blue or white. At the beginning, the player can pick a set of vertices $B$ and color them blue while others remain white. The goal is to color all vertices blue through repeated applications of the *color change rule*: If $x$ is a blue vertex and $y$ is the only white neighbor of $x$, then $y$ turns blue, denoted as $x\rightarrow y$. An initial set of blue vertices $B$ is called a *zero forcing set* if starting with $B$ one can make all vertices blue. The *zero forcing number* $Z(G)$ is the minimum cardinality of a zero forcing set. The *chronological list* of a zero forcing game records the forces $x_i\rightarrow y_i$ in the order of performance. In the following, ${\operatorname{mz}}(G)=|V(G)|-Z(G)$.
For a graph $G$ on $n$ vertices, the family $\S_\mathcal{R}(G)$ collects all $n\times n$ symmetric matrices with entries in the ring $\mathcal{R}$, whose $i,j$-entry ($i\neq j$) is nonzero whenever $i$ is adjacent to $j$ and zero otherwise. Note that the diagonal entries can be any element in the ring $\mathcal{R}$. The *minimum rank* $\mr_\mathcal{R}(G)$ of $G$ is the smallest possible rank among matrices in $\S_\mathcal{R}(G)$. Here we follow [@rankring Definition 1] and define the rank of a matrix over a commutative ring with unity as the largest $k$ such that there is a nonzero $k\times k$ minor that is not a zero divisor. In the case of $\mathcal{R}=\mathbb{Z}$, the rank over $\mathbb{Z}$ is the same as the rank over $\mathbb{R}$. In [@AIMZmr], it was proved that ${\operatorname{mz}}(G)\leq\mr_\mathcal{R}(G)$ for any field $\mathcal{R}$. And in [@alflin], it was proved that ${\operatorname{mz}}(G)\leq\gamma_\mathcal{R}(G)$ for any commutative ring $\mathcal{R}$ with unity. However, the relation between $\mr_\mathcal{R}(G)$ and $\gamma_\mathcal{R'}(G)$ depends on the rings $\mathcal{R}$ and $\mathcal{R}'$.
Let $I\subseteq \mathcal{R}[X]$ be an ideal in $\mathcal{R}[X]$. The *variety* of $I$ is defined as $$V_\mathcal{R}(I)=\left\{ {\bf a}\in \mathcal{R}^n : f({\bf a}) = 0 \text{ for all } f\in I \right\}.$$ That is, $V_\mathcal{R}(I)$ is the set of common roots between polynomials in $I$. We have that $$\langle 1\rangle \supseteq I^{\mathcal{R}}_1(G,X_G) \supseteq \cdots \supseteq I^{\mathcal{R}}_n(G,X_G) \supseteq \langle 0\rangle.$$ Thus $$\emptyset=V_\mathcal{R}(\langle 1\rangle) \subseteq V_\mathcal{R}(I^{\mathcal{R}}_1(G,X_G)) \subseteq \cdots \subseteq V_\mathcal{R}(I^{\mathcal{R}}_n(G,X_G)) \subseteq V_\mathcal{R}(\langle 0\rangle)=\mathcal{R}^n.$$ If $I^\mathcal{R}_k(G,X_G)$ is trivial, then, for all ${\bf a}\in \mathcal{R}^n$, there are $k$-minors of $L(G,{\bf a})$ which are different of 0, and $\rank(L(G,{\bf a}))\geq k$. However, it does not imply that $\mr_\mathcal{R}(G)\geq \gamma_\mathcal{R}(G)$, since matrices in $\S_\mathcal{R}(G)$ do not necessarily have only $0$ and $-1$ on the off-diagonal entries. However, if $V_\mathcal{R}(I^{\mathcal{R}}_k(G,X_G))\neq\emptyset$ for some $k$, then there exists ${\bf a}\in\mathcal{R}$ such that, for all $t \geq k$, $I^{\mathcal{R}}_{t}(G,{\bf a})=\langle 0\rangle$; that is, all $t$-minors of $L(G,{\bf a})$ are equal to $0$. Therefore, $\mr_\mathcal{R}(G)\leq k-1$. In particular, if $V_\mathcal{R}\left(I^{\mathcal{R}}_{\gamma_\mathcal{R}(G)+1}(G,X_G)\right)$ is not empty, then $\mr_\mathcal{R}(G)\leq \gamma_\mathcal{R}(G)$. Therefore, as noted in [@alflin], it follows by the Weak Nullstellensatz that if $\mathcal{R}$ is an algebraically closed field, then $\mr_\mathcal{R}(G)\leq \gamma_\mathcal{R}(G)$. That is not the case for the integers, there exist graphs for which $\mr_\mathbb{Z}(G)> \gamma_\mathbb{Z}(G)$. For the field of real numbers, it was conjectured [@alflin] that $\mr_\mathbb{R}(G)\leq\gamma_\mathbb{R}(G)$. Trying to sheed some light on this conjecture, it was proved in [@alflin] that if $G$ is a connected graph such that $\mr_{\mathbb{R}}(G)\leq 2$, then $\mr_{\mathbb{R}}(G)\leq\gamma_{\mathbb{R}}(G)$.
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$P_4$ & $\ltimes$ & [dart]{} & $K_5\setminus{P_3}$\
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$\overline{P_2\cup C_4}$ & $K_{2,2,2}$ & $K_{2,2,1,1}$
Let $\Gamma^\mathcal{R}_{\leq i}=\{G\, :\, G \text{ is a simple connected graph with } \gamma_\mathcal{R}(G)\leq i\}$. Our aim is to give a characterization of $\Gamma^\mathbb{R}_{\leq 2}$. Given a family of graphs $\mathfrak{F}$, a graph $G$ is called $\mathfrak{F}$-[*free*]{} if no induced subgraph of $G$ is isomorphic to a member of $\mathfrak{F}$. We will characterize $\Gamma^\mathbb{R}_{\leq 2}$ as the $\{P_4,\ltimes,{\sf dart},K_5\setminus{P_3},\overline{P_2\cup C_4},K_{2,2,2},K_{2,2,1,1}\}$-free graphs. Since $\mr_{\mathbb{R}}(G)\leq 2$ if and only if $G$ is {$P_4$,$K_{3,3,3}$,$\ltimes$,[dart]{}}-free, then we have that any graph $G\in\Gamma^\mathbb{R}_{\leq 2}$ has $\mr_{\mathbb{R}}(G)\leq 2$. Thus, if $G$ is a connected graph such that $\mr_{\mathbb{R}}(G)= 3$, then $\gamma(G)\geq3$. Implying that if $G$ is a connected graph such that $\mr_{\mathbb{R}}(G)\leq 3$, then $\mr_{\mathbb{R}}(G)\leq\gamma_{\mathbb{R}}(G)$.
The paper is organized as follows. In Section \[section:graphclasses\], we will give an overview of the main classifications that have been obtained for graphs with bounded ${\operatorname{mz}}$, $\mr$ and $\gamma$. We will give a characterization of the $\{P_4, \ltimes, {\sf dart}, K_5\setminus{P_3}, \overline{P_2\cup C_4}, K_{2,2,2}, K_{2,2,1,1}\}$-free graphs. In Section \[section:blowupgraphs\], we will recall a method to compute the algebraic co-rank of blowup graphs. And we will use it to prove that in fact the given characterization is of the graphs with real minimum rank at most 2.
Graph classes for bounded ${\operatorname{mz}}$, $\mr$ and $\gamma$ {#section:graphclasses}
===================================================================
It is known that algebraic co-rank, minimum rank and ${\operatorname{mz}}$ are monotone on induced subgraphs, that is, if $H$ is an induced subgraph of $G$, then $\gamma_\mathcal{R}(H)\leq \gamma_\mathcal{R}(G)$, $\mr_\mathcal{R}(H)\leq \mr_\mathcal{R}(G)$ and ${\operatorname{mz}}(H)\leq {\operatorname{mz}}(G)$. Then, it is natural to ask for classifications of graphs where these parameters are bounded from above.
Since ${\operatorname{mz}}(G)\leq \gamma_{\mathcal{R}}(G)$ and ${\operatorname{mz}}(G)\leq \mr_{\mathcal{R}}(G)$, then the family of graphs with $\gamma_{\mathcal{R}}(G)\leq k$ or $\mr_{\mathcal{R}}(G)\leq k$ are contained in the family of graphs with ${\operatorname{mz}}(G)\leq k$. However, the relation between the families of graphs with $\gamma_{\mathcal{R}}(G)\leq k$ and $\mr_{\mathcal{R}}(G)\leq k$ is still not clear.
In previous works, it was noticed in [@alfaval; @BHL04] that among all connected graphs, the complete graphs are the only graphs whose minimum rank, algebraic co-rank and ${\operatorname{mz}}$ are equal to 1. Also, in [@BHL04 Theorem 16] it was proved that for any connected graph $G$, ${\operatorname{mz}}(G)\leq2$ if and only if $G$ is $\{P_4,\ltimes,{\sf dart}\}$-free. In [@BHL04; @BHL05], there are classifications of graphs whose minimum rank is at most 2 depending on the base field. In particular for the field of real numbers, we have the following result, where $G+H$ denote the [*disjoint union*]{} of the graphs of $G$ and $H$, and $G\vee H$ denote the [*join*]{} of $G$ and $H$.
[@BF07; @BHL04] Let $G$ be a connected graph. Then, the following are equivalent:
1. $\mr(G)\leq 2$,
2. $G$ is {$P_4$,$K_{3,3,3}$,$\ltimes$,[dart]{}}-free,
3. $G=\bigvee_{i=1}^r G_i$, $r>1$, where either
1. $G_i=K_{m_i}+ K_{n_i}$ for suitable $m_i\geq 1$, $n_i\geq0$, or
2. $G_i=\overline{K_{m_i}}$ for a suitable $m_i\geq3$;
and option $(b)$ occours at most twice.
On the other hand, we have that if $\mathcal{R}'$ is a subring of $\mathcal{R}$, then $\gamma_\mathcal{R'}(G)\leq\gamma_\mathcal{R}(G)$. From which follows $\Gamma^{\mathbb{R}}_{\leq k}\subseteq\Gamma^{\mathbb{Z}}_{\leq k}$. In this sense, in [@alfaval] the connected graphs with $\gamma_{\mathbb{Z}}(G)\leq 2$ were classified.
\[thm:chacterizationZ2\] Let $G$ be a connected graph. Then, the following are equivalent:
1. $G\in\Gamma^{\mathbb{Z}}_{\leq 2}$,
2. {$P_4$, $K_{2,2,1,1}$, $K_5\setminus{P_3}$, $\ltimes$, ${\sf dart}$}-free graphs,
3. $G$ is isomorphic to $K_{n_1,n_2,n_3}$ or to $\overline{K_{n_1}}\vee(K_{n_2}+K_{n_3})$.
Few is known for graphs with minimum rank and algebraic co-rank at most 3. In [@alfaval1; @BGL09], there were obtained only partial results for the minimum rank and algebraic co-rank at most 3. And the problem still seems to be far to be completely understod. And in [@alflin; @alfavalvaz], there were characterized the digraphs whose minimum rank, algebraic co-rank and ${\operatorname{mz}}$ are equal to 1.
A graph $G$ is *forbidden* for $\Gamma^{\mathcal{R}}_{\leq k}$ when $\gamma_\mathcal{R}(G)\geq k+1$. Let ${\bf Forb}(\Gamma^{\mathcal{R}}_{\leq k})$ be the set of minimal (under induced subgraphs property) forbidden graphs for $\Gamma^\mathcal{R}_{\leq k}$. A graph $G$ is $\gamma_\mathcal{R}$-[*critical*]{} if $\gamma_\mathcal{R}(G-v)<\gamma_\mathcal{R}(G)$ for each $v\in V(G)$. Then, $G\in {\bf Forb}(\Gamma^{\mathcal{R}}_{\leq k})$ if and only if $G$ is $\gamma_\mathcal{R}$-critical such that $\gamma_\mathcal{R}(G-v)\leq k$ and $k<\gamma_\mathcal{R}(G)$ for each $v\in V(G)$. Therefore $G\in\Gamma^{\mathcal{R}}_{\leq k}$ if and only if $G$ is ${\bf Forb}(\Gamma^{\mathcal{R}}_{\leq k})$-free. Thus, characterizing ${\bf Forb}(\Gamma^{\mathcal{R}}_{\leq k})$ leads to a characterization of $\Gamma^{\mathcal{R}}_{\leq k}$.
Since $\gamma_\mathbb{Z}(G)\leq\gamma_\mathbb{R}(G)$ for any graph $G$, then we have that $P_4$, $K_{2,2,1,1}$ and $K_5\setminus{P_3}$ are forbidden graphs for $\Gamma^\mathbb{R}_{\leq 2}$. In fact we have the following.
\[lemma:free\] The graphs $P_4,\ltimes,{\sf dart},K_5\setminus{P_3},\overline{P_2\cup C_4},K_{2,2,2}$ and $K_{2,2,1,1}$ are in ${\bf Forb}(\Gamma^{\mathbb{R}}_{\leq 3})$.
This can be verified by using a Computer Algebra System like [*Macaulay2*]{}. More precisely, it can be proved that these graphs are $\gamma_\mathbb{R}$-[*critical*]{} and their real algebric co-rank is 3. At this moment it does not imply that these graphs are all the graphs in ${\bf Forb}(\Gamma^{\mathbb{R}}_{\leq 3})$.
Let us consider the Gröbner bases of the third critical ideal on $\mathbb{Z}$ of $K_{2,2,2}$: $$I_3^\mathbb{Z}\left(K_{2,2,2},X_{K_{2,2,2}}\right)=\langle x_1, x_2, x_3, x_4, x_5, x_6, 2 \rangle.$$ When we consider this ideal over the real numbers, it becomes trivial. Similarly, the Gröbner bases of the third critical ideal on $\mathbb{Z}$ of $\overline{P_2\cup C_4}$ is not trivial: $$I_3^\mathbb{Z}\left(\overline{P_2\cup C_4},X_{\overline{P_2\cup C_4}}\right)=\langle x_1 + 1, x_2 + 1, x_3 + 1, x_4 + 1, x_5, x_6, 2 \rangle,$$ where $v_5$ and $v_6$ are the vertices of degree 4. And again, when we consider this ideal over the real numbers, it becomes trivial. This is an interesting behaviour that does not happen on the rest of graphs in ${\bf Forb}(\Gamma^{\mathbb{R}}_{\leq 2})$.
We start from the characterization of ${\bf Forb}(\Gamma^{\mathbb{Z}}_{\leq 2})$, and, additionally, the induced subgraphs $K_{2,2,2}$ and $\overline{P_2\cup C_4}$ will be removed.
\[lemma:forb\] Let $G$ be a connected graph. Then, $G$ is $\{P_4,\ltimes,{\sf dart}, K_5\setminus{P_3}, \overline{P_2\cup C_4}, K_{2,2,2},K_{2,2,1,1}\}$-free if and only if $G$ is isomorphic to an induced subgraph of one of the following graphs: $K_{1,n_1,n_2}$, $\overline{K_{1}}\vee(K_{n_2}+K_{n_3})$ or $\overline{K_{n_1}}\vee(K_{1}+K_{n_3})$.
Let $G$ be $\{P_4,\ltimes,{\sf dart}, K_5\setminus{P_3}, K_{2,2,1,1}\}$-free. By Theorem \[thm:chacterizationZ2\], we have two cases, either $G$ is isomorphic to $K_{n_1,n_2,n_3}$ or to $\overline{K_{n_1}}\vee(K_{n_2}+K_{n_3})$. In the first case, since $K_{2,2,2}$ is forbidden for $G$, we have that at least one of the $n_1,n_2,n_3$ must be at most 1. In the second case, we can observe that $\overline{P_2\cup C_4}$ can be regarded as $\overline{K_{2}}\vee(K_{2}+K_{2})$. From which follows that either $n_1\leq 1$ or at least one of $n_2$ and $n_3$ is at most 1. The other direction follows since $\overline{P_2\cup C_4}$ is not and induced subgraph of $K_{1,n_2,n_3}$, and $K_{2,2,2}$ is not an induced subgraph of $\overline{K_{1}}\vee(K_{n_2}+K_{n_3})$ nor $\overline{K_{n_1}}\vee(K_{1}+K_{n_3})$.
It remains to prove that $P_4,\ltimes,{\sf dart},K_5\setminus{P_3},\overline{P_2\cup C_4},K_{2,2,2}$ and $K_{2,2,1,1}$ are in fact all the graphs in ${\bf Forb}(\Gamma^{\mathbb{R}}_{\leq k})$. This can be done by computing the algebric co-rank of the graphs in $K_{1,n_1,n_2}$, $\overline{K_{1}}\vee(K_{n_2}+K_{n_3})$ and $\overline{K_{n_1}}\vee(K_{1}+K_{n_3})$, and checking that any graph $G$ in these families has $\gamma_\mathbb{R}(G)\leq2$. That will be done in the following section.
Blowup graphs {#section:blowupgraphs}
=============
Given a graph $G=(V,E)$ and a vector ${\bf d}\in {\mathbb Z}^V$, the graph $G^{\bf d}$ is constructed as follows. For each vertex $u\in V$, associate a new vertex set $V_u$, where $V_u$ is a clique of cardinality $-{\bf d}_u$ when ${\bf d}_u$ is negative, and $V_u$ is a stable set of cardinality ${\bf d}_u$ if ${\bf d}_u$ when positive. Each vertex in $V_u$ is adjacent with each vertex in $V_v$ if and only if $u$ and $v$ are adjacent in $G$. Then the graph $G$ is called the [*underlying graph*]{} of $G^{\bf d}$.
In general, the computation of the Gröbner bases of the critical ideals is more than complicated. However, we will use a method, developed in [@alfacorrval], to decide, for $i\leq|V(G)|$, whether the $i$-[*th*]{} critical ideal of $G^{\bf d}$ is trivial or not.
For ${\bf d}\in {\mathbb Z}^{V}$, we define $\phi({\bf d})$ as follows: $$\phi({\bf d})_v =
\begin{cases}
0 & \text{ if }{\bf d}_v>1,\\
-1 & \text{ if }{\bf d}_v<-1,\\
x_v & \text{ otherwise }.
\end{cases}$$
[@alfacorrval Theorem 2.7]\[Theo:trivialcriticalidealsiff\] Let $n\geq 2$ and $G=(V,E)$ be a graph with $n$ vertices. For $1\leq j\leq n$ and ${\bf d}\in \mathbb{Z}^{V}$, the critical ideal $I^\mathcal{R}_j(G^{\bf d},X_{G^{\bf d}})$ is trivial if and only if the evaluation of $I^\mathcal{R}_j(G,X_G)$ at $X_G=\phi({\bf d})$ is trivial.
Therefore, verifying whether a family of graphs have algebraic co-rank at most $i$ becomes in an evaluation of the $i$-[*th*]{} critical ideal of the underlying graph of the family. It might be possible that such a family might be described by an infinite number of underlying graphs.
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(a) & (b)
\[lemma:gamma1\] Let $G$ be an induced subgraph of $K_{1,n_1,n_2}$, then $\gamma_\mathbb{R}(G)\leq 2$.
The underlying graph $H$ of $K_{1,n_1,n_2}$ is shown in Figure \[fig:blowupgraph\].a. We have that $$I^{\mathbb{R}}_3(H,X_H)=\langle x_1x_2x_3 - x_1 - x_2 - x_3 - 2\rangle$$ Let ${\bf d}=(0,-n_1,-n_2)$, and thus $\phi({\bf d})=(x_1,-1,-1)$. By evaluating the third critical ideal at $\phi({\bf d})$, we have $I_3(H,\phi({\bf d}))=\langle 0\rangle$. By Theorem \[Theo:trivialcriticalidealsiff\], $I^\mathbb{R}_3(H^{\bf d},X_{H^{\bf d}})$ is not trivial and $\gamma_\mathbb{R}(K_{1,n_1,n_2})\leq 2$.
\[lemma:gamma2\] Let $G$ be an induced subgraph of $\overline{K_{1}}\vee(K_{n_2}+K_{n_3})$ or $\overline{K_{n_1}}\vee(K_{1}+K_{n_3})$, then $\gamma_\mathbb{R}(G)\leq 2$.
The underlying graph $H$ of $\overline{K_{n_1}}\vee(K_{n_2}+K_{n_3})$ is shown in Figure \[fig:blowupgraph\].b. We have that $$I^{\mathbb{R}}_3(H,X_H)=\langle x_1x_2x_3 - x_2 - x_3\rangle$$ Let ${\bf d}_1=(n_1,0,-n_3)$, and thus $\phi({\bf d}_1)=(0,x_2,-1)$. By evaluating the third critical ideal at $\phi({\bf d}_1)$, we have $I_3(H,\phi({\bf d}_1))=\langle -x_2+1\rangle$. By Theorem \[Theo:trivialcriticalidealsiff\], $\gamma_\mathbb{R}\left(\overline{K_{n_1}}\vee(K_{1}+K_{n_3})\right)\leq 2$. Let ${\bf d}_2=(0,-n_2,-n_3)$, and thus $\phi({\bf d}_2)=(x_1,-1,-1)$. By evaluating the third critical ideal at $\phi({\bf d}_2)$, we have $I_3(H,\phi({\bf d}_2))=\langle x_1+2\rangle$. By Theorem \[Theo:trivialcriticalidealsiff\], $\gamma_\mathbb{R}\left(\overline{K_{1}}\vee(K_{n_2}+K_{n_3})\right)\leq 2$.
Lemmas \[lemma:free\], \[lemma:forb\], \[lemma:gamma1\] and \[lemma:gamma2\] imply our main result.
Let $G$ be a connected graph. Then, the following are equivalent.
1. $G\in\Gamma^{\mathbb{R}}_{\leq 2}$,
2. $G$ is $\{P_4,\ltimes,{\sf dart}, K_5\setminus{P_3}, \overline{P_2\cup C_4}, K_{2,2,2},K_{2,2,1,1}\}$-free,
3. $G$ is isomorphic to an induced subgraph of one of the following graphs: $K_{1,n_1,n_2}$, $\overline{K_{1}}\vee(K_{n_2}+K_{n_3})$ or $\overline{K_{n_1}}\vee(K_{1}+K_{n_3})$.
The fact that $K_{2,2,2}$ is an induced subgraph of $K_{3,3,3}$ implies that if $G\in\Gamma^{\mathbb{R}}_{\leq 2}$, then $\mr(G)\leq 2$. Therefore, if $G$ is a connected graph such that $\mr(G)= 3$, then $\gamma_{\mathbb{R}}(G)\geq3$. Which implies the following result.
If $G$ is a connected graph such that $\mr(G)= 3$, then $\mr(G)\leq\gamma_{\mathbb{R}}(G)$.
Acknowledgments {#acknowledgments .unnumbered}
===============
This research was partially supported by SNI and CONACyT.
[99]{} C.A. Alfaro and C.E. Valencia. On the sandpile group of the cone of a graph. *Linear Algebra and its Applications*, 436:1154–1176, 2012. Wai-Sin Ching. Linear equations over commutative rings. *Linear Algebra and its Applications*, 18:257–266, 1977. H. Corrales and C.E. Valencia. On the critical ideals of graphs. *Linear Algebra and its Applications*, 439:3870–3892, 2013. L.M. DeAlba, J. Grout, L. Hogben, R. Mikkelson, and K. Rasmussen. Universally optimal matrices and field independence of the minimum rank of a graph. *Electronic Journal of Linear Algebra*, 18:403–419, 2009.
[^1]: Banco de México, Mexico City, Mexico ([email protected], [email protected]).
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"pile_set_name": "ArXiv"
}
|
---
abstract: 'In this work we study the list-decoding size of Reed-Muller codes. Given a received word and a distance parameter, we are interested in bounding the size of the list of Reed-Muller codewords that are within that distance from the received word. Previous bounds of Gopalan, Klivans and Zuckerman [@GKZ08] on the list size of Reed-Muller codes apply only up to the minimum distance of the code. In this work we provide asymptotic bounds for the list-decoding size of Reed-Muller codes that apply for [*all*]{} distances. Additionally, we study the weight distribution of Reed-Muller codes. Prior results of Kasami and Tokura [@KT70] on the structure of Reed-Muller codewords up to twice the minimum distance, imply bounds on the weight distribution of the code that apply only until twice the minimum distance. We provide accumulative bounds for the weight distribution of Reed-Muller codes that apply to [*all*]{} distances.'
author:
- |
Tali Kaufman [^1]\
MIT\
[email protected]\
- |
Shachar Lovett [^2]\
Weizmann Institute of Science\
[email protected]
title: 'The List-Decoding Size of Reed-Muller Codes'
---
Introduction {#sec:intro}
============
The problem of list-decoding an error correcting code is the following: given a received word and a distance parameter find all codewords of the code that are within the given distance from the received word. List-decoding is a generalization of the more common notion of unique decoding in which the given distance parameter ensures that there can be at most one codeword of the code that is within the given distance from the received word. The notion of list-decoding has numerous practical and theoretical implications. The breakthrough results in this field are due to Goldreich and Levin [@GL] and Sudan [@Sud] who gave efficient list decoding algorithms for the Hadamard code and the Reed-Solomon code. See surveys by Guruswami [@Gur] and Sudan [@Sud2] for further details. In complexity, list-decodable codes are used to perform hardness amplification of functions [@STV]. In cryptography, list-decodable codes are used to construct hard-core predicates from one way functions [@GL]. In learning theory, list decoding of Hadamard codes implies learning parities with noise [@KM].
In this paper we study the question of list-decoding Reed-Muller codes. Specifically, we are interested in bounding the list sizes obtained for different distance parameters for the list-decoding problem.
Reed-Muller codes are very fundamental and well studied codes. $RM(n,d)$ is a linear code, whose codewords $f \in
RM(n,d):\F_2^n \to \F_2$ are evaluations of polynomials in $n$ variables of total degree at most $d$ over $\F_2$. In this work we study the code $RM(n,d)$ when $d \ll n$, and are interested in particular in the case of constant $d$.
The following facts regarding $RM(n,d)$ are straight-forward: It has block length of $2^n$, dimension $\sum_{i \leq d}{n \choose i}$ and minimum relative distance $\frac{2^{n-d}}{2^n} = 2^{-d}$. We define:
\[Relative weight of a function\] The relative weight of a function/codeword $f:\F_2^n \to \F_2$ is the fraction of non-zero elements, $$wt(f) = \frac{1}{2^n} |\{x \in \F_2^n: f(x) = 1\}|$$
A closely related definition is the distance between two functions
\[Relative distance between two functions\] The relative distance between two functions $f,g:\F_2^n \to \F_2$ is defined as $$dist(f,g) = \P_{x \in \F_2^n}[f(x) \ne g(x)]$$
The main focus of this work is in understanding the asymptotic growth of the list size in list-decoding of Reed-Muller codes, as a function of the distance parameter. Specifically we are interested in obtaining bounds on the following.
For a function $f:\F_2^n \to \F_2$ let the ball at relative distance $\alpha$ around $f$ be $$B(f,\alpha) = \{p \in RM(n,d): dist(p,f) \le \alpha \}$$
The list-decoding size of $RM(n,d)$ at distance $\alpha$, denoted by $L(\alpha)$, is the maximal size of $B(f,\alpha)$ over all possible functions $f$, i.e. $$L(\alpha) = \max_{f:\F_2^n \to \F_2} |B(f,\alpha)|$$
In a recent work Gopalan, Klivans and Zuckerman [@GKZ08] prove that for distances up to the minimal distance of the code, the list-decoding size of Reed-Muller codes remains constant.
\[thm:gkz2\] $$L(2^{-d}- \epsilon) \le O \left( (1/\epsilon)^{8d} \right)$$
Their result of bounding the list-decoding size of Reed-Muller codes is inherently limited to work up to the minimum distance of the code, since it uses a structural theorem of Kasami and Takura on Reed-Muller codes [@KT70], which implies a bound on the weight distribution of Reed-Muller codes that works up to twice the minimum distance of the code.
Additionally, the work of [@GKZ08] has developed a list-decoding algorithm for $RM(n,d)$ whose running time is polynomial in the worst list-decoding size and in the block length of the code.
\[thm:gkz:global-alg\] Given a distance parameter $\alpha$ and a received word $R:\F_2^n \to \F_2$, there is an algorithm that runs in time $poly(2^n, L(\alpha))$ and produces a list of all $p \in RM(n,d)$ such that $dist(p,R) \leq \alpha$.
Since Gopalan et al. could obtain non-trivial bounds on the list-decoding size for distance parameter $\alpha$ that is bounded by the minimum distance of the Reed-Muller code, their algorithm yields meaningful running time only for $\alpha$ that is less than twice the minimum distance of the code.
Weight distribution of Reed-Muller codes
----------------------------------------
A close notion to the list-decoding size of Reed-Muller code is the weight distribution of the code.
\[Accumulative weight distribution\] The accumulative weight distribution of $RM(n,d)$ at a relative weight $\alpha$ is the number of codewords up to this weight, i.e. $$A(\alpha) = |\{p \in RM(n,d): wt(p) \le \alpha \}|$$ where $0 \le \alpha \le 1$.
It is well-known that for any $p \in RM(n,d)$ which is not identically zero, $wt(p) \ge 2^{-d}$. Thus, $A(2^{-d} - \epsilon)=1$ for any $\epsilon>0$. Kasami and Tokura [@KT70] characterized the codewords in $RM(n,d)$ of weight up to twice the minimal distance of the code (i.e up to distance $2^{1-d}$). Based on their characterization one could conclude the following.
\[thm:gkz1\] $$A(2^{1-d} - \epsilon) \le (1/\epsilon) ^ {2(n+1)}$$
Corollary \[thm:gkz1\] and simple lower bounds (which we show later, see Lemma \[lemma:lower\_bound\]) show that $A(\alpha) =
2^{\Theta(n)}$ for $\alpha \in [2^{-d},2^{1-d}-\epsilon]$ for any $\epsilon>0$ (and constant $d$).
Our Results
-----------
Gopalan et al. [@GKZ08] left as an open problem the question of bounding the list-decoding size of Reed-Muller codes beyond the minimal distance. In particular, they ask what is the maximal $\alpha$ s.t. $L(\alpha) = 2^{O(n)}$.
In this work we answer their question. Specifically we show bounds on the list-decoding size of Reed-Muller code for distances passing the minimal distance. In fact, we show that the asymptotic behavior of $L(\alpha)$, for all $0 \le \alpha \le 1$. Our first result shows that there exist “cut-off distances”, at which the list-decoding size changes from $2^{\Theta(n^{\ell})}$ to $2^{\Theta(n^{\ell+1})}$:
\[thm:main\_list\_decode\_RM\] Let $1 \le \ell \le d-1$ be an integer, and let $\epsilon>0$. For any $\alpha \in [2^{\ell-d-1},2^{\ell-d}-\epsilon]$ $$L(\alpha) = 2^{\Theta(n^{\ell})}$$ and $L(\alpha) = 2^{\Theta(n^d)}$ for any $\alpha \ge 1/2$.
Using Theorem \[thm:main\_list\_decode\_RM\], and Theorem \[thm:gkz:global-alg\] we obtain the following algorithmic result for list-decoding Reed-Muller codes from an arbitrary distance.
\[thm:list-decode-alg\] Given a received word $R:\F_2^n \to \F_2$ that is at distance $\alpha$ from $RM(n,d)$, for $\alpha \in
[2^{\ell-d-1},2^{\ell-d}-\epsilon]$. where $1 \le \ell \le d-1$ is an integer, and $\epsilon>0$. There exists an algorithm that runs in time $poly(2^{\Theta(n^{\ell})})$ and produces a list of all $p \in
RM(n,d)$ such that $dist(p,R) \leq \alpha$
The weight distribution of $RM(n,d)$ codes beyond twice the minimum distance was widely open prior to our work. See e.g. Research Problem (15.1) in [@MS] and the related discussion in that Chapter.
In this work we provide asymptotic bounds for the weight distribution of $RM(n,d)$ that applied for all weights $2^{-d} \le
\alpha \le 1/2$. Specifically, our second main result gives exact boundaries on the range of $\alpha$ for which $A(\alpha)=2^{\Theta(n^{\ell})}$, for any $\ell=1,2,...,d$.
\[thm:main\_weight\_distrib\_RM\] Let $1 \le \ell \le d-1$ be an integer, and let $\epsilon>0$. For any $\alpha \in [2^{\ell-d-1},2^{\ell-d}-\epsilon]$ $$A(\alpha) = 2^{\Theta(n^{\ell})}$$ and $A(\alpha) = 2^{\Theta(n^d)}$ for any $\alpha \ge 1/2$.
Theorems \[thm:main\_list\_decode\_RM\] and \[thm:main\_weight\_distrib\_RM\] are asymptotically tight for constant $\epsilon>0$. For sub-constant $\epsilon$, and $\alpha \in [2^{\ell-d-1},2^{\ell-d}-\epsilon]$, our bound gives: $$A(\alpha) \le L(\alpha) \le 2^{O(n^{\ell} / \epsilon^2)}$$
We conjecture this dependency on $\epsilon$ is not optimal, and the correct dependency should be $\log(1/\epsilon)$ instead of $1/\epsilon^2$. We expand more on that in the body of the paper.
Techniques
----------
The bounds on the accumulative weight distribution of the Reed-Muller code are obtained using the following novel strategy. We show that a function $f:F_2^n \to F_2$ whose weight is bounded by $wt(f)
\leq 2^{-k}(1-\epsilon)$ can be [*computed*]{} as an expectation of its $k$th-derivatives multiplied by some bounded coefficients (Lemma \[lemma:calc\_by\_ders\]).
Using standard sampling methods we then show (Lemma \[lemma:calc\_to\_approx\_by few\]) that a function $f:F_2^n
\to F_2$ whose weight is bounded by $wt(f) \leq 2^{-k}(1-\epsilon)$ can be well approximated by a constant number $c=c(k,\epsilon)$ of its $k$th-derivatives. This implies that every $RM(n,d)$ codeword of weight up to $2^{-k}(1-\epsilon)$ can be well approximated by $c=c(k,\epsilon)$ of its $k$th-derivatives. Since the distance between every pair of $RM(n,d)$ codewords is at least $2^{-d}$, a good enough approximation of a $RM(n,d)$ codeword determines the Reed-Muller codeword uniquely. Hence, the number of $RM(n,d)$ codewords up to weight $2^{-k}(1-\epsilon)$, is bounded by the number of $k$th-derivatives to the power of $c=c(k,\epsilon)$. As $RM(n,d)$ codewords are polynomials of degree at most $d$, their $k$th-derivatives are polynomials of degree at most $d-k$. There can be at most $\Theta(2^{n^{d-k}})$ such derivatives. Thus, the number of $RM(n,d)$ codewords up to weight $2^{-k}(1-\epsilon)$, can be bounded by $O(2^{n^{d-k}})^c = O(2^{c
\cdot n^{d-k}})$. We complement these upper bound estimations with matching lower bounds.
A similar work in this line is the work of Viola and Bogdanov [@BV], which shows that a function $f:F_2^n \to F_2$ whose weight is bounded by $wt(f) \leq
1/2-\epsilon$ can be well approximated by $c=c(k,\epsilon)$ of its $1$st-derivatives. Note that approximation by $1$st-derivatives [*does not*]{} imply in general approximation by $k$th-derivatives which is crucial for obtaining our bounds here.
The bounds on the list-decoding size of Reed-Muller codes are obtained using similar techniques to the ones used for bounding the accumulative weight distributions.
Generalized Reed-Muller Codes
-----------------------------
The problems of bounding both the accumulative weight distribution and the list-decoding size can be extended to Generalized Reed-Muller, the code of low-degree polynomials over larger fields. However, our techniques fail to prove tight result in these cases. We provide some partial results for this case and make a conjecture about the correct bounds in Appendix \[sec:GRM\].
Organization
------------
Although our goal is bounding the list-decoding size of Reed-Muller codes, we first study the accumulative weight distribution of Reed-Muller codes. The techniques we develop are then easily transferred to bounding also the list-decoding size.
The paper is organized as follows. In Section \[sec:weight\_RM\] we study the weight distribution of Reed-Muller codes and we prove the Second Main Theorem (Theorem \[thm:main\_weight\_distrib\_RM\]). In Section \[sec:list\_RM\] study the list-decoding size of Reed-Muller codes. We generalize the techniques of Section \[sec:weight\_RM\] to prove the First Main Theorem (Theorem \[thm:main\_list\_decode\_RM\]). In Section \[sec:GRM\] we study similar questions for Generalized Reed-Muller code and provide non-tight bounds for these codes.
Weight distribution of Reed-Muller codes {#sec:weight_RM}
========================================
In this section we study the weight distribution of Reed-Muller codes, and we prove our Second Main Theorem (Theorem \[thm:main\_weight\_distrib\_RM\]). Let $RM(n,d)$ stand for the code of multivariate polynomials $p(x_1,...,x_n)$ over $\F_2$ of total degree at most $d$. In the following $n$ and $d$ will always stand for the number of variables and the total degree. We will assume that $d \ll n$, and study in particular the case of constant $d$.
Our Second Main Theorem (Theorem \[thm:main\_weight\_distrib\_RM\]) is a direct corollary of Theorem \[thm:weight\_distrib\_RM\], giving an upper bound on the accumulative weight at distance $2^{\ell-d}-\epsilon$, and Lemma \[lemma:lower\_bound\], giving a simple lower bound at distance $2^{\ell-d-1}$.
\[thm:weight\_distrib\_RM\] For any integer $1 \le k \le d-1$, $$A(2^{-k}(1-\epsilon)) \le c_1 2^{c_2 \frac{n^{d-k}}{\epsilon^2}}$$ where $c_1 = (1/\epsilon)^{O(d/\epsilon^2)}$ and $c_2 = O(d/(d-k)!)$. Importantly, $c_1,c_2$ are independent of $n$, and $c_2$ is independent of $\epsilon$. In particular for constant $d$ we get that $$A(2^{-k}-\epsilon) \le 2^{O(\frac{n^{d-k}}{\epsilon^2})}$$
\[lemma:lower\_bound\] For any integer $1 \le k \le d$ $$A(2^{-k}) \ge 2^{\frac{n^{d-k+1}}{(d-k+1)!}(1+o(1))}$$
In the upper bound on $A(\alpha)$, while the dependence on $n$ is tight, we believe the dependence on $\epsilon$ can be improved. For $k=d-1$ (and constant $d$), the characterization of [@KT70] shows that $$A(2^{1-d}-\epsilon) = 2^{\Theta(n \log(1/\epsilon))}$$
We conjecture that this is the correct dependence on $\epsilon$ in all the range:
Let $d$ be constant. For any integer $1 \le k \le d-1$, $$A(2^{-k}-\epsilon) = 2^{\Theta(n^{d-k} \log(1/\epsilon))}$$
We start by proving the lower bound.
Single out $k$ variables $x_1,...,x_k$, and let $q$ be any degree $d-k+1$ polynomials on the remaining $n-k$ variables. First, for any such $q$, the following degree $d$ polynomial has relative weight exactly $2^{-k}$: $$q'(x_1,...,x_n)=x_1 x_2 ... x_{k-1}(x_k + q(x_{k+1},...,x_n))$$ The number of different polynomials $q$ is $$2^{{n-k \choose d-k+1}} = 2^{\frac{n^{d-k+1}}{(d-k+1)!}(1+o(1))}$$
We will prove Theorem \[thm:weight\_distrib\_RM\] in the rest of the section. We start by defining discrete derivatives, which will be our main tool in the proof.
Let $f:\F_2^n \to \F_2$ by a function. We define the discrete derivative of $f$ in direction $a \in \F_2^n$ to be $$f_a(x) = f(x+a)+f(x)$$
We define the iterated discrete derivative of $f$ in directions $a_1,...,a_k \in \F_2^n$ to be $$f_{a_1,...,a_k}(x) = (...((f_{a_1})_{a_2})...)_{a_k}(x) = \sum_{S \subseteq [k]} f(x + \sum_{i \in S} a_i)$$
We note that usually derivatives are defined as $f_a(x)=f(x+a)-f(x)$, but since we are working over $\F_2$, we can ignore the signs.
We define another notion which is central to our proof, namely the bias of a function.
The bias of a function $f:\F_2^n \to \F_2$ is $$bias(f) = \E_{x \in \F_2^n}[(-1)^{f(x)}] = \P[f=0]-\P[f=1] = 1 - 2wt(f)$$
The following lemma will be the heart of our proof. It shows that if a function $f$ has weight less than $2^{-k}$, then it can be computed by a its iterated $k$-derivatives.
\[lemma:calc\_by\_ders\] Let $f:\F_2^n \to \F_2$ be a function s.t. $wt(f) < 2^{-k}(1-\epsilon)$. Then the function $(-1)^{f(x)}:\F_2^n \to \{-1,1\}$ can be written as $$(-1)^{f(x)} = \E_{a_1,...,a_k \in \F_2^n}[\alpha_{a_1,...,a_k} (-1)^{f_{a_1,...,a_k}(x)}]$$ where $\alpha_{a_1,...,a_k}$ are real numbers, of absolute value of at most $\frac{10}{\epsilon}$
We will first prove Theorem \[thm:weight\_distrib\_RM\] given Lemma \[lemma:calc\_by\_ders\], and then turn to prove Lemma \[lemma:calc\_by\_ders\]. We will also need the following well-known technical lemma, which shows how to transform calculation by averaging many functions, to approximation by averaging few functions.
\[lemma:calc\_to\_approx\_by few\] Let $f:\F_2^n \to \F_2$ be a function, $H=\{h_1,...,h_t\}$ a set of functions from $\F_2^n$ to $\F_2$, s.t. there exist constants $c_{h_1},...,c_{h_t}$ of absolute value at most $C$, s.t. $$(-1)^{f(x)} = \E_{i \in [t]}[c_{h_i} (-1)^{h_i(x)}]\qquad(\forall x \in \F_2^n)$$ Then $f$ can be approximated by a small number of the functions $h_1,...,h_t$. For any $\delta>0$, there exist functions $h_1,...,h_{\ell} \in H$ for $\ell = O(C^2 \log{1/\delta})$, and a function $F:\F_2^{\ell} \to \F_2$, s.t. the relative distance between $f(x)$ and $F(h_1(x),...,h_{\ell}(x))$ is at most $\delta$, i.e. $$\P_{x \in \F_2^n}[f(x) \ne F(h_1(x),...,h_{\ell}(x))] \le \delta$$ The function $F$ is a weighted majority, i.e. it is of the form: $$F(h_1(x),...,h_{\ell}(x)) = sign(\frac{\sum_{i=1}^{\ell} s_i (-1)^{h_i(x)}}{\ell})$$ where $sign(x)$ is defined by $sign(x)=1$ if $x \ge 0$ and $sign(x)=-1$ if $x<0$. Moreover, we can have $s_1,...,s_{\ell}$ to be integers of absolute value at most $C+1$.
Using Lemmas \[lemma:calc\_by\_ders\] and \[lemma:calc\_to\_approx\_by few\] we now prove Theorem \[thm:weight\_distrib\_RM\].
Fix $1 \le k \le d-1$. We will bound the number of polynomials $p \in RM(n,d)$ s.t. $wt(p) \le 2^{-k}(1-\epsilon)$. Let $p$ be any such polynomial. We apply Lemma \[lemma:calc\_by\_ders\] to $p$. We can write $(-1)^{p(x)}$ as $$(-1)^{p(x)} = \E_{a_1,...,a_k \in \F_2^n}[\alpha_{a_1,...,a_k} (-1)^{p_{a_1,...,a_k}(x)}]$$ such that $|\alpha_{a_1,...,a_k}| \le \frac{10}{\epsilon}$.
We now apply Lemma \[lemma:calc\_to\_approx\_by few\] to the set of polynomials $\{p_{a_1,...,a_k}(x): a_1,...,a_k \in \F_2^n\}$ with $\delta = 2^{-(d+2)}$. We get that there are $\ell = O(\frac{d}{\epsilon^2})$ derivatives $\{p_{a^i_1,...,a^i_k}: i \in [\ell]\}$ s.t. the distance between $p(x)$ and $F(x)$ is at most $\delta$, where $$F(x) = sign(\frac{\sum_{i=1}^{\ell} s_i (-1)^{p_{a^i_1,...,a^i_k}(x)}}{\ell})$$ and $s_1,...,s_{\ell}$ are integers of absolute value at most $O(\frac{1}{\epsilon})$.
We now make an important yet simple observation, that will let us bound the number of low weight polynomials by bounding the number of functions $F(x)$. Given any $F(x)$, there can be at most one $p \in RM(n,d)$ s.t. $dist(F,p) \le \delta$. Assume otherwise that there are two polynomials $p',p'' \in RM(n,d)$ s.t. $dist(p',F) \le \delta$ and $dist(p'',F) \le \delta$. By the triangle inequality $dist(p',p'') \le 2 \delta < 2^{-d}$, but this cannot hold if $p',p''$ are two different polynomials, since the minimum relative distance of $RM(n,d)$ is $2^{-d}$.
So, if we bound the number of different functions $F(x)$ of the above form, we will also bound the number of polynomials $p$ of relative weight at most $2^{-k}(1-\epsilon)$. Consider the terms appearing in $F$:
- We need $\ell = O(\frac{d}{\epsilon^2})$ derivatives and coefficients to describe $F$ completely.
- Any derivative $p_{a^i_1,...,a^i_k}(x)$ is a a polynomial of degree at most $d-k$, and so has at most $2^{{n \choose \le d-k}}$ possibilities.
- Any coefficient $s_i$ has $O(\frac{1}{\epsilon})$ possibilities.
Thus, the total the number of different $F$’s is at most $$\left( 2^{{n \choose \le d-k}} \cdot (1/\epsilon) \right)^{O(\frac{d}{\epsilon^2})} \le
c_1 2^{c_2 \frac{n^{d-k}}{\epsilon^2}}$$ where $c_1 = (1/\epsilon)^{O(d/\epsilon^2)}$ and $c_2 = O(d/(d-k)!)$.
We now turn to prove the Lemmas required for the proof of Theorem \[thm:weight\_distrib\_RM\]. We prove Lemma \[lemma:calc\_by\_ders\] in Subsection \[subsec:proof1\] and Lemma \[lemma:calc\_to\_approx\_by few\] in Subsection \[subsec:proof2\].
Proof of the main technical lemma: Lemma \[lemma:calc\_by\_ders\] {#subsec:proof1}
-----------------------------------------------------------------
Before proving Lemma \[lemma:calc\_by\_ders\], we need some claims regarding derivatives. The first claim shows that if a function has non-zero bias, it can be computed by an average of its derivatives.
\[claim:calc\_by\_single\_der\] Let $g:\F_2^n \to \F_2$ be a function s.t. $bias(g) \ne 0$. Then: $$(-1)^{g(x)} = \frac{1}{bias(g)} \E_{a \in \F_2^n}[(-1)^{g_a(x)}]$$ where the identity holds for any $x \in \F_2^n$.
Fix $x$. We have: $$(-1)^{g(x)} \E_{a \in \F_2^n}[(-1)^{g_a(x)}] = \E_{a \in \F_2^n}[(-1)^{g(x)-g_a(x)}] = \E_{a \in \F_2^n}[(-1)^{g(x+a)}] = bias(g)$$
The following claim shows that if a function has low weight, then derivatives of it will also have low weight, and thus large bias.
\[claim:large\_bias\_for\_der\] Let $f:\F_2^n \to \F_2$ be a function s.t. $wt(f) < 2^{-k}(1 - \epsilon)$. Let $a_1,...,a_s \in \F_2^n$ for $1 \le s \le k-1$ be any derivatives, and consider $bias(f_{a_1,...,a_s})$. Then $bias(f_{a_1,...,a_s}) \ge 1 - 2^{s+1-k}(1-\epsilon)$. In particular:
1. If $s < k-1$ then $bias(f_{a_1,...,a_s}) \ge 1 - 2^{s+1-k}$
2. If $s = k-1$ then $bias(f_{a_1,...,a_s}) \ge \epsilon$
Consider $f_{a_1,...,a_s}$ $$f_{a_1,...,a_s} = \sum_{I \subseteq [s]} f(x + \sum_{i \in I} a_i)$$
For random $x$, the probability that $f(x + \sum_{i \in I} a_i)=1$ is $wt(f)$, which is at most $2^{-k}(1-\epsilon)$. Thus by union bound, $$\P_{x \in \F_2^n}[\exists I \subseteq [s],\ f(x + \sum_{i \in I} a_i)=1] \le 2^{s-k} (1 - \epsilon)$$
In particular it implies that $$wt(f_{a_1,...,a_s}) = \P_{x \in \F_2^n}[f_{a_1,...,a_s}(x)=1] \le 2^{s-k} (1 - \epsilon)$$
and we get the bound since $bias(f_{a_1,...,a_s}) = 1 - 2 wt(f_{a_1,...,a_s})$.
We now can prove Lemma \[lemma:calc\_by\_ders\] using Claims \[claim:calc\_by\_single\_der\] and \[claim:large\_bias\_for\_der\].
Let $f:\F_2^n \to \F_2$ be a function s.t. $wt(f) \le 2^{-k}(1-\epsilon)$. Thus $bias(f) = 1 - 2 wt(f) > 0$ and by Claim \[claim:calc\_by\_single\_der\] we can write: $$(-1)^{f(x)} = \frac{1}{bias(f)} \E_{a_1 \in \F_2^n}[(-1)^{f_{a_1}(x)}]$$
If $k=1$ we are done. Otherwise by Claim \[claim:large\_bias\_for\_der\], $f_{a_1}$ also has positive bias, $$bias(f_{a_1}) \ge 1 - 2^{s+1-k}(1-\epsilon) > 0$$ and so again by Claim \[claim:calc\_by\_single\_der\] we can write $$(-1)^{f_{a_1}(x)} = \frac{1}{bias(f_{a_1})} \E_{a_2 \in \F_2^n}[(-1)^{f_{a_1,a_2}(x)}]$$
Thus we have: $$(-1)^{f(x)} = \frac{1}{bias(f)} \E_{a_1 \in \F_2^n}[\frac{1}{bias(f_{a_1})} \E_{a_2 \in \F_2^n}[(-1)^{f_{a_1,a_2}(x)}]]$$
We can continue this process as long as we can guarantee that $f_{a_1,...,a_s}$ has non-zero bias for all $a_1,...,a_s \in \F_2^n$. By Claim \[claim:large\_bias\_for\_der\] we know this happens for $s \le k-1$, and thus we have: $$(-1)^{f(x)} = \E_{a_1,...,a_k \in \F_2^n}[\alpha_{a_1,...,a_k} (-1)^{f_{a_1,...,a_k}(x)}]$$ where $$\alpha_{a_1,...,a_k} = \frac{1}{bias(f)} \frac{1}{bias(f_{a_1})} \frac{1}{bias(f_{a_1,a_2})} ... \frac{1}{bias(f_{a_1,...,a_{k-1}})}$$
We now bound $\alpha_{a_1,...,a_k}$. By Claim \[claim:large\_bias\_for\_der\] we get that: $$\alpha_{a_1,...,a_k} \le \frac{1}{\epsilon} \prod_{s=1}^{k-2} \frac{1}{1-2^{s-k+1}} \le \frac{1}{\epsilon} \prod_{r \ge 1} \frac{1}{1-2^{-r}} \le \frac{10}{\epsilon}$$
Proof of Approximation by sampling Lemma: Lemma \[lemma:calc\_to\_approx\_by few\] {#subsec:proof2}
----------------------------------------------------------------------------------
Choose $h_1,...,h_{\ell}$ uniformly and independently from $H$. Fix $x \in \F_2^n$, and let $Z_i$ be the random variable $$Z_i = c_{h_i} (-1)^{h_i(x)}$$
and let $S = \frac{Z_1 + ... + Z_{\ell}}{\ell}$. We will use the fact that if $|S - (-1)^{f(x)}| < 1$ then $sign(S) = (-1)^{f(x)}$.
We first bound the probability that $$|S - (-1)^{f(x)}| > 1/4$$
By regular Chernoff arguments for bounded independent variables, since $\E[S] = (-1)^{f(x)}$ and each $Z_i$ is of absolute value of at most $C$, we get that $$\P_{h_1,...,h_{\ell} \in H}[|S - (-1)^{f(x)}| > 1/4] \le e^{-\frac{\ell}{32 C^2}}$$ (see for example Theorem A.1.16 in [@AS00]).
In particular for $\ell = O(C^2 \log{1/\delta})$ we get that $$\P_{h_1,...,h_{\ell} \in H}[|S - (-1)^{f(x)}| > 1/4] \le \delta$$
Thus by averaging arguments, there exists $h_1,...,h_{\ell}$ s.t. $$\P_{x \in \F_2^n}[|\frac{c_{h_1} (-1)^{h_1(x)} + ... + c_{h_{\ell}} (-1)^{h_{\ell}(x)}}{\ell} - (-1)^{f(x)}| \ge 1/4] \le \delta$$
We now round each coefficient to a close rational, without damaging the approximation error. The coefficient of $(-1)^{h_i(x)}$ is $\alpha_i = \frac{c_{h_i}}{\ell}$. If we round $c_{h_i}$ to the closest integer $[c_{h_i}]$, we get that the coefficient of each $(-1)^{h_i(x)}$ is changed by at most $\frac{1}{2\ell}$, and thus the total approximation is changed by at most $1/2$. Hence we have: $$\P_{x \in \F_2^n}[|\frac{[c_{h_1}] (-1)^{h_1(x)} + ... + [c_{h_{\ell}}] (-1)^{h_{\ell}(x)}}{\ell}) - (-1)^{f(x)}| \ge 3/4] \le \delta$$
Thus we got that $$\P_{x \in \F_2^n}[sign(\frac{[c_{h_1}] (-1)^{h_1(x)} + ... + [c_{h_{\ell}}] (-1)^{h_{\ell}(x)}}{\ell}) \ne (-1)^{f(x)}] \le \delta$$
List-decoding size of Reed-Muller codes {#sec:list_RM}
=======================================
In this section we turn to the problem of bounding the list-decoding size of Reed-Muller codes, and we prove the First Main Theorem (Theorem \[thm:main\_list\_decode\_RM\]). We will see that the same techniques we used in Section \[sec:weight\_RM\] to bound the weight distribution, can be applied with minor variants to also bound the list-decoding size.
The list-decoding size of a code is at least the accumulative weight distribution, i.e. $L(\alpha) \ge A(\alpha)$. However, the list-decoding size can sometimes be much larger than the accumulative weight distribution.
Theorem \[thm:main\_list\_decode\_RM\] is a direct corollary of Theorem \[thm:list\_decode\_RM\], giving an upper bound on the list-decoding size at distance $2^{\ell-d}-\epsilon$, and the same lower bound we used to bound the accumulative weight distribution, obtained in Lemma \[lemma:lower\_bound\].
\[thm:list\_decode\_RM\] For any integer $1 \le k \le d-1$, $$L(2^{-k}(1-\epsilon)) \le c_1 2^{c_2 \frac{n^{d-k}}{\epsilon^2} + c_3 \frac{n}{\epsilon^2}}$$ where $c_1 = (1/\epsilon)^{O(d/\epsilon^2)}$, $c_2 = O(d/(d-k)!)$ and $c_3 = O(dk)$. Importantly, $c_1,c_2,c_3$ are independent of $n$, and $c_2,c_3$ are independent of $\epsilon$. In particular for constant $d$ we get that $$L(2^{-k}-\epsilon) \le 2^{O(\frac{n^{d-k}}{\epsilon^2})}$$
The proof will be similar to the proof of Theorem \[thm:weight\_distrib\_RM\]. Fix $f:\F_2^n \to \F_2$ to be any function. We will bound the number of polynomials $p$ of degree at most $d$ s.t. $dist(p,f) \le 2^{-k}(1 - \epsilon)$. Let $p \in RM(n,d)$ be such a polynomial, i.e. $dist(p,f) \le 2^{-k}(1 - \epsilon)$. Let $g(x) = p(x)-f(x)$, then $wt(g) \le 2^{-k}(1 - \epsilon)$. As in the proof of Theorem \[thm:weight\_distrib\_RM\], we use the derivatives of $g$ to approximate $g$. Set $\delta = 2^{-(d+2)}$. By Lemma \[lemma:calc\_by\_ders\] there are $\ell = O(\frac{d}{\epsilon^2})$ derivatives $\{g_{a^i_1,...,a^i_k}: i \in [\ell] \}$ s.t. the distance between $g(x)$ and $F(x)$ is at most $\delta$, where $$F(x) = sign(\frac{\sum_{i=1}^{\ell} s_i (-1)^{g_{a^i_1,...,a^i_k}(x)}}{\ell})$$
Thus we have that $F+f$ approximates $p$, since: $$dist(p,F+f) = dist(p-f,F) \le \delta$$
As in the proof of Theorem \[thm:weight\_distrib\_RM\], given $F$ (and $f$) there can be at most a single $p \in RM(n,d)$ s.t. $dist(p,F+f) \le \delta$, and so if we will bound the number of functions $F$ we will bound the number of codewords close to $f$.
Consider the derivative $g_{a^i_1,...,a^i_k}(x)$ used in the expression for $F$. By linearity of derivation it can be decomposed as $$g_{a^i_1,...,a^i_k}(x) = p_{a^i_1,...,a^i_k}(x) - f_{a^i_1,...,a^i_k}(x)$$
Each $p_{a^i_1,...,a^i_k}(x)$ is a degree $d-k$ polynomial, and so has at most $2^{{n \choose \le d-k}}$ possibilities. Each $f_{a^i_1,...,a^i_k}(x) = \sum_{S \subseteq [k]}f(x + \sum_{j \in S}a^i_j)$ can be described by the values of $a^i_1,...,a^i_k \in \F_2^n$, since we have access to $f$, and so has at most $2^{kn}$ possibilities. Each coefficient $s_i$ has $O(1/\epsilon)$ possibilities. Thus, in total the number of different $F$’s is at most $$\left( 2^{{n \choose \le d-k} + kn} \cdot (1/\epsilon) \right)^{O(\frac{d}{\epsilon^2})}
\le c_1 2^{c_2 \frac{n^{d-k}}{\epsilon^2} + c_3 \frac{n}{\epsilon^2}}$$ where $c_1 = (1/\epsilon)^{O(d/\epsilon^2)}$, $c_2 = O(d/(d-k)!)$ and $c_3 = O(kd)$.
[*Acknowledgement.*]{} The second author would like to thank his advisor, Omer Reingold, for on-going advice and encouragement. He would also like to thank Microsoft Research for their support during his internship.
[99]{}
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V. Guruswami, [*List decoding of Error-Correcting Codes*]{}, vol 3282 of Lecture notes in Computer Science, Springer 2004.
T. Kaufman and S. Lovett, [*Worst case to Average Case Reductions for Polynomials*]{}, To appear in the Proceedings of the 49th Annual Symposium on Foundations of Computer Science (FOCS), 2008.
E. Kushilevitz and Y. Mansour, [*Learning Decision Trees using the Fourier Spectrum*]{}, SIAM Journal of Computing, 22(6), (1993), pp 1331-1348.
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Generalized Reed-Muller codes {#sec:GRM}
=============================
The problems of bounding both the accumulative weight distribution and the list-decoding size can be extended to Generalized Reed-Muller, the code of low-degree polynomials over larger fields. However, our techniques fail to prove tight result in these cases. We briefly describe the reasons below, and give some partial results.
We start by making some basic definitions. Let $q$ be a prime, and let $GRM_q(n,d)$ denote the code of multivariate polynomials $p(x_1,...,x_n)$ over the field $\F_q$, of total degree at most $d$.
The relative weight of a function $f:\F_q^n \to \F_q$ is the fraction of non-zero elements, $$wt(f) = \frac{1}{q^n} |\{x \in \F_q^n: f(x) \ne 0\}|$$
The relative distance between two functions $f,g:\F_q^n \to \F_q$ is defined as $$dist(f,g) = \P_{x \in \F_q^n}[f(x) \ne g(x)]$$
The accumulative weight distribution and the list-decoding size are defined analogously for $GRM_q(n,d)$, using the appropriate definitions for relative weight and relative distance. We denote them by $A_q$ and $L_q$. For each $1 \le k \le d$, we define a distance $r_k$:
1. For $k=1$, let $d=(q-1)a + b$, where $1 \le b \le q-1$. Define $r_1 = q^{-a} (1-b/q)$.
2. For $2 \le k \le d-1$, let $d-k=(q-1)a + b$, where $1 \le b \le q-1$. Define $r_k = q^{-a} (1-b/q) (1-1/q)$.
3. For $k=d$, define $r_d = 1-1/q$.
We conjecture that both for the accumulative weight distribution and the list-decoding size, the distances $r_k$ are the thresholds for the exponential dependency in $n$:
\[conj:GRM\_asymptotics\] Let $\epsilon>0$ be constant, and consider $GRM_q(n,d)$ for constant $d$. Then:
- For $\alpha \le r_1 - \epsilon$ both $A_q(\alpha)$ and $L_q(\alpha)$ are constants.
- For $r_k \le \alpha \le r_{k+1}-\epsilon$ both $A_q(\alpha)$ and $L_q(\alpha)$ are $2^{\Theta(n^k)}$.
- For $\alpha \ge r_d$ both $A_q(\alpha)$ and $L_q(\alpha)$ are $2^{\Theta(n^d)}$.
Proving lower bounds for $A_q(r_k)$ is similar to the case of $RM(n,d)$.
\[lemma:lower\_bound\_GRM\] For any integer $1 \le k \le d$, $$A_q(r_k) \ge 2^{\Omega(n^k)}$$
The problem is proving matching upper bounds. Using directly the derivatives method we used to give upper bounds for $RM(n,d)$ gives the same bounds for $GRM_q(n,d)$, alas they are not tight for $q>2$: $$A_q(2^{-k}-\epsilon) \le 2^{O(n^{d-k})}$$ If we would like to get upper bounds closer to the lower bounds, a natural approach would be to generalize Lemma \[lemma:calc\_by\_ders\] to taking several derivatives in the same direction (which is possible over larger fields). This would give us tight results for some values of $k$, if we could also generalize Claim \[claim:calc\_by\_single\_der\] to the case of taking a multiple derivative in the same direction. However, we didn’t find a way of doing so.
Instead, we give partial results for Conjecture \[conj:GRM\_asymptotics\] in the two ends of the scale: when $\alpha \le r_1 - \epsilon$, and when $r_{d-1} \le \alpha \le r_d - \epsilon$ (when $\alpha \ge r_d$ Lemma \[lemma:lower\_bound\_GRM\] gives $L_q(\alpha)$ and $A_q(\alpha)$ are both exponential in $n^d$).
First, the minimal distance of $GRM_q(n,d)$ is known to be $r_1$. Thus, for any $\epsilon > 0$, $A_q(r_1 - \epsilon) = 1$. Gopalan, Klivans and Zuckerman [@GKZ08] prove that $L_q(r_1 - \epsilon)$ is constant when $q-1$ divides $d$:
Assume $q-1$ divides $d$. Then: $$L_q(r_1 - \epsilon) \le c(q,d,\epsilon)$$
Moving to the case of $r_{d-1} \le \alpha \le r_d - \epsilon$, we prove:
\[lemma:upper\_bound\_GRM\_r\_d\] Let $\epsilon>0$ be constant. then: $$A_q(r_d - \epsilon) \le 2^{O(n^{d-1})}$$
We now move on to prove Lemmas \[lemma:lower\_bound\_GRM\] and \[lemma:upper\_bound\_GRM\_r\_d\]. We start with Lemma \[lemma:lower\_bound\_GRM\]:
We start by proving for $2 \le k \le d-1$. Let $d-k=(q-1)a + b$, where $1 \le b \le q-1$. Single out $a+2$ variables $x_1,...,x_{a+2}$, and let $g$ be any degree $k$ polynomial on the remaining variables. The following polynomial has degree $d$ and weight exactly $q^{-a} (1-b/q) (1-1/q)$: $$g'(x_1,...,x_n) = \left( \prod_{i=1}^{a} \prod_{j=1}^{q-1} (x_i-j) \right)
\left( \prod_{j=1}^{b} (x_{a+1}-j) \right)
\left(x_{a+2} + g(x_{a+3},...,x_n)\right)$$ The number of distinct polynomial $g$ is $2^{\Omega(n^d)}$.
The proofs for $k=1$ and $k=d$ are similar: for $k=1$, let $d = (q-1)a+b$. Let $l_1(x),...,l_{a+1}(x)$ be any independent linear functions, and consider $$g'(x_1,...,x_n) = \left( \prod_{i=1}^{a} \prod_{j=1}^{q-1} (l_i(x)-j) \right)
\left( \prod_{j=1}^{b} (l_{a+1}(x)-j) \right)$$ For $k=d$, let $g$ be any degree $d$ polynomial on variables $x_2,...,x_n$, and consider $g'(x_1,...,x_n) = x_1 + g(x_2,...,x_n)$.
We now continue to prove Lemma \[lemma:upper\_bound\_GRM\_r\_d\]. We first make some necessary definitions.
The bias of a polynomial $p(x_1,...,x_n)$ over $\F_q$ is defined to be $$bias(p) = \E_{x \in \F_q^n}[\omega^p(x)]$$ where $\omega=e^{2 \pi i /q}$ is a primitive $q$-th root of unity.
Kaufman and Lovett [@KL08] prove that biased low-degree polynomials can be decomposed into a function of a constant number of lower degree polynomials:
\[thm:bias\_implies\_low\_rank\] Let $p(x_1,...,x_n)$ be a degree $d$ polynomial, s.t. $|bias(p)| \ge \epsilon$. Then $p$ can be decomposed as a function of a constant number of lower degree polynomials: $$p(x) = F(g_1(x),...,g_c(x))$$ where $deg(g_i) \le d-1$, and $c=c(q,d,\epsilon)$.
We will use Theorem \[thm:bias\_implies\_low\_rank\] to bound $A(r_d - \epsilon)$ for any constant $\epsilon>0$.
We will show that any polynomial $p \in GRM_q(n,d)$ s.t. $wt(p) \le 1-1/p - \epsilon$ can be decomposed as $$p(x) = F(g_1(x),...,g_c(x))$$ where $deg(g_i) \le d-1$, and $c$ depends only on $q,d$ and $\epsilon$. Thus the number of such polynomials is bounded by the number of possibilities to choose $c$ degree $d-1$ polynomials, and a function $F:\F_q^c \to \F_q$. The number of such possibilities is at most $2^{O(n^{d-1})}$. Let $p$ be s.t. $wt(p) \le 1-1/p-\epsilon$. We will show there exists $\alpha \in \F_q$, $\alpha \ne 0$ s.t. $bias(\alpha p) \ge \epsilon$. We will then finish by using Theorem \[thm:bias\_implies\_low\_rank\] on the polynomial $\alpha p$.
Consider the bias of $\alpha p$ for random $\alpha \in \F_q$: $$\E_{\alpha \in \F_q}[bias(\alpha p)] = \E_{\alpha \in F_q, x \in \F_q^n}[\omega^{\alpha p(x)}] = 1 - wt(p)$$ since for $x$’s for which $p(x)=0$, $\E_{\alpha \in F_q}[\omega^{\alpha p(x)}]=1$, and for $x$ s.t. $p(x) \ne 0$, $\E_{\alpha \in F_q}[\omega^{\alpha p(x)}]=0$. We thus get that: $$\E_{\alpha \in \F_q \setminus \{0\} }[bias(\alpha p)] = 1 - \frac{q}{q-1} wt(p) \ge \frac{q}{q-1} \epsilon$$ So, there must exist $\alpha \ne 0$ s.t. $bias(\alpha p) \ge \epsilon$.
[^1]: Research supported in part by NSF Awards CCF-0514167 and NSF-0729011.
[^2]: Research supported partly by the Israel Science Foundation (grant 1300/05). Research was conducted partly when the author was an intern at Microsoft Research.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'In this paper we propose an approach for monocular 3D object detection from a single RGB image, which leverages a novel disentangling transformation for 2D and 3D detection losses and a novel, self-supervised confidence score for 3D bounding boxes. Our proposed loss disentanglement has the twofold advantage of simplifying the training dynamics in the presence of losses with complex interactions of parameters, and sidestepping the issue of balancing independent regression terms. Our solution overcomes these issues by isolating the contribution made by groups of parameters to a given loss, without changing its nature. We further apply loss disentanglement to another novel, signed Intersection-over-Union criterion-driven loss for improving 2D detection results. Besides our methodological innovations, we critically review the AP metric used in KITTI3D, which emerged as the most important dataset for comparing 3D detection results. We identify and resolve a flaw in the 11-point interpolated AP metric, affecting all previously published detection results and particularly biases the results of monocular 3D detection. We provide extensive experimental evaluations and ablation studies on the KITTI3D and nuScenes datasets, setting new state-of-the-art results on object category car by large margins.'
author:
- |
Andrea Simonelli$^{{\ensuremath\vcenter{\hbox{\includegraphics[width=.5em]{compass_arrow_dark.pdf}}}},\star}$, Samuel Rota Bulò$^{{\ensuremath\vcenter{\hbox{\includegraphics[width=.5em]{compass_arrow_dark.pdf}}}}}$, Lorenzo Porzi$^{{\ensuremath\vcenter{\hbox{\includegraphics[width=.5em]{compass_arrow_dark.pdf}}}}}$, Manuel López-Antequera$^{{\ensuremath\vcenter{\hbox{\includegraphics[width=.5em]{compass_arrow_dark.pdf}}}}}$, Peter Kontschieder$^{{\ensuremath\vcenter{\hbox{\includegraphics[width=.5em]{compass_arrow_dark.pdf}}}}}$\
$^{{\ensuremath\vcenter{\hbox{\includegraphics[width=.5em]{compass_arrow_dark.pdf}}}}}$Mapillary Research, $^\star$University of Trento\
[[email protected]]{}\
title: ' Disentangling Monocular 3D Object Detection\'
---
Introduction
============
Recent developments in object recognition [@Liu_2018_DetSurvey] have led to near-human performance on monocular 2D detection tasks. For applications with given, realistic accuracy requirements or constraints on computational budget, it is possible to choose general-purpose 2D object detectors from a large pool [@Ren+15; @Liu2016; @Redmon2016; @Wu_2017_CVPR_Workshops; @Redmon_2017_CVPR; @Lin+17; @Law_2018_ECCV].
The performance situation considerably changes in the 3D object detection case. Even though there are promising methods based on multi-sensor fusion (usually exploiting LIDAR information [@Liang_CVPR_2019; @Wang_arxiv_2019; @Shin_arxiv_18; @shi2018pointrcnn] next to RGB images), 3D detection results produced from a single, monocular RGB input image lag considerably behind. This can be attributed to the ill-posed nature of the problem, where a lack of explicit knowledge about the unobserved depth dimension introduces ambiguities in 3D-to-2D mappings and hence significantly increases the task complexity.
To still enable 3D object detection from monocular images, current works usually make assumptions about the scene geometry, camera setup or the application (that cars cannot fly [@qin2019monogrnet]). The implementation of such priors determines the encoding of extent and location/rotation of the 3D boxes, the corresponding 2D projections or their 3D box center depths. The magnitudes of these parameters have different units and therefore non-comparable meanings, which can negatively affect the optimization dynamics when error terms based on them are directly combined in a loss function. As a consequence, state-of-the-art, CNN-based monocular 3D detection methods [@Manhardt_2019_CVPR; @qin2019monogrnet] report to train their networks in a stage-wise way. First the 2D detectors are trained until their performance stabilizes, before 3D reasoning modules can be integrated. While stage-wise training *per se* is not unusual in the context of deep learning, it could be an indication that currently used loss functions are yet sub-optimal.
A significant amount of recent works are focusing their experimental analyses on the KITTI3D dataset [@Geiger2012CVPR], and in particular its *Car* category [@Manhardt_2019_CVPR; @qin2019monogrnet; @Roddick18; @Xu_2018_CVPR]. The availability of suitable benchmark datasets confines the scope of experimental analyses and when only few datasets are available, progress in the research field is strongly tied to the expressiveness of used evaluation metrics. KITTI3D adopted the *11-point Interpolated Average Precision* metric [@Salton1986] used in the PASCAL VOC2007 [@Everingham2010] challenge. We found a major flaw in the metric where using a single, confident detection result per difficulty category (KITTI3D distinguishes between *easy*, *moderate* and *hard* samples) suffices to obtain AP scores of $\approx9\%$ on a dataset level, which is up to 3$\times$ higher than the performance reported by recent works [@NIPS2015_Chen; @Chen_2016_CVPR; @TongHe_2019_arxiv; @Xu_2018_CVPR].
The contributions of our paper disentangle the task of monocular 3D object detection at several levels. Our major technical contribution *disentangles* dependencies of different parameters by isolating and handling parameter groups individually at a loss-level. This overcomes the issue of non-comparability for parameter magnitudes, while preserving the nature of the final loss. Our loss disentanglement significantly improves losses on both, 2D and 3D tasks. It also enables us to effectively train the entire CNN architecture (2D+3D) together and end-to-end, without the need of hyperparameter-sensitive, stage-wise training or warm-up phases. As additional contributions we i) leverage 2D detection performance through a novel loss based on a *signed Intersection-over-Union* criterion and ii) introduce a loss term for predicting detection confidence scores of 3D boxes, learned in a self-supervised way.
Another major contribution is a critical review of the 3D metrics used to judge progress in monocular 3D object detection, with particular focus on the predominantly used KITTI3D dataset. We observe that a flaw in the definition of the 11-point, interpolated AP metric significantly biases 3D detection results at the performance level of current state-of-the-art methods. Our applied correction, despite bringing *all works evaluating on KITTI3D* back down to earth, more adequately describes their true performance.
For all our contributions, we provide ablation studies on the KITTI3D and the novel nuScenes [@Cae+19] driving datasets. Fair comparisons indicate that our work considerably improves over current monocular 3D detection methods.
Related Work
============
We review the most recent, related works from 3D object detection and group them according to the data modalities used therein. After discussing RGB-only works just like ours, we list works exploiting also depth and/or synthetic data augmentation or 3D shape information, before finalizing with a high-level summary about LIDAR and/or stereo-based approaches.
#### RGB images only.
Deep3DBox [@Mousavian_2017_CVPR] proposed to estimate full 3D pose and object dimensions from a 2D box by exploiting constraints from projective geometry. The core idea is that the perspective projection of a 3D bounding box should fit tightly to at least one side of its corresponding 2D box detection. In SSD-6D [@Kehl_2017_ICCV] an initial 2D detection hypothesis is lifted to provide 6D pose of 3D objects by using structured discretizations of the full rotational space. 3D model information is learned by only training from synthetically augmented datasets. OFTNet [@Roddick18] introduces an orthographic feature transform, mapping features extracted from 2D to a 3D voxel map. The voxel map’s features are eventually reduced to 2D (birds-eye view) by integration along the vertical dimension, and detection hypotheses are efficiently processed by exploiting integral-image representations. Mono3D [@Chen_2016_CVPR] emphasized on generation of 3D candidate boxes, scored by different features like class semantics, contour, shape and location priors. Even though at test time the results are produced based on single RGB images only, their method also requires semantic and instance segmentation results as input. The basic variant (w/o using depth) of ROI-10D [@Manhardt_2019_CVPR] proposes a novel loss to lift 2D detection, orientation and scale into 3D space that can be trained in an end-to-end fashion. FQNet [@Liu+19] infers a fitting quality criterion in terms of 3D IoU scores, allowing them to filter estimated 3D box proposals based on using only 2D object cues. MonoGRNet [@qin2019monogrnet] is the current state-of-the-art for RGB-only input, using a CNN comprised of four sub-networks for 2D detection, instance depth estimation, 3D location estimation and local corner regression, respectively. The three latter sub-networks emphasize on geometric reasoning, instance depth estimation predicts the central 3D depth of the nearest object instance, 3D location estimation seeks for the 3D bounding box center by exploiting 3D to 2D projections at given instance depth estimations, and local corner regression directly predicts the eight 3D bounding box corners in a local (or allocentric [@Kundu_2018_CVPR; @Manhardt_2019_CVPR] way). It is relevant to mention that [@qin2019monogrnet] reports that training was conducted stage-wise: First, the backbone is trained together with the 2D detector using Adam. Next, the geometric reasoning modules are trained (also with Adam). Finally, the whole network is trained end-to-end using stochastic gradient descent. The work in [@Barabanau_arXiv_2019] learns to estimate correspondences between detected 2D keypoints and 3D counterparts. However, this requires manual annotations on the surface of 3D CAD models and is limited in dealing with occluded objects.
#### Including depth.
An expansion stage of ROI-10D [@Manhardt_2019_CVPR] takes advantage of depth information provided by SuperDepth [@Pillai_2019_ICRA], which itself is learned in a self-supervised manner. In [@Xu_2018_CVPR], a multi-level fusion approach is proposed, exploiting disparity estimation results from a pre-trained module during both, the 2D box proposal generation stage as well as the 3D prediction part of their network.
#### Including 3D shape information.
3D-RCNN [@Kundu_2018_CVPR] exploits the idea of using inverse graphics for instance-level, amodal 3D shape and pose estimation of all object instances per image. They propose a differentiable Render-and-Compare loss, exploiting available 2D annotations in existing datasets for guiding optimization of 3D object shape and pose. In [@Zia_2014_CVPR], the recognition task is tackled by jointly reasoning about the 3D shape of multiple objects. Deep-MANTA [@Chabot_2017_CVPR] uses 3D CAD models and annotated 3D parts in a coarse-to-fine localization process. The work in [@Murthy_17_ICRA] encodes shape priors using keypoints for recovering the 3D pose and shape of a query object. In Mono3D++ [@TongHe_2019_arxiv], the 3D shape and pose for cars is provided by using a morphable wireframe, and it optimizes projection consistency between generated 3D hypotheses and corresponding, 2D pseudo-measurements.
#### LIDAR and/or stereo-based.
3DOP [@NIPS2015_Chen] exploits stereo images and prior knowledge about the scene to directly reason in 3D. Stereo R-CNN [@cvpr19stereorcnn] tackles 3D object detection by exploiting stereo imagery and produces stereo boxes, keypoints, dimensions and viewpoint angles, summarized in a learned 3D box estimation module. In MV3D [@Chen_2017_CVPR], a sensor-fusion approach for LIDAR and RGB images is presented, approaching 3D object proposal generation and multi-view feature fusion via individual sub-networks. Conversely, Frustrum-PointNet [@Qi_2018_CVPR] directly operates on LIDAR point clouds and aligns candidate points provided from corresponding 2D detections for estimating the final, amodal 3D bounding boxes. PointRCNN [@shi2018pointrcnn] describes a 2-stage framework where the first stage provides bottom-up 3D proposals and the second stage refines them in canonical coordinates. RoarNet [@Shin_arxiv_18] applies a 2D detector to first estimate 3D poses of objects from a monocular image before processing corresponding 3D point clouds to obtain the final 3D bounding boxes.
Task Description
================
We address the problem of monocular 3D object detection, where the input is a single RGB image and the output consists in a 3D bounding box, expressed in camera coordinates, for each object that is present in the image (see, Fig. \[fig:catchy\]). As opposed to other methods in the literature, we do *not* take additional information as input like depth obtained from LIDAR or other supervised or self-supervised monocular depth estimators. Also the training data consists solely of RGB images with corresponding annotated 3D bounding boxes. Nonetheless, we require a calibrated setting so we assume that per-image calibration parameters are available both at training and test time.
Proposed Architecture
=====================
We adopt a two-stage architecture that shares a similar structure with the state-of-the-art [@Manhardt_2019_CVPR]. It consists of a single-stage 2D detector (*first stage*) with an additional 3D detection head (*second stage*) constructed on top of features pooled from the detected 2D bounding boxes. Details of the architecture are given below.
Backbone
--------
The backbone we use is a ResNet34 [@He2015b] with a Feature Pyramid Network (FPN) [@Lin2016] built on top of it. The FPN network has the same structure as in [@Lin+17] with 3+2 scales, connected to the output of modules 3, 4 and 5 of ResNet34, corresponding to downsampling factors of $\times 8$, $\times 16$ and $\times 32$, respectively. Our ResNet34 differs from the standard one by replacing BatchNorm+ReLU layers with the synchronized version of InPlaceABN ([iABN$^\text{sync}$]{}) activated with LeakyReLU with negative slope $0.01$ as proposed in [@RotPorKon18a]. This modification does not affect the performance of the network, but allows to free up a significant amount of GPU memory, which can be exploited to scale up the batch size or input resolution. All FPN blocks depicted in Fig. \[fig:backbone\] correspond to $3\times 3$ convolutions with $256$ channels, followed by [iABN$^\text{sync}$]{}.
#### Inputs.
The input $x$ to the backbone is a single RGB image.
#### Outputs.
The backbone provides $5$ output tensors $\{f_1, \ldots, f_5\}$ corresponding to the $5$ different scales of the FPN network, covering downsampling factors of $\times 8$, $\times 16$, $\times 32$, $\times 64$, and $\times 128$, each with $256$ feature channels (see, Fig. \[fig:backbone\]).
![Backbone architecture. Rectangles in the “FPN” block represent convolutions followed by [iABN$^\text{sync}$]{}.[]{data-label="fig:backbone"}](figure02.pdf){width=".8\columnwidth"}
2D Detection Head {#sec:head2D}
-----------------
We consider the head of the single-stage 2D detector implemented in RetinaNet [@Lin+17], which applies a detection module independently to each output $f_i$ of the backbone described above. The detection modules share the same parameters but work inherently at different scales, according to the scale of the features that they receive as input. As opposed to the standard RetinaNet, we employ [iABN$^\text{sync}$]{}also in this head. The head, depicted in Fig. \[fig:head2d\], is composed of two parallel stacks of $3\times 3$ convolutions, and is parametrized by ${\ensuremath{\mathsf{n}}}_a$ reference bounding box sizes (anchors) per scale level.
#### Inputs.
The inputs are the $5$ outputs $\{f_1, \ldots, f_5\}$ of the backbone, where $f_i$ has a spatial resolution of ${\ensuremath{\mathsf{h}}}_i\times {\ensuremath{\mathsf{w}}}_i$.
#### Outputs.
For each image, and each input tensor $f_i$, the 2D detection head generates ${\ensuremath{\mathsf{n}}}_a$ bounding box proposals (one per anchor) for each spatial cell $g$ in the ${\ensuremath{\mathsf{h}}}_i\times {\ensuremath{\mathsf{w}}}_i$ grid. Each proposal for a given anchor $a$ with size $(w_a, h_a)$ is encoded as a $5$-tuple $(\zeta_{{\mathtt{2D}}}, \delta_u, \delta_v, \delta_w, \delta_h)$ such that
- $p_{{\mathtt{2D}}}=(1+e^{-\zeta_{\mathtt{2D}}})^{-1}$ gives the confidence of the 2D bounding box prediction,
- $(u_b,v_b)=(u_g+\delta_u w_a,v_g+\delta_v h_a)$ gives the center of the bounding box with $(u_g,v_g)$ being the image coordinates of cell $g$, and
- $(w_b,h_b)=(w_a e^{\delta_w}, h_a e^{\delta_h})$ gives the bounding box size.
Fig. \[fig:notation\] gives a visual description of the head’s outputs.
![2D detection module. Rectangles represent convolutions. All convolutions but the last per row are followed by [iABN$^\text{sync}$]{}.[]{data-label="fig:head2d"}](figure03.pdf){width=".8\columnwidth"}
#### Losses.
We employ the focal loss [@Lin+17] to train the bounding box confidence score. This loss takes the following form, for a given cell $g$ and anchor $a$ with target confidence $y\in\{0,1\}$ and predicted confidence $p\in[0,1]$: $$L_\mathtt{2D}^\text{conf}(p_\mathtt{2D},y) = -\alpha y(1-p_\mathtt{2D})^\gamma \log p_\mathtt{2D} - \bar\alpha \bar y p_\mathtt{2D}^\gamma\log(1-p_\mathtt{2D})\,,$$ where $\alpha\in[0,1]$ and $\gamma>0$ are hyperparameters that modulate the importance of errors and positives, respectively, $\bar \alpha=1-\alpha$ and $\bar y=1-y$. The confidence target $y$ does not depend on the regressed bounding box, but only on the cell $g$ and the anchor $a$. It takes value $1$ if the reference bounding box centered in $(u_g,v_g)$ with size $(w_a,h_a)$ exhibits an Intersection-over-Union (IoU) with a ground-truth bounding box larger than a given threshold $\tau_\text{iou}$. For each cell $g$ and anchor $a$ that matches a ground-truth bounding box ${\ensuremath{\boldsymbol{\hat b}}}$ with predicted bounding box ${\ensuremath{\boldsymbol{b}}}=(u_b-\frac{w_b}{2}, v_b-\frac{h_b}{2}, u_b+\frac{w_b}{2}, v_b+\frac{h_b}{2})$ we consider the following detection loss: $$\label{eq:det2d}
L_\mathtt{2D}^\text{bb}({\ensuremath{\boldsymbol{b}}}, {\ensuremath{\boldsymbol{\hat b}}})=1-\text{sIoU}({\ensuremath{\boldsymbol{b}}},{\ensuremath{\boldsymbol{\hat b}}})\,,$$ where $\text{sIoU}$ represents an extension of the common IoU function, which prevents gradients from vanishing in case of non-overlapping bounding boxes. We call it *signed* IoU function, as, intuitively, it creates negative intersections in case of disjoint bounding boxes (see, Appendix \[sec:siou\]). In Sec. \[sec:disentangling\], we discuss a disentangling transformation of the loss in Eq. that allows to isolate the contribution of each network’s output to the loss, while preserving the fundamental nature of the loss.
#### Output Filtering.
The dense output of the 2D head is filtered as in [@Lin+17]: first, detections with scores lower than $0.05$ are discarded, then Non-Maxima Suppression (NMS) with IoU threshold $0.5$ is performed on the $5000$ top-scoring among the remaining ones, and the best $100$ are kept.
3D Detection Head {#sec:head3D}
-----------------
The 3D detection head (Fig. \[fig:head3d\]) regresses a 3D bounding box for each 2D bounding box returned by the 2D detection head (surviving the filtering step). It starts by applying ROIAlign [@He2017] to pool features from FPN into a $14\times 14$ grid for each 2D bounding box, followed by $2\times 2$ average pooling, resulting in feature maps with shape $7\times 7\times 128$. The choice of which FPN output is selected for each bounding box ${\ensuremath{\boldsymbol{b}}}$ follows the same logic as in [@Lin2016], namely the features are pooled from the output $f_k$, where $k=\min(5,\max(1,\lfloor 2 + \log_2(\sqrt{w_bh_b}/224) \rfloor))$. On top of this, two parallel branches of fully connected layers with 512 channels compute the outputs detailed below. Each fully connected layer but the last one per branch is followed by [iABN]{}(non-synchronized).
#### Input.
The inputs are a 2D bounding box proposal ${\ensuremath{\boldsymbol{b}}}$ returned by the 2D detection head and features $f_k$ from the backbone.
#### Output.
The head returns for each 2D proposal ${\ensuremath{\boldsymbol{b}}}$ with center $(u_b, v_b)$ and dimensions $(w_b, h_b)$ a 3D bounding box encoded in terms of a $10$-tuple ${\ensuremath{\boldsymbol{\theta}}}=(\delta z, \Delta_u, \Delta_v, \delta_W, \delta_H, \delta_D, q_r, q_i, q_j, q_k)$ and an additional output $\zeta_\mathtt{3D}$ such that
- $p_\mathtt{3D|2D}=(1+e^{-\zeta_{\mathtt{3D}}})^{-1}$ represents the confidence of the 3D bounding box prediction given the 2D proposal,
- $z=\mu_z+\sigma_z\delta_z$ represents the depth of the center ${\ensuremath{\boldsymbol{C}}}$ of the predicted 3D bounding box, where $\mu_z$ and $\sigma_z$ are given, dataset-wide depth statistics,
- ${\ensuremath{\boldsymbol{c}}}=(u_b+\Delta_u, v_b+\Delta_v)$ gives the position of ${\ensuremath{\boldsymbol{C}}}$ projected on the image plane (in image coordinates),
- ${\ensuremath{\boldsymbol{s}}}=(W_0 e^{\delta_W}, H_0 e^{\delta_H},D_0 e^{\delta_D})$ is the size of the 3D bounding box, where $(W_0, H_0, D_0)$ is a given, dataset-wide reference size, and
- ${\ensuremath{\boldsymbol{q}}}=q_r+q_i{\mathtt{i}} +q_j{\mathtt{j}} +q_k{\mathtt{k}}$ is the quaternion providing the pose of the bounding box with respect to an *allocentric* [@Kundu_2018_CVPR], local coordinate system.
Fig. \[fig:notation\] gives a visual description of the head’s outputs.
![3D detection head. “FC” rectangles represent fully connected layers. All FCs except the last of each row are followed by [iABN]{}.[]{data-label="fig:head3d"}](figure04.pdf){width="\columnwidth"}
#### Losses.
Let $\theta$ be the $10$-tuple representing the regressed 3D bounding box and let $\hat B\in\mathbb R^{3\times 8}$ be the ground-truth 3D bounding box in camera coordinates. By applying the lifting transformation ${\ensuremath{\mathcal{F}}}$ introduced in [@Manhardt_2019_CVPR] and reviewed in Appendix \[sec:lifting\], we obtain the predicted 3D bounding box $B$ given the network’s output ${\ensuremath{\boldsymbol{\theta}}}$, $B={\ensuremath{\mathcal{F}}}({\ensuremath{\boldsymbol{\theta}}})$. The loss on the 3D bounding box regression is then given by $$\label{eq:det3d}
L_\mathtt{3D}^\text{bb}(B, \hat B)=\frac{1}{8}\Vert B-\hat B\Vert_\text{H}\,,$$ where $\Vert\cdot\Vert_\text{H}$ denotes the Huber loss with parameter $\delta_H$ applied component-wise to each element of the argument matrix. The loss for the confidence $p_\mathtt{3D|2D}$ about the predicted 3D bounding box is self-supervised by the 3D bounding box loss remapped into a probability range via the transformation $\hat p_\mathtt{3D|2D}=e^{-\frac{1}{T}L_\mathtt{3D}^\text{bb}(B, \hat B)}$, where $T>0$ is a temperature parameter. The confidence loss for the 3D bounding box is then the standard binary cross entropy loss: $$L_\text{3D}^\text{conf}(p_\mathtt{3D|2D}, \hat p_\mathtt{3D|2D})=-\hat p\log p -(1-\hat p)\log(1-p)\,,$$ where we have omitted the subscripts for the sake of readability. This loss allows to obtain a more informed confidence about the quality of the returned 3D bounding box than just using the 2D confidence. Akin to the 2D case, we employ also a different variant of Eq. that disentangles the contribution of groups of parameters in order to improve the stability and effectiveness of the training. Yet, the confidence computation will be steered by Eq. .
#### Output Filtering.
The final output will be filtered based on a combination of the 2D and 3D confidences, following a Bayesian rule. The 3D confidence $p_\mathtt{3D|2D}$ is implicitly conditioned on having a valid 2D bounding box and the latter probability is reflected by $p_\mathtt{2D}$. At the same time the confidence of a 3D bounding box given an invalid 2D bounding box defaults to $0$. Hence, the unconditioned 3D confidence can be obtained by the law of total probability as $$p_\mathtt{3D}=p_\mathtt{3D|2D}p_\mathtt{2D}\,.$$ This is the final confidence that our method associates to each 3D detection and that is used to filter the predictions via a threshold $\tau_\text{conf}$. We do not perform further NMS steps on the regressed 3D bounding boxes nor filtering based on 3D prior knowledge (one could reduce false positives by dropping “flying” cars).
![Visualization of the semantics of the outputs of the 2D and 3D detection heads. Left: 2D bounding box regression on image plane. Center: 3D bounding box regression. Right: allocentric angle from bird-eye view. []{data-label="fig:notation"}](figure05.pdf){width="\columnwidth"}
Disentangling 2D and 3D Detection Losses {#sec:disentangling}
========================================
In this section we propose a transformation that can be applied to the 2D bounding box loss $L_\text{2D}^{bb}$ and the 3D counterpart $L_\text{3D}^{bb}$, as well as a broader set of loss functions. We call it *disentangling* transformation because it isolates the contribution of groups of parameters to a given loss, while preserving its inherent nature. Each parameter group keeps its independent loss term, but they are all made comparable, thus sidestepping the difficulty of finding a proper weighting. While losses that combine parameters in a single term, such as those in Eq. and Eq. , are immune to the balancing issue, they might exhibit bad dynamics during the optimization as we will show with a toy experiment. The transformation we propose, instead, retains the best of both worlds.
{width=".48\textwidth"} {width=".48\textwidth"}
{width="\textwidth"}
Disentangling Transformation
----------------------------
Let $L:{\ensuremath{\mathcal{Y}}}\times{\ensuremath{\mathcal{Y}}}\to\mathbb R_+$ be a loss function defined on a space ${\ensuremath{\mathcal{Y}}}$ (the space of 3D bounding boxes) such that $L(y,\hat y)=0$ if $\hat y=y$. Let $\Theta\subset\mathbb R^{\ensuremath{\mathsf{d}}}$ be a set of possible network outputs that can be mapped to elements of ${\ensuremath{\mathcal{Y}}}$ via a function $\psi$ that we assume to be one-to-one. This property holds for 2D bounding boxes via the common 4D parametrization (center + dimensions), as well as for the 3D bounding boxes via the 10D representation described in Sec. \[sec:head3D\]. In the latter case, $\psi$ coincides with the lifting transformation ${\ensuremath{\mathcal{F}}}$. Let $\hat y$ be a fixed output element (a ground-truth bounding box) and consider a partitioning of the ${\ensuremath{\mathsf{d}}}$ dimensions of $\Theta$ into ${\ensuremath{\mathsf{k}}}$ groups. To give a concrete example, in case of 2D bounding boxes we can have $2$ groups of parameters: one for the dimensions, and one for the center. In the case of $3D$ bounding boxes we consider $4$ groups related intuitively to depth, projected center, rotation and dimensions. Given ${\ensuremath{\boldsymbol{\theta}}}\in\Theta$ we denote by ${\ensuremath{\boldsymbol{\theta}}}_j$ the sub-vector corresponding to the $j$th group and by ${\ensuremath{\boldsymbol{\theta}}}_{-j}$ the sub-vector corresponding to all but the $j$th group. Moreover, given ${\ensuremath{\boldsymbol{\theta}}},{\ensuremath{\boldsymbol{\theta}}}'\in\Theta$, we denote by $\psi({\ensuremath{\boldsymbol{\theta}}}_j,{\ensuremath{\boldsymbol{\theta}}}'_{-j})$ the mapping of a parametrization that takes the $j$th group from ${\ensuremath{\boldsymbol{\theta}}}$ and the rest of the parameters from ${\ensuremath{\boldsymbol{\theta}}}'$. The disentanglement of loss $L$ given $\hat y$, the mapping $\psi$ and a decomposition of parameters into ${\ensuremath{\mathsf{k}}}$ groups is defined as: $$L_\text{dis}(y,\hat y)=\sum_{j=1}^{\ensuremath{\mathsf{k}}} L(\psi({\ensuremath{\boldsymbol{\theta}}}_j,{\ensuremath{\boldsymbol{\hat\theta}}}_{-j}),\hat y)\,,$$ where ${\ensuremath{\boldsymbol{\theta}}}=\psi^{-1}(y)$ and ${\ensuremath{\boldsymbol{\hat \theta}}}=\psi^{-1}(\hat y)$. The idea behind the transformation is very intuitive besides the mathematical formalism. We simply replicate ${\ensuremath{\mathsf{k}}}$ times the loss $L$, each copy having only a group of parameters that can be optimized, the other being fixed to the ground-truth parametrization, which can be recovered via $\psi^{-1}$. We have applied the disentangling transformation to both the 2D loss in Eq. and to the 3D loss in Eq. and used them to conduct our experiments, unless otherwise stated.
Explanatory Toy Experiment {#sec:toy}
--------------------------
The toy experiment consists in comparing the optimization trajectories when we employ the (entangled) 3D object detection loss $L_\mathtt{3D}^{bb}$ and the disentangled counterpart, which is obtained by applying the disentangling transformation described in Sec. \[sec:disentangling\]. We took a ground-truth detection case from KITTI3D and picked an illustrative initialization for the 3D box for optimization (see, Fig. \[fig:video\] green and red boxes, respectively).
We perform the experiment using stochastic gradient descent with learning rate $0.001$, momentum $0.9$ and no weight decay. We run the experiment for $3000$ iterations. We report in Fig. \[fig:toy\] (first 4 plots from the left) the trajectories of the optimization process for each group of parameters when the entangled and disentangled losses are used. The parameter groups describe box dimensions, rotation quaternion, projected center of the 3D bounding box on the image, and the depth of the 3D bounding box center. The benefits deriving from the use of the disentangled loss can be clearly seen in the plots. Convergence is much faster and smoother. We can see that the trajectories induced by the entangled loss are suboptimal, since they explore multiple configurations of parameters before approaching the correct one, sometimes with considerable deviations (see, the dimensions of the bounding box). As an example, we report in Fig. \[fig:video\] (right) the point where the entangled version attains the largest deviation in terms of bounding box dimensions from the ground-truth, which happens at iteration $150$, while at this stage the optimization dynamics using the disentangled loss fixed already all parameters but the depth. Despite the quaternion being aligned with the ground-truth rotation axis from the beginning, the optimization dynamics with the entangled loss starts diverging from it, producing unnatural poses and sizes that are not properly penalized by the entangled loss as can be seen by the loss values reported for the two configurations. Such unstable supervision delivered by the entangled loss harms the generalization capabilities of the network. Interestingly, even though the optimization process that uses the disentangled loss does not directly optimize $L_\mathtt{3D}^{bb}$, it can minimize it more quickly than the counterpart directly optimizing it (see, Fig. \[fig:toy\] last).
We provide also a video on our project website that shows the evolution of the optimization process described above. Fig. \[fig:video\] gives an overview of the first frame (left column). For each optimized loss (entangled on top and disentangled on the bottom) we provide the ground truth 3D bounding box in green and the currently predicted one in red. The faces with thick lines and showing a cross represent the front of the car and the bottom of the car, respectively, while the white line connects the respective centers. We also show the birds-eye view, where we projected the bottom face (the one with the cross) on the ground plane. There we also report the value of the entangled loss $L_\mathtt{3D}^{bb}$ for both approaches for direct comparison and the iteration number. The video has been rendered with a logarithmic time scale in order to emphasize the initial part of the dynamics, which is also the most informative one.
Critical Review on the KITTI3D AP Metric {#sec:metric}
========================================
The KITTI3D benchmark dataset [@Geiger2012CVPR] significantly determines developments and general progress on 3D object detection, and has emerged as the most decisive benchmark for monocular 3D detection algorithms like ours. It contains a total of 7481 training and 7518 test images and has no official validation set. However, it is common practice to split the training data into 3712 training and 3769 validation images as proposed in [@NIPS2015_Chen], and then report validation results. On the official test split, there is no common agreement which of the training sets to use, but in case validation data is used for snapshot cherry-picking, it is imperative to provide test data scores from the same model.
Each 3D ground truth detection box is assigned to one out of three difficulty classes (*easy, moderate, hard*), and the used 11-point Interpolated Average Precision metric is separately computed on each difficulty class. This metric was originally proposed in [@Salton1986], and was used in the PASCAL VOC challenges [@Everingham2010] between 2007 and 2010. It approximates the shape of the Precision/Recall curve as $$\text{AP}|_R=\frac{1}{|R|}\sum_{r\in R} \rho_{interp}(r) \,,$$ averaging the precision values provided by $\rho_{interp}(r)$. In the current setting, KITTI3D applies exactly eleven equally spaced recall levels, $R_{11}=\{0, 0.1, 0.2, \ldots, 1\}$. The interpolation function is defined as $\rho_{interp}(r)=\max\limits_{r':r'\geq r} \rho(r')$, where $\rho(r)$ gives the precision at recall $r$, meaning that instead of averaging over the actually observed precision values per point $r$, the maximum precision at recall value greater or equal than $r$ is taken. The recall intervals start at $0$, which means that a single, correctly matched prediction (according to the applied IoU level) is sufficient to obtain 100% precision at the bottom-most recall bin. In other words, if for each difficulty level a single, but correct prediction is provided to the evaluation, this produces an $\text{AP}|_{R_{11}}$ score of $1/11\approx0.0909$ for the entire dataset, which as shown in our experimental section already outperforms a number of recent methods while it clearly does not properly assess the quality of an algorithm.
In light of KITTI3Ds importance, we propose a simple but effective fix that essentially exploits more of the information provided by the official evaluation server and evaluation scripts. Instead of sub-sampling 11 points from the provided 41 points, we approximate the area under the curve by simply replacing $R_{11}$ with $R_{40}=\{1/40, 2/40, 3/40, \ldots, 1\}$ thus averaging precision results on 40 recall positions but not at $0$. This eliminates the glitch encountered at the lowest recall bin, and allows to post-process all currently provided test server results on 2D and 3D AP scores.
Experiments on KITTI3D {#sec:expKITTI}
======================
We focus the validation of our method on the KITTI3D benchmark dataset that we described in Sec. \[sec:metric\], using the 0.7 IoU threshold for calculating AP.
Pre-processing {#ss:pre-processing}
--------------
We provide some observations about the annotations that can be found in the dataset, and some simple filtering steps that we have applied to the annotations of the training split defined in [@NIPS2015_Chen].
#### DontCare areas.
Besides standard classes such as *Car*, *Pedestrian* and *Cyclist*, KITTI3D provides *DontCare* annotations. This class is used to label portions of the image that potentially include positive instances which have not been labeled under the proper class for reasons such as high distance. Accordingly, we avoid harvesting negatives in the 2D detection head if an anchor has IoU above $50\%$ with those areas.
#### DontCare overlap.
Some positive bounding boxes, such as cars that were too near to the camera, have an IoU with a *DontCare* bounding box greater than $50\%$. We decided to set those bounding boxes as *DontCare*. This adjustment converted $729$ cars ($5.0\%$) to *DontCare*.
#### Full occlusion.
Some valid bounding boxes are actually fully occluded by a nearer object. Keeping those bounding boxes as positive instances might harm the learning process, so we decided to delete them. This adjustment deleted $218$ ($1.5\%$) cars.
From a total number of $14357$ cars that were annotated, the valid number of *Car* bounding boxes was $13410$ ($93.4\%$).
Implementation Details {#sec:details}
----------------------
We give more details about our implementation and instantiation of hyperparameters, in order to enable the reproducibility of our results.
#### 2D Detection Head.
For each FPN level $f_i$ and each spatial cell $g$ we employ a total of 15 anchors spanning on five aspect ratios $\{\frac{1}{3},\frac{1}{2},1,2,3\}$ and three scales $\{4 s_i 2^\frac{j}{3}\,:\,j\in\{0,1,2\}\}$, where $s_i$ is the downsampling factor of $f_i$. Each anchor is considered positive if its IoU with a ground truth instance is greater than $\tau_{iou}=0.5$.
#### 3D Detection.
We used a reference *Car* size of $W_0=1.53m$, $H_0=1.63m$, $D_0=3.88m$ and depth statistics of $\mu_z=28.01m$ and $\sigma_z=16.32m$. We filtered the final 3D detections with a score threshold of $\tau_\text{conf}=0.05$.
#### Losses.
We applied the same weighting policies in all our experiments. We set weight $1.0$ to all losses in the 2D detection head and $0.5$ to all losses in the 3D detection head. The Huber parameters is set to $\delta_H=3.0$ and the 3D confidence temperature of $T=1$.
#### Optimization.
Our training schedule is the same for all experiments, and it does not involve any multi-step or warm-up procedures. We used SGD with a learning rate set at 0.01 and apply weight decay of 0.0001 to all parameters but scale and biases of [iABN]{}. We also freeze conv1 and conv2 of ResNet34 in the backbone. We trained with batch size of 96 on 4 NVIDIA V-100 GPUs for a total of 20k iterations, scaling the learning rate by a 0.1 factor at 12k and 16k iterations. Our input resolution is set according to [@Manhardt_2019_CVPR]. We applied horizontal flipping as the only form of training-data augmentation. No augmentation was performed for test/validation.
2D Detection
------------
In a first set of experiments, we study the signed IoU loss function (Sec. \[sec:head2D\]) in isolation. To do this, we train our backbone + 2D head to perform pure 2D detection of cars in KITTI3D, comparing between the original RetinaNet regression loss, signed IoU and signed IoU with disentanglement. For this simpler task we reduce the training schedule to 3.5k iterations, with learning rate steps after 2k and 3k, while keeping all other parameters as in Sec. \[sec:details\]. As shown in Tab. \[tab:results2d\], using signed IoU leads to a modest performance increase, which improves considerably when adding disentanglement.
\[tab:results2d\]
3D Detection
------------
In this section we focus on our main task and perform a detailed ablation of our contributions, comparing the results with most relevant state-of-the-art algorithms for monocular 3D detection. Keeping the network architecture and training schedule fixed, we evaluate different loss functions and detection scoring strategies. Following the discussion in Sec. \[sec:metric\], we report both, our revised $\text{AP}|_{R_{40}}$ metric (Tab. \[tab:ablation-new-metric\]) and the original $\text{AP}|_{R_{11}}$ (Tab. \[tab:ablation-old-metric\]).
#### Ablation study.
First, we turn our attention to the *3D BB* loss in Eq. , comparing it to the direct *Regression* of the 10D parameters $\boldsymbol{\theta}$ [@Manhardt_2019_CVPR] (first two lines of both tables). Confirming the findings in [@Manhardt_2019_CVPR], we observe increased 3D detection scores when tying all parameters together in a single (entangled) loss function in metric space. Perhaps surprisingly, *3D BB* also leads to better 2D detection performance: we suppose this could be due to more informative gradients propagating from the 3D head improving the backbone features. Adding our disentangled 2D detection loss based on the signed IoU (Eq. ) and the 3D confidence prediction (Sec. \[sec:head3D\]), consistently improves performance for both *Regression* and *3D BB* (third and fourth lines in the tables). Similarly, applying disentangling to the *3D BB* loss improves 3D detection performance, and has an even larger impact on the 2D side. Bringing all our contributions together leads to noticeable performance increases under all considered metrics (*[MonoDIS]{}*). In Tab. \[tab:conf-2d\] we conduct an additional ablation study on the validation set in [@NIPS2015_Chen] to assess the importance of the 3D confidence prediction. To this end, we take our best model trained and evaluated with the 3D confidence prediction ($p_\mathtt{3D}, \text{AP}|_{R_{xx}}$) and compare against the same model when the 2D confidence is returned ($p_\mathtt{2D}, \text{AP}|_{R_{xx}}$) and when it is randomly sampled (random, $\text{AP}|_{R_{xx}}$) . The ability of computing a reliable estimation of the confidence about the prediction is of utmost importance as can be inferred by the drastic drop of performance that we get when replacing $p_\mathtt{3D}$ with $p_\mathtt{2D}$, or with a random confidence. This is a direct consequence of the important role that the returned confidence plays in the AP metric.
#### Comparison with SOTA.
In Tab. \[tab:ablation-new-metric\], \[tab:ap11-test\] and \[tab:ablation-old-metric\] we report validation and test set results, respectively, of many recent monocular 3D detection approaches. When evaluating on the validation set, we consider the split defined in [@NIPS2015_Chen], as is done in all the baselines. Please note that the works in [@Xiang_2015_CVPR; @Mousavian_2017_CVPR; @Xiang_2017_WACV] are using yet another training/validation split, rendering their results incomparable to ours while yielding numerically comparable ranges to [@Liu+19]. For the test set, we consider both the split in [@NIPS2015_Chen], which is shared with OFTNet [@Roddick18] and ROI-10D [@Manhardt_2019_CVPR], and a larger training split[^1], since the setting used for MonoGRNet [@qin2019monogrnet] is not clear. In Tab. \[tab:ablation-new-metric\] we show $\text{AP}|_{R_{40}}$ scores[^2] for the test set results, and in Tb. \[tab:ap11-test\] the corresponding $\text{AP}|_{R_{11}}$ scores. Nonetheless, we would like to stress that the $\text{AP}|_{R_{11}}$ is biased by the issue reported in Section \[sec:metric\] and we invite to rather consider $\text{AP}|_{R_{40}}$ as the reference metric for fair comparison. With a single exception, our approach beats all baselines on all 3D and bird’s eye view metrics, often by a large margin, despite the fact that some of the outperformed methods rely on additional data, such as synthetic images (ROI-10D [@Manhardt_2019_CVPR]), or a pre-trained monocular depth prediction network (ROI-10D [@Manhardt_2019_CVPR], Xu [@Xu_2018_CVPR]). Interestingly, from the validation set results in Tab. \[tab:ablation-old-metric\], many existing approaches score lower than the “single correct hypothesis” baseline (see Sec. \[sec:metric\]) on 3D detection $\text{AP}|_{R_{11}}$, highlighting the need for an improved AP metric.
#### Results on additional KITTI3D classes.
In Tab. \[tab:ped\_cyc\] we provide the $\text{AP}|_{R_{11}}$ and $\text{AP}|_{R_{40}}$ scores (at IoU treshold $0.5$, see official evaluation scripts) obtained on the validation set in [@NIPS2015_Chen] for classes *Pedestrian* and *Cyclist* (trained independently). If compared to the results on class *Car*, it can be seen that performances on these two particular classes are in general lower. The performance degradation on classes *Pedestrian* and *Cyclist* compared to *Car* is due to i) the reduced number of annotations which is $\approx6\times$ and $\approx20\times$ lower than *Car* for class *Pedestrian* and *Cyclist*, respectively, and ii) the higher impact that errors on localization have on the AP scores since the object $xz$-extent is typically smaller. For these reasons, similarly to [@Manhardt_2019_CVPR; @qin2019monogrnet; @Roddick18; @Xu_2018_CVPR], we put a larger focus on class *Car* in the main paper.
#### Qualitative results.
In Fig. \[fig:car\_viz\] we show qualitative results on a set of images taken from the validation set for the classes *Car* (top), *Pedestrian* (middle) and *Cyclist* (bottom). We also provide a video[^3] showing detection results obtained on a sequence from the validation set. The structure of the frames is similar to the one in Fig. \[fig:car\_viz\], where detections are shown on the right side and the corresponding birds-eye view on the left. For simplicity, we decided to display all the detections with the same color.
Experiments on nuScenes
=======================
We conduct additional experiments on the novel *nuScenes* dataset [@Cae+19].
#### About the dataset.
The nuScenes dataset provides multimodal, street-level data collected with a car equipped with 6 cameras, 1 LiDAR, 5 Radars and IMU. It contains 15h of driving data (242 km at average speed of 16 km/h) covering parts of the areas of Boston (Seaport and South Boston) and Singapore (One North, Holland Village and Queenstown). These two cities have been chosen due to their known dense traffic and highly challenging driving situations and driving routes are selected to capture a diverse set of locations, times and weather conditions. The dataset provides $360^\circ$, synchronized sensor coverage, calibration of sensor intrinsics and extrinsics parameters, and objects annotations for $23$ different classes from $1000$ selected scenes of $20s$ duration each. Annotated objects in the scenes come with a semantic category, 3D bounding box, tracking information, and attributes (visibility, activity and pose) for each frame they occur in.
#### Detection task.
The nuScenes detection tasks requires detecting $10$ object classes in terms of full 3D bounding boxes, attributes and velocities. In this work, we will focus on detecting the full 3D bounding box of object belonging to class *car*, because the only available baselines at the time of writing are OFTNet (monocular RGB image-based) and PointPillar [@Lang_CVPR_2019] (LiDAR-based). Fair comparison can only be made to OFTNet, where results are reported only for category *car* (see, [@Cae+19]).
#### Evaluation metric.
The authors of nuScenes propose an alternative metric called *nuScenes detection score* (NDS) that combines a measure of the detection performance with quality terms of box location (ATE, average translation error), size (ASE, average scale error), orientation (AOE, average orientation error), attributes (AAE, average attribute error) and velocity (AVE, average velocity error). The detection performance is measured in terms of Average Precision (AP), but with matches determined based on 2D center distance on the ground plane. Also the AP score is calculated as the normalized area under the precision/recall curve by excluding the $[0-10\%]$ range. The final score averages AP over matching thresholds of $\mathbb D=\{0.5,1,2,4\}$ meters and the set of classes $\mathbb C$: $$\text{mAP}=\frac{1}{|\mathbb C| |\mathbb D|}\sum_{c\in\mathbb C}\sum_{d\in\mathbb D} \text{AP}_{c,d}\,,$$ where $\text{AP}_{c,d}$ is the AP score on class $c$ with matching threshold $d$.
#### Obtained results.
We present in Fig. \[fig:nuscenesRes\] the results obtained on the car class in terms of Precision/Recall curves (for all distance thresholds in $\mathbb D$), as well as error curves for translation, scale and orientation true positive metrics (at distance threshold 2m), produced by the official nuScenes evaluation scripts. For direct comparison to available OFTNet and PointPillar results from [@Cae+19], we also provide Tab. \[tab:nuscenesTab\]. It is important to stress that direct comparison is only fair to OFTNet which is also purely image-based, unlike PointPillar, which is LiDAR-based. We are not reporting the NDS score as it also requires predictions for attributes and velocities. Since that would imply modifications of the network design it would also render results inconsistent with those obtained on KITTI3D in Sec. \[sec:expKITTI\].
The results in Tab. \[tab:nuscenesTab\] show that our approach improves by **42%** over OFTNet (in absolute terms), considering the primary AP metric at a distance threshold of 2m. In addition, [MonoDIS]{} improves on all available True Positive metrics over OFTNet and even on 2/3 metrics when compared to PointPillar (LiDAR-based). Despite obtaining better (lower) TP metrics ASE and AOE compared to PointPillar, the main advantage of LiDAR-based methods are shown in their lower translation errors (and therefore also in the corresponding AP scores at various distances). We provide some qualitative results in Fig. \[fig:nuscenesViz\], demonstrating promising 3D recognition performance without using LiDAR and therefore actively sensed depth information.
![Performance plots for class *Car* in nuScenes. Left: Precision/Recall curves for AP metric at multiple distance thresholds in $\mathbb D$. Right: Error/Recall curves for relevant TP errors metrics on translation (ATE), scale (ASE) and orientation (AOE).[]{data-label="fig:nuscenesRes"}](figure07.png "fig:"){width="0.51\columnwidth"} ![Performance plots for class *Car* in nuScenes. Left: Precision/Recall curves for AP metric at multiple distance thresholds in $\mathbb D$. Right: Error/Recall curves for relevant TP errors metrics on translation (ATE), scale (ASE) and orientation (AOE).[]{data-label="fig:nuscenesRes"}](figure08.png "fig:"){width="0.47\columnwidth"}
\[tab:nuscenesTab\]
Conclusions
===========
We proposed a new loss disentangling transformation that allowed us to effectively train a 3D object detection network end-to-end without the need of stage-wise training or warm-up phases. Our solution isolates the contribution made by groups of parameters to a given loss into separate terms that retain the same nature of the original loss, thus being compatible without the need of further, cumbersome loss balancing steps. We proposed two further loss functions where i) is based on a novel signed Intersection-over-Union criterion to improve 2D detection results and ii) is used to predict a detection confidence for the 3D bounding box predictions, learned in a self-supervised way. Besides the methodological contributions, we reveal a flaw in the primary detection metric used in KITTI3D, where a single, correctly predicted bounding box yields overall AP scores of 9.09% on validation or test splits. Our simple fix corrects performance results of previously published methods in general, and shows how significantly it was biasing monocular 3D object detection results in particular. In our extensive experimental results and ablation studies we demonstrated the effectiveness of our proposed model, and significantly improved over previous state-of-the-art on both, KITTI3D and the novel nuScenes dataset.
Signed Intersection over Union {#sec:siou}
==============================
Let ${\ensuremath{\boldsymbol{\hat b}}}=(\hat u_1,\hat v_1,\hat u_2,\hat v_2)$ and ${\ensuremath{\boldsymbol{b}}}=(u_1,v_1,u_2,v_2)$ be two bounding boxes, where $(u_1,v_1)$ denotes the top-left corner and $(u_2, v_2$) denotes the bottom-right corner. We define the signed intersection-over-union as follows: $$\label{eq:sIoU}
\text{sIoU}({\ensuremath{\boldsymbol{b}}},{\ensuremath{\boldsymbol{\hat b}}})=\frac{|{\ensuremath{\boldsymbol{b}}}\sqcap{\ensuremath{\boldsymbol{\hat b}}}|_\pm}{|{\ensuremath{\boldsymbol{b}}}|+|{\ensuremath{\boldsymbol{\hat b}}}|-|{\ensuremath{\boldsymbol{b}}}\sqcap{\ensuremath{\boldsymbol{\hat b}}}|_\pm}\,,$$ where $${\ensuremath{\boldsymbol{b}}}\sqcap{\ensuremath{\boldsymbol{\hat b}}}=
\begin{pmatrix}
\max(u_1,\hat u_1)\\
\max(v_1,\hat v_1)\\
\min(u_2,\hat u_2)\\
\min(v_2,\hat v_2)
\end{pmatrix}$$ provides an extended intersection operation between bounding boxes, $|{\ensuremath{\boldsymbol{b}}}|$ gives the area of bounding box ${\ensuremath{\boldsymbol{b}}}$ and $$|{\ensuremath{\boldsymbol{b}}}|_\pm=
\begin{cases}
+|{\ensuremath{\boldsymbol{b}}}|&\text{if } u_2>u_1\text{ and }v_2>v_1,\\
-|{\ensuremath{\boldsymbol{b}}}|&\text{otherwise,}
\end{cases}$$ gives the *signed* area of ${\ensuremath{\boldsymbol{b}}}$, which corresponds to the standard area with positive sign only if the first corner of ${\ensuremath{\boldsymbol{b}}}$ is the top-left one, while the second corner is the bottom-right one. To give a better intuition we provide some examples in Fig. \[fig:sIoU\], where green and red colors encode positive and negative areas, respectively: Left-to-right, the first two examples boil down to standard IoU yielding positive values, while the last ones are examples yielding negative values. The sIoU score is bounded in $[-1,1]$.
![Five examples of computation of the proposed signed IoU. Top: Colored areas represent the numerator of the sIoU formula, where green denotes positive area, red denotes negative area; numbers represent the corner ordering. Bottom: Areas represent the denominator, which is always positive. []{data-label="fig:sIoU"}](sIoU.pdf){width="\columnwidth"}
{width=".8\textwidth"} {width=".8\textwidth"}
{width=".8\textwidth"} {width=".8\textwidth"}
{width=".8\textwidth"} {width=".8\textwidth"}
{width=".48\textwidth"} {width=".48\textwidth"}
{width=".48\textwidth"} {width=".48\textwidth"}
{width=".48\textwidth"} {width=".48\textwidth"}
Lifting Transformation {#sec:lifting}
======================
We review the lifting transformation used in [@Manhardt_2019_CVPR]. Let ${\ensuremath{\boldsymbol{\theta}}}$ be the 10D network’s output from which we compute the depth $z$ of the 3D bounding box’s center, its projection on the image place ${\ensuremath{\boldsymbol{c}}}=(u_c,v_c)$, the dimensions of the 3D bounding box ${\ensuremath{\boldsymbol{s}}}=(W,H,D)$ and the unit quaternion ${\ensuremath{\boldsymbol{q}}}$ as described in Sec. 4.3 of the main paper. Let $K$ be the $3\times 3$ matrix of intrinsics with entries: $$K=
\begin{bmatrix}
f_x&0&c_x\\
0&f_y&c_y\\
0&0&1
\end{bmatrix}$$ and let $${\ensuremath{\boldsymbol{C}}}=
\begin{pmatrix}
\frac{u_c-c_x}{f_x}z,&
\frac{v_c-c_y}{f_y}z,&
z
\end{pmatrix}^\top=(C_x,C_y,C_z){\ensuremath{^\top}}$$ be the position of the center of the 3D bounding box. The lifting transformation is defined as: $${\ensuremath{\mathcal{F}}}({\ensuremath{\boldsymbol{\theta}}})= \frac{1}{2}R_{{\ensuremath{\boldsymbol{q}}}_{{\ensuremath{\boldsymbol{c}}}}}S\,B_0+{\ensuremath{\boldsymbol{C}}}$$ where $B_0$ holds the corners of the unit cube $[-1,1]^3$, $S$ is the diagonal matrix with entries ${\ensuremath{\boldsymbol{s}}}$, and $R_{{\ensuremath{\boldsymbol{q}}}_c}$ is the $3\times 3$ rotation matrix corresponding to quaternion $${\ensuremath{\boldsymbol{q}}}_c={\ensuremath{\boldsymbol{q}}}
\left[\cos \frac{\beta}{2} +sin \frac{\beta}{2} \mathtt j\right]$$ with $\beta=\tan^{-1}(\frac{C_x}{C_z})$.
[^1]: <https://github.com/MarvinTeichmann/KittiBox>
[^2]: We calculated these from the precision-recall values published in the KITTI3D leaderboard page.
[^3]: <https://research.mapillary.com/publication/MonoDIS>
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We consider the long time limit for the solutions of a discrete wave equation with a weak stochastic forcing. The multiplicative noise conserves the energy, and in the unpinned case also conserves the momentum. We obtain a time-inhomogeneous Ornstein-Uhlenbeck equation for the limit wave function that holds both for square integrable and statistically homogeneous initial data. The limit is understood in the point-wise sense in the former case, and in the weak sense in the latter. On the other hand, the weak limit for square integrable initial data is deterministic.'
author:
- 'Tomasz Komorowski[^1]'
- 'Stefano Olla[^2]'
- 'Lenya Ryzhik[^3]'
title: Asymptotics of the solutions of the stochastic lattice wave equation
---
Introduction {#intro}
============
Energy transport and dispersion in dynamics of oscillators in a lattice have been investigated in many situations in order to understand macroscopic thermal conductivity properties. A typical example is the Fermi-Pasta-Ulam chain under the Hamiltonian evolution corresponding to a quartic interaction potential. In the one dimension the Hamiltonian of the closed system of length $N$ with periodic boundary conditions is given by $$\label{eq:fpu}
{\mathcal}H = \sum_{y\in{{{\ensuremath{\mathbb Z}}}}/N{{{\ensuremath{\mathbb Z}}}}} \left(\frac{{\frak p}_y^2}{2m} + \frac 12
{\omega}_0^2 {\mathfrak q}_y^2 \right)
+ \sum_{y\in{{{\ensuremath{\mathbb Z}}}}/N{{{\ensuremath{\mathbb Z}}}}} \left[\frac {1}{2} ({\frak q}_y - {\frak
q}_{y-1})^2 +
{\gamma} ({\frak q}_y - {\frak q}_{y-1})^4 \right]$$ Here ${{{\ensuremath{\mathbb Z}}}}/N{{{\ensuremath{\mathbb Z}}}}$ denotes the group $\{0,\ldots,N-1\}$ with the addition modulo $N$, ${\frak q}_y$ is the displacement of the $y$-th particle from its equilibrium position, ${\frak p}_y$ is its momentum and $m$ is the mass. When $\omega_0\neq 0$, the particle is confined, this breaks translation invariance, and correspondingly the conservation of the total momentum, and we say that the chain is pinned.
When $\gamma = 0$ the Hamiltonian dynamics is given by the discrete in space linear wave equation, and the energy evolution is purely ballistic and dispersive. If $\gamma >0$ and $\omega_0\neq 0$, due to the presence of the non-linearity, wave *scattering* is expected that in turn gives a finite thermal conductivity and consequently a diffusive macroscopic evolution of the energy. If the chain is unpinned, $\omega_0 =0$, and $\gamma>0$, long waves scatter rarely, giving rise to a superdiffusive behavior of the energy [@sll].
The mathematical analysis of the macroscopic behavior of the energy is difficult in the case of deterministic nonlinear dynamics, and recently various models considering stochastic perturbations of the dynamics have been proposed. Such perturbations generate scattering qualitatively similar to the one due to the nonlinearity.
In order to mimic the nonlinear dynamics, a noisy perturbation we wish to consider should conserve energy and be local in space [@BO]. In the unpinned case it is also important that it conserve the momentum, see [@bborev; @bbo2]. The perturbations considered in these papers are given by a random exchange of momentum so that the total kinetic energy is constant (consequently, the total energy is preserved as well, since the position components are untouched by the noise) and the total momentum is also conserved. This is achieved by adding, to each triple of adjacent particles, a diffusion on the corresponding surface of constant energy and momentum. Another example of a noisy perturbation having similar properties appears in a discontinuous in time model in which momenta of pairs of adjacent particles are exchanged at independent random times that are exponentially distributed.
When the interaction is linear, the thermal diffusivity of the energy in these models can be explicitly computed – it is finite for the pinned model but diverges with the size of the system in the unpinned case (corresponding to superdiffusive energy transport for the unpinned model).
The limit dynamics for the spectral measure of the energy in these stochastic models is investigated in [@BOS], where the noise is also rescaled in such a way that there are only finitely many [wave collisions]{} in the unit macroscopic time. In a sense, this weak noise limit is similar to the regime where phonon-Boltzmann equation is valid in weakly nonlinear models (cf. [@HS]). The dynamics is defined in the following way. Consider the infinite lattice ${{{\ensuremath{\mathbb Z}}}}$ with the Hamiltonian associated to the linear evolution ($\gamma = 0$), with $N=\infty$, perturbed by a conservative noise. Formally, it is given by the solution of the stochastic differential equations: $$\label{eq:sde1}
\begin{split}
\dot{\frak q}_{y}(t)\ =&\ {\frak p}_y(t)\\
d {\frak p}_y (t) \ =&\ \left(\Delta {\frak q}_y - \omega_0^2 {\frak
q}_y\right) dt \ + d\eta_y({\epsilon}t),
\end{split}$$ where $\Delta {\frak q}_y= {\frak q}_{y+1}+ {\frak q}_{y-1}-2 {\frak
q}_y$ is the lattice Laplacian. The noise $d\eta_y({\epsilon}t)$ will be added to model random exchange of momenta between the adjacent sites so that the total kinetic energy and momentum of the system are conserved (see (\[eq:bas\]) for the precise form of the noise). The small parameter ${\epsilon}>0$ slows down its effect. The total Hamiltonian can be formally written as $$\label{eq:2}
{\mathcal}H(\frak q,\frak p) = \sum_{y\in{{{\ensuremath{\mathbb Z}}}}}\frac{{\frak p}_y^2}2 +
\sum_{x,y \in {{{\ensuremath{\mathbb Z}}}}} \alpha_{x-y} {\frak q}_x {\frak q}_y,$$ with $\alpha_0 = \frac 12 \omega_0^2 + 1$, $\alpha_{-1} = \alpha_{1} =- 1/2$, and $\alpha_y = 0$ otherwise. The dispersion relation ${\omega}(k)$ for this system is $$\label{eq:dr0}
\omega(k) := \sqrt{\hat\alpha(k)}=\left[\frac{{\omega}_0^2}{2} + 2 \sin^2(\pi k)\right]^{1/2},\quad k\in {{{\ensuremath{\mathbb T}}}}.$$ In fact we would admit a broader class of dispersion relations, requiring that $\hat\alpha(k)$ is defined as in below. Let us introduce the complex wave function $$\label{eq:wf0}
\psi_y(t) := (\check\omega * {\frak q})_y(t) + i {\frak p}_y(t),$$ where $\check\omega_y$ is the inverse Fourier transform of $\omega(k)$. Its Fourier transform $$\label{eq:fwf0}
\hat \psi(t,k) := \omega(k) \hat{{\frak q}}(k,t) + i \hat{\frak p}(t,k)$$ satisfies the equation $$\label{eq:fwf01}
d\hat \psi(t,k) = -i \omega(k) \hat \psi(t,k) dt + i d\hat\eta({\epsilon}t,k),$$ where $d\hat\eta(t,k)$ is the Fourier transform of the noise. Due to the conservation properties of the dynamics, if the initial configuration has finite total energy ${\mathcal}H(\frak
q(0), \frak p(0)) < +\infty$, then all the functions introduced in and - are well defined and $${\cal H}(\frak
q(t), \frak p(t)) = \sum_y |\psi_y(t)|^2 = \int_{{{{\ensuremath{\mathbb T}}}}} |\hat\psi(t,k)|^2 dk$$ Therefore we can identify $|\hat\psi(t,k)|^2$ with the energy density in the mode space. In the zero noise case, $|\hat\psi(t,k)|^2$ is conserved for any $k\in {{{\ensuremath{\mathbb T}}}}$ (i.e. $\partial_t |\hat\psi(t,k)|^2 =0$). The stochastic conservative perturbation mixes the energies between different modes $k$, and $|\hat\psi(t,k)|^2$ becomes a random variable. The evolution of the average energy ${\mathcal}E(t,k) :=
\mathbb E|\hat\psi(t,k)|^2$ was considered in [@BOS]. Since the stochastic perturbation is of order $\epsilon$, to have a visible effect of mixing of different modes we have to look at the time scale $\epsilon^{-1}t$. It was shown in [@BOS] that the limit $$\label{eq:bos0}
\lim_{{\epsilon}\to 0} {\mathcal}E\left(\frac t{\epsilon}, k\right) = \bar{{\mathcal}E}\left( t , k\right)$$ exists in the sense of distributions, and is the solution of the linear kinetic equation $$\label{eq:phbolhom}
\partial_t \bar{{\mathcal}E}\left( t , k\right) =
\int_{{{{\ensuremath{\mathbb T}}}}} R(k,k') \left[\bar{{\mathcal}E}\left( t , k'\right) -\bar{{\mathcal}E}\left( t , k\right) \right] dk'$$ with the initial condition $\bar{{\mathcal}E}\left(0 , k\right) = |\hat\psi(0,k)|^2$. The scattering kernel $R(k,k')$ is given by below.
The goal of the present article is to obtain a direct information on the wave function $\hat\psi(t/{\epsilon},k)$, as was done in [@BKRschr] for the Schrödinger equation, and not only for the average energy. It follows from that the unperturbed (by noise) evolution of this function is governed by the highly oscillating factor $e^{-i{\omega}(k)t/{\varepsilon}}$ (after we rescale the time). It is therefore reasonable to consider, in case of the perturbed system, the compensated wave function of the form $$\tilde\psi^{({\epsilon})}(t,k):=e^{i{\omega}(k) t/{\varepsilon}} \hat\psi(t/{\varepsilon},k).$$ We show that once we compensate for fast oscillations, the wave function converges in law to the solution a Langevin equation driven by . More precisely, we prove in Theorem \[main-thm1\] below, existence of the limit (in law and pointwise in $k$): $$\label{eq:comp0}
\lim_{{\varepsilon}\to 0} \tilde\psi^{({\epsilon})}(t,k) = \tilde\psi(t,k).$$ The limit $\tilde\psi(t,k)$ is a complex valued stochastic process satisfying the linear (time inhomogeneous) Ornstein-Uhlenbeck equation $$\label{eq:lan0}
d \tilde\psi(t,k) = - \frac{\hat\beta(k)}{4} \tilde\psi(t,k) dt
+ \sqrt{{\mathcal}R(t,k)} dw_k(t),$$ with the initial condition $\tilde\psi(0,k) = \hat\psi(0,k)$. Here $$\label{R-B}
\hat\beta(k) =2\int_{{{{\ensuremath{\mathbb T}}}}}R(k,k')dk'$$ $$\label{eq:beta0}
{\mathcal}R(t,k) = \int_{{{{\ensuremath{\mathbb T}}}}} \bar{{\mathcal}E}(t, k') R(k,k') dk',$$ and $\{w_k(t)\}$ is a family of pairwise independent standard complex valued Brownian motions parametrized by $k\in{{{\ensuremath{\mathbb T}}}}$. That is, they are complex valued, jointly Gaussian, centered processes satisfying $$\mathbb E[w_k(t) w_{k'}(s)] = 0\quad\mbox{and}\quad
\mathbb
E[w_{k'}^*(t) w_k(s)] =\delta_{k,k'} t\wedge s$$ for all $t,s\ge0$ and $k,k'\in{{{\ensuremath{\mathbb T}}}}$. Here $\delta_{k,k'}=0$ for $k\not=k'$ and $\delta_{k,k}=1$. Equation has the explicit solution $$\label{eq:3}
\tilde\psi(t,k) = e^{-\frac 14\hat\beta(k)t} \hat\psi(0,k) + \int_0^t
e^{-\frac 14\hat\beta(k)(t-s)} \sqrt{{\mathcal}R(s,k)} dw_k(s).$$ In particular, we have $$\mathbb E |\tilde\psi(t,k)|^2=
e^{-\frac12\hat\beta(k)t} |\hat\psi(0,k)|^2 + \int_0^t
e^{-\frac12\hat\beta(k)(t-s)} {\mathcal}R(s,k) ds$$ which is equivalent to , since $\bar{{\mathcal}E}(t,k) = \mathbb E |\tilde\psi(t,k)|^2$. Initial conditions such that $\int_{{{{\ensuremath{\mathbb T}}}}} |\hat\psi(0,k)|^2 dk < \infty$ correspond to a *local* perturbation of the zero temperature equilibrium. We are also interested in the macroscopic evolution of the equilibrium states at a positive temperature $T >0$, starting with random data distributed by the Gibbs measure at temperature $T$. In the mode space this is a centered, complex valued, Gaussian random field with distribution valued $\hat\psi(k)$. Its covariance is given by $$\label{eq:gibbscov}
{{\mathbb E}}[\hat\psi^*(k) \hat\psi(k')] =T \delta(k-k'), \qquad
{{\mathbb E}}[\hat\psi(k) \hat\psi(k')] = 0.$$ Here $\delta(k-k')$ is Dirac’s delta function. For any $T$, the corresponding Gibbs measure is invariant under the dynamics, due to the conservation of energy. Actually, in Section \[sec2.3.2\] we consider more general class of space homogeneous Gaussian random initial conditions whose law is not necessarily stationary in time. More precisely, we show (see Theorem \[main-thm2\]) that if the law of the initial condition is a homogeneous, centered Gaussian field with the covariance given by $${{\mathbb E}}\left[\hat\psi(k)^* \hat\psi(k')\right] = {\mathcal}E_0(k)\delta(k-k') , \qquad
{{\mathbb E}}\left[\hat\psi(k) \hat\psi(k')\right]= 0,$$ then the compensated wave function converges in law, as a continuous in time process taking values in an appropriate distribution space, to the solution of the time inhomogeneous stochastic equation: $$\label{eq:spdet1}
d \tilde\psi(t,k) = - \frac{\hat\beta(k)}{4} \tilde\psi(t,k) dt
+ \sqrt{{\mathcal}R(t,k)} dW(t,k).$$ Here, ${\mathcal}R(t,k) $ is given by and $\bar{{\mathcal}E}(t,k)$ is the solution of the deterministic equation with the initial condition $\bar{{\mathcal}E}(0,k) = {\mathcal}E_0(k)$, while $d W(t,k)$ is a white noise on ${{{\ensuremath{\mathbb R}}}}\times {{{\ensuremath{\mathbb T}}}}$, a complex valued Gaussian process with the covariance $$\mathbb E[dW(t,k)d W^*(s,k')] = \delta(k-k') \otimes\delta(t-s)dtds$$ and ${\cal R}(t,k)$ is given by . The solution of is also explicit: $\tilde\psi(t)$ is the distribution $$\tilde\psi(t) = e^{-\hat \beta t/4}\hat\psi + \int_0^t e^{-\hat \beta(t-s)/4}{\mathcal}R^{1/2}(s) dW(s) .$$ In particular, in the case of the initial condition distributed according to a Gibbs measure, the solution $\hat\psi(t,k)$ of has the same law for all times, therefore $\bar{{\mathcal}E}(t,k) = T$ for all $t\ge0$. In this case, shows that ${\cal R}(t,k)=\hat\beta(k)T/2$. Therefore, as a consequence of , the limit of the compensated wave function is the solution of the linear infinite dimensional stochastic differential equation: $$\label{eq:spde0}
d \tilde\psi(t,k) = - \frac{\hat\beta(k)}{4} \tilde\psi(t,k) dt
+\sqrt{\frac{T \hat\beta(k)}{2}} dW(t,k).$$ In the general case, when ${\mathcal}E_0(k)$ is not constant, we have $$\lim_{t\to\infty} \bar{{\mathcal}E}(t,k) = \int_{{{{\ensuremath{\mathbb T}}}}} {\mathcal}E_0(k') dk' = T,$$ hence, equation describes the asymptotic stationary regime of where the temperature is given by the average of the initial energy over all the modes $k$. Recall that the microscopic noise conserves the total energy and that the resulting temperature $T$ depends only on the law of the initial condition.
Let us also comment on the difference between the square integrable and distribution-valued initial data. While the Ornstein-Uhlenbeck equations (\[eq:lan0\]) and (\[eq:spdet1\]) look similar, there are some important differences between them. The noises appearing in (\[eq:lan0\]) are all of size $1$ and mutually independent for different $k$-s, while the noise appearing in (\[eq:spdet1\]) is $\delta$-correlated in $k$. As a result the solution of the first equation is an ensemble of mutually independent time inhomogeneous one dimensional Ornstein-Uhlenbeck processes. On the other hand, in the case of (\[eq:spdet1\]) the resulting distribution valued Ornstein-Uhlenbeck process is $\delta$-correlated in $k$. In addition, for the square integrable data, the limit equation holds point-wise in $k$. If one considers the limit in the sense of distributions (that is, integrated against a test function) for such initial data, the stochasticity is removed, due to the fact that independent random variables, representing the solution for different modes, are simply averaged out (via the law of large numbers). As a result the limit is described simply by attenuation of the initial condition by an exponential factor $e^{-\beta(k)t/4}$ (see part (ii) of Theorem \[main-thm1\]) – that is, by (\[eq:lan0\]) with no stochastic forcing. This result stands in sharp contrast with the case of spatially homogeneous initial data (note that then the energy has to be infinite) when the respective limit in the sense of distributions is stochastic, see , and fluctuations can not be averaged out by integration in $k$.
Finally, we note that the sole reason why we restrict ourselves to the case of one dimensional integer lattice is to avoid excessive complication of the notation that could obscure the main points of the argument. The technique of our proof can be straightforwardly applied in the case of lattice ${{{\ensuremath{\mathbb Z}}}}^d$. The dynamics of the corresponding perturbed system is given then by equation (45) of [@BOS] and our results contained in Theorems \[main-thm1\] and Theorem \[main-thm2\] can be easily adjusted to deal with the case of a multidimensional lattice.
The paper is organized as follows. Section \[prelim\] contains the precise mathematical formulation of the problem and necessary definitions. We formulate the results for the convergence of compensated wave function in Section \[sec3\], see Theorem \[main-thm1\] for square integrable initial data, and Theorem \[main-thm2\] for spatially homogeneous, Gaussian initial distributions. The proofs of these results are presented in Sections \[sec4\] and \[sec5\], respectively.
[**Acknowledgement.**]{} T.K. acknowledges the support of Polish Ministry of Higher Education grant NN201419139, S.O. acknowledges the support by the ERC AdG 246953 (MALADY) and by ANR-10-BLAN 0108 (SHEPI), L.R. acknowledges the support by NSF grant DMS-0908507. This work was also supported by NSSEFF fellowship by AFOSR.
Preliminaries {#prelim}
=============
Infinite system of interacting harmonic oscillators
---------------------------------------------------
The dynamics of the system of oscillators can be written formally as a system of Itô stochastic differential equations indexed by $y\in\mathbb Z$ $$\begin{aligned}
d{\frak q}_{y}(t) &=&{\frak p}_y(t)dt
\label{eq:bas}\\
&&\nonumber\\
d{\frak p}_y(t) &=& - (\alpha*{\frak q}(t))_y\ dt-\frac{{\epsilon}}{2}(\beta*{\frak p}(t))_y\ dt
+\sqrt{{\epsilon}}\sum_{z=-1,0,1}(Y_{y+z}{\frak p}_y(t))dw_{y+z}(t).\nonumber \end{aligned}$$ Here $$Y_x:=({\frak p}_x-{\frak p}_{x+1})\partial_{{\frak p}_{x-1}}+({\frak p}_{x+1}-{\frak p}_{x-1})\partial_{{\frak p}_{x}}+({\frak p}_{x-1}-{\frak p}_{x})\partial_{{\frak p}_{x+1}}$$ and $\{w_y(t),\,t\ge0\}$, $y\in{{{\ensuremath{\mathbb Z}}}}$ is a family of i.i.d. one dimensional, real valued, standard Brownian motions, that are non-anticipative over the filtered probability space $({\Omega},{\cal F},\{{\cal F}_t\},{{{{\mathbb P}}}})$. In addition, $$\label{040910}
\beta_y=\Delta\beta^{(0)}_y:=\beta^{(0)}_{y+1}+\beta^{(0)}_{y-1}-2\beta^{(0)}_y$$ with $$\beta^{(0)}_y=\left\{
\begin{array}{rl}
-4,&y=0\\
-1,&y=\pm 1\\
0, &\mbox{ if otherwise.}
\end{array}
\right.$$ Recall that the lattice Laplacian of $g:{{{\ensuremath{\mathbb Z}}}}\to\mathbb C$ is given by $\Delta g_y:=g_{y+1}+g_{y-1}-2g_y$.
To understand why we choose this particular stochastic perturbation of the Hamiltonian dynamics, let us observe that we want a (continuous) noise acting only on the velocities, as local as possible, but conserving total momentum and kinetic energy. This explains why, given a site $y$, only the momenta at sites $y+z$, $z=-1,0,1$ are exchanged randomly. For that reason we consider the vectors $Y_{x}$ that are tangent to the local energy and momentum surfaces $$\label{051207a}
{\frak p}_{x-1}^2+{\frak p}_{x}^2+{\frak p}_{x+1}^2\equiv {\rm const}$$ and $$\label{051207b}
{\frak p}_{x-1}+{\frak p}_{x}+{\frak p}_{x+1}\equiv {\rm const}.$$ The SDE defines a Markov process whose (formal) generator is given by $$\label{eq:gen}
L = A + {\epsilon}S, \qquad S=\frac12 \sum_x Y_x^2,$$ where $A$ is the Hamiltonian vector field given by the usual Poisson brackets with the Hamiltonian. In particular $-(\beta*p)_y/2 = S p_y$.
The Fourier transform of a square integrable sequence of complex numbers $\{\gamma_y,\,y\in{{{\ensuremath{\mathbb Z}}}}\}$ is defined as $$\label{fourier}
\hat \gamma(k)=\sum_{y\in{{{\ensuremath{\mathbb Z}}}}}\gamma_ye_y(k), \quad k\in{{{\ensuremath{\mathbb T}}}}.$$ Here $$e_y(k):=\exp\{-i2\pi yk\}, \quad y\in{{{\ensuremath{\mathbb Z}}}}$$ is the standard orthonormal base in $L^2({{{\ensuremath{\mathbb T}}}})$. The one dimensional torus ${{{\ensuremath{\mathbb T}}}}$ considered in this article is understood as the interval $[-1/2,1/2]$ with identified endpoints. The inverse transform is given by $$\label{inv-fourier}
\check f_y=\int_{{{{\ensuremath{\mathbb T}}}}} f(k)e^*_y(k)dk
, \quad y\in {{{\ensuremath{\mathbb Z}}}}$$ for any $f$ belonging to $L^2({{{\ensuremath{\mathbb T}}}})$ - the space of complex valued, square integrable functions. A simple calculation shows that $$\label{beta}
\hat \beta(k)=8\sin^2(\pi k)\left[1+2\cos^2(\pi k)\right].$$
We assume also (cf [@BOS]) that
- $\{\alpha_y,\,y\in{{{\ensuremath{\mathbb Z}}}}\}$ is real valued and there exists $C>0$ such that $|\alpha_y|\le Ce^{-|y|/C}$ for all $y\in {{{\ensuremath{\mathbb Z}}}}$,
- $\hat\alpha(k)$ is also real valued and $\hat\alpha(k)>0$ for $k\not=0$ and in case $\hat \alpha(0)=0$ we have $\hat\alpha''(0)>0$.
The above conditions imply that both functions $y\mapsto\alpha_y$ and $k\mapsto\hat\alpha(k)$ are even. In addition, $\hat\alpha\in C^{\infty}({{{\ensuremath{\mathbb T}}}})$ and in case $\hat\alpha(0)=0$ we have $\hat\alpha(k)=\sin^2(\pi k)\phi(k)$ for some strictly positive even function $\phi\in C^{\infty}({{{\ensuremath{\mathbb T}}}})$. Recall that the function ${\omega}(k):=\sqrt{\hat \alpha (k)}$ is [ the dispersion relation]{}.
Evolution of the wave function
------------------------------
For a given $m\in{{{\ensuremath{\mathbb R}}}}$ we define the space $H^m({{{\ensuremath{\mathbb T}}}})$ as the completion of $C^\infty({{{\ensuremath{\mathbb T}}}})$ under the norm $$\|f\|^2_{H^m({{{\ensuremath{\mathbb T}}}})}:=\sum_{y\in{{{\ensuremath{\mathbb Z}}}}}(1+y^2)^m|\check f_y|^2.$$ We shall denote by $\langle\cdot,\cdot\rangle$ the scalar product on $L^2({{{\ensuremath{\mathbb T}}}})$. By continuity it extends in an obvious way to $H^m({{{\ensuremath{\mathbb T}}}})\times H^{-m}({{{\ensuremath{\mathbb T}}}})$ for an arbitrary $m\in{{{\ensuremath{\mathbb R}}}}$.
It is convenient to introduce the wave function that, adjusted to the macroscopic time, is given by $$\label{011307}
\psi^{({\epsilon})}(t):=\check {{\omega}} * {\frak q}\left(\frac{t}{{\epsilon}}\right)+i{\frak p}\left(\frac{t}{{\epsilon}}\right).$$ Here $\{\check {\omega}_y,\,y\in{{{\ensuremath{\mathbb Z}}}}\}$ is the inverse Fourier transform of $$\label{om}
{\omega}(k):=\sqrt{\hat \alpha (k)}.$$ We shall consider the Fourier transform of the wave function $$\label{011307a}
\hat\psi^{({\epsilon})}(t,k):={\omega}(k)\hat {\frak q}\left(\frac{t}{{\epsilon}},k\right)+i\hat{\frak p}\left(\frac{t}{{\epsilon}},k\right).$$ Using as a motivation, we obtain formally, by considering the Fourier transform of , that $$\begin{aligned}
\label{basic:sde:2}
&&
d\hat\psi^{({\epsilon})}(t)=A[\hat\psi^{({\epsilon})}(t)]dt +Q[\hat\psi^{({\epsilon})}(t)]dW(t),\\
&&
\hat\psi^{({\epsilon})}(0)= \hat\psi,\nonumber
\end{aligned}$$ where $ \hat\psi\in L^2({{{\ensuremath{\mathbb T}}}})$, and mapping $A:L^2({{{\ensuremath{\mathbb T}}}})\to L^2({{{\ensuremath{\mathbb T}}}})$ is defined by $$\label{040607}
A[f](k):=-\frac{i}{{\epsilon}} {\omega}(k)f(k)- \frac{\hat\beta(k)}{4} [f_{1}(k)-f_{-1}(k)],\quad\forall\,f\in L^2({{{\ensuremath{\mathbb T}}}}).$$ Here $$\label{020906}
f_1(k):=f(k)\quad\mbox{and}\quad f_{-1}(k):=f^*(-k).$$ In addition, $Q[g]:L^2({{{\ensuremath{\mathbb T}}}})\to L^2({{{\ensuremath{\mathbb T}}}})$ is a linear mapping that for any $g\in L^2({{{\ensuremath{\mathbb T}}}})$ is given by $$\label{053009}
Q[g](f)(k):=i\int_{{{{\ensuremath{\mathbb T}}}}}r(k,k')[g_{1}(k-k')-g_{-1}(k-k')]f(k')dk',\quad\forall\,f\in L^2({{{\ensuremath{\mathbb T}}}}),$$ where $$\begin{aligned}
&&r(k,k'):=\sin(2\pi k)+\sin[2\pi(k-k')]+\sin[2\pi(k'-2k)]
\\
&&~~~~~~~~~~=4\sin(\pi k)\sin[\pi (k-k')]\sin\left[(2k-k')\pi\right]
,\quad k,k'\in {{{\ensuremath{\mathbb T}}}}.
\end{aligned}$$ The cylindrical Wiener process on $L^2({{{\ensuremath{\mathbb T}}}})$ appearing in is $dW(t):=\sum_{y\in{{{\ensuremath{\mathbb Z}}}}} e_ydw_y(t)$.
It can be easily checked that $\sum_{y\in{{{\ensuremath{\mathbb Z}}}}}\|Q[g](e_y)\|_{L^2({{{\ensuremath{\mathbb T}}}})}^2\le C\|g\|_{L^2({{{\ensuremath{\mathbb T}}}})}^2$ for some $C>0$ and all $g\in L^2({{{\ensuremath{\mathbb T}}}})$ so $Q[g]$ is Hilbert-Schmidt, which ensures that $$Q[\hat\psi^{({\epsilon})}(t)]dW(t):=\sum_{y\in{{{\ensuremath{\mathbb Z}}}}} Q[\hat\psi^{({\epsilon})}(t)](e_y)dw_y(t)$$ is summable in $L^2({{{\ensuremath{\mathbb T}}}})$, both in the $L^2$ and a.s. sense. It is also obvious that the mapping $A$ is Lipschitz. Using Theorem 7.4, p. 186, of [@DZ] one concludes therefore that there exists an $L^2({{{\ensuremath{\mathbb T}}}})$-valued, adapted process $\{\hat\psi^{({\epsilon})}(t),\,t\ge0\}$ that is a unique solution to . In addition, see Section 2 of [@BOS], the total energy is conserved: $$\label{conservation}
\|\hat\psi^{({\epsilon})}(t)\|_{L^2({{{\ensuremath{\mathbb T}}}})}={\rm const},\quad\forall\,t\ge0$$ for a.s. realization of Brownian motions and an initial condition from $L^2({{{\ensuremath{\mathbb T}}}})$.
Compensated wave function {#pseudo-wigner}
-------------------------
Let us define the compensated wave function $$\tilde \psi^{({\epsilon})}(t,k):=\hat \psi^{({\epsilon})}(t,k)\exp\left\{it\frac{{\omega}(k)}{{\epsilon}}\right\}.$$ From we obtain the following equation $$\begin{aligned}
\label{mollified-eqt}
&&d \tilde \psi^{({\epsilon})}(t,k)=
{\cal A}\left[\frac{t}{{\epsilon}},\tilde\psi^{({\epsilon})}(t)\right](k)dt +d\tilde{\cal M}^{({\epsilon})}_t(k),\nonumber\\
&&
\tilde\psi^{({\epsilon})}(0)= \hat\psi,
\end{aligned}$$ where $ \hat\psi\in L^2({{{\ensuremath{\mathbb T}}}})$, $ {\cal A}[t,\cdot]:L^2({{{\ensuremath{\mathbb T}}}}) \to L^2({{{\ensuremath{\mathbb T}}}})$ $$\label{012808}
{\cal A}[t,f](k):=-\frac{\hat\beta(k)}{4}\left[ f(k)-\exp\left\{2i{\omega}(k)t\right\}f^*(-k)\right].$$ The martingale term equals $$\label{060410}
d \tilde{\cal M}^{({\epsilon})}_t:=\tilde Q\left[\frac{t}{{\epsilon}},\tilde\psi^{({\epsilon})}(t)\right]dW(t),$$ where for any $g\in L^2({{{\ensuremath{\mathbb T}}}})$ and $t\ge0$, the operator $\tilde Q[t,g] :L^2({{{\ensuremath{\mathbb T}}}})\to L^2({{{\ensuremath{\mathbb T}}}})$, is given by $$\label{022808}
\tilde Q[t,g](f)(k):=i\sum_{{\sigma}=\pm1}{\sigma}\int_{{{{\ensuremath{\mathbb T}}}}}r(k,k')g_{{\sigma}}(k-k')f(k')
\exp\left\{i[{\omega}(k)-{\sigma}{\omega}(k-k')]t\right\}dk'.$$ Using a standard theory of S.P.D.E.-s, see [@DZ], we can show the following result.
\[prop010910\] Suppose that $-3/2<m<1$. If the initial condition $\hat \psi(\cdot)$ belongs to $H^{m}({{{\ensuremath{\mathbb T}}}})$ then there exists a unique solution $(\tilde \psi^{({\epsilon})}(t))$ of in $H^{m}({{{\ensuremath{\mathbb T}}}})$.
The proof of this result shall be presented in Appendix \[appA\]. Since the dispersion relation ${\omega}(\cdot)$ might not be differentiable in the classical sense at $0$ (but it belongs to $H^1({{{\ensuremath{\mathbb T}}}})$) we cannot guarantee better regularity of the solutions of . Recall that the classical Sobolev embedding theorem ensures that $H^m({{{\ensuremath{\mathbb T}}}})$, for $m>1/2$, is embedded in the space of continuous functions on the torus $C({{{\ensuremath{\mathbb T}}}})$, see e.g. Theorem 7.10, p. 155 of [@GT].
Convergence of the compensated process {#sec3}
======================================
Square integrable initial data {#sec3.1}
-------------------------------
Before formulating the result we introduce some auxiliaries. First, for any $k_1,k_2\in {{{\ensuremath{\mathbb T}}}}$ let us denote $${\cal K}(k_1,k_2)=\bigcup_{{\sigma}_1,{\sigma}_2,{\sigma}_3=\pm1}
[k:{\omega}(k_1)+{\sigma}_3{\omega}(k-k_1)={\sigma}_1[{\omega}(k_2)+{\sigma}_2{\omega}(k-k_2)]]$$ We shall require that: $$\hbox{ {\bf Condition} ${\omega}$)
for any $k_1\not=k_2$ the one dimensional Lebesgue measure $m_1({\cal
K}(k_1,k_2))=0$.
}$$ More detailed discussion of this condition shall be carried out in Remark 2 after Theorem \[main-thm1\] below.
Define the scattering operator ${\cal L}:L^1({{{\ensuremath{\mathbb T}}}})\to L^1({{{\ensuremath{\mathbb T}}}})$ by $$\label{L}
{\cal L}f(k):=\int_{{{{\ensuremath{\mathbb T}}}}}R(k,k')[f(k')-f(k)]dk',\quad f\in L^1({{{\ensuremath{\mathbb T}}}}),$$ where the scattering kernel is given by $$\begin{aligned}
\label{kernel}
&&
\!\!\!\!\!\!\!\!R(k,k'):=r^2(k,k-k')+r^2(k,k+k')\\
&&
=16\sin^2(\pi k)\sin^2(\pi k')\left\{\sin^2\left[\pi (k+k')\right]+\sin^2\left[\pi (k-k')\right]\right\}.\nonumber
\end{aligned}$$ Suppose that $ \hat\psi\in L^2({{{\ensuremath{\mathbb T}}}})$. Let $$\label{010810}
{\cal R}(t,k):=\int_{{{{\ensuremath{\mathbb T}}}}}R(k,k')\bar{\cal E}(t,k')dk',$$ where $\bar{\cal E}(t,k)$ is the unique solution in $C({{{\ensuremath{\mathbb R}}}},L^1({{{\ensuremath{\mathbb T}}}}))$ of an equation $$\label{100510}
\bar{\cal E}(t,k)= |\hat\psi(k)|^2+\int_0^t{\cal L}\bar{\cal E}(s,k)ds.$$ The existence and uniqueness of solutions in follows from the fact that ${\cal L}$ is clearly a bounded operator on $L^1({{{\ensuremath{\mathbb T}}}})$. The solution then is given by $\bar{\cal E}(t)=P^t\bar{\cal E}(0)$, where $\bar{\cal E}(0):=|\hat\psi|^2$ and $(P^t)$ is the contraction semigroup on $L^1({{{\ensuremath{\mathbb T}}}})$ generated by ${\cal L}$.
Assume also that $\{w_k(t),\,t\ge0\}$ is a family of pairwise independent standard, one dimensional, complex valued Brownian motions indexed by $k\in{{{\ensuremath{\mathbb T}}}}$. Our first principal result can be stated as follows.
\[main-thm1\]
Suppose that the dispersion relation $ {\omega}(\cdot)$ satisfies condition ${\omega})$. Then, the following are true:
\(i) if $ \hat\psi\in H^{m}({{{\ensuremath{\mathbb T}}}})$ for some $m>1/2$ then there exists a solution $\tilde \psi^{({\epsilon})}(t)$ of that belongs a.s. to $C({{{\ensuremath{\mathbb T}}}})$ for all $t\ge 0$. In addition, given an integer $n\ge1$ and $k_1,\ldots,k_n\in{{{\ensuremath{\mathbb T}}}}$, the processes $\{
(\tilde\psi^{({\epsilon})}(t,k_1),\ldots,\tilde\psi^{({\epsilon})}(t,k_n)),\,t\ge0\}$ converge in law over $C([0,+\infty);\mathbb C^n)$, as ${\epsilon}\to0+$, to $\{(\tilde\psi(t,k_1),\ldots,\tilde\psi(t,k_n)),\,t\ge0\}$, where $\{\tilde\psi(t,k),\,t\ge0\}$ is a complex valued, non-homogeneous in time Ornstein-Uhlenbeck process that is the solution of the equation $$\begin{aligned}
\label{limit-eqt1}
d \tilde\psi(t,k)&=&
-\frac{\hat\beta(k)}{4}\tilde \psi(t,k)dt +{\cal R}^{1/2}(t,k)dw_k(t),\nonumber\\
\tilde \psi(0,k)&=& \hat\psi(k),
\end{aligned}$$ (ii) if $ \hat\psi\in L^2({{{\ensuremath{\mathbb T}}}})$, then for any $f\in L^2({{{\ensuremath{\mathbb T}}}})$ and $t_*>0$ we have $$\label{040510}
\lim_{{\epsilon}\to0+}\sup_{t\in[0,t_*]}\left|\langle\tilde \psi^{({\epsilon})}(t)-\bar \psi(t),f\rangle\right|=0$$ in probability. Here $\bar \psi(t)$ is given by $$\label{012909}
\bar \psi(t,k):=\hat \psi_0(k)\exp\left\{-\frac{t\hat\beta(k)}{4}\right\}.$$
[**Remark 1.**]{} We claim that $$\label{020910}
\lim_{t\to+\infty}\sup_{k\in{{{\ensuremath{\mathbb T}}}}}|{\cal R}(t,k)- (\hat\beta(k)/2)T|=0,$$ where $T=\|\hat\psi_0\|_{L^2({{{\ensuremath{\mathbb T}}}})}^2$. The above easily follows from , provided we show that any solution $\bar{\cal
E}(t,k)$ of satisfies $$\label{010910}
\lim_{t\to+\infty}\|\bar{\cal E}(t)- T\|_{L^1({{{\ensuremath{\mathbb T}}}})}=0.$$ To prove recall that operator ${\cal L}$ given by is a generator of a strongly continuous semigroup $(P^t)$ of contractions on $L^1({{{\ensuremath{\mathbb T}}}})$. In fact, it is also a semigroup of contractions when restricted to any $L^p({{{\ensuremath{\mathbb T}}}})$, for $1\le p\le+\infty$, strongly continuous, provided that $p\in[1,+\infty)$. When $p=2$ generator ${\cal L}$ is symmetric (and so is each $P^t$) and $$\langle {\cal L} f,f\rangle=-\frac12\int_{{{{\ensuremath{\mathbb T}}}}^2}R(k,k')|f(k')-f(k)|^2dkdk'\le0,\quad \forall \,f\in L^2({{{\ensuremath{\mathbb T}}}}).$$ Hence $0$ is a simple eigenvalue of ${\cal L}$ in $L^2({{{\ensuremath{\mathbb T}}}})$, i.e. if $f\in L^2({{{\ensuremath{\mathbb T}}}})$ and satisfies ${\cal L}f=0$, then $f$ is a constant. This immediately implies that for $\bar{\cal E}(0)\in L^2({{{\ensuremath{\mathbb T}}}})$ with $T:=\int_{{{{\ensuremath{\mathbb T}}}}}\bar{\cal E}(0,k)dk$ we have $$\label{010910a}
\lim_{t\to+\infty}\|\bar{\cal E}(t)- T\|_{L^2({{{\ensuremath{\mathbb T}}}})}^2=\lim_{t\to+\infty}\int_0^{+\infty}e^{-{\lambda}t}\mu(d{\lambda})=0,$$ where $\mu$ is the spectral measure of $\bar{\cal E}(0)-T$ corresponding to ${\cal L}$. This in particular implies in case the initial data is square integrable. If $\bar{\cal E}(0)$ only belongs to $L^1({{{\ensuremath{\mathbb T}}}})$ we obtain approximating first $\bar{\cal E}(0)$ by square integrable functions and then using together with the fact that $(P^t)$ is a contraction semigroup on $L^1({{{\ensuremath{\mathbb T}}}})$.
From we obtain, for any $k\in {{{\ensuremath{\mathbb T}}}}$, $$\label{040510a}
\lim_{t\to +\infty}{{\mathbb E}}\left|\tilde \psi(t,k)-\tilde \psi_s(t,k)\right|^2=0,$$ where $ \tilde\psi_s(t,k)$ is a time homogeneous Ornstein-Uhlenbeck process given by $$\begin{aligned}
\label{limit-eqt1a}
d \tilde\psi_s(t,k)&=&
-\frac{\hat\beta(k)}{4}\tilde \psi_s(t,k)dt +\sqrt{\frac{\hat\beta(k)T}{2}}dw_k(t),\nonumber\\
\tilde \psi_s(0,k)&=& \hat\psi(k).
\end{aligned}$$
[**Remark 2.**]{} Let us also comment briefly on condition ${\omega})$. A similar hypothesis appears in the wave turbulence theory under the name of a [*no resonance condition*]{}, see e.g. [@lfz]. In our context we use it, among others, to prove the asymptotic (in the limit ${\epsilon}\to 0+$) independence of $\tilde\psi^{({\epsilon})}(t,k)$ for different $k$. This independence implies, in particular, the self-averaging property of the energy $ |\tilde \psi^{({\epsilon})}(t,k)|^2$ i.e. its convergence in probability to a deterministic limit, as ${\epsilon}\to0+$, in the weak topology, see Proposition \[lm013108\] below. This observation plays a crucial rôle in the proof of part (i) of Theorem \[main-thm1\]. Without lack of resonance condition of the type ${\omega}$), it is in principle possible that the second mixed moment of the energy corresponding to different modes does not vanish in the limit, as ${\epsilon}\to0+$, so that the key estimate below fails making self-averaging of energy impossible.
The following simple criterion is useful for verification of condition ${\omega})$, e.g. for dispersion relation ${\omega}(k)$ of the form . Recall that from the assumptions made we know that ${\omega}\in C^{\infty}({{{\ensuremath{\mathbb T}}}}\setminus\{0\})$.
\[lmK\] Suppose that the dispersion relation ${\omega}(\cdot)$ satisfies the following condition: for any $|a|<1/2$ and ${\sigma}=\pm1$ the set of solutions of an equation $$\label{sec-der}
{\omega}'(k)={\sigma}{\omega}'(k+a)$$ is possibly of positive Lebesgue measure in ${{{\ensuremath{\mathbb T}}}}$, only if $a=0$ and ${\sigma}=1$. Then, for any $(k_1,k_2)$ such that $k_1\not=k_2$ the hypothesis ${\omega})$ holds.
[**Proof.**]{} Fix $(k_1,k_2)$ such that $k_1\not=k_2$. To simplify we consider only the set ${\cal K}_1$ that corresponds to ${\sigma}_1={\sigma}_2={\sigma}_3=1$ and prove that: $${\cal K}_1(k_1,k_2):= [k:{\omega}(k_1)+{\omega}(k-k_2)={\omega}(k_2)+{\omega}(k-k_1)]$$ is of null Lebesgue measure. The remaining cases can be dealt with similarly. Suppose, on the contrary, that the Lebesgue measure of the set is positive. Then almost every point of ${\cal K}_1(k_1,k_2)$ is a density point of the set. In particular that means that at any such point we have $${\omega}'(k-k_2)={\omega}'(k-k_1)$$ but this would clearly contradict the assumption made in the statement of the lemma. [$\Box$]{}
It is quite straightforward to verify that the above lemma applies to the dispersion relation of the form .
Statistically homogeneous initial data {#sec2.3.2}
---------------------------------------
For a given non-negative $m$ we assume that the initial data $\hat \psi$ is an $H^{-m}({{{\ensuremath{\mathbb T}}}})$ valued Gaussian random element. More precisely, suppose that ${\cal E}_0(\cdot)$ is a non-negative function such that $$\label{011210a}
\sum_{x\in{{{\ensuremath{\mathbb Z}}}}}|\langle {\cal E}_0,e_x\rangle|<+\infty,$$ $\{\xi_y,\,y\in{{{\ensuremath{\mathbb Z}}}}\}$ are i.i.d. complex Gaussian random variables such that ${{\mathbb E}}\xi_0=0$ and ${{\mathbb E}}|\xi_0|^2=1$, and $$\label{053110}
\hat\psi(k)=\sum_{y\in{{{\ensuremath{\mathbb Z}}}}}\xi_y {\cal E}^{1/2}_0(k)e_y(k).$$ The law of $\hat \psi$ is supported in $H^{-m}({{{\ensuremath{\mathbb T}}}})$, provided that $m>1/2$. Its covariance form equals $$\label{021310aa}
{\cal C}(J_1,J_2):={{\mathbb E}}\left[\langle J_1,\hat\psi\rangle\langle J_2,\hat\psi\rangle^* \right]=\int_{{{{\ensuremath{\mathbb T}}}}}{\cal E}_0(k)J_1(k)J_2^*(k)dk$$ for any $J_1,J_2\in C^\infty({{{\ensuremath{\mathbb T}}}})$. The Gibbs equilibrium states described in the introduction correspond to ${\cal E}_0(k)\equiv \hbox{const}$. Using Proposition \[prop010910\] we conclude that equation has a unique mild solution $\{\tilde \psi^{({\epsilon})}(t),\,t\ge0\}$ whose realizations belong to $C([0,+\infty);H^{-m}({{{\ensuremath{\mathbb T}}}}))$, provided $m<3/2$.
Let ${\cal R}(t,k)$ be given by with $\bar{\cal E}(t,k)$ the solution of satisfying $\bar{\cal E}(0,k)={\cal E}_0(k)$. Observe that the operator $
f(k)\mapsto {\cal R}^{1/2}(t,k)f(k)
$ is Hilbert-Schmidt, when considered from $L^2({{{\ensuremath{\mathbb T}}}})$ to $H^{-m}({{{\ensuremath{\mathbb T}}}})$, provided $m>1/2$. Indeed $$\sum_y\|{\cal R}^{1/2}(t)e_y\|_{H^{-m}(\mathbb T)}^2=\sum_{y,y_1}(1+y_1^2)^{-m}\left|\int{\cal R}^{1/2}(t,k)e_{y-y_1}(k)dk\right|^2.$$ By Plancherel’s identity the right hand side equals $$\begin{aligned}
&&
\sum_{y_1}(1+y_1^2)^{-m}\sum_{z}\left|\int{\cal R}^{1/2}(t,k)e_{z}(k)dk\right|^2=\sum_{y_1}(1+y_1^2)^{-m}\|{\cal R}^{1/2}(t,\cdot)\|^2_{L^2(\mathbb T)}\\
&&
=\sum_{y_1}(1+y_1^2)^{-m}\|{\cal R}(t,\cdot)\|^2_{L^1(\mathbb T)}<+\infty.\end{aligned}$$
Since in addition $
f(k)\mapsto -(\hat\beta(k)/4)f(k)
$ is bounded on $H^{-m}({{{\ensuremath{\mathbb T}}}})$, the equation $$\begin{aligned}
\label{limit-eqt}
d \bar\psi_*(t,k)&=&
-\frac{\hat\beta(k)}{4} \bar \psi_*(t,k)dt +{\cal R}^{1/2}(t,k)dW(t,k),\nonumber\\
\bar \psi_*(0,k)&=& \hat\psi(k)
\end{aligned}$$ has a unique $H^{-m}({{{\ensuremath{\mathbb T}}}})$-valued mild solution, by virtue of Theorem 7.4, p. 186 of [@DZ]. It is given by the formula $$\bar\psi_*(t,k)=e^{-\hat \beta(k) t/4}\hat\psi+\int_0^te^{-\hat \beta(k) (t-s)/4}{\cal R}^{1/2}(s,k)dW(s,k).$$ We denote by $H^{-m}_w({{{\ensuremath{\mathbb T}}}})$ the Hilbert space equipped with the weak topology. Our main result is as follows.
\[main-thm2\] Suppose that $3/2>m>1/2$ and both and condition ${\omega})$ hold. Then, under the above assumptions, the processes $\{\tilde \psi^{({\epsilon})}(t),\,t\ge0\}$ converge in law over $C([0,+\infty),H^{-m}_w({{{\ensuremath{\mathbb T}}}}))$, as ${\epsilon}\to0+$, to $\{\bar\psi_*(t),\,t\ge0\}$.
[**Remark.**]{} As in the remark made after Theorem \[main-thm1\] we can also conclude that $$\label{040510b}
\lim_{t\to +\infty}{{\mathbb E}}\left|\langle\bar \psi_*(t)-\bar \psi_{s}(t),f\rangle\right|^2=0,$$ where $ \bar\psi_s(t)$ is a time homogeneous, distribution valued Ornstein-Uhlenbeck process given by $$\begin{aligned}
\label{limit-eqt1b}
d \bar\psi_s(t,k)&=&
-\frac{\hat\beta(k)}{4} \bar \psi_s(t,k)dt +\sqrt{\frac{\hat\beta(k) T}{2}}dW(t,k),\nonumber\\
\bar \psi_s(0,k)&=& \hat\psi(k),
\end{aligned}$$ where $T=\|{\cal E}_0\|_{L^1({{{\ensuremath{\mathbb T}}}})}$.
Proof of Theorem \[main-thm1\] {#sec4}
==============================
The fact that the solution of lies in $C({{{\ensuremath{\mathbb T}}}})$ for each ${\epsilon}>0$ is a direct consequence of Proposition \[prop010910\] and the embedding of $H^m({{{\ensuremath{\mathbb T}}}})$ into $C({{{\ensuremath{\mathbb T}}}})$ for $m>1/2$. We prove first the part $(i)$ of the theorem. To explain the idea of the proof assume that $n=1$ (that is, the process $\hat\psi(t,k)$ for a fixed $k$), the independence of the compensated wave function for various $k$ is handled in the same manner. Since the coefficients appearing in the stochastic differential equation describing the evolution of $\tilde \psi^{({\epsilon})}(t)$ (see ) are of the order $O(1)$, it is easy to conclude that for each $k$ the laws of the processes $\{\tilde \psi^{({\epsilon})}(t,k),\,t\ge0\}$ are tight over $C([0,+\infty);\mathbb C)$, as ${\epsilon}\to0+$. In order to identify the limit, thus proving part i) of the theorem, we have to deal with the rapidly oscillating terms. First, we show that the rapidly oscillating part of the bounded variation term in (with the factor $\exp\{2i\omega(k)t/{\epsilon}\}$ in (\[012808\])) vanishes in the limit thanks to part i) of Corollary \[cor2\] below.
Next, the limit of the martingale part $\tilde {\cal M}_t^{({\epsilon})}(k)$ in (\[mollified-eqt\]) is a complex Gaussian martingale with the quadratic variation equal to $\int_0^t{\cal R}(s,k)ds$ thanks to the following: $$\label{mar0802}
\lim_{{\epsilon}\to0+}\sup_{t\in[0,t_*]}\left|\langle\tilde {\cal M}^{({\epsilon})}(k),(\tilde {\cal M}^{({\epsilon})})^*(k)\rangle_t-\int_0^t{\cal R}(s,k)ds\right|=0,$$ where the convergence holds in probability, for any $t_*>0$. This is done in Proposition \[lm013108\]. The method of proof of (\[mar0802\]) is as follows. From , we compute the quadratic variation: $$\begin{aligned}
\label{050410}
&&
\langle \tilde{\cal M}^{({\epsilon})}(k),(\tilde {\cal M}^{({\epsilon})})^*(k)\rangle_t
\\
&&
=\sum_{{\sigma}_1,{\sigma}_2=\pm1}{\sigma}_1{\sigma}_2\int_0^tds\int_{{{{\ensuremath{\mathbb T}}}}}r^2(k,k')\hat\psi^{({\epsilon})}_{{\sigma}_1}(s,k-k')(\hat\psi^{({\epsilon})}_{{\sigma}_2})^*(s,k-k')\nonumber
dk'.\end{aligned}$$ The terms appearing in are of the following form: $$\begin{aligned}
&&{\cal V}_{\epsilon}^{(0)}(t):=\int_0^t\langle|\hat\psi^{({\epsilon})}(s)|^2,f\rangle ds,\nonumber\\
\label{mar804}&&
{\cal V}_{{\epsilon}}^{(1)}(t):=\int_0^t
\int_{{{{\ensuremath{\mathbb T}}}}}\hat\psi^{({\epsilon})}(s,k)\hat\psi^{({\epsilon})}(s,-k)f^*(k)dk \; ds.\end{aligned}$$ Here $f(k)$ is a certain explicit function related to the scattering kernel. As $\hat\psi^{({\epsilon})}(t,k)$ (without the compensation) is rapidly oscillating as $e^{-i\omega(k)t/{\epsilon}}$, therefore we expect that only ${\cal V}_{\epsilon}^{(0)}(t)$ has a non-trivial limit. This term contains no oscillation and is essentially the time integral of scattered energy $|\hat \psi^{({\epsilon})}(t,k)|^2$. It has been shown in [@BOS] that the expectation of the energy converges to the solution of . We need to strengthen this result to convergence in probability.
The proof of part ii) of the theorem uses the same ideas. Integrating against a test function results in the formula for the quadratic variation, see , containing only terms with fast oscillating factors, so the stochastic part vanishes in the limit.
We now turn to the proof of part (i) the theorem. In particular, we assume that $\hat\psi\in H^m$, $m>1/2$ so that $\tilde\psi^{({\epsilon})}(t,k)$ is continuous and point-wise evaluations in $k$ make sense. An application of the Itô formula to yields, see Theorem 4.17 of [@DZ], $$\label{wigner-eqt}
d|\hat\psi^{({\epsilon})}(t,k)|^2=\left[I_{\epsilon}(t,k)+I\!I_{\epsilon}(t,k)\right]dt
+d{\cal M}^{({\epsilon})}_t(k) + d {\cal M}^{({\epsilon})*}_t(k) ,$$ where $$\begin{aligned}
&&I_{\epsilon}(t,k):=
(A[\hat\psi^{({\epsilon})}(t)])^*\left(k\right)\hat \psi^{({\epsilon})}\left(t,k\right)\vphantom{\int_0^1}+(\hat\psi^{({\epsilon})})^*\left(t,k\right)A[\hat \psi^{({\epsilon})}(t)]\left(k\right),
\\
&&
I\!I_{\epsilon}(t,k):=\sum_{y\in{{{\ensuremath{\mathbb Z}}}}}\left|Q[\hat\psi^{({\epsilon})}(t)](e_y)\left(k\right)\right|^2,\end{aligned}$$ and $ {\cal M}^{({\epsilon})}_t $ is an ${\cal F}_t$-adapted local martingale, given by $$\begin{aligned}
&&{\cal M}^{({\epsilon})}_t(k) =\int_0^t\hat \psi^{({\epsilon})}\left(s,k\right) (Q[\hat\psi^{({\epsilon})}(s)]
dW(s))^*\left(k\right) .\end{aligned}$$ From we obtain that $$I_{\epsilon}(t,k)=-\frac{\hat\beta(k)}{2}|\hat\psi^{({\epsilon})}(t,k)|^2-\frac{\hat\beta(k)}{4}\hat\psi^{({\epsilon})}_2(t,k),$$ where $$\hat\psi^{({\epsilon})}_2(t,k):=\hat\psi^{({\epsilon})}(t,k)\hat\psi^{({\epsilon})}(t,-k)+(\hat\psi^{({\epsilon})})^*(t,k)(\hat\psi^{({\epsilon})})^*(t,-k),$$ while equation yields $$I\!I_{\epsilon}(t,k)=\int_{{{{\ensuremath{\mathbb T}}}}}R(k,k')|\hat\psi^{({\epsilon})}(t,k')|^2dk'+\frac12\int_{{{{\ensuremath{\mathbb T}}}}}R(k,k')\hat\psi^{({\epsilon})}_2(t,k')dk'.$$
Analogous equation can be derived for $d[\hat\psi^{({\epsilon})}(t,k)\hat\psi^{({\epsilon})}(t,-k)]$. The corresponding terms shall be denoted by $\tilde I_{\epsilon}(t,k)$, $\tilde{I\!I}_{\epsilon}(t,k)$ and the martingale $
{\cal N}^{({\epsilon},1)}_{t}(k) +{\cal N}^{({\epsilon},2)}_{t}(k),
$ where $$\tilde I_{\epsilon}(t,k)=-\frac{2i{\omega}(k)}{{\epsilon}}\hat\psi^{({\epsilon})}_2(t,k)+{\cal P}[\hat\psi^{({\epsilon})}(t),(\hat\psi^{({\epsilon})})^*(t)],$$ $$\label{030207ba}
I\!I_{\epsilon}(t,k)={\cal Q}[\hat\psi^{({\epsilon})}(t),(\hat\psi^{({\epsilon})})^*(t)],$$ where ${\cal P},{\cal Q}$ are second degree polynomials in $\hat\psi^{({\epsilon})}(t),(\hat\psi^{({\epsilon})})^*(t)$, and $$\begin{aligned}
&&{\cal N}^{({\epsilon},1)}_t(k) =\int_0^t\hat \psi^{({\epsilon})}\left(s,- k\right) (Q[\hat\psi^{({\epsilon})}(s)]
dW(s))\left(k\right) ,\nonumber\\
&&
\\
&&
{\cal N}^{({\epsilon},2)}_t(k) =\int_0^t\hat \psi^{({\epsilon})}\left(s,k\right)
(Q[(\hat\psi^{({\epsilon})}_{-1})^*(s)] dW(s))\left(-k\right).\nonumber\end{aligned}$$
\[lm013108\] Let $f\in L^\infty(\mathbb T)$, ${\cal V}_{\epsilon}^{(0)}(t)$ given by , and let ${\cal V}_{{\epsilon},a}^{(1)}(t)$ be defined by $$\label{eq:1}
{\cal V}_{{\epsilon},a}^{(1)}(t):=\int_0^t
\int_{{{{\ensuremath{\mathbb T}}}}} \exp\left\{\frac{isa}{{\epsilon}}
\right\}\hat\psi^{({\epsilon})}(s,k)\hat\psi^{({\epsilon})}(s,-k)f^*(k)dk \; ds, \quad
a\in {{{\ensuremath{\mathbb R}}}}.$$ Then, for any $t_*>0$ we have $$\label{013009}
\lim_{{\epsilon}\to0+}\sup_{t\in[0,t_*]}\left|{\cal V}_{\epsilon}^{(0)}(t)-\int_0^t\langle\bar{\cal E}(s),f\rangle ds\right|= 0$$ and $$\label{023009}
\lim_{{\epsilon}\to0+}\sup_{t\in[0,t_*]}\left|{\cal V}_{{\epsilon},a}^{(1)}(t)\right|= 0,\quad a\in{{{\ensuremath{\mathbb R}}}},$$ in probability.
The proof of this proposition shall be obtained at the end of a series of lemmas.
\[lm023108\] For any $p\in[2,+\infty)$ there exists $C>0$ such that, for any $t_*>0$ $$\label{110510}
\sup_{{\epsilon}\in(0,1]}{{\mathbb E}}\left[\sup_{t\in[0,t_*]}
\|\hat\psi^{({\epsilon})}(t)\|_{L^p({{{\ensuremath{\mathbb T}}}})}^p\right]\le
Ce^{Ct_*}\|\hat\psi\|_{L^p({{{\ensuremath{\mathbb T}}}})}^p ,$$ and, $$\sup_{{\epsilon}\in(0,1],k\in{{{\ensuremath{\mathbb T}}}}}{{\mathbb E}}\left[\sup_{t\in[0,t_*]}|\hat\psi^{({\epsilon})}(t,k)|^p\right]\le Ce^{Ct_*}\|\hat\psi\|^p_{L^p({{{\ensuremath{\mathbb T}}}})}.
\label{011310z}$$
[**Proof.**]{} Let $$T_t^{({\epsilon})}\hat\psi(k):=\exp\left\{-i\frac{{\omega}(k)t}{{\epsilon}}\right\}\hat\psi(k),\quad \hat \psi\in L^p({{{\ensuremath{\mathbb T}}}}),\,t\in{{{\ensuremath{\mathbb R}}}}.$$ We obviously have $$\label{013108}
\|T_t^{({\epsilon})}\hat\psi\|_{L^p({{{\ensuremath{\mathbb T}}}})}= \|\hat\psi\|_{L^p({{{\ensuremath{\mathbb T}}}})},\quad\forall\,t\ge 0. $$ Using the Duhamel formula, the solution of can be written as $$\begin{aligned}
\label{023108}
\hat\psi^{({\epsilon})}(t,k)=\hat\psi(k)+\int_0^tT_{t-s}^{({\epsilon})}B[\hat\psi^{({\epsilon})}(s)](k)ds
+\int_0^tT_{t-s}^{({\epsilon})}Q[\hat\psi^{({\epsilon})}(s)]dW(s,k),\end{aligned}$$ where $B f (k) = -\hat \beta(k) [f(k) - f^*(-k)]/4$. Hence, for a given ${\epsilon}\in(0,1]$ and $t_0>0$ to be adjusted later on, we can write $$\begin{aligned}
\label{010410}
&&{{\mathbb E}}\left[\sup_{t\in[0,t_0]} | \hat\psi^{({\epsilon})}(t,k)|^p\right]\le C\left\{|\hat\psi(k)|^p+t_0^{p-1}\int_0^{t_0}{{\mathbb E}}|\hat\psi^{({\epsilon})}(s,k)|^pds\right.\nonumber\\
&&
+\left.{{\mathbb E}}\left\{\sup_{t\in[0,t_0]}\left|\int_0^tT_{-s}^{({\epsilon})}Q[\hat\psi^{({\epsilon})}(s)]dW(s,k)\right|^p\right\}\right\}.
\end{aligned}$$ To estimate the martingale term on the right hand side we use Burkholder-Davis-Gundy inequality which allows to bound it by $$\begin{aligned}
\label{020410}
4^{p/2}{{\mathbb E}}\left(\int_0^{t_0}\int_{{{{\ensuremath{\mathbb T}}}}}R(k,k')|\hat\psi^{({\epsilon})}(s,k-k')|^2dk'ds\right)^{p/2}
\le C_1t^{p/2-1}_0\int_0^{t_0}{{\mathbb E}}\|\hat\psi^{({\epsilon})}(s)\|^p_{L^p({{{\ensuremath{\mathbb T}}}})}ds,
\end{aligned}$$ for some constant $C_1>0$. Choosing $t_0$ sufficiently small, so that $Ct_0^p+CC_1t_0^{p/2}<1/2$, we conclude that $$\label{011411}
{{\mathbb E}}\left\{\sup_{t\in[0,t_0]} \|\hat\psi^{({\epsilon})}(t)\|^p_{L^p({{{\ensuremath{\mathbb T}}}})}\right\}\le 2C\|\hat\psi\|^p_{L^p({{{\ensuremath{\mathbb T}}}})}.$$ The argument leading to can be used on each of the intervals $[jt_0, (j+1) t_0)$ for any $j\ge1$ and yields $$\label{011411j}
{{\mathbb E}}\left\{\sup_{t\in[jt_0,(j+1)t_0]} \|\hat\psi^{({\epsilon})}(t)\|^p_{L^p({{{\ensuremath{\mathbb T}}}})}\right\}\le C{{\mathbb E}}\|\hat\psi^{({\epsilon})}(jt_0)\|^p_{L^p({{{\ensuremath{\mathbb T}}}})}
\le C{{\mathbb E}}\left\{\sup_{t\in[(j-1)t_0,jt_0]} \|\hat\psi^{({\epsilon})}(t)\|^p_{L^p({{{\ensuremath{\mathbb T}}}})}\right\},$$ for some constant $C>0$ independent of $j$ and ${\epsilon}\in(0,1]$. Hence, after $j$ iterations of the above estimate, we conclude $$\label{011411ja}
{{\mathbb E}}\left\{\sup_{t\in[jt_0,(j+1)t_0]} \|\hat\psi^{({\epsilon})}(t)\|^p_{L^p({{{\ensuremath{\mathbb T}}}})}\right\}\le C^j\|\hat\psi\|^p_{L^p({{{\ensuremath{\mathbb T}}}})}$$ and follows. Combining the above result with estimates and we conclude estimate . [$\Box$]{}
Using the above lemma we conclude the following.
\[cor2\] For given $t_*>0$ and function $f\in C^1[0,t_*]$ we have the following:
- if $k\in{{{\ensuremath{\mathbb T}}}}$ and $a\in{{{\ensuremath{\mathbb R}}}}$ are such such that $-a\not={\omega}(k)$ then, $$\label{070410}
\lim_{{\epsilon}\to0+}{{\mathbb E}}\left|\sup_{t\in[0,t_*]}\int_0^t\exp\left\{-i\frac{as}{{\epsilon}}\right\}f(s)\hat\psi_{\epsilon}(s,k)ds\right|=0,$$
- if $k,k'\in{{{\ensuremath{\mathbb T}}}}$ and $a\in{{{\ensuremath{\mathbb R}}}}$ are such that $-a\not={\omega}(k)+{\omega}(k')$ then, $$\label{030410}
\lim_{{\epsilon}\to0+}{{\mathbb E}}\left\{\sup_{t\in[0,t_*]}\left|\int_0^t\exp\left\{-i\frac{as}{{\epsilon}}\right\}f(s)\hat\psi^{({\epsilon})}(s,k)\hat\psi^{({\epsilon})}(s,k')ds\right|\right\}=0,$$
- if ${\omega}(k)+a\not={\omega}(k')$ then, $$\label{030410a}
\lim_{{\epsilon}\to0+}{{\mathbb E}}\left\{\sup_{t\in[0,t_*]}\left|\int_0^t\exp\left\{-i\frac{as}{{\epsilon}}\right\}f(s)\hat\psi^{({\epsilon})}(s,k)(\hat\psi^{({\epsilon})})^*(s,k')ds\right|\right\}=0.$$
[**Proof.**]{} Using we obtain $$\begin{aligned}
\label{090510}
&&
\exp\left\{-i\frac{at}{{\epsilon}}\right\}f(t)
\hat\psi_{\epsilon}(t,k)-f(0)
\hat\psi(k)
=-i\frac{a+{\omega}(k)}{{\epsilon}}\int_0^t\exp\left\{-i\frac{sa}{{\epsilon}}\right\}f(s)\hat\psi_{\epsilon}(s,k)ds \nonumber
\\
&&
+\int_0^t{\cal P}[\hat\psi_{\epsilon}(s),(\hat\psi_{\epsilon})^*(s)](k)ds+\int_0^t\sum_{y\in{{{\ensuremath{\mathbb Z}}}}}{\cal Q}_y[\hat\psi_{\epsilon}(s),(\hat\psi_{\epsilon})^*(s)](k)w_y(ds),\end{aligned}$$ where ${\cal P}$, ${\cal Q}_y$ are first degree polynomials in $\hat\psi_{\epsilon}(s)$, $(\hat\psi_{\epsilon})^*(s)$ with bounded coefficients. Using Lemma \[lm023108\] we have $${{\mathbb E}}\left[\sup_{s\in[0,t_*]}\sum_{y\in{{{\ensuremath{\mathbb Z}}}}}|{\cal
Q}_y[\hat\psi_{\epsilon}(s),(\hat\psi_{\epsilon})^*(s)](k)|^2\right] \le
C\|\hat\psi\|_{L^2({{{\ensuremath{\mathbb T}}}})}^2.$$ Dividing both sides of by $({\omega}(k)+a)/{\epsilon}$ (possible since this factor is not equal to $0$) we calculate $$\int_0^t\exp\left\{-i\frac{sa}{{\epsilon}}\right\}f(s)\hat\psi_{\epsilon}(s,k)ds.$$ Using Lemma \[lm023108\] we can easily conclude .
The proofs of and are analogous. We use the Itô formula to express $d[\hat\psi^{({\epsilon})}(s,k)\hat\psi^{({\epsilon})}(s,k')]$ and $d[\hat\psi^{({\epsilon})}(s,k)(\hat\psi^{({\epsilon})})^*(s,k')]$. Then, we repeat the argument used above. [$\Box$]{}
The following lemma shall be crucial for us.
\[lm013009\] For any $f\in L^2({{{\ensuremath{\mathbb T}}}})$, $t_*>0$ we have $$\label{010310}
\lim_{{\epsilon}\to0+}{{\mathbb E}}\left[\sup_{t\in[0,t_*]}\left|\langle{\cal M}^{({\epsilon})}_t,f\rangle\right|^2\right]=0$$ and $$\label{020310}
\lim_{{\epsilon}\to0+}{{\mathbb E}}\left[\sup_{t\in[0,t_*]}\left|\langle{\cal N}^{(i,{\epsilon})}_{t},f\rangle\right|^2\right]=0, \quad i=1,2.$$
[**Proof.**]{} We only prove , the argument for is very similar. We write $$\begin{aligned}
\label{030207a}
&&
{{\mathbb E}}\left|\langle{\cal M}^{({\epsilon})}_t,f\rangle\right|^2
\le2\left\{ \sum_{y\in{{{\ensuremath{\mathbb Z}}}}}\int_0^tds\,{{\mathbb E}}\left|\int_{{{{\ensuremath{\mathbb T}}}}^2} r(k,k')f^*(k)(\hat\psi^{({\epsilon})})^*(s,k-k')e_y^*(k')\hat\psi^{({\epsilon})}(s,k)d{{\bf k}}\,\right|^2\right. \nonumber\\
&&\left. +\sum_{y\in{{{\ensuremath{\mathbb Z}}}}}\int_0^tds{{\mathbb E}}\left|\int_{{{{\ensuremath{\mathbb T}}}}^2} r(k,k')f^*(k)(\hat\psi^{({\epsilon})})(s,k'-k)e_y^*(k')\hat\psi^{({\epsilon})}(s,k)d{{\bf k}}\right|^2\right\} .\end{aligned}$$ Here, for abbreviation sake, we wrote $d{{\bf k}}=dkdk'$. Using the Parseval identity we can further transform the right hand side of into $$\begin{aligned}
\label{030207aa}
&&
2 \int_0^tds\int_{{{{\ensuremath{\mathbb T}}}}^3} r(k,k')r(k_1,k')f^*(k)f(k_1)\nonumber\\
&&\times
\left\{{{\mathbb E}}\left[(\hat\psi^{({\epsilon})})^*(s,k-k')\hat\psi^{({\epsilon})}(s,k)\hat\psi^{({\epsilon})}(s,k_1-k')(\hat\psi^{({\epsilon})})^*(s,k_1)\right]\right.\nonumber\\
&&\left. +{{\mathbb E}}\left[\hat\psi^{({\epsilon})}(s,k-k')\hat\psi^{({\epsilon})}(s,k)(\hat\psi^{({\epsilon})})^*(s,k_1-k')(\hat\psi^{({\epsilon})})^*(s,k_1)\right]\right\}d{{\bf k}},\end{aligned}$$ where $d{{\bf k}}=dkdk_1dk'$.
Consider the term of corresponding to the first expectation (the other can be dealt with in a similar fashion). Let $${\cal K}_{1}= [(k,k',k_1):{\omega}(k)+{\omega}(k'-k_1)={\omega}(k')+{\omega}(k-k_1)].$$ Thanks to condition ${\omega})$ the three dimensional Lebesgue measure on ${{{\ensuremath{\mathbb T}}}}^3$ of the set vanishes. We claim that for ${{\bf k}}=(k,k',k_1)\not\in {\cal K}_{1}$ we have $$\label{phi}
\lim_{{\epsilon}\to0+}\int_0^t\Psi^{({\epsilon})}(s,{{\bf k}})ds=0,$$ where $$\Psi^{({\epsilon})}(s,{{\bf k}}):={{\mathbb E}}\left[(\hat\psi^{({\epsilon})})^*(s,k-k')\hat\psi^{({\epsilon})}(s,k_1-k')\hat\psi^{({\epsilon})}(s,k)(\hat\psi^{({\epsilon})})^*(s,k_1)\right].$$ Using and Itô formula we conclude that $$\begin{aligned}
\label{010110}
&&
\frac{i}{{\epsilon}}\left[{\omega}(k-k')+{\omega}(k_1)-{\omega}(k_1-k')-{\omega}(k)\right] \int_0^t\Psi^{({\epsilon})}(s,{{\bf k}})ds\nonumber\\
&&
=\Psi^{({\epsilon})}(t,{{\bf k}})-\Psi^{({\epsilon})}(0,{{\bf k}})+\int_0^t{\cal P}[\hat\psi^{({\epsilon})}(s),(\hat\psi^{({\epsilon})})^*(s)]({{\bf k}})ds,\end{aligned}$$ where ${\cal P}$ is a fourth degree polynomial formed over the wave function $\hat\psi^{({\epsilon})}(s)$, $(\hat\psi^{({\epsilon})})^*(s)$. Dividing both sides of by the factor in front of the integral on the left hand side and subsequently using with $p=4$ we conclude . The lemma then follows, provided we can substantiate the following interchange of the limit with integral $$\begin{aligned}
&&
\lim_{{\epsilon}\to0+}\int_0^tds\int_{{{{\ensuremath{\mathbb T}}}}^3}r(k,k')r(k_1,k')f^*(k)f(k_1)\Psi^{({\epsilon})}(s,{{\bf k}})d{{\bf k}}\\
&&
=\int_{{{{\ensuremath{\mathbb T}}}}^3}r(k,k')r(k_1,k')f^*(k)f(k_1)d{{\bf k}}\left\{ \lim_{{\epsilon}\to0+}\int_0^t\Psi^{({\epsilon})}(s,{{\bf k}})ds\right\}.\end{aligned}$$ The latter however is a consequence of the Lebesgue dominated convergence theorem and . This ends the proof of . The proof of is analogous. [$\Box$]{}
Proof of Proposition \[lm013108\] {#proof-of-proposition-lm013108 .unnumbered}
----------------------------------
We first demonstrate . It is a consequence of parts ii) and iii) of Corollary \[cor2\], and the Lebesgue dominated convergence theorem. Indeed, $${{\mathbb E}}\left\{\sup_{t\in[0,t_*]}\left|{\cal V}_{{\epsilon},a}^{(1)}(t)\right|\right\}\le {{\mathbb E}}\left\{\int_{{{{\ensuremath{\mathbb T}}}}}dk\sup_{t\in[0,t_*]}\left|\int_0^t\exp\left\{\frac{isa}{{\epsilon}}\right\}\hat\psi^{({\epsilon})}(s,k)\hat\psi^{({\epsilon})}(s,-k)f^*(k)ds\right|\right\}.$$ Using condition ${\omega}$) we conclude that the expression under the integral over $k$ on the right hand side vanishes, as ${\epsilon}\to0+$, possibly outside a set of $k$-s of null Lebesgue measure. Invoking again we can substantiate exchanging of taking the limit and integration and follows.
As for , observe that from the Itô formula for $d|\hat\psi^{({\epsilon})}(t,k)|^2$ we have $$\begin{aligned}
\langle|\hat\psi^{({\epsilon})}(t)|^2,f\rangle- \langle|\hat\psi(0)|^2,f\rangle =\int_0^t\langle{\cal L} |\hat\psi^{({\epsilon})}(s)|^2,f\rangle ds
+\frac12\int_0^t\langle {\cal L}\hat\psi^{({\epsilon})}_2(s),f\rangle ds+\langle{\cal M}^{({\epsilon})}_t,f\rangle.\end{aligned}$$ Denote by $\{Q_{{\epsilon}}, \,{\epsilon}\in(0,1]\}$ the family of the laws of $\{|\hat\psi^{({\epsilon})}(t)|^2,\,t\ge0\}$ over $C([0,+\infty),L^2_w({{{\ensuremath{\mathbb T}}}}))$. Here $L^2_w({{{\ensuremath{\mathbb T}}}})$ stands for the space $L^2({{{\ensuremath{\mathbb T}}}})$ equipped with the weak topology.
Using Lemma \[lm023108\] we conclude from the above equality that for any $t_*>0$ there exists a constant $C>0$ such that $${{\mathbb E}}\left|\langle|\hat\psi^{({\epsilon})}(t)|^2,f\rangle-\langle|\hat\psi^{({\epsilon})}(s)|^2,f\rangle\right|^4\le C(t-s)^2 ,\quad\forall\,{\epsilon}\in(0,1],\,t,s\in[0,t_*].$$ This, according to Theorem 12. 3 of [@bil], implies tightness of the family of the laws of $\{\langle|\hat\psi^{({\epsilon})}(t)|^2,f\rangle,\,t\ge0\}$, as ${\epsilon}\to0+$, over $C[0,+\infty)$ equipped with the usual topology of uniform convergence on compact intervals. From the above and estimate we conclude weak pre-compactness of $Q_{{\epsilon}}$, ${\epsilon}\in(0,1]$, see Theorem 3.1, p. 276 of [@jakubowski]. Thanks to Lemma \[lm013009\] and the already proved formula we conclude that the limiting law is a $\delta$-type measure supported on $\bar{\cal E}(t)$ – the solution of . This, in particular, implies that $$\lim_{{\epsilon}\to0+} \sup_{t\in[0,t_*]}\left|\langle|\hat\psi^{({\epsilon})}(t)|^2-\bar{\cal E}(t),f\rangle\right|=0$$ in probability. Hence follows. [$\Box$]{}
Proof of part (i) of Theorem \[main-thm1\] {#proof-of-part-i-of-theorem-main-thm1 .unnumbered}
------------------------------------------
With the results proved above in hand, we return to the proof of part (i) Theorem \[main-thm1\]. Assume first that $n=1$ and we consider the process $\tilde \psi^{({\epsilon})}(t,k)$ evaluated at a single $k$. From and we conclude easily that for any $t_*>0$ there exists a constant $C>0$ such that $${{\mathbb E}}|\tilde \psi^{({\epsilon})}(t,k)-\tilde \psi^{({\epsilon})}(s,k)|^4\le C(t-s)^2,\quad\forall\,{\epsilon}\in(0,1],\,s,t\in[0,t_*].$$ This implies tightness of the laws of $\{\tilde \psi^{({\epsilon})}(t,k),\,t\ge0\}$ over $C[0,+\infty)$.
In the next step we identify the limiting law $P_k$ of $\{\tilde \psi^{({\epsilon})}(t,k),\,t\ge0\}$ over $C[0,+\infty)$. Denote by $\Pi_t(f):=f(t)$, $f\in C[0,+\infty)$ the canonical coordinate map.
Consider the complex valued martingale given by . Its quadratic variation is given by and, of course, $\langle \tilde{\cal M}^{({\epsilon})}(k),\tilde {\cal M}^{({\epsilon})}(k)\rangle_t=0.$ Using Proposition \[lm013108\] we conclude that $$\lim_{{\epsilon}\to0+}\sup_{t\in[0,t_*]}\left|\langle \tilde{\cal M}^{({\epsilon})}(k),(\tilde {\cal M}^{({\epsilon})})^*(k)\rangle_t-\int_0^t{\cal R}(s,k)ds\right|=0.$$ Then by virtue of Theorem 5.4 of [@helland] we conclude that $\{ \tilde{\cal M}^{({\epsilon})}_t,\,t\ge0\}$ converge in law over $C[0,+\infty)$ to a complex valued Gaussian process $\{ \tilde{\cal M}_t,\,t\ge0\}$ given by $$\label{010510}
\tilde{\cal M}_t(k):=\int_0^t{\cal R}^{1/2}(s,k) w(ds),$$ where $\{ w(t),\,t\ge0\}$ is a complex valued standard Brownian motion.
Assume now that $k\not=0$ and $P_k$ is a limiting law of $\{\tilde\psi^{({\epsilon})}(t,k),\,t\ge0\}$ obtained from a certain sequence ${\epsilon}_n\to0+$. Denote by $\Pi_t$ the coordinate mapping, given by $\Pi_t(g):=g(t)$ for $g\in C[0,+\infty)$. From and we infer that $$\Pi_t+\frac{\hat\beta(k)}{4}\int_0^t\Pi_sds,\quad t\ge0$$ is a $P_k$-martingale whose law coincides with that of the process described by . The conclusion extends also to the case when $k=0$ and ${\omega}(0)>0$. If, on the other hand, ${\omega}(0)=0$ we have $\hat \beta(0)=0$ and ${\cal R}^{1/2}(s,0)=0$ and therefore $\Pi_t\equiv\Pi_0$ a.s.
Suppose now that $k_1,\ldots,k_n\in {{{\ensuremath{\mathbb T}}}}$ are pairwise distinct. Denote by $Q_{\epsilon}$ the law of $$\{(\tilde \psi^{({\epsilon})}(t,k_1),\ldots,\tilde \psi^{({\epsilon})}(t,k_n)),\,t\ge0\}$$ over $C([0,+\infty),\mathbb C^n)$. Then, we claim that $$\label{101010}
\lim_{{\epsilon}\to0+}\sup_{t\in[0,t_*]}\left|\langle \tilde{\cal M}^{({\epsilon})}(k_i),(\tilde {\cal M}^{({\epsilon})})^*(k_j)\rangle_t-\delta_{i,j}\int_0^t{\cal R}(s,k_i)ds\right|=0$$ and, obviously, $$\label{111010}
\lim_{{\epsilon}\to0+}\sup_{t\in[0,t_*]}\left| \langle \tilde{\cal M}^{({\epsilon})}(k_i),\tilde {\cal M}^{({\epsilon})}(k_j)\rangle_t\right|=0,\quad \forall \,i,j=1,\ldots,n.$$ To see note that for $i\not=j$ we have $$\begin{aligned}
&&
\langle \tilde{\cal M}^{({\epsilon})}(k_i),(\tilde {\cal M}^{({\epsilon})})^*(k_j)\rangle_t=\sum_{{\sigma}_1,{\sigma}_2=\pm1}{\sigma}_1{\sigma}_2\int_0^t\exp\left\{\frac{i({\omega}(k_i)-{\omega}(k_j))s}{{\epsilon}}\right\}ds
\\
&&
\times\int_{{{{\ensuremath{\mathbb T}}}}}r(k_i,k')r(k_j,k')\hat\psi^{({\epsilon})}_{{\sigma}_1}(s,k_i-k')(\hat\psi^{({\epsilon})}_{{\sigma}_2})^*(s,k_j-k') dk',\end{aligned}$$ Using part iii) of Corollary \[cor2\] combined with condition ${\omega})$ we conclude, thanks to the fact that $k_i\not=k_j$, that $$\lim_{{\epsilon}\to0+}
\sup_{t\in[0,t_*]}\left|\int_0^t\exp\left\{\frac{i({\omega}(k_i)-{\omega}(k_j))s}{{\epsilon}}\right\}
\hat\psi^{({\epsilon})}_{{\sigma}_1}(s,k_i-k')(\hat\psi^{({\epsilon})}_{{\sigma}_2})^*(s,k_j-k')ds\right|=0$$ for a.e. $k'\in{{{\ensuremath{\mathbb T}}}}$. Using in the same way as in the proof of we can substantiate exchanging the passage to the limit with the respective integration and conclude .
Combining and with we obtain from equation that any limiting point of the family of laws of $Q_{{\epsilon}_n}$ as ${\epsilon}_n\to0+$ is a measure $P_{k_1,\ldots,k_n}$ such that $${\cal M}_t=({\cal M}_t^{(1)},\ldots,{\cal M}_t^{(n)}):= \Pi_t+\frac{\hat\beta(k)}{4}\int_0^t\Pi_sds,\quad t\ge0$$ is $\mathbb C^n$-valued martingale, whose quadratic covariation is given by $$\langle {\cal M}^{(i)},({\cal M}^{(j)})^*\rangle_t=\delta_{i,j}\int_0^t{\cal R}(s,k_j)ds$$ and $$\langle {\cal M}^{(i)},({\cal M}^{(j)})\rangle_t=0,\quad \forall \,i,j=1,\ldots,n.$$ This of course implies that $P_{k_1,\ldots,k_n}=P_{k_1}\otimes\ldots\otimes P_{k_n}$.
Proof of part ii) of Theorem \[main-thm1\] {#sec:proof-part-ii .unnumbered}
------------------------------------------
Let $f\in L^2({{{\ensuremath{\mathbb T}}}})$. We shall prove that $$\begin{aligned}
\label{050410a1}
&&
\lim_{{\epsilon}\to0+}{{\mathbb E}}|\langle \tilde{\cal M}^{({\epsilon})}_t,f\rangle|^2=0.\end{aligned}$$ Assuming this result we show how to finish the proof of part (ii). Denote $$\delta \psi^{({\epsilon})}(t):= \tilde \psi^{({\epsilon})}(t)- \bar\psi(t).$$ Using Lemma \[lm023108\] and Theorem 3.1, p. 276 of [@jakubowski] we can conclude weak pre-compactness of $P_{{\epsilon}}$, ${\epsilon}\in(0,1]$ – the family of the laws of $\{\delta \psi^{({\epsilon})}(t),\,t\ge0\}$ – in $C([0,+\infty),L^2_w({{{\ensuremath{\mathbb T}}}}))$. With the help of Corollary \[030410\] and we conclude that the limiting measure, as ${\epsilon}\to0+$, is supported on the solution of the equation $$\langle g(t),f\rangle-\frac{1}{4}\int_0^t\langle \hat \beta g(s),f\rangle ds=0,\quad\forall\,f\in L^2({{{\ensuremath{\mathbb T}}}}).$$ This of course shows that it is the $\delta$-measure supported on $g(t)\equiv0$. Hence, in particular we get $$\label{060510}
\lim_{{\epsilon}\to0+}\sup_{t\in[0,t_*]}|\langle\delta \psi^{({\epsilon})}(t),f\rangle|=0$$ in probability and follows. Coming back to the proof of note that by the definition of the martingale $\tilde{\cal M}^{({\epsilon})}_t$, see , we only need to show that $$\begin{aligned}
\label{080510}
\lim_{{\epsilon}\to0+}{{\mathbb E}}\left|\int_0^t\int_{{{{\ensuremath{\mathbb T}}}}^2}\exp\left\{i s\frac{{\omega}(k)}{{\epsilon}}\right\}r(k,k')f^*(k)\right.
\left.\vphantom{\int_0^1}\hat\psi^{({\epsilon})}_{{\sigma}}(s,k-k')dW(s,k')dk\right|^2=0\end{aligned}$$ for ${\sigma}=\pm1$. We consider only the case ${\sigma}=1$, the other one can be dealt in a similar manner. The expression under the limit in equals $$\label{080510a}
\int\limits_0^t\int\limits_{{{{\ensuremath{\mathbb T}}}}^3}\exp\left[i s\frac{{\omega}(k)-{\omega}(k_1)}{{\epsilon}}\right]\!
r(k,k')r(k_1,k')f^*(k)f(k_1)
\vphantom{\int_0^1}{{\mathbb E}}\left[\hat\psi^{({\epsilon})}(s,k-k')(\hat\psi^{({\epsilon})})^*(s,k_1-k')\right]dsd{{\bf k}},$$ with $d{{\bf k}}=dkdk_1dk'$. Using Corollary \[cor2\] and an argument identical with the one used in the proof of Lemma \[lm013009\] we conclude that $$\begin{aligned}
\lim_{{\epsilon}\to0+}\int_0^t\exp\left\{i s\frac{{\omega}(k)-{\omega}(k_1)}{{\epsilon}}\right\}
{{\mathbb E}}\left[\hat\psi^{({\epsilon})}(s,k-k')(\hat\psi^{({\epsilon})})^*(s,k_1-k')\right]ds=0\end{aligned}$$ for all $k',k,k_1$ such that ${\omega}(k-k')+{\omega}(k_1)-{\omega}(k)\not={\omega}(k_1-k')$. Since the latter inequality holds on the set of null Lebesgue measure we conclude equality in , thanks to the Lebesgue dominated convergence theorem.
Spatially homogeneous initial data {#sec5}
==================================
Tightness of the family of laws $\{\tilde\psi^{({\epsilon})}(t),\,t\ge0\}$, in the space of continuous functionals taking values in a space of distributions is again due to the fact that the evolution equation contains no terms that are large in magnitude. This is done in Sections \[sec5.1\] and \[sec5.2\]. However, we have no estimates of the $H^{-m}({{{\ensuremath{\mathbb T}}}})$ norm of $\tilde\psi^{({\epsilon})}(t)$ analogous to the ones in Lemma \[lm023108\], that have played an important role in the limit identification argument of Section \[sec4\] for square integrable data. Therefore, instead of considering the quadratic variation of the martingale term as we did in the proof of Theorem \[main-thm1\], for the proof of Theorem \[main-thm2\] we identify the limit of all moments of $\tilde\psi^{({\epsilon})}(t)$. Accordingly, we first write equations for time evolution of an arbitrary moment of $\tilde\psi^{({\epsilon})}(t)$ in Section \[sec5.3\]. Using standard averaging argument we show (see Proposition \[012410\]) the convergence of moments, as ${\epsilon}\to0+$, to a solution of the limiting equation obtained simply by discarding the oscillatory terms from the moment equation. Finally in Section \[sec5.5\] we prove that the solutions of the limiting equation coincide with the respective moments of the non-homogeneous Ornstein-Uhlenbeck equation concluding in this way the proof of Theorem \[main-thm2\].
Properties of spatially homogeneous solutions of {#sec5.1}
-------------------------------------------------
The initial data $\hat \psi$ considered in this section is random and takes values in the Hilbert space of distributions $H^{-m}({{{\ensuremath{\mathbb T}}}})$ for some $m>1/2$. In fact, in Sections \[sec5.1\]-\[sec5.4\] we shall not make any use of the assumption that the data is Gaussian and we use only the fact that it is spatially homogeneous and $$\label{031210}
{{\mathbb E}}\|\hat\psi\|^2_{H^{-m}({{{\ensuremath{\mathbb T}}}})}<+\infty.$$ Gaussianity shall be used only in Section \[sec5.5\].
Consider the random field $\{\psi_y:=\langle \hat\psi,e_y\rangle,\,y\in{{{\ensuremath{\mathbb Z}}}}\}$. The field is assumed to be spatially homogeneous, i.e. $\{\psi_{y+z},\,y\in{{{\ensuremath{\mathbb Z}}}}\}$ and $\{\psi_{y},\,y\in{{{\ensuremath{\mathbb Z}}}}\}$ have identical laws for all $z\in{{{\ensuremath{\mathbb Z}}}}$, and centered, i.e. ${{\mathbb E}}\psi_0=0$. Spatial homogeneity is equivalent to the fact that $\hat \psi(k)$ and $e_z(k)\hat \psi(k)$ are identically distributed in $H^{-m}({{{\ensuremath{\mathbb T}}}})$ for any $z\in{{{\ensuremath{\mathbb Z}}}}$. Note that, since $m>1/2$, $$\sum_{y\in{{{\ensuremath{\mathbb Z}}}}}(1+y^2)^{-m}{{\mathbb E}}|\psi_y|^2={{\mathbb E}}\|\hat\psi\|^2_{H^{-m}({{{\ensuremath{\mathbb T}}}})}<+\infty,$$ due to .
Since the covariance function of the field $$S_{x-y}:={{\mathbb E}}[\psi_x\psi^*_y],\quad\,\forall\, x,y\in{{{\ensuremath{\mathbb Z}}}}$$ is positive definite, there exists a finite measure $\hat E(dk)$ such that $$S_x=\int_{{{{\ensuremath{\mathbb T}}}}}e^{i x k}\hat E(dk),\quad\forall\,x\in{{{\ensuremath{\mathbb Z}}}}.$$ We assume that the covariance function decays sufficiently fast in space so that $$\label{conv-y}
\sum_{x\in{{{\ensuremath{\mathbb Z}}}}}(|{{\mathbb E}}[\psi_x^*\psi_0]|+|{{\mathbb E}}[\psi_x\psi_0]|)<+\infty.$$ Assumption implies, in particular, that $\hat E(dk)={\cal E}_0(k)dk$ for some non-negative energy density ${\cal E}_0\in C({{{\ensuremath{\mathbb T}}}})$ and both this function and $
{\cal Y}=\sum_{x\in{{{\ensuremath{\mathbb Z}}}}}e_x{{\mathbb E}}[\psi_x\psi_0]
$ belong to $C({{{\ensuremath{\mathbb T}}}})$. When the field $\psi_x$ is a complex valued Gaussian, as described Section \[sec2.3.2\], we have ${\cal Y}\equiv0$. This and together imply .
We note that the translation invariance of the solution persists in time. Indeed, let $\psi^{({\epsilon})}_x(t):=\langle \hat\psi^{({\epsilon})}(t),e_x\rangle$ and $z\in{{{\ensuremath{\mathbb Z}}}}$. A direct computation shows that $e_z\hat\psi^{({\epsilon})}(t)$ is also a solution of . Since the laws of the initial conditions $e_z\hat\psi$ and that of $\hat\psi$ are identical, we conclude from the uniqueness in law of solutions that the same holds for the processes $\{e_z\hat\psi^{({\epsilon})}(t),\,t\ge0\}$ and $\{\hat\psi^{({\epsilon})}(t),\,t\ge0\}$. In consequence, the laws of $\{\psi^{({\epsilon})}_x(t),\,x\in{{{\ensuremath{\mathbb Z}}}}\}$ and that of $\{\psi^{({\epsilon})}_{x+z}(t),\,x\in{{{\ensuremath{\mathbb Z}}}}\}$ are identical for any $z\in{{{\ensuremath{\mathbb Z}}}}$. We can now define the correlation functions $$S^{({\epsilon})}_{t,x}={{\mathbb E}}\left[\psi^{({\epsilon})}_x(t)(\psi^{({\epsilon})}_0)^*(t)\right]\quad\mbox{and}\quad Y^{({\epsilon})}_{t,x}={{\mathbb E}}\left[\psi^{({\epsilon})}_x(t)\psi^{({\epsilon})}_0(t)\right]$$ and introduce two distributions on $H^{-m}({{{\ensuremath{\mathbb T}}}})$ $$\langle f, \hat S^{({\epsilon})}_{t}\rangle :=\sum_{x\in{{{\ensuremath{\mathbb Z}}}}}
\check f_x(S^{({\epsilon})}_{t,x})^*\quad\mbox{and}\quad
\langle f, \hat Y^{({\epsilon})}_{t}\rangle :=\sum_{x\in{{{\ensuremath{\mathbb Z}}}}} \check f_x(Y^{({\epsilon})}_{t,x})^*.$$ We recall the following result of [@BOS].
\[lm011310\] For any ${\epsilon}\in(0,1]$ and $t\ge0$ we have $\hat S^{({\epsilon})}_{t},\hat Y^{({\epsilon})}_{t}\in L^1({{{\ensuremath{\mathbb T}}}})$. Moreover,
- $\hat S^{({\epsilon})}_{t}$ is non-negative, and for any $t_*>0$ $$\label{011310}
\sup_{{\epsilon}\in(0,1]}\sup_{t\in[0,t_*]}(\|\hat S^{({\epsilon})}_{t}\|_{L^1({{{\ensuremath{\mathbb T}}}})}+\|\hat Y^{({\epsilon})}_{t}\|_{L^1({{{\ensuremath{\mathbb T}}}})})<+\infty,$$
- for any $f\in L^\infty({{{\ensuremath{\mathbb T}}}})$ we have $$\label{021310}
\lim_{{\epsilon}\to0+}\sup_{t\in[0,t_*]}\left|\langle\hat S^{({\epsilon})}_{t}-\bar{\cal E}(t),f\rangle\right|=0,$$ where $\bar{\cal E}(t)$ is given by with the initial condition replaced by ${\cal E}_0(k)$
- for any $f$ such that $f{\omega}^{-1}\in L^\infty({{{\ensuremath{\mathbb T}}}})$ we have $$\label{021310a}
\lim_{{\epsilon}\to0+}\sup_{t\in[0,t_*]}\left|\int_0^t\langle\hat Y^{({\epsilon})}_{s},f\rangle ds\right|=0.$$
[**Proof.**]{} Parts 1) and 2) of the lemma are contained in Lemma 12 and Theorem 10 of [@BOS], respectively. Part 3) follows easily from part 1) and the arguments used in the proof of Corollary \[cor2\]. [$\Box$]{}
Tightness of solutions of {#sec5.2}
--------------------------
Given $f\in H^m({{{\ensuremath{\mathbb T}}}})$, we denote by $Q_{\epsilon}$ and $Q_{{\epsilon},f}$ the laws of the processes $\{\hat\psi^{({\epsilon})}(t),\,t\ge0\}$ and $\{\langle f, \hat\psi^{({\epsilon})}(t)\rangle,\,t\ge0\}$ over $C([0,+\infty),H^{-m}_w({{{\ensuremath{\mathbb T}}}}))$ and $C([0,+\infty),\mathbb C)$, respectively, and by $\{\tilde Q_{{\epsilon}},\,{\epsilon}\in(0,1]\}$ the family of laws of $\{\tilde\psi^{({\epsilon})}(t),\,t\ge0\}$ over $C([0,+\infty),H^{-m}_w({{{\ensuremath{\mathbb T}}}}))$. According to [@mitoma], see Remark R1, p. 997, to verify the tightness of $\tilde Q_{{\epsilon}}$, it suffices to show the following two conditions:
- for any $\sigma,M,t_*>0$ there exists a $\delta>0$ such that $${{{{\mathbb P}}}}\left[\sup_{t\in[0,t_*]}|\langle\tilde\psi^{({\epsilon})}(t),f\rangle|\ge M\right]<{\sigma},\quad\forall\,\|f\|_{H^m({{{\ensuremath{\mathbb T}}}})}<\delta,\quad {\epsilon}\in(0,1],$$ and
- for any $f\in H^{m}({{{\ensuremath{\mathbb T}}}})$ the family of the laws of the processes $
\{\langle\tilde\psi^{({\epsilon})}(t),f\rangle,\,t\in[0,t_*]\}$, ${\epsilon}\in(0,1]
$ is tight over $C[0,t_*]$ for any $t_*>0$.
As in we conclude that for any $f_1,f_2\in H^m({{{\ensuremath{\mathbb T}}}})$, where $m>1/2$, the covariance $$\label{031310}
{{\mathbb E}}\left[\langle f_1,\hat\psi^{({\epsilon})}_t\rangle\langle f_2,\hat\psi^{({\epsilon})}_t\rangle^* \right]=\int_{{{{\ensuremath{\mathbb T}}}}}\hat S_t^{({\epsilon})}(k)f_1(k)f_2^*(k)dk.$$ From and Doob’s inequality there exists a constant $C>0$ such that $$\label{041310}
{{\mathbb E}}\left[\sup_{t\in[0,t_*]}|\langle\tilde\psi^{({\epsilon})}(t),f\rangle|^2\right]\le C\left\{{{\mathbb E}}|\langle\hat\psi,f\rangle|^2\right.
\left.\!+\!\int_0^{t_*}
{{\mathbb E}}\left|\left\langle {\cal A}\left[\frac{t}{{\epsilon}},\tilde\psi^{({\epsilon})}(t)\right],f\right\rangle\right|^2dt
+{{\mathbb E}}\left|\left\langle\tilde{\cal M}^{({\epsilon})}_{t_*},f\right\rangle\right|^2\right\}.$$ Using , and the definitions of $ {\cal A}[t/{\epsilon},\cdot]$, and the martingale $\tilde{\cal M}^{({\epsilon})}_t$ (see and ) we conclude that the right hand side of can be estimated from above by $C\|f\|_\infty^2$, which can be made less than ${\sigma}>0$, provided we choose $\delta>0$ sufficiently small. To show condition (FDT) consider $\tilde Q^{(M)}_{{\epsilon},f}$ – the law of the stopped process $$\{(\langle\tilde\psi^{({\epsilon})}(t\wedge \tau_M^{({\epsilon})}),f\rangle,\langle\tilde\psi^{({\epsilon})}(t\wedge \tau_M^{({\epsilon})}), f_0\rangle)\,t\in[0,t_*]\}$$ over $C([0,t_*];\mathbb C^2)$. Here $ f_0(k):=f(-k)$ and $$\tau_M^{({\epsilon})}:=\inf[t\in[0,t_*]:|\langle\tilde\psi^{({\epsilon})}(t),f\rangle|^2+|\langle\tilde\psi^{({\epsilon})}(t), f_0\rangle|^2\ge M^2].$$ We adopt the convention that $\tau_M:=t_*$ if the set is empty. Thanks to (UC) we conclude that $\lim_{M\to+\infty}\tau_M^{({\epsilon})}=t_*,
$ a.s. for each ${\epsilon}\in(0,1]$. Denote also by $\tilde Q_{{\epsilon},f}$ the law of the process without the stopping condition.
From we conclude that for a fixed $M$ and an arbitrary non-negative function $\phi:\mathbb C^2\to{{{\ensuremath{\mathbb R}}}}$, of class $C^1_c({{{\ensuremath{\mathbb R}}}}^4),$ one can choose a constant $K_\phi$, independent of spatial translations of $\phi$, such that $$\phi(\langle\tilde\psi^{({\epsilon})}(t\wedge \tau_M^{({\epsilon})}),f\rangle,\langle\tilde\psi^{({\epsilon})}(t\wedge \tau_M^{({\epsilon})}), f_0\rangle)+K_\phi t,\,t\in[0,t_*]$$ is a non-negative submartingale. This proves tightness of $\{\tilde Q^{(M)}_{{\epsilon},f},\,{\epsilon}\in(0,1]\}$ for a fixed $M$, by virtue of Theorem 1.4.3 of [@stroock-varadhan]. Since for any ${\sigma}>0$ one can find a sufficiently large $M>0$ such that $B_M$ – the ball centered at $0$ and of radius $M$ in $ C([0,t_*];\mathbb C^2)$ – satisfies $$\tilde Q^{(M)}_{{\epsilon},f}(B^c_M)+ \tilde Q_{{\epsilon},f}(B^c_M)<{\sigma}$$ and $$\tilde Q^{(M)}_{{\epsilon},f}(B_M\cap A)=\tilde Q_{{\epsilon},f}(B_M\cap A)$$ for all Borel measurable subsets $A$ of $ C([0,t_*];\mathbb C^2)$, we conclude tightness of $\{\tilde Q_{{\epsilon},f},\,{\epsilon}\in(0,1]\}$, see step (vi) of the proof of Theorem 3 of [@kesten-papanicolaou] for details of this argument.
Evolution of moments {#sec5.3}
--------------------
To describe the evolution of moments we rewrite equation in a more compact form, as a $2 \times2$ linear system of equations with multiplicative noise. Denote by ${\bf C}(t,k)=[C_{ij}(t,{{\bf k}})]$, $i,j=\pm1$, the $2\times 2$ hermitian matrix $${\bf C}(t,k):=\left[
\begin{array}{ll}
C_{1,1}&C_{1,-1}\\
C_{-1,1}&C_{-1,-1}
\end{array}\right],$$ with the entries $$C_{p,q}(t,k):=\frac{pq\hat\beta(k)}{4}\exp\left\{ip{\omega}(k)(1-pq)t\right\}.$$ Let also ${\bf Q}(t,k,k')=[ Q_{pq}(t,k,k')]$, $p,q=\pm1$, be the $2\times 2$ matrix $$Q_{p,q}(t,k,k'):=ipqr(k,k-k')e^{ip[{\omega}(k)-pq{\omega}(k')]t}$$ and $W(t,k):=\sum_ye_y(k)w_y(t)$. Let us recall that $\tilde\psi_{-1}^{({\epsilon})}(t,k)=\tilde\psi^{({\epsilon})*}(t,-k)$. Then, equation for $$\Psi^{({\epsilon})}(t,k)=\left[
\begin{array}{c}
\tilde\psi^{({\epsilon})}(t,k)\\
\\
\tilde\psi^{({\epsilon})}_{-1}(t,k)
\end{array}\right]$$ is $$\begin{aligned}
\label{031710}
&&\!\!\!\!\!\!d\Psi^{({\epsilon})}(t,k)=-{\bf C}\left(\frac{t}{{\epsilon}},k\right)\Psi^{({\epsilon})}(t,k)dt
+\int_{{{{\ensuremath{\mathbb T}}}}}{\bf Q}\left(\frac{t}{{\epsilon}},k,k-k'\right)\Psi^{({\epsilon})}(t,k-k') W(dt,dk'),\nonumber\\
&&\!\!\!\!\!\!\Psi^{({\epsilon})}(0,k)=\Psi(k),\end{aligned}$$ with the initial data $$\Psi(k)=\left[
\begin{array}{c}
\hat\psi(k)\\
\\
\hat\psi_{-1}(k)
\end{array}\right].$$ Let $\{{\bf S}_{\epsilon}(s,t,k),\,s,t\in{{{\ensuremath{\mathbb R}}}}\}$ be the $2\times 2$ Hermitian matrices solving the deterministic system $$\begin{aligned}
&&
\frac{d{\bf S}_{\epsilon}(s,t,k)}{dt}=-{\bf C}\left(\frac{t}{{\epsilon}},k\right){\bf S}_{\epsilon}(s,t,k)\\
&&
{\bf S}_{\epsilon}(s,s,k)=I_2.\end{aligned}$$ Here $I_2$ is the $2\times 2$ identity matrix. Existence and uniqueness of solutions to in the strong sense (thus implying the result in the mild, or weak sense as well) follows from an argument used in Chapter 6 of [@DZ] (because the generators for the evolution family ${\bf S}_{\epsilon}(s,t)$ are bounded), see Proposition 6.4 there. Although the case considered here differs slightly because the coefficients are time dependent, this does not influence the results.
Given a nonnegative integer $p\ge1$, define a tensor valued distribution on $H^{-m/p}({{{\ensuremath{\mathbb T}}}}^{p})$ $$\hat M^{({\epsilon})}(t):=
\left[ \hat M_{{{\bf i}}}^{({\epsilon})}(t)\right],\,\quad {{\bf i}}=(i_1,\ldots,i_p)\in\{-1,1\}^p,$$ by $$\hat M_{{{\bf i}}}^{({\epsilon})}(t) ={{\mathbb E}}\left[\tilde \psi^{({\epsilon})}_{i_1}(t)\otimes \ldots\otimes \tilde \psi^{({\epsilon})}_{i_p}(t)\right].$$ Note that also $$\label{021910}
\hat M_{{{\bf i}}}^{({\epsilon})}(0) =\hat M_{{{\bf i}}}:={{\mathbb E}}\left[\hat \psi_{i_1}\otimes \ldots\otimes \hat \psi_{i_p}\right]$$
For a given multi-index ${{\bf i}}$ we define the multi-indices $
{{\bf i}}_{\ell}(j)=(i_1',\ldots,i_p')$, $
{{\bf i}}_{\ell,m}(j_1,j_2)=(i_1'',\ldots,i_p'')$ given by: $i_q'=i_q$ for $q\not=\ell$ and $i_\ell'=j$, and $i_q''=i_q$ for $q\not=\ell,m$ and $i_\ell''=j_1$, $i_m''=j_2$. Denote by ${\cal M}({{{\ensuremath{\mathbb T}}}}^p)$ the space of all complex valued Borel measures $\nu$ on ${{{\ensuremath{\mathbb T}}}}^p$ whose total variation norm $\|\nu\|_{\rm TV}$ is finite.
\[prop011810\] The following are true:
- $\hat M^{({\epsilon})}(t)$ is the unique solution in $H^{-m/p}({{{\ensuremath{\mathbb T}}}}^{p})$ of the system of equations $$\begin{aligned}
\label{011910}
&&\frac{d}{dt}\hat M_{{{\bf i}}}^{({\epsilon})}(t,{{\bf k}})=-\sum_{\ell=1}^p\sum_{j=\pm1}C_{i_\ell,j}\left(\frac{t}{{\epsilon}},k_\ell\right)\hat M_{{{\bf i}}_\ell(j)}^{({\epsilon})}(t,{{\bf k}})\\
&&
+\sum_{1\le \ell<m\le p}\sum_{j_1,j_2=\pm1}\int_{{{{\ensuremath{\mathbb T}}}}}{\cal R}_{i_\ell,i_m}^{j_1,j_2}\left(\frac{t}{{\epsilon}},k_\ell,k_m,k'\right) \hat M_{{{\bf i}}_{\ell,m}(j_1,j_2)}^{({\epsilon})}(t,{{\bf k}}_{\ell,m}')dk',\nonumber\end{aligned}$$ with ${{\bf i}}\in\{-1,1\}^p$ and the initial data given by . Here $${\cal R}_{i_\ell,i_m}^{j_1,j_2}\left(\frac{t}{{\epsilon}},k_\ell,k_m,k'\right):=Q_{i_\ell,j_1}\left(\frac{t}{{\epsilon}},k_\ell,k'_\ell\right)Q_{i_m,j_2}\left(\frac{t}{{\epsilon}},k_m,k'_m\right)$$ and ${{\bf k}}_{\ell,m}'=(k_1',\ldots,k_p')$, where $k_p':=k_p$ for $p\not=\ell,m$ and $k_\ell':=k_\ell-k'$, $k_m':=k_m+k'$.
- If the initial condition is from ${\cal M}({{{\ensuremath{\mathbb T}}}}^p)$ then the solution also belongs to ${\cal M}({{{\ensuremath{\mathbb T}}}}^p)$ and for any $t_*>0$ $$\label{062910}
M_*(T):=\sum_{{{\bf i}}\in\{-1,1\}^p}\sup_{{\epsilon}\in(0,1]}\sup_{t\in[0,t_*]}\|\hat M_{{{\bf i}}}^{({\epsilon})}(t)\|_{\rm TV}<+\infty.$$
[**Proof.**]{} The fact that $\hat M^{({\epsilon})}(t)$ is a solution of follows by an application of Itô formula and equation . Since the operators appearing on the right hand side of the equation in question are uniformly Lipschitz, on any compact time interval, both in $H^{-m/p}({{{\ensuremath{\mathbb T}}}}^p)$ and ${\cal M}({{{\ensuremath{\mathbb T}}}}^p)$ the proof of uniqueness of solutions in these spaces is standard. Estimate follows by an application of Gronwall’s inequality. [$\Box$]{}
Asymptotics of even moments {#sec5.4}
---------------------------
Let us now describe the limit moment equations. Assume that $p=2n$ is even, then for any $1\le \ell<m\le 2n$ let $D_{\ell,m}:=[{{\bf k}}\in{{{\ensuremath{\mathbb T}}}}^{2n}:k_\ell=-k_m]$. We define a bounded linear operator ${\cal R}_{\ell,m}:{\cal M}({{{\ensuremath{\mathbb T}}}}^{2n})\to {\cal M}({{{\ensuremath{\mathbb T}}}}^{2n})$ by $$\begin{aligned}
&&
\int_{{{{\ensuremath{\mathbb T}}}}^{2n}} fd {\cal R}_{\ell,m}\nu:=\int_{{{{\ensuremath{\mathbb T}}}}}dk\left\{\int_{D_{\ell,m}}r^2(k,k-k'_\ell)f(S({{\bf k}}',k))\nu(d{{\bf k}}')\right\}\end{aligned}$$ for any bounded, measurable $f:{{{\ensuremath{\mathbb T}}}}^{2n}\to\mathbb C$ and $\nu\in {\cal M}({{{\ensuremath{\mathbb T}}}}^{2n})$. We define $S:{{{\ensuremath{\mathbb T}}}}^{2n+1}\to{{{\ensuremath{\mathbb T}}}}^{2n}$ as follows: given ${{\bf k}}'=(k_1',\ldots,k_{2n}')\in {{{\ensuremath{\mathbb T}}}}^{2n}$ and $k\in{{{\ensuremath{\mathbb T}}}}$ we let $(k_1,\ldots,k_{2n})=S({{\bf k}}',k)$ if $k_j=k'_j$ for $j\not\in\{\ell,m\}$ and $k_\ell=k$, $k_m=-k$.
Suppose that the components of the tensor $\hat M=[\hat M_{{{\bf i}}}]$ belong to ${\cal M}({{{\ensuremath{\mathbb T}}}}^{2n})$. Similarly to part 1) of Proposition \[prop011810\] we conclude that the initial value problem $$\begin{aligned}
&&\frac{d}{dt}\hat M_{{{\bf i}}}(t)=-\frac{1}{4}\left(\sum_{\ell=1}^{2n}\hat\beta\left(k_\ell\right)\right)\hat M_{{{\bf i}}}(t)+\sum_{1\le \ell<m\le 2n}\sum_{j=\pm1}{\cal R}_{\ell,m} \hat M_{{{\bf i}}_{\ell,m}(j,-j)}(t),\nonumber
\label{031910}\\
&&
\hat M(0)=\hat M.\end{aligned}$$ possesses a unique solution in $C([0,+\infty),{\cal M}({{{\ensuremath{\mathbb T}}}}^{2n}))$.
Any partition of the set $\{1,\ldots,2n\}$ into a disjoint set of pairs is called a pairing. Define $$\mu(d{{\bf k}})=\sum_{\cal F}\prod_{(\ell,m)\in{\cal F}}\delta(k_\ell+k_m)d{{\bf k}},$$ where $d{{\bf k}}=dk_1\ldots dk_{2n}$ and the summation extends over all possible pairings of $\{1,\ldots,2n\}$. The measure is supported in ${{{\ensuremath{\mathbb H}}}}:=\bigcup_{\cal F}{{{\ensuremath{\mathbb H}}}}({\cal F})$ where $${{{\ensuremath{\mathbb H}}}}({\cal F}):=[{{\bf k}}:k_\ell+k_m=0,\,\forall\,(\ell,m)\in{\cal F}].$$ Suppose that the components of the tensor $\rho({{\bf k}})=[\rho_{{{\bf i}}}({{\bf k}})]$, ${{\bf i}}\in\{-1,1\}^{2n}$ belong to $L^1(\mu)$. Consider the initial value problem $$\begin{aligned}
&&\frac{d}{dt}\rho_{{{\bf i}}}(t,{{\bf k}})=-\frac{1}{4}\left(\sum_{\ell=1}^{2n}\hat\beta\left(k_\ell\right)\right)\rho_{{{\bf i}}}(t,{{\bf k}})\nonumber\\
&&
+\sum_{1\le \ell<m\le 2n}\sum_{j=\pm1}\int_{{{{\ensuremath{\mathbb T}}}}} r^2(k_\ell, k_\ell-k')1_{D_{\ell,m}}({{\bf k}})\rho_{{{\bf i}}_{\ell,m}(j,-j)}(t,{{\bf k}}'_{\ell,m})dk',\nonumber\\
&&
\rho_{{{\bf i}}}(0,{{\bf k}})=\rho_{{{\bf i}}}({{\bf k}}),\, {{\bf i}}\in\{-1,1\}^{2n},
\label{031910a1}\end{aligned}$$ with ${{\bf k}}'_{\ell,m}:=(k_1,\ldots,k_{\ell-1},k',\ldots,k_{m-1},-k',\ldots,k_{2n})$. It is straightforward to conclude that the above system possesses a unique continuous solution $\rho(t,{{\bf k}})=[\rho_{{{\bf i}}}(t,{{\bf k}})]$ whose components belong to $L^1(\mu)$. The next proposition gives the convergence of even moments to the solution of (\[031910\]).
\[012410\] Suppose that all the components of the tensor $[\hat M_{{{\bf i}}}(d{{\bf k}})]$ are absolutely continuous with respect to $\mu$, i.e. $
\hat M_{{{\bf i}}}(d{{\bf k}})=\rho_{{{\bf i}}}({{\bf k}})\mu(d{{\bf k}}),
$ and the dispersion relation satisfies hypothesis ${\omega})$. Then, the following are true:
- $\hat M_{{{\bf i}}}(t,d{{\bf k}})$ is absolutely continuous with respect to $\mu(d{{\bf k}})$ and $$\label{012910}
\hat M_{{{\bf i}}}(t,d{{\bf k}})=\rho_{{{\bf i}}}(t,{{\bf k}})\mu(d{{\bf k}}),\quad\forall\,{{\bf i}}\in\{-1,1\}^{2n}$$ where $\{\rho_{{{\bf i}}}(t),\,t\ge0\}$ satisfy .
- For any $T>0$ there exists a constant $C>0$ such that $$\label{063110}
\lim_{{\epsilon}\to0+}\sum_{{{\bf i}}\in\{-1,1\}^{2n}}\sup_{t\in[0,t_*]}\|\hat M_{{{\bf i}}}^{({\epsilon})}(t)-\hat M_{{{\bf i}}}(t)\|_{\rm TV}=0.$$
[**Proof.**]{} The conclusion of part 1) follows from uniqueness of solutions of and , and the fact that the right hand side of defines a solution of . From and we conclude that $$\begin{aligned}
\label{022910}
&&\|\hat M_{{{\bf i}}}^{({\epsilon})}(t)-\hat M_{{{\bf i}}}(t)\|_{\rm TV}
\le \sum_{\ell=1}^{2n}\sum_{j=\pm1}\int_0^t\left\|C_{i_\ell,j}\left(\frac{s}{{\epsilon}}\right)[\hat M_{{{\bf i}}_{\ell}(j)}^{({\epsilon})}(s)-\hat M_{{{\bf i}}_{\ell}(j)}(s)]\right\|_{\rm TV}ds\nonumber\\
&&
+\sum_{1\le \ell<m\le 2n}\sum_{j_1,j_2=\pm1}\int_0^t\left\|{\cal R}_{i_\ell,i_m}^{j_1,j_2}\left(\frac{s}{{\epsilon}}\right)[\hat M_{{{\bf i}}_{\ell,m}(j_1,j_2)}^{({\epsilon})}(s)-\hat M_{{{\bf i}}_{\ell,m}(j_1,j_2)}(s)]\right\|_{\rm TV}ds\nonumber\\
&&
+\sum_{\ell=1}^{2n}\sum_{j=\pm1}\left|\int_0^t\int_{{{{\ensuremath{\mathbb T}}}}^{2n}}E_{i_\ell,j}\left(\frac{s}{{\epsilon}},k_{\ell}\right)\rho_{{{\bf i}}_{\ell}(j)}(s,{{\bf k}})ds\mu(d{{\bf k}}) \right|\nonumber\\
&&
+\sum_{1\le \ell<m\le 2n}\sum_{j_1,j_2=\pm1}\left|\int_0^t\int_{{{{\ensuremath{\mathbb T}}}}^{2n+1}}\tilde{\cal R}_{i_\ell,i_m}^{j_1,j_2}\left(\frac{s}{{\epsilon}},{{\bf k}},k'\right)\rho_{{{\bf i}}_{\ell,m}(j_1,j_2)}(s,{{\bf k}})ds\mu(d{{\bf k}}) dk' \vphantom{\int_0^t}\right|.\nonumber\end{aligned}$$ The matrix ${\bf E}(t,k)=[E_{p,q}(t,k)]$, $p,q=\pm1$ is given by $$\label{042910}
{\bf E}(t,k):={\bf C}(t,k)-(\hat\beta(k)/4){\bf I}_2,$$ where ${\bf I}_2$ is the $2\times 2$ identity matrix. In addition, $$\tilde{\cal R}_{i_\ell,i_m}^{j_1,j_2}\left(\frac{s}{{\epsilon}},{{\bf k}},k'\right):=
{\cal R}_{i_\ell,i_m}^{j_1,j_2}\left(\frac{s}{{\epsilon}},k_\ell,k_m,k'\right)-\delta_{i_\ell}^{-i_m}
\delta_{j_1}^{-j_2}r^2(k_\ell,k_{\ell}-k') 1_{D_{\ell,m}}({{\bf k}}).$$ Denote the terms appearing on the right hand side of by $I(t)$, $I\!I(t)$, $I\!I\!I(t)$ and $I\!V(t)$ respectively. It is easy to see that $$\label{0312910}
I(t)+I\!I(t)\le C\int_0^t\sup_{{{\bf i}}\in\{-1,1\}^{2n}}\left\|\hat M_{{{\bf i}}}^{({\epsilon})}(s)-\hat M_{{{\bf i}}}(s)\right\|_{\rm TV}ds$$ for some constant $C>0$. To estimate the term $I\!I\!I$ we need to bound terms of the form $$\begin{aligned}
&&
\left|\int_0^t\int_{{{{\ensuremath{\mathbb T}}}}^{2n}}\hat\beta(k_\ell)\exp\left\{2i{\omega}(k_\ell)\frac{s}{{\epsilon}}\right\}\rho_{{{\bf i}}}(s,{{\bf k}})ds\mu(d{{\bf k}})\right|\end{aligned}$$ for some $\ell$ and ${{\bf i}}$. Integrating by parts we obtain that the expression above can be bounded from above by $$\begin{aligned}
&&
{\epsilon}\left|\int_{{{{\ensuremath{\mathbb T}}}}^{2n}}\frac{\hat\beta(k_\ell)}{2i{\omega}(k_\ell)}\left[\exp\left\{2i{\omega}(k_\ell)\frac{t}{{\epsilon}}\right\}-1\right]\rho_{{{\bf i}}}(t,{{\bf k}})1_{D_{\ell,m}}({{\bf k}})\mu(d{{\bf k}})\right|
\\
&&
+{\epsilon}\left|\int_0^t\int_{{{{\ensuremath{\mathbb T}}}}^{2n}}\frac{\hat\beta(k_\ell)}{2i{\omega}(k_\ell)}\left[\exp\left\{2i{\omega}(k_\ell)\frac{t}{{\epsilon}}\right\}-1\right]\frac{d}{ds}\rho_{{{\bf i}}}(s,{{\bf k}})1_{D_{\ell,m}}({{\bf k}})ds\mu(d{{\bf k}})\right|.\end{aligned}$$ The first term can be easily estimated by $C{\epsilon}$, due to the fact that $\sup_{k\in{{{\ensuremath{\mathbb T}}}}}\hat\beta(k){\omega}^{-1}(k)<+\infty$. To estimate the second term, we use equation . As a result,, we conclude that for any $t_*>0$ we can find a constant $C(t_*)>0$ such that $$\label{0512910}
\sup_{t\in[0,t_*]}I\!I\!I(t)\le C(t_*){\epsilon}.$$ Finally we show that $$\label{102910}
\lim_{{\epsilon}\to0+}\sup_{t\in[0,t_*]}I\!V(t)=0.$$ It implies the conclusion of part 2) of the proposition, via an application of the Gronwall’s inequality.
We write $I\!V(t)=I\!V_1(t)+I\!V_2(t)$, where the terms $I\!V_i(t)$, $i=1,2$ correspond to the integration over $D_{\ell,m}$ and its complement. In the latter case, we have to deal with terms of the form $$\begin{aligned}
&&
\left|\int_0^t\int_{{{{\ensuremath{\mathbb T}}}}^{2n+1}}1_{[k_\ell\not=-k_m]}r(k_\ell,k')r(k_m,-k')\rho_{{{\bf i}}}(s,{{\bf k}})\right.\\
&&
\left.\times\prod_{j=1}^2\exp\left\{i{\sigma}_1^{(j)}[{\omega}(k_\ell^{(j)})+{\sigma}_2^{(j)}{\omega}(k_\ell^{(j)}+(-1)^jk')]\frac{s}{{\epsilon}}\right\}ds\mu(d{{\bf k}}) dk' \vphantom{\int_0^t}\right|\end{aligned}$$ for some ${{\bf i}}\in\{-1,1\}^{2n}$, ${\sigma}_p^{(j)}\in\{-1,1\}$. Here $k_\ell^{(1)}=k_\ell$ and $k_\ell^{(2)}=k_m$. Using integration by parts over the $s$ variable we can estimate the supremum of the above expression over $t\in[0,t_*]$ by the sum of $$\begin{aligned}
\label{013110}
&&
I_{\epsilon}:=\int_{{{{\ensuremath{\mathbb T}}}}^{2n+1}}\mu(d{{\bf k}}) dk'1_{[k_\ell\not=-k_m]}|r(k_\ell,k')r(k_m,-k')|\sup_{t\in[0,t_*]}|\rho_{{{\bf i}}}(t,{{\bf k}})|\nonumber\\
&&
\times{\epsilon}\left|\sum_{j=1}^2{\sigma}_1^{(j)}[{\omega}(k_\ell^{(j)})+{\sigma}_2^{(j)}{\omega}(k_\ell^{(j)}+(-1)^jk')]\right|^{-1}
\\
&&
\times\sup_{t\in[0,t_*]}\prod_{j=1}^2\left|\exp\left\{i{\sigma}_1^{(j)}[{\omega}(k_\ell^{(j)})+\
si_2^{(j)}{\omega}(k_\ell^{(j)}+(-1)^jk')]\frac{t}{{\epsilon}}\right\}-1\right| ,\nonumber\end{aligned}$$ and $$\begin{aligned}
\label{023110}
&&
J_{\epsilon}:=\int_0^Tds\left|\int_{{{{\ensuremath{\mathbb T}}}}^{2n+1}}\mu(d{{\bf k}}) dk'1_{[k_\ell\not=-k_m]}r(k_\ell,k')r(k_m,-k')\frac{d}{ds}\rho_{{{\bf i}}}(s,{{\bf k}})\right.\nonumber\\
&&
\times{\epsilon}\left\{\sum_{j=1}^2{\sigma}_1^{(j)}[{\omega}(k_\ell^{(j)})+{\sigma}_2^{(j)}{\omega}(k_\ell^{(j)}+(-1)^jk')]\right\}^{-1}
\\
&&
\left.\times\prod_{j=1}^2\left\{\exp\left\{i{\sigma}_1^{(j)}[{\omega}(k_\ell^{(j)})
+{\sigma}_2^{(j)}{\omega}(k_\ell^{(j)}+(-1)^jk')]\frac{s}{{\epsilon}}\right\}-1\right\}\right|. \nonumber\end{aligned}$$ Using and Gronwall’s inequality, we conclude that $$\int_{{{{\ensuremath{\mathbb T}}}}^{2n}}\sup_{t\in[0,t_*]}|\rho_{{{\bf i}}}(t,{{\bf k}})|d{{\bf k}}<+\infty.$$ Using condition ${\omega})$ we conclude therefore, by virtue of Lebesgue dominated convergence theorem, that $\lim_{{\epsilon}\to0+}I_{\epsilon}=0$. Likewise, after substituting for $\rho_{{{\bf i}}}'(s,{{\bf k}})$ from , we conclude that $\lim_{{\epsilon}\to0+}J_{\epsilon}=0$. Part 2) of the proposition follows then from another application of Gronwall’s inequality. Summarizing, we have shown so far that $$\lim_{{\epsilon}\to0+}\sup_{t\in[0,t_*]}I\!V_2(t)=0.$$
We are left therefore with estimates of the term $$\begin{aligned}
\label{033110}
&&
I\!V_1(t):=\sum_{1\le \ell<m\le 2n}\sum_{j_1,j_2=\pm1}\left|\int_0^t\int_{{{{\ensuremath{\mathbb T}}}}^{2n+1}}1_{D_{\ell,m}}({{\bf k}})\right.\\
&&
\left.\times \tilde{\cal R}_{i_\ell,i_m}^{j_1,j_2}\left(\frac{s}{{\epsilon}},{{\bf k}},k'\right)\rho_{{{\bf i}}_{\ell,m}(j_1,j_2)}(s,{{\bf k}})ds\mu(d{{\bf k}}) dk' \vphantom{\int_0^t}\vphantom{\int_0^1}\right|.\nonumber\end{aligned}$$ The non-vanishing terms appearing in the above sum are of the form $$\left|\int_0^t\int_{{{{\ensuremath{\mathbb T}}}}^{2n+1}}r^2(k_\ell,k_\ell-k')1_{D_{\ell,m}}({{\bf k}})
\prod_{j=1}^2\exp\left\{i{\sigma}_1^{(j)}[{\omega}(k_\ell)+{\sigma}_2^{(j)}{\omega}(k_\ell-k')]\frac{s}{{\epsilon}}\right\}ds\mu(d{{\bf k}}) dk' \vphantom{\int_0^t}\right|,$$ with $({\sigma}_1^{(1)},{\sigma}_2^{(1)})\not=-({\sigma}_1^{(2)},{\sigma}_2^{(2)})$ and ${\sigma}_p^{(j)}\in\{-1,1\}$. To these terms we can apply the integration by parts argument as before, to conclude that $$\lim_{{\epsilon}\to0+}\sup_{t\in[0,t_*]}I\!V_1(t)=0.$$ Summarizing, we have shown that holds, and the proof of part 2 of the proposition is therefore complete. [$\Box$]{}
Proof of Theorem \[main-thm2\] {#sec5.5}
------------------------------
In this section, and in this section only, we make use of the assumption that $\hat\psi$ is Gaussian. We show that the limiting measure for $\tilde Q_{\epsilon}$, as ${\epsilon}\to0+$, coincides with the law $\tilde Q$ of the process given by proving that for any $N\ge1$, $0\le t_1<\ldots <t_N$, any non-negative integers $\ell_j,m_j$, test functions $f_j,g_j\in H^m({{{\ensuremath{\mathbb T}}}})$, $j=1,\ldots,N$ we have $$\label{011710}
\lim_{{\epsilon}\to0+}{{\mathbb E}}\left[\prod_{j=1}^N[\langle\tilde\psi^{({\epsilon})}(t_j),f_{j}\rangle^{\ell_j}(\langle\tilde\psi^{({\epsilon})}(t_j),g_{j}\rangle^*)^{m_j}]\right]
=
{{\mathbb E}}\left[\prod_{j=1}^N[\langle\bar\psi(t_j),f_{j}\rangle^{\ell_j}(\langle\bar\psi(t_j),g_{j}\rangle^*)^{m_j}]\right].$$ To simplify the notation, we prove only in the case $N=1$. The general case can be handled in the same manner, using Markov property of the process $\{\tilde\psi^{({\epsilon})}(t),\,t\ge0\}$, at the expense of some additional complications in the notation. We recall (see Section \[sec2.3.2\]) that the initial data $\{ \hat\psi(k),\,k\in{{{\ensuremath{\mathbb T}}}}\}$ is a $\delta$-correlated Gaussian random field given by . Therefore, for the odd moments we have $$\hat M^{({\epsilon})}_{{{\bf i}}}(0)=
0,\,\quad \forall\,{{\bf i}}\in\{-1,1\}^{2n-1},$$ where $n\ge 1$ is an integer. By uniqueness of solutions of we conclude that in this case $
\hat M^{({\epsilon})}(t)\equiv0$ for all $t\ge0$. When ${{\bf i}}\in \{-1,1\}^{2n}$ we can use the conclusion of Proposition \[012410\]. Define $$\bar M^{(2n)}(t):=
\left[ \bar M_{{{\bf i}}}^{(2n)}(t)\right],\,\quad {{\bf i}}=(i_1,\ldots,i_{2n})\in\{-1,1\}^{2n},$$ where $$\bar M_{{{\bf i}}}^{(2n)}(t) ={{\mathbb E}}\left[\bar \psi_{i_1}(t)\otimes \ldots\otimes \bar \psi_{i_{2n}}(t)\right]$$ and $\bar \psi_{1}(t)=\bar \psi(t)$ is the solution of and $\bar \psi_{-1}(t,k)=\bar \psi^*(t,-k)$. The conclusion of Theorem \[main-thm2\] will follow provided that we show that $\bar M^{(2n)}(t)$, satisfies . Note that for $n=1$ we obtain that $$\bar M_{i_1,i_2}^{(2)}(t,d{{\bf k}}) =\delta_{i_1,-i_2}\bar{\cal E}(t,k_1)\delta(k_1+k_2)dk_1dk_2.$$ From and Itô formula we conclude that $$\begin{aligned}
&&
\frac{d}{dt}\bar M_{{{\bf i}}}^{(2n)}(t)=-\frac{1}{4}\left(\sum_{\ell=1}^{2n}\hat\beta\left(k_\ell\right)\right)\bar M_{{{\bf i}}}^{(2n)}(t)
-\sum_{1\le \ell<m\le 2n}{\cal R}(t,k_{\ell}) \bar M_{{{\bf i}}_{\ell,m}}^{(2n-2)}(t)\otimes_{\ell,m}\Delta ,\nonumber
\label{031910a}\\
&&
\bar M(0)=\hat M.\end{aligned}$$ Here $\bar M_{{{\bf i}}_{\ell,m}}^{(2n-2)}(t)$ is the $2n-2$-nd order moment obtained from $\bar M_{{{\bf i}}}^{(2n)}(t)$ by omitting $\bar \psi_{i_\ell}(t)$ and $\bar \psi_{i_m}(t)$ and for any measure $\nu$ on ${{{\ensuremath{\mathbb T}}}}^{2n-2}$, $1\le \ell<m\le 2n$ we denote by $\nu\otimes_{\ell,m}\Delta$ a measure on ${{{\ensuremath{\mathbb T}}}}^{2n}$ given by $$\begin{aligned}
&&
\int_{{{{\ensuremath{\mathbb T}}}}^{2n}}fd(\nu\otimes_{\ell,m}\Delta)
=\int_{{{{\ensuremath{\mathbb T}}}}^{2n-2}}d{{\bf k}}\int_{{{{\ensuremath{\mathbb T}}}}}dk f(k_1,\ldots,k_{\ell-1},k,\ldots,k_{m-1},-k,\ldots,k_{2n-2})\end{aligned}$$ for all $f\in C({{{\ensuremath{\mathbb T}}}}^{2n})$. Since $$\begin{aligned}
&&
{\cal R}(t,k_{\ell})=\int_{{{{\ensuremath{\mathbb T}}}}}R(k_\ell,k')\bar{\cal E}(t,k')dk'=\int_{{{{\ensuremath{\mathbb T}}}}}[r^2(k_\ell,k_\ell-k')+r^2(k_\ell,k_\ell+k')]\bar{\cal E}(t,k')dk'
\\
&&
=\sum_{j=\pm1}\int_{{{{\ensuremath{\mathbb T}}}}^2}r^2(k_\ell,k_\ell-k'){{\mathbb E}}\left[\bar\psi_j(t,k')\otimes \bar\psi_{-j}(t,k'')\right]dk'dk''\end{aligned}$$ and $(\bar \psi_{i_1}(t),\ldots, \bar \psi_{i_{2n}}(t))$ is jointly Gaussian, we infer that the last term on the right hand side of the first equation in equals the last term on the right hand side of the first equation of . Thus the conclusion of Theorem \[main-thm2\] has been shown.
Proof of Proposition \[prop010910\] {#appA}
===================================
To prove the proposition we verify that for any $T>0$ $$\label{030910}
t\mapsto {\cal A}[t,\cdot]\quad \mbox{ is Lipschitz on }H^{m}({{{\ensuremath{\mathbb T}}}}),$$ uniformly in $t\in[0,T]$ and $\tilde Q[t,g] :L^2({{{\ensuremath{\mathbb T}}}})\to H^m({{{\ensuremath{\mathbb T}}}})$, given by is Hilbert-Schmidt for any $g\in H^m({{{\ensuremath{\mathbb T}}}})$ and its respective Hilbert-Schmidt norm satisfies $$\label{020910a}
\sup_{t\in[0,T]}\| \tilde Q[t,g_1]-\tilde Q[t,g_2]\|_{(HS)_m}\le C\|g_1-g_2\|_{H^m({{{\ensuremath{\mathbb T}}}})},\quad\forall\,g_1,g_2\in H^m({{{\ensuremath{\mathbb T}}}})$$ for some $C>0$. The conclusion of the lemma then follows from [@DZ], Theorem 7.4, p. 186.
Since $\beta_x\not=0$ only for $|x|\le 2$, see , to prove it suffices only to show that there exists $C>0$ such that $$\label{060910}
\sup_{t\in[0,T]}\|f(t)\|_{H^m({{{\ensuremath{\mathbb T}}}})}\le C\|f\|_{H^m({{{\ensuremath{\mathbb T}}}})},\quad\forall\,f\in H^m({{{\ensuremath{\mathbb T}}}}),$$ with $f(t):=\exp\left\{2i{\omega}(k)t\right\}f(k)$. Dispersion relation ${\omega}(\cdot)$ given by is bounded with its all derivatives on ${{{\ensuremath{\mathbb T}}}}\setminus\{0\}$. In addition ${\omega}'(0-)$ and ${\omega}'(0+)$ exist. Therefore $$\label{050910}
{\omega}_*:=\sup_{t\in[0,T],\,x\in{{{\ensuremath{\mathbb Z}}}}}(1+x^2)\left|\gamma_x(t)\right|<+\infty,$$ where $$\gamma_x(t):=\int_{{{{\ensuremath{\mathbb T}}}}}\exp\left\{2i{\omega}(k)t\right\}e_x^*(k)dk.$$ Note that $$\label{031010}
1+y^2\le \sup_x\frac{1+x^2}{1+(x-y)^2}\le 2(1+y^2).$$ Assume first that $m\ge0$. We can write then $$\begin{aligned}
\label{011110}
&&
\|f(t)\|_{H^m({{{\ensuremath{\mathbb T}}}})}^2=\sum_{x\in{{{\ensuremath{\mathbb Z}}}}}(1+x^2)^{m}\left|\check f_x(t)\right|^2
=\sum_{x\in{{{\ensuremath{\mathbb Z}}}}}(1+x^2)^{m}\left|\sum_{y\in{{{\ensuremath{\mathbb Z}}}}}\check f_{x-y}\gamma_y(t)\right|^2\\
&&
=\sum_{x\in{{{\ensuremath{\mathbb Z}}}}}\left|\sum_{y\in{{{\ensuremath{\mathbb Z}}}}}(1+(x-y)^2)^{m/2}\check f_{x-y}\frac{(1+x^2)^{m/2}\gamma_y(t)}{(1+(x-y)^2)^{m/2}}\right|^2.\nonumber\end{aligned}$$ Using together with we can we can estimate the utmost right hand side of by $$2{\omega}_*^2
\sum_{x\in{{{\ensuremath{\mathbb Z}}}}}\left\{\sum_{y\in{{{\ensuremath{\mathbb Z}}}}}(1+(x-y)^2)^{m/2}|\check f_{x-y}|(1+y^2)^{m/2-1}\right\}^2.$$ Using Young’s inequality $\|f*g\|_{\ell^r} \le
\|f\|_{\ell^p}\|g\|_{\ell^q}$, where $1+r^{-1} = p^{-1} + q^{-1}$, (with $r=p=2$, $q=1$) we can bound this expression by $$C\left\{\sum_{x\in{{{\ensuremath{\mathbb Z}}}}}(1+x^2)^{m}|\check f_{x}|^2\right\}
\left\{\sum_{y\in{{{\ensuremath{\mathbb Z}}}}}(1+y^2)^{m/2-1}\right\}^2$$ for some constant $C>0$. Summarizing we have shown that $$\|f(t)\|_{H^m({{{\ensuremath{\mathbb T}}}})}^2\le 2{\omega}_*^2
\|f\|_{H^m({{{\ensuremath{\mathbb T}}}})}^2\left\{\sum_y(1+y^2)^{m/2-1}\right\}^2,$$ which proves , provided $0\le m<1$.
If, on the other hand, $m<0$ we can write $$\begin{aligned}
&&
\sum_{x\in{{{\ensuremath{\mathbb Z}}}}}(1+x^2)^{m}\left|\check f_x(t)\right|^2
\le {\omega}_*^2
\sum_{x\in{{{\ensuremath{\mathbb Z}}}}}(1+x^2)^{m}\left[\sum_{y\in{{{\ensuremath{\mathbb Z}}}}}(1+y^2)^{-1}|\check f_{x-y}|\right]^2.\end{aligned}$$ By Cauchy-Schwartz inequality for any $\gamma>1/2$ the right hand side can be estimated by $$\label{011310-2012}
{\omega}_*^2
\sum_{x\in{{{\ensuremath{\mathbb Z}}}}}(1+x^2)^{m}\left[\sum_{y\in{{{\ensuremath{\mathbb Z}}}}}(1+y^2)^{-\gamma}\right]\left[\sum_{y\in{{{\ensuremath{\mathbb Z}}}}}(1+(x-y)^2)^{-(2-\gamma)}|\check f_{y}|^2\right].$$ We use the following elementary inequality: for any ${\kappa}>1/2$ there exists a constant $C>0$ such that $$\label{021110}
\sum_{x\in{{{\ensuremath{\mathbb Z}}}}}(1+x^2)^{m}(1+(x-y)^2)^{-\kappa}\le C(1+y^2)^{m\vee (-{\kappa})},\quad \forall y\in{{{\ensuremath{\mathbb Z}}}}.$$ Let ${\kappa}:=2-{\gamma}$ and ${\gamma}\in(1/2,3/2)$. We conclude from the above estimate that the expression in is less than, or equal to $
C\|f\|_{H^m({{{\ensuremath{\mathbb T}}}})}^2,
$ provided that $2+m>{\gamma}$, which is possible as long as $m>-3/2$.
To show it suffices to prove that for any functions $\phi_1,\phi_2$ that are finite combinations of the vectors from the base $(e_x)$ and $T>0$ there exists a constant $C>0$ such that $$\begin{aligned}
\label{011010}
&&
\sup_{t\in[0,T]}\sum_{x,y}(1+y^2)^m\left|\int_{{{{\ensuremath{\mathbb T}}}}^2}\phi_1(k)\phi_2(k-k')g(k-k')e_x(k') e_y^*(k)
\exp\left\{i[{\omega}(k)-{\sigma}{\omega}(k-k')]t\right\}dk dk'\right|^2\nonumber\\
&&
\le C\|g\|_{H^m({{{\ensuremath{\mathbb T}}}})}^2,\quad \forall\, g\in H^m({{{\ensuremath{\mathbb T}}}}),\, \sigma = \pm 1.\end{aligned}$$ The expression on the left hand side of can be rewritten in the form $$\label{021010}
\sum_{x,y}(1+y^2)^m\left|\psi_{y-x}^{(1)}(t)\psi_{x}^{(2)}(t)\right|^2,$$ where $$\psi_x^{(1)}(t):=\int_{{{{\ensuremath{\mathbb T}}}}}\phi_1(k) e_x^*(k)
\exp\left\{i{\omega}(k)t\right\}dk$$ and $$\psi_x^{(2)}(t):=\int_{{{{\ensuremath{\mathbb T}}}}}\phi_2(k)g(k)e_x^*(k)
\exp\left\{-i{\sigma}{\omega}(k)t\right\}dk.$$ As a consequence of for any $T>0$ there exists $C>0$ such that $$\sup_{t\in[0,T]}\sum_{x}(1+x^2)^m\left|\psi_{x}^{(2)}(t)\right|^2\le C\|g\|_{H^m({{{\ensuremath{\mathbb T}}}})}^2,\quad \forall\,g\in H^m({{{\ensuremath{\mathbb T}}}}).$$ We also have $
\sup_{t\in[0,T]}(1+x^2)|\psi_x^{(1)}(t)|<+\infty.
$ The expression in can be rewritten as $$\begin{aligned}
\label{051010}
&&
\sum_{x,z}(1+(z+x)^2)^m\left|\psi_{z}^{(1)}(t)\psi_{x}^{(2)}(t)\right|^2\\
&&
=\sum_{x,z}\frac{(1+(z+x)^2)^m}{(1+x^2)^m}(1+x^2)^m\left|\psi_{z}^{(1)}(t)\psi_{x}^{(2)}(t)\right|^2.\nonumber\end{aligned}$$ Suppose that $m\ge0$ then the right hand side can be estimated by $$\begin{aligned}
&&
2^m\sum_{x,z}(1+z^2)^m(1+x^2)^m\left|\psi_{z}^{(1)}(t)\psi_{x}^{(2)}(t)\right|^2\\
&&
=
2^m\left(\sum_{z}(1+z^2)^m\left|\psi_{z}^{(1)}(t)\right|^2\right)\left(\sum_{x}(1+x^2)^m\left|\psi_{x}^{(2)}(t)\right|^2\right)\\
&&
\le C\left(\sum_{z}(1+z^2)^{m-2}\right)\|g\|_{H^m({{{\ensuremath{\mathbb T}}}})}^2,\end{aligned}$$ which proves , provided that $0\le m<3/2$.
If, on the other hand, $m<0$ the left hand side of can be estimated by $$\begin{aligned}
\label{051010a}
&&
C\sum_{x,z}(1+(z+x)^2)^m(1+z^2)^{-2}\left|\psi_{x}^{(2)}(t)\right|^2\\
&&
\le C_1\sum_{x}(1+x^2)^{m\vee (-2)}\left|\psi_{x}^{(2)}(t)\right|^2\le C_2\|g\|_{H^m({{{\ensuremath{\mathbb T}}}})}^2,\nonumber\end{aligned}$$ provided that $m>-2$.
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[^1]: Institute of Mathematics, UMCS, pl. Marii Curie-Skłodowskiej 1, 20-031, Lublin and IMPAN, ul. Śniadeckich 8, 00-956 Warsaw, Poland, e-mail: [email protected]
[^2]: Ceremade, UMR-CNRS 7534, Université Paris Dauphine, Place Marechal Lattre de Tassigny, 75775, Paris Cedex 16, France, e-mail: [email protected]
[^3]: Department of Mathematics, Stanford University, Stanford, CA 94305, USA, e-mail: [email protected]
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: |
Reconstruction of arithmertic circuits has been heavily studied in the past few years and has connections to proving lower bounds and deterministic identity testing. In this paper we present a polynomial time randomized algorithm for reconstructing $\Sigma\Pi\Sigma(2)$ circuits over ${\mathbb{R}}$, i.e. depth$-3$ circuits with fan-in $2$ at the top addition gate and having real coefficients.
The algorithm needs only a blackbox query access to the polynomial $f \in {\mathbb{R}}[x_1,\ldots, x_n]$ of degree $d$, computable by a $\Sigma\Pi\Sigma(2)$ circuit $C$. In addition, we assume that the *“simple rank”* of this polynomial (essential number of variables after removing the gcd of the two multiplication gates) is bigger than a fixed constant. Our algorithm runs in time $poly(n, d)$ and returns an equivalent $\Sigma\Pi\Sigma(2)$ circuit(with high probability).
The problem of reconstructing $\Sigma\Pi\Sigma(2)$ circuits over finite fields was first proposed by Shpilka [@Shpilka07]. The generalization to $\Sigma\Pi\Sigma(k)$ circuits, $k = O(1)$ (over finite fields) was addressed by Karnin and Shpilka in [@KarShp09]. The techniques in these previous involve iterating over all objects of certain kinds over the ambient field and thus the running time depends on the size of the field ${\mathbb{F}}$. Their reconstruction algorithm uses lower bounds on the lengths of Linear Locally Decodable Codes with $2$ queries. In our settings, such ideas immediately pose a problem and we need new ideas to handle the case of the field ${\mathbb{R}}$.
Our main techniques are based on the use of Quantitative Syslvester Gallai Theorems from the work of Barak et.al. [@BDWY11] to find a small collection of *“nice”* subspaces to project onto. The heart of our paper lies in subtle applications of the Quantitative Sylvester Gallai theorems to prove why projections w.r.t. the *“nice”* subspaces can be ”glued”. We also use Brill’s Equations from [@GKZ94] to construct a small set of candidate linear forms (containing linear forms from both gates). Another important technique which comes very handy is the polynomial time randomized algorithm for factoring multivariate polynomials given by Kaltofen [@KalTr90].
author:
- 'Gaurav Sinha [^1]'
bibliography:
- 'cccpaper.bib'
title: 'Reconstruction of Real depth-3 Circuits with top fan-in 2'
---
Introduction
============
The last few years have seen significant progress towards interesting problems dealing with arithmetic circuits. Some of these problems include Deterministic Polynomial Identity Testing, Reconstruction of Circuits and recently Lower Bounds for Arithmetic Circuits. There has also been work connecting these three different aspects. In this paper we will primarily be concerned with the reconstruction problem. Even though it’s connections to Identity Testing and Lower Bounds are very exciting, the problem in itself has drawn a lot of attention because of elegant techniques and connections to learning. The strongest version of the problem requires that for any $f\in
{\mathbb{F}}[x_1,\ldots,x_n]$ with blackbox access given one wants to construct (roughly) most succint representation i.e. the smallest possible arithmetic circuit computing the polynomial. This general problem appears to be very hard. Most of the work done has dealt with some special type of polynomials i.e. the ones which exhibit constant depth circuits with alternating addition and multiplication gates. Our result adds to this by looking at polynomials computed by circuits of this type (alternating addition/multiplication gates but of depth $3$). Our circuits will have variables at the leaves, operations $(+,\times)$ at the gates and scalars at the edges. We also assume that the top gate has only two children and the *“simple rank”* of this polynomial (essential number of variables after removing the gcd of the two multiplication gates) is bigger than a constant. The bottom most layer has addition gates and so computes linear forms, the middle layer then multiplies these linear forms together and the top layer adds two such products. Later in Remark \[homogen\] we discuss that we may assume the linear forms computed at bottom level to be homogeneous and the in-degree of all gates at middle level to be the same $(=$ degree of $f)$. Therefore these circuits compute polynomials with the following form : $$f(x_1,\ldots,x_n) = G(x_1,\ldots,x_n)(T_0(x_1,\ldots,x_n) + T_1 (x_1,\ldots,x_n))$$
where $T_i(x_1,\ldots,x_n) = \prod\limits_{j=1}^M l_{ij}$ and $G(x_1,\ldots,x_n) = \prod\limits_{j=1}^{d-M}G_j$ with the $l_{ij}$’s and $G_j$’s being linear forms for $i\in \{0,1\}$. Also assume $gcd(T_0,T_1)=1$. Our condition about the essential number of variables (after removing gcd from the multiplication gates) is called *“simple rank”* of the polynomial and is defined as dimension of the space $$sp\{l_{ij} : i\in \{0,1\}, j\in \{1,\ldots,M\}\}$$
When the underlying field is ${\mathbb{R}}$ (i.e. the field of real numbers) we give an efficient randomized algorithm for reconstructing the circuit representation of such polynomials. Formally our main theorem reads :
\[maintheorem\]\[$\Sigma\Pi\Sigma_{\mathbb{R}}(2)$ Reconstruction Theorem\] Let $f = G(T_0+T_1) \in {\mathbb{R}}[x_1,\ldots,x_n]$ be any degree $d$, $n-$ variate polynomial (to which we have blackbox access) which can be computed by a depth $3$ circuit with top fan-in $2$ (i.e. a $\Sigma\Pi\Sigma(2)$ circuit) i.e. $G,T_i$ being products of affine forms. Assume $gcd(T_0,T_1)=1$ and $span\{l : l\mid T_0T_1\}$ is bigger than $s+1$ (a fixed constant defined below). We give a randomized algorithm which runs in time $poly(n,d)$ and computes the cicuit for $f$ with high probability.
\[rvalue\]
We fix $s$ to be any constant $> \max (C_{2k-1}+k, c_{\mathbb{R}}(4) )$ where :
1. $C_k = \frac{C^k}{\delta}$ the constant that appears in Theorem \[bdwy\].
2. $\delta$ is some fixed number in $(0,\frac{7-\sqrt{37}}{6})$.
3. $c_{\mathbb{R}}(4) = 3(4)^2 = 48$, is the rankbound needed for uniqueness of $\Sigma\Pi\Sigma(2)$ circuits as shown in Theorem \[uniqueness\].
From our discussion before the theorem about Remark \[homogen\], we can assume in the above theorem that the polynomial and all linear forms involved are homogeneous.
As per our knowledge this is the first algorithm that efficiently reconstructs such circuits (over the reals). Over finite fields, the same problem has been considered by [@Shpilka07] and our method takes inspiration from their work. They also generalized this finite field version to circuits with arbitrary (but constant) top fan-in in [@KarShp09]. However we need many new tools and techniques as their methods don’t generalize at a lot of crucial steps. For eg:
- They iterate through linear forms in a finite field which we unfortunately cannot do.
- They use lower bounds for Locally Decodable Codes given in [@DS07] which again does not work in our setup.
We resolve these issues by
- Constructing candidate linear forms by solving simultaneous polynomial equations obtained from Brill’s Equations (Chapter 4, [@GKZ94]).
- Using quantitative versions of the Sylvester Gallai Theorems given in [@BDWY11] and [@DSW12]. This new method enables us to construct $nice$ subspaces, take projections onto them and glue the projections back to recover the cicuit representation.
Previous Work and Connections
-----------------------------
Efficient Reconstruction algorithms are known for some concrete class of circuits. We list some here:
- Depth-2 $\Sigma\Pi$ circuits (sparse polynomials) in [@KS01]
- Read-once arithmetic formulas in [@SV09]
- Non-commutative ABP’s [@ArMS08]
- $\Sigma\Pi\Sigma(2)$ circuits over finite fields in [@Shpilka07], extended to $\Sigma\Pi\Sigma(k)$ circuits (over finite fields) with $k=O(1)$ in [@KarShp09].
- Random Multilinear Formular in [@GuptaKL11]
- Depth $4$ ($\Sigma\Pi\Sigma\Pi$) multilinear circuits with top fan-in $2$ in [@GuptaKL12]
- Random Arithmetic Formulas in [@GKY14]
All of the above work introduced new ideas and techniques and have been greatly appreciated.\
It’s straightforward to observe that a polynomial time deterministic reconstruction algorithm for a circuit class $C$ also implies a polynomial time Deterministic Identity Testing algorithm for the same class. From the works [@Agr05] and [@HS80] it has been established that blackbox Identity Testing for certain circuit classes imply superpolynomial circuit lower bounds for an explicit polynomial. Hence the general problem of deterministic reconstruction cannot be easier than proving superpolynomial lower bounds. So one might first try and relax the requirements and demand a randomized algorithm. Another motivation to consider the probabilistic version comes from Learning Theory. A fundamental question called the *exact learning problem using membership queries* asks the following : [**Given oracle access to a Boolean function, compute a small description for it.**]{} This problem has attracted a lot of attention in the last few decades. For eg. in [@Khar92][@OGM86] and [@KV94] a negative result stating that a class of boolean circuits containing the trapdoor functions or pseudo-random functions has no efficient learning algorithms. Among positive works [@SchSe96], [@BBB00], [@KS06] show that when $f$ has a small circuit (inside some restricted class) exact learning from membership queries is possible. Our problem is a close cousin as we are looking for exact learning algorithms for algebraic functions. Because of this connection with learning theory it makes sense to also allow randomized algorithms for reconstruction.\
Depth 3 Arithmetic Circuits
---------------------------
We will use the definitions from [@KayalSa09]. Let $C$ be an arithmetic circuit with coefficients in the field ${\mathbb{F}}$. We say $C$ is a $\Sigma\Pi\Sigma(k)$ circuit if it computes an expression of the form. $$C({\bar{x}}) = \sum\limits_{i\in [k]} \prod\limits_{j\in [d]} l_{i,j}({\bar{x}})$$
$l_{i,j}({\bar{x}}$) are linear forms of the type $l_{i,j}({\bar{x}}) = \sum
\limits_{s\in[n]}a_sx_s$ where $(a_1,\ldots,a_n)\in {\mathbb{F}}^n$ and $(x_1,\ldots,x_n)$ is an $n-$ tuple of indeterminates. For convenience we denote the multiplication gates in $C$ as $$T_i = \prod\limits_{j\in [d]} l_{i,j}({\bar{x}})$$
$k$ is the top fanin of our circuit $C$ and $d$ is the fanin of each multiplication gate $T_i$. With these definitions we will say that our circuit is of type $\Sigma\Pi\Sigma_{\mathbb{F}}(k,d,n)$. When most parameters are understood we will just call it a $\Sigma\Pi\Sigma(k)$ circuit.
#### Remark {#homogen}
Note that we are cosidering homogeneous circuits. There are two basic assumptions:
1. $l_{i,j}$’s have no constant term i.e. they are linear forms.
2. Fanin of each $T_i$ is equal to $d$.
If these are not satisfied we can homogenize our circuit by considering $Z^d(C(\frac{X_1}{Z},\ldots,\frac{X_n}{Z}))$. Now both the conditions will be taken care of by reconstructing this new homogenized circuit. We need a rank condition on our polynomial which remains essentially unchanged even after this substitution.
We say that the circuit $C$ is minimal if no strict non empty subsets of the $\Pi\Sigma$ polynomials $\{T_1,\ldots , T_k\}$ sums to zero.
A circuit $C$ is called Simple if the gcd of the $\Pi\Sigma$ polynomials $gcd(T_1,\ldots,T_k)$ is equal to $1$ (i.e. is a unit). The simplification of a $\Sigma\Pi\Sigma(k)$ circuit $C$ denoted as $Sim(C)$ is the $\Sigma\Pi\Sigma(k)$ circuit obtained by dividing each term by the gcd of all terms i.e. $$Sim(C) {\stackrel{def}{=}}\sum\limits_{i\in[k]}\frac{T_i}{gcd(T_1,\ldots,T_k)}$$
Identifying each linear form $l({\bar{x}}) = \sum\limits_{s\in [n]}a_sx_s$ with the vector $(a_1, \ldots, a_n) \in {\mathbb{F}}^n$, we define the rank of $C$ to be the dimension of the vector space spanned by the set $\{l_{i,j} | i \in [k], j \in [d]\}$.
For a $\Sigma\Pi\Sigma(k)$ circuit $C$ we define the *Simple Rank* of $C$ as the rank of the circuit $Sim(C)$.
Before we go further into the paper and explain our algorithm we state some results about uniqueness of these circuits. In a nutshell for a $\Sigma\Pi\Sigma_{\mathbb{R}}(2,d,n)$ circuit $C$, if one assumes that the *Simple rank* of $C$ is bigger than a constant ($c_{\mathbb{R}}(4) :$ defined later) then the circuit is essentially unique.
Uniqueness of Representation
----------------------------
Shpilka et. al. showed the uniqueness of circuit representation in [@Shpilka07] using rank bounds for Polynomial Identity Testing. The bound they used were from the work of Dvir et. al. in [@DS07]. It essentialy states that the rank of a simple, minimal $\Sigma\Pi\Sigma(k)$ circuit ($d\geq 2, k\geq 3$) which computes the identically zero polynomial is $\leq 2^{O(k^2)}\log^{k-2}d$. For circuits over reals improved rank bounds were given by Kayal et.al. in [@KayalSa09].
In a series of following work the rank bounds for identically zero $\Sigma\Pi\Sigma(k)$ circuits got further improved. The best known bounds over real fields were given by Saxena et. al. in [@SS10]. We rewrite Theorem 1.5 in [@SS10] here for completion.
\[rankbound\] Let $C$ be a $\Sigma\Pi\Sigma(k,d,n)$ circuit over field ${\mathbb{R}}$ that is simple, minimal and zero. Then, $rk(C) < 3k^2$.
Let $c_{{\mathbb{R}}}(k) = 3k^2$. This gives us the following version of Corollary 7, Section 2.1 in [@Shpilka07].
\[uniqueness\] Let $f({\bar{x}})\in {\mathbb{R}}[x]$ be a polynomial which exhibits a $\Sigma\Pi\Sigma(2)$ circuit $$C = G(A + B)$$ $A = \prod\limits_{j\in [M]} A_j, B=\prod\limits_{j\in [M]} B_j, G =
\prod\limits_{i\in [d-M]} G_i$, where $A_i,B_j,G_k \in Lin_{{\mathbb{R}}}[{\bar{x}}]$. $gcd(A, B)=1$, and $Sim(C) = A+B$ has rank $\geq c_{{\mathbb{R}}}(4) +1$ then the representation is unique. That is if: $$f=G(A+B) = {\tilde{G}}({\tilde{A}} + {\tilde{B}})$$ where $A,B,{\tilde{A}},{\tilde{B}}$ are $\Pi\Sigma$ polynomials over ${\mathbb{R}}$ and $gcd({\tilde{A}},{\tilde{B}})=1$ then we have $G = {\tilde{G}}$ and $(A,B)=({\tilde{A}},{\tilde{B}})$ or $({\tilde{B}},{\tilde{A}})$ (upto scalar multiplication).
*Proof.* Let $g= gcd(G, {\tilde{G}})$ and let $G=gG_1, {\tilde{G}} = g{\tilde{G_1}}$. Then $gcd(G_1,{\tilde{G_1}})=1$ and we get $$G_1 A + G_1 B - {\tilde{G_1}}{\tilde{A}} - {\tilde{G_1}} {\tilde{B}} = 0$$ This is a simple $\Sigma\Pi\Sigma(4)$ circuit with $rank$ bigger than $c_{{\mathbb{R}}}(4)+1$ and is identically $0$ so it must be not minimal. Considering the various cases one can easily prove the required equality.
Summary of Technical Ideas and Algorithms
=========================================
A General Reconstruction Technique
----------------------------------
In this section we will pictorially present the technique used to reconstruct a linear form (product of linear forms) from it’s (their) projections onto certain spaces. For details, algorithms and proofs please see Section \[Identifier\].\
Consider the linear form $l = a_1x_1+a_2x_2+a_3x_3 \in {\mathbb{R}}[x_1,x_2,x_3]$ (point $P$) with the condition that $a_3\neq 0$. Suppose we know $l\pmod{x_1}$ (point $Q$), $l\pmod{x_2}$ (point $R$) upto scalar multiplication, can we reconstruct a scalar multiple of $l$ from this data. Let us view this pictorially:\
(xyz cs:x=-10) – (xyz cs:x=4) node\[above\] [$x_1$]{}; (xyz cs:y=-4) – (xyz cs:y=8) node\[right\] [$x_3$]{}; (xyz cs:z=-8) – (xyz cs:z=5) node\[above\] [$x_2$]{}; (xyz cs:z=-4) – +(0,7) coordinate (u) – (xyz cs:y=7) – +(-10,0) – ++(xyz cs:x=-10,z=-4) coordinate (v) – +(0,-7) coordinate (w) – cycle; (u) – (v); (-10,7) – (-10,0) – (w); (0,0) –node\[anchor=east\][$L_1$]{} (-10,7,0); (0,0) – node\[anchor=west\][$L_2$]{}(0,7,-4); (0,0) – node\[anchor=north\][$L$]{}(-10,7,-4);
at (v) ; at (-10,7,0) ; at (0,7,-4) ;
Given basis $\{x_1,x_2,x_3\}$ and lines $L_1,L_2$, can we find out the line $L$ in the picture above. This is easy, we just pick points on $L_1,L_2$ with the same $x_3$ co-ordinate (i.e. same height). Then we complete the cuboid and recover our line $L$. Next suppose we have a product of linear forms $\tilde{P} = l_1\ldots l_d$ such that modulo $x_1$, all $l_i$ give the same line and modulo $x_2$, distinct (upto scalar multiplication) forms give distinct lines. Also we know the projection of $\tilde P$ onto $\{x_1=0\}$ and $\{x_2=0\}$. The property mentioned implies that we know projections of all linear forms dividing $\tilde P$. And so we can still reconstruct a scalar multiple of $\tilde P$ by using the above strategy repeatedly. In our final application $x_1$ gets replaced by a subspace $S$ and $S,\{x_2,x_3\}$ are linearly independent. The above method helps us reconstruct a scalar multiple of $\tilde P$ in this case as well.\
So when subspace $S$ and vectors $x_2,x_3$ exist with the projection property mentioned above ( i.e. on using an extension of $S\cup\{x_2,x_3\}$ as a basis, all forms dividing $\tilde P$ give the same line modulo $S$ and distinct (upto scalar multiplication) forms give distinct lines modulo $x_2$) then we can reconstruct a scalar multiple of $\tilde{P}$. Such $(S,x_2,x_3)$ exist in certain scenarios that appear during our algorithm. This is discussed in Subsection \[hardcase\] using Quantitative Sylvester Gallai Theorems from [@BDWY11]. In our application we use Corollary \[elementary\] (to the quantitative SG theorem in [@BDWY11] ) given in Section \[incidence\]. Please see Section \[incidence\] for more details about the theorem.
Algorithm Strategy
------------------
The broad structure of our algorithm is similar to that of Shpilka in [@Shpilka07] however our techniques are different. We first restrict the blackbox inputs to a low ($O(1)$) dimensional random subspace of ${\mathbb{R}}^n$ and interpolate this restricted polynomial. Next we try to recover the $\Sigma\Pi\Sigma(2)$ structure of this restricted polynomial and finally lift it back to ${\mathbb{R}}^n$. The random subspace and unique $\Sigma\Pi\Sigma(2)$ structure will ensure that the lifting is unique. Similar to [@Shpilka07] we try to answer the following questions. However our answers (algorithms) are different from theirs
1. For a $\Sigma\Pi\Sigma(2)$ polynomial $f$ over $r=O(1)$ variables, can one compute a small set of linear forms which contains all factors from both gates?
2. Let $V_0$ be a co-dimension $k$ subspace($k=O(1)$) and $V_1,\ldots,V_t$ be co-dimension $1$ subspaces of a linear space $V$. Given circuits $C_i$ ($i\in\{0,\ldots,t\}$) computing $f|_{V_i}$(restriction of $f$ to $V_i$) can we reconstruct from them a single circuit $C$ for $f|_{V}$?
3. Given co-dimension $1$ subspaces $V \subset U$ and circuits $f|_{V}$ when is the $\Sigma\Pi\Sigma(2)$ circuit representations of lifts of $f|_{V}$ to $f|_{U}$ unique?
Our first question is easily solved using Brill’s equations (See Chapter 4 [@GKZ94]). These provide a set of polynomials whose simultaneous solutions completely characterize coefficients of complex $\Pi\Sigma$ polynomials. A linear form $l = x_1-a_2x_2-\ldots-a_rx_r$ divides one of the gates of $f(x_1,\ldots,x_r)$ $\Rightarrow f(a_2x_2+\ldots+a_rx_r,x_2,\ldots,x_r)$ is a $\Pi\Sigma$ polynomial modulo $l$. When this is applied into Brill’s equation (see Corollary \[variety\]) we recover possible $l$’s which obviously include linear factors of gates. We can show that (see Claim \[candidate\]) the extra linear forms we get are not too many ($poly(d)$) and also have some special structure. We call this set ${\mathcal{C}}$ of linear forms as Candidate linear forms and non-deterministically guess from this set. It should be noted that we do all this when our polynomial is over $O(1)$ variables.\
We deal with the second question while trying to reconstruct the $\Sigma\Pi\Sigma(2)$ representation of the interpolated polynomial $f|_{V}$, where $V$ is the random low dimensional subspace. We divide the algorithm into Easy Case, Medium Case and a Hard Case.
- For the Easy Case our algorithm tries to reconstruct one of the multiplication gates of $f|_{V}$ by first looking at it’s restriction to a special co-dimension $1$ subspace $V_1$. If $f=A+B$ with $A,B$ being $\Pi\Sigma$ polynomials, the projection of one of the gates (say $A$) with respect to $V_1$ will be $0$ and the other (say $B$) will remain unchanged giving us $B$ and therefore both gates by factoring $f|_{V}-B$.
- In the Medium Case we have alteast two extra dimensions in one of the gates. This can be used to show that the only linear factors of $f_{|V}$ are those coming from $G$. Now we can recover $G$ by factoring $f$ and then use Easy Case for the remaining polynomial. An important consequence of this case is that in the Hard Case we may now assume that both gates are high dimensional which is very crucial.
- In the Hard Case we will first need $V_0$, a co-dimension $k$ (where $k=O(1)$) subspace and then iteratively select co-dimension $1$ subspaces $V_1,\ldots,V_t$. For some gate (say $B$), all pairs $(V_0,V_i)$ ($i\in [t]$) will reconstruct some linear factors of $B$. This process will either completely reconstruct $B$ or we will fall into the Easy Case. Once $B$ is known we can factor $f|_{V}-B$ to get $A$.
The restrictions that we compute always factor into product of linear forms and can be easily computed since we know $f|_{V}$ explicitly. They can then be factorized into product of linear forms using the factorization algorithms from [@KalTr90]. It is the choice of the subspaces $V_0,V_1,\ldots,V_t$ where our algorithm differs from that in [@Shpilka07] significantly. Our algorithm selects $V_0$ and iteratively selects the $V_i$’s ($i\in [t]$) such that $(V_0,V_i)$ have certain *“nice”* properties which help us recover the gates in $f|_{V}$. The existence of subspaces with *“nice”* properties is guaranteed by Quantitative Sylvester Gallai Theorems given in [@BDWY11]. To use the theorems we had to develop more machinery that has been explained later.\
The third question comes up when we want to lift our solution from the random subspace $V$ to the original space. This is done in steps. We first consider random spaces $U$ such that $V$ has co-dimension $1$ inside them. Now we reconstruct the circuits for $f|_{V}$ and $f|_{U}$. The $\Sigma\Pi\Sigma(2)$ circuits for $f|_{V}$ and $f|_{U}$ are unique since the simple ranks are high enough (because $U,V$ are random subspaces of high enough dimension) implying that the circuit for $f|_{V}$ lifts to a unique circuit for $f|_{U}$. When this is done for multiple $U$’s we can find the gates exactly.
Flowcharts for Key Algorithms
-----------------------------
This section will sketch all the key algorithms we design in the reconstruction process. Detailed explanations of algorithms, proofs and time complexity analysis can be found later in the paper in Sections \[lowdimrecon\] and \[highdimrecon\].\
Let’s define a structure called decomposition containing the information returned after a reconstruction algorithm. We assume having a data type polynomial for general polynomials and pi\_sigma for polynomials which are product of linear forms. We use C++ syntax to define our structure.
struct decomposition {
bool iscorrect; // iscorrect will be true if f = M_0 + M_1
polynomial f;
pi_sigma M_0;
pi_sigma M_1;
// Constructor when a reconstruction is found
decomposition(polynomial g, pi_sigma A, pi_sigma B){
iscorrect =true;
f=g;
M_0=A;
M_1=B;
}
// Constructor when no reconstruction is found
decomposition(){
iscorrect=false;
}
};
### Overall Algorithm :
Here is a flowchart explaining the entire algorithm:\
(start) \[startstop\] [Start]{}; (in1) \[io, left of=start, xshift=-4cm\] [Input:$f \in {\mathbb{R}}[x_1,\ldots,x_n]$ as blackbox]{}; (start)–(in1);
(pro1) \[process, below of=in1, yshift=-1cm, text width=4cm\] [Choose random basis $\{y_1,\ldots,y_n\}$ of ${\mathbb{R}}^n$, $V = sp(\{y_1,\ldots,y_s\}), V_i =
sp(\{v_1,\ldots,v_s,v_i\})$ for $i\in \{s+1,\ldots,n\}$.]{}; (in1)–(pro1); (pro1a)\[process, right of=pro1, xshift=4cm, text width=4cm\][Define $f_0(y_1,\ldots,y_s) = f_{|V}, f_i(y_1,\ldots,y_s,y_i) = f_{|V_i}$]{}; (pro1)–(pro1a); (pro1ab)\[process, right of=pro1a, xshift=4cm\][Consider sets $H\subset V,H_i\subset V_i$ with $|H|\geq d^s,|H_i|\geq d^{s+1}$ and interpolate to find $f_0,f_i$]{}; (pro1a)–(pro1ab); (pro1b) \[process, below of=pro1ab, yshift=-1.5cm, text width=4.2 cm\] [Reconstruct to get $f_0 = M_0+M_1$ and $f_i = M^i_0 + M^i_1$ with $M_0,M_1\in \Pi\Sigma[y_1,\ldots,y_s], M^i_0,M^i_1\in \Pi\Sigma[y_1,\ldots,y_s,y_i]$]{}; (pro1ab)–(pro1b); (dec1) \[decision, below of=pro1a, yshift=-1.5cm\] [If all reconstructions were successful]{}; (pro1b)–(dec1); (pro2a) \[process, left of=dec1, xshift=-5cm\] [Use $M_0,M_1,M^i_0,M^i_1$ to compute gates $N_0,N_1$ such that $f=N_0+N_1$]{}; (dec1)–node\[anchor=south\] [yes]{}(pro2a);
(dec2) \[decision, below of=pro2a, yshift=-2cm\] [If the reconstruction was successful]{}; (pro2a)–(dec2); (out1) \[io, below of=dec2, yshift=-2cm, text width = 4.5cm\] [Return : a new object of type decomposition using decomposition$(f,N_0,N_1)$]{}; (dec2)– node\[anchor=east\] [yes]{}(out1); (out2) \[io, below of=dec1, yshift = -2cm\] [Return : a new object of type decomposition using decomposition$()$]{}; (dec1) –node\[anchor=west\][no]{} (out2); (dec2) – node\[anchor=south\][no]{} (out2); (stop) \[startstop, right of=out1, xshift= 10cm\] [Stop]{}; (out1)–(stop); (out2)-|(stop);
Most steps in the above flowchart are simple and work easily in polynomial time. However there are two blocks which need explanation.\
(pro1) \[process, text width = 5 cm\] [Reconstruct to get $f_0 = M_0+M_1$ and $f_i = M^i_0 + M^i_1$ with $M_0,M_1\in \Pi\Sigma[y_1,\ldots,y_s], M^i_0,M^i_1\in \Pi\Sigma[y_1,\ldots,y_s,y_i]$]{};
(pro2) \[process, right of=pro1, xshift= 4cm\] [Use $M_0,M_1,M^i_0, M^i_1$ to compute gates $N_0,N_1$ such that $f=N_0+N_1$]{};
1. The first one corresponds to reconstructing the $\Sigma\Pi\Sigma(2)$ representations of polynomials with *simple rank* $=s$ (resp. $s+1$) over variables $\{y_1,\ldots,y_s\}$ (resp. $\{y_1,\ldots,y_s,y_i\}$). Note that our input polynomial has simple rank $\geq s+1$, therefore on projecting to a random subspace of dimension $s$ (resp. $s+1$), it’s rank becomes $s$ (resp. $s+1$) with high probability. We briefly explain in \[constrankrecon\] and give all details in Section \[lowdimrecon\].
2. The second one deals with gluing a polynomially number of such low dimensional reconstructions to get a reconstruction of the original polynomial. We discuss it in Subsubsection \[lifting\] and give all details in Section \[highdimrecon\].
### Lifting from Low to High dimension {#lifting}
Let’s Explain the second block in Picture \[blockexplain\]. So we have the reconstructions $f_0 = M_0+M_1$ and $f_i = M^i_0+M^i_1$. If we set $y_i = 0$ in $f_i$ we should get $f_0$. So ${M^i_0}_{|V} + {M^i_1}_{|V} = M_0+M_1 $. Since the simple rank of $f_0$ is $r$ this representation should be unique. The multiplication gates $M^i_0, M^i_1$ should be lifts of $M_0,M_1$. So we can just set $y_i$ to $0$ and find the correspondence between these gates. Let’s say $M^i_0|_{|V} = M_0$, this implies that the linear forms in $M^i_0$ are lifts of linear forms in $M_0$. Next notice that with high probability LI linear forms from a gate in circuit of $f$ remain LI on projecting to $V$. So LD linear forms in $M_0$ cannot have LI lifts in $M^i_0$. Now to find this lift of linear form $l$ dividing $M_0$ with multiplicity $k$, find $l_i$ in $M^i_0$ ( with multiplicity $k$) such that on setting $y_i=0$, we get $l$ i.e. ${l_i}_{|\{y_i=0\}}=l$. This gives the coefficient of $y_i$ in the lift of $l$. If we do this for all $i$ we get the lift of $l$ to ${\mathbb{R}}^n$. So we can compute lifts of all linear forms in $M_0$ and $M_1$. By uniqueness this will give us the gates $N_0,N_1$ such that $f=N_0+N_1$.
### Reconstruction for constant rank {#constrankrecon}
Let’s explain the first block now. Suppose $f = G(T_0+T_1)$ is a real $\Sigma\Pi\Sigma(2)$ polynomial with simple rank $r$ over variables $\{y_1,\ldots,y_r\}$ (in our application $r=s,s+1$) with $G,T_i$ being $\Pi\Sigma$ polynomials and $gcd(T_0,T_1)=1$. From now onwards, for a product of linear forms (called $\Pi\Sigma$ polynomial) $P$, ${\mathcal{L}}(P)$ will be the set of distict linear factors, $sp(P)$ will be the span of ${\mathcal{L}}(P)$, $dim(P)$ will be the $dim(sp(P))$. For a general polynomial $g$, $Lin(g) =$ product of all linear factors of $g$.
#### Set of candidate linear forms
For our algorithms in this section we need a small set of linear forms which contains ${\mathcal{L}}(T_i)$ ( set of linear forms dividing $T_i$ ) for $i\in\{0,1\}$. In order to compute this set we use a characterization of $\Pi\Sigma$ polynomials given by Brill’s equations (See Section \[brills\]). Our algorithm is based on Corollary \[variety\].\
(start) \[startstop\] [Start]{}; (in1) \[io, right of=start, xshift=4cm\] [Input : $f \in {\mathbb{R}}[x_1,\ldots,x_r]$ ]{}; (start)–(in1); (pro) \[process, right of=in1, xshift= 4cm\] [Compute $h = \frac{f}{Lin(f)}$ ]{}; (in1)–(pro); (pro1) \[process, below of=pro, yshift= -1cm\] [Let $l = x_1-a_2x_2-\ldots - a_rx_r$ and compute coefficients $c_{\bf a}(a_2,\ldots,a_r)$ of $h(a_2x_2+\ldots+a_rx_r,x_2,\ldots,x_r)$ corresponding to monomial ${\bf x}^{\bf a}$ ]{}; (pro)–(pro1); (pro2) \[process, left of=pro1, xshift= -4cm\] [Substitute $c_{\bf a}(a_2,\ldots,a_r)$ in polynomials $F_1,\ldots,F_m$ from Brill’s equations and solve for $(a_2,\ldots,a_r)$. Add tuples with all real $a_i$ to ${\mathcal{C}}$ ]{}; (pro1)–(pro2); (out2) \[io, left of=pro2, xshift = -4cm\] [Return : The set ${\mathcal{C}}$ of linear forms]{}; (stop) \[startstop, below of = out2, yshift= 0.5cm\] [Stop]{}; (pro2)–(out2); (out2)–(stop);
From now onwards assume we know parts of the two gates $GT_0, GT_1$ i.e. say we know polynomials $K_i\mid GT_i$, $i\in \{0,1\}$. Also define $U_i=\frac{GT_i}{K_i}$. At the beginning of the algorithm $K_0=K_1=1$. Now we will break down this low rank algorithm into three cases :
#### Easy Case
In this case we assume that one of the $T_i$’s has a linear form outside the span of the unknown part $U_{1-i}$ (of the other gate $GT_{1-i}$). On going modulo this extra dimension in $T_i$, $U_{1-i}$ remains essentially unchanged (upto a linear transformation) and we use this to recover it and complete the reconstruction.
(start) \[startstop\] [Start]{}; (in1) \[io, right of=start, xshift= 4cm\] [Input : $f \in {\mathbb{R}}[x_1,\ldots,x_r], K_0,K_1 \in \Pi\Sigma[x_1,\ldots,x_r],$ Set ${\mathcal{C}}$ ]{}; (start)–(in1); (pro) \[process, right of=in1, xshift=4cm\] [for $i\in \{0,1\}$]{}; (in1)–(pro); (pro1)\[process, below of=pro, yshift=-0.5cm\][for each $l\in {\mathcal{C}}$]{}; (pro)–(pro1); (pro2)\[process, left of=pro1, xshift=-4cm, text width = 4.5cm\][Use $f \pmod{ l}, K_i \pmod{l}$ to find $U_i$ and check if for some $i$ and $l$, $U_i \in \Pi\Sigma[x_1,\ldots,x_r]$ and $f-K_iU_i \in \Pi\Sigma[x_1,\ldots,x_r]$]{}; (pro1)–(pro2); (dec)\[decision, below of= start, yshift=-5cm\][If such an $i$ and $l$ exists then compute corresponding $U_i$]{}; (pro2) -|(dec); (out1)\[io, below of = dec, yshift = -3cm, text width = 4.5cm\][Return : a new object of type decomposition using decomposition$(f,K_iU_i, f-K_iU_i)$]{}; (dec1)\[decision, right of = dec, xshift= 4cm\][ If there is still ’$l$’ remaining in loop ]{}; (dec2)\[decision, right of = dec1, xshift= 4cm\][ If there is still ’$i$’ remaining in loop ]{}; (dec1)–node\[anchor=east\][yes]{}(pro2); (dec1)–node\[anchor=south\][no]{}(dec2);
(dec2)–node\[anchor=east\][yes]{}(pro1); (dec)–node\[anchor=east\][yes]{} (out1); (dec)–node\[anchor=south\][no]{}(dec1); (stop) \[startstop, right of = out1, xshift= 4cm\] [Stop]{}; (out2)\[io, right of = stop, xshift=4cm\][Return : a new object of type decomposition using decomposition$()$]{};
(dec2)–node\[anchor=east\][no]{}(out2); (out2)–(stop); (out1)–(stop);
#### Medium Case
We consider this case since it facilitates solving the complement (Hard Case) of the Easy Case. The assumption here is that some $T_i$ has two extra dimensions outside the span of $T_{1-i}$. This property can be used to show that the product of all linear factors of $f$ is $G$. In other words $T_0+T_1$ has no linear factors. Now we could simply use the factorization algorithm from [@KalTr90] and recover $G$. On removing $G$, $U_{1-i}=T_{1-i}$ and $T_i$ has a linear form outside it’s span enabling us to use the Easy Case algorithm from above. It’s easy to see that if we are not in this case then both $dim(sp(T_0))$ and $dim(sp(T_1))$ are $\geq r-1$ (assuming $dim(sp(T_0)+sp(T_1)) =r$). This will be very crucial in the Hard Case.\
(start) \[startstop\] [Start]{}; (in1) \[io, right of=start, xshift= 4cm\] [Input : $f \in {\mathbb{R}}[x_1,\ldots,x_r]$,Set ${\mathcal{C}}$ ]{}; (start)–(in1); (pro) \[process, right of=in1, xshift=4cm, text width = 4cm\] [Compute $L = Lin(f):=$ product of all linear factors of $f$]{}; (in1)–(pro); (out)\[io, below of = pro, yshift = -1cm\][Return : $EasyCase(f,L,L,{\mathcal{C}})$]{}; (stop) \[startstop, left of = out, xshift= -4cm\] [Stop]{}; (pro)–(out); (out)–(stop);
#### Hard Case
Fix $k=c_{\mathbb{R}}(3)+2$ (See Theorem \[rankbound\] for definition of $c_{\mathbb{R}}(m)$ and point \[candidateprop\] in Lemma \[candidate\] to see why we need it). The algorithm for this case relies on the existence of something called a Detector Pair $(S,D)$ (see Definition \[detectorset\]). $S,D$ are subsets of some ${\mathcal{L}}(T_i)$ with $|S|=k$. From the arguments given in Subsection \[hardcase\], we know that inside some ${\mathcal{L}}(T_i)$, such a pair exists with large $|D|$. The algorithm crucially depends on such a pair since projections of $f$ onto them can be glued.
1. $S$ is used to find a factor $I\mid G$ such that factors of $\frac{G}{I}$ are nice. (See Lemma \[filteredfactor\]) This factor $I$ is removed from $f$ to obtain $f^\star$. It is added after reconstruction of $f^\star$.
2. Once such an $S$, is known we can compute a set $X$ such that $D\subset X\subset {\mathcal{L}}(T_i)$ correctly using Algorithm \[overestimatedetector\]. This helps us in moving forward with the algorithm by making sure (while iterating over $X$) that linear forms are chosen from ${\mathcal{L}}(T_i)$ and the *good* linear forms (certain elements of $D$) will be chosen.
3. Finally $f^\star$ is reconstructed from $f^\star \pmod{S}$ and $f^\star \pmod{d}$, for certain $d\in D$ using the reconstructor algorithm given in Algorithm \[reconalgo\].
(start) \[startstop\] [Start]{}; (in1) \[io, right of=start, xshift= 4cm\] [Input : $f \in {\mathbb{R}}[x_1,\ldots,x_r]$, Set ${\mathcal{C}}$ ]{}; (start)–(in1); (pro) \[process, right of=in1, xshift=4cm\] [for $i\in \{0,1\}$]{}; (in1)–(pro); (pro1)\[process, below of=pro, yshift=-1cm\][for each LI $S\subset {\mathcal{C}}$ with $|S|=k$]{}; (pro)–(pro1); (pro2)\[process, left of=pro1, xshift=-4cm, text width = 6cm\]
Using $S$ do the following :
- I =IdentifyFactors$(f,S,{\mathcal{C}})$)
- $f^\star = \frac{f}{I}, K_0^\star=1, K_1^\star=1$
- X = OverestDetector$(f,S,{\mathcal{C}})$), Note $D\subseteq X\subseteq {\mathcal{L}}(T_i)$
; (pro1)–(pro2); (pro4)\[process, left of= pro2, xshift=-3.5cm\][While ($deg (K_{1-i}^\star) < deg(f^\star)$) ]{}; (pro2)–(pro4); (dec1)\[decision, below of = pro4, yshift=-3cm, text width = 5cm\][Obj=EasyCase($f^\star, K_0^\star,K_1^\star,{\mathcal{C}}$), If (obj $\rightarrow$ iscorrect) is true]{}; (pro4)–(dec1); (pro6)\[process, right of= dec1, xshift= 4cm\][Iterate over $d\in X$ to find $d$ such that projections onto $sp(S)$ and $sp(\{d\})$ can reconstruct a new linear factor and update $K_{1-i}^\star$]{}; (dec1)–node\[anchor=south\][no]{} (pro6); (dec2)\[decision, below of= pro6, yshift= -3cm\] [If such a $d$ exists]{}; (pro6)–(dec2); (dec3)\[decision, right of= pro6, xshift= 3.5cm\] [If there is $S_0$ remaining in loop ]{}; (dec2.east) –node\[anchor=south\][no]{} ++(1,0) node(lowerright) |- (dec3.west); (dec3.north) –node\[anchor=west\][yes]{} ++(0,0.5) -| (pro2.south);
(dec4)\[decision, below of= dec3, yshift= -3cm\] [If there is $i$ remaining in loop ]{}; (dec3)–node\[anchor=west\] [no]{}(dec4); (dec4.east) –node\[anchor=north\][yes]{} ++(1,0) |- (pro1.east); (dec5)\[decision, below of= out1, yshift= -2.5cm\] [If $deg(K_{1-i}^\star) < deg(f)$ ]{}; (dec2)|- node\[anchor=west\][yes]{} (dec5); (dec5.west)–node\[anchor=south\][yes]{}++(-1.2,0)|-(dec1.west); (out1)\[io,below of = dec1, yshift = -3cm, text width = 4.5cm\][Return : a new object of type decomposition using decomposition$(f,I$(obj$\rightarrow M\_0)$, $I$(obj$\rightarrow M\_1))$ ]{}; (dec6) \[decision, below of= dec4,yshift= -3cm\][If $f-IK_{1-i}^\star$ is a $\Pi\Sigma$ polynomial]{}; (dec5.south) – ++(0,-0.2)–node\[anchor=south\][no]{}++(6,0) |-(dec6.west); (dec1)–node\[anchor=west\][yes]{}(out1); (dec4)–node\[anchor=west\][no]{}(dec6); (out2)\[io,below of = dec5, yshift = -2cm, text width=4.2cm\][Let $M_0=IK_{1-i}^\star, M_1=f-IK_{1-i}^\star$. Return : a new object of type decomposition using decomposition$(f,M_0, M_1)$]{}; (out3)\[io,below of = dec6, yshift = -1.5cm, text width=4cm\][ Return : a new object of type decomposition using decomposition$( )$]{}; (dec6.east) –node\[anchor=south\][no]{} ++(1,0) |-(out3.east); (dec6.south) – ++(0,-0.25) – node\[anchor=south\][yes]{}++(-6,0) |- (out2.east);
(stop) \[startstop, left of = out3, xshift=-3.5cm, yshift=-1cm\] [Stop]{}; (out3)-|(stop); (out2)|-(stop);
Notation
========
$[n]$ denotes the set $\{1,2,\ldots,n\}$. Throughout the paper we will work over the field ${\mathbb{R}}$. Let $V$ be a finite dimensional real vector space and $S\subset V$, $sp(S)$ will denote the linear span of elements of $S$. $dim(S)$ is the dimension of the subspace $sp(S)$. If $S=\{s_1,\ldots ,s_k\}\subset V$ is a set of linearly independent vectors then $fl(S)$ denotes the affine subspace generated by points in $S$ (also called a $(k-1)-flat$ or just $flat$ when dimension is understood). In particular: $$fl(S) = \{\sum\limits_{i=1}^k \lambda_i s_i : \lambda_i\in {\mathbb{R}},
\sum\limits_{i=1}^k \lambda_i=1\}$$ Let $W\subset V$ be a subspace, then we can extend basis and get another subspace $W^\prime$ (called the complement of $W$) such that $W\oplus W^\prime = V$. Note that the complement need not be unique. Corresponding to each such decomposition of $V$ we may define orthogonal projections $\pi_W,\pi_{W^\prime}$ onto $W,W^\prime$ respectively. Let $v=w+w^\prime \in V, w\in W,w^\prime \in W^\prime$: $$\pi_{W}(v) = w, \pi_{W^\prime}(v)=w^\prime$$
$({\bar{x}})$ will be used for the tuple $(x_1,\ldots,x_n)$. $$Lin_{\mathbb{R}}[{\bar{x}}] = \{a_1x_1+\ldots + a_nx_n : a_i\in {\mathbb{R}}\}
\subset {\mathbb{R}}[{\bar{x}}]$$ is the vector space of all linear forms over the variables $(x_1,\ldots,x_n)$. For a linear form $l\in Lin_{\mathbb{R}}[{\bar{x}}]$ and a polynomial $f\in {\mathbb{R}}[x]$ we write $l\mid f$ if $l$ divides $f$ and $l\nmid f$ if it does not. We say $l^d \mid\mid f$ if $l^d\mid f$ but $l^{d+1} \nmid f$. $$\Pi\Sigma^d_{\mathbb{R}}[{\bar{x}}] = \{l_1({\bar{x}})\ldots l_d({\bar{x}}): l_i\in Lin_{\mathbb{R}}[{\bar{x}}]\}\subset {\mathbb{R}}[{\bar{x}}]$$ is the set of degree $d$ homogeneous polynomials which can be written as product of linear forms. This collection for all possible $d$ is called the set $$\Pi\Sigma_{\mathbb{R}}[{\bar{x}}] = \bigcup\limits_{d\in {\mathbb{N}}}\Pi\Sigma^d_{\mathbb{R}}[{\bar{x}}]$$ also called $\Pi\Sigma$ polynomials for convenience. Let $f({\bar{x}})\in{\mathbb{R}}[x]$ then $Lin(f)\in \Pi\Sigma_{\mathbb{R}}[{\bar{x}}]$ denotes the product of all linear factors of $f({\bar{x}})$. Let ${\mathcal{L}}(f)$ denote the set of all linear factors of $f$. For any set of polynomials $S\subset {\mathbb{C}}[{\bar{x}}]$, we denote by $\mathbb{V}(S)$, the set of all complex simultaneous solutions of polynomials in $S$ (this set is called the variety of $S$), i.e. $$\mathbb{V}(S) =\{a\in {\mathbb{C}}: \text{ for all } f\in S, f(a)=0 \}$$
Let ${\mathcal{B}}=\{b_1,\ldots,b_n\}$ be an ordered basis for $V = Lin_{\mathbb{R}}[{\bar{x}}]$. We define maps $\phi_{\mathcal{B}}: V\setminus\{0\} \rightarrow V$ as $$\phi_{{\mathcal{B}}}(a_1b_1+\ldots+a_nb_n) = \frac{1}{a_k}(a_1b_1+\ldots+a_nb_n)$$ where $k$ is such that $a_i=0$ for all $i<k$ and $a_k\neq 0$.\
A non-zero linear form $l$ is called with respect to ${\mathcal{B}}$ if $l\in \Phi_{\mathcal{B}}(V)$ i.e. the first non-zero coefficient is $1$. A polynomial $P\in \Pi\Sigma_{\mathbb{R}}[{\bar{x}}]$ is normal w.r.t. ${\mathcal{B}}$ if it is a product of normal linear forms. For two polynomials $P_1,P_2\in \Pi\Sigma_{\mathbb{R}}[{\bar{x}}]$ we define : $$gcd_{{\mathcal{B}}}(P_1,P_2) = P\in \Pi\Sigma_{\mathbb{R}}[{\bar{x}}], P \text{ normal w.r.t. } {\mathcal{B}}\text{ such that } P\mid P_1, P\mid P_2$$
When a basis is not mentioned we assume that the above definitions are with respect to the standard basis.
We can represent any linear form in $Lin_{\mathbb{R}}[{\bar{x}}]$ as a point in the vector space ${\mathbb{R}}^n$ and vice versa. To be precise we define the cannonical map $\Gamma : Lin_{{\mathbb{R}}}[{\bar{x}}] \rightarrow {\mathbb{R}}^n$ as $$\Gamma(a_1x_1+\ldots +a_nx_n) = (a_1,\ldots,a_n)$$ $\Gamma$ is a linear isomorphism of vector spaces $Lin_{\mathbb{R}}[{\bar{x}}]$ and ${\mathbb{R}}^n$. Because of this isomorphism we will interchange between points and linear forms whenever we can. We choose to represent the linear form $a({\bar{x}}) = a_1x_1+\ldots+a_nx_n$ as the point $a = (a_1,\ldots,a_n)$.\
[**LI**]{} will be the abbreviation for Linearly Independent and [**LD**]{} will be the abbreviation for Linearly Dependent.\
A non zero vector $v$ is called $standard$ with respect to basis ${\mathcal{B}}= \{b_1,\ldots,b_n\}$ if the coefficient of $b_1$ in $v$ is $1$. When a basis is not mentioned we assume we’re talking about the standard basis. (Equivalently for linear forms the coefficient of $x_1$ is $1$). A $\Pi\Sigma$ polynomial will be called $standard$ if it is a product of standard linear forms.
We close this section with a lemma telling us when can we replace the span of some vectors with the affine span or flat. We’ve used this several times in the paper.
\[spantoflat\] Let $l,l_1,\ldots,l_t \in Lin_{\mathbb{R}}[{\bar{x}}]$ be *standard* linear forms w.r.t. some basis ${\mathcal{B}}=\{b_1,\ldots,b_n\}$ such that $l\in sp(\{l_1,\ldots,l_t\})$ then $$l\in fl(\{l_1,\ldots,l_t\})$$
*Proof.* Since $l\in sp(\{l_1,\ldots,l_t\})$, we know that $l=\sum\limits_{i\in [t]}\alpha_il_i$ for some scalars $\alpha_i\in {\mathbb{R}}$. All linear forms are $standard$ w.r.t. ${\mathcal{B}}\Rightarrow$ comparing the coefficients of $b_1$ we get that $\sum\limits_{i\in [t]}\alpha_i=1$ and therefore $l\in fl(\{l_1,\ldots,l_t\})$.
Let $T\subset {\mathbb{R}}^n$, By a scaling of $T$ we mean a set where all vectors get scaled (possibly by different scalars).
Reconstruction for low rank {#lowdimrecon}
============================
Let’s recall Definition \[rvalue\] following Theorem \[maintheorem\] in Section \[introduction\].
We fix $s$ to be any constant $> \max (C_{2k-1}+k, c_{\mathbb{R}}(4) )$ where :
1. $C_k = \frac{C^k}{\delta}$ the constant that appears in Theorem \[bdwy\].
2. $\delta$ is some fixed number in $(0,\frac{7-\sqrt{37}}{6})$.
3. $c_{\mathbb{R}}(4) = 3(4)^2 = 48$, is the rankbound needed for uniqueness of $\Sigma\Pi\Sigma(2)$ circuits as shown in Theorem \[uniqueness\].
Let $r$ be any constant $\geq s$ (In our application we need $s$ and $s+1$). Our main theorem for this section therefore is:
Let $r$ be as defined above. Consider $f({\bar{x}})\in
{\mathbb{R}}[{\bar{x}}]$, a multivariate homogeneous polynomial of degree $d$ over the variables ${\bar{x}}=(x_1,\ldots,x_r)$ which can be computed by a $\Sigma\Pi\Sigma_{\mathbb{R}}(2)[{\bar{x}}]$ circuit $C$. Assume that rank of the simplification of $C$ i.e. $Sim(C)=r$. We give a $poly(d)$ time randomized algorithm which computes $C$ given blackbox access to $f({\bar{x}})$.
\[randomtransformation\]
We assume $f$ has the following $\Sigma\Pi\Sigma_{\mathbb{R}}(2)[{\bar{x}}]$ representation: $$f = \tilde G(\tilde \alpha_0\tilde T_0 + \tilde \alpha_1\tilde T_1)$$ where $\tilde G,\tilde T_i\in\Pi\Sigma_{\mathbb{R}}[{\bar{x}}]$ are $normal$ (i.e. leading non-zero coefficient is $1$ in every linear factor) and $\tilde \alpha_0,\tilde \alpha_1\in {\mathbb{R}}$ with $gcd(\tilde T_0,\tilde T_1)=1$. The $rank(Sim(C))=r$ condition then becomes $$sp({\mathcal{L}}(\tilde T_0)\cup {\mathcal{L}}(\tilde T_1)) = Lin_{\mathbb{R}}[{\bar{x}}]$$
Consider the set $T = {\mathcal{L}}(\tilde G)\cup{\mathcal{L}}(\tilde T_0)\cup {\mathcal{L}}(\tilde T_1)$. By abuse of notation we will treat these linear forms also as points in ${\mathbb{R}}^r$. Since linear factors of $\tilde G, \tilde T_i$ are normal, two linear factors of $\tilde G, \tilde T_i$ are LD iff they are same.\
#### Random Transformation and Assumptions {#assumptions}
Let $\Omega,\Lambda$ be two $r\times r$ matrices such that their entries $\Omega_{i,j}$ and $\Lambda_{i,j}$ are picked independently from the uniform distribution on $[N]$. Here $N = 2^d$. We begin our algorithm by making a few assumptions. All of these assumptions are true with very high probability and we assume them in our algorithm. Consider the standard basis of ${\mathbb{R}}^r$ given as ${\mathcal{S}}= \{e_1,\ldots,e_r\}$. Let $E_j = sp(\{e_1,\ldots,e_j\})$ and $E_j^{\prime} = sp(\{e_{j+1},\ldots,e_r\})$, clearly ${\mathbb{R}}^r = E_j\oplus E_j^\prime$. Let $\pi_{W_{E_j}}$ be the orthogonal projection onto $E_j$ w.r.t. this decomposition.
- [**Assumption 0 :** ]{} $\Omega$ is invertible. This is just the complement of event ${\mathcal{E}}_0$ in Section \[randomtransform\] and so occurs with high probability.
- [**Assumption 1 :** ]{} For all $t\in T$, $\pi_{W_{E_1}}(\Omega(t))\neq 0$ i.e. $[\Omega(t)]^1_{{\mathcal{S}}}\neq 0$ (coefficient of $e_1$ is non-zero) . This is the complement of event ${\mathcal{E}}_1$ in Section \[randomtransform\] and so occurs with high probability.
- [**Assumption 2 :** ]{} For all LI sets $\{t_1,\ldots,t_r\}\subset T$, $\{\Omega(t_1),\ldots,\Omega(t_r)\}$ is LI. This essentially means that $\Omega$ is invertible. This is the complement of ${\mathcal{E}}_2$ in Section \[randomtransform\] and so occurs with high probability.
- [**Assumption 3 :** ]{} Fix a $k<r$. For all LI sets $\{t_1,\ldots,t_r\} \subset T, \{\Omega(t_1),\ldots,\Omega(t_k),\Lambda\Omega(t_{k+1}),
\ldots , \Lambda\Omega(t_d) \}$ is LI i.e. is a basis. This is the complement of event ${\mathcal{E}}_3$ in Section \[randomtransform\] and so occurs with high probability. It’ll be used later in this chapter.
- [**Assumption 4 :** ]{} Fix a $k<r$. For all LI sets $\tilde T = \{t_1,\ldots,t_r\}\subset T$ and define the set ${\mathcal{B}}= \{\Omega(t_1),\ldots,\Omega(t_k),\Lambda\Omega(t_{k+1}),\ldots,\Lambda\Omega(t_r)\}$. By Assumption $3$ this is a basis. Consider any $t\in T$ such that $\Omega(t) \notin sp(\{\Omega(t_1),\ldots,\Omega(t_k)\})$. Then $[\Omega(t)]^{k+1}_{\mathcal{B}}\neq 0$. This event is the complement of ${\mathcal{E}}_5$ and so it occurs with high probability.
From now onwards we will assume that all the above assumptions are true. Since all of them occur with very high probability, their complements occur with very low probability and by union bound the union of their complements is a low probability event. So intersection of the above assumptions occurs with high probability and we assume all of them are true. [**Note that the assumptions will continue to be true if we scale all linear forms ( possibly different scaling for different vectors, but non-zero scalars) in $T$ i.e. if the assumptions were true for $T$ then they would have been true had we started with a scaling of $T$.**]{}\
The first step of our algorithm is to apply $\Omega$ to $f$. We have a natural identification between linear forms and points in ${\mathbb{R}}^r$. This identification converts $\Omega$ into a linear map on $Lin_{\mathbb{R}}[{\bar{x}}]$ which can be further converted to a ring homomorphism on polynomials by assuming that it preserves the products and sums of polynomials. So $\Omega$ gets applied to all linear forms in the $\Sigma\Pi\Sigma(2)$ representation of $f$. Since $f$ is a degree $d$ polynomial in $r$ variables it has atmost $poly(d^r)$ coefficients. Applying $\Omega$ to each monomial and expanding it takes $poly(d^r)$ time and gives $poly(d^r)$ terms. So computing $\Omega(f)$ takes $poly(d^r)$ time and has $poly(d^r)$ monomials.\
Now we try and reconstruct the circuit for $\Omega(f)$. If this reconstruction can be done correctly, we can apply $\Omega^{-1}$ and get back $f$. Note that [**Assumption 1**]{} tells us that the coefficient of $x_1$ in $\Omega(l)$ is non-zero for all $l$ in $T$. Let $X = \{x_1,\ldots,x_r\}$ and ${\bar{x}}$ is used for the tuple $(x_1,\ldots,x_r)$. From this discussion we know that: $$\Omega(f) = \Omega(\tilde G)(\tilde \alpha_0\Omega(\tilde T_0) + \tilde \alpha_1\Omega(\tilde T_1)) = G(\alpha_0 T_0 + \alpha_1 T_1)$$ where $\alpha_i$ are chosen such that linear factors of $G,T_i$ have their first coefficient( that of $x_1$) equal to $1$. So they are $standard$ $\Pi\Sigma$ polynomials. Note that we’ve used [**Assumption $1$**]{} here. Since we’ve moved constants to make linear forms standard we can assume $G = \lambda\Omega(\tilde G), T_i = \lambda_i \Omega(\tilde T_i)$ with $\lambda,\lambda_i \in {\mathbb{R}}$. Consider some scaling $T_{sc}$ of $T$ such that ${\mathcal{X}}= {\mathcal{L}}(G)\cup {\mathcal{L}}(T_0)\cup {\mathcal{L}}(T_1)$ is $ = \Omega(T_{sc})$. All above assumptions are true for $T_{sc}$ and so we may use the conclusions about $\Omega(T_{sc})$ i.e. ${\mathcal{X}}$. Also since $\Omega$ is invertible $gcd(T_0,T_1)=1$.\
Let $T_i = \prod\limits_{j\in [M]} l_{ij}, i=0,1$ and $G = \prod\limits_{k\in[d-M]}G_k$, with $l_{ij},G_k \in Lin_{\mathbb{R}}[{\bar{x}}]$ (so $d = deg(f)$ ).\
For simplicity from now onwards we call $\Omega(f)$ by $f$ and try to reconstruct it’s circuit. Once this is done we may apply $\Omega^{-1}$ to all the linear forms in the gates and get the circuit for $f$. This step clearly takes $poly(d^r)$ time in the same way as applying $\Omega$ took.\
Since $r$ is a constant, the steps described above take $poly(d)$ time overall.
#### Known and Unknown Parts
We also define some other $\Pi\Sigma_{\mathbb{R}}[{\bar{x}}]$ polynomials $K_i,U_i, i=0,1$ which satisfy $$K_i\mid \alpha_iGT_i, U_i=\frac{\alpha_iGT_i}{K_i}.$$ with the extra condition $$gcd(K_i,U_i)=1.$$ $K_i$ are the known factors of $\alpha_iGT_i$ and $U_i$ the unknown factors. The $gcd$ condition just means that that known and unknown parts of $\alpha_iGT_i$ don’t have common factors. In other words linear forms in $\alpha_iGT_i$ are known with full multiplicity. We initialize $K_i=1$ and during the course of the algorithm update them as and when we recover more linear forms. At the end $K_i=\alpha_iGT_i$ and so we know both gates.
Outline of the algorithm
------------------------
1. \[candidateset\][**Set ${\mathcal{C}}$ of Candidate Linear Forms :**]{}
We compute a $poly(d)$ size set ${\mathcal{C}}$ of linear forms which contains ${\mathcal{L}}(T_i), i=0,1$. We will non-deterministically guess from this set ${\mathcal{C}}$ making only a constant number of guesses everytime(thus polynomial work overall). It is important to note that the uniqueness of our circuit guarantees that our answer if computed can always be tested to be right. For more details on this please see Appendix \[findcandidate\]. We also give an algorithm to construct this set. See Algorithm \[candidatealgo\].
2. \[partknown\]
\
So $T_{1-i}$ has a linear factor $l_{(1-i)1}$ such that $$\label{eq:1}
sp(\{l_{(1-i)1}\})\cap sp(U_{i})=\{0\}$$ Let $W = sp(\{l_{(1-i)1}\})$ and extend to a basis of $V$ and in the process obtain another subspace $W^\prime\subset V$ such that $W\oplus W^\prime = V$. We can see from Equation \[eq:1\] that LI linear forms in $U_{i}$ remain LI when we project to $W^\prime$. We use this to compute $U_i$ and then since $K_iU_i=\alpha_iGT_i$ we know one of the gates. To find the other gate simply factorize $f-\alpha_iGT_i$. If it factors into a product of linear forms we have the reconstruction.
3. \[dimensiongap\] -
\
This case is just to facilitate the Hard Case. We know that $T_{1-i}$ has two linear factors $l_{(1-i)1},l_{(1-i)2}$ such that $sp(\{l_{(1-i)1},l_{(1-i)2}\})\cap sp(T_i)=\{0\}$. We show that the only linear factors of $f$ are those which appear in $G$. So we can first factorize $f$ using Kaltofen’s factoring ([@KalTr90]) and obtain $G$. Update $K_j=G, j=0,1$. So $U_j = \alpha_jT_j$ for $j=0,1$. Clearly we also have ${\mathcal{L}}(T_{1-i})\subsetneq sp(T_i) = sp(U_i)$ and we can go to [**Easy Case** ]{} above with $K_i=G$.
4. \
We know that we are not in [**Medium Case**]{} and so $dim(sp(T_0)+
sp(T_1)) - sp(T_i)\leq 1$ for $i=0,1$. Also $dim(sp(T_0)+ sp(T_1)) = r$ by assumption on the simple rank of our polynomial. So this guarantees that $dim(sp(T_{1-i})) \geq r-1 \Rightarrow$ (by the condition of this hard case) $dim(sp(U_i))
\geq r-1$ for $i=0,1$. This enables us to use the Quantitative Sylvester Gallai theorems on both sets ${\mathcal{L}}(T_i), {\mathcal{L}}(U_i)$.
- Our first step is to identify a certain *“bad”* factor $I$ of $G$ and get rid of it to get $G^\star = \frac{G}{I}$ and thus $f^\star=\frac{f}{I}$. The factors of $I$ don’t satisfy certain properties we need later and so we remove them. Thankfully we have an efficient algorithm to recover $I$. Our algorithm uses something we call a Detector Pair (See \[detectorset\]) whose existence is shown using the Quantitative Sylvester Galai Theorems mentioned above.
- So now our job is to reconstruct $f^\star$ with known (and unknown resp.) parts as $K_0^\star,K_1^\star$ ($U_0^\star,U_1^\star$ resp.).
- If $sp(U_{1-i}^\star)$ becomes low dimensional we may fall in [**Easy Case**]{} and recover the circuit for $f^\star$ directly. Otherwise the same detector pairs then provide certain *“nice”* subspaces corresponding to linear forms in $T_{i}$. Projection of $U_{1-i}^\star$ onto these subspaces can be easily glued together to recover some linear factors(with multiplicities) of $U_{1-i}^\star$, which will then be multiplied to $K_{1-i}^\star$.
- The process continues as long as $sp(U_{1-i}^\star)$ remains high dimensional. As soon as this condition fails we end up in [**Easy Case** ]{} and the gates are recovered.
We give algorithms for [**Easy**]{} and [**Medium**]{} cases. [**Hard Case**]{} will require more prepration and will be done after these subsections. From now onwards we assume that we have constructed a $poly(d)$ sized set of linear forms ${\mathcal{C}}$ which contains ${\mathcal{L}}(T_i)$ for $i=0,1$. We have other structural results about linear forms in this set. See Appendix \[findcandidate\] for more details and algorithms. Algorithm \[candidatealgo\] constructs this set in $poly(d)$ time.
Easy Case {#partknowngate}
---------
Suppose for some $i\in \{0,1\}$, ${\mathcal{L}}(T_{1-i}) \subsetneq sp(U_{i})$ then we can reconstruct $f$.
#### Explanation and Correctness Analysis
- The first for loop just guesses the gate with extra dimensions i.e. it’s not contained in span of the unknown part of the other gate.
- If for some basis $\{l_1,\ldots,l_r\}\subset {\mathcal{C}}$ the algorithm actually computes a $\Sigma\Pi\Sigma(2)$ representation in the end then it ought to be correct since the last ’if’ also checks if it is correct.
- If our guess for $i$ is correct, we show that there exists a basis $\{l_1,\ldots,l_r\} \subset {\mathcal{C}}$ for which all conditions will be satisfied and we actually arrive at a $\Sigma\Pi\Sigma(2)$ representation in the end. Since ${\mathcal{L}}(T_{1-i})\subsetneq sp(U_i)$ and ${\mathcal{L}}(T_{1-i}),{\mathcal{L}}(U_i)\subset {\mathcal{C}}$ there exists $l_1\in {\mathcal{L}}(T_{1-i})\setminus sp(U_i)\subset {\mathcal{C}}$. Choose a basis $\{l_2,\ldots,l_s\}$ of $sp(U_i)$, then $\{l_1,\ldots,l_s\}$ is an LI set. Now extend this to a basis $\{l_1,\ldots,l_s,l_{s+1},\ldots,l_r\}\subset {\mathcal{C}}$ of $V$. We go over all choices of basis in ${\mathcal{C}}$ and will arrive at the right one.
- We initialize a dummy polynomial $K_i^\prime$ to represent $K_i$ since we do not want to update $K_i$ till we actually have a solution. Let’s assume $l_1 ^t \mid\mid f$ i.e. $l_1^t\mid f$ and $l_1^{t+1}\nmid f$. We know $l_1\mid T_{1-i}\Rightarrow l_1\nmid T_i
\Rightarrow l_1\nmid \alpha_iT_i + \alpha_{1-i}T_{1-i}$. Therefore $l_1^t\mid\mid G \Rightarrow l_1^t\mid\mid \alpha_iGT_i = K_iU_i$. Also $l_1
\notin sp(U_i) \Rightarrow l_1\nmid U_i$ thus $l_1^t \mid\mid K_i \Rightarrow l_1^t\mid\mid K_i^\prime$. We remove $l_1^t$ from both $f,K_i^\prime$ to get $\tilde{f}, \tilde{K_i}$. Let $W = sp(\{l_1\})$ and $W^\prime =
sp(\{l_2,\ldots,l_r\})$, therefore $V = W\oplus W^\prime$. Note that since $l_1\in {\mathcal{L}}(T_{1-i})$ $$\pi_{W^\prime}(\tilde{f}) = \pi_{W^\prime}(U_i)\pi_{W^\prime}(\tilde{K_i})$$ Since $\pi_{W^\prime}(\tilde{K_i})\neq 0$, we get $\pi_{W^\prime}(U_i)=\frac{\pi_{W^\prime}(\tilde{f})}{\pi_{W^\prime}(\tilde{K_i})}$. If $U_i = u_1\ldots u_s$ with $u_j\in W^\prime$, we see that $\pi_{W^\prime}(U_i) =
\pi_{W^\prime}(u_1)\ldots \pi_{W^\prime}(u_s) = u_1\ldots u_s = U_i$. So we get $U_i$ and hence $\alpha_iGT_i =K_iU_i$ . Once $\alpha_iGT_i$ is known we factorize $f-\alpha_i GT_i$ to get $\alpha_{1-i} GT_{1-i}$. For the correct choice of our basis this will factorize completely into a $\Pi\Sigma$ polynomial. Now we update $K_i = K_iU_i$ and $K_{1-i}=f-K_iU_i$ and an object $decomposition(f,K_0,K_1)$. Throughout the algorithm we use Kaltofen’s factoring [@KalTr90] wherever necessary.
- If we were not able to find the $\Sigma\Pi\Sigma(2)$ representation then we return an object $decomposition()$.
#### Time Complexity -
We can see above all loops run only $poly(d)$ many times. The most expensive step is choosing $~ r$ vectors from ${\mathcal{C}}$. But recall that $r$ is a constant and so this also takes only polynomial time in $d$. Other steps like factoring polynomials (using Kaltofen’s factoring algorithm from [@KalTr90]), taking projection onto known subspaces, divding by polynomials require $poly(d)$ time ($r$ is a constant) as has been explained multiple times before.\
Medium Case {#mediumcase}
-----------
If $dim(\sfrac{sp(T_{1-i}) + sp(T_i)}{sp(T_i)} )\geq 2$ then ${\mathcal{L}}(\alpha_iT_i+\alpha_{1-i}T_{1-i})=\phi$.
*Proof.* $dim(\sfrac{sp(T_{1-i}) + sp(T_i)}{sp(T_i)} )\geq 2\Rightarrow$, there exists $l_1^\prime,l_2^\prime\in {\mathcal{L}}(T_{1-i})\setminus sp(T_{i})$ be such that $dim(\{l_1^\prime,l_2^\prime\}\cup {\mathcal{L}}(T_{i}))= dim({\mathcal{L}}(T_{i}))+2$. Assume there exist $l\in {\mathcal{L}}(\alpha_iT_i+ \alpha_{1-i}T_{1-i})$.
$$l \mid \alpha_iT_i+ \alpha_{1-i}T_{1-i} \Rightarrow l \nmid T_i \text{ and }
l\nmid T_{1-i} \text{ (since they are coprime) }$$
$$0\neq \alpha_i\prod\limits_{j\in[M]}l_{ij} =
-\alpha_{1-i}\prod\limits_{j\in[M]}l_{(1-i)j} \pmod{\{l\}}.$$
Thus there exist $l_1,l_2\in {\mathcal{L}}(T_i)$ and scalars $\gamma_j,\delta_j, j\in [2]$ such that $l = \gamma_j l_{j} + \delta_j l_{j}^\prime$. Since $l\nmid T_0,l\nmid T_1$ we get $\gamma_j,\delta_j$ are non zero.
$\delta_1,\delta_2\neq 0 \Rightarrow$, $$l_1^\prime,l_2^\prime\in sp(\{l\}\cup {\mathcal{L}}(T_{i}))\Rightarrow
dim(\{l_1^\prime,l_2^\prime\}\cup {\mathcal{L}}(T_{i})) \leq dim({\mathcal{L}}(T_{i}))+1$$ which is a contradiction. So ${\mathcal{L}}(\alpha_iT_i+ \alpha_{1-i}T_{1-i})=\phi$.
Therefore the only linear factors of $f$ are present in $G$, which can now be correctly found by using Kaltofen’s algorithm [@KalTr90] and identifying the linear factors. Update $K_j=G$ for $j=0,1$, therefore $U_j=T_j$. Also this case implies that ${\mathcal{L}}(T_{1-i})\subsetneq sp(T_i)=sp(U_i)$. and so we can use Easy Case.
So we have the following claim:
If the condition in Medium Case is true, the following algorithm reconstructs $f$, if there is a reconstruction.
$L\gets Lin(f)$ ()[EasyCase$(f,L,L,{\mathcal{C}})\rightarrow$ iscorrect]{}[\[lt\] ]{}
The above algorithm does exactly what has been explained in the preceeding paragraph. It works in $poly(d)$ time if EasyCase$(f,K_0,K_1,{\mathcal{C}})$ works in $poly(d)$ time. Kaltofen’s factoring and all other steps are $poly(d)$ time.\
Now we need to handle the [**Hard Case**]{}. This is quite technical and so we do some more preparation. We devise a technique to get rid of some factors of $f$ to get a new polynomial $f^\star$ without destroying the $\Sigma\Pi\Sigma(2)$ structure. If Easy Case holds for $f^\star$ we stop there itself. Otherwise we will use combination of different subspaces of $V$, project $f^\star$ onto them and glue projections to get gates for $f^\star$.
Detector Pair, Reducing Factors, Hard Case Preparation {#reducefactors}
------------------------------------------------------
Let’s recall: $$g = \frac{f}{G} = \alpha_0T_0+\alpha_1T_1$$
We outline an approach to identify some factors of $f$. These factors will divide $G$ but won’t divide $g$. This is going to be useful in the Hard Case. The linear factors left after removing these identified factors will have very strong structural properties and so will be instrumental in reconstruction. The main tool in this identification is a pair $(S,D)$ (defined below) inside one of the ${\mathcal{L}}(T_i)$’s. This pair will be called a *“Detector Pair”*. It will also decide the subspaces on which we take projections of $f$ and glue back to get the gates.
#### Detector Pairs $(S,D)$ {#detectorset}
Fix $k=c_{{\mathbb{R}}}(3)+2$ (See Theorem \[rankbound\] for definition of $c_{\mathbb{R}}(m)$). Let $S = \{l_{1},\ldots,l_{k}\} \subset {\mathcal{L}}(T_i)$ be an LI set of linear forms. Let $D(\neq \phi)\subseteq {\mathcal{L}}(T_i)$ .We say that $(S,D)$ is a *“Detector Pair”* in ${\mathcal{L}}(T_i)$ if the following are satisfied for all $l_{k+1}
\in D$:
- $\{l_{1},\ldots,l_{k},l_{k+1}\}$ is an LI set. Let $\mathcal{F} =
fl(\{l_{1},\ldots,l_{k},l_{k+1}\})$. $\mathcal{F}$ is elementary in ${\mathcal{L}}(T_i)$ i.e. $\mathcal{F}\cap {\mathcal{L}}(T_i) = \{l_{1},\ldots,l_{k},l_{k+1}\}$. See Definition \[elementaryset\].
- $\mathcal{F}\cap {\mathcal{L}}(T_{1-i}) \subseteq fl(\{l_{1},\ldots,l_{k}\})$ i.e. $\mathcal{F}$ contains only those points from ${\mathcal{L}}(T_{1-i})$ which lie inside $fl(\{l_{1},\ldots,l_{k}\})$.
### Identifying Some Factors Which Don’t Divide $g$ {#identification}
The two claims below give results about structure of linear forms which divide $g$. The proofs are easy but technical and so we move them to the appendix.
\[spuriousli\] Let $(S = \{l_{1}\ldots, l_{k}\},D)$ be a Detector set in ${\mathcal{L}}(T_i)$. Let $l_{k+1}\in D$. For a $standard$ linear form $l\in V$, if $l\mid g$ then $l\notin sp(\{l_{1},\ldots,l_{k}\})$ .
*Proof.* See \[spuriousliproof\] in Appendix
\[spuriousextra\] Let $l \in Lin_{\mathbb{R}}[{\bar{x}}]$ be $standard$ such that $l \mid g$ and ${\mathcal{C}}$ be the candidate set. Assume $(S = \{l_{1},\ldots,l_{k}\}, D(\neq \phi))$ is a Detector pair in ${\mathcal{L}}(T_i)$. Then $|{\mathcal{L}}(T_{1-i}) \cap (fl(S\cup\{l\}) \setminus
fl(S))|\geq 2$. That is the flat $fl(\{l_{1},\ldots,l_{k},l\})$ contains atleast two distinct points from ${\mathcal{L}}(T_{1-i})(\subseteq {\mathcal{C}})$ outside $fl(\{l_1,\ldots,l_k\})$.
*Proof.* See \[spuriousextraproof\] in Appendix
\[identifylinform\] Suppose $(S=\{l_{1},\ldots,l_{k}\},D(\neq \phi))$ is a Detector Pair in ${\mathcal{L}}(T_i)$. The following algorithm identifies some factors in ${\mathcal{L}}(G)\setminus {\mathcal{L}}(g)$. It returns the product of all linear forms identified.
$= 1$, $bool$ $flag$
*Proof.* The proof of the claim is a part of Lemma \[filteredfactor\] below.
#### Time Complexity -
Since ${\mathcal{C}}$ has size $poly(d)$ and $deg(f)=d$, the nested loops run $poly(d)$ times. $k,r$ are constants so checking linear independence of $k+1$ linear forms in $r$ variables takes constant time. Checking if some vectors belong to a $k+1$ dimensional space also takes constant time. Multiplying linear forms to [**I**]{} takes $poly(d)$ time. So overall the algorithm runs in $poly(d)$ time.\
So the above algorithm identified a factor [**I**]{} of $G$ for us. Let us define new polynomials $$G^\star = \frac{G}{{\bf I}} = \prod\limits_{t\in [N_1]}G_t$$ and $$f^\star = \frac{f}{{\bf I}} = G^\star (\alpha_0T_0 + \alpha_1T_1)$$
\[filteredfactor\] The following are true:
1. If $l\mid I$ (i.e. $l$ was identified) then $l\in {\mathcal{L}}(G)\setminus
{\mathcal{L}}(g) $.
2. \[retainedfactor\]If $l\mid G^\star$ (i.e. $l$ was retained) then $(fl(\{l_{1},\ldots,l_{k},l\})\setminus fl(\{l_{1},\ldots,l_{k}\})) \cap
({\mathcal{L}}(T_{1-i})\cup ({\mathcal{L}}(T_i)\setminus D)) \neq \phi $ that is:
$(fl(\{l_{1},\ldots,l_{k},l\})\setminus fl(\{l_{1},\ldots,l_{k}\}))$ contains a point from ${\mathcal{L}}(T_i)\setminus D$ or ${\mathcal{L}}(T_{1-i})$.
3. \[retaineddetector\] If $l\mid G^\star$ and $l_{k+1}\in D$ then $l \notin sp(\{l_{1},\ldots,l_{k},l_{k+1}\})$.
*Proof.* See \[prooffilteredfactor\] in Appendix.
### Overestimating the set $D$ of the detector pair $(S,D)$
Lemma \[filteredfactor\] is going to help us actually find an overestimate of $D$ corresponding to $S=\{l_{1},\ldots,l_{k}\}$ in the detector pair $(S,D)$ as described in the lemma below. This will be important since we need $D$ during our algorithm for the Hard Case.
\[detectorexpansion\] Let $(S=\{l_{1},\ldots,l_{k}\},D)$ be a detector in ${\mathcal{L}}(T_i)$. For each $(l,l_j) \in {\mathcal{C}}\times S$ define the space $U_{\{l,l_j\}} = sp(\{l,l_j\})$. Extend $\{l,l_j\}$ to a basis and in the process obtain $U_{\{l,l_j\}}^\prime$ such that $V = U_{\{l,l_j\}}\oplus U_{\{l,l_j\}}^\prime$. Define the set: $$X = \{l\in {\mathcal{C}}: \pi_{U^\prime_{\{l,l_j\}}}(f^\star) \neq 0, \text{ for all }
l_j\in S\}$$ Then $D\subset X\subset {\mathcal{L}}(T_i)$.
*Proof.* See \[detectorexpansionproof\] in Appendix.
This set $X$ is an overestimate of $D$ inside ${\mathcal{L}}(T_i)$ and also easy to compute. Given $S$ we may easily construct $X$ in time $poly(d)$ because of it’s simple description. Let’s give an algorithm to compute $X$ given $f^\star,S,{\mathcal{C}}$.
The following algorithm computes the overestimate $X$ of $D$ as discussed above
\[overestimatedetector\] $bool$ $flag$ Define $X\gets \phi$
#### Time Complexity -
Inside the inner for loop we look for $(r-2)$ linear forms from ${\mathcal{C}}$. $|{\mathcal{C}}| = poly(d)$ and $r$ is a constant and so this step only needs $poly(d)$ time. The nested loops run polynomially many times. Checking linear independece of $r$ linear forms and projecting to known constant dimensional subspaces also take $poly(d)$ time as has been discussed before. So the algorithm runs in $poly(d)$ time.
Hard Case {#hardcase}
----------
#### {#hardcasediss}
This Subsection will involve the most non trivial ideas. We handled $dim(\sfrac{sp(T_{1-i}) + sp(T_i)}{sp(T_i)} )\geq 2$ in the Medium Case (see Subsection \[mediumcase\]) completely, so let’s assume $dim(\sfrac{sp(T_{1-i}) + sp(T_i)}{sp(T_i)} )\leq
1\Rightarrow dim({\mathcal{L}}(T_{1-i})\cup {\mathcal{L}}(T_i))\leq dim({\mathcal{L}}(T_i))+1$ for both $i=0,1$. We already know that $rank(f)=r,$ implying $dim({\mathcal{L}}(T_i)\cup
{\mathcal{L}}(T_{1-i}))=r$. Thus for $i=0,1$; $dim({\mathcal{L}}(T_i))\geq r-1$. This works in our favour for applying the quantitative version of the Sylvester Gallai theorems given in [@BDWY11]. To be precise we will use Corollary \[bichromatic\] from Appendix \[incidence\] in this paper.
1. Our first application (See Lemma \[largedetector\]) of Quantitative Sylvester Gallai will help us prove the existence of a Detector pair $(S=\{l_{1},\ldots,l_{k}\},D)$ in ${\mathcal{L}}(T_i)$ with $k=c_{\mathbb{R}}(3)+2$ (See defn of $c_{\mathbb{R}}(.)$ in Theorem \[rankbound\]) and large size of $D$. For this we will only need $dim({\mathcal{L}}(T_i))\geq C_{2k-1}$ for $i=0,1$(See Appendix \[incidence\] for definition of $C_{2k-1}$). From Definition \[rvalue\] we know that this is true with $k =c_{\mathbb{R}}(3) +2$.
2. The above point shows the existence of a detector pair $(S,D)$ in ${\mathcal{L}}(T_i)$ with large $|D|$. So now we go back to Subsection \[reducefactors\] and remove some factors of $f$ to get $f^\star = G^\star(\alpha_0T_0 + \alpha_1T_1)$ such that linear factors of $G^\star$ satisfy properties given in Lemma \[filteredfactor\]. We also compute the overestimate $X$ of $D$ using Algorithm \[overestimatedetector\]. Let the known and unknown parts of $f^\star$ be $K_0^\star,K_1^\star$ and $U_0^\star, U_1^\star$ respectively. If for some $i\in \{0,1\}$, ${\mathcal{L}}(T_i)\subsetneq sp(U_{1-i}^\star)$ then we are in Easy Case for $f^\star$ and can recover the gates for $f^\star$. Otherwise for both $i=0,1;$ ${\mathcal{L}}(T_i)\subseteq sp(U_{1-i}^\star)
\Rightarrow dim({\mathcal{L}}(U_{1-i}^\star))\geq r-1$ and we continue with reconstruction below.
3. Next to actually reconstruct linear forms in $U_{1-i}^\star$, we will use it’s high-dimensionality ($\geq r-1\geq C_{2k-1}$) discussed above. Corollary \[bichromatic\] from Section \[incidence\] will enable us to prove the existence of a $d_1\in D$ which together with the set $S$ found above will give the existence of a *“Reconstructor”*( See Claim \[reconalgoclaim\] and Algorithm \[reconalgo\]) which recovers some linear factors of $U_{1-i}^\star$ with multiplicity (See Theorem \[foundreconstructor\]) .
### Large Size of Detector Sets
w.l.o.g. we assume $|{\mathcal{L}}(T_0)|\leq |{\mathcal{L}}(T_1)|$. First we point out a simple calculation that will be needed later. For $\delta \in (0, \frac{7-\sqrt{37}}{6})$ and $\theta\in (
\frac{3\delta}{1-\delta}, 1-3\delta )$, let $v(\delta ,\theta)$ be defined as follows: $$v(\delta,\theta) =
\begin{cases}
\hfill 1-\delta-\theta \hfill & \text{ if $|{\mathcal{L}}(T_{0})|\leq \theta
|{\mathcal{L}}(T_1)|$} \\
\hfill (1-\delta)(1+\theta)-1 \hfill & \text{ if $\theta |{\mathcal{L}}(T_1)| <
|{\mathcal{L}}(T_0)| \leq |{\mathcal{L}}(T_1)|$} \\
\end{cases}$$
\[calculation\] The following is true
$$\frac{(2-v(\delta,\theta))}{v(\delta,\theta)}\leq \frac{1-\delta}{\delta}$$
*Proof.* See \[calculationproof\] in Appendix.
\[largedetector\] Let $k=c_{{\mathbb{R}}}(3)+2$ (see defn of $c_{\mathbb{R}}(m)$ in Theorem \[rankbound\]). Fix $\delta, \theta$ in range given in Claim \[calculation\] above . Then for some $i\in \{0,1\}$ there exists a Detector $(S=\{l_{1},\ldots,l_{k}\},D)$ in ${\mathcal{L}}(T_i)$ with $|D|\geq v(\delta,\theta) \max(|{\mathcal{L}}(T_{0})|,|{\mathcal{L}}(T_{1})|)$.
*Proof.* See \[largedetectorproof\] in Appendix.
### Assuming ${\mathcal{L}}(T_i)\subseteq sp({\mathcal{L}}(U_{1-i}^\star))$ and reconstructing factors of $U_{1-i}^\star$ {#getreconstructor}
Let’s begin by stating our main reconstruction theorem for this Subsubsection. We will go through several steps to prove it:
\[foundreconstructor\] There exist pairwise disjoint LI sets $S_0,S_1,S_2$ with $S_0\cup S_1\cup
S_2$ being a basis of $V = Lin_{\mathbb{R}}[x_1,\ldots,x_r] \simeq {\mathbb{R}}^r$, and non constant polynomials $P,Q$ dividing $ U_{1-i}^\star$ such that $P\mid Q$ and $(Q,P,S_0,S_1,S_2)$ is a Reconstructor.
Once we know this result we actually recover $P$ by computing $\pi_{W_0^\prime}(Q)$ and $\pi_{W_1^\prime}(Q)$ and then using Algorithm \[reconalgo\]. We state this in the following corollary. Proof is given as Algorithm \[hardcasealgo\]
\[findprojections\] Using $f, K_{1-i}, S_0,S_1,S_2$ from above we can compute $\pi_{W_0^\prime}(Q),
\pi_{W_1^\prime}(Q)$ for $Q$ defined in the proof above.
Before going to the proof let’s do some more more preparation.
#### {#lambda}
Consider the set of linear forms (points) ${\mathcal{X}}= {\mathcal{L}}(G^\star)\cup {\mathcal{L}}(T_0)\cup {\mathcal{L}}(T_1)$. We know that $sp({\mathcal{X}}) = V = Lin_{\mathbb{R}}[{\bar{x}}]\simeq {\mathbb{R}}^r$ (By abuse of notation we will use linear forms as points in ${\mathbb{R}}^r$ wherever required). Let $(S_0 = \{l_{1},\ldots,l_{k}\},D)$ be a detector in ${\mathcal{L}}(T_i)$ with $|D|\geq
v(\delta,\theta)\max(|{\mathcal{L}}(T_{0})|, |{\mathcal{L}}(T_{1})|)$ and $W_0 = sp(S_0)$. Extend $S_0$ to a basis $\{l_1,\ldots,l_k,l_{k+1}^\prime,\ldots,l_r^\prime\}$. Now it’s time to use the other random matrix $\Lambda$. Since we had applied $\Omega$ in the beginning, $\{\Omega^{-1}(l_1),\ldots,\Omega^{-1}(l_k)\}$ are linear forms in our input polynomial for this section. By [**Assumption 3** ]{} we know that the set $\{\Omega (\Omega^{-1}l_1),\ldots , \Omega (\Omega^{-1}l_k), \Lambda \Omega(\Omega^{-1}l_{k+1}^\prime), \ldots, \Lambda \Omega(\Omega^{-1}l_{r}^\prime)
\}$ is LI. Let $l_j = \Lambda l_j^\prime, j\in \{k+1,\ldots,r\}$. So ${\mathcal{B}}= \{l_1,\ldots,l_r\}$ is a basis. and define $\tilde{W_0} = sp(\{l_{k+1},\ldots,l_r\})$. Clearly $V = W_0\oplus \tilde{W_0}$. Also by [**Assumption 4**]{} for any $l\in {\mathcal{X}}\setminus W_0$, $[l]^{k+1}_{{\mathcal{B}}} \neq 0$. We define a normalization for linear forms $l\in {\mathcal{X}}$ : $$\widehat{l} = \left\{
\begin{array}{lr}
\frac{1}{[l]^{k+1}_{\mathcal{B}}}l & : l\in W_0^c\cap {\mathcal{X}}\\
0 & : l\in W_0\cap {\mathcal{X}}\end{array}
\right.$$
i.e. normalize the $(k+1)^{th}$ co-ordinate w.r.t. the basis ${\mathcal{B}}$. For any subset $\mathcal{T}\subset {\mathcal{X}}$, we define : $$\widehat{\mathcal{T}} = \{ \widehat l : l\in \mathcal{T} \}\setminus \{0\}$$ With this notation we proceed towards detecting linear factors of the unknown parts. But first let’s show that even after projecting onto $\tilde{W_0}$, the detector is larger in size (upto a function of $\delta$) compared to one of the unknown parts.
\[findreconstructor\] The following are true:
1. $dim(\pi_{\tilde{W_0}}(\widehat{{\mathcal{L}}(U_{1-i}^\star)}))> C_4$
2. $\pi_{\tilde{W_0}}(\widehat{{\mathcal{L}}(U_{1-i}^\star)})\cap \pi_{\tilde{W_0}}(\widehat{D}) = \phi$
3. $|\pi_{\tilde{W_0}}(\widehat{{\mathcal{L}}(U_{1-i}^\star)})| \leq
\frac{1-\delta}{\delta}|\pi_{\tilde{W_0}}(\widehat{D})|$
*Proof.* See \[findreconstructorproof\] Appendix.
This Lemma enables us to apply Lemma \[bichromatic\] from Section \[incidence\]. Consider the sets $\pi_{\tilde{W_0}}(\widehat{{\mathcal{L}}(U_{1-i}^\star)})$ and $\pi_{\tilde{W_0}}(\widehat{D})$. We’ve shown above that they are disjoint, span high enough dimension and $$|\pi_{\tilde{W_0}}(\widehat{{\mathcal{L}}(U_{1-i}^\star)})| \leq
\frac{1-\delta}{\delta}|\pi_{\tilde{W_0}}(\widehat {D})|$$ Lemma \[bichromatic\] shows the existence of a line $\vec L_1$ (called a *“semiordinary bichromatic”* line) in $\tilde{W_0}$ such that $|\vec L_1\cap \pi_{\tilde{W_0}}(\widehat{{\mathcal{L}}(U_{1-i}^\star)})|=1$ and $|\vec L_1\cap
\pi_{\tilde{W_0}}(\widehat{D})|\geq 1$.\
Finally it’s time to give the proof of Theorem \[foundreconstructor\].\
*Proof of Theorem \[foundreconstructor\].* We do this in steps:
- Let $S_0,S_1,S_2$ be as defined in the discussion above.
- Let $Q$ be the largest factor of $U_{1-i}^\star$ such that for all linear forms $q\mid Q$, $\pi_{W_2}(q)\neq
0$. So $\pi_{W_2}(Q)\neq 0$ and if $u^\star\mid\frac{U_{1-i}^\star}{Q}$ is a linear form then $\pi_{W_2}(u^\star)=0$. Let $P$ be the $\Pi\Sigma$ polynomial with the largest possible degree such that for all linear factors $p$ of $P$, $\pi_{W_0^\prime}(\widetilde p) =
\pi_{W_0^\prime}(\widetilde u)$ (which was a non zero vector on $\vec L_2$). Since $\pi_{W_0^\prime}(\widetilde u)$ and $\pi_{W_0^\prime}(\widetilde d_1)$ were LI this also means that $\pi_{W_2}(u)\neq 0 \Rightarrow \pi_{W_2}(p)\neq 0$ for all $p\mid P$. Clearly $P$ is non constant since $u\mid
P$, also by definition $P\mid Q$. Then $(Q,P,S_0,S_1,S_2)$ is a *Reconstructor* (See Subsection \[Identifier\] for definition) for $P$. Let’s check this is true:
- $\pi_{W_2}(Q) \neq 0$ - By definition of $Q$ we know this for all it’s factors and therefore for $Q$ itself.
- $\pi_{W_0^\prime}(P) = \delta (\pi_{W_0^\prime}(\widetilde u))^t$, for some $\delta\in {\mathbb{R}}$ (by definition of $P$).
- Let $q\mid \frac{Q}{P}$ such that $gcd(\pi_{W_2}(P), \pi_{W_2}(q)) \neq 1 \Rightarrow$ there exists some linear factor $p\mid P$ such that $\pi_{W_2}(p), \pi_{W_2}(q)$ are LD. $\{\pi_{W_2}(p), \pi_{W_2}(q)\}$ are LD and non-zero implies $q\in sp(\{l_1,\ldots,l_k,d_1,p\})$.
$$\Rightarrow
\pi_{W_0^\prime}(q)\in
sp(\{\pi_{W_0^\prime}(d_1),\pi_{W_0^\prime}(p)\})=
sp(\{d_1,\pi_{W_0^\prime}(\widetilde u)\})$$ So clearly : $$\pi_{W_0^\prime}(\widetilde q)\in sp(\{d_1,\pi_{W_0^\prime}(\widetilde u)\})$$ Since coefficient of $d_1$ in $\pi_{W_0^\prime}(\widetilde q),d_1,$ and $\pi_{W_0^\prime}(\widetilde u)$ is $1$, it’s easy to see that $\pi_{W_0^\prime}(\widetilde q)\in fl(\{d_1,\pi_{W_0^\prime}(\widetilde u)\}) =
\vec L_2$. Since $Q\mid U_{1-i}^\star$ we have $\pi_{W_0^\prime}(\widetilde q) \in
\pi_{W_0^\prime}(\widetilde {{\mathcal{L}}(U_{1-i}^\star)})\Rightarrow \pi_{W_0^\prime}(\widetilde q) \in
\vec L_2\cap \pi_{W_0^\prime}(\widetilde{{\mathcal{L}}(U_{1-i}^\star)}) =
\{\pi_{W_0^\prime}(\widetilde u)\}$. So $\pi_{W_0^\prime}(\widetilde q) = \pi_{W_0^\prime}(\widetilde u)$ which can’t be true since $P$ is the largest polynomial dividing $Q$ where linear factors have this property and $q\nmid P$. So such a $q$ does not exist.
The $r\times r$ matrix $\Lambda$ with enries picked independently (and independent of entries of $\Omega$) from the uniform distribution on $[N]$ is also sent as an input. Fix $k = c_{\mathbb{R}}(3) +2$.\
#### **Correctness**
Let’s assume we returned an object $obj$ of type decomposition.
1. [**If $obj\rightarrow iscorrect ==true$ :**]{} then we ought to be right since we check if $obj\rightarrow f=obj\rightarrow M_0 + obj\rightarrow M_1$. Since the representation is unique this will be the correct answer.
2. [**If $obj\rightarrow iscorrect ==false$:**]{} Let’s assume $f$ actually has a $\Sigma\Pi\Sigma(2)$ representation. If we were in Easy Case or Medium Case we would have already found the circuit using their algorithms. So we are in the Hard Case. So by Lemma \[largedetector\] there exists $i$ such that ${\mathcal{L}}(T_i)$ has a detector pair $(S_0,D)$ with $|D|$ large. For this $i$ there exists such an $S_0$, so sometime during the algorithm we would have guessed the correct $i$ and the correct $S_0$. [**Now let’s analyze what happens inside the while and the third for loop when the first two guesses are correct.**]{} Note that this also implies that the $I$ we have identified is correct and now we need to solve for $$f^\star = G^\star(\alpha_0 T_0 + \alpha_1 T_1)$$ Let $K_0^\star,K_1^\star$ (initialized to $1$) be the known parts of the gates for this polynomial $f^\star$ and $U_0^\star,U_1^\star$ be the unknown parts. Note that $T_0,T_1$ are same for both polynomials so $rank(f^\star) = rank(f)$ and for $j=0,1;$ $K_j\mid G^\star T_j$.\
[**Assume till the $m^{th}$ iteration of the while loop $K_{t}^\star \mid G^\star T_{t}$ for $t\in \{0,1\}$, we show that after the $(m+1)^{th}$ iteration, this property continues to hold and $deg(K_{1-i}^\star)$ increases.**]{}
- If after the $m^{th}$ iteration of the while loop for some $j\in \{0,1\}$, ${\mathcal{L}}(T_j)\subsetneq sp({\mathcal{L}}(U_{1-j}^\star))$ we are in Easy Case for $f^\star$ . The first step in while loop is to call EasyCase$(f^\star,{\mathcal{C}},K_0^\star,K_1^\star)$. This will clearly recover the circuit for $f^\star$ and return true since $K_{t}^\star \mid G^\star T_{t}$ for $t\in \{0,1\}$. However this does not happen so for both $j=0,1$, we have ${\mathcal{L}}(T_i)\subsetneq {\mathcal{L}}(U_{1-i}^\star)$. This means that we can use the ideas in Subsection \[getreconstructor\], specifically Theorem \[foundreconstructor\].
- The first two guesses are correct imply that $D\subseteq X\subseteq {\mathcal{L}}(T_i)$.
- If $d$ gets rejected then $K_t, t\in\{0,1\}$ remain unchanged. If some $d_1$ does not get rejected then since $d_1\in {\mathcal{L}}(T_i)$, $Q_0 = \pi_{V_1^\prime}(U_{1-i}^\star)$ is a non zero $\Pi\Sigma$ polynomial. Then some factors (the ones $\in W_2^\prime$) are removed from $Q_0$. Also on projecting to $W_0^\prime$ this still remains non-zero (as $d_1$ was not rejected).
- We know that $d_1\in {\mathcal{L}}(T_i)$ and $d_1$ not getting rejected implies that $Q_1 =\pi_{W_1^\prime}(U_{1-i}^\star) $ is a non-zero $\Pi\Sigma$ polynomial. We again remove some factors (i.e. the ones in $W_2^\prime$) from $Q_1$. The non-zeroness of $Q_0,Q_1$ imply that $Q_0= \pi_{W_1^\prime}(Q), Q_1= \pi_{W_1^\prime}(Q)$ i.e. they are projections of the same polynomial $Q$ which is the largest factor of $U_{1-i}^\star$ with the property that any linear form $q\mid Q$ is not in $W_2^\prime=W_0\oplus W_1$.
- $d_1$ was not rejected implies that $Reconstructor(Q_0,Q_1,S_0,S_1,S_2)$ returned a non-trivial polynomial $P$. This has to be a factor of $Q$ by Claim \[reconalgoconverse\] following Algorithm \[reconalgo\] and therefore a factor of $U_{1-i}^\star$.
- Proof of Theorem \[foundreconstructor\] implies that in every iteration atleast some $d_1$ will not be rejected.
- So clearly the new $K_{1-i}^\star = K_{1-i}^\star \times P$ divides $G^\star T_{1-i}$. $K_i$ remains unchanged. Therefore even after the $(m+1)^{th}$ iteration $K_t\mid G^\star T_t$ for both $j=0,1$ but degree of $K_{1-i}^\star$ increases.
So the while loop cannot run more than $deg(f^\star)$ times and in the end $G^\star T_{1-i}$ will be reconstructed completely and correctly and we should have returned $obj$ with $obj\rightarrow iscorrect=true$. Therefore we have a contradiction and so $f$ did not have a $\Sigma\Pi\Sigma(2)$ circuit and we correctly returned false.
#### Running Time
- First for loop runs twice.
- Inside it chossing $r$ linear forms from ${\mathcal{C}}$ ($|{\mathcal{C}}| = poly(d)$) takes $poly(d)$ time.
- Applying $\Lambda$ to $r-k$ vectors takes $poly(r) = O(1)$ time.
- Checking if a set of size $r$ inside ${\mathbb{R}}^r$ is LI takes $poly(r)=O(1)$ time since it is equivalent to computing determinant.
- $IdentifyFactors()$ takes $poly(d)$ time and computing $f^\star$ also takes $poly(d)$ time.
- $OverestDetector()$ runs in $poly(d)$ time.
- while loop runs atmost $d$ times
- $EasyCase$ runs in $poly(d)$ time and so does polynomial multiplication.
- $X\subseteq {\mathcal{L}}(T_i)$ and $|{\mathcal{L}}(T_i)|\leq deg(f^\star)$ and so for loop runs $d$ times.
- Change of bases in ${\mathbb{R}}^r$ and application to a polynomial of degree $d$ takes $poly(d)$ time.
- Therefore projecting to subspaces also takes $poly(d)$ time.
- $Reconstructor()$ runs in $poly(d)$ time (since $r$ is a constant) and so does polynomial multiplication and factoring by [@KalTr90].
Since all of the above steps run in $poly(d)$ time, nesting them a constant number of times also takes $poly(d)$ time. Therefore the running time of our algorithm is $poly(d)$.
Algorithm including all cases :
--------------------------------
The algorithm we give here will be the final algorithm for rank $r$ $\Sigma\Pi\Sigma$ polynomials. It will use the previous three cases. Our input will be a $\Sigma\Pi\Sigma(2)$ polynomial $f(x_1,\ldots,x_r)$ and output will be a circuit computing the same.
$decomposition$ obj $(\Omega_{i,j}),(\Lambda_{i,j})$, $r\times r$ matrices with entries chosen uniformly randomly from $[N]$ $L_i({\bar{x}}) \gets \sum\limits_{k=1}^r \Omega_{i,k}x_k$ $f(x_1,\ldots,x_r) \gets f(L_1({\bar{x}}),\ldots,L_r({\bar{x}}))$ ${\mathcal{C}}\gets Candidates(f(x_1,\ldots,x_r))$ Apply $\Omega^{-1}$ to obj$\rightarrow f,$ obj$\rightarrow M_0, $ obj$\rightarrow M_1$
#### Explanation :
Here we explain every step of the given algorithm:
- The function RECONSTRUCT$(f)$ takes as input a polynomial $f\in \Sigma\Pi\Sigma_{\mathbb{R}}(2)[{\bar{x}}]$ of $rank=r$ and outputs two polynomials $K_0,K_1\in \Pi\Sigma_{\mathbb{R}}[{\bar{x}}]$ which are the two gates in it’s circuit representation.
- Steps $2,3$ picks a random matrix $\Omega$ and transforms each variable using the linear transformation this matrix defines. With high probability this will be an invertible transformation. We do the reconstruction for this new polynomial since the linear factors of it’s gates satisfy some non-degenerate conditions(because they have been randomly transformed) our algorithm needs. We apply $\Omega^{-1}$ after the reconstruction and get back our original $f$.
- The next step constructs the set of candidate linear forms ${\mathcal{C}}$. We’ve talked about the size, construction and structure of this set in Section \[findcandidate\].
- We first assume Medium Case. It that was not the case we check for Easy Case . If both did not occur we can be sure we are in the Hard case.
- We apply $\Omega^{-1}$ to polynomials in obj and return it.
Reconstruction for arbitrary $rank$ {#highdimrecon}
===================================
This section reduces the problem from $\Sigma\Pi\Sigma(2)$ Circuits with arbitrary rank $n$ ($> r$) to one with constant rank ($= r$). Also once the problem has been solved efficiently in the low rank case we use multiple instances of such solutions to lift to the general $\Sigma\Pi\Sigma(2)$ circuit. The idea is to project the polynomial to a small (polynomial) number of random subspaces of dimension $r$, reconstruct these low rank polynomials and then lift back to the original polynomial. The uniqueness of our circuit’s representation plays a major role in both the projection and lifting steps. Let $$f = G(\alpha_0 T_0 + \alpha_1 T_1)$$ $G,T_i$ are normal $\Pi\Sigma$ polynomials. All notations are borrowed from the previous section. It is almost identical to the restriction done in [@Shpilka07] except that the dimension of random subspaces is different. For more details see Section 4.2.1 and 4.2.3. in [@Shpilka07]. Since all proofs have been done in detail in [@Shpilka07] we do not spend much time here. A clear sketch with some proofs is however given.
Projection to a Random Low Dimensional Subspace {#projectrandom}
-----------------------------------------------
We explain the procedure of projecting to the random subspace below. In this low dimensional setup we can get white-box access to $\pi_V(f)$, also some important properties of $f$ are retained by $\pi_V(f)$. Proofs are simple and standard so we discuss them in the appendix at end.\
Pick $n$ vectors $v_i, i\in [n]$ with each co-ordinate chosen independently from the uniform distribution on $[N]$. Let $V = sp(\{v_i : i\in [r]\})$ and $V^\prime = sp\{v_i : i\in \{r+1,\ldots,n\}\}$. Then $V\oplus V^\prime = {\mathbb{R}}^n$ Let $\pi_V$ denote the orthogonal projection onto $V$. With high probability the following hold :
1. This set $\{v_i : i\in [n]\}$ is linearly independent (See Appendix \[linindrandom\] for proof).
2. Let $\{l_1,\ldots,l_r\}$ be a set of $r$ linearly independent linear forms in ${\mathcal{L}}(T_0)\cup
{\mathcal{L}}(T_1)$. Then $\pi_{V}(\{l_1,\ldots,l_r\})$ is linearly independent with high probability. So $rank(\pi_V(f))=r$ (See Appendix \[linindproj\] for Proof).
3. Let $l_{01}\in {\mathcal{L}}(T_0), l_{11}\in {\mathcal{L}}(T_1)$, then $\pi_V(l_{01}), \pi_V(l_{11})$ are linearly independent with high probability and so $gcd(\pi_V(T_0), \pi_V(T_1))=1$.
Pick large number of ($\geq d^{r}$) random points $p_i, i=1,\ldots,d^{r}$ in the space $V$. Use the values $\{f(p_i)\}$ and get a white-box (coefficient) representation for $\pi_V(f)$. With high probability over the choice of points lagrange interpolation will work (See Appendix \[lagrangeinterp\] for Proof). Note that the number of coefficients in $f|_{V} = O(d^r)$. Now this white box representation of $\pi_V(f)$ is reconstructed using the algorithm in Chapter \[lowdimrecon\]. A number of such reconstructions are then glued to reconstruct the original polynomial.
Lifting from the Random Low Dimensional Subspace {#liftingback}
------------------------------------------------
1. Consider spaces $V_i = V \oplus {\mathbb{R}}v_{i}$ for $i=r+1,\ldots,n$.
2. Reconstruct $\pi_{V_i}(f)$ and $\pi_V(f)$ for each $i\in \{r+1,\ldots,n\}$.
3. Let $l = \sum\limits_{i=1}^n a_iv_i $ be a linear form dividing one of the gates of $f$ say $T_0$. $\pi_V(l) = \sum\limits_{i=1}^r a_iv_i $ and $\pi_{V_i}(l) =
\sum\limits_{j=1}^{r} a_jv_j + a_iv_i$. Using our algorithm discussed in Chapter \[lowdimrecon\] we would have reconstructed $\pi_V(f)$ and $\pi_{V_i}(f)$. So we know the triples $(\pi_V(G), \pi_V(T_0),\pi_V(T_1))$ and $(\pi_{V_i}(G), \pi_{V_i}(T_0),\pi_{V_i}(T_1))$
On restricting $V_i$ to $V$ :
a\) [**Only Factors become factors**]{} with high probability so we can easily find the correspondence between $\pi_V(G)$ and $\pi_{V_i}(G)$.
b\) $\pi_V(\pi_{V_i}(T_0)) = \pi_V(T_0)$ and $\neq \pi_V (T_1)$ because of uniqueness of representation and therefore we get the correspondence between gates.
c\) Now to get correspondence between linear forms. Let $\pi_V(l)$ have multiplicity $k$ in $\pi_V(T_0)$. Then with high probability $l$ has multiplicity $k$ in $T_0$ Since two LI vectors remain LI on projecting to a random subspace of dimension $\geq 2$ (again See Appendix \[linindproj\] for proof). Therefore $\pi_{V_i}(l)$ has multiplicity $k$ and is the unique lift of $\pi_V(l)$ for all $i$. Let $\pi_{V_i}(l) = \pi_V(l) + a_iv_i$. Then $l = \pi_V(l) + \sum_{i=r+1}^n a_iv_j$. This finds $G,T_0,T_1$ for us
Time Complexity
---------------
- Interpolation to find coefficient representation $\pi_V(f)$ which is a degree $d$ polynomial over $r$ variables clearly takes $poly(d^r)$ time (accounts to solving a linear system of size $poly(d^r$)).
- Solving $n-r$ instances of the low rank problem (simple ranks $r$ and $r+1$) takes $npoly(d^r)$ time.
- The above mentioned approach to glue the linear forms in the gates clearly takes $poly(n,d)$ time.
- Overall the algorithm takes $poly(n,d)$ time since $r$ is a constant.
Conclusion and Future Work
==========================
We described an efficient randomized algorithm to reconstruct circuit representation of multivariate polynomials which exhibit a $\Sigma\Pi\Sigma(2)$ representation. Our algorithm works for all polynomials with rank(number of independent variables greater than a constant $r$). In future we would like to address the following:
- [**Reconstruction for Lower Ranks -** ]{} As can be seen in the paper, rank of the polynomial for uniqueness (i.e. $c_{{\mathbb{R}}}(4)$) and the rank we’ve assumed in the low rank reconstruction (i.e. $r$) are both $O(1)$ but $c_{{\mathbb{R}}}(4)$ is smaller than $r$. Since one would expect a reconstruction algorithm whenever the circuit is unique we would like to close this gap.
- [**$\Sigma\Pi\Sigma(k)$ circuits -** ]{} It would be interesting to consider more general top fan-in. In particular we could consider $\Sigma\Pi\Sigma(k)$ circuits with $k=O(1)$.
- [**Derandomization -** ]{} We would like to derandomize the algorithm as it was done in the finite field case in [@KarShp09].
Acknowledgements
================
I am extremely thankful to Neeraj Kayal for introducing me to this problem. Sukhada Fadnavis, Neeraj Kayal and myself started working on the problem together during my summer internship at Microsoft Research India Labs in 2011. We solved the first important case together. I’m grateful to them for all helpful discussions, constant guidance and encouragement.\
I would like to thank Zeev Dvir for communicating the most recent rank bounds on $\delta-SG_k$ configurations from [@DSW12] and his feedback on the work. This reduces the gap in the first problem we mentioned above.\
I would also like to thank Vinamra Agrawal, Pravesh Kothari and Piyush Srivastava for helpful discussions. Lastly I would like to thank Microsoft Research for giving me the opportunity to intern at their Bangalore Labs with the Applied Mathematics Group.
Easy Problem : Reconstruction of a product of linear forms
==========================================================
Before we begin the whole discussion about our algorithm let’s try to design an efficient algorithm for a much simpler problem. Consider the variables ${\bar{x}} = (x_1,\ldots,x_n)$ and a polynomial $f({\bar{x}})\in {\mathbb{R}}[{\bar{x}}]$ with the following form : $$f({\bar{x}}) = l_1({\bar{x}})l_2({\bar{x}})\ldots l_d({\bar{x}})$$ where each $l_i({\bar{x}})$ is an affine form in the $n$ variables $x_1,\ldots,x_n$. Next asssume that we are given blackbox access to $f({\bar{x}})$. Can one recover the $l_i({\bar{x}})$ with high probability? Thankfully there is an efficient and simple algorithm (in [@KalTr90]) which solves this problem as a special case. They give an efficient randomized algorithm to compute irreducible factors of a polynomial (over characteristic zero fields) given as a black-box. The factors are also computed as black-boxes. Note that an affine form can be easily reconstructed from it’s black-box representation by simply querying at appropriate points. To be precise we can reconstruct the affine form $a_1x_1 + \ldots a_nx_n + a_{n+1}$ from a black-box by simply querying at the points $\{(1,0,\ldots,0),\ldots,(0,\ldots,0,1)\} \subset {\mathbb{R}}^{n+1}$.
Unfortunately it does not solve our problem i.e. when the polynomial is a sum of two such products. But their approach does provide us with some ideas to tackle this difficult version. At a number of places in our algorithm we will need to solve this simpler problem.
Let $f({\bar{x}}) = l_1({\bar{x}})l_2({\bar{x}})\ldots l_d({\bar{x}}) \in {\mathbb{R}}[{\bar{x}}]$ be a polynomial in variables $x_1,\ldots,x_n$, such that $l_i({\bar{x}})$ is an affine form for each $i$. Then there exists an algorithm $Factorpoly(f)$ which computes the $l_i({\bar{x}})$’s with probability $\geq 1-\frac{1}{2^{poly(n,d)}}$ in time $poly(n,d)$.
$Factorpoly(f)$ will be used polynomially many times during the course of our algorithm.
Characterizing $\Pi\Sigma$ polynomials (Brill’s Equations) {#brills}
==========================================================
In this section we will explicitly compute a set of polynomials whose common solutions characterize the coefficients of all homogeneous $\Pi\Sigma_{\mathbb{C}}[x_1,\ldots,x_r]$ polynomials of degree $d$. A clean mathematical construction is given by Brill’s Equations given in Chapter $4$, [@GKZ94]. However we still need to calculate the time complexity. But before that we define some operations on polynomials and calculate the time taken by the operation along with the size of the output. Note that all polynomials are over the field of complex numbers ${\mathbb{C}}$ and all computations are also done for the complex polynomial rings.\
Let ${\bar{x}} = (x_1,\ldots,x_r)$ and ${\bar{y}} = (y_1,\ldots,y_r)$ be variables. For any homogeneous polynomial $f({\bar{x}})$ of degree $d$, define $$f_{{\bar{x}}^k}({\bar{x}},{\bar{y}}) = \frac{(d-k)!}{d!}(\sum\limits_{i}x_i\frac{\partial}{\partial y_i})^k f({\bar{y}})$$
Expanding $(\sum\limits_{i}x_i\frac{\partial}{\partial y_i})^k$ as a polynomial of differentials takes $O((r+k)^r)$ time and has the same order of terms in it. $f({\bar{y}})$ has $O((r+k)^r)$ terms. Taking partial derivatives of each term takes constant time and therefore overall computing $(\sum\limits_{i}x_i\frac{\partial}{\partial y_i})^k f({\bar{y}})$ takes $O((r+k)^{2r})$ time. Also the expression obtained will have atmost $O((r+k)^{2r})$ terms. Computing the external factor takes $poly(d)$ time and so for an arbitrary $f({\bar{x}})$ computing all $f_{{\bar{x}}^k}({\bar{x}},{\bar{y}})$ for $0\leq k\leq d$ takes $poly((r+d)^r)$ time and has $poly((r+d)^r)$ terms in it. From Section E., Chapter $4$ in [@GKZ94] we also know that $f_{{\bar{x}}^k}({\bar{x}},{\bar{y}})$ is a bihomogeneous form of degree $k$ in ${\bar{x}}$ and degree $d-k$ in ${\bar{y}}$. It is called the $k^{th}$ polar of $f$.\
Next we define an $\odot$ opeartion between homogeneous forms. Let $f({\bar{x}})$ and $g({\bar{x}})$ be homogeneous polynomials of degrees $d$, define $$(f\odot g)({\bar{x}},{\bar{y}}) = \frac{1}{d+1}\sum\limits_{k=0}^d(-1)^k{d\choose k}f_{{\bar{y}}^k}({\bar{y}},{\bar{x}})g_{{\bar{x}}^k}({\bar{x}},{\bar{y}})$$
From the discussion above we know that computing $f_{{\bar{y}}^k}({\bar{y}},{\bar{x}})g_{{\bar{x}}^k}({\bar{x}},{\bar{y}})$ takes $poly((r+d)^r)$ time and it is obvious that this product has $poly((r+d)^r)$ terms. Rest of the operations take $poly(d)$ time and therefore computing $(f\odot g)({\bar{x}},{\bar{y}})$ takes $poly((r+d)^r)$ time and has $poly((r+d)^r)$ terms. From the discussion before we may also easily conclude that the degrees of ${\bar{x}},{\bar{y}}$ in $(f\odot g)({\bar{x}},{\bar{y}})$ are $poly(d)$. The form $(f\odot g)$ is called the vertical(Young) product of $f$ and $g$. See Section G., Chapter $4$ in [@GKZ94].\
Next for $k\in \{0,\ldots,d\}$ and ${\bar{z}} = (z_1,\ldots,z_r)$ consider homogeneous forms: $$e_k = {d\choose k}f_{{\bar{x}}^k}({\bar{x}},{\bar{z}})f({\bar{z}})^{k-1}$$ Following arguments from above, it’s straightforward to see that computing $e_k$ takes $poly((r+d)^r)$ time and has $poly((r+d)^r)$ terms. Each $e_k$ is a homogeneous form in ${\bar{x}},{\bar{z}}$ and $f$. It has degree $k$ in ${\bar{x}}$, degree $k(d-1)$ in $z$, and $k$ in coefficients of $f$. See Section H. of Chapter $4$ in [@GKZ94]. Let’s define the following function of ${\bar{x}}$ with parameters $f,z$ $$P_{f,z}({\bar{x}}) = (-1)^dd\sum\limits_{i_1+2i_2+\ldots+ri_r=d}(-1)^{(i_1+\ldots+i_r)}\frac{(i_1+\ldots+i_r-1)!}{i_1!\ldots i_r!}e_1^{i_1}\ldots e_r^{i_r
}$$ Note that $\{(i_1,\ldots,i_r) :i_1+2i_2+\ldots+ri_r=d\}\subseteq \{(i_1,\ldots,i_r) : i_1+i_2+\ldots+i_r\leq d\}$ and therefore the number of additions in the above summand is $O(poly(r+d)^r)$. For every fixed $(i_1,\ldots,i_r)$ computing the coefficient $\frac{(i_1+\ldots+i_r-1)!}{i_1!\ldots i_r!}$ takes $O(poly((r+d)^r))$ time using multinomial coefficients. Each $e_k$ takes $poly((r+d)^r)$ time to compute. There are $r$ of them in each summand and so overall we take $O(poly((r+d)^r))$ time. A similar argument shows that number of terms in this polynomial is $O(poly((r+d)^r))$. Some more analysis shows that $P_{f,z}({\bar{x}})$ is a form of degree $d$ in ${\bar{x}}$ whose coefficients are homogeneous polynomials of dedgree $d$ in $f$ and degree $d(d-1)$ in ${\bar{z}}$. Let $$B_f({{\bar{x}},{\bar{y}},{\bar{z}}}) = (f\odot P_{f,z})({\bar{x}},{\bar{y}})$$
By the arguments given above calculating this form also takes time $poly((r+d)^r)$ and it has $poly((r+d)^r)$ terms. This is a homogeneous form in $({\bar{x}},{\bar{y}},{\bar{z}})$ of multidegree $(d,d,d(d-1))$ and it’s coefficients are forms of degree $(d+1)$ in the coefficients of $f$. See Section H., Chapter $4$ in [@GKZ94]. So in time $poly((r+d)^r)$ we can compute $B_f({{\bar{x}},{\bar{y}},{\bar{z}}})$ explicitly.\
Now we arrive at the main theorem
A form $f({\bar{x}}$) is a product of linear forms if and only if the polynomial $B_f({\bar{x}},{\bar{y}},{\bar{z}})$ is identically $0$.
We argued above that computing $B_f({\bar{x}},{\bar{y}},{\bar{z}})$ takes $O(poly((r+d)^r))$ time. It’s degrees in ${\bar{x}},{\bar{y}},{\bar{z}}$ are all $poly(d)$ and so the number of coefficients when written as a polynomial over the $3r$ variables
$(x_1,\ldots,x_r,y_1,\ldots,y_r,z_,\ldots,z_r)$ is $poly((r+d)^r)$. We mentioned that each coefficient is a polynomial of degree $(d+1)$ in the coefficients of $f$. Therefore we have the following corollary.
\[variety\] Let $$I{\stackrel{def}{=}}\{(\alpha_1,\ldots,\alpha_n) : \forall i : \alpha_i\geq 0,
\sum\limits_{i\in[r]}\alpha_i=d\}$$ be the set capturing the indices of all possible monomials of degree exactly $d$ in $r$ variables $(x_1,\ldots,x_r)$. Let $f_{\bf a}(y_1,\ldots,y_r) = \sum_{\alpha\in I}a_{\alpha}{\bf y}^{\alpha}$ denote an arbitrary homogeneous polynomial. The coefficient vector then becomes ${\bf a }= (a_\alpha)_{\alpha\in I}$. Then there exists an explicit set of polynomials $F_1({\bf a}),\ldots,F_m({\bf
a})$ on $poly((r+d)^r)$ variables (${\bf a} = (a_\alpha)_{\alpha\in I}$), with $m=poly((r+d)^r)$, $deg(F_i)\leq poly(d)$ such that for any particular value of ${\bf a}$, the corresponding polynomial $f_{\bf a}({\bf
y})\in \Pi\Sigma_{\mathbb{R}}^{d}[{\bar{y}}]$ if and only if $F_1({\bf a})=\ldots=F_m({\bf
a})=0$. Also this set $\{F_i, i\in [m]\}$ can be computed in time $poly((r+d)^r)$ time.
*Proof.* Clear from the theorem and discussion above.
Note that in our application $r=O(1)$ and so $poly((d+r)^r) = poly(d)$.
Tools from Incidence Geometry {#incidence}
=============================
Later in the paper we will use the quantitative version of Sylvester-Gallai Theorem from [@BDWY11]. In this subsection we do preparation for the same. Our main application will also involve a corollary we prove towards the end of this subsection.
\[elementaryset\] Let $S$ be a set of $n$ distinct points in complex space ${\mathbb{C}}^r$. A $k - flat$ is elementary if its intersection with $S$ has exactly $k+1$ points.
Let $S$ be a set of $n$ distinct points in ${\mathbb{C}}^r$. $S$ is called a $\delta -
SG_k$ configuration if for every independent $s_1,\ldots,s_k \in S$ there are atleast $\delta n$ points $t\in S$ such that either $t\in fl(\{s_1,\ldots,s_k\})$ or the $k-$flat $fl(\{s_1,\ldots,s_k,t\})$ contains a point in $S\setminus
\{s_1,\ldots,s_k,t\}$.
Let $S$ be a $\delta-SG_k$ configuration then $dim(S) \leq \frac{2^{C^k}}{\delta^2}$. Where $C>1$ is a universal constant.
This bound on the dimension of $S$ was further improved by Dvir et. al. in [@DSW12]. The latest version now states
\[bdwy\] Let $S$ be a $\delta-SG_k$ configuration then $dim(S) \leq C_k = \frac{C^k}{\delta}$. Where $C>1$ is a universal constant.
\[elementary\] Let $dim(S)> C_k$ then $S$ is not a $\delta-SG_k$ configuration i.e. there exists a set of independent points $\{s_1,\ldots,s_k\}$ and $\geq (1-\delta)n$ points $t$ such that $fl(\{s_1,\ldots,s_k,t\})$ is an elementary $k - flat$. That is:
- $t\notin fl(\{s_1,\ldots,s_k\})$
- $fl(\{s_1,\ldots,s_k,t\}) \cap S = \{s_1,\ldots,s_k,t\}$.
Right now we set $\delta$ to be a constant $ < 0.5, C_k = \frac{C^k}{\delta}$. Note that $C_i<C_{i+1}$. Using the above theorem we prove the following lemma which will be useful to us later
\[bichromatic\] Let $X$ and $Y$ be disjoint finite sets in ${\mathbb{C}}^r$ satisfying the following conditions.
1. $dim(Y)>C_4$.
2. $|Y|\leq c|X|$ with $c < \frac{1-\delta}{\delta}$.
Then there exists a line $l$ such that $|l\cap Y|=1$ and $|l\cap X|\geq 1$
*Proof.* We consider two cases:
Since $dim(Y)>C_1$, using the corollary above for $S=X\cup Y, k=1$ we can get a point $s_1 \in X\cup Y$ for which there exist $(1-\delta)(|X|+|Y|)$ points $t$ in $X\cup Y$ such that $t\notin fl\{s_1\}$ and $fl\{s_1,t\}$ is elementary. If $s_1\in X$ then $(1-\delta)(|X|+|Y|)-|X| \geq (1-2\delta)|X|>0$ of these flats intersect $Y$ and thus we get such a line $l$. If $s_1\in Y$ then $(1-\delta)(|X|+|Y|)-|Y| \geq ((1-\delta)(\frac{1}{c}+1) -1)|Y| >0$ of these flats intersect $X$ giving us the required line $l$ with $|l\cap X|=1$ and $|l\cap Y|=1$.\
Now choose a subset $X_1\subseteq X$ such that $|X_1|=|Y|$. Now using the same argument as above for $S = X_1\cup Y$ there is a point $s_1\in X_1\cup Y$ such that $(1-\delta) (|X_1|+|Y|)= 2(1-\delta) |Y| = 2(1-\delta) |X_1|$ flats through it are elementary in $X_1\cup Y$. If $s_1\in Y$ $(1-2\delta)|Y|>0$ of these flats intersect $X_1$. If $s_1\in X_1$, $(1-2\delta)|X_1| >0$ of these flats intersect $Y$. In both these above possibilities the flat intersects $Y$ and $X_1$ in exactly one point each. But it may contain more points from $X\setminus X_1$ so we can find a line $l$ such that $|l\cap Y|=1$ and $|l\cap X|\geq 1$.
A Method of Reconstructing Linear Forms {#Identifier}
=======================================
In a lot of circumstances one might reconstruct a linear form (upto scalar multiplication) inside $V=Lin_{\mathbb{R}}[{\bar{x}}]$ from it’s projections (upto scalar multiplication) onto some subspaces of $V$. For example consider a linear form $L=a_1x_1+a_2x_2+a_3x_3 (\in Lin_
{\mathbb{R}}[x_1,x_2,x_3])$ with $a_3\neq 0$, and assume we know scalar multiples of projections of $L$ onto the spaces ${\mathbb{R}}x_1$ and ${\mathbb{R}}x_2$ i.e. we know $L_1=\alpha(a_2x_2+a_3x_3)$ and $L_2=\beta(a_1x_1+a_3x_3)$ for some $\alpha,\beta\in {\mathbb{R}}$. Scale these projections to $\tilde L_1 = x_3+\frac{a_2}{a_3}x_3$ and $\tilde L_2 = x_3 + \frac{a_1}{a_3}x_3$. Using these two define a linear form $x_3+\frac{a_1}{a_3}x_1 + \frac{a_2}{a_3}x_2$. This is a scalar multiple of our original linear form $L$. We generalize this a little more below.\
Let ${\bar{x}}\equiv (x_1,\ldots,x_r)$, ${\mathcal{B}}= \{l_1,\ldots,l_r\}$ be a basis for $V=Lin_{{\mathbb{R}}}[x_1,\ldots,x_r]$. For $i\in
\{0,1,2\}$, let $S_i$ be pairwise disjoint non empty subsets of ${\mathcal{B}}$ such that $S_0\cup S_1\cup S_2 = {\mathcal{B}}$. Let $W_i=sp(S_i)$ and $W_i^\prime =
\bigoplus\limits_{j\neq i}W_j$. Clearly $V=W_0\oplus W_1\oplus W_2 = W_i\oplus
W_i^\prime, i\in \{0,1,2\} $.
\[reconlin\] Assume $L\in V$ is a linear form such that
- $\pi_{W_2}(L) \neq 0$
- For $i\in \{0,1\}, L_i=\beta_i \pi_{W_i^\prime}(L)$ are known for some non-zero scalars $\beta_i$.
Then $L$ is unique upto scalar multiplication and we can construct a scalar multiple $\tilde{L}$ of $L$.
*Proof.* Let $L=a_1l_1 + \ldots + a_rl_r, a_i\in {\mathbb{R}}$. Since $\pi_{W_2}(L)\neq 0$, there exists $l_j\in S_2$ such that $a_j\neq 0$. Let $\tilde{L} = \frac{1}{a_{j}}L$. For $i\in \{0,1\}$, re-scale $L_i $ to get $\tilde{L_i}$ making sure that coefficient of $l_j$ is $1$ in them. Thus for $i=0,1$ $$\pi_{W_i^\prime}(\tilde{L}) = \tilde{L_i}$$ Since $W_0^\prime = W_1\oplus W_2$ and $W_1^\prime = W_0\oplus W_2$ by comparing coefficients we can get $\tilde{L}$.
\[linformrecon\] Assume we know $S_0,S_1,S_2$ and therefore the basis change matrix to convert vector representations from ${\mathcal{S}}$ to ${\mathcal{B}}$. It takes $poly(r)$ time to convert $[v]_{{\mathcal{S}}}$ to $[v]_{{\mathcal{B}}}$. Given $L_i$ in the basis ${\mathcal{B}}$ it takes $poly(r)$ time(by a linear scan) to find $l_j\in S_2$ with $a_j\neq 0$. This $l_j$ has a non zero coefficient in both $L_0,L_1$. After this we just rescale $L_i$ to get $\tilde{L_i}$ such that coefficient of $l_j$ is $1$. Then since $\tilde{L_i} =
\pi_{W_i^\prime}(\tilde{L})$ the coefficient of $l_t$ in $\tilde{L}$ is as follows :
$$= \left\{
\begin{array}{lr}
\text{ coefficient of }l_t\text{ in } \tilde{L_1} & : l_t\in S_0\\
\text{ coefficient of }l_t\text{ in } \tilde{L_0} & : l_t\in S_1\\
\text{ coefficient of }l_t\text{ in } \tilde{L_0} = \text{ coefficient of
}l_t\text{ in } \tilde{L_1} & : l_t\in S_2
\end{array}
\right.$$
Finding the right coefficients using this also takes $poly(r)$ time.\
Next we try and use this to reconstruct $\Pi\Sigma$ polynomials. This case is slightly more complicated and so we demand that the projections have some special form. In particular the projections onto one subspace preserves pairwise linear independence of linear factors and onto the other makes all linear factors scalar multiples of each other.
\[polyrecon\] Let $S_i,W_i, i\in \{0,1,2\}$ be as above and $P\in \Pi\Sigma_{\mathbb{R}}[x_1,\ldots,x_r]$ such that
1. $\pi_{W_2}(P)\neq 0$
2. For $i\in \{0,1\}$ there exists $\beta_i (\neq 0) \in {\mathbb{R}}$ such that $P_0=\beta_0 \pi_{W_0^\prime}(P) = p^{t}$ and $P_1=\beta_1
\pi_{W_1^\prime}(P)=d_1\ldots d_t$. are known i.e. $p,d_j$ $(j\in [t])$ and $t$ are known.
Then $P$ is unique upto scalar multiplication and we can construct a scalar multiple $\tilde{P}$ of $P$.
*Proof.* Let $P = L_1\ldots L_t$ with $L_i\in Lin_{\mathbb{R}}[{\bar{x}}]$. There exists $\beta^j_i,
i\in \{0,1\}, j\in [t]$, such that $\beta^j_0 \pi_{W_0^\prime}(L_j)= p$ and $\beta^j_1 \pi_{W_1^\prime}(L_j) = d_j$. Since $p,d_j$ are known by above Lemma \[reconlin\] we find a scalar multiple $\tilde{L_j}=\beta^j L_j$ of $L_j$ and therefore find a scalar multiple $\tilde{P} = \tilde{L_1}\ldots\tilde{L_t}$ of $P$. Note that this method also tells us that such a $P$ is unique upto scalar multiplication. Since we’ve used the above Algorithm \[linformrecon\] at most $t$ times with $t\leq deg(P)$, it takes $poly(deg(P),r)$ time to find $\tilde P$.
This corollary is the backbone for reconstructing $\Pi\Sigma$ polynomials from their projections. But first we formally define a *“Reconstructor”*
\[Reconstructor\] Let $S_i,W_i, i\in
\{0,1,2\}$ be as above. Let $Q$ be a standard $\Pi\Sigma$ polynomial and $P$ be a standard $\Pi\Sigma$ polynomial dividing $Q$ with $Q=PR$. Then $(Q,P,S_0,S_1,S_2)$ is called a *Reconstructor* if:
- $\pi_{W_2}(P)\neq 0$.
- $\pi_{W_0^\prime}(P) = \alpha p^t$, for some linear form $p$.
- Let $l\mid R$ be a linear form and $\pi_{W_2}(l)\neq 0$ then $gcd
(\pi_{W_2}(P),\pi_{W_2}(l)) =1$.
[**Note :**]{}
Let $L_1,L_2$ be two LI linear forms dividing $P$ , then one can show $$L_1,L_2 \text{ are LI } \Leftrightarrow \pi_{W_1^\prime}(L_1),
\pi_{W_1^\prime}(L_2) \text{ are LI }$$ To see this first observe that the second bullet implies for $i\in [2], L_i\in W_0 + p \Rightarrow
sp(\{L_1,L_2\})\subseteq W_0+p$.
If $\pi_{W_1^\prime}(L_1), \pi_{W_1^\prime}(L_2)$ are LD then $$sp(\{L_1,L_2\}) \cap W_1 \neq \{0\}$$ $\Rightarrow (W_0+p) \cap W_1 \neq \{0\}$. Since $W_0\cap W_1 = \{0\}$ we get that $p\in W_0\oplus W_1 = W_2^\prime
\Rightarrow \pi_{W_2}(p)=0\Rightarrow \pi_{W_2}(P)=0$ contradicting the first bullet.\
Geometrically the conditions just mean that all linear forms dividing $P$ have LD projection ($=\gamma p$ for some non zero $\gamma \in {\mathbb{R}}$) w.r.t. the subspace $W_0^\prime$ and LI linear forms $p_1,p_2$ dividing $P$ have LI projections (w.r.t. subspace $W_1^\prime)$. Also no linear form $l$ dividing $R$ belongs to $fl(S_0 \cup S_1 \cup \{p\})$.\
We are now ready to give an algorithm to reconstruct $P$ using $\pi_{W_0^\prime}(Q)$ and $\pi_{W_1^\prime}(Q)$ by gluing appropriate projections corresponding to $P$. To be precise:\
\[reconalgoclaim\] Let $Q,P$ be standard $\Pi\Sigma$ polynomials and $P\mid Q$. Assume $(Q, P,S_0,S_1,S_2)$ is a *Reconstructor*. If we know both $\pi_{W_0^\prime}(Q)$ and $\pi_{W_1^\prime}(Q)$. Then we can reconstruct $P$.
*Proof.* Here is the algorithm:
\[overestimatedetector\]
$bool flag$, $\Pi\Sigma$ polynomial $P_0,P_1;$ Factor $\pi_{W_0^\prime}(Q) = \gamma \prod\limits_{i\in [s]} c_i^{m_i}$, $c_i$’s pairwise LI and normal, $\gamma\in {\mathbb{R}}$ Factor $\pi_{W_1^\prime}(Q) = \delta d_1\ldots d_m$, $\delta\in {\mathbb{R}}$ and $d_j$ normal Return [$1$]{}
Explanation
-----------
- The algorithm takes as input projections $\pi_{W_0^\prime}(Q)$ and $\pi_{W_1^\prime}(Q)$ along with the sets $S_i,i=0,1,2$ which form a partition of a basis ${\mathcal{B}}$. We know that there exists a polynomial $P\mid Q$ such that $(Q,P,S_0,S_1,S_2)$ is a reconstructor and so we try to compute the projections $\pi_{W_0^\prime}(P),\pi_{W_1^\prime}(P)$.
- If one assumes that $\pi_{W_0^\prime}(Q) = \gamma \prod\limits_{i\in [s]} c_i^{m_i}$ with the $c_i$’s co-prime, then by the properties of a reconstructor the projection (of a scalar multiple of $P$) onto $W_0^\prime$ say $P_0 = \beta_0\pi_{W_0^\prime}(P)$ (for some $\beta_0$) has to be equal to $c_i^{m_i}$ for some $i$. We do this assignment inside the first for loop.
- The third property of a reconstructor implies that when we project further to $W_2$, it should not get any more factors and so we check this inside the second for loop by going over all other factors $c_j$ of $\pi_{W_0^\prime}(Q)$ and checking if $c_i,c_j$ become LD on projecting to $W_2$ (i.e. by further projecting to $W_1^\prime$).
- Now to find (scalar multiple of) the other projections i.e. $P_1 = \beta_1\pi_{W_1^\prime}(P)$ (for some $\beta_1$), we go through $\pi_{W_1^\prime}(Q)$ and find $d_k$ such that $\{\pi_{W_1^\prime}(c_i) , \pi_{W_0^\prime}(d_k)\}$ are LD (i.e. they are projections of the same linear form). We collect the product of all such $d_k$’s. If the choice of $c_i$ were correct then all $d_k$’s would be obtained correctly.
- The last *“if”* statement just checks that the number of $d_k$’s found above is the same as $m_i$ since $P_0=c_i^{m_i}$ tells us that the degree of $P$ was $m_i$. We recover a scalar multiple of $P$ using the algorithm explained in Corollary \[polyrecon\] and then make it standard to get $P$.
Correctness
-----------
The corectness of our algorithm is shown by the lemma below.
\[returnreconalgo\] If $(Q,P,S_0,S_1,S_2)$ is a reconstructor for non-constant $P$, then Algorithm \[reconalgo\] returns $P$.
*Proof.* $(Q,P,S_0,S_1,S_2)$ is a reconstructor therefore
- $\pi_{W_2}(P)\neq 0$
- $\pi_{W_0^\prime}(P) = \delta p^t$
- $q\mid \frac{Q}{P} \Rightarrow gcd(\pi_{W_2}(q), \pi_{W_2}(P))=1$
1. It is clear that for one and only one value of $i$, $c_i$ divides $p$. Fix this $i$. Let $Q=PR$, if $c_i^{m_i}\nmid \pi_{W_0^\prime}(P)$ then $c_i \mid l$ for some linear form $l\mid \pi_{W_0^\prime}(R)$. Condition $3$ in definition of Reconstructor implies that $gcd(\pi_{W_2}(P),\pi_{W_2}(l))=1$ but $\pi_{W_2}(c_i)$ divides both of them giving us a contradiction. Since $\pi_{W_0^\prime}(P)$ has just one linear factor $\Rightarrow \pi_{W_0^\prime}(P)$ is a scalar multiple of $c_i^{m_i}$ for some $i$.
2. Assume the correct $c_i^{m_i}$ has been found. Now let $d_j\mid \pi_{W_1^\prime}(Q)$ such that $\{\pi_{W_2}(c_i), \pi_{W_2}(d_j)\}$ are LD. then we can show that $d_j\mid \pi_{W_1^\prime}(P)$. Assume not, then for some linear form $l\mid R = \frac{Q}{P}$, $d_j\mid \pi_{W_1^\prime}(l)$. $\pi_{W_0^\prime}(d_j)\neq 0$ (which we checked) $\Rightarrow \pi_{W_2}(l)\neq 0$. So we get $\pi_{W_2}(c_i)\mid \pi_{W_2}(l) (\neq 0)$ and so $\pi_{W_2}(c_i) \mid gcd(\pi_{W_2}(P),\pi_{W_2}(l))$ which is therefore $\neq 1$ and condition $3$ of Definiton \[Reconstructor\] is violated. So whatever $d_j$ we collect will be a factor of $\pi_{W_1^\prime}(P)$ and we will collect all of them since they are all present in $\pi_{W_1^\prime}(Q)$.
3. We know from proof of Corollary \[polyrecon\] that if we know $c_i,m_i$ and $d_j$’s correctly then we can recover a scalar multiple of $P$ correctly. But $Q,P$ are standard so we return $P$ correctly.
In fact we can show that if we return something it has to be a factor of $Q$.
\[reconalgoconverse\] If Algorithm \[reconalgo\] returns a $\Pi\Sigma$ polynomial $P$, then $P\mid Q$
- If the algorithm returns $1$ from the last return statement, we are done. So let’s assume it returns something from the previous return statement.
- So $flag$ has to be true at end $\Rightarrow$ there is an $i\in [s]$ such that $P_0 = c_i^{m_i}$ with the conditions that $\pi_{W_1^\prime}(c_i)\neq 0$ and $gcd(c_i,c_j)=1$ for $j\neq i$. It also means that for exactly $m_i$ of the $d_j$’s (say $d_1,\ldots,d_{m_i}$) $\{\pi_{W_1^\prime}(c_i), \pi_{W_0^\prime}(d_j)\}$ are LD and $P_1 = d_1\ldots d_{m_i}$.
- Since $c_i^{m_i} \mid \pi_{W_0^\prime}(Q)$, there exists a factor $\tilde P\mid Q$ of degree $m_i$ such that $\pi_{W_0^\prime}(\tilde P) = c_i^{m_i}$ and $\pi_{W_1^\prime}(c_i)\neq 0$. This $\Rightarrow \pi_{W_2}(\tilde P)\neq 0$. Clearly $\pi_{W_1^\prime}(\tilde P) \mid \pi_{W_1^\prime}(Q) = d_1\ldots d_m$, hence for all linear factors $\tilde p$ of $\tilde P$, $\pi_{W_1^\prime}(\tilde p)$ should be some $d_j$ with the condition that $\{\pi_{W_0^\prime}((\pi_{W_1}^\prime)(\tilde p)), \pi_{W_1^\prime}(c_i)\}$ should be LD. The only choice we have are $d_1,\ldots,d_{m_i}$. So $\pi_{W_0^\prime}(\tilde P) = d_1\ldots d_{m_i}$. All conditions of Corollary \[polyrecon\] are true and so $\tilde P$ is uniquely defined (upto scalar multiplication) by the reconstruction method given in Corollary \[polyrecon\]. So what we returned was actually a factor of $Q$.
Time Complexity
---------------
Factoring $\pi_{W_0^\prime}(Q),\pi_{W_1^\prime}(Q)$ takes $poly(d)$ time (using Kaltofen’s Factoring from [@KalTr90]). The nested for loops run $\leq d^3$ times. Computing projections with respect to the known decomposition $W_0\oplus W_1\oplus W_2 ={\mathbb{R}}^r$ of linear forms over $r$ variables takes $poly(r)$ time. Computing $gcd$ and linear independence of linear forms takes $poly(r)$ time. The final reconstruction of $P$ using $(P_0,P_1)$ takes $poly(d,r)$ time as has been explained in Corollary \[polyrecon\]. Multiplying linear forms to $\Pi\Sigma$ polynomial takes $poly(d^r)$ time. Therefore overall the algorithm takes $poly(d^r)$ time. In our application $r=O(1)$ and therefore the algorithm takes $poly(d)$ time.
Random Linear Transformations {#randomtransform}
=============================
This section will prove some results about linear independence and non-degeneracy under random transformations on ${\mathbb{R}}^r$. This will be required to make our input non-degenerate. From here onwards we fix a natural number $N\in {\mathbb{N}}$ and assume $0<k<r$. Let $T\subset {\mathbb{R}}^r$ be a finite set with $dim(T)= r$. Next we consider two $r\times r$ matrices $\Omega,\Lambda$. Entries $\Omega_{i,j},\Lambda_{i,j}$ are picked independently from the uniform distribution on $[N]$. For any basis ${\mathcal{B}}$ of ${\mathbb{R}}^r$ and vector $v\in {\mathbb{R}}^r$, let $[v]_{\mathcal{B}}$ denote the co-ordinate vector of $v$ in the basis ${\mathcal{B}}$. If ${\mathcal{B}}= \{b_1,\ldots,b_r\}$ then $[v]^i_{\mathcal{B}}$ denotes the $i$-th co-ordinate in $[v]_{\mathcal{B}}$. Let ${\mathcal{S}}= \{e_1,\ldots,e_r\}$ be the standard basis of ${\mathbb{R}}^r$. Let $E_j = sp(\{e_1,\ldots,e_j\})$ and $E_j^\prime = sp(\{e_{j+1},\ldots,e_r\})$, then ${\mathbb{R}}^r = E_j \oplus E_j^\prime$. Let $\pi_{W_{E_j}}$ be the orthogonal projection onto $E_j$. For any matrix $M$, we denote the matrix of it’s co-factors by $co(M)$. We consider the following events :
- ${\mathcal{E}}_0 = \{\Omega \text{ is not invertible }\}$
- ${\mathcal{E}}_1 = \{\exists t(\neq 0)\in T$ : $\pi_{W_{E_1}}(\Omega(t))= 0 \}$
- ${\mathcal{E}}_2 = \{ \exists \{t_1,\ldots,t_r\} \text{ LI vectors in } T : \{\Omega(t_1),\ldots,\Omega(t_r)\} \text{ is LD }\}$
- ${\mathcal{E}}_3 = \{\exists \{t_1,\ldots,t_r\} \text{ LI vectors in } T : \{\Omega(t_1),\ldots,\Omega(t_k),\Lambda\Omega(t_{k+1}), \ldots , \Lambda\Omega(t_r) \} \text{ is LD } \}$
- \[newmatrix\] When $t_i,\Lambda,\Omega$ are clear we define the matrix $M =[M_1 \ldots M_r]$ with columns $M_i$ given as : $$M_i= \left\{
\begin{array}{lr}
[\Omega(t_i)]_{{\mathcal{S}}} : i\leq k\\
[\Lambda\Omega(t_i)]_{{\mathcal{S}}} : i>k
\end{array}
\right.$$ $M$ corresponds to the linear map $$e_i\mapsto \Omega(t_i) \text{ for } i\leq k \text{ and } e_i\mapsto \Lambda\Omega(t_i) \text{ for } i>k$$
${\mathcal{E}}_4 = \{ \{\exists \{t_1,\ldots,t_r\} \text{ LI vectors in } T $ and $ t\in T\setminus sp(\{t_1,\ldots,t_k\}) :
[co(M)[\Omega(t)]_{{\mathcal{S}}}]^{k+1}_{\mathcal{S}}=0 \}$
- ${\mathcal{E}}_5 = {\mathcal{E}}_4 \mid {\mathcal{E}}_3^c$
Next we show that the probability of all of the above events is small. Before doing that let’s explain the events. This will give an intuition to why the events have low probabilities.
- ${\mathcal{E}}_0$ is the event where $\Omega$ is not-invertible. Random Transformations should be invertible.
- ${\mathcal{E}}_1$ is the event where there is a non-zero $t\in T$ such that the projection to the first co-ordinate (w.r.t. ${\mathcal{S}}$) of $\Omega$ applied on $t$ is $0$. We don’t expect this for a random linear transformation. Random Transformation on a non-zero vector should give a non-zero coefficient of $e_1$.
- ${\mathcal{E}}_2$ is the event such that $\Omega$ takes a basis to a LD set i.e. $\Omega$ is not invertible (random linear operators are invertible).
- ${\mathcal{E}}_3$ is the event such that for some basis applying $\Omega$ to the first $k$ vectors and $\Lambda\Omega$ to the last $n-k$ vectors gives a LD set. So this operation is not-invertible. For ranrom maps this should not be the case.
- ${\mathcal{E}}_4$ is the event that there is some basis $\{t_1,\ldots,t_r\}$ and $t$ outside $sp(t_1,\ldots,t_k)$ such that the $(k+1)^{th}$ co-ordinate of $co(M)[\Omega(t)]_{\mathcal{S}}$ w.r.t the standard basis is $0$. If $M$ were invertible, clearly the set $ {\mathcal{B}}= \{\Omega(t_1),\ldots,\Omega(t_k),\Lambda\Omega(t_{k+1}), \ldots , \Lambda\Omega(t_r) \}$ would be a basis and $co(M)$ will be a scalar multiple of $M^{-1}$. So we are asking if the $(k+1)^{th}$ co-ordinate of $\Omega(t)$ in the basis ${\mathcal{B}}$ is $0$. For random $\Omega,\Lambda$ we would expect $M$ to be invertible and this co-ordinate to be non-zero.
Now let’s formally prove everything. We will repeatedly use the popular Schawrtz-Zippel Lemma which the reader can find in [@Sax09].
$Pr[{\mathcal{E}}_1] \leq \frac{|T|}{N^r}$
*Proof.* Fix a non-zero $t = \left(\begin{array}{c} a_1\\.\\.\\a_r\end{array}\right)$ with $a_i\in {\mathbb{R}}$ and let $\Omega = (\Omega_{i,j}), 1\leq i,j\leq
r$. Then the first co-ordinate of $\Omega(t)$ is $\Omega_{1,1}a_1 + \Omega_{1,2}a_2+\ldots + \Omega_{1,r}a_r$. Since $t\neq 0$, not all $a_i$ are $0$ and this is therefore not an identically zero polynomial in $(\Omega_{1,1},\ldots,\Omega_{1,r})$. Therefore by Schwartz-Zippel lemma $ Pr[[\Omega(t)]^1_{{\mathcal{S}}}= 0] \leq \frac{1}{N^r} $. Using a union bound inside $T$ we get $ Pr[ \exists t (\neq
0)\in T :
[\Omega(t)]^1_{{\mathcal{S}}} =0] \leq
\frac{|T|}{N^r}$.
$Pr[{\mathcal{E}}_2]\leq \frac{r}{N^{r^2}}$
*Proof.* Clearly ${\mathcal{E}}_2 \subseteq {\mathcal{E}}_0$ and so $Pr[{\mathcal{E}}_2]\leq Pr[{\mathcal{E}}_0]$. ${\mathcal{E}}_0$ corresponds to the polynomial equation $det(\Omega)=0$. $det(\Omega)$ is a degree $r$ polynomial in $r^2$ variables and is also not identically zero, so using Schwartz-Zippel lemma we get $Pr[{\mathcal{E}}_2]\leq Pr[ {\mathcal{E}}_0] \leq \frac{r}{N^{r^2}}$.
$Pr[{\mathcal{E}}_3]\leq {|T|\choose r}\frac{2r}{N^{2r^2}}$
*Proof.* Fix an LI set $t_1,\ldots,t_r$. The set $\{\Omega(t_1),\ldots,\Omega(t_k), \Lambda\Omega(t_{k+1}),\ldots \Lambda\Omega(t_r)\}$ is LD iff the $r\times r$ matrix $M$ formed by writing these vectors (in basis ${\mathcal{S}}$) as columns (described in part \[newmatrix\] above) has determinant $0$. $M$ has entries polynomial (of degree $\leq 2$) in $\Omega_{i,j}$ and $\Lambda_{i,j}$ and so $det(M)$ is a polynomial in $\Omega_{i,j},\Lambda_{i,j}$ of degree $\leq 2r$. For $\Omega=\Lambda = I$ (identity matrix) this matrix just becomes the matrix formed by the basis $\{t_1,\ldots,t_r\}$ which has non-zero determinant and so $det(M)$ is not the identically zero polynomial. By Schwartz-Zippel lemma $Pr[det(M)=0]\leq \frac{2r}{N^{r^2}N^{r^2}} =
\frac{2r}{N^{2r^2}}$. Now we vary the LI set $\{t_1,\ldots,t_r\}$, there are $\leq {|T|\choose r}$ such sets and so by a union bound $Pr[{\mathcal{E}}_3]\leq {|T|\choose r} \frac{2r}{N^{2r^2}}$.
$Pr[{\mathcal{E}}_4]\leq {|T|\choose r+1}\frac{2r-1}{N^{2r^2}}$
*Proof.* Fix an LI set $t_1,\ldots,t_r$ and a vector $t\notin sp(\{t_1,\ldots,t_k\})$. Let $t = \sum\limits_{i=1}^r a_it_i$. Since $t\notin sp(\{t_1\ldots,t_k\})$, $a_s\neq 0$ for some $s\in \{k+1,\ldots,r\}$. Let ${\mathcal{B}}= \{\Omega(t_1),\ldots,\Omega(t_k), \Lambda\Omega(t_{k+1}),\ldots \Lambda\Omega(t_r)\}$. Let $M$ be the matrix whose columns are from ${\mathcal{B}}$ (Construction has been explained in part \[newmatrix\] above). We know that the co-factors of a matrix are polynomials of degree $\leq r-1$ in the matrix elements. In our matrix $M$ all entries are polynomials of degree $\leq 2$ in $\Omega_{i,j},\Lambda_{i,j}$, so all entries of $co(M)$ are polynomials of degree $\leq 2r-2$ in $\Omega_{i,j}, \Lambda_{i,j}$. Thus $[co(M)[\Omega(t)]_{\mathcal{S}}]^{k+1}_{{\mathcal{S}}} =
\sum\limits_{i=1}^r co(M)_{k+1,i}[\Omega(t)]^i_{\mathcal{S}}$ is a polynomial of degree $\leq 2r-1$. This polynomial is not identically zero. Define $\Omega$ to be the matrix (w.r.t. basis ${\mathcal{S}}$) of the linear map $\Omega(t_i) = e_i$ and $\Lambda$ to be the matrix (w.r.t. basis ${\mathcal{S}}$) of the map $$\Lambda = \left\{
\begin{array}{lr}
\Lambda(e_i)=e_i : i\notin\{s,k+1\}\\
\Lambda(e_s) = e_{k+1}\\
\Lambda(e_{k+1})=e_s\\
\end{array}
\right.$$ With these values the set ${\mathcal{B}}$ becomes $\{e_1,\ldots,e_k,e_s,e_{k+2},\ldots,e_{s-1},e_{k+1},e_{s+1},\ldots,e_r\}$. If one now looks at $M$ i.e. the matrix formed using entries of ${\mathcal{B}}$ as columns it’s just the permutation matrix that flips $e_s$ and $e_{k+1}$. This matrix is the inverse of itself and so has determinant $=\pm1$, thus $co(M) = \pm M^{-1} = \pm M$. Therefore $co(M)[\Omega(t)]_{\mathcal{S}}= \pm M \left(\begin{array}{c} a_1\\.\\.
\\a_r\end{array}\right) = \pm \left(\begin{array}{c} a_1\\.\\a_k\\a_s\\a_{k+2}\\.\\a_{s-1}\\a_{k+1}\\.a_{s+1}\\.\\a_r\end{array}\right)$. Since $a_s \neq 0$, we get $[co(M)[\Omega(t)]_{\mathcal{S}}]^{k+1}_{\mathcal{S}}\neq 0$. So the polynomial is not identically zero and we can use Schwartz-Zippel Lemma to say that $Pr[[co(M)[\Omega(t)]_{\mathcal{S}}]^{k+1}_{\mathcal{S}}= 0] \leq \frac{2r-1}{N^{r^2}N^{r^2}} = \frac{2r-1}
{N^{2r^2}}$. Now we vary $\{t_1,\ldots,t_r,t\}$ inside $T$ and use union bound to show $Pr[{\mathcal{E}}_4]\leq {|T|\choose r+1}\frac{2r-1}{N^{2r^2}}$.
Even though this is just basic probability we include the following:
$Pr[{\mathcal{E}}_5] \leq {|T|\choose r}\frac{2r-1}{N^{2r^2}-{|T|\choose r}2r}$
*Proof.* $Pr[{\mathcal{E}}_5] = Pr[{\mathcal{E}}_4\mid {\mathcal{E}}_3^c] = \frac{Pr[{\mathcal{E}}_4\cap {\mathcal{E}}_3^c]}{Pr[{\mathcal{E}}_3^c]} \leq \frac{Pr[{\mathcal{E}}_4]}{Pr[{\mathcal{E}}_3^c]}
\leq {|T|\choose r+1}\frac{\frac{2r-1}{N^{2r^2}}}{1-{|T|\choose r}\frac{2r}{N^{2r^2}}} = {|T|\choose r+1}\frac{2r-1}
{N^{2r^2}-{|T|\choose r}2r}$
In our application of the above $r = O(1), |T| = poly(d), N = 2^{d}$ and so all probabilities are very small as $d$ grows. So we will assume that none of the above events occur. By union bound that too will have small probability and so with very high probability ${\mathcal{E}}_0,{\mathcal{E}}_1,{\mathcal{E}}_2,{\mathcal{E}}_3,{\mathcal{E}}_4,{\mathcal{E}}_5$ do not occur.
Set ${\mathcal{C}}$ of Candidate Linear Forms {#findcandidate}
=============================================
This section deals with constructing a $poly(d)$ size set ${\mathcal{C}}$ which contains each $l_{ij}, (i,j)\in \{0,1\}\times [M]$. First we define the set and prove a bound on it’s size.
Structure and Size of ${\mathcal{C}}$
-------------------------------------
Let’s recall $f=G(\alpha_0T_0 + \alpha_1 T_1)$ and define two other polynomials:
$$g=\frac{f}{G} = \alpha_0T_0+\alpha_1T_1$$ $$h =\frac{f}{Lin(f)} = \frac{g}{Lin(g)}$$ Assume $deg(h) = d_h$
\[candidatedef\] Our candidate set is defined as: $${\mathcal{C}}{\stackrel{def}{=}}\{l = x_1-a_2x_2-\ldots- a_rx_r \in Lin_{{\mathbb{R}}}[{\bar{x}}] :
h(a_2x_2+\ldots + a_rx_r,x_2,\ldots,x_r) \in \Pi\Sigma^{d_h}_{\mathbb{R}}[x_2,\ldots,x_r]
\}$$ (for definition of $\Pi\Sigma^{d_h}_{\mathbb{R}}[x_2,\ldots,x_r]$ See Section \[notation\] )
In the claim below we show that linear forms dividing polynomials $T_i, i=0,1$ are actually inside ${\mathcal{C}}$ (first part of claim). The remaining linear forms in ${\mathcal{C}}$ (which we call *“spurious”*) have a nice structure (second part of claim). In the third part of our claim we arrive at a bound on the size of ${\mathcal{C}}$. Recall the definition of $c_{\mathbb{R}}(k)$ from Theorem \[rankbound\].
\[candidate\] The following are true about our candidate set ${\mathcal{C}}$.
1. ${\mathcal{L}}(T_i)\subseteq {\mathcal{C}}, i=0,1$.
2. \[candidateprop\] Let $k=c_{{\mathbb{R}}}(3)+2$ and suppose $\{ l_{j} ; j\in
[k]\} \subset {\mathcal{L}}(T_i)$ are LI . Then for any $l\in {\mathcal{C}}\setminus ({\mathcal{L}}(T_0)\cup {\mathcal{L}}(T_1))$, there exists $j\in [k]$ such that $fl(\{l,l_{j}\})\cap {\mathcal{L}}(T_{1-i}) \neq \phi$ i.e. the line joining $l$ and $l_{j}$ does not intersect the set ${\mathcal{L}}(T_{1-i})$.
3. $|{\mathcal{C}}|\leq M^4+2M\leq d^4 + 2d.$
*Proof.* Let’s first recall the definition of our candidate set $${\mathcal{C}}{\stackrel{def}{=}}\{l = x_1-a_2x_2-\ldots- a_rx_r \in Lin_{{\mathbb{R}}}[{\bar{x}}] :
h(a_2x_2+\ldots + a_rx_r,x_2,\ldots,x_r) \in \Pi\Sigma^{d_h}_{\mathbb{R}}[x_2,\ldots,x_r]
\}$$ Also recall that $$h = \frac{g}{Lin(g)} = \frac{f}{Lin(f)}$$
1. Let $l = x_1-a_2x_2-\ldots- a_rx_r \in {\mathcal{L}}(T_{1-i})$. Let’s denote the tuple $v\equiv (a_2x_2+\ldots + a_rx_r,x_2,\ldots,x_r)$. Since $gcd(T_0,T_1)=1$ and $l\mid T_{1-i}$ we know that $l\nmid T_i$ and therefore $Lin(g)(v)\neq 0$. We can then compute $$h(v) = \frac{\alpha_{i}T_{i}(v)}
{Lin(g)(v)}
= \alpha_{i}H_1(v)\ldots H_{d_h}(v)\in \Pi\Sigma^{d_h}_{{\mathbb{R}}}[x_2,\ldots,x_r]$$ where $H_j \in Lin_{{\mathbb{R}}}[x_2,\ldots,x_r]$. So ${\mathcal{L}}(T_i)\subseteq {\mathcal{C}}$ for $i=0,1$.
2. Consider $l=x_1-a_2x_2-\ldots- a_rx_r \in {\mathcal{C}}\setminus ({\mathcal{L}}(T_0)\cup {\mathcal{L}}(T_1))$ and assume that $sp(\{l,l_j\})\cap {\mathcal{L}}(T_{1-i})=\phi$ for all $j\in[k]$. We know that $$g(v) = Lin(g)(v) H_1(v)\ldots H_{d_h}(v) = \alpha_0T_0(v) + \alpha_1T_1(v)$$
Let $g^\prime $ be the following identically zero $\Sigma\Pi\Sigma(3)[x_2,\ldots,x_r]$ polynomial (with circuit ${\mathcal{C}}^\prime$)
$$g^\prime = Lin(g)(v) H_1(v)\ldots H_{d_h}(v) -\alpha_0T_0(v) -
\alpha_1T_1(v)$$
We know $${\mathcal{C}}^\prime = gcd({\mathcal{C}}^\prime) Sim({\mathcal{C}}^\prime) \Rightarrow Sim({\mathcal{C}}^\prime)\equiv
0$$ Recall that $l_j(v)\mid T_{i}(v)$, therefore the $l_j(v)$ cannot be factors of $gcd({\mathcal{C}}^\prime)$ because if they did then there exist pair $l_{j}, l_{(1-i)t}$ such that $\{l_{j}(v), l_{(1-i)t}(v)\}$ is LD or in other words $sp(\{l,l_j\})\cap {\mathcal{L}}(T_{1-i})\neq \phi$ and we have a contradiction. Also the set $\{l_j(v) : j\in [k]\}$ has dimension $\geq k-1$ since the dimension could fall only by $1$ when we go modulo a linear form (project to hyperplane). This means that $rank(Sim({\mathcal{C}}^\prime))\geq k-1\geq
c_{{\mathbb{R}}}(3)+1$.
[**If $Sim({\mathcal{C}}^\prime)$ were not minimal**]{} $\Rightarrow {\mathcal{C}}^\prime$ is not minimal $\Rightarrow$ one of it’s gates would be $0$. Since $l\notin {\mathcal{L}}(T_0)\cup{\mathcal{L}}(T_1) \Rightarrow \alpha_0T_0(v)+\alpha_1T_1(v) \equiv 0
\Rightarrow$ for every $j\in [k]$ there exist $l_{(1-i)j}\mid T_{1-i}$ such that $l_{(1-i)j}(v),l_j(v)$ are LD. $\Rightarrow sp(\{l,l_j\})\cap {\mathcal{L}}(T_{1-i}) \neq \phi$ for $j\in [k]$, a contradiction to our assumption.
[**If $Sim({\mathcal{C}}^\prime)$ were minimal**]{}, we have an identically zero simple minimal circuit $Sim({\mathcal{C}}^\prime)$ with $rank(Sim({\mathcal{C}}^\prime))\geq c_{{\mathbb{R}}}(3)+1 $ contradicting Theorem \[rankbound\].
So our assumption is wrong and $sp(\{l,l_j\})\cap {\mathcal{L}}(T_{1-i}) \neq \phi$ for some $j\in [k]$.
3. Let $l\in {\mathcal{C}}\setminus ({\mathcal{L}}(T_0)\cup {\mathcal{L}}(T_1))$. Consider a set $\{l_1,\ldots,l_{k+2}\}\subset {\mathcal{L}}(T_i)$ of $k+2$ LI linear forms. By the above argument there exist three distinct elements in this set say $l_1,l_2,l_3$ such that $sp(\{l_j,l\})\cap {\mathcal{L}}(T_{1-i})\neq \phi$ for $j\in [3]$. Let $\{l_1^\prime,l_2^\prime,l_3^\prime\} \subset {\mathcal{L}}(T_{1-i})$ such that $l_j^\prime \in sp(\{l_j,l\})$ for $j\in [3]$. Then $gcd(l_j,l_j^\prime)=1$ implies that $l\in sp(\{l_j,l_j^\prime\})$ for $j\in [3]$. Since $l,l_j,l_j^\prime$ are all standard (coefficient of $x_1$ is $1$), Lemma \[spantoflat\] tells us $$l\in fl(\{l_j,l_j^\prime\})$$ for $j\in [3]$. So $l$ lies on the lines $\vec{L_j} = fl(\{l_j,l_j^\prime\})$ for $j\in [3]$. Atleast two of these lines should be distinct otherwise $dim(\{l_1,l_2,l_3\})\leq 2$ which is a contradiction. So $l$ is the intersection of these two lines. There are $M^2$ such lines and so $M^4$ such intersections. If $l\in {\mathcal{L}}(T_0)\cup {\mathcal{L}}(T_1)$ we have $\leq 2M$ other possibilities. So $|{\mathcal{C}}|\leq M^4+2M = O(d^4)$.
Let’s now give an algorithm to construct this set.
Constructing the set ${\mathcal{C}}$
------------------------------------
Here is an algorithm to construct the set ${\mathcal{C}}$. An explanation is given in the lemma below.
\[candidatealgo\]
Define ${\mathcal{C}}=\phi;$ Use polynomial factorization from [@KalTr90] to find $Lin(f)$ Consider polynomial $h=\frac{f}{Lin(f)}$ Let $a_2,\ldots,a_r$ be variables. Compute coefficient vector [**b**]{} of $h(a_2x_2+\ldots+a_rx_r, x_2,\ldots,x_r)$. Consider the polynomials $\{F_i, i\in [m]\}$ constructed in Corollary \[variety\]. Using your favorite algorithm (eg. Buchberger’s [@Buchberger76]) to solve polynomial equations, find all complex solutions to the system $\{F_i({\bf b})=0, i\in [m]\}$. For each solution $(a_2,\ldots,a_r) \in {\mathbb{R}}^r$ do : ${\mathcal{C}}= {\mathcal{C}}\cup \{(1,a_2,\ldots,a_r)\}$
Given a polynomial $f \in {\mathbb{R}}[x_1,\ldots,x_r]$ of degree $d$ in $r$ independent variables which admits a $\Sigma\Pi\Sigma_{\mathbb{R}}(2)[x_1,\ldots,x_r]$-representation : $f =
\prod\limits_{i\in [d-M]}G_i(\alpha_0\prod\limits_{j\in[M]}l_{0j} +
\alpha_1\prod\limits_{k\in[M]}l_{1k} )$ such that $G_t,l_{ij} (t\in [d-M], i\in \{0,1\}, j\in [M])$ are standard w.r.t. the standard basis $\{x_1,\ldots,x_n\}$ then we can find in deterministic time $poly(d)$, the corresponding candidate set ${\mathcal{C}}$ (see Definition \[candidatedef\]) described above.
*Proof.* The proof also contains an explanation of the algorithm above
- Let $l = x_1-a_2x_2-\ldots -a_rx_r \in {\mathcal{C}}$ be a candidate linear form. We know that $h(a_2x_2+\ldots + a_rx_r,x_2,\ldots,x_r)\in
\Pi\Sigma^{d_h}_{\mathbb{R}}[x_2,\ldots,x_r]\subset \Pi\Sigma^{d_h}_{\mathbb{C}}[x_1,\ldots,x_r]$.
- Using Theorem \[variety\] we know that $h(a_2x_2+\ldots + a_rx_r,x_2,\ldots,x_r) \in
\Pi\Sigma^{d_h}_{\mathbb{C}}[x_2,\ldots,x_r]\Leftrightarrow$ for the coefficient vector ${\bf b}$ of $h(a_2x_2+\ldots + a_rx_r,x_2,\ldots,x_r)$ inside ${\mathbb{C}}[x_2,\ldots,x_r]$ satisifes $F_1({\bf b})=\ldots=F_m({\bf b})=0$ for the polynomials $\{F_i : i\in [m]\}$ obtained in Corollary \[variety\]. .
- For any $t\leq d_h$, computing $(a_2x_2+\ldots + a_rx_r)^t$ requires $poly(t^r)$ time and it also has $poly(t^r)$ terms and degree $t$. Multiplying such powers to other variables and adding $poly(d_h^r)$ many such expressions also requires $poly(d_h^r)$ time. Hence computing the coefficient vector [**b**]{} takes polynomial time since $r$ is a constant. Each co-ordinate of this coefficient vector is a polynomial in $r-1$ variables $(a_2,\ldots,a_r)$ of degree $poly(d_h^r)$.
- Now we think of the $a_i$’s as our unknowns and obtain them by solving the polynomial system $\{F_i({\bf b}) =0, i\in [m]\}$. The number of polynomials is $m=poly(d^r)$ and degrees are $poly(d)$. $F_i$’s are polynomials in $poly(d^r)$ variables. Expanding $F_i({\bf b})$ will clearly take $poly(d^r)$ time and now we will have $poly(d^r)$ polynomials in $r$ variables of degrees $poly(d^r)$. Note that $r=O(1)$ and so we need to solve $poly(d)$ polynomials of degree $poly(d)$ in constant many variables. Also Claim \[candidate\] implies that the number of solutions $\leq M^4+2M = O(poly(d))$. So using Buchberger’s algorithm [@Buchberger76] we can solve the system for $(a_2,\ldots,a_r)$ in $poly(d)$ time. Once we have the solutions we consider only those linear forms which are in ${\mathbb{R}}[x_1,\ldots,x_r]$ and add them to ${\mathcal{C}}$.
Proofs from Subsection \[reducefactors\]
========================================
\[spuriousliproof\] Let $(S = \{l_{1}\ldots, l_{k}\},D)$ be a Detector pair in ${\mathcal{L}}(T_i)$. Let $l_{k+1}\in D$. For a $standard$ linear form $l\in V$, if $l\mid g$ then $l\notin sp(\{l_{1},\ldots,l_{k}\})$ .
*Proof.* Assume $l\mid g$ and $l\in sp(\{l_{1},\ldots,l_{k}\})$. Let $W = sp(\{l\})$, extend it to a basis and in the process obtain $W^\prime$ such that $W\oplus
W^\prime = V$. We get $$\pi_{W^\prime}(\alpha_0T_0 + \alpha_1T_1) =0$$ $\pi_{W^\prime}(\alpha_iT_i)\neq 0$ (i.e. $l\nmid T_0T_1$), otherwise $l$ divides both $T_0,T_1$ and $gcd(T_0,T_1)$ won’t be $1$. So we have an equality of non zero $\Pi\Sigma$ polynomials $$\alpha_0\prod\limits_{j=1}^M\pi_{W^\prime}(l_{0j}) =
-\alpha_1\prod\limits_{j=1}^M\pi_{W^\prime}(l_{1j})$$
Therefore there exists a permutation $\theta : [M] \rightarrow [M]$ such that $\{\pi_{W^\prime}(l_{(1-i)j}), \pi_{W^\prime}(l_{i\theta(j)})\}$ are LD $\Rightarrow l\in sp(\{l_{(1-i)j}, l_{i\theta(j)} \})$. Since $l\nmid T_0T_1$ this also means that $l_{(1-i)j}\in sp(\{l,l_{i\theta(j)}\})$ and $l_{i\theta(j)}\in sp(\{l,l_{(1-i)j}\})$.\
In particular there is an $l_{k+1}^\prime\in {\mathcal{L}}(T_{1-i})$ such that $l_{k+1}^\prime\in
sp(\{l,l_{k+1}\})$ and $l_{k+1}\in sp(\{l,l_{k+1}^\prime\})$.\
Since $l\in sp(\{l_{1},\ldots,l_{k}\})\Rightarrow l_{k+1}^\prime\in
sp(\{l_{1},\ldots,l_{k},l_{k+1}\})$. All linear forms here are standard(i.e. coefficient of $x_1$ is $1$) and so by Lemma \[spantoflat\], $l_{k+1}^\prime \in fl(\{l_1,\ldots,l_k,l_{k+1}\})$. Below we use the definition of detector pair and get $$l_{k+1}^\prime\in fl(\{l_{1},\ldots,l_{k},l_{k+1}\})\cap {\mathcal{L}}(T_{1-i})\subseteq
fl(\{l_{1},\ldots,l_{k}\})$$ And $l_{k+1}\in sp(\{l,l_{k+1}^\prime\})\Rightarrow l_{k+1}\in
sp(\{l_1,\ldots,l_k\})$ which is a contradiction to $(S,D)$ being a detector pair..
\[spuriousextraproof\] Let $l \in Lin_{\mathbb{R}}[{\bar{x}}]$ be $standard$ such that $l \mid g$ and ${\mathcal{C}}$ be the candidate set. Assume $(S = \{l_{1},\ldots,l_{k}\}, D(\neq \phi))$ is a Detector pair in ${\mathcal{L}}(T_i)$. Then $|{\mathcal{L}}(T_{1-i}) \cap (fl(S\cup\{l\}) \setminus
fl(S))|\geq 2$. That is the flat $fl(\{l_{1},\ldots,l_{k},l\})$ contains atleast two distinct points from ${\mathcal{L}}(T_{1-i})(\subseteq {\mathcal{C}})$ outside $fl(\{l_1,\ldots,l_k\})$.
*Proof.* From the previous claim we know that $\{l_{1},\ldots,l_{k},l\}$ is an LI set. Also like above we know there exists $l_{j}^\prime \in {\mathcal{L}}(T_{1-i}), j\in [3]$ such that $l_{j} \in sp(\{l, l_{j}^\prime\}), l_{j}^\prime \in
sp(\{l, l_{j}\})$. Since $\{l_{1},l_{2},l_{3}\}$ are LI, atleast two of the $l_{j}^\prime$’s, $j\in [3]$ must be distinct, otherwise $sp(\{l_{1},l_{2},l_{3}\})\subset sp(\{l,l_{1}^\prime\})$ which is not possible as LHS has dimension $3$ and RHS has dimension $2$. Thus there exist two distinct $l_{1}^\prime, l_{2}^\prime \in sp(\{l_{1},l_{2},l_{3},l\})\subset
sp(\{l_{1},\ldots,l_{k},l\})$. Note that $l_1,\ldots,l_k,l,l_1^\prime,l_2^\prime$ are all standard (i.e. coefficient of $x_1$ is $1$) and so by Lemma \[spantoflat\] $$l_j^\prime \in fl(\{l_1,\ldots,l_k,l\})$$ for $j\in [2]$.\
If for any $j\in [2]$, $l_j^\prime \in
sp(\{l_1,\ldots,l_k\})$ then $l\in sp(\{l_j,l_j^\prime\}) \Rightarrow l\in sp(\{l_1,\ldots,l_k\})$ which is a contradiction. This also shows that $l_j^\prime \notin fl(\{l_1,\ldots,l_k\})$ for $j\in [2]$.\
From what we showed above we may conclude: $$l_j^\prime \in fl(\{l_1,\ldots,l_k,l\})\setminus fl(\{l_1,\ldots,l_k\})$$ for $j\in [2]$. Hence Proved.
\[prooffilteredfactor\] The following are true:
1. If $l\mid I$ (i.e. $l$ was identified) then $l\in {\mathcal{L}}(G)\setminus
{\mathcal{L}}(g) $.
2. \[retainedfactor\]If $l\mid G^\star$ (i.e. $l$ was retained) then $(fl(\{l_{1},\ldots,l_{k},l\})\setminus fl(\{l_{1},\ldots,l_{k}\})) \cap
({\mathcal{L}}(T_{1-i})\cup ({\mathcal{L}}(T_i)\setminus D)) \neq \phi $ that is
$(fl(\{l_{1},\ldots,l_{k},l\})\setminus fl(\{l_{1},\ldots,l_{k}\}))$ contains a point from ${\mathcal{L}}(T_i)\setminus D$ or ${\mathcal{L}}(T_{1-i})$.
3. \[retaineddetector\] If $l\mid G^\star$ and $l_{k+1}\in D$ then $l \notin sp(\{l_{1},\ldots,l_{k},l_{k+1}\})$.
*Proof.*
1. Assume $l\mid I$ (i.e. $l$ was identified) and $l\mid g$. Then by Claim \[spuriousli\] we know that $\{l_{1},\ldots,l_{k},l\}$ are LI and so the first $"if"$ condition is true. By Claim \[spuriousextra\] we know that there are two other points $\{l_1^\prime,l_2^\prime\} \subset {\mathcal{C}}\cap (fl(\{l_{1},\ldots,l_{k},l\})\setminus fl(\{l_{1},\ldots,l_{k}\}))$, so the second $"if"$ condition will also be true and thus $l$ will not be identified which is a contradiction. Therefore $l\in {\mathcal{L}}(G)\setminus{\mathcal{L}}(g)$.
2. Assume $l\mid G^\star$ (i.e. $l$ was not identified). This means both $"if"$ statements were true for $l$. Thus $\{l_{1},\ldots,l_{k},l\}$ is LI. Also there exist distinct $\{l_1^\prime, l_2^\prime\} \in {\mathcal{C}}\cap (fl(\{l_{1},\ldots,l_{k},l\})\setminus
fl(\{l_{1},\ldots,l_{k}\}))$. If $$l_1^\prime \in ({\mathcal{L}}(T_{1-i})\cup ({\mathcal{L}}(T_i)\setminus D)) \text{ or }
l_2^\prime \in ({\mathcal{L}}(T_{1-i})\cup ({\mathcal{L}}(T_i)\setminus D))$$ we are done so assume both are in $${\mathcal{C}}\setminus(({\mathcal{L}}(T_{1-i})\cup ({\mathcal{L}}(T_i)\setminus D)))) = ({\mathcal{C}}\setminus
({\mathcal{L}}(T_i)\cup {\mathcal{L}}(T_{1-i})))\cup D$$ If one of them say $l_1^\prime \in {\mathcal{C}}\setminus({\mathcal{L}}(T_i)\cup {\mathcal{L}}(T_{1-i}))$, then by Part \[candidateprop\] of Claim \[candidate\], for some $j\in [k]$, $
sp(\{l_1^\prime,l_{j}\})\cap {\mathcal{L}}(T_{1-i}) \neq \phi$. Let $\tilde l_{j}\in
sp(l_1^\prime,l_{j})\cap {\mathcal{L}}(T_{1-i}) \Rightarrow$ $$\tilde l_{j}\in sp(\{l_1^\prime ,l_{j}\})\subseteq sp(\{l_{1},\ldots,l_{k},l\})$$ Since all linear forms $\tilde l_j,l_1,\ldots,l_k,l$ are standard (coefficient of $x_1$ is $1$) by Lemma \[spantoflat\] $$\tilde l_j \in fl(\{l_{1},\ldots,l_{k},l\})$$
Also $\tilde l_{j},l_{j}$ are LI and $\tilde l_{j}\in sp(\{l_1^\prime ,l_{j}\})$ together imply $l_1^\prime\in sp(\{l_{j},\tilde l_{j}\})$. Note that $l_1^\prime \notin fl(\{l_{1},\ldots,l_{k}\})\Rightarrow l_1^\prime\notin sp(\{l_{1},\ldots,l_{k}\})$ which along with $l_1^\prime\in sp(\{l_{j},\tilde l_{j}\})$ will then give $$\tilde l_{j}\notin sp(\{l_{1},\ldots,l_{k}\})$$
So we found $\tilde l_{j}\in {\mathcal{L}}(T_{1-i})\cap
(fl(\{l_{1},\ldots,l_{k},l\})\setminus fl(\{l_{1},\ldots,l_{k}\})) $ and we are done.\
So the only case that remains now is that $l_1^\prime,l_2^\prime \in D$. Let’s complete the proof in the following steps
- $l_1^\prime\in fl(\{l_{1},\ldots,l_{k},l\})\setminus fl(\{l_{1},\ldots,l_{k}\})
\Rightarrow l\in sp(\{l_{1},\ldots,l_{k},l_1^\prime\})$
- Using the above bullet, $l_2^\prime \in fl(\{l_1,\ldots,l_k,l\})\Rightarrow l_2^\prime \in sp(\{l_{1},\ldots,l_{k},l_1^\prime\})$. Linear forms $l_2^\prime, l_1,\ldots,l_k,l$ are standard (coefficient of $x_1$ is $1$) so using Lemma \[spantoflat\], $l_2^\prime \in fl(\{l_{1},\ldots,l_{k},l_1^\prime\})$
- $l_2^\prime\in D \Rightarrow l_2^\prime\notin fl(\{l_{1},\ldots,l_{k}\})$
- The above two bullets and $\{l_1^\prime,l_2^\prime\}\subset {\mathcal{L}}(T_i)$ tell us that $fl(\{l_1,\ldots,l_k,l_1^\prime\})$ is not elementary which is a contradiction.
So atleast one of $l_1^\prime,l_2^\prime$ is inside ${\mathcal{L}}(T_{1-i})\cup ({\mathcal{L}}(T_i)\setminus D)$
3. Let $l_{k+1}\in D$ and $l\in sp(\{l_{1},\ldots,l_{k},l_{k+1}\})$. Since $l,l_1,\ldots,l_k,l_{k+1}$ are standard, by Lemma \[spantoflat\], $l\in fl(\{l_{1},\ldots,l_{k},l_{k+1}\})$. Clearly $l\notin fl(\{l_{1},\ldots,l_{k}\})$ otherwise it would get identified at the first $"if"$. Therefore $l\in fl(\{l_1,\ldots,l_k,l_{k+1}\})\setminus fl(\{l_1,\ldots,l_k\})$ By Part $2$ above let $l_1^\prime
\in (fl(\{l_{1},\ldots,l_{k},l\})\setminus fl(\{l_{1}\ldots,l_{k}\})) \cap ({\mathcal{L}}(T_{1-i})\cup
({\mathcal{L}}(T_i)\setminus D))$. So $l_1^\prime \in {\mathcal{L}}(T_{1-i})$ or $l_1^\prime \in {\mathcal{L}}(T_i)\setminus D$.\
This tells us that $l_1^\prime \in sp(\{l_{1},\ldots,l_{k},l_{k+1}\})\setminus fl(\{l_{1},\ldots,l_{k}\})$. All linear forms $l_1^\prime,l_1,\ldots,l_k,l_{k+1}$ are standard (i.e. coefficients of $x_1$ is $1$) so by Lemma \[spantoflat\] we get that $l_1^\prime \in fl(\{l_{1},\ldots,l_{k},l_{k+1}\})\setminus fl(\{l_{1},\ldots,l_{k}\})$. Now using the definition of detector pair $l_1^\prime \notin {\mathcal{L}}(T_{1-i})$ since $fl(\{l_{1},\ldots,l_{k},l_{k+1}\})\cap {\mathcal{L}}(T_{1-i})
\subseteq fl(\{l_{1},\ldots,l_{k}\})$ . The flat $fl(\{l_1,\ldots,l_k,l_{k+1}\})$ is elementary in ${\mathcal{L}}(T_i)$, so $l_1^\prime$ can belong here only if $l_1^\prime=l_{k+1}$ which is not possible since $l_1^\prime \notin D$. So we have a contradiction. Hence Proved.
\[detectorexpansionproof\] Let $(S=\{l_{1},\ldots,l_{k}\},D)$ be a detector in ${\mathcal{L}}(T_i)$. For each $(l,l_j) \in {\mathcal{C}}\times S$ define the space $U_{\{l,l_j\}} = sp(\{l,l_j\})$. Extend $\{l,l_j\}$ to a basis and in the process obtain $U_{\{l,l_j\}}^\prime$ such that $V = U_{\{l,l_j\}}\oplus U_{\{l,l_j\}}^\prime$. Define the set: $$X = \{l\in {\mathcal{C}}: \pi_{U^\prime_{\{l,l_j\}}}(f^\star) \neq 0, \text{ for all }
l_j\in S\}$$ Then $D\subset X\subset {\mathcal{L}}(T_i)$.
*Proof.* Consider $l_{k+1} \in D$. Since $D\subset
{\mathcal{L}}(T_i)
\Rightarrow l_{k+1}\in {\mathcal{C}}$. Assume $l_{k+1}\notin X$, so there exists a $j\in
[k]$ such that $\pi_{U^\prime_{\{l_{k+1},l_j\}}}(f^\star)=0$. That is: $$\pi_{U_{\{l_{k+1},l_j\}}^\prime}(G^\star(\alpha_0T_0+\alpha_1T_1))=0.$$So $$\prod\limits_{t\in
[N_1]}\pi_{U_{\{l_{k+1},l_j\}}^\prime}(G_t)(\alpha_0\prod\limits_{s\in
[M]}\pi_{U_{\{l_{k+1},l_j\}}^\prime}(l_{0s})+ \alpha_1\prod\limits_{s\in
[M]}\pi_{U_{\{l_{k+1},l_j\}}^\prime}(l_{1s})) = 0$$ Now $$l_{j}\in {\mathcal{L}}(T_i)\Rightarrow
\pi_{U_{\{l_{k+1},l_j\}}^\prime}(T_i)=0\Rightarrow
\prod\limits_{t\in [N_1]}\pi_{U_{\{l_{k+1},l_j\}}^\prime}(G_t)\prod\limits_{s\in
[M]}\pi_{U_{\{l_{k+1},l_j\}}^\prime}(l_{(1-i)s})= 0.$$ Since $G_t \mid G^\star$, by Part \[retaineddetector\] of Lemma \[filteredfactor\] $\pi_{U_{\{l_{k+1},l_j\}}^\prime}(G_t)\neq 0$ for all $t\in [N_1]$. If for some $s\in [M]$, $\pi_{U_{\{l_{k+1},l_j\}}^\prime}(l_{(1-i)s})=0$ then $l_{(1-i)s}\in sp(\{l_{j},l_{k+1}\})\Rightarrow l_{(1-i)s}\in
sp(\{l_{1},\ldots,l_{k},l_{k+1}\})
\Rightarrow l_{(1-i)s}\in sp(\{l_{1},\ldots,l_{k}\}) $ (by definition of Detector Pair in \[detectorset\]).
$$l_{(1-i)s} \in sp(\{l_j,l_{k+1}\}) \text{ and } \{l_{(1-i)s},l_j\} \text{ LI }
\Rightarrow l_{k+1}\in sp(\{l_{(1-i)s},l_j\})$$
This means $l_{k+1}\in sp(\{l_{1},\ldots,l_{k},l_{(1-i)s}\})\subset
sp(\{l_1,\ldots,l_k\})$ which is a contradiction to $l_{k+1}\in D$. So $\pi_{U_{\{l_{k+1},l_j\}}^\prime}(f^\star)\neq 0$ for all $j\in [k]
\Rightarrow l_{k+1}\in X$. Therefore $D\subset X$.\
Consider $l\in X$. We need to show $l\in
{\mathcal{L}}(T_{i})$. We already know $l\in {\mathcal{C}}$.
- If $l\in {\mathcal{L}}(T_{1-i})$, then $\pi_{U_{\{l,l_j\}}^\prime}
(f^\star) = 0$ for all $j\in [k]$ since $l\mid T_{1-i}$ and $l_j\mid T_i$. Contradiction to $l\in X$.
- If $l\in {\mathcal{C}}\setminus ({\mathcal{L}}(T_i)\cup {\mathcal{L}}(T_{1-i}))$ by Part \[candidateprop\] of Claim \[candidate\] we know that there exists $j\in [k]$ such that $sp(\{l_{j},l\})\cap {\mathcal{L}}(T_{1-i}) \neq \phi$. Let $l_{j}^\prime \in sp(\{l_{j},l\})\cap {\mathcal{L}}(T_{1-i})$. We show that $sp(\{l_j^\prime,l_j\}) = sp(\{l_j,l\}) = U_{\{l_j,l\}}$.
- $l_j^\prime \in sp(\{l_j,l\})\Rightarrow sp(\{l_j^\prime,l_j\})\subset
sp(\{l_j,l\})$.
- Let $l_j^\prime = \alpha l_j + \beta l$. We know that $\{l_j,l_j^\prime\}$ are LI since $l_j\in {\mathcal{L}}(T_i)$ and $l_j^\prime \in {\mathcal{L}}(T_{1-i})$. So $\beta\neq
0 \Rightarrow l\in sp(\{l_j^\prime,l_j\})\Rightarrow sp(\{l,l_j\}) \subset
sp(\{l_j^\prime,l_j\}) \Rightarrow sp(\{l,l_j\})=sp(\{l_j^\prime,l_j\})$.
Use the same extension for $sp(\{l,l_j\})=sp(\{l_j^\prime,l_j\})= U_{\{l_j,l\}}$ to get $\pi_{U_{\{l,l_j\}}^\prime}(f^\star)=\pi_{U_{\{l_j^\prime,l_j\}}^\prime}
(f^\star) =0$ (since $l_j^\prime \mid T_{1-i}$ and $l_j\mid T_i$). Contradiction to $l\in X$.
Therefore $l\in {\mathcal{L}}(T_i) \Rightarrow X\subset {\mathcal{L}}(T_i)$.
Proofs from Subsection \[hardcase\]
===================================
\[calculationproof\] The following is true
$$\frac{(2-v(\delta,\theta))}{v(\delta,\theta)}\leq \frac{1-\delta}{\delta}$$
*Proof.* Note that $$\frac{(2-v(\delta,\theta))}{v(\delta,\theta)} =
\begin{cases}
\hfill \frac{1+\delta + \theta}{1-\delta-\theta} \hfill & \text{ if
$|{\mathcal{L}}(T_{0})|\leq \theta |{\mathcal{L}}(T_1)|$} \\
\hfill \frac{3-(1-\delta)(1+\theta)}{(1-\delta)(1+\theta)-1} \hfill &
\text{ if $\theta |{\mathcal{L}}(T_1)| < |{\mathcal{L}}(T_0)| \leq |{\mathcal{L}}(T_1)|$} \\
\end{cases}$$ By simple computation $\delta \in (0, \frac{7-\sqrt{37}}{6})$ gives $$3\delta^2-7\delta +1 >0 \Rightarrow 0 <
\frac{3\delta}{1-\delta} < 1-3\delta < 1 \Rightarrow \frac{1+\delta +
\theta}{1-\delta-\theta}<
\frac{1-\delta}{\delta}$$
Also $$\theta > \frac{3\delta}{1-\delta} \Rightarrow
\frac{3-(1-\delta)(1+\theta)}{(1-\delta)(1+\theta)-1} <
\frac{1-\delta}{\delta}$$
\[largedetectorproof\] Let $k=c_{{\mathbb{R}}}(3)+2$ (see defn of $c_{\mathbb{R}}(k)$ in Theorem \[rankbound\]). Fix $\delta, \theta$ in range given in Claim \[calculation\] above . Then for some $i\in \{0,1\}$ there exists a Detector Pair $(S=\{l_{1},\ldots,l_{k}\},D)$ in ${\mathcal{L}}(T_i)$ with $|D|\geq v(\delta,\theta) \max(|{\mathcal{L}}(T_{0})|,|{\mathcal{L}}(T_{1})|)$.
*Proof.* We assume $|{\mathcal{L}}(T_0)|\leq {\mathcal{L}}(T_1)$. The other case gives the same result for(maybe) a different value of $i$ . We will consider linear forms as points in the space ${\mathbb{R}}^r$. Let’s consider the two cases used in the definition of $v(\delta,\theta)$.
- Since $dim({\mathcal{L}}(T_1)) \geq r-1 \geq C_{2k-1} > C_k$ (See Section \[incidence\] for definition of $C_{k}$) by Corollary \[elementary\] there exists a set $S$ of $k$ LI points say $S =\{l_{1},\ldots,l_{k}\}
\subseteq {\mathcal{L}}(T_1)$ and a set $Z \subseteq {\mathcal{L}}(T_1)$ of size $\geq (1-\delta)|{\mathcal{L}}(T_1)|$ such that for any $l_{k+1}\in Z$
- $l_{k+1}\notin fl(\{l_{1},\ldots,l_{k}\})$.
- $fl(\{l_{1},\ldots,l_{k},l_{k+1}\})$ is elementary in ${\mathcal{L}}(T_1)$.
Next we define our set $D$ according to the condition we needed in the definition of detector (See Subsection \[detectorset\]). $$D{\stackrel{def}{=}}\{ l_{k+1} \in Z : fl(\{l_{1},\ldots,l_{k},l_{k+1}\}) \cap {\mathcal{L}}(T_{0})
\subset fl(\{l_{1},\ldots,l_{k}\})\}$$ In the following lines we will show that this set $D$ has large size, to be precise: $$|D|\geq (1-\delta-\theta)|{\mathcal{L}}(T_1)|$$ We do this in steps:
1. First we define a special subset of $Z$ $$\tilde Z = \{l_{k+1}\in Z : (fl(\{l_1,\ldots,l_{k+1}\})\setminus fl(\{l_1,\ldots,l_k\})) \cap {\mathcal{L}}(T_0) \neq \phi\}$$ We claim that $Z\setminus \tilde Z \subset D$. Let $l_{k+1}\in Z\setminus \tilde{Z}\Rightarrow (fl(\{l_1,\ldots,l_{k+1}\})
\setminus fl(\{l_1,\ldots,l_k\})) \cap {\mathcal{L}}(T_0) = \phi \Rightarrow fl(\{l_1,\ldots,l_{k+1}\})\cap {\mathcal{L}}(T_0)\subset fl(\{l_1,\ldots,l_k\})$ and so $l_{k+1}\in D$.
2. Next we show that for distinct $l_{k+1},\tilde l_{k+1}\in Z (\subseteq {\mathcal{L}}(T_1))$ $$(fl(\{l_{1},\ldots,l_{k},l_{k+1}\})\setminus fl(\{l_{1},\ldots,l_{k}\}))\cap (fl(\{l_{1},\ldots,l_{k},\tilde l_{k+1}\})\setminus fl(\{l_{1},\ldots,l_{k}\})) =\phi$$ If not then there exist scalars $\mu_j,\nu_j,j\in [k+1]$ such that $$\nu_1 l_1 + \ldots \nu_k l_k + \nu_{k+1}l_{k+1} = \mu_1 l_1 + \ldots \mu_k l_k + \mu_{k+1}\tilde l_{k+1}$$ with $\nu_{k+1}\neq 0$ implying that $l_{k+1}\in sp(\{l_{1},\ldots,l_{k},\tilde l_{k+1}\})$. Since all linear forms are $standard$ this implies $l_{k+1}\in fl(\{l_{1},\ldots.l_{k},\tilde l_{k+1}\})$ (See Lemma \[spantoflat\]). Also $l_{k+1}\in Z\Rightarrow l_{k+1}\notin fl(\{l_1,\ldots,l_k\})$. Together this means that $l_{k+1}\in fl(\{l_1,\ldots,l_k,\tilde l_{k+1}\})\setminus fl(l_1,\ldots,l_k)$ and we arrive at a contradiction to $fl(\{l_1,\ldots,l_k,\tilde l_{k+1}\})$ being elementary.
3. From what we showed above every $l\in {\mathcal{L}}(T_0)$ can belong to atmost one of the sets $fl(\{l_1,\ldots,l_{k+1}\})\setminus fl(\{l_1,\ldots,l_k\})$ with $l_{k+1}\in Z$ (since intersection between two such sets is $\phi$) and therefore there can be atmost $|{\mathcal{L}}(T_0)|$ such $l_{k+1}$’s in $\tilde Z$ $\Rightarrow$ $|\tilde Z| \leq |{\mathcal{L}}(T_0)|$.
So we get : $$|D|\geq |Z|-|{\mathcal{L}}(T_{0})|\geq (1-\delta-\theta)|{\mathcal{L}}(T_1)|$$ $(S,D)$ is a detector pair in ${\mathcal{L}}(T_1)$ by the choice of $Z$ and $D$.
- Since $dim({\mathcal{L}}(T_0)\cup {\mathcal{L}}(T_{1})) = r > C_{2k-1}$, by Corollary \[elementary\] we know that there exist $2k-1$ independent points $l_1,\ldots,l_{2k-1} \in {\mathcal{L}}(T_0)\cup
{\mathcal{L}}(T_{1})$ and a set $Z\subseteq {\mathcal{L}}(T_0)\cup {\mathcal{L}}(T_{1})$ of size $\geq
(1-\delta)(|{\mathcal{L}}(T_0)|+|{\mathcal{L}}(T_{1})|) $ such that for all $l\in Z$
- $l\notin fl(\{l_1,\ldots,l_{2k-1}\})$.
- $fl(\{l_1,\ldots,l_{2k-1}, l\})$ is elementary in ${\mathcal{L}}(T_0)\cup
{\mathcal{L}}(T_{1})$.
By pigeonhole principle, $k$ of the $\{l_j\}_{j=1}^{2k-1}$ points must belong to either ${\mathcal{L}}(T_0)$ or ${\mathcal{L}}(T_{1})$. Let’s assume they belong to ${\mathcal{L}}(T_i)$ (for some $i\in \{0,1\}$) (say the points are $l_{1},\ldots,l_{k}$), then consider $D = Z\cap {\mathcal{L}}(T_i)$. Clearly for every $l\in D$, $l\notin fl(\{l_{1},\ldots,l_{k}\})$ and $fl(\{l_{1},\ldots,l_{k},l\})$ is elementary in ${\mathcal{L}}(T_0)
\cup {\mathcal{L}}(T_{1})$. This immediately tells us that $(S = \{l_{1},\ldots,l_{k}\},D)$ satisfies all properties of being a detector pair in ${\mathcal{L}}(T_i)$. We defined $D = Z\cap {\mathcal{L}}(T_i)$. Since $Z\subseteq {\mathcal{L}}(T_i)\cup {\mathcal{L}}(T_{1-i})$ we have $Z = (Z\cap {\mathcal{L}}(T_i))\cup (Z\cap {\mathcal{L}}(T_{1-i})) \subset D \cup {\mathcal{L}}(T_{1-i})$ giving $$|D| + |{\mathcal{L}}(T_{1-i})| \geq |Z| \Rightarrow |D| \geq |Z| - |{\mathcal{L}}(T_{1-i})|\geq
(1-\delta)(|{\mathcal{L}}(T_0)|+|{\mathcal{L}}(T_{1})|)-|{\mathcal{L}}(T_{1-i})|$$ $$\geq
((1-\delta)(1+\theta)-1)\max(|{\mathcal{L}}(T_0)|,|{\mathcal{L}}(T_1)|)$$
Combining the two cases we see that for some $i\in \{0,1\}$ there exists a Detector set $(S=\{l_{1},\ldots,l_{k}\},D)$ in ${\mathcal{L}}(T_i)$ with $|D|\geq v(\delta,\theta) \max(|{\mathcal{L}}(T_{0})|,|{\mathcal{L}}(T_{1})|)$.
\[findreconstructorproof\] The following are true:
1. $dim(\pi_{\tilde{W_0}}(\widehat{{\mathcal{L}}(U_{1-i}^\star)}))> C_4$
2. $\pi_{\tilde{W_0}}(\widehat{{\mathcal{L}}(U_{1-i}^\star)})\cap \pi_{\tilde{W_0}}(\widehat{D}) = \phi$
3. $|\pi_{\tilde{W_0}}(\widehat{{\mathcal{L}}(U_{1-i}^\star)})| \leq
\frac{1-\delta}{\delta}|\pi_{\tilde{W_0}}(\widehat{D})|$
*Proof.*
1. Since $dim(\widehat{{\mathcal{L}}(U_{1-i}^\star)})\geq r-1$ we get $dim(\pi_{\tilde{W_0}}(\widehat{{\mathcal{L}}(U_{1-i}^\star)})) \geq r-1-k >C_4$.
2. Assume $\exists$ $d_1 \in D, u \in {\mathcal{L}}(U_{1-i}^\star)$ such that $\pi_{\tilde{W_0}}(\widehat{d})=\pi_{\tilde{W_0}}(\widehat{u})
\Rightarrow \exists \lambda,\nu\in {\mathbb{R}}$ such that $\nu d_1+\lambda u \in
\tilde{W_0}$. Since $\pi_{\tilde W_0}(d_1)\neq 0$ both $\nu,\lambda\neq 0$. Thus $u\in sp(\{l_1,\ldots,l_k,d_1\})
\Rightarrow u \in fl(\{l_{1},\ldots,l_{k},d_1\})$ (using Lemma \[spantoflat\] since all linear forms involved are *standard* i.e. have coefficient of $x_1$ equal to $1$). Also $u\in {\mathcal{L}}(G^\star T_{1-i})\Rightarrow u\in
fl(\{l_{1},\ldots,l_{k},d_1\})\cap ({\mathcal{L}}(G^\star)\cup {\mathcal{L}}(T_{1-i}))$. We know from Part \[retainedfactor\] of Lemma \[filteredfactor\] that $fl(\{l_{1},\ldots,l_{k},d_1\})\cap {\mathcal{L}}(G^\star) = \phi \Rightarrow u \in
fl(\{l_{1},\ldots,l_{k},d_1\})\cap
{\mathcal{L}}(T_{1-i})\subseteq fl\{l_{1},\ldots,l_{k}\}$ because $(S,D)$ was a detector pair. But $u \in fl(\{l_{1},\ldots,l_{k}\})\Rightarrow d_1 \in sp(\{l_{1},\ldots,l_{k}\})$ which is a contradiction because $d_1 \in D$ and $(S,D)$ is a detector pair.
3. We first plan to show $\pi_{\tilde{W_0}}(\widehat{{\mathcal{L}}(U_{1-i}^\star)})\subset
\pi_{\tilde{W_0}}(\widehat{{\mathcal{L}}(T_{1-i})}) \cup
\pi_{\tilde{W_0}}(\widehat{{\mathcal{L}}(T_i)\setminus D})$. Clearly $U_{1-i}^\star\mid G^\star T_{1-i} \Rightarrow {\mathcal{L}}(U_{1-i}^\star)\subset
{\mathcal{L}}(G^\star T_{1-i})\Rightarrow
\pi_{\tilde{W_0}}(\widehat{{\mathcal{L}}(U_{1-i}^\star)})\subset
\pi_{\tilde{W_0}}(\widehat{{\mathcal{L}}(G^\star T_{1-i})}) \subset \pi_{\tilde{W_0}}(\widehat{{\mathcal{L}}(G^\star)})
\cup \pi_{\tilde{W_0}}(\widehat{{\mathcal{L}}(T_{1-i})})$. Now consider any $l\in {\mathcal{L}}(G^\star)$. We know that $(S_0 = \{l_{1},\ldots,l_{k}\},D)$ is a detector pair, so by Part \[retainedfactor\] of Lemma \[filteredfactor\] we get $$(fl(\{l_{1},\ldots,l_{k},l\})\setminus fl(\{l_{1},\ldots,l_{k}\})) \cap
({\mathcal{L}}(T_{1-i})\cup
({\mathcal{L}}(T_i)\setminus D)) \neq \phi$$
So there exists $l^\prime \in {\mathcal{L}}(T_{1-i})\cup ({\mathcal{L}}(T_i)\setminus D)$ such that $\pi_{\tilde{W_0}}(l), \pi_{\tilde{W_0}}(l^\prime)$ are both non-zero and are LD $\Rightarrow \pi_{\tilde{W_0}}(\widehat l)=
\pi_{\tilde{W_0}}(\widehat l^\prime)$ implying that $\pi_{\tilde{W_0}}(\widehat{{\mathcal{L}}(G^\star)})\subset
\pi_{\tilde{W_0}}({\arraycolsep=0pt\relax\begin{array}{c}
\stretchto{
\scaleto{
\scalerel*[\widthof{\ensuremath{{\mathcal{L}}(T_{1-i})\cup ({\mathcal{L}}(T_i)\setminus D)}}]{\kern-.5pt\bigwedge\kern-.5pt}
{\rule[-\textheight/2]{1ex}{\textheight}} }{\textheight} }{0.5ex}\\ {\mathcal{L}}(T_{1-i})\cup ({\mathcal{L}}(T_i)\setminus D)\\ \rule{-1ex}{0ex}
\end{array}
})$ giving us $\pi_{\tilde{W_0}}(\widehat{{\mathcal{L}}(U_{1-i}^\star)})\subset
\pi_{\tilde{W_0}}(\widehat{{\mathcal{L}}(T_{1-i})}) \cup
\pi_{\tilde{W_0}}(\widehat{{\mathcal{L}}(T_i)\setminus D})$ and therefore $$|\pi_{\tilde{W_0}}(\widehat{{\mathcal{L}}(U_{1-i}^\star)})|\leq|\pi_{\tilde{W_0}}(\widehat{{\mathcal{L}}(T_{1-i})})|
+ |\pi_{\tilde{W_0}}(\widehat{{\mathcal{L}}(T_i)\setminus D})|$$ Now we try to show $|\pi_{\tilde{W_0}}(\widehat{{\mathcal{L}}(T_i)\setminus D})| =
|\pi_{\tilde{W_0}}(\widehat{{\mathcal{L}}(T_i)})|-|D|$
1. It’s straightforward to see $\pi_{\tilde{W_0}}(\widehat{{\mathcal{L}}(T_i)}) =
\pi_{\tilde{W_0}}(\widehat D)\cup \pi_{\tilde{W_0}}(\widehat{{\mathcal{L}}(T_i)\setminus D})$. Also $\pi_{\tilde{W_0}}(\widehat{{\mathcal{L}}(T_i)\setminus D}) \cap
\pi_{\tilde{W_0}}(\widehat D)=\phi$. If not then there exists $l^\prime \in
{\mathcal{L}}(T_i)\setminus D, l^{\prime\prime} \in D$ such that $0\neq \pi_{\tilde{W_0}}(\widehat {l^{\prime\prime}}) = \pi_{\tilde{W_0}}(\widehat
l^\prime)\Rightarrow \pi_{\tilde{W_0}}(l^{\prime\prime}),\pi_{\tilde{W_0}}(l^\prime)$ are LD $\Rightarrow l^\prime \in sp\{l_{1},\ldots,l_{k},l^{\prime\prime}\}\setminus
sp\{l_{1},\ldots,l_{k}\} \Rightarrow$ (by Lemma \[spantoflat\]), $ l^\prime \in
fl\{l_{1},\ldots,l_{k},l^{\prime\prime}\}\setminus
fl\{l_{1},\ldots,l_{k}\}$ which is a contradiction to the flat being elementary inside ${\mathcal{L}}(T_i)$. So $|\pi_{\tilde{W_0}}(\widehat{{\mathcal{L}}(T_i)})| =
|\pi_{\tilde{W_0}}(\widehat D)| + |\pi_{\tilde{W_0}}(\widehat{{\mathcal{L}}(T_i)\setminus D})|$.
2. $\pi_{\tilde{W_0}}$ is injective on $\widehat D$. Let $\pi_{\tilde{W_0}}(\widehat{l^\prime})=\pi_{\tilde{W_0}}(\widehat{l^{\prime\prime}})$ for LI forms $\{l^\prime, l^{\prime\prime}\}\subset D$, then $l^\prime \in sp(\{l_1,\ldots,l_k,l^{\prime\prime}\})\Rightarrow$ (by Lemma \[spantoflat\]), $ l^\prime \in
fl(\{l_{1},\ldots,l_{k},l^{\prime\prime}\})$ and clearly $l^\prime \notin
fl\{l_{1},\ldots,l_{k}\}$ (since it’s in $D$), which is again a contradiction to the flat being elementary , thus $|\pi_{\tilde{W_0}}(\widehat D)| = |\widehat D| = |D|$ (since $D$ is a set of $normal$ linear forms ).
Combining these with Claim \[calculation\] and Lemma \[largedetector\] we get $$|\pi_{\tilde{W_0}}(\widehat{{\mathcal{L}}(U_{1-i}^\star)})|\leq 2 \max(|{\mathcal{L}}(T_0)|, |{\mathcal{L}}(T_1)|)-|D|\leq
(2-v(\delta,\theta))\max(|{\mathcal{L}}(T_0)|, |{\mathcal{L}}(T_1)|)$$ $\Rightarrow$ $$\frac{|\pi_{\tilde{W_0}}(\widehat{{\mathcal{L}}(U_{1-i}^\star)})|}{|\pi_{\tilde{W_0}}(\widehat D)|}
\leq
\frac{(2-v(\delta,\theta))}{v(\delta,\theta)}\leq \frac{1-\delta}{\delta}$$
\[newbichromaticproof\] Let $S_1 = \{d_1\}$ and $S_2 =
\{l_{k+2},\ldots,l_r\}$, $W_1=sp(S_1)$ and $W_2=sp(S_2)$. So $V = W_0 \oplus W_1
\oplus W_2$ and let $W_0^\prime=W_1\oplus W_2$. For $u\in {\mathcal{L}}(U_{1-i}^\star)$ such that $\pi_{\tilde W_0}(\widehat{u})\in
\vec L_1\cap \pi_{\tilde W_0}(\widehat {{\mathcal{L}}(U_{1-i}^\star)})$ consider the line $$\vec L_2 = fl(\{d_1, \pi_{W_0^\prime}(\widetilde{u}) \})$$ then $|\vec L_2 \cap \pi_{W_0^\prime}(\widetilde{D})|\geq 1$ and $|\vec L_2 \cap \pi_{W_0^\prime}(\widetilde{{\mathcal{L}}(U_{1-i}^\star)})|=1$, i.e. $\vec L_2$ is also a *“semiordinary bichromatic”* like $\vec L_1$.
*Proof.*
We first show the following : Let $u_2\in U_{1-i}^\star, d_2\in D$ then $$\pi_{W_0^\prime}(\widetilde{u_2})\neq \pi_{W_0^\prime}(\widetilde{d_2})$$
- Assume not, then $\exists$ $\nu, \lambda\in {\mathbb{R}}$ such that $\nu d_2+\lambda u_2 \in W_0$. $\nu,\lambda$ cannot be $0$ since this would mean $\pi_{W_0^\prime}(\widetilde{d_2})=0$. Thus $u_2\in sp(\{l_1,\ldots,l_k,d_2\})
\Rightarrow u_2 \in fl(\{l_{1},\ldots,l_{k},d_2\})$ ( using Lemma \[spantoflat\] since all linear forms involved are *standard* i.e. have coefficient of $x_1$ equal to $1$). Also $u_2\in {\mathcal{L}}(G^\star T_{1-i})\Rightarrow u_2\in
fl(\{l_{1},\ldots,l_{k},d_2\})\cap ({\mathcal{L}}(G^\star)\cup {\mathcal{L}}(T_{1-i}))$. We know from Part \[retainedfactor\] of Lemma \[filteredfactor\] that $fl(\{l_{1},\ldots,l_{k},d_2\})\cap {\mathcal{L}}(G^\star) = \phi \Rightarrow u_2 \in
fl(\{l_{1},\ldots,l_{k},d_2\})\cap
{\mathcal{L}}(T_{1-i})\subseteq fl\{l_{1},\ldots,l_{k}\}$ because $(S,D)$ was a detector pair. But $u_2 \in
fl(\{l_{1},\ldots,l_{k}\})\Rightarrow d_2 \in sp(\{l_{1},\ldots,l_{k}\})$ which is a contradiction because $d_2 \in D$ and $(S,D)$ is a detector pair.
Now let’s go back to proving this lemma.\
$|\vec L_2 \cap \pi_{W_0^\prime}(\widetilde{D})|\geq 1$ is clearly true since $d_1\in \vec L_2 \cap \pi_{W_0^\prime}(\widetilde{D})$. For the other part assume there exist $u_1\neq u$ inside ${\mathcal{L}}(U_{1-i}^\star)$ such that $\pi_{W_0^\prime}(\widetilde u), \pi_{W_0^\prime}(\widetilde u_1)$ are distinct points on $\vec L_2 \cap \pi_{W_0^\prime}
({\mathcal{L}}(U_{1-i}^\star))$ implying that the set $\{\pi_{W_0^\prime}(\widetilde u), \pi_{W_0^\prime}(\widetilde u_1), \pi_{W_0^\prime}(\widetilde d_1)=d_1\}$ is an LD set and there exist $\kappa, \nu, \theta$ with one of these non-zero such that $$\kappa\pi_{W_0^\prime}(\widetilde u) + \nu \pi_{W_0^\prime}(\widetilde u_1) + \theta \pi_{W_0^\prime}(\widetilde d_1)=0 \Rightarrow
\kappa u + \nu u_1 + \theta d_1 \in W_0$$ From what we showed at the beginning of this proof, we can conclude that $\kappa,\nu$ are non-zero. $\theta \neq 0$ since $\pi_{W_0^\prime}(\widetilde u),
\pi_{W_0^\prime}(\widetilde u_1)$ are distinct. Put $d_1 = \delta_1l_1+\ldots +\delta_kl_k + \delta_{k+1}e$ with $\delta_{k+1}\neq 0$, then the above equation becomes $$\kappa u +\nu u_1 + \theta \delta_{k+1} e \in W_0$$
Taking projection onto $\tilde W_0$ for the decomposition $W_0\oplus \tilde W_0 = V$ and normalizing their coefficients of $l_{k+1}$ when they are written in basis ${\mathcal{B}}$ $$\kappa\pi_{\tilde W_0}(\widehat u) + \nu \pi_{\tilde W_0}(\widehat u_1) + \theta \pi_{\tilde W_0}(\widehat d_1)=0$$
Since coefficient of $l_{k+1}$ is $1$ in all of them and $\nu \neq 0$ we get that $$\pi_{\tilde W_0}(\widehat u_1)\in fl(\{\pi_{\tilde W_0}(\widehat u), \pi_{\tilde W_0}(\widehat d_1)\}) = \vec L_1$$ Since $|\vec L_1\cap \pi_{\tilde W_0}(\widehat{{\mathcal{L}}(U_{1-i}^\star)})|=1 \Rightarrow \pi_{\tilde W_0}(\widehat u) = \pi_{\tilde W_0}(\widehat u_1) \neq 0
\Rightarrow \exists \delta,\psi$ both non-zero such that $\delta u + \psi u_1\in W_0$. We could eliminate $u_1$ to conclude that there exist constants $\alpha,\beta$ with $\beta\neq 0$ such that $\alpha u + \beta d_1 \in W_0 \Rightarrow
\pi_{W_0^\prime}(\widetilde d_1)=\pi_{W_0^\prime}(\widetilde u)$ which cannot happen by what we showed in the beggining of the proof or $\pi_{W_0^\prime}(d_1)=0\Rightarrow d_1\in sp(\{l_1,\ldots,l_k\})$ which is a contradiction to $(S,D)$ being a detector pair. Therefore such a $u_1$ does not exist and $|\vec{L_2}\cap \pi_{W_0^\prime}(\widetilde{{\mathcal{L}}(U_{1-i}^\star)})|=1$.
Proofs from Section \[highdimrecon\]
====================================
All random selections are done from the set $[N]=\{1,\ldots, N\}$.
\[linindrandom\] Let ${\mathbb{R}}^n$ be the $n$ dimensional vector space over ${\mathbb{R}}$. Suppose $v_i :
i=1,\ldots, n$ are $n$ vectors in ${\mathbb{R}}^n$ with each co-ordinate chosen independently from the uniform distribution on $[N]$. Consider the event $${\mathcal{E}}= \{\{v_1,\ldots,v_n\} \text{ are LI }\}$$ Then $Pr[{\mathcal{E}}]\geq 1-\frac{n}{N^{n^2}}$.
*Proof.* Each $v_i\in {\mathbb{R}}^n$ is chosen such that each co-ordinate is chosen uniformly randomly from the set $[N]$. Let $v_i$ be the vector $(V_{i,1},\ldots,V_{i,n})$. Consider the matrix $\tilde V = (V_{i,j})$. The $v_i$’s will be linearly independent if and only if $\tilde V$ is invertible i.e. $det(V_{i,j})\neq 0$. Note that $det(V_{i,j})$ is not the zero polynomial since the monomial $v_{1}^1v_2^2..v_{n}^n$ has coefficient $1$. Now we can use Schwartz-Zippel Lemma [@Sax09] on this polynomial to yield: $$Pr[det(\tilde V)=0]
\leq \frac{n}{N^{n^2}}$$ Therefore $ Pr[v_i, i=1,\ldots n \text{ are LI }] = Pr[det(\tilde V)\neq 0] \geq
1-\frac{n}{N^{n^2}}$. Therefore $Pr[{\mathcal{E}}] \geq 1-\frac{n}{N^{n^2}}$.
\[linindproj\] Assume conditions in the previous lemma. Consider the subspaces $V = sp\{v_1,\ldots,v_r\}$ and $V^\prime = sp\{v_{r+1},\ldots,v_n\}$. Let’s assume that that ${\mathcal{E}}$ occurs. So $dim(V)=r$. We know Then ${\mathbb{R}}^n = V\oplus V^\prime$. Let $\pi_{V} : {\mathbb{R}}^n \rightarrow V$ be the orthogonal projection onto $V$ under this decomposition . Let $T\subset {\mathbb{R}}^n$ be a finite set from which linear forms are chosen. Consider the event $$\mathcal{F} = \{\exists \text{ an LI set } \{l_1,\ldots,l_r\}\subset T \text{ such that } \{\pi_V(l_1),\ldots,\pi_V(l_r)\} \text{ is LD }\}$$ Then $Pr[\mathcal{F}]\leq {|T|\choose r}\{\frac{n}{N^{n^2}} + \frac{r(n-1)}{N^{n^2}}\}$
*Proof.* Fix $\{l_1,\ldots,l_r\}\subset T$ an LI set. Extend it to get a basis $\{l_1,\ldots,l_n\}$ of ${\mathbb{R}}^n$. Let $l_i = \sum\limits_{j\in [n]}L_{i,j}e_j$. Let $L$ be the matrix $(L_{i,j})_{(i,j)\in [n]\times [n]}$. From the discussion above we have $\tilde V=(V_{i,j})$. Now let $P_r$ be the $n\times n$ matrix $$P_r = \begin{bmatrix}
I_r & 0_{r,n-r} \\
0_{n-r,r} & 0_{n-r,n-r}
\end{bmatrix}$$ where $I_r$ is the $r\times r$ identity matrix and $0_{p,q}$ is the $p\times q$ matrix with all $0$ entries. Also for any $n\times n$ matrix $A$, define $M_r(A)$ to be the principal $r\times r$ minor of $A$. Consider the equation given by $$det(M_r(P_r L co(\tilde V))) =0$$ where $co(\tilde V)$ is the co-factor matrix of $\tilde V$. Since entries of $co(\tilde V)$ are polynomials in the $V_{i,j}$’s and $L$ is a fixed matrix, the entries of $P_rLco(\tilde V)$ are polynomials in $V_{i,j}$’s. So $det(M_r(P_r L co(\tilde V)))$ is a polynomial in $V_{i,j}$’s. This polynomial can’t be identically $0$. Choose $V_{i,j}=L_{i,j}$, then $\tilde V$ is invertible and $Lco(\tilde V) = det(L) I$ and so $P_rLco(\tilde V) = det(L) P_r \Rightarrow det(M_r(P_r L co(\tilde V))) = det(L)\neq 0$. Degree of the polynomial $det(M_r(P_rLco(\tilde V)))$ is clearly $\leq r(n-1)$. Therefore by Schwartz Zippel Lemma $$Pr[det(M_r(P_r L co(\tilde V))) =0] \leq \frac{r(n-1)}{N^{n^2}}$$ Consider the set $$S(\{l_1,\ldots,l_r\}) = \{(V_{i,j}) : det(\tilde V)\neq 0 , det(M_r(P_r L co(\tilde V))\neq 0\}$$ On this set $S(\{l_1,\ldots,l_r\})$, $\{v_1,\ldots,v_n\}$ is a basis and we have the following matrix equations : $$\begin{bmatrix}
v_1 \\
. \\
. \\
v_n
\end{bmatrix} = \tilde V \begin{bmatrix}
e_1 \\
. \\
. \\
e_n
\end{bmatrix} \text{ and }
\begin{bmatrix}
l_1 \\
. \\
. \\
l_n
\end{bmatrix} = L \begin{bmatrix}
e_1 \\
. \\
. \\
e_n
\end{bmatrix}
\Rightarrow
\begin{bmatrix}
l_1 \\
. \\
. \\
l_n
\end{bmatrix} = L\tilde V^{-1} \begin{bmatrix}
v_1 \\
. \\
. \\
v_n
\end{bmatrix}$$ and so $$\begin{bmatrix}
\pi_V(l_1) \\
. \\
\pi_V(l_r)
\end{bmatrix} = \frac{1}{det(\tilde V)}M_r(P_rLco(\tilde V)) \begin{bmatrix}
v_1 \\
. \\
v_r
\end{bmatrix}$$
Therefore $\{\pi_V(l_1),\ldots ,\pi_V(l_r)\}$ is an LI set. Now $S(\{l_1,\ldots,l_r\})^c = \{ (V_{i,j}) : det(\tilde V)=0$ $det(M_rLco(M))=0 \} \Rightarrow
Pr[S(\{l_1,\ldots,l_r\})^c] \leq \frac{n}{N^{n^2}} + \frac{r(n-1)}{N^{n^2}}$. Next we vary $\{l_1,\ldots,l_r\}$ and apply union bound to get $$Pr[\mathcal{F}] \leq \sum\limits_{\{l_1,\ldots,l_r\}\subset T}S(\{l_1,\ldots,l_r\})^c \leq {|T|\choose r}\{\frac{n}{N^{n^2}} + \frac{r(n-1)}{N^{n^2}}\}$$
In our application $|T|=poly(d)$ and $r$ is a constant, so we choose $N=2^{d+n}$ and make this probability very small.
\[lagrangeinterp\] Let $f|_{V}({\bar{X}}) = \sum\limits_{\{{\bar{\alpha}} :
|{\bar{\alpha}}|=d\}}a_{{\bar{\alpha}}}{\bar{X}}^{{\bar{\alpha}}}$ be a homogeneous multivariate polynomial of degree $d$ in $r$ variables $X_1,\ldots,X_r$. Let $p_i : 1\leq i
\leq {d+r-1 \choose r-1}$ be randomly chosen points in $V$ ( dimension $r$ random subspace of ${\mathbb{R}}^n$ chosen in the above lemmas). Then with high probability one can find all the $a_{{\bar{\alpha}}}$.
*Proof.* We evaluate the polynomial at each of the $p_i$’s. So we have ${d+r-1 \choose
r-1}$ evaluations. The number of coefficients is also ${d+r-1 \choose r-1}$ so we get a linear system in the coefficients where the matrix ($X$) entries are just monomials evaluated at the $p_i$’s. Since $f$ is not identically zero clearly there exist values for the points $p_i$’s such that the determinant of this matrix is non zero polynomial so it cannot be identically zero. Now the degree of the determinant polynomial is bounded by $d{d+r-1 \choose r-1} \leq poly((d+r)^{r})$. So by Schwarz Zippel lemma $$Pr[a_{{\bar{\alpha}}} \text{ is recovered correctly }] = Pr[det(X)\neq 0] \geq
1-\frac{poly(d^r)}{N^{n^2}}$$
[^1]: Department of Mathematics, California Institute of Technology, Pasadena CA 91106, USA. email : [email protected]
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'A linear flow on the torus $\mathbb{R}^d / \mathbb{Z}^d$ is uniformly distributed in the Weyl sense if the direction of the flow has linearly independent coordinates over $\mathbb{Q}$. In this paper we combine Fourier analysis and the subspace theorem of Schmidt to prove bounded error uniformity of linear flows with respect to certain polytopes if, in addition, the coordinates of the direction are all algebraic. In particular, we show that there is no van Aardenne–Ehrenfest type theorem for the mod $1$ discrepancy of continuous curves in any dimension, demonstrating a fundamental difference between continuous and discrete uniform distribution theory.'
---
[**Bounded error uniformity of the linear flow on the torus**]{}
**Bence Borda**
Alfréd Rényi Institute of Mathematics, Hungarian Academy of Sciences
1053 Budapest, Reáltanoda u. 13–15, Hungary
Email: `[email protected]`
[**Keywords:** continuous uniform distribution, set of bounded remainder, discrepancy]{}
[**Mathematics Subject Classification (2010):** 11K38, 11J87]{}
Introduction {#Introduction}
============
Arguably the simplest continuous time dynamical system on the $d$-dimensional torus $\mathbb{R}^d / \mathbb{Z}^d$ is the *linear flow*: given $\alpha \in \mathbb{R}^d$, a point $s \in \mathbb{R}^d / \mathbb{Z}^d$ is mapped to $s+t\alpha \pmod{\mathbb{Z}^d}$ at time $t \in \mathbb{R}$. We call $\alpha$ the direction of the linear flow (although we do not assume $\alpha$ to have unit norm). The classical theorem of Kronecker on simultaneous Diophantine approximation shows that the linear flow with direction $\alpha = (\alpha_1, \dots, \alpha_d) \in \mathbb{R}^d$ is minimal, that is, every orbit is dense in $\mathbb{R}^d / \mathbb{Z}^d$ if and only if the coordinates $\alpha_1, \dots, \alpha_d$ are linearly independent over $\mathbb{Q}$. A stronger result was later obtained by Weyl [@Weyl]. As an application of the famous Weyl’s criterion he proved that the linear flow with direction $\alpha$ is *uniformly distributed* if and only if the same linear independence condition holds.
To define what we mean by uniform distribution, let us work in the fundamental domain $[0,1]^d$ (where the opposite facets are identified). Fixing a starting point $s \in [0,1]^d$, the flow is thus given by the parametrized curve $(\{s_1+t \alpha_1 \}, \dots, \{ s_d +t \alpha_d \})$, $t \in \mathbb{R}$, where $\{ \cdot \}$ denotes fractional part. For a function $f: [0,1]^d \to \mathbb{R}$ let $$\label{DeltaT}
\Delta_T (s, \alpha, f)=\int_0^T f(\{ s_1+t\alpha_1 \}, \dots, \{ s_d+t\alpha_d \}) \, \mathrm{d}t - T \int_{[0,1]^d} f(x) \, \mathrm{d}x \qquad (T>0).$$ In the terminology of dynamical systems $\Delta_T(s, \alpha, f)/T$ is the difference of the “time average” and the “space average” of $f$ along the orbit of $s$. For a set $A \subseteq [0,1]^d$ let $\chi_A$ denote its characteristic function, and put $\Delta_T (s, \alpha, A)=\Delta_T (s, \alpha, \chi_A)$. We say that the linear flow with direction $\alpha$ is uniformly distributed if for any starting point $s \in [0,1]^d$ and any axis parallel box $R=\prod_{k=1}^d[a_k, b_k] \subseteq [0,1]^d$ we have $\Delta_T(s, \alpha, R)=o(T)$, i.e. $\lim_{T \to \infty} \Delta_T(s, \alpha, R)/T=0$. Note that we would have an equivalent definition by using polytopes, or even arbitrary convex sets instead of axis parallel boxes. Alternatively, we could define uniform distribution by stipulating that $\Delta_T (s, \alpha, f)=o(T)$ for any starting point $s \in [0,1]^d$ and any continuous function $f:[0,1]^d \to \mathbb{R}$. For the theory of uniform distribution of continuous curves we refer the reader to [@Drmotabook Chapter 2.3].
It is also well-known that the linear flow with direction $\alpha=(\alpha_1, \dots, \alpha_d) \in \mathbb{R}^d$ is ergodic with respect to the Haar measure on $\mathbb{R}^d / \mathbb{Z}^d$ (which coincides with the Lebesgue measure on the fundamental domain $[0,1]^d$) if and only if the coordinates $\alpha_1, \dots, \alpha_d$ are linearly independent over $\mathbb{Q}$ [@Sinai Chapter 3.1]. The ergodicity allows us to study $\Delta_T(s, \alpha, f)$ for more general test functions $f$. Most importantly, by Birkhoff’s pointwise ergodic theorem [@Sinai Chapter 1.2], for any Lebesgue integrable function $f \in L^1([0,1]^d)$ we have $\Delta_T (s, \alpha, f)=o(T)$ for almost every $s \in [0,1]^d$. In particular, for any Lebesgue measurable set $A \subseteq [0,1]^d$ we have $\Delta_T(s, \alpha, A)=o(T)$ for almost every $s \in [0,1]^d$.
The minimality, the uniform distribution and the ergodicity of a linear flow on $\mathbb{R}^d /\mathbb{Z}^d$ are thus all equivalent. This remarkable fact can actually be generalized to flows generated by a continuous one-parameter subgroup of an arbitrary compact Abelian group [@Sinai Chapter 4.1]. Moreover, the linear independence condition also has an analogue in terms of the characters of the group.
A common aspect of Weyl’s criterion and Birkhoff’s pointwise ergodic theorem is that for certain classes of test functions $f$ they only yield $\Delta_T(s, \alpha, f)=o(T)$ without an estimate on the rate of convergence. A quantitative form of ergodicity was obtained by Beck [@Beck]: given a function $f \in L^2([0,1]^d)$, for almost every unit vector $\alpha \in \mathbb{R}^d$, $|\alpha|=1$ (in the sense of the $(d-1)$-dimensional Hausdorff measure on the unit sphere in $\mathbb{R}^d$) we have $\Delta_T(0,\alpha,f)=o(T^{1/2-1/(2d-2)} \log^{3+\varepsilon} T)$ for any $\varepsilon>0$. Moreover, the estimate is almost tight in the sense that the result does not hold with $o(T^{1/2-1/(2d-2)})$. Note that the starting point is the origin. In particular, the result applies to $f=\chi_A$ with an arbitrary Lebesgue measurable set $A \subseteq [0,1]^d$. It is interesting to note that in dimension $d=2$ the estimate is simply $O(\log^{3+\varepsilon} T)$. To describe this phenomenon, i.e. uniformity with polylogarithmic error, Beck introduced the term *superuniformity*. The main message is thus that for the family of all Lebesgue measurable test sets we have superuniformity in dimension $d=2$ but not in dimensions $d \ge 3$.
For a more narrow class of test sets, however, we can improve superuniformity to *bounded error uniformity*. Such results have only been proved in dimension $d=2$ so far. Let $\left\| \cdot \right\|$ denote the distance from the nearest integer function. For the sake of simplicity, let us only consider directions of the form $\alpha=(\alpha_1, 1)$. Drmota [@Drmotapaper] showed that if there exists a constant $\eta<2$ such that the inequality $\left\| n \alpha_1 \right\|<|n|^{-\eta}$ has finitely many integer solutions $n \in \mathbb{Z}$, then for any axis parallel box $R \subseteq [0,1]^2$ we have $\Delta_T (0, \alpha, R)=O(1)$. In fact, the implied constant depends only on $\alpha$, which means that by letting $\mathcal{R}$ denote the family of axis parallel boxes in $[0,1]^2$, the *discrepancy* $\sup_{R \in \mathcal{R}} |\Delta_T (0, \alpha, R)|$ is also $O(1)$. Grepstad and Larcher [@Grepstad] considered convex polygons $P \subseteq [0,1]^2$ with no side parallel to the direction $\alpha=(\alpha_1, 1)$ as test sets. If the continued fraction representation $\alpha_1=[a_0;a_1,a_2, \dots]$ satisfies $\sum_{\ell=0}^{\infty} a_{\ell+1}/q_{\ell}^{1/2} \sum_{k=1}^{\ell+1}a_k < \infty$, where $p_{\ell}/q_{\ell}=[a_0;a_1, \dots, a_{\ell}]$ denotes the convergents to $\alpha_1$, then for any starting point $s \in [0,1]^2$ we have $\Delta_T(s, \alpha, P)=O(1)$. To make the two results easier to compare let us mention that the condition on the continued fraction holds if there exists a constant $\eta<5/4$ such that the inequality $\left\| n \alpha_1 \right\|<|n|^{-\eta}$ has finitely many integer solutions $n \in \mathbb{Z}$. Both results are tight: the estimate $O(1)$ clearly cannot be replaced by $o(1)$ in either theorem. See, however, Theorem \[theorem4\] below for an explicit bound.
It is natural to ask what the widest class of test sets is for which we have bounded error uniformity. Well, for the family of all convex test sets in $[0,1]^2$ we have superuniformity, but not bounded error uniformity. More precisely, Beck [@Beck2] proved for the direction $\alpha=(\alpha_1,1)$ that if the continued fraction representation $\alpha_1=[a_0;a_1,a_2, \dots]$ satisfies $a_{\ell}=O(1)$ (i.e. $\alpha_1$ is badly approximable), then for any convex set $C \subseteq [0,1]^2$ we have $\Delta_T (0, \alpha, C)=O(\log T)$. In fact, the implied constant depends only on $\alpha$, thus the *isotropic discrepancy* $\sup_C |\Delta_T (0, \alpha, C)|$, where the supremum is taken over all convex sets $C \subseteq [0,1]^2$ is also $O(\log T)$. Moreover, the estimate is tight. In light of Grepstad and Larcher’s theorem it is not surprising that the convex set showing that $O(\log T)$ cannot be replaced by $o(\log T)$ is a parallelogram with two sides parallel to $\alpha$.
To summarize, for arbitrary Lebesgue measurable test sets we only have metric results, that is, the estimates only hold for almost every direction $\alpha$ (but the starting point can be specified). On the other hand, for simple test sets, like boxes, polygons or convex sets in dimension $d=2$, we have quantitative uniformity results for explicit directions $\alpha$ and starting points $s$. Indeed, Beck’s result on the isotropic discrepancy holds in particular for directions $\alpha=(\alpha_1,1)$ with quadratic irrational $\alpha_1$, say $\alpha_1=\sqrt{2}$. The theorems of Drmota, and Grepstad and Larcher hold for even more general directions, e.g. for algebraic irrational $\alpha_1$: recall that the classical theorem of Roth [@Roth] states that if $\alpha_1$ is an algebraic irrational, then for any $\varepsilon>0$ the inequality $\left\| n \alpha_1 \right\| < |n|^{-1-\varepsilon}$ has finitely many integer solutions $n \in \mathbb{Z}$. Thus we have a wide class of explicit directions for which the estimates are valid.
The main purpose of this paper is to prove bounded error uniformity results in arbitrary dimensions $d \ge 2$. Our test sets will be polytopes, i.e. convex hulls of finitely many points. The $(d-1)$-dimensional faces of a polytope will be called facets; by a normal vector of a facet we mean a nonzero vector, not necessarily of unit norm, which is orthogonal to the facet. Let $|x|$ denote the Euclidean norm, and $\langle x,y \rangle=\sum_{k=1}^d x_k y_k$ the scalar product of $x,y \in \mathbb{R}^d$, and let $\lambda$ be the Lebesgue measure. The notation $f(T)=O(g(T))$ means that there exists an (implied) constant $K>0$ such that $|f(T)| \le K g(T)$ for every $T>0$. We say that $f(T)=\Omega (g(T))$ if $\limsup_{T \to \infty} |f(T)|/g(T)>0$. Similar notations are used for sequences. The following bounded error uniformity result holds for explicit directions and starting points in arbitrary dimension.
\[theorem1\] Let $d \ge 2$, and suppose that the coordinates of $\alpha=(\alpha_1, \dots, \alpha_d) \in \mathbb{R}^d$ are algebraic and linearly independent over $\mathbb{Q}$. Let $P \subseteq [0,1]^d$ be a polytope with a nonempty interior, and suppose that every facet of $P$ has a normal vector $\nu$ with algebraic coordinates and $\langle \nu, \alpha \rangle \neq 0$. For any starting point $s \in [0,1]^d$ $$\Delta_T (s, \alpha, P)=O(1)$$ with an implied constant depending only on $\alpha$ and the normal vectors of the facets of $P$.
Clearly, for any $\alpha \in \mathbb{R}^d$, any $s \in [0,1]^d$ and any polytope $P \subseteq [0,1]^d$ with $0<\lambda (P)<1$ we have $\Delta_T(s, \alpha, P)=\Omega(1)$, therefore the estimate in Theorem \[theorem1\] is best possible. It is interesting to note that the implied constant does not depend on $P$ itself, only on the normal vectors of its facets. This means that if $P$ is a polytope satisfying the conditions of Theorem \[theorem1\], then we actually have a uniform estimate for all test sets of the form $aP+b \subseteq [0,1]^d$, where $a>0$ and $b \in \mathbb{R}^d$. Furthermore, note that for axis parallel boxes the normal vectors of the facets are all $\pm 1$ times a standard basis vector of $\mathbb{R}^d$, thus we immediately obtain a corollary on the discrepancy.
\[corollary2\] Let $d \ge 2$, and suppose that the coordinates of $\alpha=(\alpha_1, \dots, \alpha_d) \in \mathbb{R}^d$ are algebraic and linearly independent over $\mathbb{Q}$. For any starting point $s \in [0,1]^d$ $$\sup_{R \in \mathcal{R}} |\Delta_T (s, \alpha, R)| =O(1)$$ with an implied constant depending only on $\alpha$, where $\mathcal{R}$ denotes the family of axis parallel boxes in $[0,1]^d$.
A comparison with the corresponding discrete problem is in order. Given $\alpha \in \mathbb{R}^d$, the discrete analogue of the linear flow with direction $\alpha$ is the *translation* with direction $\alpha$, that is, the discrete time dynamical system on $\mathbb{R}^d / \mathbb{Z}^d$ in which a point $s \in \mathbb{R}^d/\mathbb{Z}^d$ is mapped to $s+k\alpha \pmod{\mathbb{Z}^d}$ at time $k \in \mathbb{Z}$. The analogue of is of course $$\label{DN}
D_N(s, \alpha, f)=\sum_{k=0}^{N-1} f(\{s_1 + k\alpha_1 \}, \dots, \{ s_d + k\alpha_d \} ) -N \int_{[0,1]^d} f(x) \, \textrm{d}x \qquad (N \in \mathbb{N}),$$ and similarly let $D_N (s, \alpha, A)=D_N(s, \alpha, \chi_A)$. We say that the translation with direction $\alpha$ is uniformly distributed if for any starting point $s \in [0,1]^d$ and any axis parallel box $R \subseteq [0,1]^d$ we have $D_N (s, \alpha, R)=o(N)$. Again, we would get an equivalent definition by using polytopes or arbitrary convex sets instead of axis parallel boxes, or by stipulating that $D_N(s, \alpha, f)=o(N)$ for any $s\in [0,1]^d$ and any continuous function $f:[0,1]^d \to \mathbb{R}$.
Similarly to the continuous time case, the minimality, the uniform distribution and the ergodicity of a translation with direction $\alpha=(\alpha_1, \dots, \alpha_d)\in \mathbb{R}^d$ are all equivalent. The only difference is that in the discrete time case these properties hold if and only if $\alpha_1, \dots, \alpha_d, 1$ are linearly independent over $\mathbb{Q}$ [@Sinai Chapter 3.1]. Again, this fact can actually be generalized to translations on an arbitrary compact Abelian group, with the linear independence condition replaced by a condition in terms of the characters of the group [@Sinai Chapter 4.1].
The quantitative results are, however, very different from the continuous time case. Based on the analogy with the linear flow, one could think that given an arbitrary Lebesgue measurable set $A \subseteq [0,1]^d$, for almost every $\alpha \in \mathbb{R}^d$ we have $D_N(0,\alpha,A)=o(N)$. In fact, in dimension $d=1$ this was a famous, long-standing conjecture of Khinchin. Khinchin’s conjecture, however, was disproved by Marstrand [@Marstrand], who showed the existence of an open set $A \subseteq [0,1]$ for which $D_N (0,\alpha,A)=\Omega (N)$ for all $\alpha \in \mathbb{R}$. The discrete analogue of Corollary \[corollary2\] is due to Niederreiter [@Niederreiter]: if $\alpha_1, \dots, \alpha_d,1$ are algebraic and linearly independent over $\mathbb{Q}$, then $\sup_{R \in \mathcal{R}} |D_N(0,\alpha,R)|=O(N^{\varepsilon})$ for any $\varepsilon>0$.
Finally, let us mention another, arguably the most important difference between the continuous and the discrete time case. Let us generalize and as follows: for a continuous curve $g=(g_1, \dots, g_d): [0,\infty) \to \mathbb{R}^d$ let $$\Delta_T (g,f)=\int_0^T f(\{ g_1 (t)\}, \dots, \{ g_d (t) \}) \, \textrm{d}t - T\int_{[0,1]^d} f(x) \, \textrm{d}x \qquad (T>0),$$ and similarly, for a sequence $x_k=(x_{k,1}, \dots, x_{k,d}) \in \mathbb{R}^d$ let $$D_N (x_k,f)=\sum_{k=0}^{N-1} f(\{ x_{k,1} \}, \dots, \{ x_{k,d} \} )-N \int_{[0,1]^d} f(x) \, \textrm{d}x \qquad (N \in \mathbb{N}).$$ As before, for a set $A \subseteq [0,1]^d$ let $\Delta_T (g,A)= \Delta_T (g, \chi_A)$ and $D_N (x_k,A)=D_N (x_k, \chi_A)$. Note that $g$ and $x_k$ do not necessarily come from dynamical systems. The main difference between continuous and discrete uniform distribution is that bounded error uniformity is impossible in the discrete case, even for the family of axis parallel boxes as test sets. Indeed, answering a question of van der Corput, it was van Aardenne–Ehrenfest [@Aardenne] who first proved that in dimension $d=1$, for any sequence $x_k \in \mathbb{R}$ the discrepancy $\sup_{R \in \mathcal{R}} |D_N (x_k,R)|$ cannot be $O(1)$. This was later improved by Schmidt and Roth, who showed that for an arbitrary sequence $x_k \in \mathbb{R}^d$ we have $\sup_{R \in \mathcal{R}}|D_N(x_k,R)| =\Omega (\log N)$ if $d=1$, and $\sup_{R \in \mathcal{R}}|D_N(x_k,R)| =\Omega (\log^{d/2} N)$ if $d \ge 2$, with implied constants depending only on $d$ (see e.g. [@Drmotabook Chapter 1.3]). Similar lower estimates for continuous curves were considered plausible. In particular, Drmota conjectured [@Drmotapaper eq. (121)] that for any continuous curve $g: [0,\infty) \to \mathbb{R}^d$ such that the arc length $\ell_T$ of $g$ on $[0,T]$ is finite for every $T>0$ we have $\sup_{R \in \mathcal{R}} |\Delta_T(g,R)/T|=\Omega ((\log \ell_T )^{d-2-\varepsilon} / \ell_T)$ for any $\varepsilon >0$. The main message of Corollary \[corollary2\] is thus that there is no van Aardenne–Ehrenfest type theorem for continuous curves in any dimension. In particular, the conjecture of Drmota is false.
The main result
===============
For the sake of simplicity, let us consider directions $\alpha=(\alpha_1, \dots, \alpha_d) \in \mathbb{R}^d$ such that $\alpha_d=1$. The coordinates $\alpha_1, \dots, \alpha_{d-1},1$ are linearly independent over $\mathbb{Q}$ if and only if $\left\| n_1 \alpha_1 + \cdots + n_{d-1} \alpha_{d-1} \right\|>0$ for every $n \in \mathbb{Z}^{d-1}$, $n \neq 0$. Our most general result is based on the idea that by assuming a stronger, quantitative form of linear independence we can obtain a stronger, quantitative form of uniform distribution.
\[maintheorem\] Let $d \ge 2$, let $K$ be a subfield of $\mathbb{R}$, and let $\alpha \in K^d$ with $\alpha_d=1$. Suppose that for any linearly independent linear forms $L_1, \dots, L_{d-1}$ of $d-1$ variables with coefficients in $K$ there exists a constant $\gamma <1$ such that the inequality $$\left\| \alpha_1 n_1 + \cdots + \alpha_{d-1} n_{d-1} \right\| \cdot \prod_{k=1}^{d-1} \left( |L_k (n)| +1 \right) < |n|^{-\gamma}$$ has finitely many integral solutions $n \in \mathbb{Z}^{d-1}$. Let $P \subseteq [0,1]^d$ be a polytope with a nonempty interior, and suppose that every facet of $P$ has a normal vector $\nu$ with coordinates in $K$ and $\langle \nu, \alpha \rangle \neq 0$. For any starting point $s \in [0,1]^d$ $$\Delta_T (s, \alpha, P) = O(1)$$ with an implied constant depending only on $\alpha$ and the normal vectors of the facets of $P$.
In dimension $d=2$ there is only one linear form of $d-1=1$ variable up to a constant factor, while in higher dimensions there are infinitely many. This fact makes it easier to obtain an explicit bound in the case $d=2$ as follows.
\[theorem4\] Let $\alpha=(\alpha_1,1) \in \mathbb{R}^2$ be such that $0<\alpha_1<1$ is irrational, and let $P \subseteq [0,1]^2$ be a convex polygon with edges $e_1, e_2, \ldots, e_N$. Suppose that none of the edges of $P$ are parallel to $\alpha$, and for every $1 \le k \le N$ let $\phi_k$ denote the angle such that $\alpha$ rotated by $\phi_k$ in the positive direction is parallel to $e_k$. For any starting point $s \in [0,1]^2$ and any $T>0$ we have $$|\Delta_T (s, \alpha, P)| \le 2 + \frac{N+1}{\pi^2 |\alpha |} \max_{1 \le k<\ell \le N} \left| \cot \phi_k - \cot \phi_{\ell} \right| \sum_{n=1}^{\infty} \frac{1}{n^2 \| n \alpha_1 \|}.$$
By switching the coordinates if necessary, we may assume that the slope of the orbits is greater than 1, therefore the assumption $0<\alpha_1<1$ is not restrictive. The proof will clearly show that if the second coordinate of $s$ is $0$ and $T \in \mathbb{N}$, then the estimate in Theorem \[theorem4\] holds even without the first term $2$. Note that if there exists a constant $\eta<2$ such that the inequality $\| n \alpha_1 \|<|n|^{-\eta}$ has finitely many integer solutions $n \in \mathbb{Z}$, then $\sum_{n=1}^{\infty} 1/(n^2 \| n \alpha_1 \|)<\infty$.
The rest of this Section is devoted to the proofs of Theorems \[maintheorem\] and \[theorem4\], both of which are based on Fourier analysis. We deduce Theorem \[theorem1\] from Theorem \[maintheorem\] and the subspace theorem of Schmidt in Section \[section3\].
Throughout this proof the implied constants in the $O$-notation will only depend on $\alpha$ and the normal vectors of the facets of $P$. The error of replacing $s$ by $(\{s_1-\alpha_1 s_d\}, \dots , \{s_{d-1}-\alpha_{d-1} s_d\}, 0)$, and $T$ by $\lceil T \rceil$ in $\Delta_T(s, \alpha, P)$ is clearly $O(1)$, therefore we may assume $s_d=0$, and that $T$ is a positive integer. We start by reducing our $d$-dimensional, continuous time dynamical system to a $(d-1)$-dimensional, discrete time one. By breaking up the integral in the definition of $\Delta_T (s, \alpha, P)$ we get $$\Delta_T(s, \alpha, P) = \sum_{k=0}^{T-1} \left( \int_k^{k+1} \chi_P (\{ s_1+t \alpha_1 \}, \dots, \{ s_{d-1}+t \alpha_{d-1} \}, \{ t \} ) \, \mathrm{d}t - \lambda (P) \right) .$$ Applying the integral transformation $t \mapsto t+k$ we can write $\Delta_T (s, \alpha, P)$ in the form $$\label{discretedynsyst}
\Delta_T(s, \alpha, P) = \sum_{k=0}^{T-1} \left( f(s_1+ k \alpha_1, \dots, s_{d-1}+ k \alpha_{d-1}) - \lambda (P) \right) ,$$ where $f: \mathbb{R}^{d-1} \to \mathbb{R}$ is defined as $$\label{fdefinition}
f(x_1, \dots, x_{d-1}) = \int_0^1 \chi_P (\{ x_1 + t \alpha_1 \} , \dots , \{ x_{d-1} + t \alpha_{d-1} \} , t) \, \mathrm{d}t .$$ In the terminology of dynamical systems the facet $x_d=0$ of $[0,1]^d$ (which corresponds to a $(d-1)$-dimensional torus in $\mathbb{R}^d / \mathbb{Z}^d$) is a transversal, and the underlying discrete time dynamical system, the translation on $\mathbb{R}^{d-1} / \mathbb{Z}^{d-1}$ with direction $(\alpha_1, \dots, \alpha_{d-1})$ is a Poincaré map.
The geometric meaning of $f$ is the following. Consider the line segment starting at the point $(x_1, \dots, x_{d-1},0)$ parallel to $\alpha$, joining the facets $x_d=0$ and $x_d=1$ of $[0,1]^d$ (of course everything is taken modulo $\mathbb{Z}^d$, i.e. it is in fact a line segment on the torus). Then $f(x_1, \dots, x_{d-1})$ is the length of the intersection of this line segment with $P$. The crucial observation is that since $\alpha$ is not parallel to any facet of $P$, the function $f$ is continuous. This allows us to prove a nontrivial estimate for the Fourier coefficients of $f$ as follows.
\[fourierlemma\] There exists a set $\mathcal{L}$ of linearly independent linear forms $(L_1, \dots, L_{d-1})$ of $d-1$ variables with coefficients in $K$, depending only on $\alpha$ and the normal vectors of the facets of $P$, such that $|\mathcal{L}|=O(1)$ and for any $n \in \mathbb{Z}^{d-1}$, $n \neq 0$ we have $$\int_{[0,1]^{d-1}} f(x) e^{- 2 \pi i \langle n,x \rangle} \, \mathrm{d}x = O \left( \sum_{(L_1, \dots, L_{d-1}) \in \mathcal{L}} \frac{1}{|n| \prod_{k=1}^{d-1} \left( |L_k (n)|+1 \right)} \right) .$$
We start by “lifting” the line segment in the definition of $f$ from $\mathbb{R}^d / \mathbb{Z}^d$ to $\mathbb{R}^d$. For a given $x \in [0,1]^{d-1}$ let $g_x(t)=(x_1+t \alpha_1, \dots , x_{d-1}+t\alpha_{d-1}, t)$, $t \in \mathbb{R}$ denote a parametrized line. Let $M$ be a positive integer such that $|\alpha_k| \le M$ for all $1 \le k \le d$. For any $x \in [0,1]^{d-1}$ the line segment $g_x(t)$, $t \in [0,1]$ stays in $[-M,M+1]^d$. Thus it is enough to consider the translations of $P$ by the integral vectors $\varepsilon$ in the set $E=[-M, M]^d \cap \mathbb{Z}^d$. Formally, for any $x \in [0,1]^{d-1}$ we have $$\label{P+epsilon}
f(x)= \sum_{\varepsilon \in E} \int_0^1 \chi_{P + \varepsilon} (g_x(t)) \, \mathrm{d}t .$$
Note that $|E|=O(1)$. We claim that $f$ is a “piecewise linear” function. That is, there exists a decomposition of $[0,1]^{d-1}$ into polytopes $A_1, A_2, \dots, A_m$ such that $f$ is of the form $f(x)=\langle a_j, x \rangle + b_j$ on $A_j$ with some $a_j \in \mathbb{R}^{d-1}, b_j \in \mathbb{R}$.
Indeed, let $\pi : \mathbb{R}^d \to \mathbb{R}^{d-1}$ denote the projection onto the hyperplane $x_d=0$ in the direction $\alpha$, i.e. let $\pi (x_1, \dots, x_d)=(x_1-\alpha_1 x_d, x_2-\alpha_2 x_d, \dots, x_{d-1}-\alpha_{d-1} x_d)$. Consider the $(d-2)$-dimensional faces of all translates $P+\varepsilon$, $\varepsilon \in E$. Applying the projection $\pi$ to the affine hulls of these $(d-2)$-dimensional faces, we obtain affine hyperplanes in $\mathbb{R}^{d-1}$. These affine hyperplanes decompose $[0,1]^{d-1}$ into polytopes $A_1, \dots, A_m$. (The affine hyperplanes which do not intersect $[0,1]^{d-1}$ are discarded.) Observe that $m=O(1)$ and that each $A_j$ has $O(1)$ facets. More specifically, consider a $(d-2)$-dimensional face of one of the translates $P+\varepsilon$. The affine hull of this face is the set of solutions of the system $\langle \mu, x \rangle =b$, $\langle \nu , x \rangle =c$ for the normal vectors $\mu, \nu$ of two facets of $P$ and some $b,c \in \mathbb{R}$. The projection $\pi (x)=y$ satisfies $$\sum_{k=1}^{d-1} \left( \frac{\mu_k}{\langle \mu, \alpha \rangle} - \frac{\nu_k}{\langle \nu, \alpha \rangle} \right) y_k = \frac{b}{\langle \mu, \alpha \rangle}-\frac{c}{\langle \nu, \alpha \rangle}.$$ Here the coefficients of $y_k$ belong to the field $K$, and it is not difficult to check that they are not all zero. Hence the ($(d-2)$-dimensional) facets of the ($(d-1)$-dimensional) polytopes $A_1, \dots, A_m$ have normal vectors with coefficients in $K$.
For a given $x \in [0,1]^{d-1}$ the intersection of the line segment $g_x(t)$, $t \in [0,1]$ and the polytopes $P+\varepsilon$, $\varepsilon \in E$ is the union of finitely many (possibly zero) line segments with endpoints on the facets of $P+\varepsilon$, $\varepsilon \in E$. Observe that given an $A_j$, the ordered list of facets of $P+\varepsilon$, $\varepsilon \in E$ intersecting $g_x(t)$, $t \in [0,1]$ does not depend on the choice of the point $x \in A_j$.
Fix an $A_j$, and let $x \in A_j$. Consider two facets of $P+\varepsilon$, $\varepsilon \in E$ whose affine hulls have equations $\langle \mu,y \rangle = b$ and $\langle \nu, y \rangle =c$ with normal vectors $\mu, \nu$ and some $b,c \in \mathbb{R}$. The points of the line $g_x(t)$ that lie on these affine hyperplanes satisfy $$\label{ajformula}
t = \frac{b}{\langle \mu, \alpha \rangle} - \sum_{k=1}^{d-1} \frac{\mu_k}{\langle \mu, \alpha \rangle} x_k , \qquad t= \frac{c}{\langle \nu, \alpha \rangle} - \sum_{k=1}^{d-1} \frac{\nu_k}{\langle \nu, \alpha \rangle} x_k,$$ respectively. Therefore the length of the line segment on $g_x(t)$ that lies between the two given facets is an inhomogeneous linear function of $x$. Observe also that the coefficients of $x_1, \dots, x_{d-1}$ in this inhomogeneous linear function are $O(1)$. From we thus obtain that $f(x)$ is indeed of the form $f(x)=\langle a_j, x \rangle + b_j$ on $A_j$ with some $a_j \in \mathbb{R}^{d-1}$ and $b_j \in \mathbb{R}$, moreover $|a_j|=O(1)$.
We are interested in the integral of $f(x)e^{-2 \pi i \langle n,x \rangle}$, i.e. the product of an inhomogeneous linear, and an exponential function. It is therefore natural to use the divergence theorem, which is basically a multidimensional analogue of integration by parts. The key fact is that the continuity of $f$ (which follows from the assumption that $\alpha$ is not parallel to any facet of $P$) implies that the integrals over the boundaries in the divergence theorem *completely cancel*. The appearance of the extra factor $|n|$ in the denominator in Lemma \[fourierlemma\], and hence the boundedness of $\Delta_T(s, \alpha, P)$ is a consequence of this cancellation in the divergence theorem.
From now on let $n \in \mathbb{Z}^{d-1}$, $n \neq 0$ be fixed. For a given $1 \le j \le m$ let us apply the divergence theorem to the function $F: A_j \to \mathbb{R}^{d-1}$, $$F(x)=\frac{n}{2 \pi i |n|^2} f(x) e^{-2 \pi i \langle n, x \rangle} = \frac{n}{2 \pi i |n|^2} \left( \langle a_j, x \rangle + b_j \right) e^{-2 \pi i \langle n, x \rangle}$$ to obtain $$\label{divtheorem}
\int_{A_j} \left(\frac{\langle a_j, n \rangle}{2 \pi i |n|^2} e^{-2 \pi i \langle n, x \rangle} -f(x) e^{-2 \pi i \langle n, x \rangle} \right) \, \textrm{d} x = \int_{\partial A_j} \frac{\langle n, \nu (x) \rangle}{2 \pi i |n|^2} f(x) e^{-2 \pi i \langle n, x \rangle} \, \textrm{d} x.$$ Here $\partial A_j$ denotes the boundary of $A_j$, i.e. the union of its facets, and $\nu : \partial A_j \to \mathbb{R}^{d-1}$ is the outer unit normal vector. Since $f(x)$, and hence $f(x)e^{-2 \pi i \langle n, x \rangle}$ is periodic modulo $\mathbb{Z}^{d-1}$ and continuous, the sum of the right hand side of over $1 \le j \le m$ is zero. Indeed, each facet appears twice in the sum, with the same integrand except with opposite signs because the outer normals are negatives of each other. Therefore summing over $1 \le j \le m$ we obtain $$\label{sumAj}
\int_{[0,1]^{d-1}} f(x) e^{- 2 \pi i \langle n,x \rangle} \, \textrm{d}x = \sum_{j=1}^m \frac{\langle a_j, n \rangle}{2 \pi i |n|^2} \int_{A_j} e^{-2 \pi i \langle n,x \rangle} \, \textrm{d} x.$$
The sum has $m=O(1)$ terms, thus it is enough to estimate the terms separately. Let $A=A_j \subseteq [0,1]^{d-1}$ for some $1 \le j \le m$. We follow the methods of Randol [@Randol] to bound the Fourier transform of the characteristic function of the polytope $A$. An ordered tuple $\mathcal{F}=(F_{d-1}, F_{d-2}, \dots, F_k)$ is called a flag of $A$ if $0 \le k \le d-1$, $F_{\ell}$ is an $\ell$-dimensional face of $A$ for every $k \le \ell \le d-1$, and $F_{d-1} \supset F_{d-2} \supset \cdots \supset F_k$. (Note $F_{d-1}=A$.) We call $\mathcal{F}$ a complete flag if $k=0$. Recall that $A$ has $O(1)$ facets, therefore the number of flags of $A$ is also $O(1)$.
To every given flag $\mathcal{F}=(F_{d-1}, F_{d-2}, \dots, F_k)$ let us associate orthogonal vectors $v_{d-2}, v_{d-3}, \dots, v_k$ such that $v_{\ell} \in \mathbb{R}^{d-1}$ is an outer normal vector of $F_{\ell}$ in the affine hull of $F_{\ell+1}$ for every $k \le \ell \le d-2$. Note that $v_{d-2}, \dots, v_k$ can be obtained by applying the Gram–Schmidt orthogonalization procedure to the normal vectors of certain facets of $A$, therefore we can also ensure that the coordinates of $v_{d-2}, \dots, v_k$ are all in $K$ (but the vectors might not have unit length). For every $k \le \ell \le d-1$ let $\pi_{\ell} : \mathbb{R}^{d-1} \to \mathbb{R}^{d-1}$ denote the orthogonal projection onto the $\ell$-dimensional linear subspace (i.e. containing the origin) parallel to $F_{\ell}$. In particular, for a complete flag we obtain an orthogonal basis $v_{d-2}, \dots, v_0$ of $\mathbb{R}^{d-1}$, defining linearly independent linear forms $L_1(x)=\langle v_{d-2}, x \rangle, \dots, L_{d-1}(x)= \langle v_0, x \rangle$ of the variables $x=(x_1, \dots, x_{d-1})$ with coefficients in $K$. Let $\mathcal{A}=\mathcal{A}_j$ denote the set of such linearly independent linear forms $(L_1, \dots, L_{d-1})$ associated to complete flags of $A=A_j$.
Clearly $|n|=|\pi_{d-1} (n)| \ge |\pi_{d-2}(n)| \ge \cdots \ge |\pi_k (n)|$. Let us call $\mathcal{F}$ a “relevant flag” if $|\pi_k (n)| < 1$ but $|\pi_{k+1}(n)| \ge 1$. We will express $\int_A e^{-2 \pi i \langle n, x \rangle} \, \textrm{d}x$ as a sum over all relevant flags of $A$. Formally, our integral is associated to the only flag of length 1, namely $(F_{d-1})$, which is not a relevant flag.
We use the following algorithm. Let us apply the divergence theorem to $F(x)=\frac{-n}{2 \pi i |n|^2} e^{-2 \pi i \langle n,x \rangle}$ on $A$. The integral over $\partial A$ can be written as a sum over all flags $(F_{d-1}, F_{d-2})$ of length 2, with terms $$\begin{gathered}
\int_{F_{d-2}} \frac{-\langle v_{d-2}, n \rangle}{2 \pi i |v_{d-2}| |n|^2} e^{-2 \pi i \langle n,x \rangle} \, \textrm{d} x = \\ \frac{-\langle v_{d-2}, n \rangle}{2 \pi i |v_{d-2}| |n|^2} e^{-2 \pi i \langle n, w_{d-2} \rangle} \int_{\pi_{d-2}(F_{d-2})} e^{-2 \pi i \langle \pi_{d-2} (n) , x \rangle} \, \textrm{d} x.\end{gathered}$$ Here $w_{d-2} \in \mathbb{R}^{d-1}$ is the vector for which $\pi_{d-2} (F_{d-2})+w_{d-2}=F_{d-2}$. The linear subspace containing $\pi_{d-2} (F_{d-2})$ can be isometrically identified with $\mathbb{R}^{d-2}$, thus $\langle \pi_{d-2}(n), x \rangle$ is preserved in this identification. This way we obtain $$\int_A e^{-2 \pi i \langle n, x \rangle} \, \textrm{d}x = \sum_{(F_{d-1}, F_{d-2})} C_n(F_{d-1}, F_{d-2}) \int_{\pi_{d-2} (F_{d-2})} e^{-2 \pi i \langle \pi_{d-2} (n), x \rangle} \, \textrm{d}x$$ with some coefficients $|C_n (F_{d-1}, F_{d-2})| \le \frac{1}{2 \pi |\pi_{d-1}(n)|}$ (recall $\pi_{d-1}(n)=n$).
The terms indexed by relevant flags $(F_{d-1}, F_{d-2})$ are kept as they are. (Since $|\pi_{d-2}(n)|<1$, it is not worth applying the divergence theorem again.) If a term is indexed by a non-relevant flag $(F_{d-1}, F_{d-2})$, we apply the divergence theorem again and replace it by a sum over all extensions $(F_{d-1}, F_{d-2}, F_{d-3})$. We continue in a similar fashion: if a flag $(F_{d-1}, F_{d-2}, \dots, F_k)$ becomes relevant, we keep the corresponding term. If a flag is not relevant, we apply the divergence theorem again. The algorithm stops when every term in our sum is associated to a relevant flag. Note that since $|\pi_0 (n)|=0$, eventually every flag becomes relevant, and so the algorithm terminates. The algorithm yields a formula of the form $$\label{relevantflags}
\int_A e^{-2 \pi i \langle n, x \rangle} \, \textrm{d}x = \sum_{\substack{(F_{d-1}, F_{d-2}, \dots, F_k) \\ \textrm{relevant flags}}} C_n(F_{d-1}, F_{d-2}, \dots, F_k) \int_{\pi_k (F_k)} e^{-2 \pi i \langle \pi_k (n), x \rangle} \, \textrm{d}x$$ with some coefficients $|C_n(F_{d-1}, F_{d-2}, \dots, F_k)| \le \prod_{\ell=k+1}^{d-1} \frac{1}{2 \pi |\pi_{\ell} (n)|}$.
Consider a relevant flag $(F_{d-1}, F_{d-2}, \dots, F_k)$. The corresponding integral on the right hand side of is $O(1)$. If $k>0$, let us extend the relevant flag arbitrarily to a complete flag $(F_{d-1}, F_{d-2}, \dots, F_0)$. By the definition of a relevant flag we have $1>|\pi_k (n)| \ge |\pi_{k-1}(n)| \ge \cdots \ge |\pi_1 (n)|$, therefore $C_n (F_{d-1}, F_{d-2}, \dots, F_k) = O(1/\prod_{\ell=1}^{d-1} (|\pi_{\ell}(n)|+1))$. Clearly $|\pi_{\ell}(n)| \ge |\langle v_{\ell-1}, n \rangle| / |v_{\ell-1}|$ for every $1 \le \ell \le d-1$, hence we obtain the estimate $$\int_A e^{-2 \pi i \langle n, x \rangle} \, \textrm{d}x = O \left( \sum_{(L_1, \dots, L_{d-1}) \in \mathcal{A}} \frac{1}{\prod_{\ell=1}^{d-1} (|L_{\ell}(n)|+1)} \right) .$$ This holds for every $A=A_j$, therefore in light of $\mathcal{L}=\bigcup_{j=1}^m \mathcal{A}_j$ satisfies the claim of Lemma \[fourierlemma\].
Note that for any linearly independent linear forms $L_1, \dots, L_{d-1}$ of $d-1$ variables we have $$\sum_{\substack{n \in \mathbb{Z}^{d-1} \\ n \neq 0}} \frac{1}{|n| \prod_{k=1}^{d-1} (|L_k(n)|+1)} < \infty .$$ Lemma \[fourierlemma\] thus implies, in particular, that the Fourier series of $f$ is absolutely convergent. It follows (see e.g. [@Grafakos Proposition 3.2.5.]) that the Fourier series converges pointwise to $f$, i.e. $f(x)=\sum_{n \in \mathbb{Z}^{d-1}} \hat{f}(n) e^{2 \pi i \langle n,x \rangle}$ for every $x \in \mathbb{R}^{d-1}$, where $\hat{f}(n)=\int_{[0,1]^{d-1}} f(x)e^{-2\pi i \langle n,x \rangle} \, \mathrm{d}x$. It is not difficult to see from Fubini’s theorem that $$\hat{f}(0)=\int_{[0,1]^{d-1}} f(x) \, \mathrm{d}x = \lambda (P).$$ Replacing $f$ by its Fourier series in , and switching the order of summation we thus obtain with $s^*=(s_1, \dots, s_{d-1})$ and $\alpha^*=(\alpha_1, \dots, \alpha_{d-1})$ that $$\Delta_T (s, \alpha, P)=\sum_{k=0}^{T-1} \sum_{\substack{n \in \mathbb{Z}^{d-1}\\ n \neq 0}} \hat{f}(n) e^{2 \pi i \langle n, s^* +k \alpha^* \rangle} = \sum_{\substack{n \in \mathbb{Z}^{d-1}\\ n \neq 0}} \hat{f}(n) e^{2 \pi i \langle n, s^* \rangle} \frac{1-e^{2 \pi i \langle n, \alpha^* \rangle T}}{1-e^{2 \pi i \langle n, \alpha^* \rangle}} .$$ Using the general estimate $|1-e^{2 \pi i z}|=2|\sin (\pi z)| \ge 4 \left\| z \right\|$, $z \in \mathbb{R}$, we get $$\label{fourierbound}
|\Delta_T (s, \alpha, P)| \le \sum_{\substack{n \in \mathbb{Z}^{d-1}\\ n \neq 0}} |\hat{f}(n)| \cdot \frac{1}{2 \left\| n_1 \alpha_1 + \cdots + n_{d-1} \alpha_{d-1} \right\|} .$$ In light of Lemma \[fourierlemma\] it is thus enough to prove that for any linearly independent linear forms $L_1, \dots, L_{d-1}$ of $d-1$ variables with coefficients in $K$ we have $$\label{enoughtosee}
\sum_{\substack{n \in \mathbb{Z}^{d-1}\\ n \neq 0}} \frac{1}{|n| \prod_{k=1}^{d-1} (|L_k (n)|+1) \left\| n_1 \alpha_1 + \cdots + n_{d-1} \alpha_{d-1} \right\|} < \infty .$$
We know that $\left\| n_1 \alpha_1 + \cdots + n_{d-1} \alpha_{d-1} \right\| \prod_{k=1}^{d-1} (|L_k (n)|+1) \ge C |n|^{-\gamma}$ for every $n \in \mathbb{Z}^{d-1}$, $n \neq 0$ with some constants $C>0$ and $\gamma <1$. For any integers $\ell_1, \dots, \ell_{d-1} \ge 0$ and $\ell \ge 0$ let $S_{\ell} (\ell_1, \dots, \ell_{d-1})$ denote the set of all $n \in \mathbb{Z}^{d-1}$, $n \neq 0$ such that $2^{\ell} \le |n| < 2^{\ell +1}$ and $2^{\ell_k} \le |L_k(n)|+1 < 2^{\ell_k +1}$ for all $1 \le k \le d-1$. Let $g: S_{\ell}(\ell_1, \dots, \ell_{d-1}) \to (-1/2, 1/2]$ be the function $g(n)=n_1 \alpha_1 + \cdots +n_{d-1} \alpha_{d-1} \pmod{1}$.
Let $H=\lceil C^{-1} 2^{(\ell_1+2) + \cdots + (\ell_{d-1}+2)} 2^{\gamma (\ell+2)} \rceil$. For every $n \in S_{\ell}(\ell_1, \dots, \ell_{d-1})$ we have $$|g(n)| = \left\| n_1 \alpha_1 + \cdots + n_{d-1} \alpha_{d-1} \right\| \ge \frac{1}{H} .$$ Moreover, for any $n,m \in S_{\ell}(\ell_1, \dots, \ell_{d-1})$, $n \neq m$ we have $|L_k (n-m)|+1 \le |L_k (n)| + |L_k (m)| +1 < 2^{\ell_k+2}$ for every $k$ and $|n-m| < 2^{\ell+2}$, and hence $$|g(n)-g(m)| \ge \left\| (n_1-m_1) \alpha_1 + \cdots + (n_{d-1}-m_{d-1}) \alpha_{d-1} \right\| > \frac{1}{H} .$$ In other words, $g(n) \not\in (-1/H,1/H)$ for any $n$, and every interval of the form $[h/H, (h+1)/H)$ and $(-(h+1)/H,-h/H]$, $h \ge 1$ contains $g(n)$ for at most one $n$. Since $|g(n)| \le 1/2$, we therefore obtain $$\begin{split} \sum_{n \in S_{\ell} (\ell_1, \dots, \ell_{d-1})} \frac{1}{|g(n)|} \le 2 \sum_{1 \le h \le H/2} \frac{H}{h} &= O(H \log H) \\ &=O \left( 2^{\ell_1 + \cdots + \ell_{d-1}} 2^{\gamma \ell} (\ell_1 + \cdots + \ell_{d-1} + \ell) \right) , \end{split}$$ and consequently $$\begin{gathered}
\sum_{n \in S_{\ell} (\ell_1, \dots, \ell_{d-1})} \frac{1}{|n| \prod_{k=1}^{d-1} (|L_k (n)|+1) \left\| n_1 \alpha_1 + \cdots + n_{d-1} \alpha_{d-1} \right\|} \\ = O \left( 2^{(\gamma -1)\ell} (\ell_1 + \cdots + \ell_{d-1} + \ell) \right) .\end{gathered}$$
Note that $|L_k (n)|+1=O(|n|)$ shows $\ell_k = O(\ell)$, unless $S_{\ell} (\ell_1, \dots, \ell_{d-1})$ is empty. Summing over $0 \le \ell_1, \dots, \ell_{d-1} = O(\ell)$ we get $$\sum_{2^{\ell} \le |n| <2^{\ell+1}} \frac{1}{|n| \prod_{k=1}^{d-1} (|L_k (n)|+1) \left\| n_1 \alpha_1 + \cdots + n_{d-1} \alpha_{d-1} \right\|} = O \left( 2^{(\gamma -1)\ell} \ell^d \right) .$$ Finally, summing over $\ell \ge 0$ shows that indeed holds. The proof of Theorem \[maintheorem\] is thus complete.
We use the notation and follow the proof of Theorem \[maintheorem\]. From the definition of $\Delta_T (s, \alpha, P)$ it is easy to deduce that $$\begin{split} |\Delta_T (s, \alpha, P) - \Delta_{T+s_2} ((\{ s_1-\alpha_1 s_2\} ,0), \alpha, P)| &\le 1, \\ |\Delta_T (s, \alpha, P) - \Delta_{\lceil T \rceil} (s, \alpha, P)| &\le 1 . \end{split}$$ In other words, the error of replacing $s_2$ by $0$, and $T$ by $\lceil T \rceil$ is at most $2$. From now on we will assume $s_2=0$ and that $T \in \mathbb{N}$, and will prove the estimate in the claim without the first term $2$.
Let $f: \mathbb{R} \to \mathbb{R}$ be as in , and for any $x \in [0,1]$ let $g_x(t)=(x+t\alpha_1, t)$, as before. Since $0<\alpha_1<1$, the line segment $g_x(t)$, $t \in [0,1]$ stays in $[0,2] \times [0,1]$, and so it can only intersect the translates $P$ and $P+(1,0)$. That is, for any $x \in [0,1]$ $$\label{P,P+(1,0)}
f(x)=\int_0^1 \chi_P (g_x(t)) \, \mathrm{d}t + \int_0^1 \chi_{P+(1,0)} (g_x(t)) \, \mathrm{d}t .$$ Again, $f$ is a piecewise linear function. Indeed, by applying a projection in the direction $\alpha$, that is, the map $\pi : \mathbb{R}^2 \to \mathbb{R}$, $\pi (x_1,x_2)=x_1-\alpha_1 x_2$ to the vertices of $P$ and $P+(1,0)$, we obtain a partition $0=c_0<c_1<\cdots<c_m=1$ of the interval $[0,1]$. (The projections outside $[0,1]$ are discarded.) Note that $m \le N+1$ since a pair of corresponding vertices of $P$ and $P+(1,0)$ have projections at distance $1$ from each other. For a given $1 \le j \le m$, as $x$ runs in $[c_{j-1},c_j]$ the line segment $g_x(t)$, $t \in [0,1]$ either does not intersect $P$, or intersects the same pair of edges $e_k, e_{\ell}$ of $P$ with some $1 \le k<\ell \le N$ depending on $j$. Thus the first term in is of the form $a_j'x+b_j'$ on $[c_{j-1},c_j]$. As observed in , either $a_j'=0$ or $$|a_j'| = |\alpha| \left| \frac{\nu_{k,1}}{\langle \nu_k, \alpha \rangle} - \frac{\nu_{\ell,1}}{\langle \nu_{\ell}, \alpha \rangle} \right| = |\alpha| \left| \frac{\nu_{k,1} \nu_{\ell,2} -\nu_{k,2} \nu_{\ell,1} }{\langle \nu_k, \alpha \rangle \cdot \langle \nu_{\ell}, \alpha \rangle} \right| ,$$ where $\nu_k=(\nu_{k,1}, \nu_{k,2}), \nu_{\ell}=(\nu_{\ell,1},\nu_{\ell,2})$ are normal vectors of $e_k$, $e_{\ell}$, respectively. Using the angles $\phi_k, \phi_{\ell}$ in the latter case we have $$|a_j'|= |\alpha| \frac{|\nu_k| \cdot |\nu_{\ell}| \cdot |\sin (\phi_k - \phi_{\ell})|}{|\nu_k| \cdot |\alpha| \cdot |\cos \left( \frac{\pi}{2}-\phi_k \right)| \cdot |\nu_{\ell}| \cdot |\alpha| \cdot |\cos \left( \frac{\pi}{2}-\phi_{\ell} \right)|} = \frac{\left| \cot \phi_k - \cot \phi_{\ell} \right|}{|\alpha|} .$$ Note that although the angles formed by $\alpha$, $\nu_k$ and $\nu_{\ell}$ are not well-defined functions of $\phi_k, \phi_{\ell}$, the absolute value of the trigonometric functions in the formula above are well-defined. Similarly, the second term in is of the form $a_j''x+b_j''$ on $[c_{j-1},c_j]$ with either $a_j''=0$ or $|a_j''|=|\cot \phi_p- \cot \phi_q|/|\alpha|$ with some $1 \le p<q\le N$ depending on $j$.
Thus $f(x)$ is of the form $f(x)=a_jx+b_j$ on $[c_{j-1},c_j]$, where $a_j=a_j'+a_j''$. Consider the Fourier coefficients $\hat{f}(n)=\int_0^1 f(x)e^{-2 \pi i n x} \, \mathrm{d}x$, $n \in \mathbb{Z}$. It is easy to see from Fubini’s theorem that $\hat{f}(0)=\lambda (P)$. For $n \neq 0$ we can apply integration by parts to obtain $$\begin{split} \hat{f}(n) &= \sum_{j=1}^m \left( \int_{c_{j-1}}^{c_j} (a_jx+b_j)e^{-2 \pi i nx} \, \mathrm{d}x \right) \\ &= \sum_{j=1}^m \left( f(c_j)\frac{e^{-2 \pi i nc_j}}{-2 \pi i n}-f(c_{j-1}) \frac{e^{-2 \pi i nc_{j-1}}}{-2 \pi i n} \right) -\sum_{j=1}^m a_j\int_{c_{j-1}}^{c_j} \frac{e^{-2 \pi i n x}}{-2 \pi i n} \, \mathrm{d}x . \end{split}$$ Here the first sum is $0$ because $f$ is continuous and $1$-periodic, hence $$|\hat{f}(n)| \le \sum_{j=1}^m \frac{|a_j'| + |a_j''|}{2 \pi^2 n^2} \le \frac{N+1}{\pi^2 |\alpha| n^2} \max_{1 \le k<\ell \le N} \left| \cot \phi_k - \cot \phi_{\ell} \right| .$$ From we finally deduce $$|\Delta_T (s, \alpha, P)| \le \sum_{\substack{n \in \mathbb{Z} \\ n \neq 0}} \frac{|\hat{f}(n)|}{2 \| n \alpha_1 \|} \le \frac{N+1}{\pi^2 |\alpha |} \max_{1 \le k<\ell \le N} \left| \cot \phi_k - \cot \phi_{\ell} \right| \sum_{n=1}^{\infty} \frac{1}{n^2 \| n \alpha_1 \|} .$$
The proof of Theorem \[theorem1\] {#section3}
=================================
We now prove Theorem \[theorem1\]. By applying a simple integral transformation in the definition of $\Delta_T(s, \alpha, P)$, we may assume $\alpha_d=1$. Choosing $K$ to be the field of algebraic reals, it is thus enough to show that if $\alpha_1, \ldots, \alpha_{d-1},1$ are algebraic and linearly independent over $\mathbb{Q}$, then $\alpha$ satisfies the Diophantine condition of Theorem \[maintheorem\]. The celebrated subspace theorem of Schmidt [@Schmidt] shows that this Diophantine condition is in fact satisfied with any $\gamma >0$. In other words, we do not even need the full power of the subspace theorem. Unfortunately, most monographs on simultaneous Diophantine approximation prove this condition only for the linear forms $L_1(x)=x_1, \dots, L_{d-1}(x)=x_{d-1}$. For the sake of completeness, we include a proof for arbitrary linearly independent linear forms with real algebraic coefficients. Nevertheless, the following theorem can still be considered to be a form of the subspace theorem of Schmidt.
Let $d \ge 2$, and let the algebraic reals $\alpha_1, \dots, \alpha_{d-1}, 1$ be linearly independent over $\mathbb{Q}$. Let $L_1, \dots, L_{d-1}$ be linearly independent linear forms of $d-1$ variables with real algebraic coefficients. For any $\varepsilon >0$ the inequality $$\label{Schmidtbound}
\left\| \alpha_1 n_1 + \cdots + \alpha_{d-1} n_{d-1} \right\| \cdot \prod_{k=1}^{d-1} \left( |L_k (n)|+1 \right) < |n|^{-\varepsilon}$$ has finitely many integer solutions $n \in \mathbb{Z}^{d-1}$.
We derive the theorem from two different versions of Schmidt’s subspace theorem. First, a special case of the subspace theorem [@Schmidt Corollary 1] says that for any $\varepsilon >0$ the inequality $\left\| \alpha_1 n_1 + \cdots + \alpha_{d-1} n_{d-1} \right\| < |n|^{-(d-1)-\varepsilon}$ has finitely many integer solutions $n \in \mathbb{Z}^{d-1}$. Therefore it will be enough to consider $n \in \mathbb{Z}^{d-1}$ such that, say, $\left\| \alpha_1 n_1 + \dots + \alpha_{d-1} n_{d-1} \right\| \ge |n|^{-d}$.
Let $c_0 \le c_1 \le \dots \le c_{d-1}$ be reals such that $\sum_{k=0}^{d-1} c_k =0$, and let $M_0, M_1, \dots$, $M_{d-1}$ be linear forms of $d$ variables with real algebraic coefficients. We call $$\label{generalroth}
( M_0,M_1, \dots, M_{d-1} ; c_0, c_1, \dots , c_{d-1} )$$ a *general Roth system* if for every $\delta >0$ there exists a $Q_1>0$ such that for any real $Q \ge Q_1$ the system of inequalities $$\label{systemofineq}
|M_k (m)| \le Q^{c_k-\delta} \qquad (0 \le k \le d-1)$$ has no integer solution $m \in \mathbb{Z}^d, m \neq 0$. For a linear subspace $S$ of $\mathbb{R}^d$ of dimension $r>0$, define $c(S)$ the following way. If the rank of the forms $M_0, M_1, \dots, M_{d-1}$ on $S$ is less than $r$, then let $c(S)=\infty$. Otherwise, let $k_1$ be the smallest index such that $M_{k_1}$ is not constant zero on $S$. Let $k_2>k_1$ be the smallest index such that $M_{k_1}, M_{k_2}$ have rank 2 on $S$ etc, and define $c(S)=c_{k_1} + \cdots + c_{k_r}$. A general version of the subspace theorem [@Schmidt Theorem 2] states that is a general Roth system if and only if $c(S) \le 0$ for every rational linear subspace $S \neq 0$ of $\mathbb{R}^d$.
Fix an $\varepsilon >0$, and let us choose a positive integer $p$ such that $1/p < \varepsilon /(3d^2)$. Let $M_0 (x) = x_0+ \alpha_1 x_1 + \cdots + \alpha_{d-1} x_{d-1}$, and let $M_k (x) = L_k (x_1, \dots, x_{d-1})$, $1 \le k \le d-1$ be linear forms of the variables $x=(x_0, x_1, \dots, x_{d-1})$. We wish to apply the subspace theorem to $M_0, M_1, \dots, M_{d-1}$ with $\delta = 1/p$, $c_0=-1$ and $c_1, \dots, c_{d-1}$ all of the form $j/p$ for some integer $1 \le j \le p$ such that $\sum_{k=0}^{d-1}c_k=0$. The forms $M_0, M_1, \dots, M_{d-1}$ are clearly linearly independent, because $x_0$ appears only in $M_0$, and $L_1, \dots, L_{d-1}$ are linearly independent. Hence on any rational subspace $S$ of $\mathbb{R}^d$ of dimension $r>0$ the rank of $M_0, M_1, \dots, M_{d-1}$ is $r$. Moreover, choosing a nonzero rational vector $v \in S$ we have $M_0 (v) \neq 0$. Therefore in the definition of $c(S)$ we have $k_1=0$, and so $$c(S) = c_{k_1} + \dots + c_{k_r} \le -1 + \sum_{k=1}^{d-1} c_k =0 .$$ According to the subspace theorem we thus have a general Roth system. Since there are finitely many ways to choose such $c_1, \dots, c_{d-1}$, there exists a $Q_1 >0$ depending only on $\alpha_1, \dots, \alpha_{d-1}$, $L_1, \dots, L_{d-1}$ and $\varepsilon$ such that for any real $Q \ge Q_1$ and any such $c_1, \dots, c_{d-1}$ the system of inequalities has no integral solution $m \in \mathbb{Z}^d$, $m \neq 0$.
Consider now an integer solution $n \in \mathbb{Z}^{d-1}$, $n \neq 0$ of such that $$\left\| \alpha_1 n_1 + \cdots + \alpha_{d-1} n_{d-1} \right\| \ge |n|^{-d}.$$ Let the integer $Q>0$ be such that $$\frac{1}{(Q+1)^{1+\delta}} < \left\| \alpha_1 n_1 + \cdots + \alpha_{d-1} n_{d-1} \right\| \le \frac{1}{Q^{1+\delta}} .$$ From we have $1/(Q+1)^{1+\delta} \le |n|^{-\varepsilon}$, and so for a given integer $Q>0$ there are finitely many such solutions $n$. It will therefore be enough to show that $Q < Q_1$.
Choosing $m_k=n_k$ for $1 \le k \le d-1$ and $m_0$ to be the integer closest to $\alpha_1 n_1 + \cdots + \alpha_{d-1} n_{d-1}$, we have $|M_0 (m)| \le Q^{c_0-\delta}$. Note $Q \le |n|^d$. From we have $$-(1+\delta) \log (Q+1) + \sum_{k=1}^{d-1} \log (|L_k (n)| +1) < - \varepsilon \log |n| \le - \frac{\varepsilon}{d} \log Q .$$ Since $1+\delta - \varepsilon /d < 1+\varepsilon / (3d^2) -\varepsilon /d$, we have, for $Q$ large enough, that $$\sum_{k=1}^{d-1} \frac{\log (|L_k (n)|+1)}{\log Q} < 1+\frac{\varepsilon}{3d^2} -\frac{\varepsilon}{d} .$$ Let $c_k'$ be the number of the form $j/p$ for some integer $j \ge 1$, such that $c_k'-2/p < \frac{\log (|L_k (n)|+1)}{\log Q} \le c_k'-1/p$. Then we clearly have $$\sum_{k=1}^{d-1} c_k' \le \sum_{k=1}^{d-1} \left( \frac{\log (|L_k (n)|+1)}{\log Q} + \frac{2}{p} \right) < 1+\frac{\varepsilon}{3d^2} -\frac{\varepsilon}{d} + \frac{2d}{p} <1 .$$ By increasing $c_k'$ we can find numbers $c_k \ge c_k'$ of the form $j/p$ for some integer $1 \le j \le p$ such that $\sum_{k=0}^{d-1} c_k = -1 + \sum_{k=1}^{d-1} c_k =0$. From $\frac{\log (|L_k (n)|+1)}{\log Q} \le c_k'-1/p$ we have $$|M_k (m)| \le |L_k (n)|+1 \le Q^{c_k' - \delta} \le Q^{c_k - \delta} \qquad (1 \le k \le d-1) .$$ Therefore $Q < Q_1$, and we are done.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We present the first survey of electric field data using the *ARTEMIS* spacecraft in the solar wind to study inertial range turbulence. It was found that the average perpendicular spectral index of the electric field depends on the frame of measurement. In the spacecraft frame it is $-5/3$, which matches the magnetic field due to the large solar wind speed in Lorentz transformation. In the mean solar wind frame, the electric field is primarily due to the perpendicular velocity fluctuations and has a spectral index slightly shallower than $-3/2$, which is close to the scaling of the velocity. These results are an independent confirmation of the difference in scaling between the velocity and magnetic field, which is not currently well understood. The spectral index of the compressive fluctuations was also measured and found to be close to $-5/3$, suggesting that they are not only passive to the velocity but may also interact nonlinearly with the magnetic field.'
author:
- 'C. H. K. Chen, S. D. Bale, C. Salem, and F. S. Mozer'
title: Frame Dependence of the Electric Field Spectrum of Solar Wind Turbulence
---
Introduction
============
The solar wind is a plasma that is observed to be turbulent with fluctuations at a broad range of scales [@tu95; @goldstein95; @horbury05; @bruno05a; @petrosyan10]. It is usually modeled as a cascade of energy from large scales [e.g., @wicks10a], where the energy is injected, to small scales [e.g., @chen10b], where kinetic processes dissipate the energy [e.g., @schekochihin09]. The inertial range fluctuations are thought to be primarily [Alfvénic]{} in nature, with [Alfvén]{}-wave-like polarizations [@belcher71] and phase speeds close to the [Alfvén]{} speed [@bale05].
There are various theories of [Alfvénic]{} turbulence, based on interacting packets of [Alfvén]{} waves. The theory of @goldreich95, based on critical balance, predicts that the [Alfvénic]{} fluctuations have a perpendicular one-dimensional energy spectrum $E(k_\perp)\sim k_\perp^{-5/3}$, where $k_\perp$ is the wavevector perpendicular to the magnetic field. The theory of @boldyrev06, which in addition assumes scale-dependent alignment, predicts that their spectrum is $E(k_\perp)\sim k_\perp^{-3/2}$. Similar predictions also exist for the multitude of imbalanced theories [e.g., @lithwick07; @beresnyak08; @chandran08; @perez09; @podesta10c; @podesta11b].
In the solar wind at 1 AU, it has been shown that the spectral index of the magnetic field is close to $-5/3$ on average but that the spectral index of the velocity is closer to $-3/2$ [e.g., @mangeney01; @podesta07; @salem09; @tessein09; @podesta10d; @wicks11]. This difference between the two fields is not consistent with any of the current theories of [Alfvénic]{} turbulence and is one of the currently unsolved problems of solar wind turbulence.
Past measurements of the electric field spectrum in the frame of the spacecraft found it to closely match the magnetic field [@bale05; @sahraoui09]. These measurements used single intervals of data but it has been shown [e.g., @tessein09] that the velocity and magnetic field have a large spread of spectral indices and many intervals are needed to determine the average behavior.
In this Letter, we present a survey of electric field measurements in the solar wind using many intervals of data. We explain why the electric field in the spacecraft frame follows the magnetic field and make new measurements of the electric field in the mean solar wind frame. In Section \[sec:data\], we describe the data set, in Section \[sec:results\] we discuss our results and in Section \[sec:conclusions\] we present our conclusions.
Data Set {#sec:data}
========
We used data from the *ARTEMIS* mission [@angelopoulos10], which is an extension of the *THEMIS* mission [@angelopoulos08]. During late 2010, the two *ARTEMIS* spacecraft (*P1* and *P2*) moved from equatorial Earth orbits to Lunar Lagrange orbits ($\sim$ 60 $R_{\text{E}}$ from the Earth). Periods of solar wind data were selected in which each spacecraft was upstream of the Moon, out of Earth’s ion foreshock and the required instruments were operational. The selected days are: days 245–257, 308–310, 316–318, 337–343 of 2010 and days 1–3, 40–42 of 2011 for *P1*; days 217–230, 275–284, 304–307, 361–364 of 2010 and days 25–28 of 2011 for *P2*. The same day in both spacecraft was avoided so that the intervals are independent. All of the data from these days were split into 6 hr sections resulting in 272 intervals, 98% of which were in slow solar wind ($<$500 km s$^{-1}$).
Spin resolution ($\sim$3 s) electric field data, [$\mathbf{E}_{\text{sc}}$]{}, from the electric field instrument [EFI; @bonnell08] was used, along with spin resolution magnetic field data, $\mathbf{B}$, from the fluxgate magnetometer [FGM; @auster08] and varying resolution ion velocity, $\mathbf{v}$, and ion number density, $n$, onboard moments from the electrostatic analyzer [ESA; @mcfadden08]. A despun spacecraft coordinate system (DSL) was used, in which $z$ is the spacecraft spin axis. The DSL system for *ARTEMIS* is approximately the same as the geocentric solar ecliptic (GSE) system with the sign of the $y$- and $z$- axes reversed. The wire boom electric field antennas are in the $x$–$y$ plane and extend a few Debye lengths from the spacecraft. Data with the currently most recent calibrations (v01) were used for all instruments.
For [$\mathbf{E}_{\text{sc}}$]{} it was found that some extra calibration was needed. A least-squares fit, varying the [$E_{y,\text{sc}}$]{} offset $O_{E_y}$ and the [$\mathbf{E}_{\text{sc}}$]{} scaling factor $F$, was performed to minimize the difference between [$E_{y,\text{sc}}$]{} and the $y$-component of $-\mathbf{v}\times\mathbf{B}$ for each interval (this technique assumes ideal MHD). Each 6 hr interval was corrected using these empirically determined values. The mean value of $O_{E_y}$ was found to be $-0.17$ mV m$^{-1}$ for *P1* and $-0.23$ mV m$^{-1}$ for *P2*; the mean value of $F$ was found to be 1.02 for *P1* and 0.99 for *P2*. An alternative fit using a $B_z$ offset instead of an [$E_{y,\text{sc}}$]{} offset was also tried, resulting in $B_z$ offsets $\sim$ 0.6 nT. The results of this Letter, however, are not significantly affected by either of these additional calibration methods.
The velocity and density data was cleaned up by removing unphysical spikes and other spurious data that was present. This was done by linearly interpolating over data points more than 4 standard deviations from the mean in each 6 hr interval (this process was repeated three times for each interval). Any data gaps in the 3 s resolution data were also linearly interpolated over to produce time series with consistent 3 s resolution. After this process, occasional small spikes, sometimes seen in all three instruments, remained in the time series of some intervals. These are likely due to noise but did not affect this analysis (excluding these intervals did not significantly change the results of this Letter).
The electric field was measured in the frame of the spacecraft, [$\mathbf{E}_{\text{sc}}$]{}, and converted into the frame of the mean solar wind velocity, [$\mathbf{E}_{\text{sw}}$]{}, using the Lorentz transformation, $$\label{eq:lorentz}
\mathbf{E}_{\text{sw}}=\mathbf{E}_{\text{sc}}+\mathbf{v}_{\text{sw}}\times\mathbf{B},$$ where $\mathbf{v}_{\text{sw}}$ is the mean solar wind velocity relative to the spacecraft over each interval ($\mathbf{v}={\ensuremath{\mathbf{v}_{\text{sw}}}}+\delta\mathbf{v}$). Since $\mathbf{B}$ is a fluctuating quantity, it was linearly interpolated onto the times of [$\mathbf{E}_{\text{sc}}$]{} so that the transformation could be done for each electric field measurement.
The power spectrum of each component of $\mathbf{B}$ and $\mathbf{v}$, of the $x$- and $y$- components of [$\mathbf{E}_{\text{sc}}$]{} and [$\mathbf{E}_{\text{sw}}$]{}, of the magnetic field magnitude $|\mathbf{B}|$, and of $n$ was calculated for each of the intervals. The multitaper method with time-bandwidth product $NW=4$ and 7 eigentapers [@percival93] was used (using a standard Fourier transform does not affect the results to within errors). Typical power spectra for a longer interval (14 days) are shown in Figure \[fig:spectra\]: the trace of the magnetic field spectrum and the $y$-component of the electric field spectrum in both frames. Since the solar wind fluctuations are anisotropic with $k_\perp>k_{\parallel}$ [e.g., @chen10a], these are measurements of the perpendicular spectrum $E(k_\perp)$ (to measure $E(k_{\parallel})$ a local field tracking technique would be needed [e.g., @horbury08; @chen11a]).
Each spectral index was determined from the gradient of the best-fit line to the power spectrum in log–log space over the spacecraft frequency range $1\times10^{-3}$ Hz to $2\times10^{-2}$ Hz (marked as dotted lines in Figure \[fig:spectra\]). Applying Taylor’s hypothesis [@taylor38] since the solar wind is super-[Alfvénic]{}, this range corresponds approximately to scales 18,000 km to 350,000 km and a perpendicular wavevector range $0.0018<k_\perp\rho_i<0.036$, where $\rho_i\approx100$ km is the typical ion gyroradius. This is in the middle of the inertial range and was chosen because good power laws exist here in all intervals. The results of the analysis are described in the next section.
Results {#sec:results}
=======
Histograms of the spectral indices for the magnetic and electric fields are shown in Figure \[fig:histograms\]. The magnetic field trace spectral index histogram can be seen to peak close to $-5/3$, in agreement with previous results [e.g., @smith06a; @tessein09]. The histogram of the $y$-component of the electric field in the spacecraft frame also peaks near $-5/3$ but the histogram of the same component in the mean solar wind frame peaks closer to $-3/2$.
The mean spectral indices for each field are given in Table \[tab:scaling\], along with the standard error of the mean $\sigma/\sqrt{N}$, where $\sigma$ is the sample standard deviation and $N$ is the number of intervals. The mean velocity and density spectral indices were calculated from only 117 and 120 of the intervals, respectively. These are the intervals for which 3 s onboard moment data was available and no more than 5% of the data was missing. This explains the larger error for these fields. The same analysis was also tried with 24 hr intervals (not shown here), resulting in a smaller spread of spectral indices but the same mean values to within 2 standard errors.
[cc]{} $B_{\text{trace}}$ & $-1.67\pm 0.01$\
$v_{\text{trace}}$ & $ -1.50\pm0.02$\
[$E_{y,\text{sc}}$]{}& $-1.66\pm 0.01$\
[$E_{x,\text{sc}}$]{}& $-1.45\pm 0.01$\
[$E_{y,\text{sw}}$]{}& $-1.40\pm 0.01$\
[$E_{x,\text{sw}}$]{}& $-1.39\pm 0.01$\
$|\mathbf{B}|$ & $-1.64\pm0.01$\
$n$ & $-1.63\pm0.02$
The fact that the scaling of [$E_{y,\text{sc}}$]{} matches $B_{\text{trace}}$ can be shown to be due to the Lorentz transformation. In ideal MHD, the three fields are related to each other in the mean solar wind frame by $\mathbf{E}_{\text{sw}}=-\delta\mathbf{v}\times\mathbf{B}$ and putting this into Equation (\[eq:lorentz\]) gives $$\label{eq:ideallorentz}
\mathbf{E}_{\text{sc}}=-\delta\mathbf{v}\times\mathbf{B}-\mathbf{v}_{\text{sw}}\times\mathbf{B}.$$ Since the mean solar wind speed is much larger than the fluctuations, $|\mathbf{v}_{\text{sw}}|>|\delta\mathbf{v}|$, and is mostly in the $x$ (radial) direction, [$E_{y,\text{sc}}$]{} is dominated by the magnetic field fluctuations convected by the mean solar wind flow and therefore follows their scaling. The amplitudes of the spectra in Figure \[fig:spectra\] agree with this interpretation: the [$E_{y,\text{sc}}$]{} spectrum is an order of magnitude larger than the [$E_{y,\text{sw}}$]{} spectrum, showing that for the $y$-component, the second term on the right-hand side of Equation (\[eq:ideallorentz\]) is larger than the first. The $x$-component of Eq. \[eq:ideallorentz\] does not depend on the radial solar wind velocity, so scaling of [$E_{x,\text{sc}}$]{} does not depend only on the scaling of $\mathbf{B}$ and indeed is different to that of $B_{\text{trace}}$.
[$\mathbf{E}_{\text{sw}}$]{} has a scaling closer to that of $v_{\text{trace}}$, which can also be shown to be consistent with [Alfvénic]{} fluctuations in ideal MHD. Splitting the magnetic field into a constant mean value plus fluctuations, $\mathbf{B}=\mathbf{B}_0+\delta\mathbf{B}$, the electric field in the mean solar wind frame is given by $$\mathbf{E}_{\text{sw}}=-\delta\mathbf{v}\times\left(\mathbf{B}_0+\delta\mathbf{B}\right).$$ The mean value of $|\delta\mathbf{B}|/|\mathbf{B}_0|$ is between 0.1 and 0.4 for the range of scales to which spectral indices were fitted. Since at these small scales in the solar wind $\mathbf{B}_0>\delta\mathbf{B}$, the electric field spectrum in the mean solar wind frame is dominated by the velocity fluctuations, and therefore has a similar scaling. Similar arguments can be made for the [Alfvénic]{} fluctuations in gyrokinetic theory [@schekochihin09]. The fact that we observe a spectral index close to $-3/2$ in [$E_{x,\text{sw}}$]{} and [$E_{y,\text{sw}}$]{} also suggests that the perpendicular velocity component has this scaling, which is in agreement with the results of @chapman07. The electric field scaling is also in agreement with previous measurements of the velocity trace spectral index [e.g., @tessein09; @podesta10d].
The scaling of the compressive fluctuations ($|\mathbf{B}|$ and $n$) is close to $-5/3$, matching the trace magnetic field spectrum, rather than the velocity spectrum. Previous observations [e.g., @marsch90b; @bellamy05; @issautier10] could not distinguish between $-5/3$ and $-3/2$ in the compressive fluctuations so this scaling is consistent with those observations. The compressive fluctuations are mainly due to the slow mode [@howes11b] and are sometimes thought to be passive to the [Alfvénic]{} turbulence. Since their scaling matches the magnetic field, rather than the velocity, the nonlinearity cannot be due solely to passive convection and may include nonlinearities with the magnetic field. This supports the theories of compressible reduced MHD and kinetic reduced MHD [@schekochihin09], in which the compressive fluctuations interact nonlinearly with both the magnetic field and velocity.
To test the significance of the difference between the mean spectral index values in Table \[tab:scaling\], the $t$-test was applied. This is appropriate since the spectral indices appear to be normally distributed and are independent measurements. The $t$ value for differentiating between the spectral indices of $B_{\text{trace}}$ and [$E_{y,\text{sc}}$]{} is $t=0.41$. This is smaller than the 95% value of 1.96 for infinite degrees of freedom, showing that there is no statistically significant difference between the scaling of these two fields. For differentiating between the spectral indices of [$E_{y,\text{sw}}$]{} and [$E_{y,\text{sc}}$]{} the $t$ value is $t=15$, larger than the 95% value, showing that these two fields have significantly different spectral indices. This confirms that the $-5/3$ and $-3/2$ difference is a statistically robust result.
To examine the cause of the spread of spectral index values, the correlation between the different spectral indices was measured. The linear correlation coefficients, calculated from various pairs of sets of the 272 spectral index values of each field, are shown in Table \[tab:correlations\]. It can be seen that the spectral indices of most pairs of fields are poorly correlated, having correlation coefficients lower than 0.4. This suggests that the spread of values is mostly due to random, rather than systematic, variation, although the fact that the correlation coefficients are all slightly positive suggests perhaps some small underlying systematic variation. The exceptions are correlations between $B_{\text{trace}}$ and [$E_{y,\text{sw}}$]{} and between [$E_{x,\text{sc}}$]{} and [$E_{x,\text{sw}}$]{}, which have correlation coefficients larger than 0.8. This is due to the reasons discussed above: the [$E_{y,\text{sw}}$]{} spectrum is essentially a measure of the $B_{\text{trace}}$ spectrum because the $y$-component of the last term in Equation (\[eq:ideallorentz\]) is large and [$E_{x,\text{sc}}$]{} and [$E_{x,\text{sw}}$]{} are similar because the $x$-component of the last term in Equation (\[eq:ideallorentz\]) is not large (since [$\mathbf{v}_{\text{sw}}$]{} is mostly in the $x$-direction).
[ccc]{} $B_{\text{trace}}$ & [$E_{y,\text{sc}}$]{}& 0.82\
$B_{\text{trace}}$ & [$E_{y,\text{sw}}$]{}& 0.15\
$B_{\text{trace}}$ & [$E_{x,\text{sc}}$]{}& 0.24\
$B_{\text{trace}}$ & [$E_{x,\text{sw}}$]{}& 0.08\
[$E_{y,\text{sc}}$]{}& [$E_{y,\text{sw}}$]{}& 0.17\
[$E_{y,\text{sc}}$]{}& [$E_{x,\text{sc}}$]{}& 0.24\
[$E_{y,\text{sc}}$]{}& [$E_{x,\text{sw}}$]{}& 0.14\
[$E_{y,\text{sw}}$]{}& [$E_{x,\text{sc}}$]{}& 0.38\
[$E_{y,\text{sw}}$]{}& [$E_{x,\text{sw}}$]{}& 0.36\
[$E_{x,\text{sc}}$]{}& [$E_{x,\text{sw}}$]{}& 0.86
Summary and Conclusions {#sec:conclusions}
=======================
We have performed the first survey of electric field data in the solar wind to measure the perpendicular spectrum of inertial range fluctuations. It was found that there is a spread of spectral index values but that the average spectral index depends on the frame of measurement. In the spacecraft frame, the $y$-component of the electric field is primarily due to the magnetic field fluctuations being convected past the spacecraft at the average solar wind speed. It, therefore, has the same average spectral index (to within errors) as the magnetic field of $-1.66\pm0.01$. This is consistent with previous single interval electric field measurements in the spacecraft frame [@bale05; @sahraoui09]. In the mean solar wind frame, the electric field is primarily due to velocity fluctuations in a mean magnetic field and has a spectral index of $-1.40\pm0.01$, which is closer to the velocity spectral index than the magnetic field spectral index, although not the same to within errors. The compressive fluctuations ($|\mathbf{B}|$ and $n$) were found to have the same spectral index as the magnetic field and not the velocity.
The difference between the scaling of the electric field in the spacecraft frame and the mean solar wind frame provides independent confirmation of the difference in scaling between the velocity and magnetic field. This difference is not expected for [Alfvénic]{} fluctuations, since $\delta\mathbf{v}$ is proportional to $\delta\mathbf{B}$ in an [Alfvén]{} wave, and is not predicted by any of the current theories of [Alfvénic]{} turbulence (although see recent work by @boldyrev11 and @wang11). Recently, @roberts10 found that further out into the heliosphere, past 5 AU, the velocity spectral index evolves toward $-5/3$ to match the magnetic field. Although an important result, this does not explain the difference at 1 AU. Possible reasons for the difference include the effects of scale-dependent alignment, imbalance and residual energy and these will be investigated in a future paper.
This work was supported by NASA grant NNX09AE41G. We acknowledge the *THEMIS* team and NASA contract NAS5-02099. We thank J. Bonnell, J. McFadden, A. Schekochihin, and D. Sundkvist for useful conversations and an anonymous referee for helpful comments.
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\#1[\#1|]{} \#1[| \#1]{} tempcntc citex\[\#1\]\#2[@fileswauxout tempcnta@tempcntb@neciteacite[forciteb:=\#2citeo]{}[\#1]{}]{} citeo[tempcnta>tempcntbciteacitea[,]{} tempcnta=tempcntbtempcnta]{}
[NEUTRINOLESS DOUBLE BETA DECAY AND ITS “INVERSE"]{}\
\
[*Santa Cruz Institute for Particle Physics*]{}\
[*University of California, Santa Cruz*]{}\
\
[*Institute for Theoretical Physics*]{}\
[*University of Bern, Switzerland*]{}\
ABSTRACT
Recent considerations by these authors pointed out the attractive features which a search for the exchange of heavy Majorana neutrinos could have for solving the mass and the lepton number puzzles for all neutrinos, in TeV-level electron-electron scattering. In the present note, we show that, contrary to subsequently published arguments, non-observation of neutrinoless double beta decay has, to date, no bearing on the promise of this important task for future linear electron colliders.
Recent developments in the planning for electron colliders in the TeV energy region have renewed interest in the possibility to investigate the reaction $$e^-e^- \to W^-W^-, % (1)
\label{eqone}$$ which can proceed by means of the exchange of a Majorana neutrino. While previous work on this reaction (Refs. ) had not led to promising experimental prospects, we showed that the new generation of presently projected linear colliders can in fact deliver luminosities compatible with the production of convincing signals for reaction (\[eqone\]), or, failing that, important new limits on the masses and couplings of heavy Majorana neutrinos. This is predicated on the availability of highly polarized electron beams at center-of-mass energies upward of 500 GeV, where left-handed electrons will be able to interact [*via*]{} the exchange of heavy left-handed Majorana singlet neutrino states with masses $\ge 1$ TeV. For the case of two longitudinal $W^-$ in the final state, the relevant cross-section increases $\sim s^2$ in the kinematic region $2m_W < \sqrt s < m_N$. The existence of two or more such singlets follows naturally from SO(10) decomposition, and could be the key ingredient for our understanding of both the observed very light masses of the three known neutrino states, and of the meaning of lepton number and its conservation or non-conservation.
Several authors [@belanger; @pantis] have argued that our calculations cannot serve as the basis for a possible successful observation of reaction (1): they maintain that existing limits on the related process involving neutrinoless double beta decay \[hereafter $\beta\beta_{0\nu}$\] already exclude it. In so doing, the authors of Ref. [@belanger] explicitly refer to reaction (\[eqone\]) as “inverse neutrinoless double beta decay".
The fallacy of their argument can, in a nutshell, be gleaned from this misnomer: in the present note, we show that only a profound misreading of the $\beta\beta_{0\nu}$ reaction mediated by heavy Majorana neutrinos can lead to the conclusions of Refs. [@belanger; @pantis]. Let us consider the diagrams in Fig. 1: we ask ourselves whether the lack of observation of graph 1b can serve to impose tight constraints on the observability of graph 1a.
\[fig:one\]
The best present evidence on the non-observation of the $\beta\beta_{0\nu}$ reaction is from the experimental limit on the process $$^{76}Ge \to ^{76}Se\ e^-e^-, %(2)
\label{eqtwo}$$ with $\tau_{1/2} > 5 \times10^{24}$ years. How do we translate this into a limit on reaction (\[eqone\])? Literally, the inverse to reaction (\[eqtwo\]) is
$$e^-e^-\ ^{76}Se \to\ ^{76}Ge \label{eqthree-a} \\ % (3a)$$
$$e^-e^- \to ^{76}Ge\ ^{76}\overline{Se}. \label{eqthree-b}\\ % (3b)$$
Neither of these two reactions are experimentally realizable, but the argument points up a troubling question: Can the (sub-)reactions that lead to the decays $nn \to ppe^-e^-$ occur freely inside the nuclei $^{76}$Ge and $^{76}$Se?
At this point, we have to remark that graph 1b symbolizes a process that acts over distances of order $(m_W)^{-1}$ or about $10^{-16}$cm. If, on the other hand, the exchanged neutrinos had the masses of the known light varieties, $m_\nu <1$ eV, the range over which the interaction extends would be $>10^{11}$ times larger. In the heavy ($\sim$1 TeV) neutrino exchange case, the subprocess we have to study is $$\begin{aligned}
dd &\to& uue^-e^- , \nonumber \\ % (4)
(nn)& &(pp) \\
((^{76}Ge))& & ((^{76}Se)) \nonumber\end{aligned}$$ where the symbols in single and double parentheses stand for the constraining configurations within which the interacting particles of the lines above have to be considered, respectively: the hadronic systems must be treated on the quark level, but are heavily constrained by nucleon-nucleon forces, and the latter are in turn constrained by the specific wave functions of their host nuclei. It is worth noting that, in this context, the study of $\beta\beta_{0\nu}$ [*via*]{} heavy $N$ exchange can be seen as a unique probing of nuclear structure on the quark correlation level, at distances $<10^{-16}$cm [@heusch]. What then is our chance of observing $dd$ overlap at these distances, within the constraints of eq. (4)? Let us first determine the constraints we can easily establish:
1. The final-state electrons have to emerge in an overall $S$ state, as a spin singlet. This, in turn, imposes a spin singlet configuration on the dd wave function.
2. To achieve an overall antisymmetric wave function for the $l=0\ dd$ system, the product of space and SU3$_c$ wave functions has to be symmetrical. This leaves only the 6 representation of SU3$_c$, imposing a suppression factor of 2/3.
3. To evaluate the strong Hamiltonian density involved in the $dd \to
uue^-e^-$ subgraphs of Fig. 2a, we have to keep the constraints imposed by the surrounding nuclear and nucleon environment in mind, as schematically shown in Fig. 2b. This leads to two further suppression factors to be determined: one is due to the color Coulomb repulsion of the $d$ quarks, the other to the collective pull which the saturated nucleon configurations of two neutrons exert on each quark that may be drawn into an interaction with a quark from another nucleon.
4. Finally, the resulting Hamiltonian density operator will have to include the leptonic weak current operator, integrated over the appropriate interaction volume, and then sandwiched between the mother and daughter nuclei’s wave functions.
\[fig:two\]
Each of these effects will lead to a suppression factor. In an attempt to write down the Hamiltonian density of the highly local quark-quark Hamiltonian density in the overall expression $$H = G^2_F \left[ \bar e_\alpha \bar u^{c\alpha} d^c_\gamma
(x_2) \bar e_\beta \bar u^{b\beta} d^{b\gamma}
(x_1)\right] _{x_2\to x_1} %% (5)$$ we can write, using the customary lepton-hadron factorized expression, $$H \propto U_{eN}^2 / m_N
\left \lbrack \bar{e}_{\alpha} \bar{e}_{\beta} \right\rbrack\
\left \lbrack \bar{u}^{c \alpha}
d^{c \gamma}\bar{u}^{b \beta} d^{b}_{\gamma}
\right\rbrack$$ $U_{eN}$ is the mixing angle for electron/heavy neutrino $N$ with mass $M$, the $b,c$ are color indices. The high degree of locality that governs the interaction involving both sub-diagrams of Fig. 2a permits us to rewrite the hadronic Hamiltonian density in eq. (6) such as to pair like-flavor quarks: $$H_{q} (x) = \left[ \bar{u}^{b}_{\alpha}
\bar{u}^{c \alpha} (x) \right]
\ \left[ d^{c \gamma} d^{b}_{\gamma} \ (x)\right]\ .$$
This density operator can then conveniently be sandwiched between the nuclear state vectors for mother $(A,Z)$ and daughter $(A,Z+2)$ nucleus, for a hadronic matrix element $$\bra{A,Z+2;p_{2} } H_{q} (x) \ket{A,Z;p_{1}} =
V^{-1} e^{i q x} \varrho_{21}\ ; \quad
q = p_2 - p_1 \ .$$
$V$ is a normalizing volume, $\rho$ is a density matrix that contains the Fermi and Gamow-Teller structure of the interaction when expressed in the current–current form (VV and AA on the quark level). It can be saturated with a complete set of the possible intermediate states with the requisite energy and momentum [@noteone]. We will narrow our interest down to two-nucleon correlations; they will dominate the small-distance behavior in the case of $N_M$ exchange. Recall that $H_q$ in eq. (8) is a four-quark operator: the $H_q$ operator does not “see" the Ge, As, Se nuclei; rather, its interaction involves the 76 nucleons in terms of their 228 quarks.
We therefore have to try and evaluate the quark-quark suppression factors in the density matrix $\rho_{21}$ of eq. (8). They are a function of the relative distance $r_{12}$ of any two interacting quarks, at $r_{12}$ values in the region of the “hard-core", $r_{12}<0.3$fm. First, there is the 2/3 factor due to the spin singlet requirement (see above). Second, the repulsive color sextet interaction can be reasonably estimated by a WKB method for an evaluation of the color Coulomb barrier. A straightforward relativistic treatment leads to a barrier penetration/inhibition factor $$F_B = e^{- \pi\alpha_s/3},$$ irrespective of the nuclear environment.
Third, as Fig. 2b indicates, there are similar inhibition factors to be expected from the interaction of the remaining two quarks in the two interacting neutrons, two each that are not directly overlapping. We illustrate this schematically in Fig. 2c: each of the two central $d$ quarks is being “pulled on" by a $u$ and a $d$ quark from its “own" neutron, trying to keep the straying companion in the color singlet configuration. Although the relevant Clebsch Gordan coefficient may make these inhibition factors somewhat stronger, we approximate them by a joint ansatz of $$F_{nn} \approx F_B^2 = e^{-2/3 \pi \alpha_s}.$$ This factor of order 1/9 is very conservatively estimated, given that the color force increases considerably above the $r^{-2}$ level known for the small-distance behavior valid for the initial $dd$ interaction at $r_{12}$ values of $>~1/3$fm, for this somewhat longer-range color singlet restoration force. We feel justified in regarding this suppression as amply supported by such evidence as the loose binding of the deuteron and the non-existence of bound $nn$ and $pp$ states. Note that this repulsive vector interaction can also be modeled in terms of omega meson exchange between the two neutrons, leading to an inhibition factor stronger than the one resulting from eq. (10).
Lastly, let us recall that the entire process thus inhibited by a factor bounded from above by $2/3 \times (1/3)^3 = 2/81$, or about 0.025, but likely to be considerably smaller, has to happen inside the “mother-daughter" nuclear system. For the $^{76}$Ge $\beta\beta_{0\nu}$ decay with the best present limits, this means that the above suppression modifies the nuclear wave function overlap symbolized by the projection operator $$\hat P \equiv \ket{^{76}Se^{-2u}} \bra{^{76}Ge^{-2d}} , %% 11$$ which effectively sums over all intermediate states in the density matrix $\rho_{21}$ that contain two $d$ quarks less than $^{76}$Ge, two $u$ quarks less than $^{76}$Se [@PM].
Finally, we compare our conservative estimate of the overall inhibition factor with the recent literature: it reduces the exclusion zone for observable heavy Majorana neutrino masses from the estimate made by Pantis et al. [@pantisetal] from $(1/m_N)_L^{-1} \gsim 6.7\times10^3$ TeV to $${(m_N)_L\over |U_{eN}|^2} = 0.025 \times 6.7 \times 10^3\ {\rm TeV}.$$ With the mixing parameter $|U_{eN}|^2 = (2-40) 10^{-4}$ Ref. [@heuschmink1], this puts $$(m_N)_L > \left\{ {0.67\ {\rm TeV}\atop 0.033\ {\rm TeV}} \right\}\
{\rm for\ the}\ \left\{ {{\rm upper}\atop {\rm lower}} \right\}
\ {\rm limit\ on}\ |U_{eN}|^2 .
%eq.(13)$$ Similarly, it moves the exclusion zone advocated in Ref. [@belanger] for the possible observation of process (1) in the face of existing evidence from neutrinoless double beta decay searches, as drawn in the $U_{eN}^2$ vs. $m_N$ plane, well out of danger’s way [@notetwo]. We conclude that established limits on the observation of neutrinoless double beta decay do not in any way preclude the observability of process (1), and thereby the possible discovery of TeV-level Majorana masses in electron-electron scattering. The next generation of electron linear colliders thus has a highly attractive chance of unraveling the major mystery that shrouds our understanding of the observed lepton spectrum and forces an illogical treatment of the lepton sector on the Standard Model.
\#1&\#2&\#3&[*Phys. Rev. Lett. **\#1, \#2 (19\#3)***]{} \#1&\#2&\#3&[*Phys. Rev. **\#1, \#2 (19\#3)***]{} \#1&\#2&\#3&[*Phys. Rep. **\#1, \#2 (19\#3)***]{} \#1&\#2&\#3&[*Nucl. Phys. **\#1, \#2 (19\#3)***]{} \#1&\#2&\#3&[*Phys. Lett. **\#1, \#2 (19\#3)***]{}
[00]{} K. Ushio, [*Lett. Nuovo Cimento*]{} [**35**]{}, 121 (1992). T.G. Rizzo, B116&23&82&; D50&5602&94&. D. London, G. Belanger, J.N. Ng, B188&155&87&. J. Maalampi, A. Pietilä, J. Vuori, B279&327&92&. G. Belanger, F. Boudjema, D. London, H. Nadeau, D53&6292&96&. G. Pantis, J.G. Vergados, 242&285&94&. C.A. Heusch, Proceedings of the 7th International Workshop on Neutrino Telescopes, February 27–March 1, 1996, M. Baldo Ceolin ed., SCIPP 96/18, to be published. In the case of 0+ ground state transitions between $^{76}$Ge and $^{76}$Se, this means all bound states of $^{76}$As plus many hadronic combinations with the same quantum numbers. P. Minkowski, [*Nucl. Phys.*]{} [**B201**]{}, 269 (1982). G. Pantis, A. Faessler, W.A. Kaminski, J.A. Vergados, [*J. Phys.*]{} [**G18**]{}, 605 (1992). C.A. Heusch, P. Minkowski, B416&3&94&. Note that Fig. 5 of Ref. [@belanger] has to be corrected for the presence of a high degree of polarization in both incident electron beams. With $P_e=90$%, this raises the sensitivity by a factor 3.24 at the same time that the lower limits to the exclusion zone are decreased by the factor 0.025.
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=1
Introduction
============
In this paper we give explicit closed forms for the semi-reproducing kernels associated with thinplate spline interpolation on the sphere. Polyharmonic or thinplate splines for $\RR^d$ were introduced by Duchon in his classic papers [@Duchon_76; @Duchon_77] and have become a widely used tool in myriad applications. The analogues for ${{{\mathbb S}^{d-1}}}\subset {{\mathbb R}}^d$ are the thinplate splines for the sphere. The topic was first discussed by Wahba [@Wahba81; @Wahba82] in the early 1980’s, for the ${{\mathbb S}}^2$ case. Wahba presented the associated semi-reproducing kernels as infinite series. These semi-reproducing kernels play a central role in expressions for the solution of the associated spline interpolation and smoothing problems.
The main aims of the current paper are to give a recurrence for these semi-reproducing kernels, and also to use the recurrence to obtain explicit closed form expressions. Here we are building on previous work of Martinez-Morales [@Ma05]. Unfortunately, there are errors in the theory presented in [@Ma05] and consequently many of the expressions given there for the kernels are incorrect. The closed form expressions given here will usually be significantly faster to evaluate than the series expansions. This will enhance the practicality of using the thinplate splines for the sphere in computations.
The paper is laid out as follows. Section \[sec:reproducing\_kernels\] discusses the central role played by semi-reproducing kernels in the solution of both interpolation and penalized least squares fitting problems. Section \[sec:Thinplate\_splines\_on\_the\_sphere\] develops semi-reproducing kernels associated with the thinplate splines on the sphere, that is, semi-reproducing kernels associated with minimum energy interpolation and penalized least squares fitting problems with a particular choice of energy. The energy chosen being that naturally associated with iterated Laplace–Beltrami operators. These semi-reproducing kernels are given in this section as infinite series. Section \[sec:Thinplate\_splines\_on\_the\_sphere\] also recalls known results concerning Fourier–Gegenbauer expansions that will be needed later. Section \[sec:the\_operators\_T\_and\_T\*\] motivates the construction of an operator $T$ and its adjoint $T^*$. It also presents some fundamental properties of these operators. These operators were initially developed in Martinez-Morales [@Ma05]. Section \[sec:recurrence\] gives a recurrence for the various thinplate spline kernels $K_{d,m}(x,y)$, where $d$ indicates the dimension and $m$ is the power of the associated differential operator. More precisely, it gives a recurrence for the related functions $k_{d,m}\colon [-1,1]\rightarrow {{\mathbb R}}$ where $k_{d,m}(\langle x, y \rangle)= K_{d,m}(x,y)$. Sections \[sec:explicit\_forms\] and \[sec:m\_equal\_1\] give short closed form expressions for many of the functions $k_{d,m}$.
For ease of access to the relevant background we will base our notation on that used in Dai and Xu [@DaiXu2013]. Occasionally, when the value for the dimension $d$ is particularly important, we will supplement the symbol they use with a $d$.
Reproducing kernels and approximation {#sec:reproducing_kernels}
=====================================
Let us begin with summarising the main ideas of reproducing kernels in indefinite inner product spaces and their role in both interpolation and penalized least squares fitting problems. These results show the central role of reproducing kernels in the solution of these problems. We focus on the specific case of reproducing kernels[^1] for semi-Hilbert spaces, or simply semi-reproducing kernels, as this is the appropriate framework for thinplate spline approximation. For further definitions and basic properties of semi-reproducing kernels we refer to [@BerlinetAgnan; @BezaevValisenko; @CheneyLight_2000; @MosamamKent2010]. Our treatment of the relevant interpolation and penalized least squares problems is based on that of Strauss [@Strauss_2002].
Let $\CD$ be a subset of $\RR^d$. Consider approximation from a vector space over the reals $\CH \subset C(\CD)$. Assume the space $\CH$ is endowed with a semi-inner product $( \cdot, \cdot)$, that is, the inner product is lacking definiteness. Thus, there are non-zero vectors $f\in\CH$ with $(f,f)=0$. Further assume that the kernel $\CH_0$ of the semi-inner product $(\cdot,\cdot)$ is finite-dimensional, i.e., $\dim\CH_0=m<\infty$, and $(f,f)=0$ if and only if $f\in \CH_0$.
A standard approach to deal with semi-inner products is to supplement the semi-inner product with an inner product on $\CH_0$ thereby obtaining a definite inner product on the space $\CH$ (see [@Bognar1974]). Towards this aim, we need to decompose the space $\CH$ into a direct sum $\CH_0\oplus\CH_1$, such that the given semi-inner product provides a definite inner product on the subspace $\CH_1$. Given $m$ linearly independent functionals spanning the dual of $\CH_0$, we define $\CH_1$ as the space of functions in $\CH$ which are mapped onto zero by all these functionals.
Let us recall this approach using point evaluations. Nevertheless, it is important to note that there are many choices for such a set of functionals. Clearly, the decomposition obtained for the space $\CH$ depends upon the choice made for the $m$ functionals.
A set of distinct points $\CX $ is said to be unisolvent for $\CH_0$ if the only function in $\CH_0$ which is zero at all points of $\CX$ is the zero function.
Given a unisolvent set $\CX=\{z_1, \ldots,x_m\}$ for $\CH_0$, where $m=\dim (\CH_0)$, the set of point evaluation functionals $\{\delta_{x}\colon x\in\CX \}$ is linearly independent on $\CH_0$. Hence, we can find a Lagrange basis $u_1,\dots, u_m$ of $\CH_0$ with respect to $\CX$, i.e., $u_i(x_j)=\delta_{ij}$, $1\leq i,j \leq m$. Then $$\begin{gathered}
[f,g]_0 = \sum_{j=1}^m f(x_j)g(x_j), \qquad f,g\in\CH_0,\end{gathered}$$ defines an inner product on $\CH_0$. The basis $u_1,\dots, u_m$ is orthogonal with respect to the inner product $[\cdot,\cdot]_0$. Furthermore, the mapping $$\begin{gathered}
P_0\colon \ \CH \to \CH_0, \qquad f\mapsto f_0=\sum_{j=1}^m f(x_j) u_j\end{gathered}$$ is a projection of $\CH$ onto $\CH_0$. The definition of inner product for the full space $\CH$ which follows will make $P_0$ the orthogonal projection onto $\CH_0$. The subspace $\CH_1$ can then be defined via the projector $P_1=I-P_0$, i.e., $$\begin{gathered}
\CH_1 = \{ f\in\CH \colon f(x)=0 \ \text{for all} \ x\in\CX\}.\end{gathered}$$ Since $\CH_0$ is the kernel of the semi-inner product $(\cdot,\cdot)$, i.e., $(f,f)=0$ iff $f\in\CH_0$, the semi-inner product is definite on $\CH_1$ via construction. Therefore, $$\begin{gathered}
[f,g] = [P_0f, P_0 g]_0 + (P_1f, P_1g), \qquad f,g \in \CH,\end{gathered}$$ defines a definite inner product on $\CH$ (see [@Bognar1974] for further details).
We are interested in Hilbert spaces carrying the special property of being reproducing kernel spaces. There is a one-to-one correspondence between reproducing kernel Hilbert spaces and positive definite kernels. A similar relation holds true for semi-reproducing kernel Hilbert spaces.
Given $n\in\NN$, a pair $(\CX,{{\boldsymbol{a}}})$ with $\CX=\{x_1,\dots,x_n\}$ a set of distinct points from $\CD$ and ${{\boldsymbol{a}}}=(a_1,\dots,a_n)^{\rm T}\in\RR^n$ is called an $\CH_0$-increment if $$\begin{gathered}
\sum_{j=1}^n a_j f(x_j) = 0 \qquad\text{for all} \ f\in\CH_0.\end{gathered}$$ The set of all $\CH_0$-increments is denoted by $\CH_0^\perp$.
Note that an $\CH_0$-increment can naturally be identified with a linear functional $$\begin{gathered}
\lambda_{\CX,{{\boldsymbol{a}}}}(f) = \sum_{j=1}^n a_j \delta_{x_j}(f), \qquad f\in \CH,\end{gathered}$$ vanishing on $\CH_0$.
The structure of semi-reproducing kernels already shows that we can expect the reproducing kernel for a semi-Hilbert space to be positive definite only on a suitable subspace.
Let $\CD \subset \RR^d$, $\CH_0$ be a finite dimensional subset of $C(\CD)$ and $K\colon \CD \times \CD \rightarrow \RR$ be a symmetric function. $K$ is called conditionally positive definite with respect to $\CH_0$ if for all $n\in\NN$ $$\begin{gathered}
\label{eq:cpd_of_P_wrt_U}
\sum_{i=1}^n\sum_{j=1}^n a_i a_j K(x_i,x_j) \geq 0,\end{gathered}$$ for all $\CH_0$-increments $(\CX,{{\boldsymbol{a}}})$. $K$ is said to be strictly conditionally positive definite with respect to $\CH_0$ if the inequality in is strict whenever in addition ${{\boldsymbol{a}}}$ is nonzero.
There is a correspondence between conditionally positive definite kernels and semi-reproducing spaces (see [@Atteia; @Meinguet; @MosamamKent2010]). The associated reproducing property can again be stated in terms of $\CH_0$-increments.
\[def:semi-rep-kernel\]Let $\CH\subset C(\CD)$ be a semi-Hilbert space, i.e., $\CH$ is a semi-inner product space with semi-inner product $(\cdot, \cdot)$ the kernel $\CH_0$ of which is finite-dimensional, and $\CH$ is complete with respect to the induced semi-norm. A symmetric kernel $K\colon \CD \times \CD \rightarrow \RR$ is called a semi-reproducing kernel for $\CH$ if $(\cdot, \cdot)$ reproduces $\CH_0$-increments, i.e., for all $\lambda_{\CX,{{\boldsymbol{a}}}}$ annihilating $\CH_0$ the following two properties hold $$\begin{gathered}
\sum_{j=1}^n a_j K(\cdot, x_j) \in \CH, \label{eq:first_prop_semi_repro_kernel}
\end{gathered}$$ and $$\begin{gathered}
\left( f, \sum_{i=1}^n a_i K(\cdot, x_i) \right) = \sum_{j=1}^n a_j f(x_j) \qquad \text{for all} \ f \in \CH. \label{eq:second_prop_semi_repro_kernel}
\end{gathered}$$
Given $m=\dim\CH_0$ linearly independent functionals $\lambda_1, \dots, \lambda_m$ on $\CH_0$ and the corresponding Lagrange basis $$\begin{gathered}
u_1,\dots, u_m \qquad\mbox{ such that }\quad \lambda_j(u_i)=\delta_{ji}, \quad 1\leq i,j\leq m,\end{gathered}$$ the kernel $$\begin{gathered}
K_0(x,y) = \sum_{j=1}^m u_j(x)u_j(y), \qquad x,y\in \CD,\end{gathered}$$ obviously provides a reproducing kernel for $\CH_0$ with respect to the inner product $$\begin{gathered}
[f,g]_0 = \sum_{j=1}^m \lambda_j(f)\lambda_j(g), \qquad f,g\in \CH_0.\end{gathered}$$ Using the orthogonal projection $$\begin{gathered}
P_0f = \sum_{j=1}^m \lambda_j(f) u_j, \qquad f\in\CH,\end{gathered}$$ we can again define $\CH_1=(I-P_0)\CH$. By construction, $$\begin{gathered}
\CH_1 = \{ f\in\CH\colon \lambda_j(f)=0,\, 1\leq j\leq m \}.\end{gathered}$$
Furthermore, if $K$ is a semi-reproducing kernel of $\CH$ the kernel $$\begin{gathered}
K_1(x,y) = K(x,y) - \sum_{j=1}^m u_j(y)\lambda_j\big(K(x,\cdot)\big) - \sum_{j=1}^m u_j(x) \lambda_j(K\big(\cdot,y)\big) \\
\hphantom{K_1(x,y) =}{} + \sum_{i=1}^m\sum_{j=1}^m u_i(x)u_j(y) \lambda_i^1\lambda_j^2\big(K(\cdot,\cdot)\big),\end{gathered}$$ is the reproducing kernel of $\CH_1$. Here, the superindex in the last term indicates the functional operating on the first and second variable, respectively, Thus, the space $\CH$ is a reproducing kernel Hilbert space itself with reproducing kernel $$\begin{gathered}
K_\CH(x,y) = K_1(x,y) + K_0(x,y), \qquad x,y\in \CD.\end{gathered}$$ See [@CheneyLight_2000; @MosamamKent2010] for details. Note that if $K$ is given such that $P_0K(\cdot,x)=0$ for all $x\in\CD$, then the projection in the above expression vanishes, i.e. $K_1=K$.
The framework of reproducing kernel spaces allows us to consider regularized interpolation problems in broad mathematical generalities (see [@BerlinetAgnan Chapter 2.1]). The rather beautiful result of Strauss [@Strauss_2002] concerning mixed interpolation and regularized least squares problems provides a special case of [@BerlinetAgnan Theorem 59].
The following notation will be used. For a set of points $X=\{x_1,\dots,x_n\}\subset\CD$ and a function $f$ on $\CD$ we write $f_X$ for the vector $(f(x_1),\dots,f(x_n))^{\rm T}$, $K_X$ for the $n\times n$ matrix with $ij$-entry $K(x_i,x_j)$, and $C_X$ for the $m\times n$ matrix $\big(u_i(x_j)\big)$, where $u_1,\dots,u_m$ is a basis of $\CH_0$. Furthermore, $W^\dagger$ denotes the pseudo-inverse of the matrix $W$ appearing in the assumptions of the following theorem. Under the assumptions of the theorem $W^\dagger = \left[\begin{smallmatrix} R^{-1} & O \\ O & O \end{smallmatrix}\right]$.
\[thm:Interpolation\_and\_smoothing\] Let $\CH$ and $(\cdot, \cdot)$ be as in Definition [\[def:semi-rep-kernel\]]{}, and let $K$ be a semi-reproducing kernel for $(\CH, (\cdot, \cdot))$ with respect to $\CH_0$. Further suppose that $K$ is strictly conditionally positive definite with respect to $\CH_0$. Let $\mu > 0$, $n \geq m=\text{dim } \CH_0$, $ 0\leq p \leq n$, and $W$ be an $n \times n$ matrix of the form $$\begin{gathered}
W=\begin{bmatrix} R & O\\
O & O \end{bmatrix},\end{gathered}$$ where $R$ is $p \times p$ and symmetric positive definite. Then given any set $X=\{x_1, \ldots, x_n\}$ of $n$ distinct points in $\CD$ which is unisolvent for $\CH_0$, and $n$ corresponding values $y_i\in\RR$, there is a unique member of the space $\CH$ minimizing the quadratic functional $$\begin{gathered}
(f_{X} - {{\boldsymbol{y}}})^{\rm T} W^\dagger (f_{X}-{{\boldsymbol{y}}}) +\mu (f,f),
\end{gathered}$$ over those functions in $\CH$ which satisfy the interpolation conditions $$\begin{gathered}
f(x_i) = y_i, \qquad p+1 \leq i \leq n.
\end{gathered}$$ This function can be written in the form $$\begin{gathered}
s = \sum_{i=1}^n a_i K(\cdot, x_i) + \sum_{i=1}^m b_i u_i,
\end{gathered}$$ where the coefficients ${{\boldsymbol{a}}}=(a_1,\dots,a_n)^{\rm T}$ and ${{\boldsymbol{b}}}=(b_1,\dots,b_m)^{\rm T}$ are the solution of the system $$\begin{gathered}
(K_{X} + \mu W ) {{\boldsymbol{a}}}+C_{X} ^{\rm T} {{\boldsymbol{b}}}= {{\boldsymbol{y}}}, \qquad C_{X} {{\boldsymbol{a}}}= {{\boldsymbol{0}}}.
\end{gathered}$$
Note that the statement reduces to the well-known result concerning the solution of the smoothest interpolation problem when $p=0$, and to a known expression for the smoothing spline when $p=n$.
Series representations for thinplate spline kernels\
on the sphere {#sec:Thinplate_splines_on_the_sphere}
====================================================
For functions on $\RR^d$ interpolating and smoothing with thinplate/polyharmonic splines associated with the energy $$\begin{gathered}
E_\kappa(f)= \int_{\RR^d} \sum_{i_1=1,i_2=1,\ldots, i_\kappa=1}^d\left( \frac{\partial}{\partial x_{i_1}} \frac{\partial}{\partial x_{i_2}} \cdots \frac{\partial}{\partial x_{i_\kappa}} f(x)\right)^2 {\rm d}x,\end{gathered}$$ are very successful approximation methods. For sufficiently smooth functions $f$, decaying sufficiently fast at infinity, integration by parts gives $$\begin{gathered}
E_\kappa(f)= (-1)^\kappa \int_{\RR^d} f(x) (\triangle^\kappa f)(x) {\rm d}x,\end{gathered}$$ where $\triangle$ is the Laplacian. Analogues for the sphere come from considering instead of the Laplacian the Laplace–Beltrami operator $ \triangle^\star$, and working on the “Fourier” side since the spherical harmonics $\{Y_{nj}\}$ are both a complete orthonormal system for $L^2\big({{{\mathbb S}^{d-1}}}\big)$ and also eigenfunctions of the Laplace–Beltrami operator. More precisely, $$\begin{gathered}
\triangle^\star Y_{j}^n = -n(n+d-2) Y_{j}^n, \qquad j=1,\ldots , N_{d,n}, \qquad n\in\NN_0.\end{gathered}$$ For the explicit value of the dimension $N_{d,n}$ see (\[eq:Ndn\]), below.
This section will consider corresponding spaces of functions on the sphere and the relevant semi-reproducing kernels $K_{d,m}$. The central role played by these semi-reproducing kernels in minimal energy interpolation and regularised least squares fitting is clear from the discussion in Section \[sec:reproducing\_kernels\], and in particular Theorem \[thm:Interpolation\_and\_smoothing\]. This topic was first considered by Wahba [@Wahba81] in the $\SS^2$ case. [See also Wahba’s monograph [@Wabha90].]{} The reader can find valuable additional relevant material in Freeden, Gervens and Schreiner [@Fr98 Chapter 5], [Levesley, Light, Ragozin and Sun [@Lev98]]{}, and Cheney and Light [@CheneyLight_2000 Chapter 32]. The material in these references differs somewhat from what appears here. Usually this is due to a treatment based on reproducing kernels rather than semi-reproducing kernels, or to a different choice of the energy. Gneiting [@Gneiting2013] gives an excellent survey of recent work concerning kernels for the sphere.
Let $\CH^d_n$ denote the space of real harmonic polynomials homogeneous of degree $n$ on ${{\mathbb R}}^d$. The spherical harmonics are the restrictions of these to the sphere ${{{\mathbb S}^{d-1}}}$. In a slight abuse of notation the space of spherical harmonics of degree $n$ on ${{{\mathbb S}^{d-1}}}$ is also written $\CH^d_n$. The dimension of the space is $$\begin{gathered}
\label{eq:Ndn}
N_{d,n} = \dim \CH^d_n = \binom{n+d-1}{n} -\binom{n+d-3}{n-2}.\end{gathered}$$ Spherical harmonics are a complete orthogonal system on $L^2\big({{{\mathbb S}^{d-1}}}\big)$ with respect to the inner product $$\begin{gathered}
\label{eq:inner_product_sdmone}
[ f, g ]_{{{{\mathbb S}^{d-1}}}} = \frac{1}{\sigma_d} \int_{{{{\mathbb S}^{d-1}}}} f(x)g(x) {\rm d}\sigma(x),\end{gathered}$$ where $$\begin{gathered}
\sigma_d = \frac{2\pi^{\frac d2}}{\Gamma\left(\frac d2\right)} =\frac{2\pi^{\lambda+1}}{\Gamma(\lambda+1)}\end{gathered}$$ is the surface area of ${{{\mathbb S}^{d-1}}}$ (for $\lambda$ see below).
For the set $\{ Y_{j}^n \colon 1 \leq j \leq N_{d,n} \}$, being an orthonormal basis for $\CH^d_n$, the addition formula $$\begin{gathered}
\label{eq:addition_formula}
\sum_{j=1}^{N_{d,n}} Y_{j}^n(x) Y_{j}^n(y) =\frac{n+\lambda}{\lambda} C^\lambda_n\big( x^{\rm T} y\big)= N_{d,n}W^\lambda_n\big( x^{\rm T} y\big),\qquad \lambda = \frac{d-2}2,\end{gathered}$$ shows that the reproducing kernels of the spaces $\CH^d_n$ are the zonal polynomials $W_n^\lambda$ which are Gegenbauer polynomials normalized so that $W^\lambda_n(1)=1$. The *Gegenbauer polynomial* of order $\lambda\geq 0$ and degree $n\in\NN_0$ is defined as the hypergeometric polynomial $$\begin{gathered}
C_n^\lambda(x) = \frac{\Gamma(n+2\lambda)}{n!\Gamma(2\lambda)}\, {{
{_2F_1}\left[ \left.
\begin{matrix}
-n , n+2\lambda\\
\lambda+\frac 12 \\
\end{matrix} \right| \frac{1-x}2 \right] }}, \qquad x\in[-1,1].\end{gathered}$$ The Gegenbauer polynomials are orthogonal with respect to the inner product $$\begin{gathered}
\label{eq:inner_product}
[f,g]_\lambda = \int_{-1}^1 f(x) g(x) \big(1-x^2\big)^{\lambda-\frac 12} {\rm d}x.\end{gathered}$$ Indeed, $$\begin{gathered}
\label{eq:orthogonality}
\int_{-1}^1 C_n^\lambda(x) C_m^\lambda(x)\big(1-x^2\big)^{\lambda-\frac 12} {\rm d}x =h_n^\lambda \delta_{nm},\end{gathered}$$ where $$\begin{gathered}
h_n^\lambda =\frac{\pi\Gamma(2\lambda+n)} {2^{2\lambda-1}n!(\lambda+n)\Gamma^2(\lambda)}=\frac{\sigma_d}{\sigma_{d-1}}\frac{\lambda}{\lambda+n} C_n^\lambda(1).\end{gathered}$$ Note that since the convolution of zonal functions on the sphere remains zonal, the inner product reduces to for zonal functions, where $\lambda=\frac{d-2}2$. The remaining weight function has the integral $$\begin{gathered}
\int_{-1}^1 \big(1-x^2\big)^{\lambda-\frac 12} {\rm d}x= h^\lambda_0 = \frac{\sigma_d}{\sigma_{d-1}} C^\lambda_0(1).\end{gathered}$$ Although, Gegenbauer polynomials provide a complete orthogonal system for all $\lambda>-\frac 12$, we will fix $\lambda=\frac{d-2}2$ throughout this paper.
The constant $C_n^\lambda(1)$ relates to the dimension $N_{d,n}$ given in ; indeed, $$\begin{gathered}
C_n^\lambda(1) =\binom{n+d-3}{d-3}=\frac{\lambda}{\lambda+n} N_{d,n}.\end{gathered}$$ Since $\{Y_{j}^n\colon 1\leq j \leq N_{d,n},\, n\in\NN_0 \}$ is a complete orthonormal system for $L^2\big({{{\mathbb S}^{d-1}}}\big)$, we can consider Fourier series $$\begin{gathered}
f ~ \sim \sum_{n=0}^\infty \sum_{j=1}^{N_{d,n}} a_{nj}Y_{j}^n ,
\end{gathered}$$ where $$\begin{gathered}
a_{nj}= \langle f, Y_{j}^n\rangle ,\end{gathered}$$ converges to $f\in L^2\big({{{\mathbb S}^{d-1}}}\big)$ in the $L^2$-sense.
Let $\CF^d_m$ be the subspace of $L^2\big({{{\mathbb S}^{d-1}}}\big)$ formed by the functions $f\in L^2\big({{{\mathbb S}^{d-1}}}\big)$ such that $$\begin{gathered}
\sum_{n=1}^\infty [n(n+d-2)]^m \sum_{j=1}^{N_{d,n}} a_{nj}^2 < \infty.$$ Further, $\CF^{d,\ell}_m$ is the space of all functions $f\in \CF^d_m$ with Fourier coefficients $a_{nj}=0$, for all $1\leq j \leq N_{d,n}$ and $0\leq n\leq \ell$. In what follows we will consider approximations $s$ to $f$ whose smoothness is measured by an inner product on $\CF^{d,\ell}_m$ with an additional spherical polynomial part of degree $\ell$ viewed as a trend. The case most frequently occurring in the literature is that of $\CF^{d,0}_m$.
For $f,g \in \CF^{d,0}_m$ with $f \sim \sum\limits_{n=1}^\infty \sum\limits_{j=1}^{N_{d,n}} a_{nj} Y^n_j$ and $g \sim \sum\limits_{n=1}^\infty \sum\limits_{j=1}^{N_{d,n}} b_{nj} Y^n_j$ and $m$ even, $$\begin{gathered}
\frac{1}{\sigma_d} \int_{{{{\mathbb S}^{d-1}}}}
\big(\triangle_{0,d}^{m/2} f\big)(x) \big(\triangle_{0,d}^{m/2} g\big)(x) {\rm d}\sigma(x) \label{eq:semi_inner_energy_form} \\
= \frac{1}{\sigma_d} \int_{{{{\mathbb S}^{d-1}}}} \! \left( \sum_{n=1}^\infty \sum_{j=1}^{N_{d,n}} [n(n+d-2)]^{m/2}
a_{nj}Y^n_j(x)\right)\!\left( \sum_{n=1}^\infty \sum_{j=1}^{N_{d,n}} [n(n+d-2)]^{m/2} b_{nj}Y^n_j(x)\right)\!
{\rm d}x \nonumber \\
= \sum_{n=1}^\infty [n(n+d-2)]^{m} \sum_{j=1}^{N_{d,n}} a_{nj} b_{nj},\label{eq:energy_semi_inner_coeff_form}\end{gathered}$$ by the extended Parseval identity in the space $L^2\big({{{\mathbb S}^{d-1}}}\big)$. In view of the equality between expressions and define an “energy” semi-inner product for $\CF^{d}_m$ by $$\begin{gathered}
(f,g)_{m,\ell}= \sum_{n=\ell+1}^\infty [n(n+d-2)]^{m} \sum_{j=1}^{N_{d,n}} a_{nj} b_{nj} . \label{eq:energy_semi_inner_prod}\end{gathered}$$ It is clear from and that this is an analogue of the usual semi-inner product associated with smoothing splines on $\RR^d$.
$(\cdot,\cdot)_{m,\ell}$ is an inner product for $\CF^{d,\ell}_m$. It is easy to show that $\CF^{d,\ell}_m$ with norm $\|f\|_{m,\ell}=\sqrt{(f,f)_{m,\ell}}$ is a Hilbert space. A proof could be based on the arguments in [@CheneyLight_2000 pp. 247–250].
Now for $x,y\in {{{\mathbb S}^{d-1}}}$ define $$\begin{gathered}
K_{d,m,\ell}(x,y) =\sum_{n=\ell+1}^\infty [n(n+d-2)]^{-m} \sum_{j=1}^{N_{d,n}} Y^n_j(x)Y^n_j(y) \nonumber \\
\hphantom{K_{d,m,\ell}(x,y}{} = \sum_{n=\ell+1}^\infty [n(n+d-2)]^{-m} N_{d,n}W^\lambda_n\big(x^{\rm T} y\big), \label{series_for_K}
\end{gathered}$$ by the addition formula . $K_{d,m,\ell}$ is clearly a zonal kernel since it depends only on the cosine of the angle between $x$ and $y$. We will use the notation $K_{d,m}$ for the kernels $K_{d,m,0}$, and call these kernels the [*thinplate spline kernels for the sphere*]{} ${{\mathbb S}}^{d-1}$. The $K_{2,m}$ and $K_{3,m}$ kernels were initially introduced by Wahba [@Wahba81]. Since $K_{d,m,\ell}(x,y)$ is zonal we can sensibly define functions $k_{d,m,\ell}$ by $$\begin{gathered}
\label{eq:kernels_and_functions}
k_{d,m,\ell}(\xi)= K_{d,m,\ell}(x,y), \qquad \text{where}\quad x,y \in {{\mathbb S}}^{d-1}\quad \text{and}\quad \xi=x^{\rm T} y.
\end{gathered}$$ Explicitly, $$\begin{gathered}
k_{d,m,\ell}(\xi) = \sum_{n=\ell+1}^\infty [n(n+d-2)]^{-m} N_{d,n}W^\lambda_n(\xi), \label{series_for_k}
\end{gathered}$$ We will refer to the functions $k_{d,m,\ell}$ as [*semi-reproducing functions*]{}, and the functions $k_{d,m}=k_{d,m,0}$ as [*thinplate spline functions*]{}.
\[Le:rk\_for\_truncFS\] Let $\ell$ be a non-negative integer and $2m \geq d \geq 2$.
- $K_{d,m,\ell}(x,y)$ is the reproducing kernel for the Hilbert space $\CF^{d,\ell}_m$.
- $K_{d,m,\ell}(x,y)$ considered as a function in $C\big({{\mathbb S}}^{d-1} \times {{\mathbb S}}^{d-1}\big)$ is strictly positive definite.
[*Proof of part $(a)$.*]{} Assume $2m\geq d \geq 2$. To show $K=K_{d,m,\ell}$ is the reproducing kernel for $\CF^{d,\ell}_m$ it suffices to show that the following two properties hold, [@Davis75 p. 317].
- For each fixed $y \in {{{\mathbb S}^{d-1}}}$, $ f(\cdot)=K(\cdot,y)$ is in $\CF^{d,\ell}_m$.
- For each function $f \in \CF^{d,\ell}_m$ the reproducing property $$\begin{gathered}
\big( f(\cdot), K(\cdot,y) \big)_{m,\ell} =f(y),
\end{gathered}$$ holds.
To show property (i) let $y$ be some fixed point in ${{{\mathbb S}^{d-1}}}$ and define $k_y(\cdot)=K(\cdot,y)$. From the definition of $K$, $k_y$ has Fourier coefficients $$\begin{gathered}
\label{eq:fc_kernel}
a_{nj}=[n(n+d-2)]^{-m} Y^n_j(y), \qquad n \geq \ell+1 \quad \text{and}\quad 1\leq j \leq N_{d,n}.
\end{gathered}$$ Hence, $$\begin{gathered}
(k_y,k_y)_{m,\ell} = \sum_{n=\ell+1}^\infty [n(n+d-2)]^m \sum_{j=1}^{N_{d,n}} (a_{nj})^2 = \sum_{n=\ell+1}^\infty [n(n+d-2)]^{-m} \sum_{j=1}^{N_{d,n}} \left( Y^n_j(y) \right)^2 \\
\hphantom{(k_y,k_y)_{m,\ell}}{} = \sum_{n=\ell+1}^\infty [n(n+d-2)]^{-m} N_{d,n} W^\lambda_n(1)=\sum_{n=\ell+1}^\infty [n(n+d-2)]^{-m} N_{d,n} ,
\end{gathered}$$ where in the second to last step the addition formula has been used. The estimate $N_{d,n}={\mathcal O}\big(n^{d-2}\big)$, holding for $d>1$, shows that the sum above is finite when $2m\geq d$, and hence $k_y \in \CF^{d,\ell}_m$ as required.
To show the reproducing property let $f$ be any function in $\CF^{d,\ell}_m$. Suppose $f$ has Fourier series $ \sum\limits_{n=\ell+1}^\infty \sum\limits_{j=1}^{N_{d,n}} a_{nj} Y^n_j$. Then, from and $$\begin{gathered}
( f, k_y )_{m,\ell} = \sum_{n=\ell+1}^\infty [n(n+d-2)]^{m} \sum_{j=1}^{N_{d,n}} a_{nj} [n(n+d-2)]^{-m} Y^n_j(y) \\
\hphantom{( f, k_y )_{m,\ell}}{} = \sum_{n=\ell+1}^\infty \sum_{j=1}^{N_{d,n}} a_{nj} Y^n_j(y) = f(y). $$ That is the reproducing property holds.
[*Proof of part $(b)$.*]{} In view of the characterisations of strict positive definiteness of zonal kernels given by Chen, Menegatto and Sun [@Ch03] for $d>2$, and by Menegatto [@Me06] for $d=2$, part (b) follows from the signs of the Gegenbauer coefficients of $K_{d,m}$ displayed in equation .
Above we have shown $K_{d,m,\ell}$ is the reproducing kernel for the space $\CF^{d,\ell}_m$ which arises from using a Fourier projection onto spherical polynomials to split the space $\CF^d_m$ into a direct sum $\CH_0 \oplus\CF^{d,\ell}_m$, where $\CH_0 = \cup_{n=0}^\ell\CH^d_n$ is the space of spherical polynomials of degree at most $\ell$.
For interpolation problems it is natural to use instead a projection onto polynomials interpolating at a certain finite set of points, and this results in a different direct sum decomposition. Fortunately, here the semi-reproducing kernel approach becomes especially convenient as $K_{d,m,\ell}$ is a semi-reproducing kernel for the space $\CF^{d,\ell}_m$ with respect to the space of polynomials $\CH_0$. This is the content of the following easily shown lemma whose proof is included for the sake of completeness.
\[Le:rk\_for\_semi-inner\_product\] Let $\ell$ be a nonnegative integer and $2m \geq d \geq 2$. The reproducing kernel $K_{d,m,\ell}$ for the space $\CF^{d,\ell}_m$ with semi-inner product $(\cdot, \cdot)_{m,\ell}$, defined above, is one choice of semi-reproducing kernel for the space $\CF^{d}_m$ with respect to the space of spherical polynomials of degree $\ell$, $\CH_0 = \cup_{n=0}^\ell \CH^d_n$.
$\CF^{d}_m = \CH $ is considered as a semi-inner product space with semi-inner product $(\cdot,\cdot)_{m,\ell}$. Choose $P_0$ as Fourier projection onto the kernel $\CH_0$ of the semi-inner product, which is the space of spherical polynomials of degree not exceeding $\ell$. Set $\CH_1=(I-P_0) \CF^{d}_m=\CF^{d,\ell}_m$. Then clearly $\CH= \CF^{d}_m = \CH_0\oplus \CF^{d,\ell}_m = \CH_0\oplus \CH_1$. Also, for each $x\in{{{\mathbb S}^{d-1}}}$, $K_{d,m,\ell}( \cdot,x) \in \CH_1=\CF^{d,\ell}_m \subset \CF^{d}_m=\CH$ by Lemma \[Le:rk\_for\_truncFS\](a). Therefore, considering an $\CH_0$-increment $(\CX, {{\boldsymbol{a}}})$ $$\begin{gathered}
\sum_{i=1}^n a_i K_{d,m,\ell}(\cdot,x_i) \in \CF^{d,\ell}_m \subset \CF^{d}_m,\end{gathered}$$ the first property, i.e., property , of a semi-reproducing kernel. Also, given any $f\in K^d_m$ the direct sum splitting allows us to write $f=f_0+f_1$ where $f_0 \in \CH_0$ and $f_1 \in \CF^{d,\ell}_m$. Therefore, considering the $\CH_0$-increment $(\CX,{{\boldsymbol{a}}})$ $$\begin{gathered}
\left( f(\cdot), \sum_{i=1}^n a_i K_{d,m,\ell}(\cdot,x_i) \right)_{m,\ell}=
\left( f_0(\cdot)+f_1(\cdot), \sum_{i=1}^n a_i K_{d,m,\ell}(\cdot,x_i) \right)_{m,\ell} \\
=\sum_{i=1}^n a_i \big( f_0(\cdot), K_{d,m,\ell}(\cdot,x_i) \big)_{m,\ell}
+\sum_{i=1}^n a_i \left( f_1(\cdot), \sum_{i=1}^n K_{d,m,\ell}(\cdot,x_i) \right)_{m,\ell} = 0 +\sum_{i=1}^n a_i f_1(x_i) ,
\end{gathered}$$ which follows from $( h_0, h_1 )_{m,\ell}=0$ for all $h_0\in \CH_0$ and $h_1 \in \CF^{d,\ell}_m$, and also from $K_{d,m,\ell}$ being the reproducing kernel for $\CF^{d,\ell}_m$. Continuing, using the vanishing property of $\CH_0$-increments, $$\begin{aligned}
\left( f(\cdot), \sum_{i=1}^n a_i K_{d,m,\ell}(\cdot,x_i) \right)_{m,\ell}
&= 0 +\sum_{i=1}^n a_i f_1(x_i)\\
&= \sum_{i=1}^n a_i f_0(x_i) + \sum_{i=1}^n a_i f_1(x_i) = \sum_{i=1}^n a_i f(x_i),\end{aligned}$$ the second property, i.e., property , of a semi-reproducing kernel. Therefore, $K_{d,m,\ell}$ is a semi-reproducing kernel for $\CF^d_m$ with semi-inner product $(\cdot,\cdot)_{m,\ell}$, as required.
In view of Lemma \[Le:rk\_for\_semi-inner\_product\], Theorem \[thm:Interpolation\_and\_smoothing\] concerning the solution of interpolation and penalized least squares fitting problems applies to interpolation and smoothing problems on the sphere. Applying the theorem the semi-reproducing kernel $K=K_{d,m,\ell}$, defined in equation , plays a central role in interpolation and penalized least squares fitting problems posed in the space $\CH=\CF^d_m$, with semi-inner product $(\cdot,\cdot)_{m,\ell}$ with respect to the finite dimensional subspace $ \CH_0 = \cup_{n=0}^\ell \CH^d_n$. In this section we have given series expansions for these kernels. In Section \[sec:recurrence\], Theorem \[thm:recurrence\_for\_TPS\], below, we will provide a recurrence relation for the particularly important thinplate spline kernels, $K_{d,m,}$, via a recurrence for the corresponding functions $k_{d,m}$. In Sections \[sec:explicit\_forms\] and \[sec:m\_equal\_1\] the recurrence relation will be used to give short explicit expressions for many of the thinplate spline kernels.
The operator $\boldsymbol{T}$ and its adjoint $\boldsymbol{T^*}$ {#sec:the_operators_T_and_T*}
================================================================
In this section we discuss an operator $T$, and its adjoint $T^*$, which will be crucial parts of the recurrence for the thinplate spline functions $k_{d,m}$. These operators were defined by Martinez-Morales in [@Ma05].
In view of the series expansions for the kernels $k_{d,m}$ given in equation a multiplier operator with Fourier multiplier of $(n(n+2\lambda))^{-1}$ would transform $k_{d,m}$ into $k_{d,m+1}$. The operators $T$ and $T^*$, are discussed below have some, but not quite all, the desired properties.
Note that the differential equation for the Gegenbauer polynomials is given by [@Abramowitz equation (22.6.5)] or [@DLMF Table 18.8.1] $$\begin{gathered}
\big(1-x^2\big)y'' -(2\lambda+1)x y' + n(n+2\lambda)y = 0.\end{gathered}$$ Rewriting this we obtain $$\begin{gathered}
\big(1-x^2\big)^{-\lambda+\frac 12} \frac{{\rm d}}{{\rm d}x} \left( \big(1-x^2\big)^{\lambda+\frac 12} \frac{{\rm d}}{{\rm d}x} \right) y = -n(n+2\lambda) y.\end{gathered}$$
Call the operator on the left hand side of the equation $D_\lambda$. Then, for $\lambda>0$ and $n\in{\mathbb N}$, $$\begin{gathered}
D_\lambda C_n^\lambda(x) = -n(n+2\lambda) C_n^\lambda(x).\end{gathered}$$ Let us formally invert the equation $D_\lambda f = g$ using averages centered at 1. Dealing with the outer derivative in $D_\lambda$ we obtain that $$\begin{gathered}
\int_x^1 \big(1-y^2\big)^{\lambda-\frac 12} g(y) {\rm d}y = \left. \big(1-y^2\big)^{\lambda+\frac 12}\frac{{\rm d}}{{\rm d}y}f(y) \right|_x^1.\end{gathered}$$ If $$\begin{gathered}
\label{eq:condition1}
f'(y) \ \text{is continuous at}\ y=1,\end{gathered}$$ the term on the right hand side for $y=1$ vanishes. We can then proceed obtaining $$\begin{gathered}
\int_x^1 \big(1-y^2\big)^{-\lambda-\frac 12} \int_y^1 \big(1-z^2\big)^{\lambda-\frac 12} g(z) {\rm d}z {\rm d}y =f(x) - f(1).\end{gathered}$$ Setting $f(x)=C_n^\lambda(x)$ and $g(x)=-n(n+2\lambda)C_n^\lambda(x)$, condition (\[eq:condition1\]) is clearly satisfied. We therefore obtain the following statement.
\[prop:eigenfunction\_Tadjoint\]For $\lambda>0$ and $n\in{\mathbb N}$ $$\begin{gathered}
\int_x^1 \big(1-y^2\big)^{-\lambda-\frac 12} \int_y^1 \big(1-z^2\big)^{\lambda-\frac 12} C_n^\lambda(z) {\rm d}z {\rm d}y = \frac{C_n^\lambda(1)-C_n^{\lambda}(x)}{n(n+2\lambda)}.\end{gathered}$$
We would like to give a direct proof of the statement. Towards this goal, we use an integral given in [@Abramowitz equation (22.13.2)] (or [@DLMF equation (18.17.1)] for the general Jacobi case) $$\begin{gathered}
\int_0^x \big(1-y^2\big)^{\lambda-\frac 12} C_n^\lambda(y) {\rm d}y =\frac{2\lambda}{n(2\lambda+n)}\big[ C_{n-1}^{\lambda+1}(0) - \big(1-x^2\big)^{\lambda+\frac 12}C_{n-1}^{\lambda+1}(x) \big].\end{gathered}$$ Decomposing the integral over $[0,1]$ into two integrals over $[0,x]$ and $[x,1]$, respectively, we can use the formula to obtain that $$\begin{gathered}
\int_y^1 \big(1-z^2\big)^{\lambda-\frac 12} C_n^\lambda(z) {\rm d}z =
\frac{2\lambda}{n(2\lambda+n)} \big(1-y^2\big)^{\lambda+\frac 12} C_{n-1}^{\lambda+1}(y).\end{gathered}$$ Note that the function on the right hand side vanishes at $y=1$. Towards the claim, it remains to integrate the polynomial $C_{n-1}^{\lambda+1}$ which readily follows from [@DLMF equation (18.9.19)] $$\begin{gathered}
\frac{{\rm d}}{{\rm d}x} C_n^\lambda(x) = 2\lambda C_{n-1}^{\lambda+1}(x)\quad \Leftrightarrow \quad C_n^{\lambda}(x) = 2\lambda \int C_{n-1}^{\lambda+1}(y) {\rm d}y,\end{gathered}$$ completing the proof.
Similarly, we could have treated the average centered at $-1$ giving the following result.
\[prop:eigenfunction\_T\] For $\lambda>0$ and $n\in{\mathbb N}$ we have that $$\begin{gathered}
\int_{-1}^x \big(1-y^2\big)^{-\lambda-\frac 12} \int_{-1}^y \big(1-z^2\big)^{\lambda-\frac 12} C_n^\lambda(z) {\rm d}z {\rm d}y =\frac{C_n^\lambda(-1)-C_n^{\lambda}(x)}{n(n+2\lambda)}.\end{gathered}$$
The operators $$\begin{gathered}
T_\lambda f(x) = - \int_{-1}^x \big(1-y^2\big)^{-\lambda-\frac 12} \int_{-1}^y \big(1-z^2\big)^{\lambda-\frac 12} f(z) {\rm d}z {\rm d}y\end{gathered}$$ and $$\begin{gathered}
T_\lambda^*f(x) = - \int_{x}^1 \big(1-y^2\big)^{-\lambda-\frac 12} \int_{y}^1 \big(1-z^2\big)^{\lambda-\frac 12} f(z) {\rm d}z {\rm d}y\end{gathered}$$ have been defined in [@Ma05], showing that $T_\lambda^*$ is the adjoint of $T_\lambda$ with respect to the inner product (\[eq:inner\_product\]). To be precise, the following statement holds true (cf. [@Ma05 Theorem 3]).
\[thm:Morales\_theorem3\] Let $f\in C[-1,1)\cap L^1_\lambda[-1,1]$ and $g\in C(-1,1]\cap L^1_\lambda[-1,1]$. Then $gT_\lambda f, fT_\lambda^*g\in L^1_\lambda[-1,1]$ and $$\begin{gathered}
[T_\lambda f,g]_\lambda = [f,T_\lambda^*g]_\lambda.\end{gathered}$$
The proof exploits the fact that with the weight $\big(1-z^2\big)^{\lambda-\frac 12}$ in the inner integral, continuity of $f$ suffices to cope with the singularity introduced by the weight $\big(1-y^2\big)^{-\lambda-\frac 12}$ in the outer integral of $T_\lambda$. Based on this observation, the theorem follows from Fubini’s theorem.
Propositions \[prop:eigenfunction\_Tadjoint\] and \[prop:eigenfunction\_T\] above thus show that the polynomials $C_n^\lambda$ are basically – up to a constant – eigenfunctions of $T^*$ and $T$, respectively. This is somewhat obvious from the fact that both $T$ and $T^*$ invert $D_\lambda$. Both propositions have been derived in [@Ma05 Lemma 1] via a different proof employing the Rodriguez formula of the Gegenbauer polynomials.
A recurrence for the thinplate spline functions for the sphere {#sec:recurrence}
==============================================================
This section concerns a recurrence for the thinplate spline functions $k_{d,m}$ for ${{\mathbb S}}^{d-1}$. The recurrence will be used in Sections \[sec:explicit\_forms\] and \[sec:m\_equal\_1\] to give short explicit forms for many of the functions $k_{d,m}$. It is important to note that the corresponding recurrence given in [@Ma05 Theorem 4] is incorrect and does not yield the thinplate spline functions $k_{d,m}$. Consequently, many of the explicit formulas claimed for the functions $k_{d,m}$ in the paper [@Ma05] are also incorrect.
\[thm:recurrence\_for\_TPS\] Let $d\geq 2$, $\lambda=\frac{d-2}2$, $x\in[-1,1]$, and $e_0(x)=1$ for all $x\in[-1,1]$. The thinplate spline functions, $k_{d,m}$, $m\in{{\mathbb N}}$, for the sphere ${{\mathcal S}}^{d-1}$, defined via the series , are alternatively generated by the recurrence $$\begin{gathered}
k_{d,m}(x) = \begin{cases} \dfrac{[ e_0, T_\lambda e_0]_\lambda}{[ e_0,e_0]_\lambda} -(T_\lambda e_0)(x) ,&\text{when}\ m=1, \\
(T_\lambda k_{d,m-1} )(x) - \dfrac{[ e_0, T_\lambda k_{d,m-1}]_\lambda}{[ e_0,e_0]_\lambda}, & \text{when}\ m>1.
\end{cases}\end{gathered}$$
Fix $d\geq 2$ and define a sequence of functions $(f_1, f_2, \ldots)$ by the recurrence $$\begin{gathered}
\label{eq:recursion_stated_for_f}
f_{m}(x) = \begin{cases} \dfrac{[ e_0, T_\lambda e_0]_\lambda}{[ e_0,e_0]_\lambda} -(T_\lambda e_0)(x) ,&\text{when}\ m=1,\\
( T_\lambda f_{m-1})(x) - \dfrac{[ e_0, T_\lambda f_{m-1}]_\lambda}{[ e_0,e_0]_\lambda}, & \text{when}\ m>1,
\end{cases}\end{gathered}$$ this recursion mirroring the one in the statement of the theorem.
Throughout this proof we view the series definition as an orthogonal expansion $$\begin{gathered}
g \sim \sum_{n=0}^\infty \widehat{g}_n C^\lambda_n\end{gathered}$$ in terms of the Gegenbauer polynomials $C^\lambda_n$. The uniqueness theorem tells us that functions with the same coefficients are identical.
First, observe that the constant term in the definition of $f_m$ ensures that the zeroth Fourier coefficient $\widehat{(f_m)}_0$ is zero. Therefore, consider in what follows Fourier coefficients of index $n\geq 1$.
Clearly, $e_0\in C[-1,1]\cap L^1_\lambda[-1,1]$. It therefore follows from Theorem \[thm:Morales\_theorem3\] that $f_1$ is continuous on $[-1,1]$ and $f_1\in L^1_\lambda[-1,1]$. Using induction on $m$ we can then conclude, again using Theorem \[thm:Morales\_theorem3\], that $f_m$ is continuous on $[-1,1]$ and $f_m\in L^1_\lambda[-1,1]$ for $m>1$.
For a function $f\in L^1_\lambda[-1,1]$, and $n\geq 1$, Theorem \[thm:Morales\_theorem3\] gives that $$\begin{gathered}
\widehat{T_\lambda f}_n = \frac 1{h_n^\lambda}\int_{-1}^1 T_\lambda f(x) C_n^\lambda(x) \big(1-x^2\big)^{\lambda-\frac 12} {\rm d}x
= \frac 1{h_n^\lambda} \int_{-1}^1 f(x) T_\lambda^* C_n^\lambda(x) \big(1-x^2\big)^{\lambda-\frac 12} {\rm d}x,\end{gathered}$$ which by Proposition \[prop:eigenfunction\_Tadjoint\] yields $$\begin{gathered}
\widehat{T_\lambda f}_n = \frac 1{n(n+2\lambda)} \frac{1}{h_n^\lambda} \int_{-1}^1 f(x) C_n^{\lambda}(x)\big(1-x^2\big)^{\lambda-\frac 12} {\rm d}x\nonumber\\
\hphantom{\widehat{T_\lambda f}_n=}{} - \frac 1{n(n+2\lambda)} \frac{C_n^\lambda(1)}{h_n^\lambda}
\int_{-1}^1 f(x) \big(1-x^2\big)^{\lambda-\frac 12} {\rm d}x. \label{eq:critical_Fourier_coeff}\end{gathered}$$
In the special case of $f=e_0$, the first integral vanishes due to orthogonality (\[eq:orthogonality\]). Furthermore, $$\begin{gathered}
\int_{-1}^1 e_0(x)\big(1-x^2\big)^{\lambda-\frac 12} {\rm d}x= \int_{-1}^1 \big(1-x^2\big)^{\lambda-\frac 12} {\rm d}x = [ e_0, e_0 ]_\lambda = h_0^\lambda.\end{gathered}$$ We therefore obtain that for $n\geq 1$, $$\begin{gathered}
\widehat{(T_\lambda e_0)}_n = - \frac{1}{n(n+2\lambda)} \frac{C_n^\lambda(1)}{h_n^\lambda} h^\lambda_0 = -\frac{1}{n(n+2\lambda)} \frac{n+\lambda}\lambda = -\frac{1}{n(n+2\lambda)} \frac{N_{d,n}}{C_n^\lambda(1)}.\end{gathered}$$ Thus, the function $f_1$ generated as specified in equation has the same Fourier coefficients as the thinplate spline function $k_{d,1}$ of equation . Thus, by uniqueness, it is $k_{d,1}$.
Now returning to general functions $f$ we can rewrite as $$\begin{gathered}
\label{eq:fourier_coefficient_of_Tf}
\widehat{T_\lambda f}_n = \frac 1{n(n+2\lambda)} {\widehat f}_n- \frac 1{n(n+2\lambda)} C_n^\lambda(1)\frac{h^\lambda_0}{h_n^\lambda} {\widehat f}_0.\end{gathered}$$
From (\[eq:fourier\_coefficient\_of\_Tf\]) and the definition of $f_m$ we have that for $m\geq 2$ and $n \geq 1$ $$\begin{gathered}
\big(\widehat{f_m}\big)_n = \frac{1}{n(n+2\lambda)} \big(\widehat{f_{m-1}}\big)_n -\frac 1{n(n+2\lambda)} C_n^\lambda(1)\frac{h_0^\lambda}{h_n^\lambda} \big(\widehat{f_{m-1}}\big)_0= \frac{1}{n(n+2\lambda)} \big(\widehat{f_{m-1}}\big)_n,\end{gathered}$$ since $\big(\widehat{f_{m-1}}\big)_0=0$. Thus, by induction $$\begin{gathered}
\big(\widehat{f_m}\big)_n =
\begin{cases}
0, & \text{when}\ n=0,\\
(n(n+2\lambda) )^{-m} \dfrac{N_{d,n}}{C^\lambda_n(1)}, & \text{when} \ n \geq 1.
\end{cases}\end{gathered}$$ Hence $f_m$ has the same Fourier coefficients as $k_{d,m}$. Therefore, by uniqueness, it is $k_{d,m}$. That is the thinplate spline functions, $k_{d,m}$, are generated by the recursion of the theorem.
Explicit forms for some of the thinplate spline functions $\boldsymbol{k_{d,m}}$ {#sec:explicit_forms}
================================================================================
The reader will recall the correspondence between the zonal thinplate spline kernels $K_{d,m}$ and the associated functions $k_{d,m}$, see . The recurrences of the previous section yield explicit formulas for many of the thinplate spline functions $k_{d,m}$. A sample of these explicit expressions is presented below, thereby correcting the expressions given in [@Ma05].
In the formulas that follow $u= \frac{1-x}{2}$ and $v = \frac{\pi}{2}+\arcsin(x) $. Note that in angular coordinates $x=\cos\theta$ and $v=\pi-\theta$.
Functions for $\boldsymbol{{{\mathbb S}}^1}$
--------------------------------------------
$$\begin{gathered}
k_{2,1} = \frac{1}{2}v^2 -\frac{1}{6}\pi^2, \qquad
k_{2,2} = -\frac{1}{24} v^4 + \frac{1}{12}\pi^2 v^2 -\frac{7}{360}\pi^4, \\
k_{2,3} =\frac{1}{720}v^6 -\frac{1}{144} \pi^2 v^4 + \frac{7}{720}\pi^4 v^2 -\frac{31}{15120}\pi^6,\\
k_{2,4}= -\frac{1}{40320}v^8 +\frac{1}{4320}\pi^2 v^6 -\frac{7}{8640}\pi^4 v^4 +\frac{31}{30240}\pi^6v^2 -\frac{127}{60480}\pi^8.\end{gathered}$$
See Wahba [@Wabha90 p. 22] for explicit forms of these functions in terms of Bernoulli polynomials.
Functions for $\boldsymbol{{{\mathbb S}}^2}$
--------------------------------------------
$$\begin{gathered}
k_{3,1} =-\ln (u) -1, \qquad k_{3,2} = \Li_2(1-u) +1 -\frac{\pi^2}{6}, \\
k_{3,3} = -2\Li_3(u) -\Li_2(1-u)+\ln (u)\Li_2(u) + 2\zeta(3)+\frac{\pi^2}{6}-2.\end{gathered}$$
Functions for $\boldsymbol{{{\mathbb S}}^3}$
--------------------------------------------
$$\begin{gathered}
k_{4,1} =\frac{1}{2}\frac{x v}{\sqrt{1-x^2}} -\frac{1}{4}, \qquad k_{4,2} = \frac{1}{8}v^2 +\frac{1}{16}-\frac{\pi^2}{24}.\end{gathered}$$
Functions for $\boldsymbol{{{\mathbb S}}^4}$
--------------------------------------------
$$\begin{gathered}
k_{5,1} = -\frac{1}{3}\ln(u) +\frac{1}{6u} -\frac{7}{9}, \qquad
k_{5,2} = \frac{1}{9}\Li_2(1-u) -\frac{2}{9}\ln(u) +\frac{\ln(u)}{9(x+1)} +\frac{1}{81}-\frac{\pi^2}{54}.\end{gathered}$$
Some functions for higher dimensional spheres
---------------------------------------------
$$\begin{gathered}
k_{6,1} = xv\left( \frac{1}{4\sqrt{1-x^2} }+\frac{1}{8\big(1-x^2\big)^{3/2}} \right) +\frac{1}{8\big(1-x^2\big)} - \frac{5}{16},\\
k_{8,1} = xv \left( \frac{1}{6\sqrt{1-x^2}} +\frac{1}{12\big(1-x^2\big)^{3/2}}+\frac{1}{16\big(1-x^2\big)^{5/2}} \right)\\
\hphantom{k_{8,1} =}{} +\frac{1}{16\big(1-x^2\big)} +\frac{1}{16\big(1-x^2\big)^2} -\frac{5}{18},\end{gathered}$$
and in general, as is shown in Section \[subsection8\_1\] below, for $d=2\lambda+2$ even, i.e., when $\lambda$ is an integer, $$\begin{gathered}
k_{2\lambda+2,1} (x) = x v \sum_{j=1}^\lambda c_j^\lambda \big(1 -x^2\big)^{-j+\frac{1}{2}}+ \sum_{j=1}^{\lambda-1} d_j^\lambda \big(1 -x^2\big)^{-j} -C_\lambda .\end{gathered}$$
Also, $$\begin{gathered}
k_{7,1} = -\frac{1}{5} \ln(u) +\frac{1}{10 u}+\frac{1}{60 u^2} -\frac{43}{75},\\
k_{9,1} = -\frac{1}{7} \ln(u) +\frac{1}{14 u} +\frac{1}{70u^2}+\frac{1}{420u^3} -\frac{337}{735},\\
k_{11,1} = -\frac{1}{9}\ln(u) +\frac{1}{18u} +\frac{1}{84u^2}+\frac{1}{378u^3}+\frac{1}{2520u^4}-\frac{1091}{2835},\end{gathered}$$ and in general, as is shown in Section \[subsection8\_2\] below, for $d=2\kappa+3$ odd, $$\begin{gathered}
k_{2\kappa+3,1}(x) =\frac{-1}{2\kappa+1} \ln(u) + \sum_{\nu=1}^\kappa g^\lambda_\nu(1-x)^{-\nu} -D_\lambda,\end{gathered}$$ with $g^\lambda_\nu$ as given in .
Explicit formulas for the thinplate spline functions $\boldsymbol{k_{d,1}}$, $\boldsymbol{d>2}$ {#sec:m_equal_1}
===============================================================================================
In the section simple explicit formulas will be obtained for the thinplate spline functions $k_{d,1}$. As explained at the end of Section \[sec:Thinplate\_splines\_on\_the\_sphere\] the function $k_{d,1}$ is associated with approximation problems on ${{{\mathbb S}^{d-1}}}$.
Theorem \[thm:recurrence\_for\_TPS\] and the formula for the operator $T_\lambda$ given in Proposition \[prop:eigenfunction\_T\] lead to a method of calculating $k_{d,1}$. Define $$\begin{gathered}
G_\beta^\alpha (y)= \int_{-1}^y \left(1-z^2\right)^{\beta -{\ensuremath{\frac{1}{2}}}} (1-z)^{-\alpha} {\rm d}z, \qquad \text{where}\quad \alpha \geq 0\quad \text{and} \quad \beta >\alpha -{\ensuremath{\frac{1}{2}}}, \label{eq:Glambda_def}\end{gathered}$$ $G_\beta=G_\beta^0$, and $$\begin{gathered}
F_\beta(x) = \int_{-1}^1 \big(1-y^2\big)^{-\beta -{\ensuremath{\frac{1}{2}}}} G_\beta(y) {\rm d}y.
\end{gathered}$$ Then, with $\lambda = (d-2)/2$, $$\begin{gathered}
\label{eq:express1_for_kd1}
k_{d,1}(x) = F_\lambda (x) - \frac{[ F_\lambda, e_0]_\lambda}{[e_0,e_0]_\lambda} .
\end{gathered}$$
Actually it is somewhat easier to deal with the indefinite integral $$\begin{gathered}
\label{eq:Slambda}
S_\lambda (y) = \int \big(1-y^2\big)^{-(2\lambda +1)/2} G_\lambda (y) {\rm d}y,\end{gathered}$$ where here we mean any fixed representative value of the indefinite integral, since the second term in deletes the constant term. Then, from equation , $$\begin{gathered}
\label{eq:alternative_formula_for_k_d1}
k_{d,1}(x) = S_\lambda (x) - \frac{[S_\lambda, e_0]_\lambda}{[e_0,e_0]_\lambda}.
\end{gathered}$$
The functions $\boldsymbol{k_{d,1}}$ when $\boldsymbol{d}$ is even {#subsection8_1}
------------------------------------------------------------------
This subsection considers the functions $k_{d,1}$ when $d$ is even. Thus in this subsection $\lambda$ is a positive integer.
$$\begin{gathered}
G_0 (y) =\frac{\pi}{2} + \arcsin(y), \label{eq:G0}\end{gathered}$$
and $$\begin{gathered}
G_\beta(y) = \frac{1}{2\beta} y \big(1-y^2\big)^{(2\beta-1)/2} + \frac{2\beta-1}{2\beta} G_{\beta-1}(y), \qquad \beta > 1/2.\label{eq:Glambda_recur}\end{gathered}$$ Explicitly, for $\lambda \in {{\mathbb N}}$, $$\begin{gathered}
G_\lambda (y) =a_0^\lambda \left(\frac{\pi}{2} + \arcsin (y) \right) +a_1^\lambda y \big(1-y^2\big)^{1/2}+ a_2^\lambda y\big(1-y^2\big)^{3/2} + \cdots\nonumber\\
\hphantom{G_\lambda (y) =}{} +a_\lambda^\lambda y \big(1-y^2\big)^{(2\lambda-1)/2},\label{eq:Glambda_form}\end{gathered}$$ where $$\begin{gathered}
a_j^\lambda = \begin{cases} \dfrac{(2j-2)!!}{(2j-1)!!} \dfrac{(2\lambda-1)!!}{(2\lambda)!!}, &1 \leq j \leq \lambda,\\
a_1^\lambda,& j=0,\\
0, & \text{otherwise}.
\end{cases}\end{gathered}$$
Equation follows immediately from the definition . The recurrence follows from the definition via an easy integration by parts.
The general form of the explicit expression for $G_\lambda(y)$, given in equation , is clear from the expression for $G_0(y)$ and the recurrence . Consider now the expression for the coefficients $a_j^\lambda$ in equation . Considering $G_j(y)$ it is clear from the form of $G_{j-1}(y)$, and the recurrence, that the coefficient $a_j^j$ of $y\big(1-y^2\big)^{(2j-1)/2}$ is $1/(2j)$. Now applying the recurrence $\lambda-j$ times to yield $G_{j+1}(y)$, $G_{j+2}(y)$, up to $G_\lambda(j)$, in turn, it follows that $$\begin{gathered}
a^\lambda_j = \frac{1}{2j} \frac{2j+1}{2j+2} \frac{2j+3}{2j+4} \cdots \frac{2\lambda-1}{2\lambda}
= \frac{1}{2j} \frac{(2\lambda-1)!!}{(2j-1)!!} \frac{(2j)!!}{(2\lambda)!!}\\
\hphantom{a^\lambda_j}{} = \frac{(2\lambda-1)!!}{(2j-1)!!} \frac{(2j-2)!!}{(2\lambda)!!} , \qquad 1 \leq j \leq \lambda.\end{gathered}$$ Further, the expression for $G_0(y)$ and the recurrence, yield the explicit expression $$\begin{gathered}
G_1(y) = \frac{1}{2} \left(\frac{\pi}{2}+\arcsin(y)\right) + \frac{1}{2} y \big(1-y^2\big)^{1/2}.\end{gathered}$$ Therefore, $a_0^1= a_1^1 = 1/2$. Consequently, the recurrence implies that $a_0^\lambda = a_1^\lambda$ for all $\lambda \in {{\mathbb N}}$.
Analogously, define the indefinite integral $$\begin{gathered}
\label{eq:defn_of_Ht}
H_\beta(y)= \int \big(1 -y^2\big)^{-(2\beta+1)/2} {\rm d}y, \qquad \beta > -1/2.\end{gathered}$$ This indefinite integral is often well defined when the corresponding definite integral $G_{-\beta} $ of equation is not.
Some representatives of the indefinite integral $H_\beta$ are $$\begin{gathered}
H_0 (y) = \int \frac{1}{\sqrt{1-y^2}} {\rm d}y = \frac{\pi}{2} + \arcsin(y), \label{eq:H0}\end{gathered}$$ and $$\begin{gathered}
H_1(y) = \int \big(1 -y^2\big)^{-3/2} {\rm d}y = \frac{y}{\sqrt{1-y^2}}. \label{eq:H1}\end{gathered}$$ Families of representatives may be generated by the recurrence $$\begin{gathered}
H_\beta(y) = \frac{1}{2\beta-1}y \big(1-y^2\big)^{-(2\beta-1)/2} + \frac{2\beta-2}{2\beta-1} H_{\beta-1}(y), \qquad t \geq 1.
\label{eq:Ht_recur}\end{gathered}$$ Explicitly, for $\lambda \in {{\mathbb N}}$, starting from $H_1$, as given by equation , the recurrence generates representatives of the form $$\begin{gathered}
H_\lambda(y) = b^\lambda_1 y \big(1-y^2\big)^{-1/2}\! +b^\lambda_2 y \big(1-y^2\big)^{-3/2}\! + \cdots +
b^\lambda_\lambda y \big(1-y^2\big)^{-(2\lambda-1)/2}, \qquad\! \lambda \in {{\mathbb N}}.\!\! \label{eq:Hk_form}\end{gathered}$$ where $$\begin{gathered}
b_j^\lambda = \begin{cases} \dfrac{1}{2j-1} \dfrac{(2j-1)!!}{(2j-2)!!} d\dfrac{(2\lambda-2)!!}{(2\lambda-1)!!},
& 1 \leq j \leq \lambda,\\
0, & \text{otherwise}.
\end{cases}\end{gathered}$$
Equations and follow immediately from the definition . The recurrence follows from the definition via an easy integration by parts. The general form of the explicit expression for $H_\lambda(y)$, $\lambda\in {{\mathbb N}}$, given in equation , is clear from the expression for $H_1(y)$ and the recurrence . Consider now the expression for the coefficients $b_j^\lambda$ ocurring in equation . From the recurrence and the formula for $H_1 $ the term involving $ y \big(1-y^2\big)^{-(2j-1)/2}$ first appears for $\lambda=j$, where it has the value $b_j^j =1/(2j-1)$. This term is then propagated to the functions $H_\lambda$, with $\lambda>j$, via the recurrence. Hence, $$\begin{gathered}
b^\lambda_j = \frac{1}{2j-1} \frac{2j}{2j+1} \frac {2j+2}{2j+3} \cdots \frac{2\lambda-2}{2\lambda-1}
= \frac{1}{2j-1} \frac{(2\lambda-2)!!}{(2j-2)!!} \frac {(2j-1)!!}{(2\lambda-1)!!}, \qquad 1 \leq j \leq \lambda.\tag*{\qed}\end{gathered}$$
Given the formula for $G_\lambda(y)$ when $\lambda$ is a positive integer, the definition of $H_\lambda$ and the definition of $S_\lambda$, $$\begin{gathered}
S_\lambda(y) = \int \big(1 -y^2\big)^{-(2\lambda +1)/2} \bigg[ a_0^\lambda \left( \frac{\pi}{2} + \arcsin(y) \right) \\
\hphantom{S_\lambda(y) =}{} +a_1^\lambda y \big(1 -y^2\big)^{1/2} + \cdots +
a_\lambda^\lambda y \big(1-y^2\big)^{(2\lambda -1)/2} \bigg] {\rm d}y\\
\hphantom{S_\lambda(y)}{} = \int a_0^\lambda \left( \frac{\pi}{2} +\arcsin(y) \right) {\rm d} H_\lambda(y) \\
\hphantom{S_\lambda(y) =}{}
+\int a_1^\lambda y \big(1 -y^2\big)^{-\lambda} + a_2^\lambda y \big(1 -y^2\big)^{-\lambda +1} + \cdots +
a_\lambda^\lambda y \big(1 -y^2\big)^{-1} {\rm d}y= I_1 + I_2,\end{gathered}$$ where $I_1$ and $I_2$ are the first and second indefinite integrals, respectively. Ignoring the constant parts in the indefinite integrals a representative value of $I_2$ is $$\begin{gathered}
I_2 = a_1^\lambda \frac{\big(1 -y^2\big)^{-(\lambda-1)}}{2(\lambda-1)} + a_2^\lambda \frac{\big(1-y^2\big)^{-(\lambda-2)} }{2(\lambda-2)}
+ \cdots + a_{\lambda-1}^\lambda\frac{\big(1-y^2\big)^{-1}}{2} - a_\lambda^\lambda \frac{\ln \big(1 -y^2\big)}{2} .\end{gathered}$$ A representative value of $I_1$ is $$\begin{gathered}
I_1 = a_0^\lambda \left[ \left( \frac{\pi}{2} + \arcsin(y) \right) H_\lambda(y) - \int H_\lambda (y) \frac{1}{\sqrt{1 -y^2}} {\rm d}y \right] \\
\hphantom{I_1}{} = a_0^\lambda \left( \frac{\pi}{2} + \arcsin(y)\right) H_\lambda (y)\\
\hphantom{I_1=}{}
- a_0^\lambda \int b_1^\lambda y \big(1 -y^2\big)^{-1} +b_2^\lambda y \big(1 -y^2\big)^{-2} + \cdots +b_\lambda^\lambda y \big(1 -y^2\big)^{-\lambda} {\rm d}y \\
\hphantom{I_1}{} = a_0^\lambda \left( \frac{\pi}{2} + \arcsin(y)\right) H_\lambda (y) \\
\hphantom{I_1=}{} +a_0^\lambda \left[ \frac{b_1^\lambda \ln \big(1 -y^2\big)}{2} + \frac{b_2^\lambda \big(1 -y^2\big)^{-1}}{2}
+ \frac{b_3^\lambda \big(1-y^2\big)^{-2} }{4} + \cdots + b_\lambda^\lambda \frac{\big(1-y^2\big)^{-(\lambda-1)}}{2\lambda -2} \right].\end{gathered}$$
Now note that $$\begin{gathered}
a_0^\lambda b_1^\lambda = \frac{(2\lambda-1)!!}{(2\lambda)!!} \frac{ (2\lambda -2)!!}{(2\lambda-1)!!} = \frac{1}{2\lambda} = a_\lambda^\lambda .\end{gathered}$$ Hence, the terms in $I_1$ and $I_2$ involving $\ln \big(1 -y^2\big)$ have coefficients of equal magnitude and opposite sign. Thus, we conclude that a representative value of $S_\lambda (y)$ is $$\begin{gathered}
S_\lambda(y) = \left(\frac{\pi}{2} + \arcsin(y) \right)\sum_{j=1}^\lambda c_j^\lambda y \big(1 -y^2\big)^{-j+\frac{1}{2}} + \sum_{j=1}^{\lambda-1} d_j^\lambda \big(1 -y^2\big)^{-j} ,\end{gathered}$$ where $$\begin{gathered}
c_j^\lambda = a_0^\lambda b_j^\lambda = \frac{1}{2 \lambda } \frac{1}{2j-1} \frac{(2j-1)!!}{(2j-2)!!} ,\end{gathered}$$ and $$\begin{gathered}
d_j^\lambda =\frac{1}{2j} \big( a_{\lambda-j}^\lambda - c_{j+1}^\lambda \big)\\
\hphantom{d_j^\lambda}{} =\frac{1}{2j} \left[ \frac{(2\lambda-1)!!}{ (2\lambda-2j-1)!!} \frac{ (2\lambda-2j-2)!!}{ ( 2\lambda )!!} - \frac{1}{2\lambda} \frac{(2j-1)!!}{(2j)!!} \right], \qquad 1 \leq j \leq \lambda-1.\end{gathered}$$
Now recall from equation that $$\begin{gathered}
k_{d,1} = S_\lambda(y) -\frac{[S_\lambda, e_0]_\lambda}{[e_0,e_0]_\lambda},\end{gathered}$$ where $\lambda=(d-2)/2$. To calculate this quantity first define $f_\mu$ as the Beta integral $$\begin{gathered}
f_\mu = \int_{-1}^1 \big(1 -y^2\big)^\mu {\rm d}y = \frac{\sqrt{\pi} \Gamma(\mu+1)}{\Gamma(\mu+3/2)},\qquad \mu > -1.\end{gathered}$$ Then $$\begin{gathered}
[S_\lambda,e_0]_\lambda = \int_{-1}^1 \sum_{j=1}^{\lambda-1} d_j^\lambda \big(1-y^2\big)^{-j} \big(1-y^2\big)^{\lambda -\frac{1}{2}} {\rm d}y \\
\hphantom{[S_\lambda,e_0]_\lambda =}{} + \int_{-1}^1 \sum_{j=1}^\lambda c_j^\lambda \left( \frac{\pi}{2} + \arcsin(y) \right) y
\big(1 -y^2\big)^{-j+\frac{1}{2}} \left(1 -y^2 \right)^{\lambda -\frac{1}{2}} {\rm d}y \\
\hphantom{[S_\lambda,e_0]_\lambda}{}= \sum_{j=1}^{\lambda-1} d_j^\lambda f_{\lambda -j -\frac{1}{2}}
+ \sum_{j=1}^\lambda c_j^\lambda
\int_{-1}^1 \left( \frac{\pi}{2} + \arcsin(y) \right) {\rm d} \left( \frac{ - \big(1 -y^2\big) ^{\lambda -j +1}}{2 (\lambda -j +1)} \right) \\
\hphantom{[S_\lambda,e_0]_\lambda}{} = \sum_{j=1}^{\lambda-1} d_j^\lambda f_{\lambda -j -\frac{1}{2}} + \sum_{j=1}^\lambda
c_j^\lambda \int_{-1}^1 \frac{ \big(1-y^2\big)^{\lambda-j+1}}{2(\lambda-j+1)} \frac{1}{\sqrt{1-y^2}} {\rm d}y \\
\hphantom{[S_\lambda,e_0]_\lambda}{}= \sum_{j=1}^\lambda
\frac{1}{2(\lambda-j+1)} c_j^\lambda f_{\lambda-j +\frac{1}{2}} + \sum_{j=1}^{\lambda-1} d_j^\lambda f_{\lambda-j -\frac{1}{2}}.
\end{gathered}$$ Therefore, substituting the various quantities into equation $$\begin{gathered}
k_{d,1} (y) = \left(\frac{\pi}{2} + \arcsin(x) \right)\sum_{j=1}^\lambda c_j^\lambda y \big(1 -y^2\big)^{-j+\frac{1}{2}}+ \sum_{j=1}^{\lambda-1} d_j^\lambda \big(1 -y^2\big)^{-j} -C_\lambda ,
\end{gathered}$$ where $$\begin{gathered}
C_\lambda = \frac{1}{f_{\lambda-\frac{1}{2}}}\left(\sum_{j=1}^\lambda \frac{1}{2(\lambda-j+1)} c_j^\lambda f_{\lambda-j +\frac{1}{2}} + \sum_{j=1}^{\lambda-1} d_j^\lambda f_{\lambda-j -\frac{1}{2}}\right) .
\end{gathered}$$
The functions $\boldsymbol{k_{d,1}}$ when $\boldsymbol{d>1}$ is odd {#subsection8_2}
-------------------------------------------------------------------
Now turn to the calculation of the functions $k_{d,1}$ when $d>1$ is odd. Then the Gegenbauer parameter $\lambda = (d-2)/2 = \kappa +{\ensuremath{\frac{1}{2}}}$, for some nonnegative integer $\kappa$.
We will need the following technical lemmas.
\[lem:technical1\]Let $\alpha$ be an integer and $\beta$ a nonnegative integer. If $\alpha$ is nonnegative further suppose $\beta < a$. Then $$\begin{gathered}
J_{\beta,\alpha} =\sum_{\nu=0}^\beta \binom{\beta}{\nu} \frac{(-1)^\nu}{\alpha-\nu}=\frac{(-1)^\beta \beta!}{\alpha(\alpha-1) \cdots (\alpha-\beta)}.\end{gathered}$$
The identity will be proven by induction on $\beta$. For $\beta=0$ the result is immediate. Now assume that the identity is true for $\beta=\kappa-1$ where $\kappa \in {{\mathbb N}}$. Then $$\begin{gathered}
J_{\kappa,a} = \binom{\kappa}{0}\frac{1}{\alpha} +\binom{\kappa}{\kappa} \frac{(-1)^\kappa}{\alpha-\kappa}+ \sum_{\nu=1}^{\kappa-1}\left\{ \binom{\kappa-1}{\nu}+\binom{\kappa-1}{\nu-1}\right\} \frac{(-1)^\nu}{\alpha-\nu}\\
\hphantom{J_{\kappa,a}}{} = \sum_{\nu=0}^{\kappa-1} \binom{\kappa-1}{\nu} \frac{(-1)^\nu}{\alpha-\nu} + \left\{ \frac{(-1)^\kappa}{\alpha-\kappa}+ \sum_{\ell=0}^{\kappa-2} \binom{\kappa-1}{\ell}
\frac{(-1)^{\ell+1}}{\alpha-1-\ell} \right\}\\
\hphantom{J_{\kappa,a}}{} = \sum_{\nu=0}^{\kappa-1} \binom{\kappa-1}{\nu} \frac{(-1)^\nu}{\alpha-\nu} - \sum_{\ell=0}^{\kappa-1} \binom{\kappa-1}{\ell} \frac{ (-1)^\ell }{\alpha-1-\ell} .\end{gathered}$$ Applying the induction hypothesis twice $$\begin{gathered}
J_{\kappa,\alpha} = \frac{(-1)^{\kappa-1} (\kappa-1)!}{\alpha(\alpha-1) \cdots (\alpha-\kappa+1)}
- \frac{ (-1)^{\kappa-1} (\kappa-1)!}{(\alpha-1)(\alpha-2)\cdots (\alpha-\kappa )}=\frac{(-1)^\kappa \kappa! }{\alpha (\alpha-1) \cdots (\alpha-\kappa)},\end{gathered}$$ showing that the identity also holds for $\beta=\kappa$.
\[lem:technical2\]Let $\kappa\in {{\mathbb N}}_0$. Then $$\begin{gathered}
\int_{-1}^1 \left(1-x^2\right)^\kappa \ln \left( \frac{1-x}{2} \right) {\rm d}x = 2^{2\kappa+1}(-1)^{\kappa+1} \sum_{\nu=0}^\kappa \binom{\kappa}{\nu} \frac{(-1)^\nu}{(2\kappa-\nu+1)^2}.\end{gathered}$$
$$\begin{gathered}
I = \int_{-1}^1 \left(1-x^2\right)^\kappa \ln \left( \frac{1-x}{2} \right) {\rm d}x = \sum_{\nu=0}^\kappa \binom{\kappa}{\nu} 2^\nu (-1)^\nu \int_{-1}^1 \ln\left( \frac{1-x}{2} \right) (1-x)^{2\kappa-\nu} {\rm d}x
\nonumber \\
\hphantom{I} =\sum_{\nu=0}^\kappa \binom{\kappa}{\nu} 2^{2\kappa} (-1)^\nu \int_{-1}^1 \ln\left( \frac{1-x}{2} \right) \left( \frac{1-x}{2} \right)^{2\kappa-\nu} {\rm d}x. \label{eq:tech_lem2_I}\end{gathered}$$
Now $$\begin{gathered}
\int_{-1}^1 \ln \left( \frac{1-x}{2} \right) \left( \frac{1-x}{2} \right)^{2\kappa-\nu} {\rm d}x = 2 \int_{0}^1 \ln( t) t^{2\kappa-\nu} {\rm d}t= \frac{ -2}{(2\kappa-\nu+1)^2}.\end{gathered}$$ Substituting into equation yields the result.
We now turn to the development of an expression for $G_\lambda$ which will be particularly convenient for the evaluation of the indefinite integral $S_\lambda (y)$ of equation in this $\lambda=\kappa+{\ensuremath{\frac{1}{2}}}$ case. Recall the definition of $G_\lambda^\alpha$. In this section we restrict ourselves to the case where $\alpha$ is a nonnegative integer, with $\alpha \leq \kappa$. Substituting $1-z=2-(1+z)$ into the expression for $G_\lambda^\alpha$ yields $$\begin{gathered}
G_\lambda^\alpha (y) = \int_{-1}^y \big(1-z^2\big)^\kappa (1-z)^{-\alpha} {\rm d}z \nonumber\\
\hphantom{G_\lambda^\alpha (y)}{} = \int_{-1}^y (1+z)^\kappa \sum_{\gamma=0}^{\kappa-\alpha} \binom{\kappa-\alpha}{\gamma} 2^\gamma (-1)^{\kappa-\alpha-\gamma} (1+z)^{\kappa-\alpha-\gamma} {\rm d}z \nonumber\\
\hphantom{G_\lambda^\alpha (y)}{}= \sum_{\gamma=0}^{\kappa-\alpha} \binom{\kappa-\alpha}{\gamma} \frac{2^\gamma (-1)^{\kappa-\alpha-\gamma}}{2\kappa-\alpha -\gamma+1} (1+y)^{2\kappa-\alpha-\gamma+1}.
\label{special_form_of_G_lambda_alpha}\end{gathered}$$ In particular, for $\alpha=0$, $$\begin{gathered}
G_\lambda (y) = (1+y)^{\kappa+1} \widetilde{G}_\kappa (y),\end{gathered}$$ where $$\begin{gathered}
\widetilde{G}_\lambda (y) = \sum_{\gamma=0}^\kappa \binom{\kappa}{\gamma} \frac{2^\gamma (-1)^{\kappa-\gamma}}{2\kappa-\gamma+1} (1+y)^{\kappa-\gamma}\\
\hphantom{\widetilde{G}_\lambda (y)}{} = \sum_{\gamma=0}^\kappa \binom{\kappa}{\gamma} \frac{2^\gamma (-1)^{\kappa-\gamma}}{2\kappa-\gamma+1}
\sum_{\ell=0}^{\kappa-\gamma} \binom{\kappa-\gamma}{\ell} 2^\ell (-1)^{\kappa-\gamma-\ell} (1-y)^{\kappa-\gamma-\ell}.\end{gathered}$$ Now substituting $\nu=\gamma+\ell$ and noting that $ \binom{\kappa}{\gamma} \binom{\kappa-\gamma}{\nu-\gamma}=\binom{\kappa}{\nu}\binom{\nu}{\gamma}$ $$\begin{gathered}
\widetilde{G}_\lambda (y) = \sum_{\gamma=0}^\kappa \sum_{\nu=\gamma}^\kappa \binom{\kappa}{\gamma}\binom{\kappa-\gamma}{\nu-\gamma} \frac{2^\nu (-1)^{\nu-\gamma} }{2\kappa-\gamma+1} (1-y)^{\kappa-\nu} \\
\hphantom{\widetilde{G}_\lambda (y)}{} = \sum_{\nu=0}^\kappa 2^\nu \binom{\kappa}{\nu} \left\{
\sum_{\gamma=0}^\nu \binom{\nu}{\gamma} (-1)^{\nu-\gamma} \frac{1}{2\kappa-\gamma+1} \right\} (1-y)^{\kappa-\nu}.\end{gathered}$$ Applying Lemma \[lem:technical1\] finally gives $$\begin{gathered}
\widetilde{G}_\lambda (y)= \sum_{\nu=0}^\kappa 2^\nu \binom{\kappa}{\nu} \frac{ \nu! (2\kappa-\nu)!}{(2\kappa+1)!} (1-y)^{\kappa-\nu}.\end{gathered}$$ Therefore, $$\begin{gathered}
G_\lambda (y) = (1+y)^{\kappa+1} \sum_{\nu=0}^\kappa 2^\nu \binom{\kappa}{\nu}\frac{ \nu! (2\kappa-\nu)!}{(2\kappa+1)!} (1-y)^{\kappa-\nu},\end{gathered}$$ and hence a representative value of $S_\lambda$ is $$\begin{gathered}
S_\lambda(y) = \int \big(1-y^2\big)^{-\kappa-1} G_\lambda (y) {\rm d}y = \sum_{\nu=0}^\kappa 2^\nu \binom{\kappa}{\nu}
\frac{ \nu! (2\kappa-\nu)!}{(2\kappa+1)!}\int (1-y)^{-1-\nu } {\rm d}y \\
\hphantom{S_\lambda(y)}{}= g^\lambda_0 \ln \left(\frac{1-y}{2}\right) + \sum_{\nu=1}^\kappa g^\lambda_\nu(1-y)^{-\nu},
\end{gathered}$$ where $$\begin{gathered}
\label{eq:def_g_lambda_nu}
g^\lambda_\nu = \begin{cases} \dfrac{-1}{2\kappa+1}, &\text{when}\ \nu=0,\\
\dfrac{1}{2\kappa+1},& \text{when}\ \nu=1,\\
\displaystyle 2^\nu \binom{\kappa}{\nu}
\frac{ (\nu-1)! (2\kappa-\nu)!}{(2\kappa+1)!},& \text{when}\ 1<\nu\leq \kappa.
\end{cases}\end{gathered}$$ To complete the calculation of $k_{d,1}$ we need to evaluate the constant $$\begin{gathered}
[ S_\lambda , e_0]_\lambda = \int_{-1}^1 \big(1-x^2\big)^\kappa g^\lambda_0 \ln\left(\frac{1-x}{2}\right) {\rm d}y
+ \int_{-1}^1 \big(1-x^2\big)^\kappa \sum_{\nu=1}^\kappa g^\lambda_\nu(1-x)^{-\nu} {\rm d}x = I_1 +I_2.\end{gathered}$$ The integral $I_1$ is given in Lemma \[lem:technical2\]. To compute $I_2$ we apply equation which implies that for $0 \leq \nu \leq \kappa$ $$\begin{gathered}
\int_{-1}^1 \big(1-x^2\big)^\kappa (1-x)^{-\nu} {\rm d}x =G^\nu_\lambda(1) = \sum_{\gamma=0}^{\kappa-\nu} \frac{\binom{\kappa-\nu}{\gamma} 2^\gamma (-1)^{\kappa-\nu-\gamma} 2^{2\kappa-\nu-\gamma+1}}{2\kappa-\nu-\gamma+1} \\
\hphantom{\int_{-1}^1 \big(1-x^2\big)^\kappa (1-x)^{-\nu} {\rm d}x}{} = 2^{2\kappa-\nu+1} (-1)^{\kappa-\nu} \sum_{\gamma=0}^{\kappa-\nu} \frac{ \binom{\kappa-\nu}{\gamma} (-1)^\gamma}{2\kappa-\nu -\gamma+1} .
\end{gathered}$$ An application of Lemma \[lem:technical1\] then shows $$\begin{gathered}
G^\nu_\lambda(1) = \frac{2^{2\kappa-\nu +1} \kappa! (\kappa-\nu)!}{(2\kappa-\nu +1)!}.
\end{gathered}$$ Putting all these things together, for $d=2\kappa+1 >1$ odd, and $\lambda=(d-2)/2$, $$\begin{gathered}
k_{2\kappa+1,1}(x) =\frac{-1}{2\kappa+1} \ln\left(\frac{1-x}{2}\right) + \sum_{\nu=1}^\kappa g^\lambda_\nu(1-x)^{-\nu} -D_\lambda,
\end{gathered}$$ where $$\begin{gathered}
D_\lambda =\frac{ [S_\lambda,e_0]_\lambda}{[e_0,e_0]_\lambda} = \frac{1}{f_{\lambda-{\ensuremath{\frac{1}{2}}}}} \left(\sum_{\nu=1}^{\kappa} g^\lambda_\nu
G_\lambda^\nu(1) +\frac{1}{2\kappa+1} 2^{2\kappa+1} (-1)^\kappa \sum_{\nu=0}^\kappa \frac{\binom{\kappa}{\nu} (-1)^\nu}{(2\kappa-\nu+1)^2} \right).
\end{gathered}$$
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Wahba G., Spline models for observational data, *CBMS-NSF Regional Conference Series in Applied Mathematics*, Vol. 59, Society for Industrial and Applied Mathematics, Philadelphia, PA, 1990.
[^1]: Note that reproducing kernels for semi-Hilbert spaces also appear as *semi-kernels* [@BerlinetAgnan; @Laurent1991] or *increment reproducing kernels* [@MosamamKent2010].
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{
"pile_set_name": "ArXiv"
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abstract: 'An inverted planar pendulum with horizontally moving pivot point is considered. It is assumed that the law of motion of the pivot point is given and the pendulum is moving in the presence of dry friction. Sufficient conditions for the existence of solutions along which the pendulum never falls below the horizontal positions are presented. The proof is based on the fact that solutions of the corresponding differential inclusion are right-unique and continuously depend on initial conditions, which is also shown in the paper.'
---
<span style="font-variant:small-caps;">Ivan Polekhin$^*$</span>
(Communicated by ...)
Mechanical model
================
Consider an inverted planar pendulum in a gravitational field with its pivot point moving along a horizontal line in the plane of the pendulum. The law of motion $\xi(t)$ of the pivot point is given and the pendulum is moving in the presence of dry friction. Let $l$ be the length of the pendulum, $m$ be its mass. Let $r = (r_x, r_y)$ be the radius vector of the massive point of the pendulum, and $r_x$, $r_y$ are its components in axes of an orthogonal coordinate system $Oxy$, where $Oy$ is the vertical axis. The general equation of motion has the form $$m \ddot r = F_{grav} + F_{fric} + N.$$ Here $F_{grav} = -mg\cdot e_y$ is the applied force of gravity, $F_{fric}$ is the force of dry friction, and $N$ is the force of constraint that appears from the holonomic constraint $(r_x - \xi(t))^2 + r_y^2 = l^2$. By $g$ we denote the gravitational acceleration. We assume that the force of dry friction is the Coulomb friction. In our model we consider the case when the Stribeck effect can be ignored, and the difference between the dynamic and static friction coefficients is negligibly small. Therefore, $F_{fric}$ is as follows [@popov2010; @ivanov2009bifurcations] $$F_{fric} = -\mu |N| \frac{v}{|v|}, \mbox{ if } v \ne 0, \quad |F_{fric}| \leqslant \mu |N|,
\mbox{ if } v = 0.$$ Here $\mu > 0$ is the dry friction coefficient, $N$ is the normal reaction force, and $v$ is the relative velocity of the massive point: $v = (\dot r_x - \dot \xi)e_x + \dot r_y e_y$.
Let $q$ be the angle between the horizontal line and the rod ($q = 0$ or $q = \pi$ are the horizontal positions). It is not hard to obtain that $|N| = m |\ddot \xi \cos q -l\dot q^2 + g\sin q|$. Therefore, for $v \ne 0$, the equation of motion can be presented as follows $$\label{eq1}
\begin{aligned}
&\dot q = p,\\
&\dot p = \frac{\ddot\xi}{l}\sin q -\frac{\mu}{l}\left| \ddot\xi \cos q - lp^2 + g\sin q \right|\frac{p}{|p|} - \frac{g}{l}\cos q.
\end{aligned}$$ When $p = 0$, $|F_{fric}|$ can be any value between zero and $\mu |N|$. Therefore, the motion of the system cannot be described by an ordinary differential equation. One of the possible and convenient solution to this problem is to consider a differential inclusion corresponding to the considered model of dry friction, which we do in the next section.
The system of a one-dimensional inverted pendulum — being a simple yet strongly non-linear system — has been considered by many authors (see, for instance, [@kapitsa1954pendulum; @bardin1995stability; @butikov2001dynamic; @seyranian2006stability]). Many of these results deal with the smooth system of a pendulum with vertically oscillating pivot. Unlike these cases, we consider a non-smooth system with dry friction and the pivot point moving horizontally and show that there always exists a solution along which the inverted pendulum never falls below the horizontal line.
Main result
===========
The system (\[eq1\]) and similar systems can be presented in the following form $$\label{eq2}
\dot x = f(x, t),$$ where $f$ is a piecewise continuous function in a domain $G \subset \mathbb{R}^{n+1}$ and $M \subset G$ is a set of measure zero of points of discontinuity of $f$. Following [@filippov2013differential], consider a differential inclusion associated with the above equation (\[eq2\]) $$\label{eq3}
\dot x \in F(x,t),$$ where $F \colon G \to 2^{\mathbb{R}^{n}}$ is a set-valued function defined as follows: for any point $(x,t) \in G$, the set $F(x,t)$ is the smallest convex closed set containing all the limit values of $f(x^*,t)$, $(x^*,t) \notin M$, $x^* \to x$.
A solution of the differential inclusion (\[eq3\]) is an absolutely continuous function $x \colon I \to \mathbb{R}^n$ defined on an interval or on a segment $I$ for which (\[eq3\]) is satisfied almost everywhere.
Below, by the solution of (\[eq1\]) we mean the solution of the corresponding differential inclusion with the right-hand side denoted by $\Phi = \Phi(q,p,t)$ (in our case, $M$ is the plane $p = 0$). We also assume that $\ddot \xi $ in (\[eq1\]) is a Lipschitz function.
First, we show that the existence of solutions and their continuous dependence on initial data follow directly from the general properties of $\Phi$.
Let $A, B \subset \mathbb{R}^n $ be non-empty closed sets. Then $$\beta(A,B) = \sup\limits_{a \in A}\rho(a, B).$$ Here $\rho(a, B) = \inf\limits_{b \in B} \rho(a,b)$ and $\rho(a,b)$ is the Euclidean distance in $\mathbb{R}^n$.
A set-valued function $F$ is called upper semicontinuous at $x \in \mathbb{R}^n$, if $\beta(F(y),F(x)) \to 0$ as $y \to x$. A function is called upper semicontinuous on a set $G$ if it is upper semicontinuous at each point of $G$.
It is not hard to see that $\Phi$ is upper semicontinuous in $\mathbb{R} / 2\pi\mathbb{Z} \times \mathbb{R} \times \mathbb{R}$.
[@filippov2013differential] Let $F$ be an upper semicontinuous set-valued function in a domain $G \subset \mathbb{R}^{n+1}$, and for all $(x, t) \in G$, $F(x, t)$ is a non-empty, bounded, closed and convex set. Then for any point $(x_0, t_0) \in G$ there exists a local solution of the problem $$\dot x \in F(x,t), \quad x(t_0) = x_0.$$ Moreover, if $G$ is closed and bounded, then every solution can be continued up to the boundary of $G$.
From this theorem, it follows that for the system (\[eq1\]), solutions exist for all $t > t_0$. Indeed, set-valued function $\Phi$ is periodic in $q$ and it can be shown that if $p > 0$ and $$p^2 > p_*^2 = \left( g + \max\limits_{t \in [t_0,t_1]}|\ddot \xi| \right) \left(1 + \frac{1}{\mu} \right)\frac{1}{l},$$ then $\dot p < 0$ for any $t \in [t_0, t_1]$. Similarly, for $p<0$ and large $|p|$, we have $\dot p > 0$. Therefore, solutions can be continued to an arbitrarily long time interval. From the below result, it also follows that solutions of (1) depend continuously on initial data.
[@filippov2013differential] Let $F$ be an upper semicontinuous set-valued function in a domain $G \subset \mathbb{R}^{n+1}$, and for all $(x, t) \in G$, $F(x, t)$ is a non-empty, bounded, closed and convex set; $t_0 \in [a, b]$, let all the solutions of the problem $$\dot x \in F(x,t), \quad x(t_0) = x_0$$ exist for $a \leqslant t \leqslant b$ and their graphs lie in $G$. Then for any $\varepsilon > 0$ there exists $\delta > 0$, such that for any $t_0^* \in [a,b]$ and $x_0^*$, $|t_0^* - t_0|<\delta$ and $|x_0^* - x_0|<\delta$, each solution with initial conditions $t = t_0^*$, $x = x_0^*$ exists and differs (w.r.t. the uniform norm) from some solution with initial conditions $t = t_0$, $x = x_0$ by not more than $\varepsilon$.
We say that (\[eq3\]) has a right-unique solution at a point $(x_0, t_0)$ if there exists $t_1 > t_0$ such that each two solutions of the differential inclusion satisfying the condition $x(t_0) = x_0$ coincide on $[t_0, t_1]$.
Let us now show that, for given initial conditions, the solution of (\[eq2\]) is right-unique. The following result was also proved in [@filippov2013differential]
Let a function $f(t,x)$ in a domain $G$ be discontinuous only on a set of measure zero. Let there exist a summable function $l(t)$ such that for almost all points $(x,t)$ and $(y,t)$ of the domain $G$ we have $f(x,t) \leqslant l(t)$ and for $|x - y| < \varepsilon_0$, $\varepsilon_0 > 0$, the following holds $$\label{eq4}
(x-y)\cdot(f(x,t) - f(y,t)) \leqslant l(t) |x-y|^2.$$ Then any solution of the corresponding differential inclusion (\[eq3\]) is right-unique in the domain $G$.
As usual, we say that function $l \colon \mathbb{R} \to \mathbb{R}$ is summable if it is Lebesgue integrable and $$\int\limits_K|l(t)|\, dt < \infty,$$ For any compact $K$. Below we consider only constant functions $l(t) = l$ that are always summable.
The solutions of (\[eq1\]) are right-unique.
Let $G$ be a bounded domain in $\mathbb{R}^{3}$, by $G^+$ we denote $\{ p > 0 \} \cap G$. Similarly, $G^- = \{ p < 0 \} \cap G$. By $f(q,p,t)$ we denote the right-hand side of the system (1). Let $f^+(q,0,t)$ and $f^-(q,0,t)$ be the limiting values of the function $f$ at the point $(q,0,t)$, from $G^+$ and $G^-$, correspondingly. Let $n$ be a vector directed toward increasing values of $p$. From (\[eq1\]) we have $$n \cdot f^+(q,0,t) = \frac{\ddot\xi}{l}\sin q -\frac{\mu}{l}\left| \ddot\xi \cos q - lp^2 + g\sin q \right| - \frac{g}{l}\cos q,$$ and $$n \cdot f^-(q,0,t) = \frac{\ddot\xi}{l}\sin q + \frac{\mu}{l}\left| \ddot\xi \cos q - lp^2 + g\sin q \right| - \frac{g}{l}\cos q.$$ Therefore, $n \cdot f^+(q,0,t) \leqslant n \cdot f^-(q,0,t)$. Let us now show that for almost all points $(q_1,p_1,t)$ and $(q_2, p_2,t)$, inequality (4) holds.
If both points are in $G^+$ or in $G^-$, then the inequality follows from the fact that the right-hand side is Lipschitz continuous (in this case, we can put $l(t)$ to be a constant). Now suppose that $(q_1,p_1,t) \in G^+$ and $(q_2, p_2,t) \in G^-$. By $(q, 0, t)$ we denote the point of intersection of the line segment connecting $(q_1,p_1,t)$ and $(q_2, p_2,t)$ with the plane $p = 0$. Since $f$ is Lipschitz continuous in $G^-$ and $G^+$, then for some constant $l$, we have the following inequalities. $$\begin{aligned}
&|f(q_1, p_1, t) - f^+(q, 0, t)| \leqslant l |(q_1, p_1,t) - (q, 0,t)|,\\
&|f^-(q, 0, t) - f(q_2, p_2, t)| \leqslant l |(q_2, p_2,t) - (q, 0,t)|.
\end{aligned}$$ From these inequalities and the fact that the points $(q_1,p_1,t)$, $(q,0,t)$, $(q_2,p_2,t)$ are on the same line, we have $$\begin{aligned}
|f(q_1, p_1, t) - f^+(q, 0, t) + f^-(q, 0, t) - f(q_2, p_2, t)| \leqslant l |(q_1, p_1,t) - (q_2, p_2,t)|.
\end{aligned}$$ Therefore, $$\begin{aligned}
((q_1, p_1,t) - (q_2, p_2,t))&\cdot (f(q_1, p_1, t) - f^+(q, 0, t) + f^-(q, 0, t) - f(q_2, p_2, t)) \\&\leqslant l |(q_1, p_1,t) - (q_2, p_2,t)|^2.
\end{aligned}$$ Note that $f^+(q, 0, t) - f^-(q, 0, t)$ is parallel to $n$ or equals zero, therefore $$((q_1, p_1,t) - (q_2, p_2,t)) \cdot (f^+(q, 0, t) - f^-(q, 0, t)) \leqslant 0.$$ Finally, if we sum the last two inequalities, we obtain (\[eq4\]). When $G$ is not a bounded region, it can be presented as a union of bounded sets, in which the solutions are right-unique.
We note, that the presented proof is similar to the one in [@filippov2013differential], yet it covers a wider class of functions (in [@filippov2013differential], $f$ is assumed to be twice-differentiable almost everywhere).
We have shown that any solution of (\[eq1\]) exists for all $t \geqslant t_0$, this solution is right-unique and continuously depends on initial conditions. From these properties, we obtain the following result.
\[propo1\] There exist $q_0 \in [0, \pi]$, $p_0$ such that for the solution $(q(t), p(t))$ of (\[eq1\]) with the corresponding initial conditions $q(t_0) = q_0$, $p(t_0) = p_0$, the following holds $q(t) \in [0, \pi]$ for all $t > t_0$.
First, consider (\[eq1\]) in the domain $G = \{ 0 < q < \pi \}$. Any solution leaving $G$ can be continued up to the boundary of $G$. At the same time, the solution cannot leave $G$ at the points where $q = 0$ and $p > 0$ or where $q = \pi$ and $p < 0$. Therefore, for any solution starting in $G $ there are three possibilities: it can never leave $G$; it can leave $G$ through the set $q = 0$, $p < 0$ or through the set $q = \pi$, $p > 0$; it can leave $G$ through the set $q = 0$, $p = 0$ or $q = \pi$, $p = 0$.
Let us now consider a continuous curve $p = \sigma(q)$, $t = t_0$, $0 \leqslant q \leqslant \pi$, where $\sigma$ is a continuous function and $\sigma(0) < 0$, $\sigma(\pi) > 0$. Consider all the solutions starting at this curve. Suppose that all these solutions leave $G$.
If some solution leaves $G$ through the set $q = 0$, $p < 0$ or $q = \pi$, $p > 0$, then all the solutions starting from close initial conditions also leave $G$ through close boundary points. It follows from the continuous dependence on the initial data.
Now consider the case when some solution, starting at the considered curve, reaches the line $q = 0$, $p = 0$ for the first time at moment $t = t^*$. This solution either stays in $q = 0$, $p = 0$ for all $t \geqslant t^*$ or leaves it at some $t = t^{**}$. If it stays in the line for all $t \geqslant t^*$, then it is the required solution. However, above we have supposed that all solutions leave $G$, i.e., our solution leaves line $q = 0$, $p = 0$ at $t = t^{**}$, where $t^{**} = t^* + \sup \{ \Delta t \geqslant 0 \colon q(t^* + t) = 0, \, p(t^* + t) = 0,\quad \forall\, 0 \leqslant t \leqslant \Delta t \}$. There are two possibilities: the pendulum can start moving either outside or inside the set $G$. If it moves inside $G$, then the map between the curve and the boundary $\partial G$ may become discontinuous because then there is a possibility for two close solutions to leave $G$ through the different components of the boundary ($q = 0$ and $q = \pi$). Below we prove that it is not the case.
For small $|q|$ and $p > 0$, we have $\dot p < 0$. Therefore, our solution can leave the line only to the set where $p \leqslant 0$, i.e., there exists $t^{***} > t^{**}$ such that $p(t) \leqslant 0$, for all $t \in [t^{**}, t^{***}]$. Moreover, for some $t \in [t^{**}, t^{***}]$ we have $p(t) < 0$ (if it is not true, then the solution do not leave the line at $t = t^{**}$). Since $\dot q = p$, we obtain that our solution, and solutions close to it, leave $G \cup \partial G$. Similarly, one can prove that if some solution reaches the line $p = 0$, $q = \pi$, it either stay in it forever or leaves $G \cup \partial G$.
Consider the continuous map from $\partial G$ to the points $q = 0$, $p = \sigma(0)$ and $q = \pi$, $p = \sigma(\pi)$ that maps two connected components of the boundary to these two points, respectively (to be more precise, it maps the plane $q = 0$ into the point $q = 0$, $p = \sigma(0)$, $t = 0$ and the plane $q = \pi$ into the point $q = \pi$, $p = \sigma(\pi)$ $t = 0$). We supposed that all solutions starting at the curve leave $G \cup \partial G$. Previously, we have shown that in this case there exists a continuous map between the curve $\sigma$ and the set $\partial G$. This map This map is a correspondence that assigns to every point of $\sigma$ the point of the first exit of the corresponding solution from $G$. Now we can consider the composition of the above two continuous maps. Finally, we obtain a continuous map between the curve and its boundary points. This contradiction proves the proposition.
Note that from the proof it also follows that there exist infinitely many solutions without falling. Indeed, a one-parameter family of such solutions can be obtained if we consider a family of non-intersecting curves $\sigma(q)$.
Similarly, one can prove the following result, which contains sufficient conditions for the existence of a solution staying in $(0, \pi)$ for all $t \geqslant t_0$.
Suppose that $\mu|\ddot \xi| < g$ for all $t \geqslant t_0$. Then there exist $q_0 \in (0, \pi)$, $p_0$ such that for the solution $(q(t), p(t))$ of (1) with the corresponding initial conditions $q(t_0) = q_0$, $p(t_0) = p_0$, the following holds $q(t) \in (0, \pi)$ for all $t > t_0$.
The proof is analogous to the previous one. The only difference is that it is possible to show that solutions starting in $G$ cannot leave this set at the points where $q = 0$ or $q = \pi$ and $p = 0$. Indeed, suppose that for the solution $(q(t), p(t))$, for some $t^*$, we have $q(t^*) = 0$ and $p(t^*) = 0$. Since for all $t$, we have $-g/2l - \mu|\ddot \xi|/2l \leqslant -g/2l + \mu|\ddot \xi|/2l < 0$, then any solution which reaches the plane $p=0$ at the point $q = 0$, leaves this plane. Moreover, we can conclude that this solution can reach the plane only from the region where $p > 0$. Taking into account the above inequalities, for small $\Delta t < 0$, we have $$q(t^* + \Delta t) = -\frac{g}{2l}\Delta t^2 - \frac{\mu}{2l}|\ddot \xi|\Delta t^2 + o(\Delta t^2) < 0.$$ We obtain that the considered solution reaches the point $q = 0$ and $p = 0$ from the outside of $G$. Similarly, one can prove that solutions cannot leave $G$ at the point $q = \pi$, $p = 0$.
Conclusion
==========
The presented proof is based on the topological ideas of so-called Ważewski method (see [@wazewski1948principe], [@reissig1963qualitative]), which can be also applied to other pendulum-like systems. For instance, in [@polekhin2014examples], it was proved that, if the motion of the pendulum is frictionless, then for any $\ddot \xi$, there exists a solution without falling. Note, that this result agrees with Proposition \[propo1\] from this paper if we put $\mu = 0$. A good overview of the attempts to prove the existence of a solution that always remains above the horizontal line in the system without friction can be found in [@Srzednicki2017]. However, the system with dry friction is qualitatively different comparing to the frictionless case: in the latter case, there are no equilibrium points when $|\ddot\xi|\ne 0$. At the same time, if $\mu \ne 0$ and $|\ddot \xi|$ is relatively small, then there is a set of equilibrium points in a vicinity of $q = \pi/2$. These points can be considered as solutions without falling, yet from the proof it can be seen that there also exists at least one non-constant solution that never falls.
The ideas that are used in the presented paper can also be used for systems with viscous friction and for more complex systems where the massive point is moving on a two-dimensional surface. Further development of the ideas of Ważewski method can also be used to prove the existence of periodic solutions without falling in pendulum-like and general mechanical systems [@polekhin2015forced], [@polekhin2016forced], [@bolotin2015calculus]. Similar methods have been found useful in studying global controllability of an inverted pendulum [@Polekhin2017gsi; @Polekhin2018].
In conclusion, we would like to note that the above results hold for a wider class of friction models. In particular, it is possible to consider various sufficiently smooth Stribeck curves. Therefore, we may expect the existence of a falling free motion (however, possibly unstable) in a real system of an inverted pendulum with horizontally moving pivot point.
[99]{}
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A. F. Filippov, *Differential equations with discontinuous righthand sides*, Vol. 18, Springer Science & Business Media, 2013.
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Received xxxx 20xx; revised xxxx 20xx.
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{
"pile_set_name": "ArXiv"
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abstract: 'We study whether hierarchical galaxy formation in a concordance $\Lambda$CDM universe can produce enough massive and red galaxies compared to the observations. We implement a semi-analytical model in which the central black holes gain their mass during major mergers of galaxies and the energy feedback from active galaxy nuclei (AGN) suppresses the gas cooling in their host halos. The energy feedback from AGN acts effectively only in massive galaxies when supermassive black holes have been formed in the central bulges. Compared with previous models without black hole formation, our model predicts more massive and luminous galaxies at high redshift, agreeing with the observations of K20 up to $z\sim 3$. Also the predicted stellar mass density from massive galaxies agrees with the observations of GDDS. Because of the energy feedback from AGN, the formation of new stars is stopped in massive galaxies with the termination of gas cooling and these galaxies soon become red with color $R-K>$5 (Vega magnitude) , comparable to the Extremely Red Objects (EROs) observed at redshift $z\sim$1-2. Still the predicted number density of very EROs is lower than observed at $z\sim 2$, and it may be related to inadequate descriptions of dust extinction, star formation history and AGN feedback in those luminous galaxies.'
author:
- 'X. Kang$^{1,2}$, Y. P. Jing$^{1}$, J. Silk$^{2}$'
title: Massive and Red Objects predicted by a semianalytical model of galaxy formation
---
Introduction
============
There are many recent observations of high-redshift galaxies that probe the star formation history of the Universe. The finding of many massive galaxies, especially massive Extreme Red Objects (EROs), at high redshift is particularly interesting. These observations show that some EROs are passive ellipticals, and were already in place at redshift z$\sim 2$. It is usually argued that in a Cold Dark Matter (CDM) universe, structures form via a hierarchical formation process in which small galaxies form first at early times, and massive galaxies form later through the continuous mergers of the smaller systems. With representative semi-analytical models (SAMs; Kauffmann et al. 1999, Somerville & Primack 1999, Cole et al. 2000), it was found that in the concordance $\Lambda$CDM universe, it is difficult to produce enough massive and red galaxies that look like those observed(e.g. Cimatti et al. 2002a, Glazebrook et al. 2004). On the other hand, the existence of the observed massive galaxies at high redshift is not necessarily in conflict with the concordance $\Lambda$CDM model, because the conversion of just ten percent of baryons in dark matter halos of mass $M >10^{13}M_{\odot}$ to stars is sufficient to produce the number of observed massive galaxies (Somerville 2004a).
Many authors have studied the formation of these massive, red objects using SAMs or Smoothed Particle Hydrodynamics (SPH) simulations. It was shown that the SAMs (Kauffmann et al. 1999, Somerville & Primack 1999, Cole et al. 2000) cannot produce enough massive/red objects at redshift $z>1$ (e.g. Firth et al 2002, Somerville et al. 2004b, Daddi et al. 2005). The SPH simulations (e.g. Nagamine et al. 2004, 2005) have succeeded in producing massive and red galaxies at high redshift, but at the cost of introducing more uncertainties. First, it is unknown if these SPH simulations can produce the local galaxy luminosity function. It seems that these simulations produce too many bright galaxies at $z=0$ (Nagamine et al. 2004). Secondly, Nagamine et al. (2005) used a high dust extinction for the entire galaxy population, but the observations show that some EROs are passive ellipticals with little dust extinction (Cimatti et al. 2002b).
The main reason that the SAMs fail to produce enough massive and luminous galaxies at high redshift is that the gas cooling and star formation in early massive halos is over-suppressed. In previous SAMs, the gas cooling in massive halos is switched off in order not to produce more luminous central galaxies than observed at redshift $z=0$. The suppression of gas cooling is also motivated by the X-ray observations that massive cooling flows are not observed in groups and clusters (e.g. Peterson et al. 2003). But as the consequence, the gas cooling may be over-suppressed at high redshift if a simplified prescription is used for the cooling cutoff. For example, in the Munich group model and also in Kang et al. (2005), the gas cooling is shut off by hand in halos with the virial velocity greater than $350km/s$. Since the halo mass is much lower at high redshift than at the present for a given virial velocity, the gas cooling is suppressed in this model for halos with the virial mass greater than 2.5$\times
10^{12}M_{\odot}$ at z $=$ 3. This artificial cooling switch-off seems to be the main reason that these models do not produce as many massive galaxies as observed.
In this paper, we implement a new model in which the energy from AGN is used to suppress the cooling of hot gas in halos. Following Kauffmann & Haehnelt (2000) we use a simple model wherein black holes gain most of their mass during major mergers. Our implementation of the feedback from AGN is very similar to that used recently by Croton et al. (2006) and Bower et al. (2005), and resembles a combination of their models. In our model, the total energy from the AGN is proportional to the Eddington luminosity of the central black hole and the efficiency of reheating the gas is proportional to a power of the virial velocity of the galaxy. Then the energy compensates for the radiative energy of the cooling gas, and the actual cooling rate is determined by the ratio between the two energies. The cooling is totally suppressed if the energy from AGN is larger than the energy radiated by the cooling gas. Compared with the previous model used by Kang et al. (2005) with an artificial cut-off of the gas cooling in the halos with the virial velocity larger than $350km/s$, the gas cooling and AGN feedback in the new model are treated in a more self-consistent way. The $M_{\rm bh}$-$\sigma$ relation of black hole mass $M_{\rm bh}$ and the bulge velocity dispersion $\sigma$ implies that massive black holes are present only in massive spheroids. In our present model, the energy feedback from AGN indeed is efficient in galaxies with a massive spheroid. We also require that the star formation rate in quiescent disks is reduced at high redshift as motivated by the observed evolution of cosmological cold gas content with redshift (Keres et al. 2005); thus the gas-rich mergers result in earlier formation of supermassive black holes in massive central bulges. Once the energy feedback is enough to suppress the gas cooling, the termination of new star formation will soon make the galaxies red. We will compare the model prediction of the number density of luminous galaxies with the K20 survey, and find that good agreement holds up to z$\sim$3, beyond which there is little observational data. Compared with previous SAMs, our present model can also produce some very red ($R-K>5$, magnitudes are given in the Vega system unless otherwise stated) passive ellipticals which are observed by the Great Observatories Origins Deep Survey (GOODS) at z$\sim 1-2$.
We arrange our paper as follows. In section 2, we briefly introduce our new model with AGN feedback and compare our model predictions with the local galaxy population. In section 3, we give the model predictions and compare them with the observations at high redshift. Finally, we discuss our results and conclude our work in section 4.
Model
=====
The SAM that we use here was described in detail by Kang et al. (2005) who studied the local galaxy population. The merger tree is constructed based on a high-resolution N-body simulation (Jing & Suto 2002) of 512$^{3}$ particles in a box of 100$h^{-1}{\rm Mpc}$. The cosmological parameters adopted there are $\Omega_{m} = 0.3$, $\Omega_{\Lambda} = 0.7$, $h=0.7$, $\sigma_{8} = 0.9$. Here we still use this simulation, but the SAM model is modified in two ways.
1\. We adopt a star formation efficiency $\alpha \sim (1+z)^{-1}$ in a quiescent disk that was shown to give a better match with the evolution of cosmological cold gas content with redshift (Kauffmann & Haehnelt 2000, P$\acute{\rm e}$roux et al. 2003, Keres et al. 2005). In the recent model of Durham group (Baugh et al. 2005, Bower et al. 2005), they adopt a constant star formation timescale for the disk. The star formation timescale used in our model is the dynamical time of the disk which scales with redshift as $(1+z)^{-1.5}$. So the star formation rate ($\dot{M_{\ast}}=\alpha M_{cold}/t_{dyn}$) of our model differs from that of the Durham model only slightly. Note that the relatively lower star formation rate in quiescent disks leaves more cold gas which helps to produce massive black holes during galaxy mergers at high redshift.
2\. We include a model for the growth of black holes and for the energy feedback from AGN to suppress the gas cooling. As the $M_{\rm
bh}$-$\sigma$ relation indicates that the central black holes grow with the growth of the spheroid components, it is plausible that the black holes get their mass through major mergers. But it is far from clear about the exact way that the black holes accrete the surrounding material. Here following Kauffmann & Haehnelt (2000), we use a simple parameterised form to describe the cold gas accreted by the black hole during a major merger, $$\Delta M_{bh} = F_{acc} \frac {M_{cold}} {1+(280km/s/V_{vir})^{2}}$$ where $M_{cold}$ is the total cold gas in merging galaxies, and $V_{vir}$ is the virial velocity of the post-merger host halo. We normalize the parameter $F_{acc}$ by best matching the observed $M_{bulge}-M_{bh}$ relation at z=0 (Häring & Rix 2004). During the gas accretion by black holes, part of the gravitational energy will be converted into radiations which in turn will heat the surrounding cold gas. But it is again unclear in a quantitative way about how much the radiation is produced and how efficiently the cold gas is re-heated. Croton et al. (2006) use a simple phenomenological model to describe the accretion rate which depends on the hot gas fraction and circular velocity of the halo, but the efficiency of heating the gas by AGN are the same in all halos of different mass. Sijacki & Springel (2006) have shown that heating efficiency from a AGN bubble is lower in low mass halos. Here we simply assume that the energy from the central AGN is proportional to the Eddington luminosity $L_{edn}$ and the heating efficiency is proportional to a power of the virial velocity of the host halo. Thus the heating rate ejected into the gas is taken as, $$L_{reheat}=F_{0}(V_{vir}/V_{\star})^{n}L_{edn}\,.$$ If we denote the cooling rate in a halo of gas temperature $T$ by $\dot{M}_{0,cool}$ in the case of no AGN feedback, then the cooling rate $\dot{M}_{cool}$ in the presence of AGN feedback is: $$\frac {\dot{M}_{cool}} {\dot{M}_{0,cool}} = 1 - \frac {L_{reheat}} {\frac {3} {4}\dot{M}_{0,cool}V_{vir}^{2}}.$$ If the heating rate from AGN $L_{reheat}/\frac{3}{4}V_{vir}^2$ is larger than the radiative cooling rate ${\dot{M}_{0,cool}}$, the gas cooling is totally suppressed. We normalize the parameters $F_{0}$, $V_{\star}$ to get a good match to the galaxy luminosity function at z=0. In our model we obtain $F_{0}=2\times 10^{-5}$ and $V_{\star}=200km/s$ and $n=4$.
In Fig. \[fig:Bh\_Bulge\] we plot the relation between the bulge mass and the black hole mass. The data points show for the model galaxies and the solid line the best fit to the observations by H$\ddot{\rm
a}$ring & Rix (2004). Here $F_{acc}$ is taken to be $0.01$. It is seen that a simple model of black hole growth with a free parameter can reproduce the observed $M_{bulge}-M_{bh}$ relation. After the black hole mass is normalized, we then tune the parameters in equation 2 to get good fits to the local galaxy luminosity functions. In Fig. \[fig:LF\_z0\] we show the luminosity function at B$_{j}$ and K bands. The upper panel shows a comparison with the 2dFGRS at B$_{j}$ band. The solid circles show the observational data of 2dFGRS, and the thick solid histogram associated with Poisson errors is our model prediction.
The lower panel shows the comparison at K band where the circles are from Cole et al. (2001) and squares are the observations by Huang et al. (2003). We find that the new model can produce the local galaxy luminosity functions at blue and near-IR bands which are respectively sensitive to the current star formation rate and the total stellar mass in the galaxies. It has been shown (Croton et al. 2006, Bower et al. 2005) that without an effective energy feedback, the predicted luminosity functions at the bright end are too flat with many more luminous galaxies predicted than observed. Note that here our model predictions at high luminosity ends are still slightly higher than observed. This might point to the fact that a more detailed model is needed for AGN heating in massive halos which we will address in future work.
Results at high redshift
========================
As discussed in Section 1, the gas cooling in our new model is not suppressed artificially but by heating due to the energy injected from AGN in the galaxy center. So compared to previous SAMs without AGN, the gas cooling and star formation can continues until a massive spheroid forms at the galaxy center. It is expected that this model can produce more massive and luminous galaxies at high redshift. In Fig. \[fig:K20\_LF\] we show the predicted rest-frame K band luminosity function at z$\sim 1.5$. The squares with error bars are the observational results from K20 (Pozzetti et al. 2003). The solid circles are the predictions by the new model and the triangles show the results predicted by Kang et al. (2005) where they adopted a artificial shut off of gas cooling in galaxies with $V_{vir}>350km/s$. We also re-plot the results of K band luminosity function at z=0 by the solid line, taken from from lower panel of Fig.\[fig:LF\_z0\]. It is clearly seen from the plot that the new model produces more massive galaxies and the agreement with the observations is very good. Also note that the good agreement holds for faint galaxies as well, whereas it was reported previously that SAM models produce more faint galaxies than observed (Pozzetti et al. 2003).
Another test, firstly proposed by Kauffmann & Charlot (1998), is the evolution of the surface number density of galaxies at a fixed limiting magnitude, which also widely used to constrain the models. There are plenty of data from GOODS that are already publicly available (Giavalisco et al. 2004). In Fig. \[fig:GOODS\_num\] we show the predicted redshift surface number density of galaxies with K$<20$. The square points show the results of K20 and triangles are the data from GOODS. The new model predictions are shown as the solid line, and the dashed line shows the prediction by the model of Kang et al. (2005). Here we find that compared with Somerville et al. (2004b) who predicted much fewer luminous galaxies at $z>1.5$, the agreement between our model and the observations holds much better up to z$\sim
3$. Here we also show how dust extinction will change the result. The dotted line is the new model with the simple dust extinction model of Calzetti et al. (2000) with $E(B-V)=0.1$. Clearly dust extinction has no significant effect on the predicted number of galaxies in the observed-frame K band up to z=3.
Though the predicted numbers of luminous galaxies agree with the observations, it would be interesting to check the predicted color distributions. The color is dependent on the star formation history and on the dust extinction. At high redshift the galaxy mergers are very frequent and the dust extinction is significant in these starburst galaxies, but no reliable model of dust extinction is available for such galaxies. Observations show that at z$\sim 1-2$ the EROs have contributions both from passive ellipticals with little dust and from dust-enshrouded starburst galaxies (Cimatti et al. 2002b, Cimatti et al. 2003, Yan & Thompson 2003, Yan et al. 2004, Moustakas et al. 2004). Because there are significant uncertainties in the dust extinction modelling for the starburst galaxies, we think that the predicted number density of passive ellipticals should set a more meaningful constraint on the galaxy formation model. Here we take a simple model of dust extinction. We classify the galaxies with starbursts produced during the major mergers in the past 0.1Gyr as young starburst galaxies and those otherwise as passive galaxies. We then use the Calzetti et al. (2000) reddening law to model the dust extinction effect on the galaxy color. The amount of dust in passive and young starburst galaxies is difficult to assess, and here we simply assume a small reddening $E(B-V)=0.05$ for the passive galaxies. The dust extent in young starburst galaxy is expected to be high. Observations of EROs show that some extremely red galaxies have heavy dust extinction with $E(B-V)=0.4$. But the average extinction should be lower. Here we assume a Gaussian distribution of $E(B-V)$ with a mean of 0.1 and a dispersion of 0.05 for the young starburst galaxies. Our main motivation is to see if a simple dust reddening model can produce the main features of the observed color distribution.
In Fig. \[fig:GOODS\_color\] we show the observed $R-K$ (both in the AB magnitude system, $(R-K)_{AB} \simeq (R-K)_{Vega}-1.65$) color distribution with a comparison with the data which are from the GOODS Southern field in an area of 160 arcmin$^{2}$ (Somerville et al. 2004b). The upper panel shows the GOODS data, which is from Figure 2 of Somerville et al. (2004b). The model galaxies are selected using the magnitude cut and are normalized to the same area of 160 arcmin$^{2}$. The total number of galaxies selected in our model is 1595 which is $6\%$ higher than the GOODS data points used here. The lower panel shows the model predictions. In each panel we also show the evolution track of single burst stellar populations with solar metallicity, the Salpeter IMF, and the ages (at $z=0$) of 13.35 and 11.7 Gyrs (i.e. $z_{f}=26, 2.6$) based on the model of BC03 (Bruzual & Charlot 2003). From the figure, our model can reproduce the main features of the observed galaxies: 1) many extremely red galaxies ($R-K>4$) at $z>1$; 2) the bimodal color distribution, red passive and young starburst galaxies at $z>1.5$. Still there are some discrepancies. The predicted numbers of blue galaxies are too prominent at z$<1.5$ and this might be due to the inadequate treatment of star formation rate, stellar initial mass function, or the dust extinction model. Also the predicted number of extremely red galaxies with $(R-K)_{AB}>3.35$ at $z \sim 2$ is still lower than observed. In our model there are enough luminous galaxies but insufficient number of very red galaxies, which means that the star formation (at $\sim 2$) in the current model are still high. There are two possible reasons for this discrepancy. First the star formation is not strong enough in the past in our model, as we do not include any star formation during minor mergers which are also frequent at early times. Second the energy from central AGN is not high enough to suppress the hot gas cooling. Observations have shown that there are already massive black holes ($\sim 10^{9}M_{\odot}$) at $z \sim 6$ (Fan et al. 2001), so the growth of black holes in massive galaxies might be much quicker at early time than in our model in which the fraction of cold gas accreted by black hole is constant with time. We will address this in a forthcoming paper (Kang et al. 2006).
Glazebrook et al. (2004) used the Gemini Deep Deep Survey (GDDS) to obtain the stellar mass distribution from $z \simeq$ 0.7 to 2. The evolution of stellar mass density does place important constraints on the formation model of massive spheroids. But due to the uncertainties in fitting the multi-broad band colors of high redshift galaxies including those of the IMF and dust extinction, the constraints are weak. In Fig. \[fig:stellar\_GDDS\], we show the stellar mass density of galaxies with stellar mass above certain limits. The lines show the predicted stellar density in galaxies with stellar mass in the range indicated in the plot. Black lines are for this model and blue lines are from the model of Kang et al. (2005) where they used an artificial cut of gas cooling in the halos with $V_{vir} > 350 km/s$. We can still see a good match between the model and the data. Although it seems that the stellar mass density with $M_{\star}> 10^{10.46}M_{\odot}$ is higher than the data points, it agrees with the integral of the star formation rate (see figure 4 of Glazebrook et al. 2004). Note that galaxies with $M_{\star}>10^{11}M_{\odot}$ are in the sharply declining tail of the mass function, therefore a small uncertainty in the estimated stellar mass can introduce a very large uncertainty in the number density. The hexagon in the plot shows the stellar mass density of massive galaxies with $M_{\star}>10^{11}M_{\odot}$ recently obtained by van Dokkum et al. (2006) making use of the deep multi-wavelength GOODS, FIRES and MUSYC surveys. It is seen from the black dashed lines that our model prediction is slight lower than the data by a factor of 2. At high redshift the cosmic variance is so large in the observed catalogs (about $60\%$, Somerville et al. 2004c) that the discrepancy might not be serious.
Discussion
==========
Here we have implemented a new semi-analytical model in which the energy from AGN suppresses hot gas cooling in massive halos. The growth of black holes and bulges, and the gas cooling, are determined in a self-consistent way. In our description, the AGN feedback becomes efficient in massive galaxies after a massive black hole is formed in the galaxy center. The AGN feedback model has drawn much recent attentions. The main motivation is that in massive groups and clusters cooling flows are not observed. There should be some physical process to reheat the cooling region, and the energy from AGN has been proposed as an effective source (e.g. B$\ddot{\rm o}$hringer et al. 2002, Begelmen et al. 2002, Sijacki & Springel 2006). At the same time, the AGN feedback models have also been incorporated into the SAMs recently and it has been shown that AGN feedback can produce a break of the luminosity function at the bright end and produce the color-magnitude relation observed in SDSS (Croton et al. 2006, Bower et al. 2005). Our model of AGN feedback is very similar to theirs in spirit, but the detailed prescription is different. In this paper we use this model to address some issues about the number distribution and color distribution of galaxies at high redshift. We compare the model predictions with the K20 and GOODS surveys. Our conclusions are as follows.
- The predicted number distribution of $K<20$ galaxies matches well with that of the GOODS and K20 galaxies up to a redshift of z $\sim$ 3;
- The predicted color distribution is similar to that observed in the surveys and many extremely red galaxies ($R-K_{AB}>4$) are produced, which has not been seen in previous models (Somerville et al. 2004b). At $z > 1.5$ the galaxy population already displays a bimodal color distribution;
- The predicted stellar mass density can marginally agree with the GDDS observation even with the uncertainties in the IMFs;
These results demonstrate that it is not difficult to produce massive and red galaxies at z $\sim$ 1-2 in the concordance CDM universe. The stellar mass in galaxy centers continues to grow until the energy from central AGN is high enough to suppress the gas cooling. In our model the black holes acquire most of their mass during major mergers, so the AGN energy feedback is expected to be effective after the last major merger which led to massive bulge formation at galactic centers. In our model we can produce some of those passive ellipticals at z$\sim 1-2$ with extremely red colors $(R-K)_{AB}>4$.
Many observations have shown that the star formation rate was higher in massive galaxies at high redshift and these support the “downsizing” formation scenario (Cowie et al. 1996). It is often argued that hierarchical galaxy formation cannot reproduce the downsizing formation process. But recent works (de Lucia et al. 2005, Bower et al. 2005, Scannapieco et al. 2005) have shown that models with AGN feedback in the hierarchical universe can reproduce the downsizing process in which the massive galaxies forms earlier. In this paper, we also find that the predicted luminous and massive galaxies are increased to the degree that is in agreement with the observations, though the predicted number of red galaxies may still be fewer than observed. Once more observations are available on the dust extinction in these galaxies, the number density and evolution of red passive ellipticals will put more stringent constraints on the galaxy formation models. It is also possible that a new ingredient is needed, such as the star formation induced by AGN feedback prior to disruption of the cold gas supply (Silk 2005), in order to make bulge formation more efficient and to account for the chemical evolution of massive early-type galaxies.
We thank Mashiro Nagashima for kindly providing the GOODS data, and Manfred Georg Kitzbichler for the binned data of GOODS and K20. Xi Kang acknowledge support from the Royal Society China Royal Fellowship Fellowship scheme. This work is supported in part by NSFC(No. 10373012, 10533030) and by Shanghai Key Projects in Basic research (04jc14079, 05xd14019).
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|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'The temperature in the optically thick interior of protoplanetary discs is essential for the interpretation of millimeter observations of the discs, for the vertical structure of the discs, for models of the disc evolution and the planet formation, and for the chemistry in the discs. Since large icy grains have a large albedo even in the infrared, the effect of scattering of the diffuse radiation in the discs on the interior temperature should be examined. We have performed a series of numerical radiation transfer simulations including isotropic scattering by grains with various typical sizes for the diffuse radiation as well as for the incident stellar radiation. We also have developed an analytic model including isotropic scattering to understand the physics concealed in the numerical results. With the analytic model, we have shown that the standard two-layer approach is valid only for grey opacity (i.e. grain size $\ga10$ ) even without scattering. A three-layer interpretation is required for grain size $\la10$ . When the grain size is 0.1–10 , the numerical simulations show that isotropic scattering reduces the temperature of the disc interior. This reduction is nicely explained by the analytic three-layer model as a result of the energy loss by scatterings of the incident stellar radiation and of the warm diffuse radiation in the disc atmosphere. For grain size $\ga10$ (i.e. grey scattering), the numerical simulations show that isotropic scattering does not affect the interior temperature. This is nicely explained by the analytic two-layer model; the energy loss by scattering in the disc atmosphere is exactly offset by the “green-house effect” due to scattering of the cold diffuse radiation in the interior.'
author:
- |
Akio K. Inoue$^{1}$[^1], Akinori Oka$^{2}$, and Taishi Nakamoto$^{2}$\
$^{1}$College of General Education, Osaka Sangyo University, 3-1-1, Nakagaito, Daito, Osaka 574-8530, Japan\
$^{2}$Department of Earth and Planetary Sciences, Tokyo Institute of Technology, Ookayama, Meguro, Tokyo 152-8551, Japan
title: 'Effects of scattering and dust grain size on the temperature structure of protoplanetary discs: A three-layer approach'
---
\[firstpage\]
dust, extinction — methods: analytical — methods: numerical — planetary systems: protoplanetary discs — radiative transfer — scattering
Introduction
============
Protoplanetary discs are planet formation sites. We observe the electro-magnetic radiation from the discs to understand their physical conditions, and then, to know the planet formation, especially, its beginning. Property of the radiation from a disc is essentially determined by the structure of the disc. The structure follows the response of the disc to the radiation from the central star. To interpret the radiation from protoplanetary discs, therefore, we should have a robust link between the radiation from the central star and the disc structure, in particular, the temperature structure.
The temperature structure of the discs is also essential for chemical reactions in the discs. The evolution of various molecules in the discs and the exchange of these molecules between the gas phase and the solid phase on grains will be discussed in detail with the ALMA in near future [e.g., @nom08]. The condensation front of icy molecules, so-called ’snow line’, is also determined by the temperature structure of the discs [e.g., @sas00; @oka09]. The location of the snow line is very important because it significantly enhances the amount of solid materials to make planetary cores and affects the supply of water to rocky planets. The temperature structure also affects the evolution of the discs themselves. The accretion activity in the discs is supposed to be driven by the magnetorotational instability [e.g., @san00]. This requires a certain degree of the ionization of disc materials which depends on the structure of the discs.
A milestone in the research of the vertical temperature structure of protoplanetary discs is the work by [@cg97] (hereafter CG97). They proposed a two-layer model consisting of the super-heated layer directly exposed by the stellar radiation and the interior warmed by the super-heated layer. This model explained the shape of the spectral energy distribution of the discs very well. The computational cheapness of the analytic approach makes the CG97 model very useful to compare with a large sample of the discs observed. Therefore, some attempts to refine the simple model of CG97 were performed [@chi01; @dul01; @dul03a; @raf06; @gar07].
Despite a great success of the CG97 model, numerical simulations of the radiation transfer in the discs showed a significant decrement of the equatorial temperature relative to the prediction by the CG97 model [@dul03a]. The internal energy loss by the radiation at a long wavelength where the optical depth of the disc is relatively small is suggested as a cause of the decrement [@dul02; @dul03a]. In the CG97 model and refined ones, the wavelength dependence of dust opacity was taken into account in terms of mean opacity. [@dul02] argued the importance of using full wavelength dependent opacity. However, we propose an alternative approach in this paper: a three-layer model with mean opacity which reproduces the temperature reduction quite well. In addition, we show that the two-layer approximation in the CG97 model is valid only when the dust opacity is “grey” which is expected if the size of dust grains is larger than about 10 .
Effect of scattering on the vertical temperature structure was not considered in the literature very much. Scattering of the stellar radiation was taken into account analytically by [@cal91] and numerically by [@dul03] who showed that the temperature of the disc interior is slightly reduced by the scattering. How about scattering of the diffuse disc radiation? As the size of grains increases, the scattering albedo increases. In particular, the albedo of large icy grains is close to unity even for the infrared wavelength. This may affect the disc structure significantly. Nevertheless, it has not been examined so far.
This paper discusses the effect of scattering of the diffuse radiation as well as that of the stellar radiation on the vertical temperature structure of protoplanetary discs. Since the albedo depends on the grain size, we examine the scattering effect as a function of the grain size. Although there are 2-D/3-D numerical radiation transfer codes available publicly, most of them have a serious difficulty in solving the radiation equilibrium in very high optical depth ($\tau\sim10^6$) found in protoplanetary discs [@pas04; @ste06]. The RADICAL developed by [@dul00] can solve such a problem without any difficulty thanks to a variable Eddington tensor method. However, it treats scattering of only the stellar radiation. We present a variable Eddington factor code with both scatterings of the stellar radiation and of the diffuse radiation but in a 1-D geometry. We also present an analytic model to interpret the numerical results. This simple model would be very useful to understand the physics determining the temperature structure of protoplanetary discs.
The rest of this paper consists of three sections; in section 2, we develop a numerical radiation transfer code taking into account both of the scatterings but only for isotropic case and show the obtained numerical solutions. In section 3, we construct an analytic model to interpret the numerical solutions and discuss the physical mechanism determining the temperature structure in protoplanetary discs. In the final section, we summarise our findings.
Numerical radiation transfer with scattering
============================================
Our method is an extension of the variable Eddington factor method developed by [@dul02]; we include isotropic scattering of the diffuse radiation as well as that of the stellar radiation. A disc is divided into many annuli in which the transfer of the diffuse radiation is treated one-dimensionally along the normal axis of each annulus with neglecting the radiation energy transport among annuli. This approximation, so-called 1+1D approximation, would be reasonable in an optically thick disc, but not at the near of the disc inner edge nor in a self-shadowing region [e.g., @dul01]. The radiation from the central star is separated from the diffuse radiation and is treated with the so-called grazing angle recipe. In this paper, we only consider some single annulus cases in order to feature the effect of the scattering on the temperature structure along the normal axis of the annulus. Therefore, we assume a grazing angle $\alpha=0.05$ radian throughout of the paper. The density structure along the normal axis of the annulus is solved to be consistent with the obtained temperature structure assuming the hydrostatic equilibrium. In Appendix A, we describe how to obtain the numerical solution of the diffuse radiation transfer with isotropic scattering in each annulus in detail.
A simple dust model
-------------------
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We adopt a very simple dust model in order to feature the effect of scattering on the temperature structure. The absorption and scattering cross sections per unit gas mass at the wavelength $\lambda$ are assumed to be $$\kappa_\lambda^{\rm abs} =
\cases{
\kappa_0^{\rm abs} & ($\lambda \le \lambda_{\rm c}$) \cr
\kappa_0^{\rm abs} \left(\frac{\lambda}{\lambda_{\rm c}}\right)^{-1}
& ($\lambda > \lambda_{\rm c}$) \cr
}\,,$$ and $$\kappa_\lambda^{\rm sca} =
\cases{
\kappa_0^{\rm sca} & ($\lambda \le \lambda_{\rm c}$) \cr
\kappa_0^{\rm sca} \left(\frac{\lambda}{\lambda_{\rm c}}\right)^{-4}
& ($\lambda > \lambda_{\rm c}$) \cr
}\,,$$ respectively. The single scattering albedo at the wavelength $\lambda$ is $$\omega_\lambda = \frac{\kappa_\lambda^{\rm sca}}
{\kappa_\lambda^{\rm abs} + \kappa_\lambda^{\rm sca}}\,.$$ The critical wavelength $\lambda_{\rm c}$ may be related to a typical grain radius $a$ as $\lambda_{\rm c} = 2\pi a$. If we consider a spherical grain composed of uniform material, the absorption cross section is expressed as $\kappa_\lambda^{\rm abs}=(3{\cal D}Q_\lambda^{\rm abs})/(4\rho_{\rm d}a)$, where $Q_\lambda^{\rm abs}$ is the absorption cross section normalised by the geometrical cross section $\pi a^2$, $\rho_{\rm d}$ is the grain material density, and $\cal D$ is the dust-to-gas mass ratio. With the values of ${\cal D}=10^{-2}$ (Solar system nebula), $\rho_{\rm d}=3$ g cm$^{-3}$ (silicate), and $Q_\lambda^{\rm abs} \to 1$ ($\lambda \to 0$), we obtain the absorption cross section for small wavelengths as $$\kappa_0^{\rm abs} = 250~{\rm cm^{2}~g^{-1}}
~\left(\frac{\rm 0.1~\micron}{a}\right)\,.$$ The scattering cross section can be given by the single scattering albedo for small wavelengths: $\omega_0=\kappa_0^{\rm sca}/(\kappa_0^{\rm abs}+\kappa_0^{\rm sca})$. In this paper, we consider three cases of $\omega_0=0$ (no scattering), $0.9$, or $0.99$. The values of $\omega_0$ for the last two cases may be extreme but such a large albedo is expected for icy grains in some wavelengths.
Figure 1 shows Planck means of the absorption cross section and the scattering albedo assumed in this paper as a function of the temperature input into the Planck function. In the panels, we show seven cases of grain size from 0.01 to 1 cm. We note that the absorption cross section and the scattering albedo become independent of the temperature, i.e. “grey”, when the temperature exceeds a critical one which depends on the grain size, corresponds to the critical wavelength $\lambda_{\rm c}$, and is roughly expressed as $T_{\rm c}\sim10^3(1~\micron/a)$ K. In this paper, we do not consider the size distribution of the dust grains. Thus, the “grain size” of this paper means a typical grain size averaged over a size distribution function with a weight.
Numerical results: Temperature structure
----------------------------------------
We here show the results of the annulus with the radius of 1 AU obtained from our numerical radiation transfer in Figures 2 and 3. The results with other radii have been confirmed to be the same qualitatively. The gas column density is assumed to be $10^3(R/{\rm AU})^{-1}$ g cm$^{-2}$, where $R$ is the radial distance from the central star. The properties of the central star assumed are the effective temperature $T_*=3,000$ K, the radius $R_*=2.0$ $R_\odot$, and the mass $M_*=0.5$ $M_\odot$. Other assumed parameters are as follows: the grazing angle $\alpha=0.05$, the visible fraction of the stellar photosphere at the annuli $f_{\rm vis}=0.5$, and the mean molecular weight $\mu_{\rm m}=7/3$.
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Figure 2 shows the vertical temperature structures of annuli with 1 AU radius with various grain sizes. We take a coordinate of the Planck mean extinction optical depth with the stellar effective temperature as the horizontal axis. Note that the maximum optical depth in each curve occurs the equatorial plane. The grain sizes assumed are shown in each panel. The solid, dotted, and dashed curves are the cases of no scattering (i.e. $\omega_0=0$), $\omega_0=0.9$, and $\omega_0=0.99$, respectively.
For no scattering cases (solid curves), the so-called two-layer structure proposed by CG97 is confirmed. The dust temperature near the surface is enhanced due to the direct stellar radiation: “super-heated layer”. The thickness of the super-heated layer is well expressed by the Planck mean extinction optical depth as $\tau_{\rm P,*}^{\rm ext}\simeq\alpha=0.05$ (grazing angle). The temperature rapidly decreases if $\tau_{\rm P,*}^{\rm ext}>\alpha$. If the interior is optically thick against its own radiation, then, the interior reaches the thermal equilibrium and becomes isothermal. As the grain size becomes larger, the temperature of the super-heated layer becomes lower. In contrast, the temperature of the interior becomes higher. The physical reason of this phenomenon will be discussed in section 3 with two analytic models: the standard two-layer model like CG97 and a newly developed three-layer model. Here, we just mention the fact that the numerical results agree with the prediction by the three-layer model for the grain size of 0.01–1 , whereas the results agree with that by the two-layer model for the size $\ga10$ .
When there is scattering, some differences appear. For $a=0.01$ (panel \[a\]), the scattering albedo $\omega$ is negligible in the wavelength interest (e.g., an effective temperature less than $T_*=3,000$ K in Figure 1). Thus, scattering virtually has no effect. For $a=0.1$ (panel \[b\]), $\omega$ for the stellar radiation is significant, but that for the diffuse radiation in the annulus (its effective temperature is less than about 300 K) is still negligible (see Figure 1). In this case, the temperature at the equatorial plane becomes slightly lower than that in the no scattering case, which is consistent with [@dul03]. For $a=1$–10 (panels \[c,d\]), $\omega$ becomes significant for the radiation of the super-heated layer. In this case, we observe a plateau like structure at around $\tau_{\rm P,*}^{\rm ext}\sim1$ and a significant reduction of the equatorial temperature. For $a\ga100$ –1 mm (panels \[e,f\]), finally, $\omega$ becomes “grey” for all the radiation considered here. In this case, the temperature structure with scattering becomes indistinguishable from that without scattering; a “grey” scattering has no effect on the temperature structure in the optical depth coordinate. The physical reasons of these features will be discussed in section 3 with an analytic model.
Even when the “grey” scattering, we find a difference of the temperature structures with/without scattering if we take the physical height as the coordinate as shown in Figure 3. The diamonds in Figure 3 indicate the lower boundary of the super-heated layer, at which the Planck mean extinction optical depth with the stellar effective temperature ($T_*=3,000$ K) is equal to the grazing angle. We find that with scattering, the height of the super-heated layer is always enhanced; scattering causes more flaring disc [see also @dul03]. This suggests that the scattering may affect the global structure of the disc, which will be discussed in a future work.
Three-layer analytic model with scattering
==========================================
In order to understand the numerical results presented in the previous section, we here develop an analytic model as an extension of the seminal two-layer model by CG97: three-layer model with scattering. To describe fluxes across the boundaries of the layers, we adopt a two-stream Eddington approximation with isotropic scattering. The notations in this section are summarised in Table 1.
Notation Meaning Remarks
------------------------ ------------------------------------------------------------------------- -------------------------------
$\alpha$ Grazing angle of the entering stellar radiation
$\Omega_*$ Solid angle of the stellar photosphere
$W_*$ Dilution factor of the stellar radiation $\Omega_*/4\pi$
$H_*^{\rm in}$ Stellar input flux
$H_x^{\rm up}$ Upwards flux from the $x$ layer
$H_x^{\rm down}$ Downwards flux from the $x$ layer
$H_{\rm input}$ Total downwards flux from the super-heated layer
$T_x$ Temperature of the $x$ layer
$B_x$ Frequency integrated Planck function with $T_x$ $(\sigma_{\rm SB}/\pi)T_x^4$
$\Sigma_x$ Gas mass column density of the $x$ layer
$\kappa^{\rm abs}_x$ Absorption cross section per unit gas mass for the radiation with $T_x$
$\kappa^{\rm ext}_x$ Extinction cross section per unit gas mass for the radiation with $T_x$
$\tau^{\rm ext}_{x,y}$ Extinction optical depth of the $x$ layer for the radiation with $T_y$ $\kappa^{\rm ext}_y \Sigma_x$
$\omega_x$ Scattering albedo for the radiation with $T_x$
$\chi_x$ Square-root of the thermal coefficient $1-\omega_x$ $\sqrt{1-\omega_x}$
$I^{\rm up}_{x,y}$ Upwards intensity of the radiation with $T_y$ from the $x$ layer
$I^{\rm down}_{x,y}$ Downwards intensity of the radiation with $T_y$ from the $x$ layer
$a_{x,y}$ Thermal coefficient in the $x$ layer for the radiation with $T_y$ eq. (B13)
$b_{x,y}$ Reflection coefficient in the $x$ layer for the radiation with $T_y$ eq. (B14)
$c_{x,y}$ Transmission coefficient in the $x$ layer for the radiation with $T_y$ eq. (B15)
$\Phi_{\rm input}$ Ratio of $H_{\rm input}$ to $H_*^{\rm in}$ eq. (20)
$\Phi_{\rm i(2)}$ Reduction factor of $B_{\rm i}$ by scattering in the two-layer model eq. (23)
$\Phi_{\rm i(3)}$ Reduction factor of $B_{\rm i}$ by scattering in the three-layer model eq. (29)
Subscripts $x$ and $y$ are \* for stellar quantities, s for super-heated layer quantities, m for middle layer quantities, or i for interior quantities.\
$\sigma_{\rm SB}$ is the Stefan-Boltzmann constant.
Model description
-----------------
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Suppose two or three layers in an annulus as shown in Figure 4. We assume that each layer is isothermal with the temperature determined by the radiation equilibrium in the layer. Then, we consider that each layer emits the radiation characterised by its temperature and other layers just work as absorption and scattering media for the radiation. The radiation from the central star is characterised by the stellar effective temperature. The characterisation of the radiation is done by the Planck mean and the characteristic frequencies are denoted by each subscript such as “\*” for the stellar radiation (see the caption of Figure 4 and Table 1). We denote, for example, the extinction cross section characterised by the stellar effective temperature as $\kappa_*^{\rm ext}$. Note that all the quantities depending on the frequency are Planck averaged in this section.
We call the two or three layers super-heated layer, middle layer, and interior as shown in Figure 4. The super-heated layer is defined as only the layer exposed by the direct stellar radiation entering into the annulus with a small grazing angle $\alpha$. The optical thickness of the layer is $\approx\alpha$ as shown by the numerical solutions in §2.2. Thus, we define the thickness of the super-heated layer as $\tau_{\rm s,*}^{\rm ext}\equiv\kappa_*^{\rm ext}\Sigma_{\rm s}=\alpha$, where $\Sigma_{\rm s}$ is the gas column density of the layer. The interior represents the isothermal part found in the numerical solutions. Thus, the boundary can be defined by the photosphere of its own radiation. However, we here simply define the interior as the part other than the super-heated and the middle layers.
The middle layer is introduced by the following consideration. When the opacity coefficient decreases as the wavelength increases, the absorption of the radiation from the warm super-heated layer occurs well above the photosphere of the cold interior radiation. In this case, the interior is not warmed directly by the super-heated layer but by the “middle” layer where the radiation of the super-heated layer is effectively absorbed. We here define the thickness of the middle layer as $\tau_{\rm m,s}^{\rm ext}\equiv\kappa_{\rm s}^{\rm ext}\Sigma_{\rm m}=1$ with the gas column density of the middle layer $\Sigma_{\rm m}$ although this definition is rather arbitrary. On the other hand, when the opacity coefficient is grey, the middle layer with the above thickness is optically thick for its own radiation. Thus, the middle layer reaches the thermal equilibrium and is merged into the isothermal interior. In this case, we do not need to consider the middle layer. Therefore, we have two cases: the three-layer model when $\tau_{\rm m,m}^{\rm ext}\equiv\kappa_{\rm m}^{\rm ext}\Sigma_{\rm m}<1$ and the two-layer model when $\tau_{\rm m,m}^{\rm ext}\simeq1$ (or $\tau_{\rm m,i}^{\rm ext}\equiv\kappa_{\rm i}^{\rm ext}\Sigma_{\rm m}\simeq1$ because the middle layer is merged into the interior). In other words, three layers are needed when $\kappa_{\rm m}^{\rm ext}/\kappa_{\rm s}^{\rm ext}<1$ and two layers are sufficient when $\kappa_{\rm i}^{\rm ext}/\kappa_{\rm s}^{\rm ext}\simeq1$. Tables 2 and 3 are summaries of the thickness of the upper two layers and the condition of the two or three-layer models. Importantly, the seminal two-layer model is valid only when the opacity coefficient is grey in the frequencies interest. This fact has not seemed to be known well so far.
-------------------- ------------------------------------------------------------------------------------
Super-heated layer $\tau^{\rm ext}_{\rm s,*}=\alpha$, i.e. $\Sigma_{\rm s}=\alpha/\kappa^{\rm ext}_*$
Middle layer $\tau^{\rm ext}_{\rm m,s}=1$, i.e. $\Sigma_{\rm m}=1/\kappa^{\rm ext}_{\rm s}$
-------------------- ------------------------------------------------------------------------------------
: Thickness of the upper two layers.
------------------- ------------------------------------------------------------
Three-layer model $\kappa^{\rm ext}_{\rm m}/\kappa^{\rm ext}_{\rm s}<1$
Two-layer model $\kappa^{\rm ext}_{\rm i}/\kappa^{\rm ext}_{\rm s}\simeq1$
------------------- ------------------------------------------------------------
: Condition of the two or three-layer models.
### Stellar fluxes
When the grazing angle $\alpha$ is small, the stellar flux at the top of the annulus is $$H_*^{\rm in} = \alpha W_* B_*\,,$$ with the integrated Planck function $B_*=(\sigma_{\rm SB}/\pi)T_*^4$ and the dilution factor $W_*=\Omega_*/4\pi$, where $\Omega_*$ is the solid angle of the stellar photosphere from the top of the annulus. If only the fraction $f_{\rm vis}$ of the stellar photosphere is visible because of the optically thick disc, the solid angle becomes $\Omega_*=f_{\rm vis}\pi(R_*/R)^2$, where $R_*$ is the stellar radius and $R$ is the radius of the annulus.
When there is scattering, a part of the incident stellar flux is reflected upwards and downwards by the super-heated layer. [@cal91] presented an analytic expression of the scattered flux for isotropic scattering. From equation (5) in [@cal91], the outbound fluxes at the upper and lower boundaries of the super-heated layer (see Figure 4) become $$H_*^{\rm up} = \alpha W_* B_*
\left[\frac{\omega_*}{1+\chi_*}\right]\,,$$ and $$H_*^{\rm down} = \alpha W_* B_*
\left[\frac{\omega_* \chi_*}{1+\chi_*}\right]\,,$$ where $\omega_*$ is the single scattering albedo at the stellar frequency and $\chi_*=\sqrt{1-\omega_*}$. In the derivation of equations (6) and (7), we have assumed $\tau_{\rm s,*}^{\rm ext}\equiv\kappa_*^{\rm ext}\Sigma_{\rm s}=\alpha\ll1$, adopted a different upper boundary condition from [@cal91][^2], and neglected the term $e^{-1}$ for the downwards flux. Note that the total scattered flux is $H^{\rm up}_* + H^{\rm down}_*
= \omega_* \alpha W_* B_* = \omega_* H^{\rm in}_*$ and $H_*^{\rm up}=H_*^{\rm down}=0$ if $\omega_*=0$ (no scattering case).
### Super-heated layer fluxes
The radiation characterised by the temperature of the super-heated layer $T_{\rm s}$ is produced only in the super-heated layer. In other layers, this radiation is not produced but is absorbed or scattered. The super-heated layer is vertically optically thin for its own radiation as $\tau_{\rm s,s}^{\rm ext}\equiv\kappa_{\rm s}^{\rm ext}\Sigma_{\rm s}
=(\kappa_{\rm s}^{\rm ext}/\kappa_*^{\rm ext})\alpha\ll1$ because the grazing angle $\alpha$ is small and we have $(\kappa_{\rm s}^{\rm ext}/\kappa_*^{\rm ext})\le1$. On the other hand, the total optical depth of other layers is very large. Thus, we consider a geometry that a thin isothermal layer lies on a semi-infinite absorption and scattering slab.
The above of the super-heated layer is assumed to be vacuum; there is no downwards input radiation with the temperature $T_{\rm s}$ at the top of the layer. However, there is upwards input radiation at the bottom of the layer because of the reflection by the semi-infinite interior below the layer. The upwards and downwards radiation intensities from the super-heated layer ($I^{\rm up}_{\rm s,s}$ and $I^{\rm down}_{\rm s,s}$, respectively) in the two-stream Eddington approximation (the cosine of the angle between the stream lines and the normal of the layer is set to be $\pm1/\sqrt{3}$) become from equations (B12) and (B16) $$I^{\rm up}_{\rm s,s} = a_{\rm s,s} B_{\rm s}
+ c_{\rm s,s} I^{\rm up}_{\rm i,s}\,,$$ and $$I^{\rm down}_{\rm s,s} = a_{\rm s,s} B_{\rm s}
+ b_{\rm s,s} I^{\rm up}_{\rm i,s}\,,$$ where $I^{\rm up}_{\rm i,s}$ is the upwards input intensity from the interior, and $a_{\rm s,s}$, $b_{\rm s,s}$, and $c_{\rm s,s}$ are the thermal, reflection, and transmission coefficients in the super-heated layer for the radiation characterised by the temperature $T_{\rm s}$ (see eqs. \[B13–B15\]). The reflected intensity from the interior becomes $$I^{\rm up}_{\rm i,s}=b_{\rm i,s}I^{\rm down}_{\rm s,s}\,,$$ where $b_{\rm i,s}$ is the reflection coefficient in the interior for the radiation with $T_{\rm s}$. Note that the interior itself does not emit radiation with $T_{\rm s}$.
We can always obtain the unique exact solution of $I^{\rm up}_{\rm s,s}$, $I^{\rm down}_{\rm s,s}$, and $I^{\rm up}_{\rm i,s}$ from equations (8–10), whereas we derive an approximate solution of them in this paper. As found equation (B18), for a semi-infinte slab, we have $b_{\rm i,s}\approx(1-\chi_{\rm s})/(1+\chi_{\rm s})$, where $\chi_{\rm s}=\sqrt{1-\omega_{\rm s}}$ with $\omega_{\rm s}$ being the scattering albedo for the radiation with $T_{\rm s}$. Since the super-heated layer is optically thin for its own radiation, we can approximate $b_{\rm s,s}\ll1$ and $c_{\rm s,s}\approx1$. We also find $a_{\rm s,s}\approx\sqrt{3}\chi_{\rm s}^2\tau_{\rm s,s}^{\rm ext}
=\sqrt{3}\chi_{\rm s}^2(\kappa_{\rm s}^{\rm ext}/\kappa_*^{\rm ext})\alpha$ from equation (B13) for a small optical depth. Then, we obtain $I^{\rm up}_{\rm s,s}\approx (1+b_{\rm i,s})a_{\rm s,s}B_{\rm s}$, $I^{\rm down}_{\rm s,s}\approx a_{\rm s,s}B_{\rm s}$, and $I^{\rm up}_{\rm i,s}\approx b_{\rm i,s}a_{\rm s,s}B_{\rm s}$. The upwards and downwards fluxes from the super-heated layer are $H^{\rm up}_{\rm s}=I^{\rm up}_{\rm s,s}/(2\sqrt{3})$ and $H^{\rm down}_{\rm s}=(I^{\rm down}_{\rm s,s}-I^{\rm up}_{\rm i,s})/(2\sqrt{3})$. Therefore, we obtain $$H_{\rm s}^{\rm up} = \alpha B_{\rm s}
\left(\frac{\kappa_{\rm s}^{\rm ext}}{\kappa_*^{\rm ext}}\right)
\left[\frac{\chi_{\rm s}^2}{1+\chi_{\rm s}}\right]\,,$$ and $$H_{\rm s}^{\rm down} = \alpha B_{\rm s}
\left(\frac{\kappa_{\rm s}^{\rm ext}}{\kappa_*^{\rm ext}}\right)
\left[\frac{\chi_{\rm s}^3}{1+\chi_{\rm s}}\right]\,.$$
### Middle layer fluxes
As discussed in the section 3.1, we consider the middle layer only when the layer is optically thin for its own radiation. Although the middle layer is sandwiched between the super-heated layer and the interior, we neglect the effect of the super-heated layer because the optical depth of the layer is very small (see above). The optical depth of the interior is so large that we can regard it as a semi-infinite medium. Thus, we have the same setting as the super-heated layer, other than the optical thickness of the layer, $\tau_{\rm m,m}^{\rm ext}
=(\kappa_{\rm m}^{\rm ext}/\kappa_{\rm s}^{\rm ext})$. Following the section 3.1.2, we have $$H_{\rm m}^{\rm up} = B_{\rm m}
\left(\frac{\kappa_{\rm m}^{\rm ext}}{\kappa_{\rm s}^{\rm ext}}\right)
\left[\frac{\chi_{\rm m}^2}{1+\chi_{\rm m}}\right]\,,$$ and $$H_{\rm m}^{\rm down} = B_{\rm m}
\left(\frac{\kappa_{\rm m}^{\rm ext}}{\kappa_{\rm s}^{\rm ext}}\right)
\left[\frac{\chi_{\rm m}^3}{1+\chi_{\rm m}}\right]\,,$$ where $B_{\rm m}$ is the integrated Planck function with the temperature of the middle layer $T_{\rm m}$, $\omega_{\rm m}$ is the scattering albedo for the radiation of the middle layer, and $\chi_{\rm m}=\sqrt{1-\omega_{\rm m}}$.
### Interior flux
We always consider the interior to be optically thick for its own radiation. On the other hand, other layers above the interior are considered to be always optically thin. If we neglect the effect of the upper layers, the interior is regarded as a semi-infinite isothermal medium without incident flux of the radiation with the temperature $T_{\rm i}$. In this case, based on equation (B12), the upwards outbound flux becomes $H_{\rm i}^{\rm up} = a_{\rm i,i} B_{\rm i} / (2\sqrt{3})$ with $a_{\rm i,i}$ being the thermal coefficient in the interior for the radiation with $T_{\rm i}$ and $B_{\rm i}$ being the integrated Planck function with $T_{\rm i}$. For a semi-infinite medium, $a_{\rm i,i}\approx2\chi_{\rm i}/(1+\chi_{\rm i})$, where $\chi_{\rm i}=\sqrt{1-\omega_{\rm i}}$ with $\omega_{\rm i}$ being the single scattering albedo for the interior radiation (eq. \[B17\]). Therefore, we have $$H_{\rm i}^{\rm up}=\frac{B_{\rm i}}{\sqrt{3}}
\left[\frac{\chi_{\rm i}}{1+\chi_{\rm i}}\right]\,.$$ Note that the factor in $[~~~]$ is equal or less than 1/2. The equal is true without scattering. Therefore, in general, scattering reduces the radiation energy loss from the interior, i.e. the green-house effect.
Temperature of the super-heated layer
-------------------------------------
The radiation energy conservation in the super-heated layer is $$H_*^{\rm in} - H_*^{\rm up} - H_*^{\rm down}
= H_{\rm s}^{\rm up} + H_{\rm s}^{\rm down}\,.$$ The left-hand side is the net input flux of the stellar radiation and becomes $\alpha W_* B_* (1-\omega_*)$. When there is scattering, the input flux is reduced by a factor of $1-\omega_*$. The right-hand side is the net output flux of the super-heated layer and becomes $\alpha B_{\rm s}(\kappa_{\rm s}^{\rm ext}/\kappa_*^{\rm ext})
(1-\omega_{\rm s})$. Thus we obtain $$B_{\rm s} = B_* W_*
\left(\frac{\kappa_*^{\rm abs}}{\kappa_{\rm s}^{\rm abs}}\right)\,,$$ or $$T_{\rm s} = T_* W_*^{1/4}
\left(\frac{\kappa_*^{\rm abs}}{\kappa_{\rm s}^{\rm abs}}\right)^{1/4}\,,$$ where we have applied $\kappa^{\rm abs}=(1-\omega) \kappa^{\rm ext}$. Therefore, we expect $T_{\rm s}$ to be independent of the existence of scattering. This result can be recovered by a microscopic consideration for each dust grain in the radiation equilibrium with the stellar radiation field in the super-heated layer. We also expect that $T_{\rm s}$ shows two asymptotic values since $(\kappa_*^{\rm abs}/\kappa_{\rm s}^{\rm abs})=T_*/T_{\rm s}$ for a small grain size and $(\kappa_*^{\rm abs}/\kappa_{\rm s}^{\rm abs})={\rm constant}$ for a large grain size as shown in Figure 1.
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Figure 5 shows the temperature of the super-heated layer as a function of grain size. The numerical solutions presented in section 2 are shown by symbols: circles for the no scattering case, triangles for the $\omega_0=0.9$ case, and squares for the $\omega_0=0.99$ case. The analytic model presented in equation (18) is shown by the solid lines. To obtain the solution $T_{\rm s}$ of equation (18), we need an iterative procedure because $\kappa^{\rm abs}_{\rm s}$ depends on $T_{\rm s}$. Figure 5 first shows that the scattering hardly affect $T_{\rm s}$ as expected by the analytic model. Indeed, the cases of the three different albedos are superposed for almost all grain sizes, although a temperature enhancement of a few percent is observed in the scattering cases at a grain size of 0.1–1 . Figure 5 second shows that the numerical solutions can be divided into two cases: higher temperature for smaller grain size ($\la 0.1$ ) and lower temperature for larger grain size ($\ga 10$ ), which is also expected by the analytic model. The agreement between the numerical solutions and the analytic model is excellent although the numerical solutions give about 3% higher temperature than the analytic model. This small difference is probably caused by neglecting the absorption of the interior radiation in the super-heated layer in the analytic model.
Temperature of the interior
---------------------------
We can define the radiation flux input into the interior (including the middle layer) of the annulus as $$H_{\rm input} \equiv H_*^{\rm down} + H_{\rm s}^{\rm down}
= \alpha W_* B_* \Phi_{\rm input}\,,$$ where $$\Phi_{\rm input} = \frac{\omega_*\chi_*}{1+\chi_*}
+ \frac{\chi_*^2 \chi_{\rm s}}{1+\chi_{\rm s}}\,,$$ and we have eliminated $B_{\rm s}$ by equation (17). If there is no scattering ($\omega_*=\omega_{\rm s}=0$, i.e. $\chi_*=\chi_{\rm s}=1$), we have $\Phi_{\rm input}=1/2$. That is, $H_{\rm input}=\alpha W_* B_*/2=H_*^{\rm in}/2$; a half of the radiation energy received at the top of the annulus is input into the interior [@cg97]. For the isotropic scattering case, we always have $H_{\rm input}<H_*^{\rm in}/2$; isotropic scattering always reduces the energy input into the interior relative to the no scattering case [@dul03].
### Two-layer model
The energy conservation in the interior under the two-layer model is $$H_{\rm input} = H_{\rm i}^{\rm up}\,.$$ From equations (15) and (19), we obtain $$B_{\rm i} = \sqrt{3} \alpha W_* B_* \Phi_{\rm i(2)}\,,$$ where $$\Phi_{\rm i(2)} = \left(\frac{1+\chi_{\rm i}}{\chi_{\rm i}}\right)
\Phi_{\rm input}\,.$$ The interior temperature becomes $$T_{\rm i} = T_* \left[\sqrt{3}\alpha W_* \Phi_{\rm i(2)}\right]^{1/4}\,.$$ Therefore, $T_{\rm i}$ is proportional to a factor of $\Phi_{\rm i(2)}^{1/4}$. When there is no scattering ($\Phi_{\rm input}=1/2$ and $\chi_{\rm i}=1$), we have $\Phi_{\rm i(2)}=1$.
As discussed in section 3.1, the two-layer model is valid if the opacity coefficient is regarded as grey at all the frequencies interest: $\kappa_{\rm i}^{\rm ext}/\kappa_{\rm s}^{\rm ext}\simeq1$. If this condition is satisfied, the scattering is also grey; $\omega_*=\omega_{\rm s}=\omega_{\rm i}$ (i.e. $\chi_*=\chi_{\rm s}=\chi_{\rm i}$). In this case, we have $$\Phi_{\rm i(2)} = 1~~~{\rm (no/grey~scattering)}\,.$$ Importantly, the factor $\Phi_{\rm i(2)}$ for grey isotropic scattering is independent of the albedo and equal to the case without scattering. This is caused by the fact that the reduction of the flux input into the interior by scattering ($\Phi_{\rm input}$ in eq.\[19\]) is completely offset by the reduction of the flux outbound from the interior by scattering ($(1+\chi_{\rm i})/\chi_{\rm i}$ in eq.\[15\]) for grey and isotropic scattering. Therefore, we expect the same interior temperature for grey and isotropic scattering case as that for no scattering case in the two-layer model.
### Three-layer model
The energy conservations in the middle layer and the interior under the three-layer model are $$H_{\rm input} = H_{\rm m}^{\rm up} + H_{\rm m}^{\rm down}\,,$$ and $$H_{\rm m}^{\rm down} = H_{\rm i}^{\rm up}\,.$$ Since $H_{\rm m}^{\rm down} = \chi_{\rm m} H_{\rm m}^{\rm up}$ from equations (13) and (14), we obtain $H_{\rm i}^{\rm up}=\chi_{\rm m}/(1+\chi_{\rm m}) H_{\rm input}$. Thus, $$B_{\rm i} = \sqrt{3} \alpha W_* B_* \Phi_{\rm i(3)}\,,$$ where $$\Phi_{\rm i(3)} = \left(\frac{\chi_{\rm m}}{\chi_{\rm i}}\right)
\left(\frac{1+\chi_{\rm i}}{1+\chi_{\rm m}}\right) \Phi_{\rm input}\,.$$ The interior temperature becomes $$T_{\rm i} = T_* \left[\sqrt{3}\alpha W_* \Phi_{\rm i(3)}\right]^{1/4}\,.$$ Therefore, $T_{\rm i}$ is proportional to a factor of $\Phi_{\rm i(3)}^{1/4}$.
From the discussion in section 3.1, the three layer model is valid when $\kappa_{\rm m}^{\rm ext}/\kappa_{\rm s}^{\rm ext}<1$. This condition is satisfied when the grain size, $a$, is smaller than 10–100 . When $a\la0.01$ , the scattering albedo is negligible at the all frequencies interest. In this case or the no scattering case ($\omega_x=0$, i.e. $\chi_x=1$), we have $$\Phi_{\rm i(3)} = \frac{1}{2}~~~{\rm (no~scattering)}\,.$$ Comparing this with the two-layer model, we find that $T_{\rm i}$ in the three-layer model is reduced by a factor of $(1/2)^{1/4}$ relative to that in the two-layer model even for no scattering case.
When $0.01\la a \la0.1$ , $\omega_{\rm i}\approx\omega_{\rm m}\approx\omega_{\rm s}\approx0$ ($\chi_{\rm i}\approx\chi_{\rm m}\approx\chi_{\rm s}\approx1$) but $\omega_*>0$ ($\chi_*<1$). In this case, we have $$\Phi_{\rm i(3)} = \frac{\chi_*}{2}(2-\chi_*)\,.$$ Thus, we find $\Phi_{\rm i(3)}<1/2$; we expect a reduction of $T_{\rm
i}$ relative to that without scattering. When $0.1\la a \la1$–10 , $\omega_{\rm i}\approx\omega_{\rm m}\approx0$ ($\chi_{\rm i}\approx\chi_{\rm m}\approx1$) and $\omega_{\rm s}\approx\omega_*>0$ ($\chi_{\rm s}\approx\chi_*<1$). In this case, we have $$\Phi_{\rm i(3)} = \frac{\chi_*}{1+\chi_*}\,.$$ When 1–$10\la a \la10$–100 , $\omega_{\rm i}\approx0$ ($\chi_{\rm i}\approx1$) and $\omega_{\rm m}\approx\omega_{\rm s}\approx\omega_*>0$ ($\chi_{\rm m}\approx\chi_{\rm s}\approx\chi_*<1$). Then, we have $$\Phi_{\rm i(3)} = 2\left(\frac{\chi_*}{1+\chi_*}\right)^2\,.$$ When $a\ga10$–100 , the opacity becomes almost grey, thus, the three-layer model is no longer valid. We should choose the two-layer model in this case.
{width="7cm"}
Figure 6 shows the scattering reduction factor $\Phi_{\rm i,(3)}$ given by equations (32–34) as a function of the albedo for the stellar radiation. We find that the factor decreases from equation (32) to (34), in other words, as a function of the grain size. The factor $\Phi_{\rm i,(3)}^{1/4}$ gives the reduction of the interior temperature, $T_{\rm i}$, in the three-layer model by isotropic scattering relative to that in the two-layer model without scattering. For typical grain sizes found in protoplanetary discs of 0.1–10 , equation (33) would give a good approximation for the reduction factor. If $\omega_*\approx1$, equations (32) and (33) are reduced to $\approx\chi_*=\sqrt{1-\omega_*}$. In this case, we expect the reduction factor of $T_{\rm i}$ to be $\approx(1-\omega_*)^{1/8}$ which is found in Figure 7 later.
### Comparison between numerical and analytic results
{width="7cm"}
Figure 7 shows the temperature of the interior as a function of the grain size. The numerical solutions presented in section 2 are shown by symbols: circles for the no scattering case, triangles for the $\omega_0=0.9$ case, and squares for the $\omega_0=0.99$ case. The analytic models are shown by the solid lines. In analytic models, we always assume the optically thick interior. Most of the numerical solutions shown in Figure 7 are really optically thick. However, when the grain size is larger than about 100 , because of the reduction of $\kappa_0^{\rm abs}$ given by equation (4), some cases without scattering and with $\omega_0=0.9$ become optically thin. In such cases, we find relatively high temperatures.
The solutions of the analytic models are obtained as follows: for the two-layer model, we solved equation (24) to obtain $T_{\rm i}$. We need an iterative procedure because the term $\Phi_{\rm i(2)}$ depends on $T_{\rm i}$. For the three-layer model, we obtained $T_{\rm i}$ from equation (30) after obtaining $T_{\rm m}$ from $$T_{\rm m} = \left[\alpha W_* B_*
\left(\frac{\kappa_{\rm s}^{\rm ext}}{\kappa_{\rm m}^{\rm abs}}\right)
\Phi_{\rm input}\right]^{1/4}\,,$$ which is derived from equations (13), (14), and (26). We again need an iterative procedure to obtain both of $T_{\rm m}$ and $T_{\rm i}$.
As discussed in §3.1, the two-layer model is valid only when $\kappa_{\rm i}^{\rm ext}/\kappa_{\rm s}^{\rm ext}\approx1$. Otherwise, we should adopt the three-layer model. Here, we connect these two models at the grain size where $\kappa_{\rm i}^{\rm ext}/\kappa_{\rm s}^{\rm ext}=0.8$ in the two-layer model, for example; we adopt the two-layer model for a larger grain size than it and adopt the three-layer model for a smaller grain size than it. This threshold is rather arbitrary, but we find that this choice is good as seen in Figure 7 where the connected analytic models reproduce the numerical solutions reasonably well. Note that a sudden jump on the solid lines in Figure 7 is caused by this connection and numerical solutions also show relatively rapid change of the temperature around there.
When there is no scattering, we find two asymptotic values of $T_{\rm i}$. Interestingly, $T_{\rm i}$ for smaller grain size is lower than that for larger grain size, whereas $T_{\rm s}$ show the opposite trend in Figure 5. Note that the radiation flux input from the super-heated layer is in fact independent of $T_{\rm s}$ as shown in equation (20) in the no scattering case. Nevertheless, the numerical solutions show a factor of $(1/2)^{1/4}$ reduction of $T_{\rm i}$ for small grain cases. This is excellently explained by the three-layer model as found in equation (31). The physical explanation is as follows: when the grain size is small, the radiation flux from the super-heated layer is absorbed by the middle layer once. Then, the middle layer emits a half of the absorbed energy downwards but the rest of the half goes upwards. Therefore, the interior does not receive a half of the stellar radiation energy absorbed by the super-heated layer but receive only a quarter of the energy when the grain size is small.
When there is isotropic scattering, we find further reduction of $T_{\rm i}$ in a range of the grain size $a=0.1$–10 or 100 . For $a\sim0.01$ , the scattering effect is not observed because of negligible albedo. For $a\ga10$–100 , the scattering effect is not observed, either (but except for the optically thin cases). This is nicely explained by the two-layer model with grey isotropic scattering as found in equation (25). The physical reason is that the grey isotropic scattering equally reduces the downwards flux from the super-heated layer (eq. \[19\]) and the upwards flux from the interior (eq. \[15\]). The reduction of $T_{\rm i}$ due to scattering for $a=0.1$–10 is explained by the three-layer model as summarised in equations (32), (33), and (34). The physical reason of the $T_{\rm i}$ reduction is that the scattering reduces only the downwards fluxes from the super-heated layer and from the middle layer because the scattering albedo at the frequency of the interior radiation is still negligible.
Conclusion
==========
We have examined the effect of scattering of the diffuse radiation on the vertical temperature structure of protoplanetary discs. This is motivated by the fact that scattering albedo increases as the size of dust grains grows in the discs. In particular, large icy grains have a significant albedo even in the infrared wavelength. For this aim, we have developed a 1D plane-parallel numerical radiation transfer code including isotropic scattering of the diffuse radiation as well as that of the incident radiation. We have also developed an analytic model with isotropic scattering both of the diffuse and the incident radiations in order to interpret the solutions obtained from the numerical simulation. All results of the numerical simulations has been nicely reproduced by the analytic model.
The analytic model presented in this paper is an extension of the seminal two-layer model by [@cg97]; we have introduced a new layer between the super-heated surface layer and the disc interior. This middle layer (or disc atmosphere) is required when the absorption of the radiation from the super-heated layer occurs well above the photosphere of the opaque isothermal interior. This situation is realised if the dust opacity is negatively proportional to the wavelength. Thus, we should consider three layers rather than two layers if the grain size is smaller than about 10 . On the other hand, for grey opacity, which is realised if the grain size is $\ga10$ , the standard treatment with two layers is justified.
We have found from the numerical simulation that the dust temperature of the disc surface is almost not affected by scattering. This is because the temperature is determined by the radiation equilibrium of grains in the incident radiation field locally. The grain size has an effect on the surface temperature via the wavelength dependence of the dust opacity. There are two asymptotic temperatures: the higher temperature is realised by small grains ($\la0.1$ ) and the lower temperature is realised by large grains ($\ga1$–10 ). This is because the emission efficiency relative to absorption efficiency of the small grains is smaller than that of the large grains which have grey opacity. This trend of the numerical solutions is excellently reproduced by the analytic model of the super-heated layer.
The numerical simulations without scattering show that the dust temperature of the optically thick interior also has two asymptotic values: the lower temperature for small grains ($\la0.1$ ) and the higher temperature for large grains ($\ga10$–100 ). Thus, the trend is opposite from the surface temperature. The higher asymptotic temperature for large grains is well matched with the prediction of the two-layer model. On the other hand, the lower asymptotic temperature for small grains is a factor of $(1/2)^{1/4}$ lower than the prediction of the two-layer model. In fact, the flux input from the super-heated layer is always same although the interior temperature is different as a function of the grain size. This phenomenon has been already found by [@dul03a] who attributed it to the energy loss from the interior by the radiation at a long wavelength [see also @dul02]. We have proposed another interpretation by the middle layer between the super-heated layer and the interior; the super-heated layer gives a half of the absorbed energy of the incident radiation to the middle layer which gives a half of the obtained energy to the interior. This three-layer model exactly predicts a factor of $(1/2)^{1/4}$ reduction of the interior temperature for the small grain case.
The numerical simulations with isotropic scattering also show two asymptotic temperatures of the interior. Interestingly, these asymptotic temperatures of no scattering and isotropic scattering cases are the same. For small grains ($\la0.1$ ), since the scattering albedo is negligible for all wavelengths interest, the same temperature is trivial. The same temperature for large grains ($\ga10$–100 ) has been nicely explained by the two-layer model with grey opacity. The physical mechanism is the exact offset between the reduction of the flux input into the interior by scattering in the super-heated layer and the reduction of the flux output from the interior by scattering in itself (i.e. green-house effect).
For grain sizes of 0.1–10 , which are expected by a moderate growth of the size in the discs, we have found a further reduction of the interior temperature in isotropic scattering cases relative to that without scattering from the numerical simulations. This reduction has been well explained by the three-layer analytic model. The physical mechanism is the wavelength dependence of albedo; the flux input into the interior is reduced by scattering in the super-heated and middle layers, whereas the flux output from the interior is not reduced because of negligible (or weak) scattering. Note that the interior flux has a longer wavelength typically, thus, the albedo for the radiation is smaller.
In conclusion, the scattering of the diffuse radiation can affect the vertical temperature structure of protoplanetary discs significantly when the grain size grows to be about 1–10 . We need to investigate the effect in a global disc model in future. The analytic model presented in this paper could be useful to understand the physics determining the temperature structure in the discs.
Acknowledgments {#acknowledgments .unnumbered}
===============
We would appreciate comments from the anonymous referee which were very useful for us to improve the quality of this paper. AKI is grateful to all members of the Department of Physics, Nagoya University, especially the $\Omega$ Laboratory led by Tsutomu T.Takeuchi, for their hospitality during this work. AKI is supported by KAKENHI (the Grant-in-Aid for Young Scientists B: 19740108) by The Ministry of Education, Culture, Sports, Science and Technology (MEXT) of Japan.
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Variable Eddington factor method with isotropic scattering in plane-parallel slab
=================================================================================
We here present our numerical radiation transfer method in a plane-parallel medium in detail. The method is based on that developed by [@dul02], but is extended to treat isotropic scatterings of both of the incident radiation (from the central star) and the diffuse radiation (from dust grains in the medium). We find a solution, in which the radiation field, the temperature structure, and the density structure are consistent with each other, iteratively as the following procedure:
1. assuming an initial temperature structure
2. solving the density structure consistent with the given temperature structure under the hydrostatic equilibrium
3. solving the transfer of the incident (stellar) radiation with the grazing angle recipe
4. solving the transfer of the diffuse radiation with a variable Eddington factor method and obtaining the temperature structure under the radiation equilibrium
5. checking the convergence of the temperature structure and if not going back to the step (ii)
In the following we describe the set of equations and assumptions and the result of a benchmark test.
Hydrostatic equilibrium
-----------------------
Suppose an annulus clipped from a protoplanetary disk to be a plane-parallel medium. We set the coordinate $z$ as the vertical height of the medium. The origin $z=0$ is the equatorial plane of the annulus and we set a mirror boundary condition there. Assuming the vertical hydrostatic equilibrium, we obtain the density of gas in the medium $\rho(z)$ consistent with the temperature structure $T(z)$ which is assumed as an initial guess or is obtained by the previous step of the iteration. We assume that the gas temperature is the same as the dust temperature which is determined by the radiation equilibrium. This assumption is usually well established in the protoplanetary disc because the collision between gas particles and dust grains occurs enough frequently.
The vertical hydrostatic equilibrium is given by $$\frac{dP}{dz} = -\rho g\,,$$ where $P$ is the gas pressure and $g$ is the gravitational acceleration. In a protoplanetary disc, the self-gravity of the disc is negligible relative to the gravity of the central star. Thus, we have $g=GM_*z/R^3$, where $G$ is the gravitational constant, $M_*$ is the mass of the central star, and $R$ is the distance from the star (or radius of the annulus considered). We have assumed $R\gg z$. The gas pressure is given by the equation of state for the ideal gas as $P=(\rho k_{\rm B}T)/(\mu_{\rm m} m_{\rm p})$, where $\mu_{\rm m}$ is the mean molecular weight, $m_{\rm p}$ is the proton mass, and $k_{\rm B}$ is the Boltzmann constant. Then, equation (A1) is reduced to $$\frac{d\ln \rho}{dz} = -\left(\frac{\mu m_{\rm p}}{k_{\rm B}T}\right)
\left(\frac{GM_*}{R^3}\right)z + \frac{d\ln T}{dz}\,.$$ If we integrate equation (A2) from $z=0$ with a given $T(z)$, we obtain the functional shape of $\rho(z)$. The absolute value of $\rho(z)$ is scaled by $$\Sigma = 2\int_0^{z_{\rm max}} \rho(z) dz\,,$$ where $\Sigma$ is the gas column density and $z_{\rm max}$ is the maximum height for the numerical calculation. We set $z_{\rm max}=R$ in our calculation and we set the minimum value of $\rho=10^{-25}$ g cm$^{-3}$ just for avoiding a too small value of the density.
We set the optical depth coordinate $\tau$ for the radiation transfer from the obtained $\rho(z)$ as $$\tau_\nu(z) = \int_z^{z_{\rm max}} \rho(z) \kappa_\nu^{\rm ext} dz\,,$$ where $\kappa_\nu^{\rm ext}$ is the extinction cross section by dust grains per unit gas mass at the frequency $\nu$.
In fact, we can obtain the temperature structure as a function of the optical depth $T(\tau)$ by the radiation transfer without the density structure as a function of the vertical height $\rho(z)$. The reason why we calculate $\rho(z)$ and the relation between the optical depth and the vertical height $\tau(z)$ is to see $T(z)$ as shown in Figure 3. Another reason, which may be more important, is to determine the grazing angle consistent with the disc global structure for future calculations.
Transfer of the incident radiation
----------------------------------
Let us consider an incident radiation beam entering the plane-parallel medium. The angle between the incident ray and the surface of the medium is called the grazing angle, $\alpha$. In a protoplanetary disc, the grazing angle $\alpha$ is usually as small as $\sim0.05$ radian [e.g., @dal06]. Thus, we use the approximation $\sin\alpha\approx\alpha$. The optical depth along the incident ray becomes $\tau_\nu(z)/\alpha$. Thus, the mean intensity of the (direct) incident radiation is given by $$J^*_\nu(z) = J^{*{\rm max}}_\nu e^{-\tau_\nu(z)/\alpha}\,,$$ where $J^{*{\rm max}}_\nu$ is the mean intensity at the top of the medium (i.e. $z=z_{\rm max}$). For the incident radiation from the central star, $J^{*{\rm max}}_\nu=B_\nu(T_*) \Omega_*/4\pi$, where $B_\nu(T_*)$ is the Planck function with the stellar effective temperature $T_*$ and $\Omega_*=\pi(R_*/R)^2 f_{\rm vis}$ is the solid angle of the stellar photosphere visible from the top of the medium ($f_{\rm vis}$ is the visible fraction of the stellar photosphere). Finally, we give the absorbed and extincted energy density of the incident radiation per unit time interval at the height $z$ as $$q_{\rm abs}(z)=\int_0^\infty \rho(z)\kappa_\nu^{\rm abs}4\pi J_\nu^* d\nu\,,$$ and $$q_{\rm ext}(z)=\int_0^\infty \rho(z)\kappa_\nu^{\rm ext}4\pi J_\nu^* d\nu\,,$$ where $\kappa_\nu^{\rm abs}$ and $\kappa_\nu^{\rm ext}$ are the absorption and the extinction cross section by dust per unit gas mass at the frequency $\nu$.
Transfer of the diffuse radiation
---------------------------------
The transfer equation of the diffuse radiation (or the radiation reprocessed by dust) in a plane-parallel medium is $$\mu\frac{dI_{\nu\mu}}{dz} =
-\rho\kappa_\nu^{\rm ext}I_{\nu\mu}
+\rho\kappa_\nu^{\rm ext}S_\nu\,,$$ where $\mu$ is the cosine of the angle between the ray and the $z$ coordinate, $I_{\nu\mu}$ is the specific intensity at the frequency $\nu$ towards the direction $\mu$, and $S_\nu$ is the source function which is given by $$S_\nu = (1-\omega_\nu) B_\nu(T) + \omega_\nu J_\nu + \omega_\nu J_\nu^*\,,$$ where $\omega_\nu$ is the single scattering albedo at the frequency $\nu$, $B_\nu(T)$ is the Planck function with the dust temperature $T$, and $J_\nu$ is the mean intensity of $I_{\nu\mu}$, that is, $\displaystyle J_\nu=\frac{1}{2}\int_{-1}^1 I_{\nu\mu}d\mu$. We have assumed that the scattering is isotropic for the simplicity. Note that we consider the scattering of the diffuse radiation as the second term in equation (A9). The third term accounts for the scattering of the incident radiation. To determine the dust temperature $T$, we assume the radiation equilibrium as $$\int_0^\infty \rho \kappa_\nu^{\rm abs} B_\nu(T) d\nu
= \int_0^\infty \rho \kappa_\nu^{\rm abs} J_\nu d\nu
+ \frac{q_{\rm abs}}{4\pi}\,.$$
To obtain the mean intensity $J_\nu$, we adopt a variable Eddington factor method. The first and second moments of equation (A8) are $$\frac{dH_\nu}{dz} = \rho \kappa_\nu^{\rm abs} (B_\nu - J_\nu)
+ \rho \kappa_\nu^{\rm sca} J_\nu^*\,,$$ and $$\frac{dK_\nu}{dz} = - \rho \kappa_\nu^{\rm ext} H_\nu\,,$$ where $\displaystyle H_\nu=\frac{1}{2}\int_{-1}^1I_{\nu\mu}\mu d\mu$ and $\displaystyle K_\nu=\frac{1}{2}\int_{-1}^1I_{\nu\mu}\mu^2d\mu$, and we have used equation (A9) to eliminate $S_\nu$. Note that $\kappa_\nu^{\rm abs}=(1-\omega_\nu)\kappa_\nu^{\rm ext}$ and the scattering cross section per unit gas mass $\kappa_\nu^{\rm sca}=\omega_\nu \kappa_\nu^{\rm ext}$. If we integrate equations (A11) and (A12) over the frequency, we obtain $$\frac{dH}{dz} = \frac{q_{\rm ext}}{4\pi}\,,$$ and $$\frac{dK}{dz} = -\int_0^\infty \rho \kappa_\nu^{\rm ext} H_\nu d\nu\,,$$ where $\displaystyle H=\int_0^\infty H_\nu d\nu$ and $\displaystyle K=\int_0^\infty K_\nu d\nu$, and we have used the radiation equilibrium (eq.\[A10\]) and equations (A6) and (A7) in equation (A13). Finally, we introduce the Eddington factor as the closure equation: $$f_{\rm E} = \frac{K}{J}\,,$$ where $\displaystyle J=\int_0^\infty J_\nu d\nu$.
We do not assume $f_{\rm E}=1/3$ (constant) as in the usual Eddington approximation, but obtain $f_{\rm E}$ directly from $I_{\nu\mu}$ which is calculated by the formal solution of equation (A8). The integration of the formal solution is performed with a parabolic interpolation of $S_\nu$ among three successive spatial points [@ols87]. As described in [@dul02], we alternate the integration of the formal solution (i.e. ray-tracing) with the integration of the moment equations (eqs. \[A13\]–\[A15\]) and the determination of $T$ by equation \[A10\] until we reach a convergence in $T$ (the difference between the two successive iterations becomes less than 0.1%). In addition, we adopt the acceleration algorithm by [@ng74] for a rapid convergence. For no scattering case, the convergence is very rapid independent of the total optical depth of the medium, typically 20 iterations. On the other hand, a factor of $\sim10$ times more iterations depending on the albedo are needed for isotropic scattering case.
Benchmark test
--------------
{width="7cm"}
Figure A1 shows the result of the benchmark test No.3 proposed by C. P. Dullemond on his web page [^3]. The settings are as follows: the stellar temperature $T_*=3,000$ K, the stellar radius $R_*=2.0$ $R_\odot$, the distance from the star $R=1$ AU, the grazing angle $\alpha=0.05$, the visible fraction of the stellar photosphere $f_{\rm vis}=0.5$, the total visual optical depth of the disc $\tau_{0.55\micron}=10^4$ (i.e. $\tau_{0.55\micron}=5\times10^3$ at the equatorial plane), and no scattering. We used the dust opacity model downloaded from the web page which is the same as that assumed in the calculation by C. P. Dullemond. The figure shows an excellent agreement between both results.
Analytic expression of radiation field in isothermal, absorption, and isotropic scattering medium
=================================================================================================
Here, we derive an analytic expression of the radiation field in an isothermal medium with absorption and isotropic scattering. In the derivation, we adopt the Eddington approximation with two-stream lines [e.g., @ryb79]. We consider a diffuse incident radiation.
{width="7cm"}
{width="7cm"}
Suppose a plane-parallel medium (see Figure B1). We set the extinction (absorption+scattering) optical depth coordinate $\tau$ along the normal of the medium. The total extinction optical depth of the medium is set to be $\tau_{\rm L}$. Then, suppose that the single scattering albedo in this medium is $\omega$ and this medium is isothermal and in the thermal equilibrium. The thermal radiation is denoted as $B$. Let us consider two-stream lines with the direction of $\mu=\pm1/\sqrt{3}$, where $\mu$ is the cosine of the angle between the ray and the $\tau$ coordinate. When the direction of the rays is denoted by the subscript of $+$ or $-$, the equation of the radiation transfer becomes $$\pm\frac{1}{\sqrt{3}} \frac{dI_{\pm}}{d\tau} = S_{\pm} - I_{\pm}\,.$$ The source function can be expressed as $$S_{\pm}=(1-\omega)B + \omega J\,,$$ where $J$ is the mean intensity. In the two-stream approximation, we can define the mean intensity $J$, the mean flux $H$, and the mean radiation pressure $K$ as $$J = \frac{1}{2}(I_+ + I_-)\,,$$ $$H = \frac{1}{2\sqrt{3}}(I_+ - I_-)\,,$$ and $$K = \frac{1}{6}(I_+ + I_-) = \frac{1}{3}J\,.$$
The first and second moments of (B1) with the source function (B2) are $$\frac{dH}{d\tau} = (1-\omega)(B-J)\,,$$ and $$\frac{1}{3}\frac{dJ}{d\tau} = - H\,.$$ With an effective optical depth $\tau_{\rm e}\equiv \tau\sqrt{3(1-\omega)}$, we have $$\frac{d^2 J}{d\tau_{\rm e}^2} = J - B\,,$$ from equations (B6) and (B7).
If we have incident radiations at the upper and lower boundaries as $I_0^{\rm in}$ and $I_{\rm L}^{\rm in}$, respectively, the boundary conditions are $$J(0)=I_0^{\rm in}+\sqrt{1-\omega}
\left(\frac{dJ}{d\tau_{\rm e}}\right)_{0}\,,$$ and $$J(\tau_{\rm e}^{\rm L}) =
I_{\rm L}^{\rm in}-\sqrt{1-\omega}
\left(\frac{dJ}{d\tau_{\rm e}}\right)_{\rm L}\,,$$ where $\tau_{\rm e}^{\rm L}=\tau_{\rm L}\sqrt{3(1-\omega)}$. The solution of equation (B8) with the boundary conditions (B9) and (B10) is $$\begin{aligned}
J(\tau_{\rm e}) & = & B
\left[1-\frac{e^{-\tau_{\rm e}}+e^{-\tau_{\rm e}^{\rm L}+\tau_{\rm e}}}
{1+\chi+(1-\chi)e^{-\tau_{\rm e}^{\rm L}}}\right]
\cr
& + & I_0^{\rm in}
\frac{(1+\chi)e^{-\tau_{\rm e}}
-(1-\chi)e^{-2\tau_{\rm e}^{\rm L}+\tau_{\rm e}}}
{(1+\chi)^2 - (1-\chi)^2 e^{-2\tau_{\rm e}^{\rm L}}}
\cr
& + & I_{\rm L}^{\rm in}
\frac{(1+\chi)e^{-\tau_{\rm e}^{\rm L}+\tau_{\rm e}}
-(1-\chi)e^{-\tau_{\rm e}^{\rm L}-\tau_{\rm e}}}
{(1+\chi)^2 - (1-\chi)^2 e^{-2\tau_{\rm e}^{\rm L}}}
\,,\end{aligned}$$ where $\chi=\sqrt{1-\omega}$. In the case without incident radiation (i.e. $I_0^{\rm in}=I_{\rm L}^{\rm in}=0$), this solution is exactly same as equation (28) of [@miy93].
Then, let us consider the outbound intensity at the surface of the medium. At $\tau=0$, the outbound intensity can be $$I_{0}^{\rm out} = a B + b I_{0}^{\rm in} + c I_{\rm L}^{\rm in}\,,$$ where $a$, $b$, and $c$ can be called as “thermal”, “reflection”, and “transmission” coefficients, respectively. Since $J(0)=(1/2)(I_0^{\rm in}+I_0^{\rm out})$, we obtain from equation (B11) $$a = \frac{2\chi(1-e^{-\tau_{\rm e}^{\rm L}})}
{1+\chi + (1-\chi)e^{-\tau_{\rm e}^{\rm L}}}\,,$$ $$b = \frac{(1-\chi)(1+\chi)
(1-e^{-2\tau_{\rm e}^{\rm L}})}
{(1+\chi)^2 - (1-\chi)^2
e^{-2\tau_{\rm e}^{\rm L}}}\,,$$ and $$c = \frac{4\chi\,e^{-\tau_{\rm e}^{\rm L}}}
{(1+\chi)^2 - (1-\chi)^2
e^{-2\tau_{\rm e}^{\rm L}}}\,.$$ At $\tau=\tau_{\rm L}$, we obtain symmetrically $$I_{\rm L}^{\rm out} = a B + b I_{\rm L}^{\rm in} + c I_{0}^{\rm in}\,.$$
Figure B2 shows the three coefficients for an isotropic case as a function of the effective optical depth of the medium $\tau_{\rm e}^{\rm L}$. As limiting values, we obtain $$a \to \cases{
\displaystyle\frac{2\chi}{1+\chi}
& ($\tau_{\rm L}\to\infty$) \cr
0 & ($\tau_{\rm L}\to 0$) \cr
}\,,$$ $$b \to \cases{
\displaystyle\frac{1-\chi}{1+\chi}
& ($\tau_{\rm L}\to\infty$) \cr
0 & ($\tau_{\rm L}\to 0$) \cr
}\,,$$ and $$c \to \cases{
0 & ($\tau_{\rm L}\to\infty$) \cr
1 & ($\tau_{\rm L}\to 0$) \cr
}\,.$$
\[lastpage\]
[^1]: E-mail: [email protected]
[^2]: [@cal91] adopted $J(0)=2H(0)$, where $J(0)$ and $H(0)$ are the mean intensity and the mean flux of the scattered stellar radiation at the top of the medium. On the other hand, we adopted $J(0)=\sqrt{3}H(0)$. This difference is due to the choice of the angle of the stream line.
[^3]: http://www.mpia-hd.mpg.de/homes/dullemon/radtrans/benchmarks/
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UdeM-GPP-TH-11-196\
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$a$: [*Département de physique, Cégep de Baie-Comeau,*]{}\
[*537 boulevard Blanche, Baie-Comeau, QC, Canada G5C 2B2*]{}\
$b$: [*Physique des Particules, Université de Montréal,*]{}\
[*C.P. 6128, succ. centre-ville, Montréal, QC, Canada H3C 3J7*]{}\
() 0.5cm [Abstract\
]{} 3truemm
Introduction
============
In the standard model (SM), CP violation is due to the presence of a nonzero complex phase in the Cabibbo-Kobayashi-Maskawa (CKM) quark mixing matrix $V$. This phase information is elegantly displayed in the unitarity triangle, in which the CP-violating interior angles are $\alpha$, $\beta$ and $\gamma$ [@pdg]. By measuring these CP phases in many different ways, one can test the SM.
Much theoretical work has gone into elucidating clean methods for obtaining $\alpha$, $\beta$ and $\gamma$ from $B$ decays. In 1999, it was pointed out that, apart from CKM matrix elements, the amplitudes for the decays $\bd\to\pi^+\pi^-$ and $\bs\to K^+ K^-$ are equal under U-spin symmetry ($d\leftrightarrow s$) [@Fleischer99]. With one additional piece of information, the phase $\gamma$ can be obtained. Subsequently, all $B$ decay pairs that are related by U spin were tabulated [@GroUspin], and another method for extracting weak-phase information using a different U-spin pair ($\bs \to \pi^+
K^-$ and $\bd \to \pi^- K^+$) was proposed [@GRBsKpi].
In order to determine the theoretical uncertainty of a particular method, it is necessary to address the issue of U-spin breaking. In general, theoretical input is used. However, one of the purposes of the present paper is to note that, in fact, this can be experimentally measured. The point is that, under U-spin symmetry, four of the experimental observables – the branching ratios and direct CP asymmetries of the two decays – are related, i.e. they are not independent. Thus, the experimental values of these observables, and the extent to which the relation among them is not satisfied, gives a measure of U-spin breaking. Note: this is not a completely new result. The relation among the four observables already appears in a number of papers. However, in general it is used as a theoretical constraint, rather than an experimental result.
In addition, one can go further. If one neglects annihilation- and exchange-type diagrams (which are expected to be small) in the $B$ decay amplitudes, there are other pairs of amplitudes which are equal, apart from CKM matrix elements [@GHLR]. In this case, it is not U spin that is assumed, but rather full flavor SU(3) symmetry[^1]. Here there are many more pairs whose amplitudes are related. And because the relation among the four observables holds in the SU(3) limit, it is possible to measure SU(3)-breaking effects using any of these decay pairs.
In fact, there are a number of two-body $B$ decay pairs for which this information is presently available. Furthermore, in such decays, the factorizable contribution to the breaking is often under good theoretical control. If this is taken into account, the measurement of U-spin/SU(3) breaking then tells us the size of nonfactorizable effects. In most cases, the data shows that such effects are small. However, as we show below, there is one decay pair – $\bd \to
\pi^+\pi^-$ and $\bd \to\pi^-K^+$ – which exhibits large nonfactorizable breaking. Although this is just one data point, so that no strong conclusions should be drawn, it does raise questions about analyses which neglect nonfactorizable U-spin/SU(3) breaking.
We begin in Sec. 2 with a discussion of U spin and U-spin breaking as it applies to a pair of charmless $\btod$ and $\btos$ decays . We show how the measurement of the branching ratios and direct CP asymmetries of these two decays allows one to experimentally measure the breaking. In Sec. 3, we turn to an examination of two-body $B$ decays. We present lists of 5 U-spin pairs and 11 additional SU(3) pairs whose U-spin/SU(3) breaking can be measured using this method. We show the latest data for five of these pairs. For two of these, the measurements are reasonably accurate, and one pair shows signs of significant nonfactorizable U-spin/SU(3) breaking. Finally, we discuss several pairs of decays whose amplitudes are equal, including CKM factors, within SU(3). A measure of SU(3) breaking is given by comparing the branching ratios of the two decays, as well as the direct CP asymmetries.
We discuss three-body decays in Sec. 4. There are 7 decay pairs whose amplitudes are related by U spin – the amount of breaking can be measured experimentally using the above method. In passing, we note that the pure-penguin decay $\bs \to \kbar \kbar K^0$ is interesting. Given that the final state is a CP eigenstate, the measurement of the indirect CP asymmetry in this decay cleanly probes the $\bs$-$\bsbar$ mixing phase, and might be easier experimentally than what is done at the moment. We also present the list of an additional 24 decay pairs whose amplitudes are related by SU(3). In this case, all final-state particles are identical, and so permutations of these particles must be considered. We show that, in (almost) all cases, the amplitudes are equal only for the totally symmetric final state $|S\rangle$, so that this state must be isolated experimentally in order to measure SU(3) breaking. We also point out the decay pairs whose amplitudes are equal, including CKM factors, within SU(3) for $|S\rangle$. In principle, these can also give information about SU(3) breaking. We conclude in Sec. 5.
U Spin and U-Spin Breaking {#Uspinbreak}
==========================
Consider charmless $\btod$ and $\btos$ decays[^2]. Their amplitudes can be written as A() &=& A\_u \_u\^[(d)]{} + A\_c \_c\^[(d)]{} + A\_t \_t\^[(d)]{} ,\
A() &=& A’\_u \_u\^[(s)]{} + A’\_c \_c\^[(s)]{} + A’\_t \_t\^[(s)]{} , \[amps\] where the $A_i$ and $A'_i$ ($i=u,c,t$) each represent a linear combination of diagrams, and $\lambda_p^{(q)}=V^*_{pb} V_{pq}$. Using the unitarity of the CKM matrix ($\lambda_u^{(q)}+\lambda_c^{(q)}+\lambda_t^{(q)}=0$), we can reduce the number of terms in the amplitudes from three to two. For instance, if the $\lambda_c^{(q)}$ piece is eliminated, we have A() &=& (A\_u - A\_c) \_u\^[(d)]{} + (A\_t - A\_c) \_t\^[(d)]{}\
&=& (A\_u - A\_c)\
&=& C ,\
A() &=& (A’\_u - A’\_c) \_u\^[(s)]{} + (A’\_t - A’\_c) \_t\^[(s)]{}\
&=& (A’\_u - A’\_c)\
&=& C’ , \[fullamps\] where $C \equiv (A_u - A_c)$, $C' \equiv (A'_u - A'_c)$, $r e^{i
\delta_r} \equiv (A_t - A_c)/(A_u - A_c)$ and $r' e^{i \delta'_r}
\equiv (A'_t - A'_c)/(A'_u - A'_c)$. Above we have explicitly written the weak-phase dependence, including the minus sign from $V_{tb}^*
V_{ts}$.
If the two amplitudes are given by a similar combination of diagrams, then under U-spin symmetry, which exchanges $d$ and $s$ quarks, we have $A'_i=A_i$, so that C’ = C , r’ = r , ’\_r = \_r , \[Uspinrels\] and the two amplitudes are described by four unknown theoretical parameters: $\gamma$, $|C|$, $r$, $\delta_r$ ($\beta$ has been measured quite accurately through the indirect CP asymmetry in $\bd\to
J/\psi \ks$ [@pdg], and is therefore taken to be known).
In general, there are four observables in the $\btod$ and $\btos$ processes: B\_d &=& |A(|b |d)|\^2+|A(b d)|\^2 ,\
B\_s &=& |A(|b |s)|\^2+|A(b s)|\^2 ,\
A\_d &=& ,\
A\_s &=& . \[ABdefs\] $B_d$ and $B_s$ are related to the CP-averaged $\btod$ and $\btos$ decay rates, while $A_d$ and $A_s$ are direct CP asymmetries. The CP-conjugate amplitude $A(\bar b \to \bar q)$ is obtained from $A(b
\to q)$ by changing the signs of the weak phases.
Since there are four unknown theoretical parameters in the amplitudes in the U-spin limit, one might imagine that these can be determined from measurements of $B_{d,s}$ and $A_{d,s}$. However, this is not true. It is straightforward to show that, in this limit, $X=1$, where X - . \[Xdef\] Thus, there are only three independent observables. This implies that - = 1 .
Explicitly, we have - = . Now, the sine law associated with the unitarity triangle gives = = . We therefore have - & = &\
&=&\
&=& , \[Xdef2\] where $|V_{us}| |V_{tb}| |V_{ts}|/|V_{ud}| |V_{cb}| |V_{cd}| = 1$. The above ratio equals 1 only in the U-spin limit. Thus, $(X-1)$ is a measure of U-spin breaking.
Until now, when this breaking was taken into account, it was only through theoretical estimates (e.g. see Refs. [@Fleischer99; @Fleischer]). However, in fact it can be obtained from the experimental data. This can be combined with the theoretical calculations to look for large nonfactorizable corrections (we will see this explicitly in Sec. \[nonfactmeasure\]). Furthermore, if the theoretical prediction of U-spin breaking is accurate, one can use the measurement of $(X-1)$ to search for new physics [@NSL].
Two-Body Decays
===============
U-spin pairs {#2bodyUspin}
------------
We begin with $B\to PP$ decays ($P =$ pseudoscalar), focusing on those $\btod$ and $\btos$ processes that are related by U spin. (It is straightforward to extend our analysis to other two-body decays, such as $B\to VP$ ($V =$ vector).) There are five U-spin pairs:
1. $\bd \to \pi^+ \pi^-$ and $\bs \to K^+ K^-$,
2. $\bs \to \pi^+ K^-$ and $\bd \to \pi^- K^+$,
3. $B^+ \to K^+ \bar K^0$ and $B^+ \to \pi^+ K^0$,
4. $\bd \to K^0 \kbar$ and $\bs \to \kbar K^0 $,
5. $\bd \to K^+ K^-$ and $\bs \to \pi^+ \pi^-$.
The first (second) decay is $\btod$ ($\btos$). In all cases, the two decays within a pair are related by U-spin reflection ($d
\leftrightarrow s$). This applies not only to the particles in the process (e.g. $\pi^+ \leftrightarrow K^+$, $\bd \leftrightarrow \bs$, etc.), but also to the individual diagrams involved. For any pair, one can measure the two branching ratios and direct CP asymmetries in order to obtain $X$ \[Eq. (\[Xdef\])\], and measure U-spin breaking.
SU(3) pairs {#2bodynonUspin}
-----------
U-spin pairs have been discussed at some length in Refs. [@GroUspin; @NSL]. However, one can go further. First, one pair which is not included in the list in Sec. \[2bodyUspin\], but appears in Refs. [@GroUspin; @NSL], is $\bs \to \pi^0 \kbar$ and $\bd \to \pi^0K^0$. The reason it is not included is that the two decays are not related by U spin. There are a number of ways to see this. First, $\pi^0 = (d{\bar d} - u{\bar u})/\sqrt{2}$, so that it does not transform into itself under U spin. Second, one diagram that contributes to $\bs \to \pi^0 \kbar$ is the penguin $P$, involving the quark-level transition ${\bar b} \to {\bar d} d {\bar d}$. Under U-spin reflection, this becomes ${\bar b} \to {\bar s} s {\bar s}$, which does not contribute to $\bd \to \pi^0K^0$. What is going on is the following: it is true that the amplitudes for $\bs \to \pi^0
\kbar$ and $\bd \to \pi^0K^0$ have the same diagrammatic decomposition [@GHLR], and so they satisfy Eq. (\[Uspinrels\]). However, the diagrams assume isospin invariance in addition to U spin, so that the symmetry is really flavor SU(3). Thus, $\bs \to \pi^0 \kbar$ and $\bd
\to \pi^0K^0$ is not a U-spin pair, but is in fact an SU(3) pair.
Second, it is standard to express the amplitudes for $B\to PP$ decays in terms of diagrams [@GHLR]. Certain of these diagrams – those of annihilation- and exchange-type – are expected to be smaller than the others. If these diagrams are neglected, then there are additional pairs of decays which satisfy Eq. (\[Uspinrels\]). These are not related by U spin, but are instead related by SU(3). The complete list of SU(3) pairs (which includes some U-spin pairs) is
- ($\bd \to \pi^+\pi^-$, $\bs \to \pi^+K^-$) and ($\bd \to
\pi^-K^+$, $\bs \to K^+K^-$),
- ($\bd \to \pi^0\pi^0$, $\bs \to \pi^0 \kbar$, $\bs \to \eta_8 \kbar$) and ($\bd \to \pi^0K^0$, $\bd \to \eta_8 K^0$),
- ($\bd \to K^0\kbar$, $B^+ \to K^+\kbar$, $\bd \to \pi^0 \eta_8$) and ($B^+ \to \pi^+K^0$, $\bs \to K^0\kbar$).
(Here, $\eta_8$ is a member of the octet of SU(3). The physical $\eta$ and $\eta'$ are linear combinations of $\eta_8$ and the SU(3) singlet, $\eta_0$.) The decays in the first (second) parentheses are $\btod$ ($\btos$) transitions. (Note that, depending on the pair, there may be an additional factor (e.g. $\sqrt{2}$) in relating the $\btod$ and $\btos$ decays.) From this list, we see that there are, in fact, 16 possible pairs of decays rather than the 5 of Sec. \[2bodyUspin\].
If $(X-1)$ is obtained using a pair from Sec. \[2bodyUspin\], then U-spin breaking is measured. However, if an SU(3) pair is used, then what is probed is not U-spin breaking, but rather SU(3) breaking. Interestingly, we have data for a number of the decays in the above list, so that it is possible to get $X$, and obtain an experimental measurement of U-spin/SU(3) breaking in these decays. This is done in Sec. \[numerical\].
Estimates of $A_{s,d}$ {#estimates}
----------------------
As described above, one can measure U-spin/SU(3) breaking through $X$. This quantity involves the direct CP asymmetries $A_{d,s}$, which arise due to the interference of two amplitudes with different weak and strong phases. The maximal size of $A_{d,s}$ is roughly equal to the ratio of the magnitudes of the two interfering amplitudes.
In two-body decays, the $\btos$ diagrams[^3] are expected to obey the approximate hierarchy [@GHLR] 1 &:& |P’\_[tc]{}| ,\
[|]{} &:& |T’| , |P’\_[EW]{}| ,\
[|]{}\^2 &:& |C’| , |P’\_[uc]{}| , |P\^[C]{}\_[EW]{}| , \[btoshierarchy\] where ${\bar\lambda} \simeq 0.2$. Since all $\btos$ decays in the list in Sec. \[2bodynonUspin\] receive contributions from $P'_{tc}$, $A_s$ is sizeable ($\lsim O(\bar\lambda) \sim 20\%$) only if the decay amplitude also includes $T'$. If there is no $T'$, but only $C'$ or $P'_{uc}$, then $A_s$ is small ($\lsim O(\bar\lambda^2) \sim 5\%$). In this case, the relative experimental error will necessarily be large, which will then translate into a large error on $(X-1)$.
The expected approximate hierarchy[^4] of the $\btod$ diagrams is [@GHLR] 1 &:& |T| ,\
[|]{} &:& |C| , |P\_[tc]{}| , |P\_[uc]{}| ,\
[|]{}\^2 &:& |P\_[EW]{}| ,\
[|]{}\^3 &:& |P\^C\_[EW]{}| . \[btodhierarchy\] Since all $\btod$ decays in the list in Sec. \[2bodynonUspin\] receive penguin contributions, $A_d$ is always sizeable (at least $\lsim O(\bar\lambda) \sim 20\%$).
Thus, the most promising pairs for measuring U-spin/SU(3) breaking are those whose $\btos$ decay amplitude includes a $T'$. These are given in the first entry in the list in Sec. \[2bodynonUspin\].
There are two types of contributions to U-spin/SU(3) breaking – factorizable and nonfactorizable. The factorizable effects depend essentially on form factors and decay constants, and can be reliably calculated. It has been shown that factorization holds well for $T$/$T'$ diagrams [@BBNS]. Thus, for those decay pairs which include these diagrams – i.e. the most promising for measuring $X$ – the ratio $|C'/C|$ is dominated by factorizable U-spin/SU(3) breaking.
The U-spin relations $r' = r$ and $\delta'_r = \delta_r$ are not affected by factorizable breaking effects[^5], as the various form factors and decay constants cancel [@Fleischer99; @Fleischer]. On the other hand, they could be altered by nonfactorizable effects, and these cannot be calculated theoretically. Still, it is thought that nonfactorizable U-spin/SU(3) breaking is not large, being higher-order in $1/m_b$. As we show below, this can be checked experimentally through the measurement of $(X-1)$.
Numerical analysis {#numerical}
------------------
The four quantities required for the measurement of $X$ are $B_{d,s}$ and $A_{d,s}$ \[Eq. (\[ABdefs\])\]. The $B_{d,s}$’s are related to the branching ratios by \_[(q)]{} p\_[c(q)]{} B\_[q]{} = 8 m\_[B(q)]{}\^2 \_[(q)]{} , where, for a $\bar b \to \bar q$ process ($q=d,s$), $\tau_{(q)}$ is the $B$-meson lifetime, $p_{c(q)}$ is the momentum of the final-state mesons in the $B$ rest frame, $m_{B(q)}$ is the rest mass of the $B$ meson, and $\mathcal{B}_{(q)}$ is the CP-averaged branching ratio. The $A_{d,s}$’s are equal to $-C_{CP}$, where $C_{CP}$ is the direct CP asymmetry in a given decay.
At present, there are five different pairs of two-body decays for which we have the data required by the method of Sec. \[Uspinbreak\] for measuring U-spin/SU(3) breaking:
1. $\bd \to \pi^+ \pi^-$ and $\bd \to \pi^- K^+$,
2. $\bs \to \pi^+ K^-$ and $\bd \to \pi^- K^+$,
3. $B^+ \to K^+ \bar K^0$ and $B^+ \to \pi^+ K^0$,
4. $\bd \to K^0 \bar K^0$ and $B^+ \to \pi^+ K^0$,
5. $\bd \to \pi^0 \pi^0$ and $\bd \to \pi^0 K^0$.
The current experimental values are given in Table \[tab:inputs1\]. The values of the $B$ masses and lifetimes can be found in Ref. [@pdg].
Decay $\mathcal{B}$ \[$\times 10^6$\] $-C_{CP}$ $p_c$ \[MeV/$c$\]
----------------------------- --------------------------------- ---------------------------- -------------------
$\bd \to \pi^+ \pi^-$ $5.16 \pm 0.22$ $0.38 \pm 0.06$ $2636$
$\bd \to \pi^- K^+$ $19.4 \pm 0.6$ $-0.098^{+0.012}_{-0.011}$ $2615$
$\bs \to \pi^+ K^-$ $5.0 \pm 1.1$ $0.39 \pm 0.17$ $2659$
$B^+ \to K^+ \bar K^0$ $1.36^{+0.29}_{-0.27}$ $0.12^{+0.17}_{-0.18}$ $2593$
$B^+ \to \pi^+ K^0$ $23.1 \pm 1.0$ $0.009 \pm 0.025$ $2614$
$\bd \to K^0 \bar K^0$ $0.96^{+0.21}_{-0.19}$ $0.06 \pm 0.26$ $2592$
$\bd \to \pi^0 \pi^0$ $1.55 \pm 0.19$ $0.43^{+0.25}_{-0.24}$ $2636$
$\bd \to \pi^0 K^0$ (BaBar) $10.1 \pm 0.6 \pm 0.4$ $-0.13 \pm 0.13 \pm 0.03$ $2615$
$\bd \to \pi^0 K^0$ (Belle) $8.7 \pm 0.5 \pm 0.6$ $0.14 \pm 0.13 \pm 0.06$
: Input values for the experimental quantities [@pdg; @hfag]. For asymmetric error bars, we take the average of both errors and assume a gaussian distribution.[]{data-label="tab:inputs1"}
With these inputs, one can compute the value of $(X-1)$ obtained for each of the five decay pairs using Eq. (\[Xdef\]). The results are shown in Table \[tab:outputs\]. Note that, as described in Sec. \[estimates\], the direct CP asymmetries in $B^+ \to \pi^+ K^0$ and $\bd \to \pi^0 K^0$ are expected to be quite small, leading to a very large error on $(X-1)$. This is indeed what is found \[pairs (3), (4) and (5)\].
Decay pair $(X-1)$
------------ ------------------------
(1) $-0.02 \pm 0.18$
(2) $-0.08 \pm 0.42$
(3) $-2.3 \pm 3.6$
(4) $-4 \pm 16$
(5) $1.0 \pm 2.1$ (BaBar)
$-2.8 \pm 2.0$ (Belle)
: Output values for the quantity $(X-1)$ for the five pairs of decays.[]{data-label="tab:outputs"}
Finally, the decays $\bd \to \pi^+ \pi^-$ and $\bs \to K^+K^-$ form a U-spin pair. From the updated QCD light-cone sum-rule calculation of Ref. [@LCSR2], we have \_[fact]{} = ( ) = 1.41\^[+0.20]{}\_[-0.11]{} . Here and below, we take $f^+(M_K^2) \simeq f^+(M_\pi^2) \simeq f^+(0)$ since the variation of the form factors over this range of $q^2$ falls well within the errors of their calculation [@LCSR1]. Thus, using the data from Table \[tab:inputs1\] and Eq. (\[BdpiKBsKKdata\]) below, we expect A\_[CP]{}(K\^+K\^-) & = & - \^2\_[fact]{} A\_[CP]{}(\^+ \^-)\
& = & -0.16 0.05 . Similarly, the decays $\bs \to \pi^+ K^-$ and $\bs \to K^+K^-$ form an SU(3) pair, so that A\_[CP]{}(K\^+K\^-) & = & - \^2\_[fact]{} A\_[CP]{}(\^+ K\^-)\
& = & -0.12 0.06 , where $|C'/C|_{fact} = f_K/f_\pi$. These predictions are in agreement with one another and will be tested when $A_{CP}(\bs \to K^+K^-)$ is measured.
Measurement of nonfactorizable SU(3) breaking {#nonfactmeasure}
---------------------------------------------
The theoretical expression for $X$ is given in Eq. (\[Xdef2\]). As discussed above, within factorization, only the ratio $|C'/C|$ contributes to $X$. Therefore, given an experimental measurement of $X$ and a theoretical calculation of $|C'/C|_{fact}$, one can obtain = \_[fact]{}\^2 X and see whether it is consistent with 1 (small nonfactorizable U-spin/SU(3) breaking).
For the first two pairs of the previous subsection, which yield reasonably precise measurements of $X$, we have & : & \_[fact]{} = ( ) = 1.20 ,\
& : & \_[fact]{} = ( ) = 1.01\^[+0.07]{}\_[-0.15]{} . \[C’/C\] The ratio $f_K/f_\pi$ and the value in the second line are taken from Ref. [@LCSR2]. (We have neglected small errors in $f_K/f_\pi$.) These give & : & = 0.68 0.13 ,\
& : & = 0.90 0.43 .
For pair (2), the theoretical prediction for factorizable U-spin breaking is consistent with the experimental measurement of Table \[tab:outputs\]. However, for pair (1), there is a $2.5 \sigma$ deviation of the value of $|C/C'|_{fact}^2 X$ from 1. Now, as it is just one data point, one cannot draw any firm conclusions – it could simply be a statistical fluctuation. However, it does hint at a large nonfactorizable SU(3)-breaking correction. (Or, if one is certain that such nonfactorizable effects are small, it could be suggestive of new physics.) All of this illustrates the importance of measuring $X$ experimentally, and this in as many different decay pairs as possible.
This result does call into question any analysis which does not include nonfactorizable corrections. However, it is straightforward to take this into account. Within U-spin/SU(3) symmetry, the four observables $B_{d,s}$ and $A_{d,s}$ are not independent. However, if one allows U-spin/SU(3) breaking, this no longer holds. If one assumes that $\delta'_r = \delta_r$, i.e. the phase is unaffected by the breaking, and takes $|C'/C|$ from factorization, then nonfactorizable U-spin/SU(3) breaking contributes only to $r'/r$. That is, there is one additional theoretical parameter ($r'/r$), but there is one additional measurement, so that the nonfactorizable breaking can be obtained. This is essentially just the measurement of $X$.
Now, pair (1) is useful for another reason. As detailed previously, it is not possible to obtain the theoretical parameters in the amplitudes solely from the measurements of $B_{d,s}$ and $A_{d,s}$ – additional input is needed. This has been discussed for two of the U-spin pairs. For $\bd \to \pi^+ \pi^-$ and $\bs \to K^+ K^-$, it has been noted that $\gamma$ can be extracted through the additional measurement of the indirect CP asymmetry in $\bd \to \pi^+ \pi^-$ [@Fleischer99; @Fleischer]. Similarly, $\gamma$ can be obtained from $\bs \to
\pi^+ K^-$ and $\bd \to \pi^- K^+$ with the added information coming from the measurement of the branching ratio of $B^+ \to \pi^+ K^0$ [@GRBsKpi].
Both of these pairs appear in the list in Sec. \[2bodyUspin\]. However, if one expands the symmetry from U spin to SU(3), they can be combined, producing the pair $\bd \to \pi^+\pi^-$ and $\bd
\to\pi^-K^+$ (pair (1), in the list in Sec. \[2bodynonUspin\]). $\gamma$ can then be extracted using the method of Ref. [@Fleischer99], taking $B_d$, $A_d$ and $A^{CP}_{ind}$ from $\bd \to \pi^+\pi^-$, and $B_s$ from $\bd \to \pi^-K^+$ instead of $\bs \to K^+ K^-$. Since [@hfag; @Tonelli] (\^-K\^+) &=& (19.4 0.6) 10\^[-6]{} ,\
(K\^+ K\^-) &=& (23.9 3.9) 10\^[-6]{} , \[BdpiKBsKKdata\] one sees that the first (experimental) error is smaller than the second one. Thus, the error on $\gamma$ is also smaller. Alternatively, suppose that the technique of Ref. [@GRBsKpi] is used, taking $B_s$ and $A_s$ from $\bd \to \pi^- K^+$, and $B_d$ from $\bd \to \pi^+\pi^-$ instead of $\bs \to \pi^+ K^-$ (information from $\mathcal{B}(B^+ \to \pi^+ K^0)$ is added). The error on $\gamma$ will still be smaller since [@hfag] (\^+\^-) &=& (5.16 0.22) 10\^[-6]{} ,\
(\^+ K\^-) &=& (5.0 1.1) 10\^[-6]{} . The point is that the branching ratios of $\bd$ decays are measured much more accurately than those of $\bs$ decays, so that the extracted value of $\gamma$ is more precise if pair (1) is used, rather than either U-spin pair.
In fact, this method was proposed many years ago, in 1995 [@GR95]. In this paper, information from both $A^{CP}_{ind}(\bd
\to \pi^+\pi^-)$ and $\mathcal{B}(B^+ \to \pi^+ K^0)$ is added simultaneously. In addition, perfect SU(3) symmetry is not imposed, so there are a total of 6 independent measurements. It is assumed that $|C'/C| = f_K/f_\pi$ \[Eq. (\[C’/C\])\] and that $\delta'_r =
\delta_r$, but $r'$ and $r$ are left as independent. This means that the amplitudes are written in terms of 4 hadronic theoretical parameters and two weak phases. In Ref. [@GR95], it is argued that both weak phases can be extracted. However, this method can be modified: if we assume that $\beta$ is known from $A^{CP}_{ind}(\bd\to
J/\psi \ks)$, then we have the freedom to take $\delta'_r$ and $\delta_r$ as independent. Now there are 6 equations in 6 unknowns ($C$, $r'$, $r$, $\delta'_r$, $\delta_r$, $\gamma$), so that one can solve for the theoretical parameters (numerically, if necessary). This analysis was partially performed in Ref. [@GR2007]. We must stress here that no assumption about the size of nonfactorizable effects in $r'/r$ and $\sin\delta'_r/\sin\delta_r$ is made here – this information is taken from the experimental data.
Other signals of SU(3) breaking
-------------------------------
There are pairs of decays whose amplitudes are equal at the quark level, including CKM factors, under SU(3). At the meson level, the processes are those within parentheses in the list in Sec. \[2bodynonUspin\]. The amplitudes for the two decays can be written A\_i = C\_i\^[()]{} , where $i=1,2$ and $q=d,s$ (the hadronic parameters have primes for $q=s$). Assuming only factorizable SU(3) breaking, $r^{(\prime)}_1 =
r^{(\prime)}_2$ and $\delta_{r,1}^{(\prime)} =
\delta_{r,2}^{(\prime)}$. We therefore expect the branching ratios and direct CP asymmetries for the two decays to satisfy \_2 & = & \_[fact]{}\^2 \_1 ,\
A\_[CP,2]{} & = & A\_[CP,1]{} . (We neglect any mass and lifetime differences between the two decaying $B$ mesons.) Any deviation from these relations is a sign of nonfactorizable SU(3) breaking.
The pairs or amplitude relations are (all experimental data is taken from Ref. [@hfag]):
1. $\bd \to \pi^-K^+$ and $\bs \to K^+K^-$: \_[fact]{} = ( ) = 0.85\^[+0.07]{}\_[-0.12]{} . \[C1/C2\] (This is based on the results of Ref. [@LCSR2].) The data for the branching ratios for these decays are given in Eq. (\[BdpiKBsKKdata\]). We expect \_[fact]{}\^2 to be consistent with 1. It equals $1.12 \pm 0.26$, so there is no evidence of nonfactorizable SU(3) breaking.
We also expect that A\_[CP]{}(K\^+K\^-) = A\_[CP]{}(\^-K\^+) = -0.098\^[+0.012]{}\_[-0.011]{} .
2. $\bd \to \pi^+\pi^-$ and $\bs \to \pi^+K^-$: here, $\left\vert
C_1/C_2 \right\vert_{fact} = 0.85^{+0.07}_{-0.12}$, as in Eq. (\[C1/C2\]). The experimental data is: $\mathcal{B}(\bd \to
\pi^+\pi^-) = (5.16 \pm 0.22) \times 10^{-6}$, $\mathcal{B}(\bs \to
\pi^+K^-) = (5.0 \pm 1.1) \times 10^{-6}$. We expect \_[fact]{}\^2 to be consistent with 1. It equals $1.43 \pm 0.40$. We also expect the direct CP asymmetries to be equal. It is found that $A_{CP}(\bd \to
\pi^+\pi^-) = 0.38 \pm 0.06$, $A_{CP}(\bs \to \pi^+K^-) = 0.39 \pm
0.17$, which are in good agreement with one another. We therefore conclude that there is no evidence for nonfactorizable SU(3) breaking in this decay pair.
3. $A(\bd \to \pi^0 K^0) = \sqrt{3} A(\bd \to \eta_8 K^0)$: we expect $\mathcal{B}(\bd \to \pi^0 K^0) = \left\vert C'_1/C'_2
\right\vert_{fact}^2 \\ 3 \mathcal{B}(\bd \to \eta_8 K^0)$ and $A_{CP}(\bd \to \pi^0 K^0) = A_{CP}(\bd \to \eta_8 K^0)$.
4. $A(\bd \to \pi^0 \pi^0) = A(\bs \to \pi^0 \kbar) = \sqrt{3}
A(\bs \to \eta_8 \kbar)$: this leads to the prediction that $A_{CP}(\bs \to \pi^0 \kbar) = A_{CP}(\bs \to \eta_8 \kbar) =
0.43^{+0.25}_{-0.24}$. Also, we expect that $\mathcal{B}(\bs \to
\pi^0 \kbar) = (1.55 \pm 0.16) \times 10^{-6}$, $\mathcal{B}(\bs \to
\eta_8 \kbar) = (0.52 \pm 0.05) \times 10^{-6}$, modulo factorizable SU(3) corrections.
5. $A(B^+ \to \pi^+ K^0) = A(\bs \to K^0 \kbar)$, so that the direct CP asymmetries are expected to be equal for these decays, and we expect $\mathcal{B}(B^+ \to \pi^+ K^0) = \left\vert C'_1/C'_2
\right\vert_{fact}^2 \mathcal{B}(\bs \to K^0 \kbar)$.
6. $A(\bd \to K^0 \kbar) = A(B^+ \to K^+ \kbar) = \sqrt{3} A(\bd
\to \pi^0 \eta_8)$: we expect the direct CP asymmetries for these three decays to be equal. Also, we expect that $\mathcal{B}(\bd \to
K^0 \kbar) = \mathcal{B}(B^+ \to K^+ \kbar) = 3 \mathcal{B}(\bd \to
\pi^0 \eta_8)$, modulo factorizable SU(3) corrections.
Note: it would not be a surprise to see evidence of significant nonfactorizable effects in the decays in items 4-6, as these are dominated by diagrams for which factorization is not expected to hold.
Three-Body Decays
=================
We now turn to $B\to PPP$ decays. In the past, such decays were little studied – most of the theoretical work looking at clean methods for obtaining the weak phases focused on two-body $B$ decays. This is essentially for two reasons: (i) final states such as $\psi \ks$, $\pi^+\pi^-$, etc. are CP eigenstates, and (ii) when there is a second decay amplitude, with a different weak phase, it has been possible to find methods to remove this “pollution,” and cleanly extract weak-phase information.
Things are not the same in the case of three-body $B$ decays. First, because there are three particles, final states such as $\ks
\pi^+\pi^-$ are not CP eigenstates – the value of its CP depends on whether the relative $\pi^+\pi^-$ angular momentum is even or odd. And second, even if it were possible to distinguish the states of CP $+$ and $-$, one still has the problem of removing the pollution due to additional decay amplitudes. For these reasons, the conventional wisdom has been that it is not possible to obtain clean weak-phase information from three-body decays.
Recently, it was shown that, by doing a diagrammatic analysis, one can address these two problems [@diagramspaper]. First, a Dalitz-plot analysis can be used to experimentally separate the CP $+$ and $-$ final states. Second, one can often remove the pollution of additional diagrams and cleanly measure the CP phases. In Ref. [@Kpipipaper], the procedure for extracting $\gamma$ from $B\to K\pi\pi$ decays was described in detail. Thus, in fact, it [*is*]{} possible to use three-body decays to obtain weak-phase information and search for new physics.
In this paper, the goal is to find pairs of $\btod$ and $\btos$ decays which satisfy Eq. (\[Uspinrels\]) and permit the measurement of $X$. As we will see, in order to do this with three-body decays, the diagrammatic decomposition of Ref. [@diagramspaper] is necessary.
U-spin pairs {#u-spin-pairs}
------------
As with $B\to PP$ decays (Sec. \[2bodyUspin\]), we look for pairs of $\btos$ and $\btod$ decays that are related by U-spin reflection. We find that there are seven such pairs of three-body decays:
1. $\bs \to K^+ K^- \kbar$ and $\bd \to K^0 \pi^+ \pi^-$,
2. $\bs \to \kbar \pi^+\pi^-$ and $\bd \to K^+ K^0 K^-$,
3. $\bd \to K^0 K^- \pi^+$ and $\bs \to K^+\kbar \pi^-$,
4. $\bd \to K^+ \kbar \pi^-$ and $\bs\to K^0K^-\pi^+$,
5. $B^+ \to \pi^+ \pi^+ \pi^-$ and $B^+ \to K^+ K^+ K^-$,
6. $B^+ \to K^+ K^- \pi^+$ and $B^+ \to K^+ \pi^+ \pi^-$,
7. $\bs \to \kbar \kbar K^0$ and $\bd \to K^0 K^0 \kbar$.
In order to show that these pairs do indeed satisfy Eq. (\[Uspinrels\]), one has to compare the amplitudes of the decays within a pair.
Under U spin, the $d$ and $s$ quarks are in a doublet, as are ${\bar
s}$ and $-{\bar d}$. Thus, $K^+$ and $\pi^+$, and $K^-$ and $\pi^-$, are considered to be identical particles. We therefore see that the final states of pairs 1-4 contain no identical particles. One can straightforwardly compare the amplitudes of the decays within these pairs. We refer to Ref. [@diagramspaper] for a description of the diagrams; here we label each diagram $D$ by an index $q$ ($q=u,d,s$) denoting the flavor of the quark pair “popped” from the vacuum. Under isospin symmetry, $D_u = D_d$, under U spin, $D_d =
D_s$, and under full SU(3), $D_u = D_d = D_s$. We have:
pair 1: A(K\^+ K\^- ) &=& - T\_[1,s]{} e\^[i]{} - C\_[1,s]{} e\^[i]{} - [P]{}\_[a;uc]{} e\^[i]{}\
&& -1.5truein - [P]{}\_[a;tc]{} e\^[-i]{} - 23 P\_[EW1,s]{} e\^[-i]{} + 13 P\_[EW1,u]{} e\^[-i]{} - 23 P\_[EW1,s]{}\^C e\^[-i]{} + 13 P\_[EW2,u]{}\^C e\^[-i]{} ,\
A(K\^0 \^+ \^-) &=& -T’\_[1,d]{} e\^[i]{}-C’\_[1,d]{} e\^[i]{}-[P]{}’\_[a;uc]{} e\^[i]{}\
&& -0.5truecm + [P]{}’\_[a;tc]{} + 23 P’\_[EW1,d]{} - 13 P’\_[EW1,u]{} + 23 P\^[C]{}\_[EW1,d]{} - 13 P\^[C]{}\_[EW2,u]{} , pair 2: A(|K\^0 \^+ \^-) &=& -T\_[2,d]{} e\^[i]{} - C\_[1,d]{} e\^[i]{} - [P]{}\_[b;uc]{} e\^[i]{}\
&& -1.5truein - [P]{}\_[b;tc]{} e\^[-i]{} - 23 P\_[EW1,d]{} e\^[-i]{} + 13 P\_[EW1,u]{} e\^[-i]{} + 13 P\_[EW1,u]{}\^C e\^[-i]{} - 23 P\_[EW2,d]{}\^C e\^[-i]{} ,\
A(K\^+ K\^0 K\^-) &=& -T’\_[2,s]{} e\^[i]{}-C’\_[1,s]{} e\^[i]{} -[P]{}’\_[b;uc]{} e\^[i]{}+ [P]{}’\_[b;tc]{}\
&& 0.5truecm + 23 P’\_[EW1,s]{} - 13 P’\_[EW1,u]{} - 13 P\^[C]{}\_[EW1,u]{} + 23 P\^[C]{}\_[EW2,s]{} , pair 3: A(K\^0 K\^- \^+) &=& - T\_[2,s]{} e\^[i ]{} -[P]{}\_[b;uc]{} e\^[i ]{}\
&& - [P]{}\_[b;tc]{} e\^[-i]{} + 13 P\^C\_[EW1,u]{}e\^[-i]{} - 23 P\^C\_[EW2,s]{} e\^[-i]{} ,\
A(K\^+\^-) &=& - T’\_[2,d]{} e\^[i ]{} -[P]{}’\_[b;uc]{} e\^[i ]{} + [P]{}’\_[b;tc]{} - 13 P\_[EW1,u]{}\^[’C]{} + 23 P\_[EW2,d]{}\^ [’C]{} , pair 4: A(K\^+\^-) & = & -T\_[1,s]{} e\^[i]{} - [P]{}\_[a;uc]{} e\^[i]{}\
&& - [P]{}\_[a;tc]{} e\^[-i]{} - 23 P\^C\_[EW1,s]{} e\^[-i]{} + 13 P\^C\_[EW2,u]{} e\^[-i]{} ,\
A(K\^0K\^-\^+) &=& - T’\_[1,d]{} e\^[i ]{} -[P]{}’\_[a;uc]{} e\^[i ]{} + [P]{}’\_[a;tc]{} + 23 P\_[EW1,d]{}\^[’C]{} - 13 P\_[EW2,u]{}\^[’C]{} , where \_a P\_[1,d]{} +P\_[2,u]{} & , & [P]{}\_b P\_[1,u]{} +P\_[2,d]{} ,\
[P]{}\_a P\_[1,s]{} + P\_[2,u]{} & , & [P]{}\_b P\_[1,u]{} + P\_[2,s]{} . \[Pdefs1\] For $\btod$ transitions, the diagrams are written without primes; for $\btos$ transitions, they are written with primes. (The overall signs of the amplitudes assume ${\bar u}$ is negative, as with isospin. If one takes ${\bar d}$ to be negative, as with U spin, one may obtain a different overall sign. But the physics does not change.)
There are two truly identical particles in the final states in pair 5 ($\pi^+$ in $B^+ \to \pi^+ \pi^+ \pi^-$ and $K^+$ in $B^+ \to K^+ K^+
K^-$), so the overall wavefunction must be symmetric with respect to the exchange of these two particles: A(B\^+\^+\^+\^-)\_[sym]{} &=& -T\_[2,d]{} e\^[i]{} - C\_[1,d]{} e\^[i]{} - [P]{}\_[b;uc]{} e\^[i]{}\
&& -1.5truein - [P]{}\_[b;tc]{} e\^[-i]{} - 23 P\_[EW1,d]{} e\^[-i]{} + 13 P\_[EW1,u]{} e\^[-i]{} + 13 P\^[C]{}\_[EW1,u]{} e\^[-i]{} - 23 P\^[C]{}\_[EW2,d]{} e\^[-i]{} ,\
A(B\^+ K\^+ K\^+ K\^-)\_[sym]{} &=& -T’\_[2,s]{} e\^[i]{} -C’\_[1,s]{} e\^[i]{} -[P]{}’\_[b;uc]{} e\^[i]{}\
&& -0.8truein + [P]{}’\_[b;tc]{} + 23 P’\_[EW1,s]{} - 13 P’\_[EW1,u]{} - 13 P\^[C]{}\_[EW1,u]{} + 23 P\^[C]{}\_[EW2,s]{} . The penguin diagrams are defined in Eq. (\[Pdefs1\]).
The final states of pair 6 contain the identical particles (under U spin) $K^+$ and $\pi^+$. The overall wavefunction of the final $K^+\pi^+$ pair must be symmetrized with respect to the exchange of these two particles. If the relative angular momentum is even (odd), the U-spin state must be symmetric (antisymmetric): A(B\^+ K\^+K\^-\^+)\_[sym]{} &=& -T\_[2,s]{} e\^[i]{} - C\_[1,s]{} e\^[i]{} - [P]{}\_[b;uc]{} e\^[i]{} - [P]{}\_[b;tc]{} e\^[-i]{}\
&& -2.2truecm + 13 P\_[EW1,u]{} e\^[-i]{} -23 P\_[EW1,s]{} e\^[-i]{} + 13 P\^C\_[EW1,u]{} e\^[-i]{} - 23 P\^C\_[EW2,s]{} e\^[-i]{} ,\
A(B\^+ K\^+K\^-\^+)\_[anti]{} &=& T\_[2,s]{} e\^[i]{} + C\_[1,s]{} e\^[i]{} + [P]{}\_[b;uc]{} e\^[i]{} + [P]{}\_[b;tc]{} e\^[-i]{}\
&& -2.2truecm + 13 P\_[EW1,u]{} e\^[-i]{} -23 P\_[EW1,s]{} e\^[-i]{} + 13 P\^C\_[EW1,u]{} e\^[-i]{} - 23 P\^C\_[EW2,s]{} e\^[-i]{} ,\
A(B\^+ K\^+\^+\^-)\_[sym]{} &=& -T’\_[2,d]{} e\^[i]{}-C’\_[1,d]{} e\^[i]{}-[P]{}’\_[b;uc]{} e\^[i]{}+ [P]{}’\_[b;tc]{}\
&& 0.2truecm - 13 P’\_[EW1,u]{} + 23 P’\_[EW1,d]{} - 13 P\^[C]{}\_[EW1,u]{} + 23 P\^[C]{}\_[EW2,d]{} ,\
A(B\^+ K\^+\^+\^-)\_[anti]{} &=& -T’\_[2,d]{} e\^[i]{}-C’\_[1,d]{} e\^[i]{}-[P]{}’\_[b;uc]{} e\^[i]{}+ [P]{}’\_[b;tc]{}\
&& 0.2truecm + 13 P’\_[EW1,u]{} - 23 P’\_[EW1,d]{} + 13 P\^[C]{}\_[EW1,u]{} - 23 P\^[C]{}\_[EW2,d]{} , where, for the antisymmetric amplitudes, diagrams with the $K^+$ above (below) the $\pi^+$ are multiplied by $+1$ ($-1$). The penguin diagrams are defined in Eq. (\[Pdefs1\]).
Both the $\kbar$ and $K^0$ are contained in a U-spin triplet, and so these are considered as identical particles. Thus, the final states of the decays in pair 7 contain three identical particles and the group $S_3$ must be used to describe their permutations. Fortunately, for these decays, the situation is less complicated. For $K^0 K^0
\kbar$, in any diagram, the position of the $\kbar$ cannot change, so that only exchanges of the two $K^0$’s need be considered. Things are similar for $\kbar \kbar K^0$. Thus, in order to show that these decays do indeed form a pair which respects Eq. (\[amps\]), it is sufficient to examine the amplitudes which are symmetric in the exchange of the two truly identical particles. We have A(K\^0)\_[sym]{} &=& [P]{}\_[a;uc]{} e\^[i]{} + [P]{}\_[a;tc]{} e\^[-i]{} - 13 P\_[EW1,s]{} e\^[-i]{} - 13 P\_[EW1,d]{} e\^[-i]{}\
&& 0.8truecm - 13 P\_[EW1,s]{}\^C e\^[-i]{} - 13 P\_[EW2,d]{}\^C e\^[-i]{} ,\
A(K\^0 K\^0 )\_[sym]{} &=& [P]{}’\_[b;uc]{} e\^[i]{}- [P]{}’\_[b;tc]{} + 13 P’\_[EW1,s]{} + 13 P’\_[EW1,d]{}\
&& 0.8truecm + 13 P\^[C]{}\_[EW1,d]{} + 13 P\^[C]{}\_[EW2,s]{} , where \_a P\_[1,s]{} +P\_[2,d]{} , [P]{}\_b P\_[1,d]{} +P\_[2,s]{} . \[Pdefs2\]
Now, under U spin, primed diagrams are equal to unprimed diagrams with the exchange $d \leftrightarrow s$, i.e. they differ only by $\lambda_p^{(d)} \leftrightarrow \lambda_p^{(s)}$. Thus, $D'_s \sim
D_d$, $D'_d \sim D_s$, $D'_u \sim D_u$, ${\tilde P}'_a \sim {\hat
P}_a$, ${\tilde P}_a \sim {\hat P}'_a$, ${\tilde P}'_b \sim {\hat
P}_b$, ${\tilde P}_b \sim {\hat P}'_b$, and ${\mathcal P}_a \sim
{\mathcal P}_b$. We therefore see that (almost all) the amplitudes for the $\btod$ and $\btos$ decays in pairs 1-7 have the same form, modulo CKM factors (recall that the $\btos$ amplitudes include the minus sign from $V_{tb}^* V_{ts}$ \[$P'_{tc}$ and EWP’s\]). The single exception is $A(B^+ \to K^+K^-\pi^+)_{anti}$ and $A(B^+ \to
K^+\pi^+\pi^-)_{anti}$ in pair 6. Here, recall that the contribution of a diagram is positive (negative) if the $K^+$ is above (below) the $\pi^+$ in that diagram. However, since the U-spin transformation switches $K^+ \leftrightarrow \pi^+$, we expect the antisymmetric amplitudes to have a relative $-$ sign, and this is indeed what is found. We therefore see that, in all pairs, the amplitudes of the $\btod$ and $\btos$ decays respect Eq. (\[Uspinrels\]), so that $(X-1)$ (U-spin breaking) can be measured using these processes.
Previously, in discussing two-body decays, we noted that the U-spin/SU(3) corrections could be separated into two types – factorizable and nonfactorizable – and that the factorizable corrections could be reliably calculated. As such, the measurement of $X$ can be translated into a determination of the nonfactorizable corrections. In principle, this can be applied to three-body decays. In practice, however, things are more complicated. In particular, while the $T^{(\prime)}$ diagram in two-body decays is, within factorization, proportional to the product of a decay constant and a form factor, in three-body decays, new structures appear. The $T_1^{(\prime)}$ diagram is proportional to the product of a $\langle
2 ~{\rm particles} | (V-A) | 0 \rangle$ matrix element and a form factor, and the $T_2^{(\prime)}$ diagram is proportional to the product of a decay constant and a $\langle 2 ~{\rm particles} | (V-A)
| B \rangle$ matrix element. To date, there have been no definitive calculations of these matrix elements. They have been studied in Ref. [@3fact], but more work is clearly needed.
To this end, the measurement of $X$ can help. Given that nonfactorizable U-spin/SU(3) breaking is expected to be subdominant compared to factorizable breaking, $X$ as measured in the above decay pairs can be considered to be a factorizable correction (especially pairs 1-6, which have $T_1^{(\prime)}/T_2^{(\prime)}$ contributions). The knowledge of the precise values of such factorizable effects will guide the calculation of the new matrix elements.
Finally, a natural question is whether clean weak-phase information can be extracted from these decays. For example, the pair $\bs \to K^+ K^- \kbar$ and $\bd \to K^0 \pi^+ \pi^-$ is the three-body equivalent of $\bd \to \pi^+ \pi^-$ and $\bs \to K^+
K^-$. Can one adapt the method of Ref. [@Fleischer99] to obtain $\gamma$? Unfortunately, the answer is no. In two-body decays, additional information is provided by the measurement of the indirect CP asymmetry in $\bd \to \pi^+ \pi^-$. Here, however, because $\bd \to
K^0 \pi^+ \pi^-$ is a three-body decay, the relative $\pi^+\pi^-$ angular momentum is not fixed, and so the final state is not a CP eigenstate. Thus, the measurement of the indirect CP asymmetry in this decay does not give clean information. The situation is the same for the second pair, $\bs \to \kbar \pi^+\pi^-$ and $\bd \to K^+ K^0 K^-$.
In a similar vein, $\bd \to K^0 K^- \pi^+$ and $\bs \to K^+\kbar
\pi^-$ is the three-body equivalent of $\bs \to \pi^+ K^-$ and $\bd
\to \pi^- K^+$. Can the method of Ref. [@GRBsKpi], in which additional information comes from $\mathcal{B}(B^+ \to \pi^+ K^0)$, be adapted to this situation? Unfortunately, here too the answer is no. Unlike the two-body situation, here there is no other three-body decay which provides the appropriate additional information. This holds as well for pairs 4-6.
On the other hand, pair 7, $\bs \to \kbar \kbar K^0$ and $\bd \to K^0
K^0 \kbar$ is intriguing. The key point here is that, because there are truly identical particles in the final state, their relative angular momentum is even, and so the final state [*is*]{} a CP eigenstate. Now, the diagram contributing to the $e^{i\gamma}$ piece of the $\btos$ amplitude is ${\mathcal P}'_{b;uc}$. In Sec. \[estimates\], we noted that $|P'_{uc}|$ is expected to be small in two-body decays, and so a direct CP asymmetry which is proportional to this diagram will also be small. If the same property holds in three-body decays, the measurement of the indirect CP asymmetry in the pure-penguin decay $\bs \to \kbar \kbar K^0$ cleanly probes the $\bs$-$\bsbar$ mixing phase (experimentally, this might be easier than performing the angular analysis in $\bs\to J/\psi \phi$, which is presently done). However, if ${\mathcal P}'_{b;uc}$ is not small, as could happen if there are significant rescattering effects, then $A_s$ is not negligible, and the method of Ref. [@Fleischer99] can be applied to this pair to obtain $\gamma$. Here, U-spin symmetry is assumed, but, as noted above, it is possible to measure $X$, which gives the size of U-spin breaking.
SU(3) pairs {#3bodynonUspin}
-----------
Unlike two-body decays, with three-body decays one cannot obtain additional pairs satisfying Eq. (\[Uspinrels\]) by simply neglecting annihilation- and exchange-type diagrams. However, there is another possibility. If, as in the two-body case, one takes isospin into account in addition to U-spin symmetry, one effectively assumes full flavor SU(3) symmetry. Under this symmetry, $\pi$’s and $K$’s are identical particles, so that the final state in all decays contains three identical particles. In this case, the six permutations of these particles (the group $S_3$) must be considered. This was analyzed in Ref. [@diagramspaper]. For a given decay, there are six possibilities for the $S_3$ state of the three particles: a totally symmetric state $\ket{S}$, a totally antisymmetric state $\ket{A}$, or one of four mixed states $\ket{M_i}$ ($i=1$-4). The states are defined as follows. The final-state particles are numbered 1, 2, 3, so that the six possible orders are 123, 132, 312, 321, 231, 213. Under $S_3$, && ( + + + + + ) ,\
&& ( 2 + 2 - - - - ) ,\
&& ( - - + ) ,\
&& ( - - + + ) ,\
&& ( 2 - 2 - + - + ) ,\
&& ( - + - + - ) . \[SU3states\]
One can show that certain pairs of decays, related by SU(3) and not by U spin, satisfy Eq. (\[amps\]), but only for the state $\ket{S}$ (in most cases). This applies to the following SU(3) pairs[^6] (as is standard, we neglect annihilation- and exchange-type diagrams):
- ($B^+ \to \pi^+ K^- K^+$, $B^+ \to \pi^+ \pi^0 \pi^0$, $B^+ \to
\pi^+ \pi^+ \pi^-$, $\bs \to \kbar \pi^+\pi^-$) and ($\bd \to
K^+K^-K^0$, $B^+ \to K^+K^+K^-$, $B^+ \to K^+ \pi^+\pi^-$),
- ($B^+ \to K^+ \kbar \pi^0$, $B^+ \to \kbar K^+ \eta_8$) and ($B^+ \to K^0 \pi^+ \pi^0$, $\bd
\to K^+ \pi^- \pi^0$, $B^+ \to K^0 \pi^+ \eta_8$),
- $\bs \to \kbar \kbar K^0$ and ($B^+ \to K^+ K^0 \kbar$, $\bd \to
K^0 K^0 \kbar$),
- $\bd \to \pi^0 \pi^0 \pi^0$ and $\bd \to K^0 \pi^0 \pi^0$,
- ($\bd \to K^- K^+ \pi^0$, $\bd \to K^- K^+ \eta_8$) and ($\bs \to \pi^0 \pi^0 \eta_8$, $\bs \to \pi^+ \pi^- \eta_8$),
- $\bs \to \kbar \pi^0 \pi^0$ and $\bd \to K^0 \pi^0 \pi^0$,
- $\bs \to \kbar \pi^0 \eta_8$ and $\bd \to K^0 \pi^0 \eta_8$,
- $\bs \to \kbar \eta_8 \eta_8$ and $\bd \to K^0 \eta_8 \eta_8$.
The decays in the first (second) parentheses are $\btod$ ($\btos$) transitions.
In order to establish which states are the same (modulo CKM factors) for the decays within a pair, one writes the amplitudes for each decay in terms of diagrams, noting the order of the final-state particles for each diagram. It is this order which determines which $S_3$ states are common to both decays. The state $\ket{S}$ is symmetric in all possible orders. Thus, as long as the two amplitudes are comprised of the same diagrams, the final-state order is unimportant, and the two decays are related by SU(3) for $\ket{S}$. For $\ket{A}$, if the first decay amplitude contains the diagram $D$ with the order $ijk$, the second decay amplitude must contain $D$ with a cyclic permutation of $ijk$, or $-D$ with a anticyclic permutation of $ijk$.
The mixed states are more complicated. The six elements of $S_3$ are: $I$ (identity), $P_{12}$ (exchanges particles 1 and 2), $P_{13}$ (exchanges particles 1 and 3), $P_{23}$ (exchanges particles 2 and 3), $P_{cyclic}$ (cyclic permutation of particle numbers, i.e. $1\to 2$, $2\to 3$, $3\to 1$), $P_{anticyclic}$ (anticyclic permutation of particle numbers, i.e. $1\to 3$, $2\to 1$, $3\to 2$). The point is that, under the group transformations, $\ket{M_1}$ and $\ket{M_3}$ transform among themselves. Writing ( ) , ( ) , we can represent each group element by a $2\times 2$ matrix: & I = ( ) , P\_[12]{} = ( ) , P\_[13]{} = ( ) , &\
& P\_[23]{} = ( ) , P\_[cyclic]{} = ( ) , P\_[anticyclic]{} = ( ) . & \[matrices\] Similarly, if we write ( ) , ( ) , the $S_3$ matrices take the same form, showing that $\ket{M_2}$ and $\ket{M_4}$ also transform among themselves.
From the above matrices, we see that the first rows of the matrices are the same for ($I$, $P_{23}$), ($P_{12}$, $P_{cyclic}$) and ($P_{13}$, $P_{anticyclic}$). This indicates that the symmetric mixed states ($\ket{M_1}$ and $\ket{M_2}$) are the same for the two decays if the particle orders for a given diagram are \[(123) or (132)\], \[(213) or (231)\], or \[(321) or (312)\]. For the antisymmetric mixed states ($\ket{M_3}$ and $\ket{M_4}$), things are the same, except that there is an additional minus sign if the particle order is anticyclic (this can be seen from the second rows of the matrices).
To demonstrate how this works, we present several examples. First, consider the decays $B^+ \to \pi^+ K^- K^+$ and $\bd \to K^+K^-K^0$. For $B^+ \to \pi^+ K^- K^+$ we take particle 1 as $\pi^+$, particle 2 as $K^-$, and particle 3 as $K^+$. The amplitude is A(B\^+ \^+ K\^- K\^+) &=& -T\_[2,s]{}e\^[i]{}(123) - C\_[1,s]{}e\^[i]{}(132) - [P]{}\_[b;uc]{} e\^[i]{} (123)\
&& -0.8truecm - [P]{}\_[b;tc]{} e\^[-i]{} (123) + 13 P\_[EW1,u]{}e\^[-i]{}(231) -23 P\_[EW1,s]{}e\^[-i]{}(321)\
&& -0.8truecm + 13 P\^C\_[EW1,u]{}e\^[-i]{}(321) - 23 P\^C\_[EW2,s]{}e\^[-i]{}(321) , where the particle order for each diagram (top to bottom) is given in parentheses. We have continued to label each diagram by an index denoting the flavor of the popped quark pair, but under SU(3), these are all equal. For $\bd \to K^+K^-K^0$, we take particle 1 as $K^+$, particle 2 as $K^-$, particle 3 as $K^0$. The amplitude is A(K\^+K\^0K\^-) &=& -T’\_[2,s]{}e\^[i]{}(123) - C’\_[1,s]{}e\^[i]{}(312) -[P]{}’\_[b;uc]{} e\^[i]{}(123) + [P]{}’\_[b;tc]{}(123)\
&& -2.8truecm - 13 P’\_[EW1,u]{}(213) + 23 P’\_[EW1,s]{}(123) - 13 P\^[C]{}\_[EW1,u]{}(321) + 23 P\^[C]{}\_[EW2,s]{}(321) . The penguin diagrams for the two decays are defined in Eq. (\[Pdefs1\]). Comparing the two amplitudes, we see that, due to $C_{1,s}$ and $P_{EW1,s}$, $\ket{A}$ and the mixed states are not common. Therefore, the two decays are related only for $\ket{S}$.
Consider $\bs \to K^0 \kbar \kbar$ and $B^+ \to K^+ K^0 \kbar$. For $\bs \to K^0 \kbar \kbar$, particle 1 is $K^0$, particles 2 and 3 are $\kbar$ (consistent with the choice of mixed states above). $\ket{M_3}
= \ket{M_4} = \ket{A} = 0$. The amplitude is A(K\^0) &=& [P]{}\_[a;uc]{} e\^[i]{} (213) + [P]{}\_[a;tc]{} e\^[-i]{} (213) - 13 P\_[EW1,s]{} e\^[-i]{} (123)\
&& -2.5truecm - 13 P\_[EW1,d]{} e\^[-i]{} (213) - 13 P\_[EW1,s]{}\^C e\^[-i]{} (213) - 13 P\_[EW2,d]{}\^C e\^[-i]{} (213) . For $B^+ \to K^+ K^0 \kbar$, we take particle 1 as $K^+$, particle 2 as $K^0$, and particle 3 as $\kbar$. The amplitude is A(B\^+ K\^+ K\^0 ) &=& [P]{}’\_[b;uc]{} e\^[i]{} (231) - [P]{}’\_[b;tc]{} (231) + 13 P’\_[EW1,s]{} (231)\
&& -2truecm + 13 P’\_[EW1,d]{} (321) + 13 P\^[C]{}\_[EW2,s]{} (132) + 13 P\^[C]{}\_[EW1,d]{} (132) . The penguin diagrams for the two decays are defined in Eq. (\[Pdefs2\]). Due to the EWP’s, we see that the two decays are related only for $\ket{S}$.
Consider $B^+ \to \pi^0 \kbar K^+$ and $B^+ \to \pi^0 K^0\pi^+$. For $B^+ \to \pi^0 \kbar K^+$, we take particle 1 as $\pi^0$, particle 2 as $\kbar$, and particle 3 is $K^+$. The amplitude is A(B\^+ K\^+\^0) &=& -T\_[1,s]{}e\^[i]{} (321)- C\_[2,s]{}e\^[i]{} (321)\
&& -3truecm + [P]{}\_[b;uc]{}e\^[i]{} (123) - [P]{}\_[a;uc]{}e\^[i]{} (231) + [P]{}\_[b;tc]{}e\^[-i]{} (123) - [P]{}\_[a;tc]{}e\^[-i]{} (231)\
&& -3truecm - P\_[EW2,s]{}e\^[-i]{} (123) - 13 P\^C\_[EW1,d]{}e\^[-i]{} (321) - 23 P\^C\_[EW1,s]{} e\^[-i]{}(132)\
&& -3truecm + 13 P\^C\_[EW2,u]{} e\^[-i]{}(132) - 13 P\^C\_[EW2,s]{} e\^[-i]{}(321) . \[BKKpiamp\] The penguin diagrams are defined in Eqs. (\[Pdefs1\]) and (\[Pdefs2\]). For $B^+ \to \pi^0K^0\pi^+$, we take particle 1 as $\pi^0$, particle 2 as $K^0$, and particle 3 is $\pi^+$. The amplitude is A(B\^+ K\^0\^+\^0) &=& -T’\_[1,d]{}e\^[i]{} (321) -C’\_[2,d]{}e\^[i]{} (321) + P’\_[EW2,d]{}(123)\
&& + 13 P\^[C]{}\_[EW1,u]{} (312) + 23 P\^[ C]{}\_[EW1,d]{}(132) . Under SU(3), ${\mathcal P}_{b} = {\hat P}_{a}$ Thus, in order for the gluonic-penguin contribution to cancel in Eq. (\[BKKpiamp\]) above, we need a state which is symmetric in $(123) \leftrightarrow
(231)$. This is $\ket{S}$ or $\ket{A}$ – mixed states are excluded. However, $\ket{A}$ is itself excluded by the $P^C_{EW1}$ contribution – apart from CKM factors, it has the same sign in the two amplitudes, despite the particle order being cyclic in one case and anticyclic in the other. Thus, the two decay amplitudes are related only for $\ket{S}$.
Finally, consider $B^+ \to \pi^- \pi^+ \pi^+$ and $B^+ \to \pi^- K^+
\pi^+$. For $B^+\to \pi^-\pi^+\pi^+$, particle 1 is $\pi^-$, particles 2 and 3 are $\pi^+$. This implies that $\ket{M_3} = \ket{M_4} =
\ket{A} = 0$. The amplitude is A(B\^+\^-\^+\^+) &=& - T\_[2,d]{} e\^[i]{}(213) - C\_[1,d]{}e\^[i]{} (231) - [P]{}\_[b;uc]{} e\^[i]{}(213)\
&& -3truecm - [P]{}\_[b;tc]{} (213) + 13 P\_[EW1,u]{} (123) -23 P\_[EW1,d]{} (213)\
&& -3truecm + 13 P\^C\_[EW1,u]{} (213) - 23 P\^C\_[EW2,d]{} (213) . For $B^+ \to \pi^- K^+ \pi^+$, take particle 1 as $\pi^-$, particle 2 as $K^+$, and particle 3 is $\pi^+$. All six $S_3$ states allowed. The amplitude is A(B\^+\^- K\^+ \^+) &=& - T’\_[2,d]{} e\^[i]{}(213) - C’\_[1,d]{} e\^[i]{} (231) - [P]{}’\_[b;uc]{} e\^[i]{} (213)\
&& -3truecm + [P]{}’\_[b;tc]{} (213) - 13 P’\_[EW1,u]{} (132) +23 P’\_[EW1,d]{} (312)\
&& -3truecm - 13 P\^[,C]{}\_[EW1,u]{} (312) + 23 P\^[,C]{}\_[EW2,d]{} (312) . The penguin diagrams for the two decays are defined in Eq. (\[Pdefs1\]). All states with $2\leftrightarrow 3$ symmetry are allowed. Thus, unlike the above cases, the two decay amplitudes are related for $\ket{S}$, $\ket{M_1}$ and $\ket{M_2}$. This is a special case. Here, the processes are identical, save for the flavor of the decay quark ($d$ or $s$). As a result, the amplitudes are equal for all nonzero states. There is one other pair like this – $B^+ \to K^-
\pi^+ K^+$ and $B^+ \to K^- K^+ K^+$. For all other pairs, the two decay amplitudes are related only for $\ket{S}$ (or for all $S_3$ states in the case of U-spin pairs).
Now, in Refs. [@diagramspaper; @Kpipipaper] it was shown how the $S_3$ states can be determined experimentally. Below we review the method, focussing on the state $\ket{S}$. Consider the decay $B^+ \to
\pi^+ K^- K^+$. The Dalitz-plot events can be described by $s_+ =
\left( p_{\pi^+} + p_{K^+} \right)^2$ and $s_- = \left( p_{\pi^+} +
p_{K^-} \right)^2$, so that the decay amplitude, ${\cal M}(s_+,s_-)$, can be extracted. We introduce the third Mandelstam variable, $s_0 =
\left( p_{K^+} + p_{K^-} \right)^2$. It is related to $s_+$ and $s_-$ as follows: s\_+ + s\_- + s\_0 = m\_B\^2 + m\_\^2 + 2m\_K\^2 . The totally symmetric SU(3) decay amplitude is then given by &=& . The state $\ket{S}$ can be determined for the other decays similarly. With this, the size of U-spin/SU(3) breaking can be found through the measurement of $X$ using any of the SU(3) pairs.
Other signals of SU(3) breaking
-------------------------------
Finally, we note that there are certain decays which have identical amplitudes for the totally symmetric state $\ket{S}$. They are given by the processes within parentheses in the list in Sec. \[3bodynonUspin\]. For these, the branching ratios and direct CP asymmetries should be equal in the SU(3) limit. Thus, by obtaining the state $\ket{S}$ for these decays, the measurement of these quantities constitutes a further test of SU(3) breaking.
Conclusions
===========
Within U-spin symmetry ($d\leftrightarrow s$), the amplitudes for certain charmless $\btod$ and $\btos$ decays are equal, apart from CKM matrix elements. Using this, two methods were proposed for extracting weak-phase information from measurements of particular U-spin decay pairs. The theoretical uncertainty of these methods must include the issue of U-spin breaking. In general, theoretical input is used to address this. However, one of the points of the present paper is that this breaking can be measured experimentally. Under U spin, the branching ratios and direct CP asymmetries of the two decays are not independent – there is a relation among them. Thus, one can determine U-spin breaking by measuring the four observables, and seeing the extent to which this relation is not satisfied.
Furthermore, if one neglects annihilation- and exchange-type diagrams, there are additional pairs of $B$ decays whose amplitudes are equal, apart from CKM matrix elements. In this case, the symmetry is flavor SU(3). Here, too, the relation among the four observables holds in the SU(3) limit, so that SU(3)-breaking effects can be determined from the measurements of these quantities.
In this paper, we present the list of two-body $B$ decay pairs from which the size of the breaking can be obtained. In fact, there are five such pairs for which these measurements have been done. We present this data, along with the determination of U-spin/SU(3) breaking. In many such decays, the calculation of the factorizable contribution to the breaking is reliable. Taking this into account, one can measure the size of nonfactorizable effects. It is expected that these are small. However, there is one decay pair – $\bd \to
\pi^+\pi^-$ and $\bd \to\pi^-K^+$ – which exhibits large ($\sim
2.5\sigma$) nonfactorizable breaking. With only one data point, one cannot draw any firm conclusions. However it does perhaps provide an interesting hint, and raises questions about analyses which neglect nonfactorizable U-spin/SU(3) breaking.
We also present the list of three-body $B$ decay pairs whose amplitudes are the same, apart from CKM factors. However, here the situation is more complicated. Under SU(3), the final-state particles are all identical, and the equality of amplitudes holds (almost always) only for the totally symmetric final state $|S\rangle$. Thus, this state must be isolated experimentally in order to measure SU(3) breaking, and we describe how to do this.
We discuss the decay pairs whose amplitudes are equal, including CKM factors, within SU(3). For two-body decays, the size of SU(3) breaking is indicated by comparing the branching ratios and direct CP asymmetries of the two decays. For three-body decays, once again the equality of amplitudes holds only for $|S\rangle$, so that this state must be distinguised in order to probe SU(3) breaking.
Finally, we note in passing that the pure-penguin decay $\bs \to \kbar
\kbar K^0$ is particularly interesting. Here the final state is a CP eigenstate. Thus, given that the direct CP asymmetry is expected to be small, the measurement of the indirect CP asymmetry in this decay cleanly probes the $\bs$-$\bsbar$ mixing phase. This might be easier experimentally than performing the angular analysis in $\bs\to J/\psi
\phi$, which is what is done at present.
[**Acknowledgments**]{}: We thank A. Datta, M. Duraisamy, M. Gronau, J. Rosner and D. Tonelli for helpful communications. This work was financially supported by NSERC of Canada and FQRNT of Québec.
[99]{}
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[^1]: Note: because isospin is a good symmetry, in practice there is little difference between U spin and SU(3).
[^2]: Much of the material in this section can be found in Refs. [@GroUspin; @NSL].
[^3]: The diagrams include the magnitudes of the associated CKM matrix elements.
[^4]: $C'$ and $C$ in Eqs. (\[btoshierarchy\]) and (\[btodhierarchy\]) represent color-suppressed tree diagrams, and are not the parameters in Eq. (\[fullamps\]).
[^5]: Decays such as $\bd\to\pi^0 K^0$ constitute an exception to this rule, as they can be factorized in two different ways. However, there are very few such decays.
[^6]: Note: this list includes some U-spin pairs. These pairs are related for all $S_3$ states.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We exhibit path connected metric spaces and a map $f:X\rightarrow Y$ such that $f$ induces an isomorphism on homotopy groups but such that, with natural topologies, $X$ and $Y$ have nonhomeomorphic fundamental groups. Consequently we can conclude $X$ and $Y$ have distinct homotopy types despite the failure of the Whitehead theorem in this context.'
author:
- |
Paul Fabel\
Department of Mathematics & Statistics\
Mississippi State University
title: Topological fundamental groups can distinguish spaces with isomorphic homotopy groups
---
Introduction
============
Given CW complexes $X$ and $Y,$ the Whitehead theorem ([@hatch]) asserts that a map $f:X\rightarrow Y$ is a homotopy equivalence provided $f$ induces an isomorphism on homotopy groups. However the result can fail in the context of path connected metric spaces. For example the standard Warsaw circle has trivial homotopy groups but fails to have the homotopy type of a point. This note aims to show the *topological fundamental group* can help counterbalance the general failure of the Whitehead theorem.
For a general space $X$ work of Biss [@Biss] initiates the development of a theory whose fundamental notion is the following. Endowed with the quotient topology inherited from the path components of based loops in $X$, the familiar based fundamental group $\pi _{1}(X,p)$ of a topological space $X$ becomes a *topological group*. For example if $X$ is locally contractible then loops in $X$ are homotopically invariant under small perturbation, and consequently the fundamental group $\pi _{1}(X,p)$ has the discrete topology. For spaces that are complicated both locally and globally, the topology of $\pi _{1}(X,p)$ can be more interesting ([@fab] [@fab2] [@fab5]). An important feature of the theory is that if $X$ and $Y$ have the same homotopy type then $\pi _{1}(X,p)$ and $\pi _{1}(Y,p)$ are isomorphic *and* homeomorphic (Proposition 3.3 [@Biss]).
These facts motivate the question of whether the added topological structure on $\pi _{1}(X,p)$ can ever succeed in distinguishing the homotopy type of spaces $X$ and $Y$ in instances when the hypotheses of the Whitehead theorem are satisfied.
In fact this is so and in this note we exhibit aspherical spaces $X$ and $Y$ such that inclusion $j:X\rightarrow Y$ induces an isomorphism on homotopy groups. However $\pi _{1}(X)$ and $\pi _{1}(Y)$ fail to be homeomorphic, and thus we can conclude that $X$ and $Y$ do not have the same homotopy type despite the failure of the Whitehead theorem for this pair of examples.
The theory of topological fundamental groups is still in the early stages of development ([@fab3] [@fab4] [@fab6]) and it is hoped this note will be seen as promoting its utility and helping to motivate its continued investigation. For example the space $Y$ constructed in this paper is not locally path connected. This suggests the following.
**Question.** Suppose $Y$ is an aspherical (metric) Peano continuum and $X\subset Y$ is aspherical and path connected. Suppose inclusion $j:X\hookrightarrow Y$ induces an isomorphism $j^{\ast }:\pi
_{1}(X,p)\rightarrow \pi _{1}(Y,p).$ Must $j^{\ast }$ be a homeomorphism? If $j^{\ast }$ is a homeomorphism must $j$ be a homotopy equivalence?
Definitions and Preliminaries
=============================
All definitions are compatible with those found in Munkres [@Munk]. If $X $ is a metrizable space and $p\in X$ let $C_{p}(X)=\{f:[0,1]\rightarrow X$ such that $f$ is continuous and $f(0)=f(1)=p\}.$ Endow $C_{p}(X)$ with the topology of uniform convergence.
The **topological fundamental group** $\pi _{1}(X,p)$ is the set of path components of $C_{p}(X)$ endowed with the quotient topology under the canonical surjection $q:C_{p}(X)\rightarrow \pi _{1}(X,p)$ satisfying $q(f)=q(g)$ if and only if $f$ and $g$ belong to the same path component of $C_{p}(X).$ Thus a set $U\subset \pi _{1}(X)$ is open if and only if $q^{-1}(U)$ is open in $C_{p}(Y).$
The topological fundamental group $\pi _{1}(X,p)$ is a topological group under concatenation of paths. (Proposition 3.1[@Biss]). A map $f:X\rightarrow Y$ determines a continuous homomorphism $f^{\ast }:\pi
_{1}(X,p)\rightarrow \pi _{1}(Y,f(p))$ via $f^{\ast }([\alpha ])=[f(\alpha
)] $ (Proposition 3.3 [@Biss]). If $X$ and $Y$ have the same homotopy type then $\pi _{1}(X)$ is homeomorphic and isomorphic to $\pi _{1}(Y)$ (Corollary 3.4 [@Biss]) For the remainder of this paper all fundamental groups will be considered topological groups.
The space $X$ is **semilocally simply connected** at $p$ if there exists an open set $U\subset X$ such that inclusion $j:U\hookrightarrow X$ induces the trivial homomorphism $j^{\ast }:\pi _{1}(U,p)\rightarrow \pi
_{1}(X,p).$ The space $Z$ is **discrete** if each one point subset of $Z $ is open.
\[rem2\]The main result of [@fab] shows that if $X$ is locally path connected then $\pi _{1}(X,p)$ is discrete if and only if $\pi _{1}(X,p)$ is semilocally simply connected.
Main result
===========
There exist path connected aspherical separable metric spaces $X$ and $Y$ such that $X\subset Y$ and inclusion $j:X\hookrightarrow Y$ induces an isomorphism $j^{\ast }:\pi _{1}(X,p)\rightarrow \pi _{1}(Y,p)$. Thus $(X,Y,j) $ satisfies the hypothesis of the Whitehead theorem. However the topological fundamental groups $\pi _{1}(X,p)$ and $\pi _{1}(Y,p)$ are not homeomorphic. Hence the **topology** of fundamental groups has the capacity to distinguish the homotopy type of $X$ and $Y$ when the algebra fails to do so.
The basic idea is to let $X$ denote the countable union of a sequence of large simple closed curves $C_{1}\cup C_{2}...$ joined at a common point $p.$ Such a space is sometimes called a bouquet of infinitely many loops. In particular $X$ is locally contractible and should not be mistaken for the Hawaiian earring. The space $Y$ is a compactification of $X$ obtained by attaching a line segment $\alpha $ based at $p$ such that the curves $C_{n}$ converge to $\alpha $ in the Hausdorff metric.
Since each of $X$ and $Y$ is path connected and 1 dimensional, if $n\neq 1$ then $\pi _{n}(X,p)=\pi _{n}(Y,p)=1.$ Thus, to show that $(X,Y,j)$ satisfies the hypothesis of the Whitehead theorem it suffices to show that $j^{\ast
}:\pi _{1}(X,p)\rightarrow \pi _{1}(Y,p)$ is an isomorphism.
Formally for $n\geq 2$ let $C_{n}\subset R^{2}$ denote boundary of the convex hull of the following 3 point set: $\{(0,0),(\frac{1}{n},1),(\frac{1}{n},1)+\frac{1}{10^{(10n)}}(n,-1)\}.$ Then for each $n\geq 2$ $C_{n}$ is the boundary of a triangle and in particular $C_{n}$ is a simple closed curve. Let $p=(0,0).$ Note $C_{n}\cap C_{m}=p$ if $n\neq m.$ Let $\alpha $ denote the line segment $[(0,0),(0,1)]\subset R^{2}.$ Let $X=\cup _{n=2}^{\infty
}C_{n}$ and let $Y=\overline{X}.$ Note $X\cup \alpha .$ Note the path connected spaces $X$ and $Y$ are 1 dimensional and hence aspherical ([@Fort]). We will show inclusion $j:X\hookrightarrow Y$ induces an isomorphism $j^{\ast }:\pi _{1}(X,p)\rightarrow \pi _{1}(Y,p).$
To prove $j^{\ast }$ is one to one suppose $f:\partial D^{2}\rightarrow X$ is inessential in $Y$ and suppose $f(1)=p.$ Let $F:D^{2}\rightarrow Y$ satisfy $F_{\partial D^{2}}=f.$ Let $U=F^{-1}(\alpha \backslash p).$ Since $D^{2}$ is locally path connected, and since $\alpha \backslash p$ is a component of $X\backslash p$ the set $U$ is open. Suppose $x\in \overline{U}\backslash U.$ Then $F(x)=p$ since $\alpha =\overline{\alpha \backslash p}.$ Thus, we may redefine $F$ to be $p$ on the set $U$ and obtain a continuous function $G:D^{2}\rightarrow X$ such that $G_{\partial D^{2}}=f.$ This proves $j^{\ast }$ is one to one.
To prove $j^{\ast }$ is a surjection suppose $\beta \in C_{p}(Y).$ We must show there exists $\gamma \in C_{p}(X)$ such that $\gamma $ and $\beta $ are path homotopic in $Y.$ Since $im(\beta )$ is a Peano continuum $im(\beta )$ is locally path connected. Thus we may choose $N$ such that $im(\beta )\cap
(\{\frac{1}{N},\frac{1}{N+1},..\}\times \{1\})=\emptyset .$ Let $A=im(\beta
)\cap (\alpha \cup C_{N}\cup C_{N+1}...).$ Let $B=C_{1}\cup C_{2}...\cup
C_{N-1}.$ Note $A$ is a contractible Peano continuum such that $p\in A.$ Moreover $B$ is a strong deformation retract of $B\cup A.$ Thus there exists a homotopy $h_{t}:A\cup B\rightarrow B$ such that $h_{0}=id_{A\cup B}$ and $h_{t}$ fixes $B$ pointwise. Thus the homotopy $h_{t}(\beta )$ determines that $\beta $ is path homotopic in $Y$ to $\gamma =h_{1}(\beta ).$ Note $im(\gamma )\subset X.$ Hence $j^{\ast }$ is a surjection and therefore $j^{\ast }$ is an isomorphism.
Since the space $X$ is locally contractible $\pi _{1}(X,p)$ has the discrete topology (Remark \[rem2\]). On the other hand $\pi _{1}(Y,p)$ does not have the discrete topology, since there exists an inessential loop $f\in
C_{p}(Y)$ which is the uniform limit of inessential loops. (Let the (inessential) map $f$ go up and down once on $\alpha $ and let $f_{n}$ be an (essential) loop going once around $C_{n}$). Thus the path component of the constant map is not open in $C_{p}(Y)$ and thus $\pi _{1}(Y,p)$ cannot have the discrete topology. Thus $\pi _{1}(X,p)$ and $\pi _{1}(Y,p)$ are not homeomorphic and hence $X$ and $Y$ do not have the same homotopy type.
The space $Y$ constructed is semilocally simply connected. However $\pi
_{1}(Y,p)$ does not have the discrete topology. Consequently $Y$ is a counterexample to the (false) Theorem 5.1 [@Biss] which asserts that $\pi _{1}(Y,p)$ is discrete if and only if $\pi _{1}(Y,p)$ is semilocally simply connected.
[99]{} Biss, Daniel K. *The topological fundamental group and generalized covering spaces.* Topology Appl. 124 (2002), no. 3, 355–371.
Curtis, M. L.; Fort, M. K., Jr. *Homotopy groups of one-dimensional spaces.* Proc. Amer. Math. Soc. 8 (1957), 577–579.
Fabel, Paul. *A characterization of spaces with discrete topological fundamental group.* Preprint. http://front.math.ucdavis.edu/math.GN/0502249
Fabel, Paul. *The fundamental group of the harmonic archipelago.* Preprint.http://front.math.ucdavis.edu/math.AT/0501426
Fabel, Paul. *The topological Hawaiian earring group does not embed in the inverse limit of free groups.* Preprint. http://front.math.ucdavis.edu/math.GN/0501482
Fabel, Paul *A retraction theorem for topological fundamental groups with applications to the Hawaiian earring.* Preprint.http://front.math.ucdavis.edu/math.AT/0502218
Fabel, Paul *The Hawaiian earring group is topologically incomplete.* Preprint. http://front.math.ucdavis.edu/math.GN/0502148
Fabel, Paul *A monomorphism theorem for the inverse limit of nested retracts.* Preprint. http://front.math.ucdavis.edu/math.AT/0502275
Hatcher, Allen. *Algebraic topology*. Cambridge University Press, Cambridge, 2002.
Munkres, James R., *Topology: a first course.* Prentice-Hall, Inc., Englewood Cliffs, N.J., 1975.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'For a given nonnegative integer $\alpha$, a matrix $A_n$ of size $n$ is called $\alpha$-Toeplitz if its entries obey the rule $A_n=\left[a_{r-\alpha s}\right]_{r,s=0}^{n-1}$. Analogously, a matrix $A_n$ again of size $n$ is called $\alpha$-circulant if $A_n=\left[a_{(r-\alpha s)\ {\rm mod}\, n}\right]_{r,s=0}^{n-1}$. Such kind of matrices arises in wavelet analysis, subdivision algorithms and more generally when dealing with multigrid/multilevel methods for structured matrices and approximations of boundary value problems. In this paper we study the singular values of $\alpha$-circulants and we provide an asymptotic analysis of the distribution results for the singular values of $\alpha$-Toeplitz sequences in the case where $\{a_k\}$ can be interpreted as the sequence of Fourier coefficients of an integrable function $f$ over the domain $(-\pi,\pi)$. Some generalizations to the block, multilevel case, amounting to choose $f$ multivariate and matrix valued, are briefly considered.'
author:
- 'Eric Ngondiep, Stefano Serra-Capizzano, and Debora Sesana [^1]'
title: 'Spectral features and asymptotic properties for $\alpha$-circulants and $\alpha$-Toeplitz sequences: theoretical results and examples'
---
[**Keywords:**]{} circulants, Toeplitz, $\alpha$-circulants, $\alpha$-Toeplitz, spectral distributions, multigrid methods.\
[**AMS SC:**]{} 65F10, 15A18.
Introduction {#sec1}
============
A matrix $A_n$ of size $n$ is called $\alpha$-Toeplitz if its entries obey the rule $A_n=\left[a_{r-\alpha
s}\right]_{r,s=0}^{n-1}$, where $\alpha$ is a nonnegative integer. As an example, if $n=5$ and $\alpha=3$ then $$\begin{aligned}
A_n\equiv T_{n,\alpha}=\left[\begin{array}{ccccc}
a_{0} & a_{-3} & a_{-6} & a_{-9} & a_{-12} \\
a_{1} & a_{-2} & a_{-5} & a_{-8} & a_{-11} \\
a_{2} & a_{-1} & a_{-4} & a_{-7} & a_{-10} \\
a_{3} & a_{0} & a_{-3} & a_{-6} & a_{-9} \\
a_{4} & a_{1} & a_{-2} & a_{-5} & a_{-8}
\end{array}\right].\end{aligned}$$
Along the same lines, a matrix $A_n$ of size $n$ is called $\alpha$-circulant if $A_n=\left[a_{(r-\alpha s)\ {\rm mod}\,
n}\right]_{r,s=0}^{n-1}$. For instance if $n=5$ and $\alpha=3$ then we have $$\begin{aligned}
A_n\equiv C_{n,\alpha}=\left[\begin{array}{ccccc}
a_{0} & a_{2} & a_{4} & a_{1} & a_{3} \\
a_{1} & a_{3} & a_{0} & a_{2} & a_{4} \\
a_{2} & a_{4} & a_{1} & a_{3} & a_{0} \\
a_{3} & a_{0} & a_{2} & a_{4} & a_{1} \\
a_{4} & a_{1} & a_{3} & a_{0} & a_{2}
\end{array}\right].\end{aligned}$$
Such kind of matrices arises in wavelet analysis [@wave] and subdivision algorithms or, equivalently, in the associated refinement equations, see [@subd] and references therein. Furthermore, it is interesting to remind that Gilbert Strang [@strang] has shown rich connections between dilation equations in the wavelets context and multigrid methods [@Hack; @Trot], when constructing the restriction/prolongation operators [@FS2; @ADS] with various boundary conditions. It is worth noticing that the use of different boundary conditions is quite natural when dealing with signal/image restoration problems or differential equations, see [@model-tau; @Sun].
In this paper we address the problem of characterizing the singular values of $\alpha$-circulants and of providing an asymptotic analysis of the distribution results for the singular values of $\alpha$-Toeplitz sequences, in the case where the sequence of values $\{a_k\}$, defining the entries of the matrices, can be interpreted as the sequence of Fourier coefficients of an integrable function $f$ over the domain $(-\pi,\pi)$. As a byproduct, we will show interesting relations with the analysis of convergence of multigrid methods given, e.g., in [@mcirco; @ADS]. Finally we generalize the analysis to the block, multilevel case, amounting to choose the symbol $f$ multivariate, i.e., defined on the set $(-\pi,\pi)^d$ for some $d>1$, and matrix valued, i.e., such that $f(x)$ is a matrix of given size $p\times q$.
The paper is organized as follows. In Section \[sec:tools\] we report useful definitions, well-known results in the standard case of circulants and Toeplitz that is when $\alpha=1$ (or $\alpha=e$, $e=(1,\ldots,1)$, in the multilevel setting), and a preliminary analysis of some special cases. Section \[sec:circ\] deals with the singular value analysis of $\alpha$-circulants while in Section \[sec:toep\] we treat the $\alpha$-Toeplitz case in an asymptotic setting, and more precisely in the sense of the Weyl spectral distributions. Section \[sec:multigrid\] is devoted to sketch useful connections with multigrid methods, while in Section \[sec:gen\] we report the generalization of the results when we deal the multilevel block case. Section \[sec:fin\] is aimed to draw conclusions and to indicate future lines of research.
General definitions and tools {#sec:tools}
=============================
For any $n\times n$ matrix $A$ with eigenvalues $\lambda_j(A)$, $j=1,\ldots,n$, and for any $m\times n$ matrix $B$ with singular values $\sigma_j(B)$, $j=1,\ldots,l$, $l=\min\{m,n\}$, we set $${\rm Eig}(A)= \{\lambda_1(A), \lambda_2(A),\ldots,\lambda_n(A)\},
\ \ \ \ {\rm Sgval}(B)= \{\sigma_1(B),
\sigma_2(B),\ldots,\sigma_l(B)\}.$$
The matrix $B^{*}B$ is positive semidefinite, since $x^{*}(B^{*}B)x=\|Bx\|_{2}^{2}\geq 0$ for all $x\in \mathbb{C}^{n}$, with $^*$ denoting the transpose conjugate operator. Moreover, it is clear that the eigenvalues $\lambda_{1}(B^*B)\geq
\lambda_{2}(B^*B)\geq\cdots\geq \lambda_{n}(B^*B)\geq 0$ are nonnegative and can therefore be written in the form $$\label{1}
\lambda_{j}(B^*B)=\sigma_{j}^{2},$$ with $\sigma_{j}\geq0$, $j=1,\ldots,n$. The numbers $\sigma_{1}\geq\sigma_{2}\geq\cdots\geq\sigma_{l}\geq0$, $l=\min\{m,n\}$, are called singular values of $B$, i.e., $\sigma_{j}=\sigma_{j}(B)$ and if $n>l$ then $\lambda_{j}(B^*B)=0$, $j=l+1,\ldots,n$. A more general statement is contained in the singular value decomposition theorem (see e.g. [@GV]).
\[2\] Let $B$ be an arbitrary (complex) $m\times n$ matrix. Then:
\(a) There exists a unitary $m\times m$ matrix $U$ and a unitary $n\times n$ matrix $V$ such that $U^{*}BV=\Sigma$ is an $m\times n$ “diagonal matrix” of the following form: $$\Sigma=\begin{bmatrix}
D & 0 \\
0 & 0 \\
\end{bmatrix}, \text{\,\,}
D:={\rm diag}(\sigma_{1},\ldots,\sigma_{r}),\text{\,\,\,}
\sigma_{1}\geq\sigma_{2}\geq\cdots\geq\sigma_{r}>0.$$
Here $\sigma_{1},\ldots,\sigma_{r}$ are the nonvanishing singular values of $B$, and $r$ is the rank of $B.$
\(b) $\;\,$The nonvanishing singular values of $B^{*}$ are also precisely the number $\sigma_{1},\ldots,\sigma_{r}.$\
The decomposition $B=U\Sigma V^{*}$ is called “the singular value decomposition of $B$”.
For any function $F$ defined on ${\mathbb R}^+_0$ and for any $m\times n$ matrix $A$, the symbol $\Sigma_{\sigma}(F,A)$ stands for the mean $$\begin{aligned}
\label{sigmaFA}
\Sigma_{\sigma}(F,A):= {\frac 1 {\min\{n,m\}}
\sum_{j=1}^{\min\{n,m\}} F\left(\sigma_j(A)\right)}={\frac 1
{\min\{n,m\}} \sum_{\sigma\in {\rm Sgval}(A)} F(\sigma)}.\end{aligned}$$
Throughout this paper we speak also of [*matrix sequences*]{} as sequences $\{A_k\}$ where $A_k$ is an $n(k)\times m(k)$ matrix with $\min\{n(k),m(k)\}\rightarrow \infty$ as $k\rightarrow \infty$. When $n(k)=m(k)$ that is all the involved matrices are square, and this will occur often in the paper, we will not need the extra parameter $k$ and we will consider simply matrix sequences of the form $\{A_n\}$.
Concerning the case of matrix-sequences an important notion is that of spectral distribution in the eigenvalue or singular value sense, linking the collective behavior of the eigenvalues or singular values of all the matrices in the sequence to a given function (or to a measure). The notion goes back to Weyl and has been investigated by many authors in the Toeplitz and Locally Toeplitz context (see the book by Böttcher and Silbermann [@BS] where many classical results by the authors, Szegö, Avram, Parter, Widom Tyrtyshnikov, and many other can be found, and more recent results in [@jacobi-GOL; @ku-ser; @zabroda; @tyrtL1; @Tillinota; @tillicomplex]). Here we report the definition of spectral distribution only in the singular value sense since our analysis is devoted to singular values. The case of eigenvalues will be the subject of future investigations.
\[def-distribution\]
Let $\mathcal C_0({\mathbb R}^+_0)$ be the set of continuous functions with bounded support defined over the nonnegative real numbers, $d$ a positive integer, and $\theta$ a complex-valued measurable function defined on a set $G\subset\mathbb R^d$ of finite and positive Lebesgue measure $\mu(G)$. Here $G$ will be often equal to $(-\pi,\pi)^d$ so that $e^{i\overline{G}}={\mathbb T}^d$ with ${\mathbb T}$ denoting the complex unit circle. A matrix sequence $\{A_k\}$ is said to be [*distributed $($in the sense of the singular values$)$ as the pair $(\theta,G)$,*]{} or to [*have the distribution function $\theta$*]{} ($\{A_k\}\sim_{\sigma}(\theta,G)$), if, $\forall F\in \mathcal
C_0({\mathbb R}^+_0)$, the following limit relation holds $$\label{distribution:sv-eig}
\lim_{k\rightarrow
\infty}\Sigma_{\sigma}(F,A_k)=\frac1{\mu(G)}\,\int_G F(|\theta(t)|)\,
dt,\qquad t=(t_{1},\ldots,t_{d}).$$
When considering $\theta$ taking values in ${\cal {M}}_{pq}$, where ${\cal {M}}_{pq}$ is the space of $p \times q$ matrices with complex entries and a function is considered to be measurable if and only if the component functions are, we say that $\{A_k\}\sim_{\sigma}
(\theta,G)$ when for every $F\in \mathcal C_0({\mathbb R}^+_0)$ we have $$\lim_{k\rightarrow \infty}\Sigma_{\sigma}(F,A_k)= \frac {1}
{\mu(G)}\,\int_G \frac {\sum_{j=1}^{\min\{p,q\}}
\left(F(\sigma_j(\theta(t)))\right)} {\min\{p,q\}}\, dt,\qquad t=(t_{1},\ldots,t_{d}),$$ with $\sigma_j(\theta(t))=\sqrt{\lambda_j(\theta(t)\theta^*(t))}=\lambda_j(\sqrt{\theta(t)\theta^*(t)})$. Finally we say that two sequences $\{A_k\}$ and $\{B_k\}$ are [*equally distributed*]{} in the sense of singular values ($\sigma$) if, $\forall F\in \mathcal C_0({\mathbb R}^+_0)$, we have $$\lim_{k\rightarrow\infty}[\Sigma_{\sigma}(F,B_k)-\Sigma_{\sigma}(F,A_k)]=0.$$
Here we are interested in explicit formulae for the singular values of $\alpha$-circulants and in distribution results for $\alpha$-Toeplitz sequences. In the latter case, following what is known in the standard case of $\alpha=1$ (or $\alpha=e$ in the multilevel setting), we need to link the coefficients of the $\alpha$-Toeplitz sequence to a certain symbol.
Let $f$ be a Lebesgue integrable function defined on $(-\pi,\pi)^d$ and taking values in ${\cal {M}}_{pq}$, for given positive integers $p$ and $q$. Then, for $d$-indices $r=(r_1,\ldots,r_d),
j=(j_1,\ldots,j_d), n=(n_1,\ldots,n_d)$, $e=(1,\ldots,1)$, $\underline{0}=(0,\ldots,0)$, the Toeplitz matrix $T_n(f)$ of size $p\hat n\times q\hat n$, $\hat n=n_1\cdot n_2\cdots
n_d$, is defined as follows $$\begin{aligned}
T_{n}(f)=[\tilde{f}_{r-j}]_{r,j=\underline{0}}^{n-e},\end{aligned}$$ where $\tilde{f}_{k}$ are the Fourier coefficients of $f$ defined by equation $$\label{defcoeff}
\tilde{f}_j=\tilde{f}_{(j_1, \ldots, j_d)} (f) = \frac 1 {(2\pi)^d}
\int_{{[-\pi,\pi]}^d} f(t_1,\ldots,t_d)e^{-i(j_1t_1 + \cdots +
j_dt_d)}\, dt_1 \cdots dt_d,\quad \quad i^2=-1,$$ for integers $j_{\ell}$ such that $-\infty < j_{\ell} < \infty$ for $1 \le \ell \le d$. Since $f$ is a matrix-valued function of $d$ variables whose component functions are all integrable, then the $(j_1, \ldots, j_d)$-th Fourier coefficient is considered to be the matrix whose $(u,v)$-th entry is the $(j_1, \ldots, j_d)$-th Fourier coefficient of the function $(f(t_1,\ldots,t_d))_{u,v}$.
According to this multi-index block notation we can define general multi-level block $\alpha$-Toeplitz and $\alpha$-circulants. Of course, in this multidimensional setting, $\alpha$ denotes a $d$-dimensional vector of nonnegative integers that is $\alpha=(\alpha_1,\ldots,\alpha_d)$. In that case $A_n=\left[a_{r-\alpha \circ s}\right]_{r,s=\underline{0}}^{n-e}$ where the $\circ$ operation is the componentwise Hadamard product between vectors or matrices of the same size. A matrix $A_n$ of size $p\hat
n\times q\hat n$ is called $\alpha$-circulant if $A_n=\left[a_{(r-\alpha \circ s)\ {\rm mod}\,
n}\right]_{r,s=\underline{0}}^{n-e}$, where $$\begin{aligned}
(r-\alpha\circ s)\textrm{ mod $n$}=\left((r_{1}-\alpha_{1}s_{1})\textrm{ mod $n_{1}$},(r_{2}-\alpha_{2}s_{2})\textrm{ mod $n_{2}$},\ldots,(r_{d}-\alpha_{d}s_{d})\textrm{ mod $n_{d}$}\right).
\end{aligned}$$
The extremal cases where $\alpha=\underline{0}$ or $\alpha=e$, and the intermediate cases
-----------------------------------------------------------------------------------------
We consider a $d$-level setting and we analyze in detail the case where $\underline{0}\le
\alpha\le e$ and with $\le$ denoting the componentwise partial ordering between real vectors. When $\alpha$ has at least a zero component, the analysis can be reduced to the positive one as studied in Subsection \[nonnegative-vs-positive\].
### $\alpha=e$
In the literature the only case deeply studied is the case of $\alpha=e$ (standard shift in every level). Here for multilevel block circulants $A_{n}=[a_{(r-\alpha\circ s)\textrm{ mod $n$}}]_{r,s=\underline{0}}^{n-e}$ the singular values are given by those of $$\begin{aligned}
\sigma_{k}(A_{n})=\sum_{j=\underline{0}}^{n-e} a_j e^{i2\pi(j_1 k_1/n_1 + \cdots
+ j_d k_d/n_d)},\qquad k=(k_{1},\ldots,k_{d}),\end{aligned}$$ for any $k_{\ell}$ such that $0\le k_{\ell}\le n_{\ell}-1$, $\ell=1,\ldots,d$. Of course when the coefficients $a_j$ comes from the Fourier coefficients of a given Lebesgue integrable function $f$, i.e. $\tilde{f}_j=a_{j\ {\rm mod}\, n}$, $j=-n/2,\ldots,n/2$ ($n/2=(n_{1}/2,n_{2}/2,\ldots,n_{d}/2)$), the singular values are those of $n/2$-th Fourier sum of $f$ evaluated at the grid points $$2\pi k/n=2\pi\left(k_1/n_1,\ldots,k_d/n_d\right),$$ $0\le k_j\le n_j-1$, $j=1,\ldots,d$. Moreover the explicit Schur decomposition is known. For $d=p=q=1$ any standard circulant matrix can be written in the form $$\begin{aligned}
\label{iV}
A_{n}\equiv C_{n} =F_{n}D_{n}F_{n}^{\ast},\end{aligned}$$ where $$\begin{aligned}
\label{V}
\notag F_{n} &=&\frac{1}{\sqrt{n}}\left[e^{-\frac{2\pi ijk}{n}}\right]_{j,k=0}^{n-1},\text{\,\,Fourier matrix,} \\
D_{n} &=& {\rm diag}(\sqrt{n}F_{n}^{\ast}\underline{a}), \\
\notag \underline{a} &=&\left[a_{0},a_{1},\ldots,a_{n-1}\right]^{T}, \text{\,\,\,
first column of the matrix $A_{n}$}.\end{aligned}$$
Of course for general $d,p,q$ the formula generalizes as $$A_{n} =(F_{n}\otimes I_p) D_{n}(F_{n}^{\ast}\otimes I_q),$$ with $F_n=F_{n_1}\otimes F_{n_2} \otimes \cdots \otimes F_{n_d}$ $D_{n} = {\rm diag}(\sqrt{\hat n}(F_{n}^{\ast}\otimes
I_p)\underline{a})$, where $\hat{n}=n_{1}\cdot n_{2}\cdots n_{d}$ and $\underline{a}$ being the first “column” of $A_n$ whose entries $a_j$, $j=(j_1,\ldots,j_d)$, ordered lexicographically, are blocks of size $p\times q$.
For multilevel block Toeplitz sequences $\{T_{n}(f)\}$ generated by an integrable $d$ variate and matrix valued symbol $f$ the singular values are not explicitly known but we know the distribution in the sense of Definition \[def-distribution\]; see [@Tillinota]. More precisely we have $$\label{szego-tyrty}
\{T_n(f)\}\sim_{\sigma}
(f,Q^d), \quad \ \ Q=(-\pi,\pi).$$
### $\alpha=\underline{0}$ {#alphazero}
The other extreme is represented by the case where $\alpha$ is the zero vector. Here the multilevel block $\alpha$-circulant and $\alpha$-Toeplitz coincide when $\alpha=\underline{0}$ and are both given by $$\begin{aligned}
A_{n}=[a_{(r-\underline{0}\circ s)\textrm{ mod $n$}}]_{r,s=\underline{0}}^{n-e}=[a_{r\textrm{ mod $n$}}]_{r,s=\underline{0}}^{n-e}=
[a_{r}]_{r,s=\underline{0}}^{n-e}=\left[\begin{array}{ccc}
a_{\underline{0}} & \cdots & a_{\underline{0}}\\
\vdots & & \vdots \\
a_{n-e} & \cdots & a_{n-e}
\end{array}\right].\end{aligned}$$
A simple computation shows that all the singular values are zero except for few of them given by $\sqrt{\hat{n}}\sigma$, where $\hat{n}=n_{1}\cdot n_{2}\cdots n_{d}$ and $\sigma$ is any singular value of the matrix $(\sum_{j=\underline{0}}^{n-e}
a_j^*a_j)^{1/2}$. Of course in the scalar case where $p=q=1$ the choice of $\sigma$ is unique and by the above formula it coincides with the Euclidean norm of the first column $\underline{a}$ of the original matrix. In that case it is evident that $$\begin{aligned}
\{A_n\}\sim_{\sigma}
(0,G),\end{aligned}$$ for any domain $G$ satisfying the requirements of Definition \[def-distribution\].
### When some of the entries of $\alpha$ vanish {#nonnegative-vs-positive}
The content of this subsection reduces to the following remark: the case of a nonnegative $\alpha$ can be reduced to the case of a positive vector so that we are motivated to treat in detail the latter in the next section. Let $\alpha$ be a $d$-dimensional vector of nonnegative integers and let ${\cal{N}}\subset \{1,\ldots,d\}$ be the set of indices such that $j\in \cal N$ if and only if $\alpha_j=0$. Assume that $\cal N$ is nonempty, let $t\ge 1$ be its cardinality and $d^+=d-t$. Then a simple calculation shows that the singular values of the corresponding $\alpha$-circulant matrix $A_{n}=[a_{(r-\alpha\circ s)\textrm{ mod $n$}}]_{r,s=\underline{0}}^{n-e}$ are zero except for few of them given by $\sqrt{\hat n[0]} \sigma$ where $$\hat n[0]=\prod_{j\in \cal{N}} n_j,\quad \quad
n[0]=(n_{j_1},\ldots,n_{j_t}),\quad {\cal{N}}=\{j_1,\ldots,j_t\},$$ and $\sigma$ is any singular value of the matrix $$\label{eq-2-1-3}
\left(\sum_{j=\underline{0}}^{n[0]-e}C_j^*C_j\right)^{1/2}.$$
Here $C_j$ is a $d^+$-level $\alpha^+$-circulant matrix with $\alpha^+=(\alpha_{k_1},\ldots,\alpha_{k_{d^+}})$ and of partial sizes $n[>0]=(n_{k_1},\ldots,n_{k_{d^+}})$, ${\cal{N}}^C=\{k_1,\ldots,k_{d^+}\}$, and whose expression is $$C_j=\left[a_{(r-\alpha \circ s)\ {\rm mod}\,
n}\right]_{r',s'=\underline{0}}^{n[>0]-e},$$ where $(r-\alpha\circ s)_k=j_k$ for $\alpha_k=0$ and $r'_i=r_{k_i}$, $s'_i=s_{k_i}$, $i=1,\ldots,d^+$. Taking into account the above notation, for the $\alpha$-Toeplitz $A_{n}=[a_{r-\alpha\circ s}]_{r,s=\underline{0}}^{n-e}$ the same computation shows that all the singular values are zero except for few of them given by $\sqrt{\hat n[0]} \sigma$ where $\sigma$ is any singular value of the matrix $$\label{eq-2-1-3-bis}
\left(\sum_{j=\underline{0}}^{n[0]-e}T_j^*T_j\right)^{1/2}.$$
Here $T_j$ is a $d^+$-level $\alpha^{+}$-Toeplitz matrix with $\alpha^+=(\alpha_{k_1},\ldots,\alpha_{k_{d^+}})$ and of partial sizes $n[>0]=(n_{k_1},\ldots,n_{k_{d^+}})$, ${\cal{N}}^C=\{k_1,\ldots,k_{d^+}\}$, and whose expression is $$T_j=\left[a_{(r-\alpha \circ s)}\right]_{r',s'=\underline{0}}^{n[>0]-e},$$ where $(r-\alpha\circ s)_k=j_k$ for $\alpha_k=0$ and $r'_i=r_{k_i}$, $s'_i=s_{k_i}$, $i=1,\ldots,d^+$. Also in this case, since most of the singular values are identically zero, we infer that $$\begin{aligned}
\{A_n\}\sim_{\sigma}
(0,G),\end{aligned}$$ for any domain $G$ satisfying the requirements of Definition \[def-distribution\].
Singular values of $\alpha$-circulant matrices {#sec:circ}
==============================================
Of course the aim of this paper is to give the general picture for any nonnegative vector $\alpha$. Since the notations can become quite heavy, for the sake of simplicity, we start with the case $d=p=q=1$. Several generalizations, including also the degenerate case in which $\alpha$ has some zero entries is treated in Section \[sec:gen\] via the observations in Subsection \[nonnegative-vs-positive\], which imply that the general analysis can be reduced to the case where all the entries of $\alpha$ are positive, that is $\alpha_j>0$, $j=1,\ldots,d$.
In the following, we denote by $(n,\alpha)$ the greater common divisor of $n$ and $\alpha$. i.e., $(n,\alpha)=\gcd(n,\alpha)$, by $n_{\alpha}=\frac{n}{(n,\alpha)}$, by $\check{\alpha}=\frac{\alpha}{(n,\alpha)}$, and by $I_t$ the identity matrix of order $t$.
If we denote by $C_{n}$ the classical circulant matrix (i.e. with $\alpha=1$) and by $C_{n,\alpha}$ the $\alpha$-circulant matrix generated by its elements, for generic $n$ and $\alpha$ one verifies immediately that $$\begin{aligned}
\label{0}
C_{n,\alpha} = C_{n}Z_{n,\alpha},\end{aligned}$$ where $$\begin{aligned}
\label{i}
Z_{n,\alpha}=\left[\delta_{r-\alpha s}\right]_{r,s=0}^{n-1},\qquad\delta_{k}=
\left\{\begin{array}{cl} 1 & \textrm{if $k\equiv 0\textrm{ (mod $n$)}$,}\\0 &
\textrm{otherwise.}\end{array} \right.\end{aligned}$$
Let $n$ be any integer greater than $2$ then $$\begin{aligned}
\label{18}
Z_{n,\alpha}=\underbrace{\left[\widetilde{Z}_{n,\alpha}|\widetilde{Z}_{n,\alpha}|\cdots|\widetilde{Z}_{n,\alpha}\right]}_{(n,\alpha)\textrm{ times}},\end{aligned}$$ where $Z_{n,\alpha}$ is the matrix defined in $(\ref{i})$ and $\widetilde{Z}_{n,\alpha}\in \mathbb{C}^{n\times n_{\alpha}}$ is the submatrix of $Z_{n,\alpha}$ obtained by considering only its first $n_{\alpha}$ columns, that is $$\begin{aligned}
\label{z-n-alfa}
\widetilde{Z}_{n,\alpha} = Z_{n,\alpha}
\left[\begin{array}{c}
I_{n_{\alpha}} \\ 0 \end{array}\right].\end{aligned}$$
Setting $\widetilde{Z}_{n,\alpha}^{(0)}=\widetilde{Z}_{n,\alpha}$ and denoting by $\widetilde{Z}_{n,\alpha}^{(j)}\in\mathbb{C}^{n\times n_{\alpha}}$ the $(j+1)$-th block-column of the matrix $Z_{n,\alpha}$ for $j=0,\ldots,(n,\alpha)-1,$ we find $$\begin{aligned}
Z_{n,\alpha}=\left[\underbrace{\widetilde{Z}_{n,\alpha}^{(0)}}_{n\times n_{\alpha}}|
\underbrace{\widetilde{Z}_{n,\alpha}^{(1)}}_{n\times n_{\alpha}}|\cdots|
\underbrace{\widetilde{Z}_{n,\alpha}^{((n,\alpha)-1)}}_{n\times n_{\alpha}}\right].\end{aligned}$$
For $r=0,1,\ldots,n-1$ and $s=0,1,\ldots,n_{\alpha}-1$, we observe that $$(\widetilde{Z}_{n,\alpha}^{(j)})_{r,s}=
(Z_{n,\alpha})_{r,jn_{\alpha}+s},$$ and $$\begin{aligned}
(Z_{n,\alpha})_{r,jn_{\alpha}+s}&=&\delta_{r-\alpha(jn_{\alpha}+s)}\\
&=&\delta_{r-j\alpha n_{\alpha}-\alpha s}\\
&{\underset{\rm (a)}=}&\delta_{r-\alpha s}\\
&=&(\widetilde{Z}_{n,\alpha}^{(0)})_{r,s}=(\widetilde{Z}_{n,\alpha})_{r,s},\end{aligned}$$ where $n_{\alpha}=\frac{n}{(n,\alpha)}$ and (a) is a consequence of the fact that $\frac{\alpha}{(n,\alpha)}$ is an integer greater than zero and so $j\alpha
n_{\alpha}=j\frac{\alpha}{(n,\alpha)}n\equiv 0 $ (mod $n$). Thus we conclude that $\widetilde{Z}_{n,\alpha}^{(j)}=\widetilde{Z}_{n,\alpha}^{(0)}=\widetilde{Z}_{n,\alpha}$ for $j=0,\ldots,(n,\alpha)-1$.
Another useful fact is represented by the following equation $$\begin{aligned}
\label{3bis}
\widetilde{Z}_{n,\alpha}=\widetilde{Z}_{n,(n,\alpha)}Z_{n_{\alpha},\check{\alpha}},\end{aligned}$$ where $Z_{n_{\alpha},\check{\alpha}}$ is the matrix defined in $(\ref{i})$ of dimension $n_{\alpha}\times n_{\alpha}$. Therefore $$\begin{aligned}
\label{pna}
Z_{n_{\alpha},\check{\alpha}}=\left[\widehat{\delta}_{r-\check{\alpha}
s}
\right]_{r,s=0}^{n_{\alpha}-1},\qquad\widehat{\delta}_{k}=\left\{\begin{array}{cl}
1 & \textrm{if $k\equiv 0 \textrm{ (mod $n_{\alpha}$),}$}\\0 &
\textrm{otherwise.}\end{array} \right.\end{aligned}$$
Relation $(\ref{3bis})$ will be used later.
(of relation $(\ref{3bis}).$) For $r=0,1,\ldots,n-1$ and $s=0,1,\ldots,n_{\alpha}-1,$ we find $$\begin{aligned}
(\widetilde{Z}_{n,\alpha})_{r,s}&=&\delta_{r-\alpha s}\\
&=&\delta_{(r-\alpha s)\textrm{ mod $n$}},\end{aligned}$$ and $$\begin{aligned}
(\widetilde{Z}_{n,(n,\alpha)}Z_{n_{\alpha},\check{\alpha}})_{r,s} &=&
\sum_{l=0}^{n_{\alpha}-1}(\widetilde{Z}_{n,(n,\alpha)})_{r,l}(Z_{n_{\alpha},\check{\alpha}})_{l,s} \\
&=& \sum_{l=0}^{n_{\alpha}-1}\delta_{r-(n,\alpha) l}\widehat{\delta}_{l-\check{\alpha} s} \\
&{\underset{\rm (a)}=}& \delta_{r-(n,\alpha)\cdot(\check{\alpha} s)\textrm{ mod $n_{\alpha}$}}\\
&=& \delta_{r-(n,\alpha)\cdot\left(\frac{\alpha}{(n,\alpha)} s\right)\textrm{ mod $n_{\alpha}$}} \\
&{\underset{\rm (b)}=}& \delta_{r-(\alpha s)\textrm{ mod $n$}}\\
&=& \delta_{(r-(\alpha s)\textrm{ mod $n$})\textrm{ mod $n$}}\\
&=& \delta_{(r-\alpha s)\textrm{ mod $n$}},\end{aligned}$$ where
- holds true since there exists a unique $l\in\{0,1,\ldots,n_{\alpha}-1\}$ such that $ l-\check{\alpha} s\equiv0\textrm{ (mod $n_{\alpha}$)}$, that is, $l\equiv \check{\alpha} s\textrm{ (mod $n_{\alpha}$)}$ and hence $\delta_{r-(n,\alpha)l}=\delta_{r-(n,\alpha)\cdot(\check{\alpha}
s)\textrm{ mod $n_{\alpha}$}}$;
- is due to the following property: if we have three integer numbers $\rho,\,\theta,$ and $\gamma$, then $$\begin{aligned}
\rho(\theta\textrm{ mod $\gamma)$}=(\rho\theta)\textrm{ mod $\rho\gamma$}.\end{aligned}$$
\[amag\] If $\alpha\geq n$ then $Z_{n,\alpha}=Z_{n,{\alpha}^\circ}$ where ${\alpha}^\circ$ is the unique integer which satisfies $\alpha=tn+{\alpha}^\circ$ with $0\leq{\alpha}^\circ<n$ and $t\in\mathbb{N}$; $Z_{n,\alpha}$ is defined in $(\ref{i})$.
One can define ${\alpha}^\circ$ by: ${\alpha}^\circ:=\alpha\,\textrm{mod $n$}$.
From $(\ref{i})$ we know that $$\begin{aligned}
Z_{n,\alpha}=\left[\delta_{r-\alpha c}\right]_{r,c=0}^{n-1},\qquad\delta_{k}=\left\{\begin{array}{cl} 1 & \textrm{if $k\equiv 0 \textrm{ (mod $n$)}$,}\\0 & \textrm{otherwise.}\end{array} \right.\end{aligned}$$
For $r,s=0,1,\ldots,n-1$, one has $$\begin{aligned}
(Z_{n,\alpha})_{r,s}=\delta_{r-\alpha s}=\delta_{r-(tn+{\alpha}^\circ)s}=
\delta_{r-{\alpha}^\circ s}=(Z_{n,{\alpha}^\circ})_{r,s},\end{aligned}$$ since $tns\equiv 0\textrm{ (mod $n$)}$. Whence $Z_{n,\alpha}=Z_{n,{\alpha}^\circ}$.
The previous lemma tells us that, for $\alpha$-circulant matrices, we can consider only the case where $0\leq\alpha<n$. In fact, if $\alpha\geq n$, from $(\ref{0})$ we infer that $$\begin{aligned}
C_{n,\alpha}=C_{n}Z_{n,\alpha}=C_{n}Z_{n,{\alpha}^\circ}=C_{n,{\alpha}^\circ}.\end{aligned}$$
Finally, it is worth noticing that the use of $(\ref{iV})$ and $(\ref{0})$ implies that $$\begin{aligned}
\label{Vi}
C_{n,\alpha} = F_{n}D_{n}F_{n}^{\ast}Z_{n,\alpha}.\end{aligned}$$
Formula $(\ref{Vi})$ plays an important role for studying the singular values of the $\alpha$-circulant matrices.
A characterization of $Z_{n,\alpha}$ in terms of Fourier matrices
-----------------------------------------------------------------
\[l1\] Let $F_{n}$ be the Fourier matrix of order $n$ defined in (\[V\]) and let $\widetilde{Z}_{n,\alpha}\in \mathbb{C}^{n\times
n_{\alpha}}$ be the matrix represented in $(\ref{z-n-alfa})$. Then $$\begin{aligned}
\label{4}
F_{n}\widetilde{Z}_{n,\alpha} =
\frac{1}{\sqrt{(n,\alpha)}}I_{n,\alpha}F_{n_{\alpha}}Z_{n_{\alpha},\check{\alpha}},\end{aligned}$$ where $I_{n,\alpha}\in \mathbb{C}^{n\times n_{\alpha}}$ and $$\begin{aligned}
I_{n,\alpha}=\left.\left[\begin{array}{c}
I_{n_{\alpha}}\\
\hline
I_{n_{\alpha}}\\
\hline
\vdots\\
\hline
I_{n_{\alpha}}
\end{array}\right]\right\}\textrm{$(n,\alpha)$ times,}\end{aligned}$$ with $I_{n_{\alpha}}$ being the identity matrix of size $n_{\alpha}$ and $Z_{n_{\alpha},\check{\alpha}}$ as in $(\ref{pna})$.
$n= n_{\alpha}\cdot(n,\alpha).$
(of Lemma $\ref{l1}$.) Rewrite the Fourier matrix as $$\begin{aligned}
F_{n}=\frac{1}{\sqrt{n}}\left[\begin{array}{c|c|c|c|c}
f_{0} & f_{1} & f_{2} & \cdots & f_{n-1}
\end{array}\right],\end{aligned}$$ where $f_{k},$ $k=0,1,2,\ldots,n-1,$ is the $k-th$ column of the Fourier matrix of order $n$: $$\begin{aligned}
\label{6}
f_{k}=\left[e^{-\frac{2\pi ikj}{n}}\right]_{j=0}^{n-1}=\left[\begin{array}{c}
e^{-\frac{2\pi ik\cdot0}{n}}\\
e^{-\frac{2\pi ik\cdot1}{n}}\\
e^{-\frac{2\pi ik\cdot2}{n}}\\
\vdots\\
e^{-\frac{2\pi ik\cdot(n-1)}{n}}\\
\end{array}\right].\end{aligned}$$
From $(\ref{3bis})$, we find $$\label{7}
F_{n}\widetilde{Z}_{n,\alpha}=F_{n}\widetilde{Z}_{n,(n,\alpha)}Z_{n_{\alpha},\check{\alpha}}=
\frac{1}{\sqrt{n}}\left[\begin{array}{c|c|c|c|c}
f_{0} & f_{1\cdot(n,\alpha)} & f_{2\cdot(n,\alpha)} & \cdots & f_{(n_{\alpha}-1)\cdot(n,\alpha)}
\end{array}\right]Z_{n_{\alpha},\check{\alpha}}\in \mathbb{C}^{n\times
n_{\alpha}}.$$
Indeed, for $k=0,1,\ldots,n_{\alpha}-1$, $j=0,1,\ldots,n-1,$ one has $$\label{8}
\left(F_{n}\widetilde{Z}_{n,(n,\alpha)}\right)_{j,k}=\overset{n-1}
{\underset{l=0}\sum}(F_{n})_{j,l}(\widetilde{Z}_{n,(n,\alpha)})_{l,k}=
\overset{n-1}{\underset{l=0}\sum}
\delta_{l-(n,\alpha)k}e^{-\frac{2\pi ijl}{n}},$$ and, since $0\leq (n,\alpha)k\leq n-(n,\alpha),$ there exists a unique $l_{k}\in \{0,1,2,\ldots,n-1\}$ such that $l_{k}-(n,\alpha)k\equiv0$ (mod $n$), so $l_{k}=(n,\alpha)k$. Consequently relation $(\ref{8})$ implies $$\left(F_{n}\widetilde{Z}_{n,(n,\alpha)}\right)_{j,k}=\delta_{l_{k}-(n,\alpha)k}
e^{-\frac{2\pi ijl_{k}}{n}}=e^{-\frac{2\pi ij
(n,\alpha)k}{n}}=\left(f_{(n,\alpha)k}\right)_{j},$$ for all $0\leq j\leq n-1$ and $0\leq k\leq n_{\alpha}-1,$ and hence $$F_{n}\widetilde{Z}_{n,(n,\alpha)}=
\frac{1}{\sqrt{n}}\left[\begin{array}{c|c|c|c|c}
f_{0} & f_{1\cdot(n,\alpha)} & f_{2\cdot(n,\alpha)} & \cdots & f_{(n_{\alpha}-1)\cdot(n,\alpha)}
\end{array}\right].$$
For $k=0,1,2,\ldots,n_{\alpha}-1,$ we deduce $$f_{(n,\alpha)k}=\left[e^{-\frac{2\pi ij(n,\alpha)k}{n}}\right]_{j=0}^{n-1}=\left[e^{-\frac{2\pi ijk}{n_{\alpha}}}\right]_{j=0}^{n-1},$$ and then, taking into account the equalities $n=(n,\alpha)\frac{n}{(n,\alpha)}=(n,\alpha)n_{\alpha},$ we can write $$\begin{aligned}
\label{9}
f_{(n,\alpha)k} =\left[\begin{array}{l}
\left[e^{-\frac{2\pi ikj}{n_{\alpha}}}\right]_{j=0}^{n_{\alpha}-1} \\
\left[e^{-\frac{2\pi ikj}{n_{\alpha}}}\right]_{j=n_{\alpha}}^{2n_{\alpha}-1} \\
\qquad\;\vdots \\
\left[e^{-\frac{2\pi ikj}{n_{\alpha}}}\right]_{j=((n,\alpha)-1)n_{\alpha}}^{(n,\alpha)
n_{\alpha}-1}
\end{array}\right],\end{aligned}$$ where $$\begin{aligned}
\label{10}
\left[e^{-\frac{2\pi ikj}{n_{\alpha}}}\right]_{j=0}^{n_{\alpha}-1} =
\begin{bmatrix}
e^{-\frac{2\pi ik\cdot0}{n_{\alpha}}} \\
e^{-\frac{2\pi ik\cdot1}{n_{\alpha}}} \\
e^{-\frac{2\pi ik\cdot2}{n_{\alpha}}} \\
\vdots \\
e^{-\frac{2\pi ik\cdot(n_{\alpha}-1)}{n_{\alpha}}}
\end{bmatrix}.\end{aligned}$$
According to formula $(\ref{6}),$ one observes that the vector in $(\ref{10})$ is the $k-th$ column of the Fourier matrix $F_{n_{\alpha}}$. Furthermore, for $l=0,1,2,\ldots,(n,\alpha)-1$, we find $$\label{11}
\left[e^{-\frac{2\pi ikj}{n_{\alpha}}}\right]_{j=ln_{\alpha}}^{(l+1)n_{\alpha}-1}
=
\begin{bmatrix}
e^{-\frac{2\pi ikln_{\alpha}}{n_{\alpha}}} \\
e^{-\frac{2\pi ik(ln_{\alpha}+1)}{n_{\alpha}}} \\
e^{-\frac{2\pi ik(ln_{\alpha}+2)}{n_{\alpha}}} \\
\vdots \\
e^{-\frac{2\pi ik(ln_{\alpha}+n_{\alpha}-1)}{n_{\alpha}}}
\end{bmatrix}
=e^{-2\pi ikl}
\begin{bmatrix}
e^{-\frac{2\pi ik\cdot0}{n_{\alpha}}} \\
e^{-\frac{2\pi ik\cdot1}{n_{\alpha}}} \\
e^{-\frac{2\pi ik\cdot2}{n_{\alpha}}} \\
\vdots \\
e^{-\frac{2\pi ik\cdot(n_{\alpha}-1)}{n_{\alpha}}}
\end{bmatrix}
=\left[e^{-\frac{2\pi ikj}{n_{\alpha}}}\right]_{j=0}^{n_{\alpha}-1}.$$
Using $(\ref{11})$, the expression of the vector in $(\ref{9})$ becomes $$\begin{aligned}
\label{12}
f_{(n,\alpha)k}=\left.
\begin{bmatrix}
\left[e^{-\frac{2\pi ikj}{n_{\alpha}}}\right]_{j=0}^{n_{\alpha}-1} \\
\left[e^{-\frac{2\pi ikj}{n_{\alpha}}}\right]_{j=0}^{n_{\alpha}-1} \\
\!\!\!\!\!\vdots \\
\left[e^{-\frac{2\pi ikj}{n_{\alpha}}}\right]_{j=0}^{n_{\alpha}-1}
\end{bmatrix}
\right\}\textrm{ $(n,\alpha)$ times.}\end{aligned}$$
Setting $\widetilde{f}_{r}=\left[e^{-\frac{2\pi irj}
{n_{\alpha}}}\right]_{j=0}^{n_{\alpha}-1},$ for $0\leq r\leq n_{\alpha}-1$, the Fourier matrix $F_{n_{\alpha}}$ of size $n_{\alpha}$ takes the form $$\begin{aligned}
\label{13}
F_{n_{\alpha}}=\frac{1}{\sqrt{n_{\alpha}}}\left[\begin{array}{c|c|c|c|c}
\widetilde{f}_{0} & \widetilde{f}_{1} & \widetilde{f}_{2} & \cdots & \widetilde{f}_{n_{\alpha}-1}
\end{array}\right].\end{aligned}$$
From formula $(\ref{10}),$ the relation $(\ref{12})$ can be expressed as $$\begin{aligned}
f_{(n,\alpha)k}=\left.\left[\begin{array}{c}
\widetilde{f}_{k}\\
\widetilde{f}_{k}\\
\vdots\\
\widetilde{f}_{k}
\end{array}\right]\right\}\textrm{ $(n,\alpha)$ times,} \qquad k=0,\ldots,n_{\alpha}-1,\end{aligned}$$ and, as a consequence, formula $(\ref{7})$ can be rewritten as $$\begin{aligned}
F_{n}\widetilde{Z}_{n,\alpha}=F_{n}\widetilde{Z}_{n,(n,\alpha)}Z_{n_{\alpha},\check{\alpha}} &=&
\frac{1}{\sqrt{n}}\left[\begin{array}{c|c|c|c|c}
\widetilde{f}_{0} & \widetilde{f}_{1} & \widetilde{f}_{2} & \cdots & \widetilde{f}_{n_{\alpha}-1}\\
\widetilde{f}_{0} & \widetilde{f}_{1} & \widetilde{f}_{2} & \cdots & \widetilde{f}_{n_{\alpha}-1}\\
\vdots & \vdots & \vdots & \vdots & \vdots \\
\widetilde{f}_{0} & \widetilde{f}_{1} & \widetilde{f}_{2} & \cdots & \widetilde{f}_{n_{\alpha}-1}
\end{array}\right]Z_{n_{\alpha},\check{\alpha}}\\
&=&\frac{1}{\sqrt{(n,\alpha)n_{\alpha}}}\left[\begin{array}{c}
\sqrt{n_{\alpha}}F_{n_{\alpha}}\\
\hline
\sqrt{n_{\alpha}}F_{n_{\alpha}}\\
\hline
\vdots \\
\hline
\sqrt{n_{\alpha}}F_{n_{\alpha}}
\end{array}\right]Z_{n_{\alpha},\check{\alpha}}\\
&=&\frac{1}{\sqrt{(n,\alpha)}}\left[\begin{array}{c}
F_{n_{\alpha}}\\
\hline
F_{n_{\alpha}}\\
\hline
\vdots \\
\hline
F_{n_{\alpha}}
\end{array}\right]Z_{n_{\alpha},\check{\alpha}}\\
&=&\frac{1}{\sqrt{(n,\alpha)}}\left[\begin{array}{c}
I_{n_{\alpha}}\\
\hline
I_{n_{\alpha}}\\
\hline
\vdots \\
\hline
I_{n_{\alpha}}
\end{array}\right]F_{n_{\alpha}}Z_{n_{\alpha},\check{\alpha}}\\
&=& \frac{1}{\sqrt{(n,\alpha)}}I_{n,\alpha}F_{n_{\alpha}}Z_{n_{\alpha},\check{\alpha}}.\end{aligned}$$
In the subsequent subsection, we will exploit Lemma $\ref{l1}$ in order to characterize the singular values of the $\alpha$-circulant matrices $C_{n,\alpha}$. Here we conclude the subsection with the following simple observations.
In Lemma $\ref{l1}$, if $(n,\alpha)=\alpha$, we have $n_{\alpha}=\frac{n}{(n,\alpha)}=\frac{n}{\alpha}$ and $\check{\alpha}=\frac{\alpha}{(n,\alpha)}=1$; so the matrix $Z_{n_{\alpha},\check{\alpha}}=Z_{n_{\alpha},1}$, appearing in $(\ref{4})$, is the identity matrix of dimension $\frac{n}{\alpha}\times\frac{n}{\alpha}$. The relation $(\ref{4})$ becomes $$\begin{aligned}
F_{n}\widetilde{Z}_{n,\alpha}=\frac{1}{\sqrt{\alpha}}I_{n,\alpha}F_{n_{\alpha}}.\end{aligned}$$
The latter equation with $\alpha=2$ and even $n$ appear (and is crucial) in the multigrid literature; see [[@mcirco]]{}, equation (3.2), page 59 and, in slightly different form for the sine algebra of type I, see [[@FS1]]{}, Section 2.1.
If $(n,\alpha)=1$, Lemma $\ref{l1}$ is trivial, because $n_{\alpha}=\frac{n}{(n,\alpha)}=n$, $\check{\alpha}=\frac{\alpha}{(n,\alpha)}=\alpha$, and so $\widetilde{Z}_{n,\alpha}=Z_{n,\alpha}$. The relation $(\ref{4})$ becomes $$\begin{aligned}
F_{n}\widetilde{Z}_{n,\alpha}=F_{n}Z_{n,\alpha}&=&I_{n,\alpha}F_{n_{\alpha}}Z_{n_{\alpha},\check{\alpha}}\\
&=&F_{n}Z_{n,\alpha},
\end{aligned}$$ since the matrix $I_{n,\alpha}$ reduces by its definition to the identity matrix of order $n$.
Lemma $\ref{l1}$ is true also if, instead of $F_{n}$ and $F_{n_{\alpha}}$, we put $F_{n}^{*}$ and $F_{n_{\alpha}}^{*}$, respectively, because $F_{n}^{*}=\overline{F_{n}}$. In fact there is no transposition, but only conjugation.
Characterization of the singular values of the $\alpha$-circulant matrices
--------------------------------------------------------------------------
Now we link the singular values of $\alpha$-circulant matrices with the eigenvalues of its circulant counterpart $C_n$. This is nontrivial given the multiplicative relation $C_{n,\alpha}= C_n
Z_{n,\alpha}$.
Having in mind the definition of the diagonal matrix $D_n$ given in (\[V\]), we start by setting $$\begin{aligned}
\label{DJ}
\notag&&D_{n}^{\ast}D_{n}={\rm diag}(|D_{n}|^{2}_{s,s};\,\,s=0,1,\ldots,n-1)=
{\rm diag}(d_{s};\,\,s=0,1,\ldots,n-1)=
\overset{(n,\alpha)}{\underset{l=1}\oplus}\Delta_{l},\\
&&J_{(n,\alpha)}\otimes I_{n_{\alpha}} =\underset{(n,\alpha)\text{\,\,}times}{
\underbrace{\left[I_{n,\alpha}|I_{n,\alpha}|\cdots|I_{n,\alpha}\right]}}=
\left.\left[\begin{array}{c|c|c|c}
I_{n_{\alpha}} & I_{n_{\alpha}} & \cdots & I_{n_{\alpha}}\\
\hline
I_{n_{\alpha}} & I_{n_{\alpha}} & \cdots & I_{n_{\alpha}}\\
\hline
\vdots & \vdots & \vdots & \vdots\\
\hline
I_{n_{\alpha}} & I_{n_{\alpha}} & \cdots & I_{n_{\alpha}}
\end{array}\right]\right\}\textrm{ $(n,\alpha)$ times,}\end{aligned}$$ where $$\begin{aligned}
\label{ds}&&d_{s}=|D_{n}|_{s,s}^{2}=(D_{n})_{s,s}\cdot\overline{(D_{n})_{s,s}},
\quad\textrm{$D_{n}$ defined in $(\ref{V})$, $s=0,1,\ldots,n-1$,}\\
\notag&&\Delta_{l}=\left[\begin{array}{cccc}
d_{(l-1)n_{\alpha}} & & & \\
& d_{(l-1)n_{\alpha}+1} & & \\
& & \ddots & \\
& & & d_{(l-1)n_{\alpha}+n_{\alpha}-1}
\end{array}\right]\in \mathbb{C}^{n_{\alpha}\times
n_{\alpha}};\text{\,\,}l=1,2,\ldots,(n,\alpha),\\
\label{19bis}&& J_{(n,\alpha)} =\left.\left[\begin{array}{cccc}
1 & 1 & \cdots & 1\\
1 & 1 & \cdots & 1\\
\vdots & \vdots & \vdots & \vdots\\
1 & 1 & \cdots & 1
\end{array}\right]
\right\}\textrm{ $(n,\alpha)$ times.}\end{aligned}$$
We now exploit relation $(\ref{18})$ and Lemma $\ref{l1}$, and we obtain that $$\begin{aligned}
\label{fnz}
\notag F_{n}Z_{n,\alpha} &=& F_{n}\left[\widetilde{Z}_{n,\alpha}|\widetilde{Z}_{n,\alpha}
|\cdots|\widetilde{Z}_{n,\alpha}\right] \\
\notag&=& \left[F_{n}\widetilde{Z}_{n,\alpha}|F_{n}\widetilde{Z}_{n,\alpha}|\cdots
|F_{n}\widetilde{Z}_{n,\alpha}\right] \\
\notag&=& \frac{1}{\sqrt{(n,\alpha)}}\left[I_{n,\alpha}F_{n_{\alpha}}Z_{n_{\alpha},\check{\alpha}}
|I_{n,\alpha}F_{n_{\alpha}}Z_{n_{\alpha},\check{\alpha}}|\cdots|I_{n,\alpha}F_{n_{\alpha}}Z_{n_{\alpha},\check{\alpha}}\right] \\
\notag&=& \frac{1}{\sqrt{(n,\alpha)}}\left[I_{n,\alpha}|I_{n,\alpha}|\cdots|I_{n,\alpha}\right]
\left.\left[\begin{array}{cccc}
F_{n_{\alpha}}Z_{n_{\alpha},\check{\alpha}} & & & \\
&\!\!\!\!\! F_{n_{\alpha}}Z_{n_{\alpha},\check{\alpha}} & & \\
& & \!\!\!\!\!\ddots & \\
& & & \!\!\!\!\!F_{n_{\alpha}}Z_{n_{\alpha},\check{\alpha}} \\
\end{array}\right]\right\}\textrm{ $(n,\alpha)$ times}\\
&=&\frac{1}{\sqrt{(n,\alpha)}}\left[I_{n,\alpha}|I_{n,\alpha}|\cdots
|I_{n,\alpha}\right]\left(I_{(n,\alpha)}\otimes F_{n_{\alpha}}Z_{n_{\alpha},\check{\alpha}}\right),
\end{aligned}$$ where $I_{(n,\alpha)}$ is the identity matrix of order $(n,\alpha).$ Furthermore, $$\begin{aligned}
\label{CastC}
\notag C_{n,\alpha}^{\ast}C_{n,\alpha} &=&(F_{n}D_{n}F_{n}^{\ast}Z_{n,\alpha})^{\ast}(F_{n}D_{n}F_{n}^{\ast}Z_{n,\alpha}) \\
&=&
\notag Z_{n,\alpha}^{\ast}F_{n}D_{n}^{\ast}F_{n}^{\ast}F_{n}D_{n}F_{n}^{\ast}Z_{n,\alpha} \\
&=&
\notag Z_{n,\alpha}^{\ast}F_{n}D_{n}^{\ast}D_{n}F_{n}^{\ast}Z_{n,\alpha} \\
&=&
(F_{n}^{\ast}Z_{n,\alpha})^{\ast}D_{n}^{\ast}D_{n}F_{n}^{\ast}Z_{n,\alpha}.\end{aligned}$$
From $(\ref{fnz})$ and $(\ref{DJ})$, we plainly infer the following relations $$\begin{aligned}
(F_{n}^{\ast}Z_{n,\alpha})^{\ast}&=&\left(\frac{1}{\sqrt{(n,\alpha)}}\left[I_{n,\alpha}|I_{n,\alpha}|\cdots
|I_{n,\alpha}\right]\left(I_{(n,\alpha)}\otimes F_{n_{\alpha}}^{*}Z_{n_{\alpha},\check{\alpha}}\right)\right)^{*}\\
&=&\frac{1}{\sqrt{(n,\alpha)}}\left(I_{(n,\alpha)}\otimes F_{n_{\alpha}}^{*}Z_{n_{\alpha},\check{\alpha}}\right)^{*}\left(J_{(n,\alpha)}\otimes I_{n_{\alpha}}\right)\\
&=&\frac{1}{\sqrt{(n,\alpha)}}\left(I_{(n,\alpha)}\otimes Z_{n_{\alpha},\check{\alpha}}^{*}F_{n_{\alpha}}\right)\left(J_{(n,\alpha)}\otimes I_{n_{\alpha}}\right),\\
F_{n}^{\ast}Z_{n,\alpha}&=&\frac{1}{\sqrt{(n,\alpha)}}\left[I_{n,\alpha}|I_{n,\alpha}|\cdots
|I_{n,\alpha}\right]\left(I_{(n,\alpha)}\otimes F_{n_{\alpha}}^{*}Z_{n_{\alpha},\check{\alpha}}\right)\\
&=&\frac{1}{\sqrt{(n,\alpha)}}\left(J_{(n,\alpha)}\otimes
I_{n_{\alpha}}\right)\left(I_{(n,\alpha)}\otimes
F_{n_{\alpha}}^{*}Z_{n_{\alpha},\check{\alpha}}\right).\end{aligned}$$
Hence $$\begin{aligned}
C_{n,\alpha}^{\ast}C_{n,\alpha}=\!\left(I_{(n,\alpha)}\otimes
Z_{n_{\alpha},\check{\alpha}}^{*}F_{n_{\alpha}}\right)\!\left(J_{(n,\alpha)}\otimes
I_{n_{\alpha}}\right)\!\frac{1}{(n,\alpha)}D_{n}^{\ast}D_{n}\left(J_{(n,\alpha)}\otimes
I_{n_{\alpha}}\right)\!\left(I_{(n,\alpha)} \otimes
F_{n_{\alpha}}^{*}Z_{n_{\alpha},\check{\alpha}}\right).\end{aligned}$$
Now using the properties of the tensorial product $$\begin{aligned}
(I_{(n,\alpha)}\otimes Z_{n_{\alpha},\check{\alpha}}^{*}F_{n_{\alpha}})\!\!\!\!\!\!\!\!&&\!\!\!\!\!\!\!\!(I_{(n,\alpha)}\otimes F_{n_{\alpha}}^{*}Z_{n_{\alpha},\check{\alpha}})\\
&=&I_{(n,\alpha)}I_{(n,\alpha)}\otimes Z_{n_{\alpha},\check{\alpha}}^{*}F_{n_{\alpha}}F_{n_{\alpha}}^{*}Z_{n_{\alpha},\check{\alpha}}\\
&=&I_{(n,\alpha)}I_{(n,\alpha)}\otimes Z_{n_{\alpha},\check{\alpha}}^{*}Z_{n_{\alpha},\check{\alpha}}\\
&=&I_{(n,\alpha)}I_{(n,\alpha)}\otimes I_{n_{\alpha}}=I_{n},\end{aligned}$$ and from a similarity argument, one deduces that the eigenvalues of $C_{n,\alpha}^{\ast}C_{n,\alpha}$ are the eigenvalues of the matrix $$\begin{aligned}
&&\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\left(J_{(n,\alpha)}\otimes I_{n_{\alpha}}\right)\frac{1}{(n,\alpha)}D_{n}^{\ast}D_{n}\left(J_{(n,\alpha)}\otimes I_{n_{\alpha}}\right)\\
&=&\frac{1}{(n,\alpha)}\left[\begin{array}{c|c|c|c}
I_{n_{\alpha}} & I_{n_{\alpha}} & \cdots & I_{n_{\alpha}}\\
\hline
I_{n_{\alpha}} & I_{n_{\alpha}} & \cdots & I_{n_{\alpha}}\\
\hline
\vdots & \vdots & \vdots & \vdots\\
\hline
I_{n_{\alpha}} & I_{n_{\alpha}} & \cdots & I_{n_{\alpha}}
\end{array}\right]
\left[\begin{array}{cccc}
\Delta_{1} & & & \\
& \Delta_{2} & & \\
& & \ddots & \\
& & & \Delta_{(n,\alpha)}
\end{array}\right]
\left[\begin{array}{c|c|c|c}
I_{n_{\alpha}} & I_{n_{\alpha}} & \cdots & I_{n_{\alpha}}\\
\hline
I_{n_{\alpha}} & I_{n_{\alpha}} & \cdots & I_{n_{\alpha}}\\
\hline
\vdots & \vdots & \vdots & \vdots\\
\hline
I_{n_{\alpha}} & I_{n_{\alpha}} & \cdots & I_{n_{\alpha}}
\end{array}\right]\\
&=&\frac{1}{(n,\alpha)}\left[\begin{array}{c|c|c|c}
I_{n_{\alpha}} & I_{n_{\alpha}} & \cdots & I_{n_{\alpha}}\\
\hline
I_{n_{\alpha}} & I_{n_{\alpha}} & \cdots & I_{n_{\alpha}}\\
\hline
\vdots & \vdots &\vdots & \vdots\\
\hline
I_{n_{\alpha}} & I_{n_{\alpha}} &\cdots & I_{n_{\alpha}}
\end{array}\right]
\left[\begin{array}{c|c|c|c}
\Delta_{1} & \Delta_{1} & \cdots & \Delta_{1} \\
\hline
\Delta_{2} & \Delta_{2} & \cdots & \Delta_{2} \\
\hline
\vdots & \vdots & \vdots & \vdots \\
\hline
\Delta_{(n,\alpha)} & \Delta_{(n,\alpha)} & \cdots & \Delta_{(n,\alpha)}
\end{array}\right]\\
&=&\frac{1}{(n,\alpha)}\left[\begin{array}{c|c|c|c}
\overset{(n,\alpha)}{\underset{l=1}\sum}\Delta_{l} & \overset{(n,\alpha)}{\underset{l=1}\sum}\Delta_{l} & \cdots & \overset{(n,\alpha)}{\underset{l=1}\sum}\Delta_{l} \\
\hline
\overset{(n,\alpha)}{\underset{l=1}\sum}\Delta_{l} & \overset{(n,\alpha)}{\underset{l=1}\sum}\Delta_{l} & \cdots & \overset{(n,\alpha)}{\underset{l=1}\sum}\Delta_{l} \\
\hline
\vdots & \vdots & \vdots & \vdots \\
\hline
\overset{(n,\alpha)}{\underset{l=1}\sum}\Delta_{l} & \overset{(n,\alpha)}{\underset{l=1}\sum}\Delta_{l} & \cdots & \overset{(n,\alpha)}{\underset{l=1}\sum}\Delta_{l}
\end{array}\right]\\
&=&\frac{1}{(n,\alpha)}\underbrace{\left[\begin{array}{cccc}
1 & 1 & \cdots & 1\\
1 & 1 & \cdots & 1\\
\vdots & \vdots & \vdots & \vdots\\
1 & 1 & \cdots & 1
\end{array}\right]}_{\textrm{$(n,\alpha)$ times}}\otimes\left(\sum_{l=1}^{(n,\alpha)}\Delta_{l}\right).\end{aligned}$$
Therefore, from $(\ref{19bis})$, we infer that $$\label{eigg}
{\rm Eig}(C_{n,\alpha}^{\ast}C_{n,\alpha})=\frac{1}{(n,\alpha)}{\rm Eig}\left(J_{(n,\alpha)}
\otimes \overset{(n,\alpha)}{\underset{l=1}\sum}\Delta_{l}\right),$$ where $$\begin{aligned}
\label{19}
\frac{1}{(n,\alpha)}{\rm Eig}(J_{(n,\alpha)}) = \{0,1\}.\end{aligned}$$
Here we must observe that $\frac{1}{(n,\alpha)}J_{(n,\alpha)}$ is a matrix of rank 1, so it has all eigenvalues equal to zero except one eigenvalue equal to 1. In fact note that the trace of a matrix is, by definition, the sum of its eigenvalues: in our case the trace is $(n,\alpha)\cdot\frac{1}{(n,\alpha)}=1$ and hence the only nonzero eigenvalue is necessarily equal to 1. Moreover $$\begin{aligned}
\overset{(n,\alpha)}{\underset{l=1}\sum}\Delta_{l}&=&
\overset{(n,\alpha)}{\underset{l=1}\sum}{\rm diag}(d_{(l-1)n_{\alpha}+j};\text{\,\,}j=0,1,\ldots,n_{\alpha}-1)\\
&=&{\rm diag}\left(\overset{(n,\alpha)}{\underset{l=1}\sum}d_{(l-1)n_{\alpha}+j};\text{\,\,}
j=0,1,\ldots,n_{\alpha}-1\right).
\end{aligned}$$
Consequently, since $\overset{(n,\alpha)}{\underset{l=1}\sum}\Delta_{l}$ is a diagonal matrix, we have $$\begin{aligned}
\label{20}
{\rm Eig}\left(\overset{(n,\alpha)}{\underset{l=1}\sum}\Delta_{l}\right)&=&
\left\{\overset{(n,\alpha)}{\underset{l=1}\sum}d_{(l-1)n_{\alpha}+j};\text{\,\,}
j=0,1,\ldots,n_{\alpha}-1\right\},
\end{aligned}$$ where $d_{k}$ are defined in $(\ref{ds})$.
Finally, by exploiting basic properties of the tensor product, we know that the eigenvalues of a tensor product of two square matrices $A\otimes B$ are given by all possible products of eigenvalues of $A$ of order $p$ and of eigenvalues of $B$ of order $q$, that is $\lambda(A\otimes B)=\lambda_{j}(A)\lambda_{k}(B)$ for $j=1,\ldots,p$ and $k=1,\ldots,q$. Therefore, by taking into consideration $(\ref{eigg})$, $(\ref{19})$, and $(\ref{20})$, we find $$\begin{aligned}
\label{21}
\lambda_{j}(C_{n,\alpha}^{\ast}C_{n,\alpha})&=&\overset{(n,\alpha)}
{\underset{l=1}\sum}d_{(l-1)n_{\alpha}+j}, \quad j=0,1,\ldots,n_{\alpha}-1,\\
\label{21bis}\lambda_{j}(C_{n,\alpha}^{\ast}C_{n,\alpha})&=& 0,\qquad j=n_{\alpha},\ldots,n-1.\end{aligned}$$
From $(\ref{21})$, $(\ref{21bis})$ and $(\ref{1}),$ one obtains that the singular values of an $\alpha$-circulant matrix $C_{n,\alpha}$ are given by $$\begin{aligned}
\label{22}
\sigma_{j}(C_{n,\alpha})&=&\sqrt{\overset{(n,\alpha)}
{\underset{l=1}\sum}d_{(l-1)n_{\alpha}+j}}, \quad j=0,1,\ldots,n_{\alpha}-1,\\
\notag\sigma_{j}(C_{n,\alpha})&=& 0,\qquad j=n_{\alpha},\ldots,n-1,\end{aligned}$$ where the values $d_{k}$, $k=0,\ldots,n-1$, are defined in $(\ref{ds})$.
Special cases and observations {#special-case}
------------------------------
In this subsection we consider some special cases and we furnish a further link between the eigenvalues of circulant matrices and the singular values of $\alpha$-circulants. In the case where $(n,\alpha)=1,$ we have $n_{\alpha}=\frac{n}{(n,\alpha)}=n$. Hence the formula $(\ref{22})$ becomes $$\begin{aligned}
\sigma_{j}(C_{n,\alpha})=\sqrt{d_{j}}, \quad j=0,1,\ldots,n-1.\end{aligned}$$
In other words the singular values of $C_{n,\alpha}$ coincide with those of $C_n$ (this is expected since $Z_{n,\alpha}$ is a permutation matrix) and in particular with the moduli of the eigenvalues of $C_n$.
Concerning the eigenvalues of circulant matrices it should be observed that formula (\[V\]) can be interpreted in function terms as the evaluation of a polynomial at the grid points given by the $n$-th roots of the unity. This is a standard observation because the Fourier matrix is a special instance of the classical Vandermonde matrices when the knots are exactly all the $n$-th roots of the unity.
Therefore, defining the polynomial $p(t)=\sum_{k=0}^{n-1} a_k e^{ikt}$, it is trivial to observe that the eigenvalues of $C_n= F_n D_n
F_n^*$ are given by $$\lambda_j(C_n)=p\left(\frac{2\pi j}{n}\right), \ \ \ j=0,\ldots,n-1.$$
The question that naturally arises is how to connect the expression in (\[22\]) of the nontrivial singular values of $C_{n,\alpha}$ with the polynomial $p$. The answer is somehow intriguing and can be resumed in the following formula which could be of interest in the multigrid community (see Section \[sec:multigrid\]) $$\begin{aligned}
\label{sv-nontrivial-symbol-p}
\sigma_{j}(C_{n,\alpha}) & = & \sqrt{\overset{(n,\alpha)-1}
{\underset{l=0}\sum} |p|^2\left(\frac{x_j+2\pi l}{(n,\alpha)} \right)},
\quad x_j=\frac{2\pi j}{n_{\alpha}}, \ \ j=0,1,\ldots,n_{\alpha}-1.\end{aligned}$$
In addition if $\alpha$ is fixed and a sequence of integers $n$ is chosen so that $(n,\alpha)>1$ for $n$ large enough, then $\{C_{n,\alpha}\}\sim_\sigma (0,G)$ for a proper set $G$. If the sequence of $n$ is chosen so that $n$ and $\alpha$ are coprime for all $n$ large enough, then the existence of the distribution is related to the smoothness properties of a function $f$ such that $\{a_k\}$ can be interpreted as the sequence of its Fourier coefficients (see e.g. [@appr-mult]). From the above reasoning it is clear that, if $n$ is allowed to be vary among all the positive integer numbers, then $\{C_{n,\alpha}\}$ does not possess a joint singular value distribution.
Singular values of $\alpha$-Toeplitz matrices {#sec:toep}
=============================================
For $p=q=d=1$, we recall that the $\alpha$-Toeplitz matrices of dimension $n\times n$ are defined as $$\begin{aligned}
\label{tnalpha}
T_{n,\alpha}=[a_{r-\alpha c}]_{r,c=0}^{n-1},\end{aligned}$$ where the quantities $r-\alpha s$ are not reduced modulus $n$. In analogy with the case of $\alpha=1$, the elements $a_{j}$ are the Fourier coefficients of some function $f$ in $L^{1}(Q)$, with $Q=(-\pi,\pi)$, i.e., $a_j=\tilde f_j$ as in (\[defcoeff\]) with $d=1$. If we denote by $T_{n}$ the classical Toeplitz matrix generated by the function $f\in L^{1}(Q)$, $T_{n}=[a_{r-c}]_{r,c=0}^{n-1}$, $a_{j}=\tilde f_j$ defined as in $(\ref{defcoeff})$, and by $T_{n,\alpha}$ the $\alpha$-Toeplitz matrix generated by the same function, one verifies immediately for $n$ and $\alpha$ generic that $$\begin{aligned}
\label{T}
T_{n,\alpha}=\left[\widehat{T}_{n,\alpha}|\mathcal{T}_{n,\alpha}\right]=
\left[T_{n}\widehat{Z}_{n,\alpha}|\mathcal{T}_{n,\alpha}\right],\end{aligned}$$ where $\widehat{T}_{n,\alpha}\in\mathbb{C}^{n\times \mu_{\alpha}}$, $\mu_{\alpha}=\left\lceil \frac{n}{\alpha}\right\rceil$, is the matrix $T_{n,\alpha}$ defined in $(\ref{tnalpha})$ by considering only the $\mu_{\alpha}$ first columns, $\mathcal{T}_{n,\alpha}\in\mathbb{C}^{n\times (n-\mu_{\alpha})}$ is the matrix $T_{n,\alpha}$ defined in $(\ref{tnalpha})$ by considering only the $n-\mu_{\alpha}$ last columns, and $\widehat{Z}_{n,\alpha}$ is the matrix defined in $(\ref{i})$ by considering only the $\mu_{\alpha}$ first columns.
(of relation $(\ref{T}).$) For $r=0,1,\ldots,n-1$ and $s=0,1,\ldots,\mu_{\alpha}-1,$ one has $$\begin{aligned}
(\widehat{T}_{n,\alpha})_{r,s}&=&(T_{n})_{r,\alpha s},\\
(\widehat{Z}_{n,\alpha})_{r,s}&=&\delta_{r-\alpha s},\end{aligned}$$ and $$\begin{aligned}
(T_{n}\widehat{Z}_{n,\alpha})_{r,s} &=& \overset{n-1}
{\underset{l=0}\sum}(T_{n})_{r,l}(\widehat{Z}_{n,\alpha})_{l,s} \\
&=& \overset{n-1}{\underset{l=0}\sum}\delta_{l-\alpha s}(T_{n})_{r,l} \\
&{\underset{\rm (a)}=}& (T_{n})_{r,\alpha s} \\
&=& (\widehat{T}_{n,\alpha})_{r,s},\end{aligned}$$ where (a) follows because there exists a unique $l\in\{0,1,\ldots,n-1\}$ such that $l-\alpha s\equiv0\textrm{ (mod
$n$)}$, that is, $l\equiv \alpha s\textrm{ (mod $n$)}$, and, since $0\leq\alpha s\leq n-1$, we obtain $l=\alpha s$.
If we take the matrix $\widehat{T}_{n,\alpha}$ of size $n\times(\mu_{\alpha}+1)$, then relation $(\ref{T})$ is no longer true. In reality, looking at the $(\mu_{\alpha}+1)$-th column of the $\alpha $-Toeplitz we observe Fourier coefficients with indices which are not present (less or equal to $-n$) in the Toeplitz matrix $T_{n}$. More precisely, $$\begin{aligned}
(T_{n,\alpha})_{0,\mu_{\alpha}}=a_{0-\alpha\mu_{\alpha}}=
a_{-\alpha\mu_{\alpha}},\qquad\textrm{and $\,-\alpha\mu_{\alpha}\leq -n$.}\end{aligned}$$
It follows that $\mu_{\alpha}$ is the maximum number of columns for which relation $(\ref{T})$ is true.
Some preparatory results
------------------------
We begin with some preliminary notations and definitions.
\[appr:seq\] Suppose a sequence of matrices $\{A_n\}_{n}$ of size $d_n$ is given. We say that $\{\{B_{n,m}\}_{n}:m\ge 0\}$, $B_{n,m}$ of size $d_n$, $m\in\mathbb{N}$, is an approximating class of sequences $(a.c.s.)$ for $\{A_n\}_{n}$ if, for all sufficiently large $m\in\mathbb{N}$, the following splitting holds: $$\label{spli1}
A_n=B_{n,m}+R_{n,m}+N_{n,m}\quad \mbox{for all}\ n> n_m,$$ with $$\label{propri1} {\rm Rank}(R_{n,m}) \leq d_n\, c(m), \quad {\left\Vert
N_{n,m} \right\Vert} \leq \omega(m),$$ where $\|\cdot\|$ is the spectral norm (largest singular value), $n_m$, $ c(m)$ and $\omega(m)$ depend only on $m$ and, moreover, $$\label{propri1bis}
\lim_{m\to\infty} \omega(m)=0, \ \ \lim_{m\to\infty} c(m)=0.$$
[[@algebra]]{} \[lem\] Let $\{d_n\}_n$ be an increasing sequence of natural numbers. Suppose a sequence formed by matrices $\{A_n\}_n$ of size $d_n$ is given such that $\{\{B_{n,m}\}_n:\ m\ge 0\}$, $m\in\hat {\mathbb{N}}
\subset \mathbb{N}$, $\# \hat {\mathbb{N}}=\infty$, is an $a.c.s.$ for $\{A_n\}_n$ in the sense of Definition $\ref{appr:seq}$. Suppose that $\{B_{n,m}\}_n\sim_\sigma (\theta_m, G)$ and that $\theta_m$ converges in measure to the measurable function $\theta$ over $G$. Then necessarily $$\label{l7} \{A_n\}_n\sim_\sigma (\theta,G),$$ (see Definition $\ref{def-distribution}$).
\[fgbis\][[@algebra; @ser-glt]]{} If $\{A_{n}\}_{n}$ and $\{B_{n}\}_{n}$ are two sequences of matrices of strictly increasing dimension, such that $\{A_{n}\}_{n}\sim_{\sigma}(\theta,G)$ and $\{B_{n}\}_{n}\sim_{\sigma} (0,G)$, then $$\begin{aligned}
\{A_{n}+B_{n}\}_{n}\sim_{\sigma} (\theta,G).\end{aligned}$$
\[fg\][[@algebra]]{} Let $f,\,g\in L^{1}(Q^d)$, $Q=(-\pi,\pi)$, and let $\{T_{n}(f)\}_{n}$ and $\{T_{n}(g)\}_{n}$ be the two sequences of Toeplitz matrices generated by $f$ and $g$, respectively. The following distribution result is true $$\begin{aligned}
\{T_{n}(f)T_{n}(g)\}_{n}\sim_{\sigma}(fg,Q^d).\end{aligned}$$
\[terzo\] Let $f$ be a measurable complex-valued function on a set $K$, and consider the measurable function $\sqrt{|f|}:K\rightarrow\mathbb{R}^{+}$. Let $\{A_{n,m}\}$, with $A_{n,m}\in\mathbb{C}^{d_{n}\times d'_{n}}$, $d'_{n}\leq d_{n}$, be a sequence of matrices of strictly increasing dimension: $d'_{n}<d'_{n+1}$ and $d_{n}\leq d_{n+1}$. If the sequence of matrices $\{A_{n,m}^{*}A_{n,m}\}$, with $A_{n,m}^{*}A_{n,m}\in\mathbb{C}^{d'_{n}\times d'_{n}}$ and $d'_{n}<d'_{n+1}$, is distributed in the singular value sense as the function $f$ over a proper set $G\subset K$ in the sense of Definition $\ref{def-distribution}$, then the sequence $\{A_{n,m}\}$ is distributed in the singular value sense as the function $\sqrt{|f|}$ over the same $G$.
From the singular value decomposition ($SVD$), we can write $A_{n,m}$ as $$\begin{aligned}
A_{n,m}=U\Sigma V^{*}=U\left[\begin{array}{cccc}
\sigma_{1} & & &\\
& \sigma_{2} & & \\
& & \ddots & \\
& & & \sigma_{d'_{n}}\\
\hline
& 0 & &
\end{array}\right]V^{*},\end{aligned}$$ with $U$ and $V$ unitary matrices $U\in\mathbb{C}^{d_{n}\times d_{n}}$, $V\in\mathbb{C}^{d'_{n}\times d'_{n}}$ and $\Sigma\in\mathbb{R}^{d_{n}\times d'_{n}}$, $\sigma_{j}\geq 0$; by multiplying $A_{n,m}^{*}A_{n,m}$ we obtain: $$\begin{aligned}
\label{asa}
\notag A_{n,m}^{*}A_{n,m}=V\Sigma^{T} U^{*}U\Sigma V^{*}&=&V\Sigma^{T}\Sigma V^{*}=V\Sigma^{(2)} V^{*}\\
&=&V\left[\begin{array}{cccc}
\sigma_{1}^{2} & & &\\
& \sigma_{2}^{2} & & \\
& & \ddots & \\
& & & \sigma_{d'_{n}}^{2}
\end{array}\right]V^{*},\end{aligned}$$ with $V$ unitary matrix $V\in\mathbb{C}^{d'_{n}\times d'_{n}}$ and $\Sigma^{(2)}\in\mathbb{R}^{d'_{n}\times d'_{n}}$, $\sigma_{j}^{2}\geq
0$; we observe that $(\ref{asa})$ is an $SVD$ for $A_{n,m}^{*}A_{n,m}$, that is, the singular values $\sigma_{j}(A_{n,m}^{*}A_{n,m})$ of $A_{n,m}^{*}A_{n,m}$ are the square of singular values $\sigma_{j}(A_{n,m})$ of $A_{n,m}$. Since $\{A_{n,m}^{*}A_{n,m}\}\sim_{\sigma} (f,G)$, by definition it hold that for every $F\in {{{\cal C}_0}}(\mathbb{R}_{0}^{+})$ $$\begin{aligned}
\label{disasa}
\notag\lim_{n\to\infty} \frac{1}{d'_{n}}\sum_{i=1}^{d'_{n}}
F\left(\sigma_{i}(A_{n,m}^{*}A_{n,m})\right)&=&
\frac{1}{\mu(G)}\int_{G}
F\left(|f(t)|\right)\, dt\\
&=& \frac{1}{\mu(G)}\int_{G}H\left(\sqrt{|f(t)|}\right)\, dt,\end{aligned}$$ where $H$ is such that $F=H\circ\sqrt{\cdot}$; but, owing to $\sigma_{j}(A_{n,m})=\sqrt{\sigma_{j}(A_{n,m}^{*}A_{n,m})}$ we obtain $$\begin{aligned}
\label{disasa1}
\notag\lim_{n\to\infty} \frac{1}{d'_{n}}\sum_{i=1}^{d'_{n}}F\left(\sigma_{i}(A_{n,m}^{*}A_{n,m})\right)&=&
\lim_{n\to\infty} \frac{1}{d'_{n}}\sum_{i=1}^{d'_{n}}F\left(\sigma_{i}^{2}(A_{n,m})\right)\\
&=&\lim_{n\to\infty} \frac{1}{d'_{n}}\sum_{i=1}^{d'_{n}}
H\left(\sigma_{i}(A_{n,m})\right).\end{aligned}$$
From $(\ref{disasa})$ and $(\ref{disasa1})$ we obtain $$\begin{aligned}
\lim_{n\to\infty} \frac{1}{d'_{n}}\sum_{i=1}^{d'_{n}}
H\left(\sigma_{i}(A_{n,m})\right)=\frac{1}{\mu(G)}\int_{G}
H\left(\sqrt{|f(t)|}\right)\, dt,\end{aligned}$$ for every $H\in {{{\cal C}_0}}(\mathbb{R}_{0}^{+})$, so $\{A_{n,m}\}\sim_{\sigma}(\sqrt{|f(t)|},G)$.
\[primo\] Let $\{A_{n}\}_{n}$ and $\{Q_{n}\}_{n}$ be two sequences of matrices of strictly increasing dimension ($A_{n},Q_{n}\in\mathbb{C}^{d_{n}\times d_{n}}$, $d_{n}<d_{n+1}$), where $Q_{n}$ are all unitary matrices ($Q_{n}Q_{n}^{*}=I$). If $\{A_{n}\}_{n}\sim_{\sigma}(0, G)$ then $\{A_{n}Q_{n}\}_{n}\sim_{\sigma} (0,G)$ and $\{Q_{n}A_{n}\}_{n}\sim_{\sigma} (0,G)$.
Putting $B_{n}=A_{n}Q_{n}$, assuming that $$\begin{aligned}
A_{n}=U_{n}\Sigma_{n}V_{n},\end{aligned}$$ is an $SVD$ for $A_{n}$, and taking into account that the product of two unitary matrices is still a unitary matrix, we deduce that the writing $$\begin{aligned}
B_{n}=A_{n}Q_{n}=U_{n}\Sigma_{n}V_{n}Q_{n}=U_{n}\Sigma_{n}\widehat{V}_{n},\end{aligned}$$ is an $SVD$ for $B_{n}$. The latter implies that $A_{n}$ and $B_{n}$ have exactly the same singular values, so that the two sequences $\{A_{n}\}_{n}$ and $\{B_{n}\}_{n}$ are distributed in the same way.
\[secondo\] Let $\{A_{n}\}_{n}$ and $\{Q_{n}\}_{n}$ be two sequences of matrices of strictly increasing dimension ($A_{n},Q_{n}\in\mathbb{C}^{d_{n}\times
d_{n}}$, $d_{n}<d_{n+1}$). If $\{A_{n}\}_{n}\sim_{\sigma} (0,G)$ and $\|Q_{n}\| \le M$ for some nonnegative constant $M$ independent of $n$, then $\{A_{n}Q_{n}\}_{n}\sim_{\sigma} (0,G)$ and $\{Q_{n}A_{n}\}_{n}\sim_{\sigma} (0,G)$.
Since $\{A_{n}\}_{n}\sim_{\sigma} (0,G)$, then $\{0_{n}\}_{n}$ (sequence of zero matrices) is an $a.c.s.$ for $\{A_{n}\}_{n}$; this means (by Definition $(\ref{appr:seq})$) that we can write, for every $m$ sufficiently large, $m\in\mathbb{N}$ $$\begin{aligned}
\label{A}
A_{n}=0_{n}+R_{n,m}+N_{n,m},\qquad \forall n>n_{m},\end{aligned}$$ with $$\begin{aligned}
{\rm Rank}(R_{n,m})\leq d_{n}c(m),\qquad\|N_{n,m}\|\leq\omega(m),\end{aligned}$$ where $n_{m}\geq 0$, $c(m)$ and $\omega(m)$ depend only on $m$ and, moreover $$\begin{aligned}
\lim_{m\rightarrow\infty}c(m)=0,\qquad\lim_{m\rightarrow\infty}\omega(m)=0.\end{aligned}$$
Now consider the matrix $A_{n}Q_{n}$; from $(\ref{A})$ we obtain $$\begin{aligned}
A_{n}Q_{n}=0_{n}+R_{n,m}Q_{n}+N_{n,m}Q_{n},\qquad \forall n>n_{m},\end{aligned}$$ with $$\begin{aligned}
&&{\rm Rank}(R_{n,m}Q_{n})\leq\min\{{\rm Rank}(R_{n,m}),{\rm Rank}(Q_{n})\}\leq
{\rm Rank}(R_{n,m})\leq d_{n}c(m),\\
&&\|N_{n,m}Q_{n}\|\leq\|N_{n,m}\|\|Q_{n}\|\leq M\omega(m),\end{aligned}$$ where $$\begin{aligned}
\lim_{m\rightarrow\infty}c(m)=0,\qquad\lim_{m\rightarrow\infty}M\omega(m)=0,\end{aligned}$$ then $\{0_{n}\}_{n}$ is an $a.c.s.$ for the sequence $\{A_{n}Q_{n}\}_{n}$ and, by Proposition $\ref{lem}$, $\{A_{n}Q_{n}\}_{n}\sim_{\sigma} (0,G)$.
Singular value distribution for the $\alpha$-Toeplitz sequences
---------------------------------------------------------------
As stated in formula $(\ref{T})$, the matrix $T_{n,\alpha}$ can be written as $$\begin{aligned}
\label{dist}
\notag T_{n,\alpha}&=&\left[T_{n}\widehat{Z}_{n,\alpha}|\mathcal{T}_{n,\alpha}\right]\\
&=&\left[\begin{array}{c|c}
T_{n}\widehat{Z}_{n,\alpha} & 0
\end{array}\right]+\left[\begin{array}{c|c}
0 & \mathcal{T}_{n,\alpha}
\end{array}\right].\end{aligned}$$
To find the distribution in the singular value sense of the sequence $\{T_{n,\alpha}\}_{n}$, the idea is to study separately the distribution of the two sequences $\{[T_{n}\widehat{Z}_{n,\alpha}|0]\}_{n}$ and $\{[0|\mathcal{T}_{n,\alpha}]\}_{n}$, to prove $\{[0|\mathcal{T}_{n,\alpha}]\}_{n}\sim (0,G)$, and then apply Proposition $\ref{fgbis}$.
### Singular value distribution for the sequence $\{[T_{n}\widehat{Z}_{n,\alpha}|0]\}_{n}$
Since $T_{n}\widehat{Z}_{n,\alpha}\in\mathbb{C}^{n\times
\mu_{\alpha}}$ and $[T_{n}\widehat{Z}_{n,\alpha}|0]\in\mathbb{C}^{n\times n}$, the matrix $[T_{n}\widehat{Z}_{n,\alpha}|0]$ has $n-\mu_{\alpha}$ singular values equal to zero and the remaining $\mu_{\alpha}$ equal to those of $T_{n}\widehat{Z}_{n,\alpha}$; to study the distribution in the singular value sense of this sequence of non-square matrices, we use Lemma $\ref{terzo}$: consider the $\alpha$-Toeplitz matrix “truncated” $\widehat{T}_{n,\alpha}=T_{n}(f)\widehat{Z}_{n,\alpha}$, where the elements of the Toeplitz matrix $T_{n}(f)=[a_{r-c}]_{r,c=0}^{n-1}$ are the Fourier coefficients of a function $f$ in $L^{1}(Q)$, $Q=(-\pi,\pi)$, then we have $$\begin{aligned}
\label{ff}
\notag\widehat{T}_{n,\alpha}^{*}\widehat{T}_{n,\alpha}&=&(T_{n}(f)\widehat{Z}_{n,\alpha})^{*}T_{n}(f)\widehat{Z}_{n,\alpha}=
\widehat{Z}_{n,\alpha}^{*}T_{n}(f)^{*}T_{n}(f)\widehat{Z}_{n,\alpha}\\
&=&\widehat{Z}_{n,\alpha}^{*}T_{n}(\overline{f})T_{n}(f)\widehat{Z}_{n,\alpha}.\end{aligned}$$
We provide in detail the analysis in the case where $f\in
L^{2}(Q)$. The general setting in which $f\in L^{1}(Q)$ can be obtained by approximation and density arguments as done in [@algebra]. From Proposition $\ref{fg}$ if $f\in L^{2}(Q)\subset
L^{1}(Q)$ (that is $|f|^{2}\in L^{1}(Q)$), then $\{T_{n}(\overline{f})T_{n}(f)\}_{n}\sim_{\sigma}(|f|^{2},Q)$. Consequently, for every $m$ sufficiently large, $m\in\mathbb{N}$, the use of Proposition $\ref{lem}$ implies $$\begin{aligned}
T_{n}(\overline{f})T_{n}(f)=T_{n}(|f|^{2})+R_{n,m}+N_{n,m},\qquad \forall n>n_{m},\end{aligned}$$ with $$\begin{aligned}
{\rm Rank}(R_{n,m})\leq nc(m),\qquad\|N_{n,m}\|\leq\omega(m),\end{aligned}$$ where $n_{m}\geq 0$, $c(m)$ and $\omega(m)$ depend only on $m$ and, moreover $$\begin{aligned}
\lim_{m\rightarrow\infty}c(m)=0,\qquad\lim_{m\rightarrow\infty}\omega(m)=0.\end{aligned}$$
Therefore $(\ref{ff})$ becomes $$\begin{aligned}
\label{tst}
\notag\widehat{T}_{n,\alpha}^{*}\widehat{T}_{n,\alpha}
&=&\widehat{Z}_{n,\alpha}^{*}(T_{n}(|f|^{2})+R_{n,m}+N_{n,m})\widehat{Z}_{n,\alpha}\\
\notag&=&\widehat{Z}_{n,\alpha}^{*}T_{n}(|f|^{2})\widehat{Z}_{n,\alpha}+
\widehat{Z}_{n,\alpha}^{*}R_{n,m}\widehat{Z}_{n,\alpha}+\widehat{Z}_{n,\alpha}^{*}N_{n,m}\widehat{Z}_{n,\alpha}\\
&=&\widehat{Z}_{n,\alpha}^{*}T_{n}(|f|^{2})\widehat{Z}_{n,\alpha}+\widehat{R}_{n,m,\alpha}+\widehat{N}_{n,m,\alpha},\end{aligned}$$ with $$\begin{aligned}
\label{1bis}
&&{\rm Rank}(\widehat{R}_{n,m,\alpha})\leq\min\{{\rm Rank}(\breve{Z}_{n,\alpha}),{\rm Rank}(R_{n,m})\}\leq
{\rm Rank}(R_{n,m})\leq nc(m),\\
\label{2bis}&&\|\widehat{N}_{n,m,\alpha}\|\leq2\|\breve{Z}_{n,\alpha}\|\|N_{n,m}\|\leq2\omega(m),\end{aligned}$$ and $$\begin{aligned}
\lim_{m\rightarrow\infty}c(m)=0,\qquad\lim_{m\rightarrow\infty}2\omega(m)=0,\end{aligned}$$ where in $(\ref{1bis})$ and $(\ref{2bis})$, $\breve{Z}_{n,\alpha}=[\widehat{Z}_{n,\alpha}|0]\in\mathbb{C}^{n\times
n}$. In other words $\breve{Z}_{n,\alpha}$ is the matrix $\widehat{Z}_{n,\alpha}$ supplemented by an appropriate number of zero columns in order to make it square. Furthermore, it is worth noticing that $\|\widehat{Z}_{n,\alpha}\|=\|\widehat{Z}_{n,\alpha}^*\|=1$, because $\widehat{Z}_{n,\alpha}$ is a submatrix of the identity: we have used the latter relations in $(\ref{2bis})$.
Now, consider the matrix $\widehat{Z}_{n,\alpha}^{*}T_{n}(|f|^{2})\widehat{Z}_{n,\alpha}
\in\mathbb{C}^{\mu_{\alpha}\times\mu_{\alpha}}$, with $\mu_{\alpha}=\left\lceil \frac{n}{\alpha}\right\rceil$, $f\in
L^{2}(Q)\subset L^{1}(Q)$ (so $|f|^{2}\in L^{1}(Q)$). From $(\ref{T})$, setting $T_{n}=T_{n}(|f|^{2})=[\tilde{a}_{r-c}]_{r,c=0}^{n-1}$, with $\tilde{a}_{j}$ being the Fourier coefficients of $|f|^{2}$, and setting $T_{n,\alpha}$ the $\alpha$-Toeplitz generated by the same function $|f|^{2}$, it is immediate to observe $$\begin{aligned}
\label{brc}
T_{n}\widehat{Z}_{n,\alpha}=\widehat{T}_{n,\alpha}\in
\mathbb{C}^{n\times \mu_{\alpha}},\qquad \textrm{with}
\quad (\widehat{T}_{n,\alpha})_{r,c}=\tilde{a}_{r-\alpha c},\end{aligned}$$ for $r=0,\ldots,n-1$ and $c=0,\ldots,\mu_{\alpha}-1$. If we compute $\widehat{Z}_{n,\alpha}^{*}\widehat{T}_{n,\alpha}\in\mathbb{C}^{\mu_{\alpha}\times\mu_{\alpha}}$, where $Z_{n,\alpha}^{*}=[\delta_{c-\alpha r}]_{r,c=0}^{n-1}$ ($\delta_{k}$ defined as in $(\ref{i})$) and $\widehat{Z}_{n,\alpha}^{*}\in\mathbb{C}^{\mu_{\alpha}\times n}$ is the submatrix of $Z_{n,\alpha}^{*}$ obtained by considering only the $\mu_{\alpha}$ first rows, for $r,c=0,\ldots,\mu_{\alpha}-1$, we obtain $$\begin{aligned}
(\widehat{Z}_{n,\alpha}^{*}T_{n}(|f|^{2})\widehat{Z}_{n,\alpha})_{r,c}&=&
(\widehat{Z}_{n,\alpha}^{*}\widehat{T}_{n,\alpha})_{r,c}\\
&=&\sum_{\ell=0}^{n-1}(\widehat{Z}_{n,\alpha}^{*})_{r,\ell}(\widehat{T}_{n,\alpha})_{\ell,c}\\
&{\underset{\rm (a)}=}&(\widehat{T}_{n,\alpha})_{\alpha r,c}\\
&{\underset{\rm from\,(\ref{brc})}=}& \widehat{a}_{\alpha r-\alpha c},\end{aligned}$$ where (a) follows from the existence of a unique $\ell\in\{0,1,\ldots,n-1\}$ such that $\ell-\alpha
r\equiv0\textrm{ (mod $n$)}$, that is, $\ell\equiv \alpha
r\textrm{ (mod $n$)}$, and, since $0\leq\alpha r\leq n-1$, we find $\ell=\alpha r$.
Therefore $$\begin{aligned}
\widehat{Z}_{n,\alpha}^{*}T_{n}(|f|^{2})\widehat{Z}_{n,\alpha}&=&
[\tilde{a}_{\alpha r-\alpha c}]_{r,c=0}^{\mu_{\alpha}-1}\\
&=&T_{\mu_{\alpha}}(\widehat{|f|^{(2)}}),\end{aligned}$$ where $\widehat{|f|^{(2)}}\in L^{1}(Q)$ is given by $$\begin{aligned}
\label{f2t}
\widehat{|f|^{(2)}}(x)&=&\frac{1}{\alpha}\sum_{j=0}^{\alpha-1}|f|^{2}\left(\frac{x+2\pi j}{\alpha}\right),\\
\label{f2tbis}|f|^{2}(x)&=&\sum_{k=-\infty}^{+\infty}\tilde{a}_{k}e^{ikx}.\end{aligned}$$
(of relation $(\ref{f2t}).$) We denote by $\textrm{a}_{j}$ the Fourier coefficients of $\widehat{|f|^{(2)}}$. We want to show that for $r,c=0,\ldots,\mu_{\alpha}-1$, $\textrm{a}_{r-c}=\tilde{a}_{\alpha
r-\alpha c}$, where $\tilde{a}_{k}$ are the Fourier coefficients of $|f|^{2}$. From $(\ref{defcoeff})$, $(\ref{f2t})$ and $(\ref{f2tbis})$, we have $$\begin{aligned}
\textrm{a}_{r-c}&=&\frac{1}{2\pi}\int_{-\pi}^{\pi}\frac{1}{\alpha}\sum_{j=0}^{\alpha-1}
\sum_{k=-\infty}^{+\infty}\tilde{a}_{k}e^{ik\left(\frac{x+2\pi j}{\alpha}\right)}e^{-i(r-c)x}dx\\
&=&\frac{1}{2\pi\alpha}\int_{-\pi}^{\pi}\sum_{k=-\infty}^{+\infty}\tilde{a}_{k}\left(\sum_{j=0}^{\alpha-1}e^{\frac{i2\pi kj}{\alpha}}\right)
e^{\frac{ikx}{\alpha}}e^{-i(r-c)x}dx.\end{aligned}$$ Some remarks are in order:
- if $k$ is a multiple of $\alpha$, $k=\alpha t$ for some value of $t$, then we have that $\overset{\alpha-1}{\underset{j=0}\sum}e^{\frac{i2\pi kj}{\alpha}}=\overset{\alpha-1}{\underset{j=0}\sum}e^{\frac{i2\pi\alpha tj}{\alpha}}=
\overset{\alpha-1}{\underset{j=0}\sum}e^{i2\pi tj}=\overset{\alpha-1}{\underset{j=0}\sum}1=\alpha$.
- if $k$ is not a multiple of $\alpha$, then $e^{\frac{i2\pi k}{\alpha}}\neq 1$ and therefore $\overset{\alpha-1}{\underset{j=0}\sum}e^{\frac{i2\pi kj}{\alpha}}=\overset{\alpha-1}{\underset{j=0}\sum}\left(e^{\frac{i2\pi k}{\alpha}}\right)^{j}$ is a finite geometric series whose sum is given by $$\begin{aligned}
\sum_{j=0}^{\alpha-1}\left(e^{\frac{i2\pi k}{\alpha}}\right)^{j}=\frac{1-e^{\frac{i2\pi k\alpha}{\alpha}}}{1-e^{\frac{i2\pi k}{\alpha}}}=
\frac{1-e^{i2\pi k}}{1-e^{\frac{i2\pi k}{\alpha}}}=\frac{1-1}{1-e^{\frac{i2\pi k}{\alpha}}}=0.
\end{aligned}$$
Finally, taking into account the latter statements and recalling that $\frac{1}{2\pi}\int_{-\pi}^{\pi} e^{i\ell x} dx=
\left\{\begin{smallmatrix} 1 & \textrm{ if $\ell=0$}\\ 0
&\textrm{otherwise} \end{smallmatrix}\right.$, we find $$\begin{aligned}
\textrm{a}_{r-c}&=&\frac{1}{2\pi\alpha}\int_{-\pi}^{\pi}\sum_{t=-\infty}^{+\infty}\tilde{a}_{\alpha t}
\alpha e^{\frac{i\alpha t x}{\alpha}}e^{-i(r-c)x}dx\\
&=&\sum_{t=-\infty}^{+\infty}\tilde{a}_{\alpha t}\frac{1}{2\pi}\int_{-\pi}^{\pi}e^{ix(t-(r-c))}dx\\
&=&\tilde{a}_{\alpha(r-c)}.\end{aligned}$$
In summary, from $(\ref{tst})$ we have $$\begin{aligned}
\widehat{T}_{n,\alpha}^{*}\widehat{T}_{n,\alpha}=
T_{\mu_{\alpha}}(\widehat{|f|^{(2)}})+\widehat{R}_{n,m,\alpha}+\widehat{N}_{n,m,\alpha},\end{aligned}$$ with $\{T_{\mu_{\alpha}}(\widehat{|f|^{(2)}})\}_{n}\sim_{\sigma}(\widehat{|f|^{(2)}},Q)$. We recall that, owing to $(\ref{f2t})$, the relation $|f|^{2}\in
L^{1}(Q)$ implies $\widehat{|f|^{(2)}}\in L^{1}(Q)$. Consequently Proposition $\ref{lem}$ implies that $\{\widehat{T}_{n,\alpha}^{*}\widehat{T}_{n,\alpha}\}_{n}\sim_{\sigma}(\widehat{|f|^{(2)}},Q)$. Clearly $\widehat{|f|^{(2)}}\in L^{1}(Q)$ is equivalent to write $\sqrt{\widehat{|f|^{(2)}}}\in L^{2}(Q)$: therefore, from Lemma $\ref{terzo}$, we infer $\{\widehat{T}_{n,\alpha}\}_{n}\sim_{\sigma}(\sqrt{\widehat{|f|^{(2)}}},Q)$.
Now, as mentioned at the beginning of this section, by Definition $\ref{def-distribution}$, we have $$\begin{aligned}
\lim_{n\rightarrow\infty} \frac{1}{n}\sum_{j=1}^{n}F\left(\sigma_j([\widehat{T}_{n,\alpha}|0])\right)&=&
\lim_{n\rightarrow\infty}\frac{1}{n}\sum_{j=1}^{\mu_{\alpha}}F\left(\sigma_j([\widehat{T}_{n,\alpha}|0])\right)+
\lim_{n\rightarrow\infty}\frac{1}{n}\sum_{j=\mu_{\alpha}+1}^{n}F(0)\\
&=&\lim_{n\rightarrow\infty}\frac{\mu_{\alpha}}{n}\sum_{j=1}^{\mu_{\alpha}}\frac{F\left(\sigma_j([\widehat{T}_{n,\alpha}|0])\right)}{\mu_{\alpha}}+
\lim_{n\rightarrow\infty}\frac{n-\mu_{\alpha}}{n}F(0)\\
&=&\frac{1}{\alpha}\frac{1}{2\pi}\int_{-\pi}^{\pi}F\left(\sqrt{\widehat{|f|^{(2)}}(x)}\right)\,dx+
\left(1-\frac{1}{\alpha}\right)F(0),\end{aligned}$$ which results to be equivalent to the following distribution formula $$\begin{aligned}
\label{dist1}
\{[T_{n}\widehat{Z}_{n,\alpha}|0]\}_{n}\sim_{\sigma}(\theta, Q\times[0,1]),\end{aligned}$$ where $$\begin{aligned}
\label{teta}
\theta(x,t)=\left\{\begin{array}{cl}
\sqrt{\widehat{|f|^{(2)}}(x)} & \textrm{for $t\in\left[0,\frac{1}{\alpha}\right]$,}\\
0 & \textrm{for $t\in\left(\frac{1}{\alpha},1\right]$}.
\end{array}\right.\end{aligned}$$
### Singular value distribution for the sequence $\{[0|\mathcal{T}_{n,\alpha}]\}_{n}$
In perfect analogy with the case of the matrix $[T_{n}\widehat{Z}_{n,\alpha}|0]$, we can observe that $\mathcal{T}_{n,\alpha}\in\mathbb{C}^{n\times (n-\mu_{\alpha})}$ and $[0|\mathcal{T}_{n,\alpha}]\in\mathbb{C}^{n\times n}$. Therefore the matrix $[0|\mathcal{T}_{n,\alpha}]$ has $\mu_{\alpha}$ singular values equal to zero and the remaining $n-\mu_{\alpha}$ equal to those of $\mathcal{T}_{n,\alpha}$. However, in this case we have additional difficulties with respect to the matrix $\widehat{T}_{n,\alpha}=T_{n}\widehat{Z}_{n,\alpha}$, because it is not always true that $\mathcal{T}_{n,\alpha}$ can be written as $T_{n}\mathcal{Z}_{n,\alpha}$, where $\mathcal{Z}_{n,\alpha}$ is the matrix obtained by considering the $n-\mu_{\alpha}$ last columns of $Z_{n,\alpha}$. Indeed, in $\mathcal{T}_{n,\alpha}$ there are Fourier coefficients with index, in modulus, greater than $n$: the Toeplitz matrix $T_{n}=[a_{r-c}]_{r,c=0}^{n-1}$ has coefficients $a_j$ with $j$ ranging from $1-n$ to $n-1$, while the $\alpha$-Toeplitz matrix $T_{n,\alpha}=[a_{r-\alpha
c}]_{r,c=0}^{n-1}$ has $a_{n-1}$ as coefficient of maximum index and $a_{-\alpha(n-1)}$ as coefficient of minimum index, and, if $\alpha\geq 2$, we have $-\alpha(n-1)<-(n-1)$.
Even if we take the Toeplitz matrix $T_{n}$, which has as its first column the first column of $\mathcal{T}_{n,\alpha}$ and the other generated according to the rule $(T_{n})_{j,k}=a_{j-k}$, it is not always true that we can write $\mathcal{T}_{n,\alpha}=T_{n}P$ for a suitable submatrix $P$ of a permutation matrix, indeed, if the matrix $T_{n}=[\beta_{r-c}]_{r,c=0}^{n-1}$ has as first column the first column of $\mathcal{T}_{n,\alpha}$, we find that $\beta_{0}=(\mathcal{T}_{n,\alpha})_{0,0}=(T_{n,\alpha})_{0,\mu_{\alpha}}=a_{-\alpha\mu_{\alpha}}$. As a consequence, $T_{n}$ has $\beta_{-(n-1)}=a_{-(n-1)-\alpha\mu_{\alpha}}$ as coefficient of minimum index, while $\mathcal{T}_{n,\alpha}$ has $a_{-\alpha(n-1)}$ as coefficient of minimum index. Therefore $$\begin{aligned}
-(n-1)\alpha-(-(n-1)-\alpha\mu_{\alpha})&=&(1-\alpha)(n-1)+\alpha\mu_{\alpha}\qquad\;\;\; n\leq\alpha\mu_{\alpha}=\alpha\left\lceil \frac{n}{\alpha}\right\rceil\leq (n+\alpha-1)\\
&\leq &(1-\alpha)(n-1)+(n+\alpha-1)\\
&=&(1-\alpha)(n-1)+(n-1)+\alpha\\
&=&(n-1)(1-\alpha+1)+\alpha\\
&=&(2-\alpha)(n-1)+\alpha< 0\qquad\; \textrm{for $\alpha>2$ and
$n>4$}.\end{aligned}$$
Thus, if $\alpha>2$ and $n>4$ we have $-(n-1)\alpha<-(n-1)-\alpha\mu_{\alpha}$ and the coefficient of minimum index $a_{-\alpha(n-1)}$ of $\mathcal{T}_{n,\alpha}$ is not contained in the matrix $T_{n}$ that has $a_{-(n-1)-\alpha\mu_{\alpha}}$ as coefficient of minimum index.
Then we proceed in another way: in the first column of $\mathcal{T}_{n,\alpha}\in\mathbb{C}^{n\times (n-\mu_{\alpha})}$ (and consequently throughout the matrix) there are only coefficients with index $< 0$, indeed coefficient with the largest index of $\mathcal{T}_{n,\alpha}$ is $(\mathcal{T}_{n,\alpha})_{n-1,0}=(T_{n,\alpha})_{n-1,\mu_{\alpha}}=a_{n-1-\alpha\mu_{\alpha}}$ and $n-1-\alpha\mu_{\alpha}\leq n-1-n<0$ and the coefficient with smallest index is $(\mathcal{T}_{n,\alpha})_{0,n-\mu_{\alpha}-1}=(T_{n,\alpha})_{0,n-\mu_{\alpha}-1+\mu_{\alpha}}=
(T_{n,\alpha})_{0,n-1}=a_{-\alpha(n-1)}$. Consider therefore a Toeplitz matrix $T_{d_{n,\alpha}}$ of dimension $d_{n,\alpha}$ with $d_{n,\alpha}>\frac{\alpha(n-1)}{2}+1$, defined in this way: $$\begin{aligned}
\label{tdn}
T_{d_{n,\alpha}}=\left[\begin{array}{ccccc}
a_{-d_{n,\alpha}+1} & a_{-d_{n,\alpha}} & a_{-d_{n,\alpha}-1} & \cdots & a_{-2d_{n,\alpha}+2} \\
a_{-d_{n,\alpha}+2} & a_{-d_{n,\alpha}+1} & \ddots & \ddots & a_{-2d_{n,\alpha}+3} \\
\vdots & \ddots & \ddots & \ddots & \vdots\\
a_{-1} & a_{-2} & \ddots & \ddots & a_{-d_{n,\alpha}}\\
a_{0} & a_{-1} & a_{-2} & \cdots & a_{-d_{n,\alpha}+1}
\end{array}\right]=\left[a_{r-c-d_{n,\alpha}+1}\right]_{r,c=0}^{d_{n,\alpha}-1}.\end{aligned}$$
Since the coefficient with smallest index is $a_{-2d_{n,\alpha}+2}$, we find $$\begin{aligned}
-2d_{n,\alpha}+2<-2\left(\frac{\alpha(n-1)}{2}+1\right)+2=-\alpha(n-1)-2+2=-\alpha(n-1).\end{aligned}$$
As a consequence, we obtain that all the coefficients of $\mathcal{T}_{n,\alpha}$ are “contained” in the matrix $T_{d_{n,\alpha}}$. In particular, if $$\begin{aligned}
d_{n,\alpha}>(\alpha-1)(n-1)+2,\end{aligned}$$ (this condition ensures $d_{n,\alpha}>\frac{\alpha(n-1)}{2}+1$, that all the subsequent inequalities are correct, and that the size of all the matrices involved are non-negative), then it can be shown that $$\begin{aligned}
\label{hat}
\mathcal{T}_{n,\alpha}=\left[0_{1}|I_{n}|0_{2}\right]T_{d_{n,\alpha}}\mathcal{Z}_{d_{n,\alpha},\alpha},\end{aligned}$$ where $\mathcal{Z}_{d_{n,\alpha},\alpha}\in\mathbb{C}^{d_{n,\alpha}\times
(n-\mu_{\alpha})}$ is the matrix defined in (\[i\]), of dimension $d_{n,\alpha}\times d_{n,\alpha}$, by considering only the $n-\mu_{\alpha}$ first columns and $\left[0_{1}|I_{n}|0_{2}\right]\in\mathbb{C}^{n\times d_{n,\alpha}}$ is a block matrix with $0_{1}\in\mathbb{C}^{n\times
(d_{n,\alpha}-\alpha\mu_{\alpha}-1)}$ and $0_{2}\in\mathbb{C}^{n\times (\alpha\mu_{\alpha}-n+1)}$.
(of relation $(\ref{hat}).$) First we observe that:
- for $r=0,1,\ldots,n-1$ and $s=0,1,\ldots,n-\mu_{\alpha}-1$ we have $$\begin{aligned}
\label{eq1}
(\mathcal{T}_{n,\alpha})_{r,s}=(T_{n,\alpha})_{r,s+\mu_{\alpha}}=a_{r-\alpha
s-\alpha\mu_{\alpha}};
\end{aligned}$$
- for $r=0,1,\ldots,n-1$ and $s=0,1,\ldots,d_{n,\alpha}-1$ we have $$\begin{aligned}
\label{eq2}
(\left[0_{1}|I_{n}|0_{2}\right])_{r,s}=\left\{\begin{array}{cl} 1 &
\textrm{if $s=r+d_{n,\alpha}-\alpha\mu_{\alpha}-1$},\\
0 & \textrm{otherwise};\end{array}\right.
\end{aligned}$$
- for $r,s=0,1,\ldots,d_{n,\alpha}-1$ we have $$\begin{aligned}
(T_{d_{n,\alpha}})_{r,s}=a_{r-s-d_{n,\alpha}+1};
\end{aligned}$$
- for $r=0,1,\ldots,d_{n,\alpha}-1$ and $s=0,1,\ldots,n-\mu_{\alpha}-1,$ we have $$\begin{aligned}
(\mathcal{Z}_{d_{n,\alpha},\alpha})_{r,s}=\delta_{r-\alpha s}.
\end{aligned}$$
Since $T_{d_{n,\alpha}}\mathcal{Z}_{d_{n,\alpha},\alpha}\in\mathbb{C}^{d_{n,\alpha}\times
(n-\mu_{\alpha})}$, for $r=0,1,\ldots,d_{n,\alpha}-1$ and $s=0,1,\ldots,n-\mu_{\alpha}-1,$ it holds $$\begin{aligned}
\label{eq3}
\notag(T_{d_{n,\alpha}}\mathcal{Z}_{d_{n,\alpha},\alpha})_{r,s} &=& \overset{d_{n,\alpha}-1}
{\underset{l=0}\sum}(T_{d_{n,\alpha}})_{r,l}(\mathcal{Z}_{d_{n,\alpha},\alpha})_{l,s} \\
\notag&=& \overset{d_{n,\alpha}-1}{\underset{l=0}\sum}\delta_{l-\alpha s}a_{r-l-d_{n,\alpha}+1}\\
& {\underset{\rm (a)}=} & a_{r-\alpha s-d_{n,\alpha}+1},\end{aligned}$$ where (a) follows from the existence of a unique $l\in\{0,1,\ldots,d_{n,\alpha}-1\}$ such that $l-\alpha
s\equiv 0\textrm{ (mod $d_{n,\alpha}$)}$, that is, $l\equiv \alpha
s\textrm{ (mod $d_{n,\alpha}$)}$, and, since $0\leq\alpha s\leq d_{n,\alpha}-1$, we have $l=\alpha s$. Since $\left[0_{1}|I_{n}|0_{2}\right]T_{d_{n,\alpha}}\mathcal{Z}_{d_{n,\alpha},\alpha}\in\mathbb{C}^{n\times
(n-\mu_{\alpha})}$, for $r=0,1,\ldots,n-1$ and $s=0,1,\ldots,n-\mu_{\alpha}-1,$ we find $$\begin{aligned}
(\left[0_{1}|I_{n}|0_{2}\right]T_{d_{n,\alpha}}\mathcal{Z}_{d_{n,\alpha},\alpha})_{r,s} &=&
\overset{d_{n,\alpha}-1}{\underset{l=0}\sum}(\left[0_{1}|I_{n}|0_{2}\right])_{r,l}
(T_{d_{n,\alpha}}\mathcal{Z}_{d_{n,\alpha},\alpha})_{l,s} \\
& {\underset{\rm (d)}=}&a_{r+d_{n,\alpha}-\alpha\mu_{\alpha}-1-\alpha s-d_{n,\alpha}+1}\\
&=&a_{r-\alpha\mu_{\alpha}-\alpha s}\\
&{\underset{\rm from\,(\ref{eq1})}=}&(\mathcal{T}_{n,\alpha})_{r,s},\end{aligned}$$ where (d) follows from $(\ref{eq3})$, $(T_{d_{n,\alpha}}\mathcal{Z}_{d_{n,\alpha},\alpha})_{l,s}=a_{l-\alpha
s-d_{n,\alpha}+1}$, and from the following fact: using $(\ref{eq2})$, we find $(\left[0_{1}|I_{n}|0_{2}\right])_{r,l}=1$ if and only if $l=r+d_{n,\alpha}-\alpha\mu_{\alpha}-1$.
We can now observe immediately that the matrix $T_{d_{n,\alpha}}$ defined in $(\ref{tdn})$ can be written as $$\begin{aligned}
\label{flip}
T_{d_{n,\alpha}}=JH_{d_{n,\alpha}},\end{aligned}$$ where $J$ is the “flip” matrix of dimension $d_{n,\alpha}\times d_{n,\alpha}$: $$\begin{aligned}
J=\left[\begin{array}{cccc}
& & & 1\\
& & 1 & \\
& {\cdot^{\textstyle\cdot^{\textstyle\cdot}}}& & \\
1 & & &
\end{array}\right],\end{aligned}$$ and $H_{d_{n,\alpha}}$ is the Hankel matrix of dimension $d_{n,\alpha}\times d_{n,\alpha}$: $$\begin{aligned}
H_{d_{n,\alpha}}=\left[\begin{array}{ccccc}
a_{0} & a_{-1} & a_{-2} & \cdots & a_{-d_{n,\alpha}+1} \\
a_{-1} & a_{-2} & {\cdot^{\textstyle\cdot^{\textstyle\cdot}}}& {\cdot^{\textstyle\cdot^{\textstyle\cdot}}}& a_{-d_{n,\alpha}}\\
\vdots & {\cdot^{\textstyle\cdot^{\textstyle\cdot}}}& {\cdot^{\textstyle\cdot^{\textstyle\cdot}}}& {\cdot^{\textstyle\cdot^{\textstyle\cdot}}}& \vdots\\
a_{-d_{n,\alpha}+2} & a_{-d_{n,\alpha}+1} & {\cdot^{\textstyle\cdot^{\textstyle\cdot}}}& {\cdot^{\textstyle\cdot^{\textstyle\cdot}}}& a_{-2d_{n,\alpha}+3} \\
a_{-d_{n,\alpha}+1} & a_{-d_{n,\alpha}} & a_{-d_{n,\alpha}-1} & \cdots & a_{-2d_{n,\alpha}+2}
\end{array}\right].\end{aligned}$$
If $f(x)\in L^{1}(Q)$, $Q=(-\pi,\pi)$, is the generating function of the Toeplitz matrix $T_{n}=T_{n}(f)=[a_{r-c}]_{r,c=0}^{n-1}$ in $(\ref{T})$, where the $k$-th Fourier coefficient of $f$ is $a_k$, then $f(-x)\in L^{1}(Q)$ is the generating function of the Hankel matrix $H_{d_{n,\alpha}}=[a_{-r-c}]_{r,c=0}^{d_{n,\alpha}-1}$; by invoking Theorem 6, page 161 of [@FasTi], the sequence of matrices $\{H_{d_{n,\alpha}}\}$ is distributed in the singular value sense as the zero function: $\{H_{d_{n,\alpha}}\}\sim_{\sigma}
(0,Q)$. From Lemma $\ref{primo}$, by $(\ref{flip})$, since $J$ is a unitary matrix, we have $\{T_{d_{n,\alpha}}\}\sim_{\sigma} (0,Q)$ as well.
Consider the decomposition in $(\ref{hat})$: $$\begin{aligned}
\mathcal{T}_{n,\alpha}=\left[0_{1}|I_{n}|0_{2}\right]T_{d_{n,\alpha}}\mathcal{Z}_{d_{n,\alpha},\alpha}=
Q_{d_{n,\alpha}}T_{d_{n,\alpha}}\mathcal{Z}_{d_{n,\alpha},\alpha}.\end{aligned}$$
If we complete the matrices $Q_{d_{n,\alpha}}\in\mathbb{C}^{n\times
d_{n,\alpha}}$ and $\mathcal{Z}_{d_{n,\alpha},\alpha}\in\mathbb{C}^{d_{n,\alpha}\times
(n-\mu_{\alpha})}$ by adding an appropriate number of zero rows and columns, respectively, in order to make it square $$\begin{aligned}
\mathbf{Q}_{d_{n,\alpha}}&=&\left[\begin{array}{c}
Q_{d_{n,\alpha}}\\
\hline
0
\end{array}\right]\in\mathbb{C}^{d_{n,\alpha}\times d_{n,\alpha}},\\
\mathbf{Z}_{d_{n,\alpha},\alpha}&=&\left[\begin{array}{c|c}
\mathcal{Z}_{d_{n,\alpha},\alpha} & 0
\end{array}\right]\in\mathbb{C}^{d_{n,\alpha}\times d_{n,\alpha}},\end{aligned}$$ then it is immediate to note that $$\begin{aligned}
\mathbf{Q}_{d_{n,\alpha}}T_{d_{n,\alpha}}\mathbf{Z}_{d_{n,\alpha},\alpha}=\left[\begin{array}{c|c}
\mathcal{T}_{n,\alpha} & 0 \\
\hline
0 & 0
\end{array}\right]=\mathbf{T}_{n,\alpha}\in\mathbb{C}^{d_{n,\alpha}\times d_{n,\alpha}}.\end{aligned}$$
From Lemma $\ref{secondo}$, since $\|\mathbf{Q}_{d_{n,\alpha}}\|=\|\mathbf{Z}_{d_{n,\alpha},\alpha}\|=1$ (indeed they are both “incomplete” permutation matrices), and since $\{T_{d_{n,\alpha}}\}\sim_{\sigma} (0,Q)$, we infer that $\{\mathbf{T}_{n,\alpha}\}\sim_{\sigma} (0,Q)$.
Recall that $\mathbf{T}_{n,\alpha}\in\mathbb{C}^{d_{n,\alpha}\times
d_{n,\alpha}}$ with $d_{n,\alpha}>(\alpha-1)(n-1)+2$; then we can always choose $d_{n,\alpha}$ such that $\alpha
n=d_{n,\alpha}>(\alpha-1)(n-1)+2$ (if $n,\alpha\geq2$). Now, since $\{\mathbf{T}_{n,\alpha}\}\sim_{\sigma} (0,Q)$, it holds that the sequence $\{\mathbf{T}_{n,\alpha}\}$ is weakly clustered at zero in the singular value sense, i.e., $\forall\epsilon>0$, $$\begin{aligned}
\label{sigmabft}
\sharp\{j:\sigma_{j}(\mathbf{T}_{n,\alpha})>\epsilon\}=o(d_{n,\alpha})=o(\alpha n)=o(n).\end{aligned}$$
The matrix $\mathbf{T}_{n,\alpha}$ is a block matrix that can be written as $$\begin{aligned}
\mathbf{T}_{n,\alpha}=\left[\begin{array}{c|c}
\mathcal{T}_{n,\alpha} & 0 \\
\hline
0 & 0
\end{array}\right]=\left[\begin{array}{c|c}
[\mathcal{T}_{n,\alpha}|0] & 0 \\
\hline
0 & 0
\end{array}\right],\end{aligned}$$ where $\mathcal{T}_{n,\alpha}\in\mathbb{C}^{n\times (n-\mu_{\alpha})}$ and $[\mathcal{T}_{n,\alpha}|0]\in\mathbb{C}^{n\times n}$. By the singular value decomposition we obtain $$\begin{aligned}
\mathbf{T}_{n,\alpha}=\left[\begin{array}{c|c}
[\mathcal{T}_{n,\alpha}|0] & 0 \\
\hline
0 & 0
\end{array}\right]=\left[\begin{array}{c|c}
U_{1}\Sigma_{1} V_{1}^{*} & 0 \\
\hline
0 & U_{2}0V_{2}^{*}
\end{array}\right]=\left[\begin{array}{c|c}
U_{1} & 0 \\
\hline
0 & U_{2}
\end{array}\right]\left[\begin{array}{c|c}
\Sigma_{1} & 0 \\
\hline
0 & 0
\end{array}\right]\left[\begin{array}{c|c}
V_{1} & 0 \\
\hline
0 & V_{2}
\end{array}\right]^{*},\end{aligned}$$ that is, the singular values of $\mathbf{T}_{n,\alpha}$ that are different from zero are the singular values of $[\mathcal{T}_{n,\alpha}|0]\in\mathbb{C}^{n\times n}$. Thus $(\ref{sigmabft})$ can be written as follows: $\forall\epsilon>0$, $$\begin{aligned}
\sharp\{j:\sigma_{j}([\mathcal{T}_{n,\alpha}|0])>\epsilon\}=o(d_{n,\alpha})=o(\alpha
n)=o(n).\end{aligned}$$
The latter relation means that the sequence $\{[\mathcal{T}_{n,\alpha}|0]\}_{n}$ is weakly clustered at zero in the singular value sense, and hence $\{[\mathcal{T}_{n,\alpha}|0]\}_{n}\sim_{\sigma} (0,Q)$. If we now consider the matrix $$\begin{aligned}
\hat G=\left[\begin{array}{c|c}
0 & I_{n-\mu_{\alpha}}\\
\hline
0 & 0
\end{array}\right]\in\mathbb{C}^{n\times n},\end{aligned}$$ where $I_{n-\mu_{\alpha}}$ is the identity matrix of dimension $(n-\mu_{\alpha})\times(n-\mu_{\alpha})$, then $[\mathcal{T}_{n,\alpha}|0]\hat G=[0|\mathcal{T}_{n,\alpha}]$, and since $\|\hat G\|=1$ and $\{[\mathcal{T}_{n,\alpha}|0]\}_{n}\sim_{\sigma} (0,Q)$, from Lemma $\ref{secondo}$ we find $$\begin{aligned}
\label{dist2}
\{[0|\mathcal{T}_{n,\alpha}]\}_{n}\sim_{\sigma} (0,Q).\end{aligned}$$
In conclusion: from the relations $(\ref{dist})$, $(\ref{dist1})$ and $(\ref{dist2})$, using Proposition $\ref{fgbis}$ with $G=Q\times
[0,1]$, we obtain that $$\begin{aligned}
\{T_{n,\alpha}\}_{n}\sim_{\sigma}(\theta, Q\times [0,1]),\end{aligned}$$ where $\theta$ is defined in $(\ref{teta})$. Notice that for $\alpha=1$ the symbol $\theta(x,t)$ coincides with $|f|(x)$ on the extended domain $Q\times [0,1]$. Hence the Szegö-Tilli-Tyrtyshnikov-Zamarashkin result is found as a particular case. Indeed $\theta(x,t)=|f|(x)$ does not depend on $t$ and therefore this additional variable can be suppressed i.e. $\{T_{n,\alpha}\}_{n}\sim_{\sigma}(f,Q)$ with $T_{n,\alpha}=T_n(f)$. The fact that the distribution formula is not unique should not surprise since this phenomenon is inherent to the measure theory because any measure-preserving exchange function is a distribution function if one representative of the class is.
Some remarks on multigrid methods {#sec:multigrid}
=================================
In the design of multigrid methods for large positive definite linear systems one of the key points is to maintain the structure (if any) of the original matrix in the lower levels. This means that at every recursion level the new projected linear system should retain the main properties of the original matrix (e.g. bandedness, the same level of conditioning, the same algebra/Toeplitz/graph structure etc.). Here for the sake of simplicity the example that has to be considered is the one-level circulant case. Following [@ADS; @mcirco], if $A_n=C_n$ is a positive circulant matrix of size $n$ with $n$ power of $2$, then the projected matrix $A_k$ with $k=n/2$ is defined as $$\label{proj}
A_k=\widetilde{Z}_{n,2}^T P_n^* A_n P_n \widetilde{Z}_{n,2},$$ where $P_n$ is an additional circulant matrix. It is worth noticing that the structure is kept since for every circulant $P_n$ the matrix $A_k$ is a circulant matrix of size $k=n/2$. The features of the specific $P_n$ have to be designed in such a way that the convergence speed of the related multigrid is as high as possible (see [@FS2; @ADS] for a general strategy). We observe that the eigenvalues of $A_k$ are given by $$\label{eig-proj}
\frac{1}{2}\overset{1}
{\underset{l=0}\sum} g\left(\frac{x_j+2\pi l}{2} \right),
\quad x_j=\frac{2\pi j}{k}, \ \ j=0,1,\ldots,k-1,\ k=n/2,$$ where $g$ is the polynomial associated with the circulant matrix $P_n^* A_n P_n$ in the sense of Subsection \[special-case\]. Therefore the singular values of $(P_n^* A_n
P_n)^{1/2}\widetilde{Z}_{n,2}$ are given by $$\label{sv-sqrt-proj}
\frac{1}{\sqrt{2}}\sqrt{\overset{1}
{\underset{l=0}\sum} g\left(\frac{x_j+2\pi l}{2} \right)},
\quad x_j=\frac{2\pi j}{k}, \ \ j=0,1,\ldots,k-1,\ k=n/2.$$
Notice that the latter formula is a special instance of (\[sv-nontrivial-symbol-p\]) for $|p|^2=g$ ($g$ is necessarily nonnegative since it can be written a $|q|^2 f$ where $q$ is the polynomial associated with $P_n$ and $f$ the nonnegative polynomial associated with $A_n$), for $\alpha=2$ and $n$ even number so that $(n,2)=2$. Therefore, according to (\[sv-nontrivial-symbol-p\]), the numbers in (\[sv-sqrt-proj\]) identify the nontrivial singular values of the $2$-circulant matrix $(P_n^* A_n P_n)^{1/2}Z_{n,2}$ up to a scaling factor. In other words $\alpha$-circulant matrices arise naturally in the design of fast multigrid solvers for circulant linear systems and, along the same lines, $\alpha$-Toeplitz matrices arise naturally in the design of fast multigrid solvers for Toeplitz linear systems; see [@FS2; @ADS; @Sun].
Conversely, we now can see clearly that formula (\[sv-nontrivial-symbol-p\]) furnishes a wide generalization of the spectral analysis of the projected matrices, by allowing a higher degree of freedom: we can choose $n$ divisible by $\alpha$ with $\alpha\neq 2$, we can choose $n$ not divisible by $\alpha$. Such a degree of freedom is not just academic, but could be exploited for devising optimally convergent multigrid solvers also in critical cases emphasized e.g. in [@ADS; @Sun]. In particular, if $x_0$ is an isolated zero of $f$ (the nonnegative polynomial related to $A_n=C_n$) and also $\pi+x_0$ is a zero for the same function, then due to special symmetries, the associated multigrid (or even two-grid) method cannot be optimal. In other words, for reaching a preassigned accuracy, we cannot expect a number of iterations independent of the order $n$. However these pathological symmetries are due to the choice of $\alpha=2$, so that a choice of a projector as $P_n\widetilde{Z}_{n,\alpha}$ for a different $\alpha\neq 2$ and a different $n$ could completely overcome the latter drawback.
Generalizations {#sec:gen}
===============
First of all we observe that the requirement that the symbol $f$ is square integrable can be removed. In [@algebra] it is proven that the singular value distribution of $\{T_{n}(f)T_n(g)\}_{n}$ is given by $h=fg$ with $f,g$ being just Lebesgue integrable and with $h$ that is only measurable and therefore may fail to be Lebesgue integrable. This fact is sufficient for extending the proof of the relation $\{T_{n,\alpha}\}_{n}\sim_{\sigma}(\theta,Q\times [0,1])$ to the case where $\theta(x,t)$ is defined as in $(\ref{teta})$ with the original symbol $f\in L^1$.
Now we consider the general multilevel case. When $\alpha$ is a positive vector, we have $$\begin{aligned}
\label{dist1-d}
\{T_{n,\alpha}\}_{n}\sim_{\sigma}(\theta,Q^d\times[0,1]^d),\end{aligned}$$ where $$\begin{aligned}
\label{teta-d}
\theta(x,t)=\left\{\begin{array}{cl}
\sqrt{\widehat{|f|^{(2)}}(x)} & \textrm{for $t\in\left[\underline{0},\frac{1}{\alpha}\right]$,}\\
0 & \textrm{for $t\in\left(\frac{1}{\alpha},e\right]$},
\end{array}\right.\end{aligned}$$ with $$\begin{aligned}
\label{f2t-d}
\widehat{|f|^{(2)}}(x)&=&\frac{1}{\hat{\alpha}}\sum_{j=\underline{0}}^{\alpha-e}|f|^{2}\left(\frac{x+2\pi j}{\alpha}\right),
$$ and where all the arguments are modulus $2\pi$ and all the operations are intended componentwise that is $t\in\left[\underline{0},\frac{1}{\alpha}\right]$ means that $t_k\in
[0,1/\alpha_k]$, $k=1,\ldots,d$, $t\in\left(\frac{1}{\alpha},e\right]$ means that $t_k\in
(1/\alpha_k,1]$, $k=1,\ldots,d$, the writing $\frac{x+2\pi
j}{\alpha}$ defines the $d$-dimensional vector whose $k$-th component is $(x_j+2\pi j_k)/\alpha_k$, $k=1,\ldots,d$, and $\hat{\alpha}=\alpha_{1}\alpha_{2}\cdots\alpha_{d}$.
### Examples of $\alpha$-circulant and $\alpha$-Toeplitz matrices when some of the entries of $\alpha$ vanish
We start this subsection with a brief digression on multilevel matrices. A $d$-level matrix $A$ of dimension $\hat{n}\times\hat{n}$ with $n=(n_{1},n_{2},\ldots,n_{d})$ and $\hat{n}=n_{1}n_{2}\cdots n_{d}$ can be viewed as a matrix of dimension $n_{1}\times n_{1}$ in which each element is a block of dimension $n_{2}n_{3}\cdots n_{d}\times n_{2}n_{3}\cdots n_{d}$; in turn, each block of dimension $n_{2}n_{3}\cdots n_{d}\times n_{2}n_{3}\cdots n_{d}$ can be viewed as a matrix of dimension $n_{2}\times n_{2}$ in which each element is a block of dimension $n_{3}n_{4}\cdots n_{d}\times n_{3}n_{4}\cdots n_{d}$, and so on. So we can say that $n_{1}$ is the most “outer” dimension of the matrix $A$ and $n_{d}$ is the most “inner” dimension. If we multiply by an appropriate permutation matrix $P$ the $d$-level matrix $A$, we can exchange the “order of dimensions” of $A$, namely $P^{T}AP$ becomes a matrix again of dimension $\hat{n}\times\hat{n}$ but with $n=(n_{p(1)},n_{p(2)},\ldots,n_{p(d)})$ and $\hat{n}=n_{p(1)}n_{p(2)}\cdots n_{p(d)}=n_{1}n_{2}\cdots n_{d}$ (where $p$ is a permutation of $d$ elements) and $n_{p(1)}$ is the most “outer” dimension of the matrix $A$ and $n_{p(d)}$ is the most “inner” dimension.
This trick helps us to understand what happens to the singular values of $\alpha$-circulant and $\alpha$-Toeplitz $d$-level matrices, especially when some of the entries of the vector $\alpha$ are zero; indeed, as we observed in Subsection $\ref{alphazero}$, if $\alpha=\underline{0}$, the $d$-level $\alpha$-circulant (or $\alpha$-Toeplitz) matrix $A$ is a block matrix with constant blocks on each row, so if we order the vector $\alpha$ (which has some components equal to zero) so that the components equal to zero are in the top positions, $\alpha=(0,\ldots,0,\alpha_{k},\ldots,\alpha_{d})$, the matrix $P^{T}AP$ (where $P$ is the permutation matrix associated with $p$) becomes a block matrix with constant blocks on each row and with blocks of dimension $n_{k}\cdots n_{d}\times n_{k}\cdots n_{d}$; with this “new” structure, formulas $(\ref{eq-2-1-3})$ and $(\ref{eq-2-1-3-bis})$ are even more intuitively understandable, as we shall see later in the examples.
\[scambio\] Let $A$ be a 2-level Toeplitz matrix of dimension $\hat{n}\times\hat{n}$ with $n=(n_{1},n_{2})$ and $\hat{n}=n_{1}n_{2}$, $$\begin{aligned}
A=\left[\left[a_{(j_{1}-k_{1},j_{2}-k_{2})}\right]_{j_{2},k_{2}=0}^{n_{2}-1}\right]_{j_{1},k_{1}=0}^{n_{1}-1}.\end{aligned}$$ There exists a permutation matrix $P$ such that $$\begin{aligned}
P^{T}AP=\left[\left[a_{(j_{1}-k_{1},j_{2}-k_{2})}\right]_{j_{1},k_{1}=0}^{n_{1}-1}\right]_{j_{2},k_{2}=0}^{n_{2}-1}.\end{aligned}$$
Let $n=(n_{1},n_{2})=(2,3)$ and consider the 2-level Toeplitz matrix $A$ of dimension $6\times6$ $$\begin{aligned}
A=\left[
\begin{array}{ccc|ccc}
a_{(0,0)} & a_{(0,-1)} & a_{(0,-2)} & a_{(-1,0)} & a_{(-1,-1)} & a_{(-1,-2)}\\
a_{(0,1)} & a_{(0,0)} & a_{(0,-1)} & a_{(-1,1)} & a_{(-1,0)} & a_{(-1,-1)}\\
a_{(0,2)} & a_{(0,1)} & a_{(0,0)} & a_{(-1,2)} & a_{(-1,1)} & a_{(-1,0)}\\
\hline
a_{(1,0)} & a_{(1,-1)} & a_{(1,-2)} & a_{(0,0)} & a_{(0,-1)} & a_{(0,-2)}\\
a_{(1,1)} & a_{(1,0)} & a_{(1,-1)} & a_{(0,1)} & a_{(0,0)} & a_{(0,-1)}\\
a_{(1,2)} & a_{(1,1)} & a_{(1,0)} & a_{(0,2)} & a_{(0,1)} & a_{(0,0)}
\end{array}\right].\end{aligned}$$
This matrix can be viewed as a matrix of dimension $2\times 2$ in which each element is a block of dimension $3\times 3$. If we take the permutation matrix $$\begin{aligned}
P=\left[\begin{array}{cccccc}
1 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 1 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 1 & 0 \\
0 & 1 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 1 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 1
\end{array}\right],\end{aligned}$$ then it is plain to see that $$\begin{aligned}
P^{T}AP=\left[
\begin{array}{cc|cc|cc}
a_{(0,0)} & a_{(-1,0)} & a_{(0,-1)} & a_{(-1,-1)} & a_{(0,-2)} & a_{(-1,-2)}\\
a_{(1,0)} & a_{(0,0)} & a_{(1,-1)} & a_{(0,-1)} & a_{(1,-2)} & a_{(0,-2)}\\
\hline
a_{(0,1)} & a_{(-1,1)} & a_{(0,0)} & a_{(-1,0)} & a_{(0,-1)} & a_{(-1,-1)}\\
a_{(1,1)} & a_{(0,1)} & a_{(1,0)} & a_{(0,0)} & a_{(1,-1)} & a_{(0,-1)}\\
\hline
a_{(0,2)} & a_{(-1,2)} & a_{(0,1)} & a_{(-1,1)} & a_{(0,0)} & a_{(-1,0)}\\
a_{(1,2)} & a_{(0,2)} & a_{(1,1)} & a_{(0,1)} & a_{(1,0)} & a_{(0,0)}
\end{array}\right],\end{aligned}$$ and now $P^{T}AP$ can be naturally viewed as a matrix of dimension $3\times 3$ in which each element is a block of dimension $2\times 2 $.
\[scambiod\] Let $A$ be a $d$-level Toeplitz matrix of dimension $\hat{n}\times\hat{n}$ with $n=(n_{1},n_{2},\ldots,n_{d})$ and $\hat{n}=n_{1}n_{2}\cdots n_{d}$, $$\begin{aligned}
A=\left[\left[\cdots\left[a_{(j_{1}-k_{1},j_{2}-k_{2},\ldots,j_{d}-k_{d})}\right]_{j_{d},k_{d}=0}^{n_{d}-1}\cdots\right]_{j_{2},k_{2}=0}^{n_{2}-1}
\right]_{j_{1},k_{1}=0}^{n_{1}-1}.\end{aligned}$$ For every permutation $p$ of $d$ elements, there exists a permutation matrix $P$ such that $$\begin{aligned}
P^{T}AP=\left[\left[\cdots\left[a_{(j_{1}-k_{1},j_{2}-k_{2},\ldots,j_{d}-k_{d})}\right]_{j_{p(d)},k_{p(d)}=0}^{n_{p(d)}-1}\cdots\right]_{j_{p(2)},k_{p(2)}=0}^{n_{p(2)}-1}\right]_{j_{p(1)},k_{p(1)}=0}^{n_{p(1)}-1}.\end{aligned}$$
Lemma $\ref{scambio}$ and Corollary $\ref{scambiod}$ also apply to $d$-level $\alpha$-circulant and $\alpha$-Toeplitz matrices.
Now, let $\alpha=(\alpha_{1},\alpha_{2},\ldots,\alpha_{d})$ be a $d$-dimensional vector of nonnegative integers and $t=\sharp\{j:\alpha_{j}=0\}$ be the number of zero entries of $\alpha$. If we take a permutation $p$ of $d$ elements such that $\alpha_{p(1)}=\alpha_{p(2)}=\ldots=\alpha_{p(t)}=0$, (that is, $p$ is a permutation that moves all the zero components of the vector $\alpha$ in the top positions), then it is easy to prove that formulas $(\ref{eq-2-1-3})$ and $(\ref{eq-2-1-3-bis})$ remain the same for the matrix $P^{T}AP$ (where $P$ is the permutation matrix associated with $p$) but with $n[0]=(n_{p(1)},n_{p(2)},\ldots,n_{p(t)})$ and where $C_j$ and $T_j$ are a $d^+$-level $\alpha^+$-circulant and $\alpha^+$-Toeplitz matrix, respectively, with $\alpha^+=(\alpha_{p(t+1)},\alpha_{p(t+2)},\ldots,\alpha_{p(d)})$, of partial sizes $n[>0]=(n_{p(t+1)},n_{p(t+2)},\ldots,n_{p(d)})$, and whose expressions are $$\begin{aligned}
C_j&=&\left[\left[\cdots\left[a_{(r-\alpha \circ s)\ {\rm mod}\,
n}\right]_{r_{p(d)},s_{p(d)}=0}^{n_{p(d)}-1}\cdots\right]_{r_{p(t+2)},s_{p(t+2)}=0}^{n_{p(t+2)}-1}\right]_{r_{p(t+1)},s_{p(t+1)}=0}^{n_{p(t+1)}-1},\\
T_j&=&\left[\left[\cdots\left[a_{(r-\alpha \circ s)}\right]_{r_{p(d)},s_{p(d)}=0}^{n_{p(d)}-1}\cdots\right]_{r_{p(t+2)},s_{p(t+2)}=0}^{n_{p(t+2)}-1}\right]_{r_{p(t+1)},s_{p(t+1)}=0}^{n_{p(t+1)}-1},\end{aligned}$$ with $(r_{p(1)},r_{p(2)},\ldots,r_{p(t)})=j$. Obviously ${\rm
Sgval}(A)={\rm Sgval}(P^{T}AP)$.
We recall that if $B$ is a matrix of size $n \times n$ positive semidefinite, that is $B^{*}=B$ and $x^{*}Bx\geq 0$ $\forall x\neq
0$, then ${\rm Eig}(B)={\rm Sgval}(B)$. Moreover, if $B=U\Sigma
U^{*}$ is a $SVD$ for $B$ (which coincides with the Schur decomposition of $B$) with $\Sigma={\begin{smallmatrix}\vspace{-0.5ex}\textrm{\normalsize diag}\\\vspace{-0.8ex}j=1,\ldots,n\end{smallmatrix}}(\sigma_{j})$, then $$\begin{aligned}
\label{bmezzi}
B^{1/2}=U\Sigma^{1/2}U^{*},\end{aligned}$$ where $\Sigma^{1/2}={\begin{smallmatrix}\vspace{-0.5ex}\textrm{\normalsize diag}\\\vspace{-0.8ex}j=1,\ldots,n\end{smallmatrix}}(\sqrt{\sigma_{j}})$.
We proceed with two detailed examples: a 3-level $\alpha$-circulant matrix with $\alpha=(\alpha_{1},\alpha_{2},\alpha_{3})=(1,2,0)$, and a 3-level $\alpha$-Toeplitz with $\alpha=(\alpha_{1},\alpha_{2},\alpha_{3})=(0,1,2)$, which helps us to understand what happens if the vector $\alpha$ is not strictly positive. Finally we will propose the explicit calculation of the singular values of a $d$-level $\alpha$-circulant matrix in the particular case where the vector $\alpha$ has only one component different from zero.
Consider a 3-level $\alpha$-circulant matrix $A$ where $\alpha=(\alpha_{1},\alpha_{2},\alpha_{3})=(1,2,0)$ $$\begin{aligned}
A&=&\left[\left[\left[a_{((r_{1}-1\cdot s_{1})\textrm{ mod $n_{1}$},(r_{2}-2\cdot s_{2})\textrm{ mod $n_{2}$},(r_{3}-0\cdot s_{3})\textrm{ mod $n_{3}$})}
\right]_{r_{3},s_{3}=0}^{n_{3}-1}\right]_{r_{2},s_{2}=0}^{n_{2}-1}
\right]_{r_{1},s_{1}=0}^{n_{1}-1} \\
&=&\left[\left[\left[a_{((r_{1}-s_{1})\textrm{ mod $n_{1}$},(r_{2}-2s_{2})\textrm{ mod $n_{2}$},r_{3})}
\right]_{r_{3}=0}^{n_{3}-1}\right]_{r_{2},s_{2}=0}^{n_{2}-1}
\right]_{r_{1},s_{1}=0}^{n_{1}-1}.\end{aligned}$$
If we choose a permutation $p$ of 3 elements such that $$\begin{aligned}
&&(p(1),p(2),p(3))=(3,2,1),\\
&&(\alpha_{p(1)},\alpha_{p(2)},\alpha_{p(3)})=(0,2,1), \\
&&(n_{p(1)},n_{p(2)},n_{p(3)})=(n_{3},n_{2},n_{1}),\end{aligned}$$ and if we take the permutation matrix $P$ related to $p$, then $$\begin{aligned}
P^{T}AP\equiv\hat{A}=\left[\left[\left[a_{((r_{1}-s_{1})\textrm{ mod $n_{1}$},(r_{2}-2s_{2})\textrm{ mod $n_{2}$},r_{3})}
\right]_{r_{1},s_{1}=0}^{n_{1}-1}\right]_{r_{2},s_{2}=0}^{n_{2}-1}
\right]_{r_{3}=0}^{n_{3}-1}.\end{aligned}$$
Now, for $r_{3}=0,1,...,n_{3}-1$, let us set $$\begin{aligned}
C_{r_{3}}=\left[\left[a_{((r_{1}-s_{1})\textrm{ mod $n_{1}$},(r_{2}-2s_{2})\textrm{ mod $n_{2}$},r_{3})}
\right]_{r_{1},s_{1}=0}^{n_{1}-1}\right]_{r_{2},s_{2}=0}^{n_{2}-1}.\end{aligned}$$ As a consequence, $C_{r_{3}}$ is a 2-level $\alpha^+$-circulant matrix with $\alpha^+=(2,1)$ and of partial sizes $n[>0]=(n_{2},n_{1})$ and the matrix $\hat{A}$ can be rewritten as $$\begin{aligned}
\hat{A}=\left[\begin{array}{cccc}
C_{0} & C_{0} & \cdots & C_{0} \\
C_{1} & C_{1} & \cdots & C_{1} \\
\vdots & \vdots & \vdots & \vdots \\
C_{n_{3}-1} & C_{n_{3}-1} & \cdots & C_{n_{3}-1}
\end{array}\right],\end{aligned}$$ and this is a block matrix with constant blocks on each row. From formula $(\ref{1})$, the singular values of $\hat{A}$ are the square root of the eigenvalues of $\hat{A}^{*}\hat{A}$: $$\begin{aligned}
\hat{A}^{*}\hat{A} &=&\left[\begin{array}{cccc}
C_{0}^{\ast} & C_{1}^{\ast} & \cdots & C_{n_{3}-1}^{\ast} \\
C_{0}^{\ast} & C_{1}^{\ast} & \cdots & C_{n_{3}-1}^{\ast} \\
\vdots & \vdots & \vdots & \vdots \\
C_{0}^{\ast} & C_{1}^{\ast} & \cdots & C_{n_{3}-1}^{\ast} \\
\end{array}\right]\left[\begin{array}{cccc}
C_{0} & C_{0} & \cdots & C_{0} \\
C_{1} & C_{1} & \cdots & C_{1} \\
\vdots & \vdots & \vdots & \vdots \\
C_{n_{3}-1} & C_{n_{3}-1} & \cdots & C_{n_{3}-1}
\end{array}\right] \\
&=&\left[\begin{array}{cccc}
\overset{n_{3}-1}{\underset{j=0}\sum}C_{j}^{\ast}C_{j} &
\overset{n_{3}-1}{\underset{j=0}\sum}C_{j}^{\ast}C_{j} & \cdots &
\overset{n_{3}-1}{\underset{j=0}\sum}C_{j}^{\ast}C_{j} \\
\overset{n_{3}-1}{\underset{j=0}\sum}C_{j}^{\ast}C_{j} &
\overset{n_{3}-1}{\underset{j=0}\sum}C_{j}^{\ast}C_{j} & \cdots &
\overset{n_{3}-1}{\underset{j=0}\sum}C_{j}^{\ast}C_{j} \\
\vdots & \vdots & \vdots & \vdots\\
\overset{n_{3}-1}{\underset{j=0}\sum}C_{j}^{\ast}C_{j} &
\overset{n_{3}-1}{\underset{j=0}\sum}C_{j}^{\ast}C_{j} & \cdots &
\overset{n_{3}-1}{\underset{j=0}\sum}C_{j}^{\ast}C_{j} \\
\end{array}\right] \\
&=& \underbrace{\left[\begin{array}{cccc}
1 & 1 & \cdots& 1 \\
1 & 1 & \cdots& 1 \\
\vdots & \vdots & \vdots & \vdots \\
1 & 1 &\cdots& 1 \\
\end{array}\right]}_{\textrm{$n_{3}$ times}}\otimes\overset{n_{3}-1}{\underset{j=0}\sum}C_{j}^{\ast}C_{j}\\
&=& J_{n_{3}}\otimes\overset{n_{3}-1}{\underset{j=0}\sum}C_{j}^{\ast}C_{j}.\end{aligned}$$
Therefore $$\label{AstarA}
{\rm Eig}(\hat{A}^{*}\hat{A})={\rm Eig}\left(J_{n_{3}}\otimes\overset{n_{3}-1}{\underset{j=0}\sum}C_{j}^{\ast}C_{j}\right),$$ where $$\begin{aligned}
\label{eigJ}
{\rm Eig}(J_{n_{3}}) = \{0,n_{3}\},\end{aligned}$$ because $J_{n_{3}}$ is a matrix of rank 1, so it has all eigenvalues equal to zero except one eigenvalue equal to ${\rm tr}(J_{n_{3}})=n_{3}$ (${\rm tr}$ is the trace of a matrix). If we put $$\begin{aligned}
\lambda_{k}=\lambda_{k}\left(\overset{n_{3}-1}{\underset{j=0}\sum}C_{j}^{\ast}C_{j}\right),\qquad k=0,\ldots,n_{1}n_{2}-1,\end{aligned}$$ by exploiting basic properties of the tensor product and taking into consideration $(\ref{AstarA})$ and $(\ref{eigJ})$ we find $$\begin{aligned}
\label{eigAA1}
\lambda_{k}(\hat{A}^{*}\hat{A})&=&n_{3}\lambda_{k},\qquad k=0,\ldots,n_{1}n_{2}-1,\\
\label{eigAA2}\lambda_{k}(\hat{A}^{*}\hat{A})&=&0,\qquad k=n_{1}n_{2},\ldots,n_{1}n_{2}n_{3}-1.\end{aligned}$$
From $(\ref{eigAA1})$, $(\ref{eigAA2})$ and $(\ref{1}),$ and recalling that ${\rm Sgval}(\hat{A})={\rm Sgval}(A)$, one obtains that the singular values of $A$ are given by $$\begin{aligned}
\sigma_{k}(A)&=&\sqrt{n_{3}\lambda_{k}},\qquad k=0,\ldots,n_{1}n_{2}-1,\\
\sigma_{k}(A)&=&0,\qquad k=n_{1}n_{2},\ldots,n_{1}n_{2}n_{3}-1,\end{aligned}$$ and, since $\overset{n_{3}-1}{\underset{j=0}\sum}C_{j}^{\ast}C_{j}$ is a positive semidefinite matrix, from $(\ref{bmezzi})$ we can write $$\begin{aligned}
\sigma_{k}(A)&=&\sqrt{n_{3}}\widetilde{\sigma}_{k},\qquad k=0,\ldots,n_{1}n_{2}-1,\\
\sigma_{k}(A)&=&0,\qquad k=n_{1}n_{2},\ldots,n_{1}n_{2}n_{3}-1,\end{aligned}$$ where $\widetilde{\sigma}_{k}$ are the singular values of $\left(\overset{n_{3}-1}{\underset{j=0}\sum}C_{j}^{\ast}C_{j}\right)^{1/2}$.
Regarding the distribution in the sense of singular values, let $F\in C_{0}(\mathbb{R}_{0}^{+})$, continuous function over $\mathbb{R}_{0}^{+}$ with bounded support, then there exists $a\in\mathbb{R}^{+}$ such that $$\label{222}
\left|F(x)\right|\leq a \text{\,\,\,}\forall x\in\mathbb{R}_{0}^{+}.$$
From formula $(\ref{sigmaFA})$ we have $$\begin{aligned}
\Sigma_{\sigma}(F,A_{n}) &=& \frac{1}{n_{1}n_{2}n_{3}}\overset{n_{1}n_{2}n_{3}-1}{\underset{k=0}\sum}
F(\sqrt{n_{3}}\widetilde{\sigma}_{k}) \\
&=& \frac{n_{1}n_{2}(n_{3}-1)F(0)}{n_{1}n_{2}n_{3}}+ \frac{1}{n_{1}n_{2}n_{3}}
\overset{n_{1}n_{2}-1}{\underset{k=0}\sum}F(\sqrt{n_{3}}\widetilde{\sigma}_{k}) \\
&=& \left(1-\frac{1}{n_{3}}\right)F(0)+ \frac{1}{n_{1}n_{2}n_{3}}
\overset{n_{1}n_{2}-1}{\underset{k=0}\sum}F(\sqrt{n_{3}}\widetilde{\sigma}_{k}).\end{aligned}$$
According to $(\ref{222}),$ we find $$-an_{1}n_{2}\leq\overset{n_{1}n_{2}-1}{\underset{k=0}\sum}F(\sqrt{n_{3}}\widetilde{\sigma}_{k})\leq
an_{1}n_{2}.$$ Therefore $$-\frac{a}{n_{3}}\leq\frac{1}{n_{1}n_{2}n_{3}}
\overset{n_{1}n_{2}-1}{\underset{k=0}\sum}F(\sqrt{n_{3}}\widetilde{\sigma}_{k})\leq\frac{a}{n_{3}},$$ so that $$\left(1-\frac{1}{n_{3}}\right)F(0)-\frac{a}{n_{3}}\leq\Sigma_{\sigma}(F,A_{n})\leq\left(1-\frac{1}{n_{3}}\right)F(0)+
\frac{a}{n_{3}}.$$
Now, recalling that the writing $n\rightarrow\infty$ means $\min_{1\leq j\leq 3}n_{j}\rightarrow\infty$, we obtain $$F(0)\leq\underset{n\rightarrow\infty}{\lim}\Sigma_{\sigma}(F,A_{n})\leq F(0),$$ which implies $$\underset{n\rightarrow\infty}{\lim}\Sigma_{\sigma}(F,A_{n})=F(0).$$
Whence $$\{A_{n}\}\sim_{\sigma}(0,G),$$ for any domain $G$ satisfying the requirements of Definition $\ref{def-distribution}$.
Consider a 3-level $\alpha$-Toeplitz matrix $A$ where $\alpha=(\alpha_{1},\alpha_{2},\alpha_{3})=(0,1,2)$ $$\begin{aligned}
A&=&\left[\left[\left[a_{(r_{1}-0\cdot s_{1},r_{2}-1\cdot s_{2},r_{3}-2\cdot s_{3})}
\right]_{r_{3},s_{3}=0}^{n_{3}-1}\right]_{r_{2},s_{2}=0}^{n_{2}-1}
\right]_{r_{1},s_{1}=0}^{n_{1}-1}\\
&=&\left[\left[\left[a_{(r_{1},r_{2}-s_{2},r_{3}-2s_{3})}
\right]_{r_{3},s_{3}=0}^{n_{3}-1}\right]_{r_{2},s_{2}=0}^{n_{2}-1}
\right]_{r_{1}=0}^{n_{1}-1}.\end{aligned}$$
The procedure is the same as in the previous example of an $\alpha$-circulant matrix, but in this case we do not need to permute the vector $\alpha$ since the only component equal to zero is already in first position. For $r_{1}=0,1,...,n_{1}-1$, let us set $$\begin{aligned}
T_{r_{1}}=\left[\left[a_{(r_{1},r_{2}-s_{2},r_{3}-2s_{3})}
\right]_{r_{3},s_{3}=0}^{n_{3}-1}\right]_{r_{2},s_{2}=0}^{n_{2}-1},\end{aligned}$$ then $T_{r_{1}}$ is a 2-level $\alpha^+$-Toeplitz matrix with $\alpha^+=(1,2)$ and of partial sizes $n[>0]=(n_{2},n_{3})$ and $$\begin{aligned}
A=\left[\begin{array}{cccc}
T_{0} & T_{0} & \cdots & T_{0} \\
T_{1} & T_{1} & \cdots & T_{1} \\
\vdots & \vdots & \vdots & \vdots \\
T_{n_{1}-1} & T_{n_{1}-1} & \cdots & T_{n_{1}-1}
\end{array}\right].\end{aligned}$$ The latter is a block matrix with constant blocks on each row. From formula $(\ref{1})$, the singular values of $A$ are the square root of the eigenvalues of $A^{*}A$: $$\begin{aligned}
A^{*}A &=&\left[\begin{array}{cccc}
T_{0}^{\ast} & T_{1}^{\ast} & \cdots & T_{n_{1}-1}^{\ast} \\
T_{0}^{\ast} & T_{1}^{\ast} & \cdots & T_{n_{1}-1}^{\ast} \\
\vdots & \vdots & \vdots & \vdots \\
T_{0}^{\ast} & T_{1}^{\ast} & \cdots & T_{n_{1}-1}^{\ast} \\
\end{array}\right]\left[\begin{array}{cccc}
T_{0} & T_{0} & \cdots & T_{0} \\
T_{1} & T_{1} & \cdots & T_{1} \\
\vdots & \vdots & \vdots & \vdots \\
T_{n_{1}-1} & T_{n_{1}-1} & \cdots & T_{n_{1}-1}
\end{array}\right] \\
&=&\left[\begin{array}{cccc}
\overset{n_{1}-1}{\underset{j=0}\sum}T_{j}^{\ast}T_{j} &
\overset{n_{1}-1}{\underset{j=0}\sum}T_{j}^{\ast}T_{j} & \cdots &
\overset{n_{1}-1}{\underset{j=0}\sum}T_{j}^{\ast}T_{j} \\
\overset{n_{1}-1}{\underset{j=0}\sum}T_{j}^{\ast}T_{j} &
\overset{n_{1}-1}{\underset{j=0}\sum}T_{j}^{\ast}T_{j} & \cdots &
\overset{n_{1}-1}{\underset{j=0}\sum}T_{j}^{\ast}T_{j} \\
\vdots & \vdots & \vdots & \vdots\\
\overset{n_{1}-1}{\underset{j=0}\sum}T_{j}^{\ast}T_{j} &
\overset{n_{1}-1}{\underset{j=0}\sum}T_{j}^{\ast}T_{j} & \cdots &
\overset{n_{1}-1}{\underset{j=0}\sum}T_{j}^{\ast}T_{j} \\
\end{array}\right] \\
&=& \underbrace{\left[\begin{array}{cccc}
1 & 1 & \cdots& 1 \\
1 & 1 & \cdots& 1 \\
\vdots & \vdots & \vdots & \vdots \\
1 & 1 &\cdots& 1 \\
\end{array}\right]}_{\textrm{$n_{1}$ times}}\otimes\overset{n_{1}-1}{\underset{j=0}\sum}T_{j}^{\ast}T_{j}\\
&=& J_{n_{1}}\otimes\overset{n_{1}-1}{\underset{j=0}\sum}T_{j}^{\ast}T_{j}.\end{aligned}$$
Therefore $$\label{AstarAhat}
{\rm Eig}(A^{*}A)={\rm Eig}\left(J_{n_{1}}\otimes\overset{n_{1}-1}{\underset{j=0}\sum}T_{j}^{\ast}T_{j}\right),$$ where $$\begin{aligned}
\label{eigJ1}
{\rm Eig}(J_{n_{1}}) = \{0,n_{1}\},\end{aligned}$$ because $J_{n_{1}}$ is a matrix of rank 1, so it has all eigenvalues equal to zero except one eigenvalue equal to ${\rm tr}(J_{n_{1}})=n_{1}$ (${\rm tr}$ is the trace of a matrix). If we put $$\begin{aligned}
\lambda_{k}=\lambda_{k}\left(\overset{n_{1}-1}{\underset{j=0}\sum}T_{j}^{\ast}T_{j}\right),\qquad k=0,\ldots,n_{3}n_{2}-1,\end{aligned}$$ by exploiting basic properties of the tensor product and taking into consideration $(\ref{AstarAhat})$ and $(\ref{eigJ1})$ we find $$\begin{aligned}
\label{eigAAhat1}
\lambda_{k}(A^{*}A)&=&n_{1}\lambda_{k},\qquad k=0,\ldots,n_{3}n_{2}-1,\\
\label{eigAAhat2}\lambda_{k}(A^{*}A)&=&0,\qquad k=n_{3}n_{2},\ldots,n_{3}n_{2}n_{1}-1.\end{aligned}$$
From $(\ref{eigAAhat1})$, $(\ref{eigAAhat2})$ and $(\ref{1}),$ one obtains that the singular values of $A$ are given by $$\begin{aligned}
\sigma_{k}(A)&=&\sqrt{n_{1}\lambda_{k}},\qquad k=0,\ldots,n_{3}n_{2}-1,\\
\sigma_{k}(A)&=&0,\qquad k=n_{3}n_{2},\ldots,n_{3}n_{2}n_{1}-1.\end{aligned}$$ and, since $\overset{n_{1}-1}{\underset{j=0}\sum}T_{j}^{\ast}T_{j}$ is a positive semidefinite matrix, from $(\ref{bmezzi})$ we can write $$\begin{aligned}
\sigma_{k}(A)&=&\sqrt{n_{1}}\widetilde{\sigma}_{k},\qquad k=0,\ldots,n_{3}n_{2}-1,\\
\sigma_{k}(A)&=&0,\qquad k=n_{3}n_{2},\ldots,n_{3}n_{2}n_{1}-1,\end{aligned}$$ where $\widetilde{\sigma}_{k}$ denotes the generic singular value of $\left(\overset{n_{1}-1}{\underset{j=0}\sum}T_{j}^{\ast}T_{j}\right)^{1/2}$.
Regarding the distribution in the sense of singular values, by invoking exactly the same argument as in the above example for $\alpha$-circulant matrix, we deduce that $$\{A_{n}\}\sim_{\sigma}(0,G),$$ for any domain $G$ satisfying the requirements of Definition $\ref{def-distribution}$.
Let us see what happens when the vector $\alpha$ has only one component different from zero. Let $n=(n_{1},n_{2},\ldots,n_{d})$ and $\alpha=(0,\ldots,0,\alpha_{k},0,\ldots,0)$, $\alpha_{k}>0$; in this case we can give an explicit formula for the singular values of the $d$-level $\alpha$-circulant matrix. For convenience and without loss of generality we take $\alpha=(0,\ldots,0,\alpha_{d})$ (with all zero components in top positions, otherwise we use a permutation). From $\ref{nonnegative-vs-positive}$, the singular values of $A_{n}=[a_{(r-\alpha\circ s)\textrm{ mod $n$}}]_{r,s=\underline{0}}^{n-e}$ are zero except for few of them given by $\sqrt{\hat n[0]} \sigma$ where, in our case, $\hat n[0]=n_{1}n_{2}\cdots n_{d-1}$, $n[0]=(n_{1},n_{2},\ldots,n_{d-1})$, and $\sigma$ is any singular value of the matrix $$\left(\sum_{j=\underline{0}}^{n[0]-e}C_j^*C_j\right)^{1/2},$$ where $C_j$ is an $\alpha_{d}$-circulant matrix of dimension $n_{d}\times n_{d}$ whose expression is $$\begin{aligned}
C_j=\left[a_{(r-\alpha \circ s)\ {\rm mod}\,
n}\right]_{r_{d},s_{d}=0}^{n_{d}-1}&=&\left[a_{(r_{1},r_{2},\ldots,r_{d-1},(r_{d}-\alpha_{d}s_{d})\ {\rm mod}\,
n_{d})}\right]_{r_{d},s_{d}=0}^{n_{d}-1}\\
&=&\left[a_{(j,(r_{d}-\alpha_{d}s_{d})\ {\rm mod}\,
n_{d})}\right]_{r_{d},s_{d}=0}^{n_{d}-1},\end{aligned}$$ with $(r_{1},r_{2},\ldots,r_{d-1})=j$. For $j=\underline{0},\ldots,n[0]-e$, if $C_{n_{d}}^{(j)}$ is the circulant matrix which has as its first column the vector $a^{(j)}=[a_{(j,0)},a_{(j,1)},\ldots,a_{(j,n_{d}-1)}]^{T}$ (which is the first column of the matrix $C_{j}$), $C_{n_{d}}^{(j)}=[a_{(j,(r-s)\textrm{ mod $n_{d}$})}]_{r,s=0}^{n_{d}-1}=F_{n_{d}}D_{n_{d}}^{(j)}F_{n_{d}}^{*}$, with $D_{n_{d}}^{(j)}=diag(\sqrt{n_{d}}F_{n_{d}}^{*}a^{(j)})$, then, from $(\ref{CastC})$, $(\ref{0})$, and $(\ref{Vi})$, it is immediate to verify that $$\begin{aligned}
\sum_{j=\underline{0}}^{n[0]-e}C_j^*C_j&=&\sum_{j=\underline{0}}^{n[0]-e}(F_{n_{d}}D_{n_{d}}^{(j)}F_{n_{d}}^{*}Z_{n_{d},\alpha_{d}})^{*}
(F_{n_{d}}D_{n_{d}}^{(j)}F_{n_{d}}^{*}Z_{n_{d},\alpha_{d}})\\
&=&\sum_{j=\underline{0}}^{n[0]-e}(F_{n_{d}}^{*}Z_{n_{d},\alpha_{d}})^{*}(D_{n_{d}}^{(j)})^{*}D_{n_{d}}^{(j)}(F_{n_{d}}^{*}Z_{n_{d},\alpha_{d}})\\
&=&(F_{n_{d}}^{*}Z_{n_{d},\alpha_{d}})^{*}\left(\sum_{j=\underline{0}}^{n[0]-e}(D_{n_{d}}^{(j)})^{*}D_{n_{d}}^{(j)}\right)(F_{n_{d}}^{*}Z_{n_{d},\alpha_{d}}).\end{aligned}$$
Now, if we put $n_{d,\alpha}=\frac{n_{d}}{(n_{d},\alpha_{d})}$ and $$\begin{aligned}
q_{s}^{(j)}&=&|D_{n_{d}}^{(j)}|_{s,s}^{2}=(D_{n_{d}}^{(j)})_{s,s}\cdot\overline{(D_{n_{d}}^{(j)})_{s,s}},
\quad s=0,1,\ldots,n_{d}-1,\\
\Delta_{l}&=&\left[\begin{array}{cccc}
\overset{n[0]-e}{\underset{j=\underline{0}}\sum} q_{(l-1)n_{d,\alpha}}^{(j)} & & & \\
& \overset{n[0]-e}{\underset{j=\underline{0}}\sum} q_{(l-1)n_{d,\alpha}+1}^{(j)} & & \\
& & \ddots & \\
& & & \overset{n[0]-e}{\underset{j=\underline{0}}\sum} q_{(l-1)n_{d,\alpha}+n_{d,\alpha}-1}^{(j)}
\end{array}\right]\in \mathbb{C}^{n_{d,\alpha}\times
n_{d,\alpha}},\\\end{aligned}$$ for $l=1,2,\ldots,(n_{d},\alpha_{d})$, then, following the same reasoning employed for proving formula $(\ref{eigg})$, we infer $${\rm Eig}\left(\sum_{j=\underline{0}}^{n[0]-e}C_j^*C_j\right)=\frac{1}{(n_{d},\alpha_{d})}{\rm Eig}\left(J_{(n_{d},\alpha_{d})}
\otimes \overset{(n_{d},\alpha_{d})}{\underset{l=1}\sum}\Delta_{l}\right),$$ where $$\begin{aligned}
J_{(n_{d},\alpha_{d})}&=&\underbrace{\left[\begin{array}{cccc}
1 & 1 & \cdots & 1\\
1 & 1 & \cdots & 1\\
\vdots & \vdots & \vdots & \vdots\\
1 & 1 & \cdots & 1
\end{array}\right]}_{\textrm{$(n_{d},\alpha_{d})$ times}},\\
\frac{1}{(n_{d},\alpha_{d})}{\rm Eig}(J_{(n_{d},\alpha_{d})}) &=& \{0,1\},\end{aligned}$$ and $$\begin{aligned}
\overset{(n_{d},\alpha_{d})}{\underset{l=1}\sum}\Delta_{l}&=&
\overset{(n_{d},\alpha_{d})}{\underset{l=1}\sum}{\rm diag}\left(\overset{n[0]-e}{\underset{j=\underline{0}}\sum}q_{(l-1)n_{d,\alpha}+k}^{(j)};\text{\,\,}k=0,1,\ldots,n_{d,\alpha}-1\right)\\
&=&{\rm diag}\left(\overset{(n_{d},\alpha_{d})}{\underset{l=1}\sum}
\overset{n[0]-e}{\underset{j=\underline{0}}\sum}q_{(l-1)n_{d,\alpha}+k}^{(j)};\text{\,\,}
k=0,1,\ldots,n_{d,\alpha}-1\right).\end{aligned}$$
Consequently, since $\overset{(n_{d},\alpha_{d})}{\underset{l=1}\sum}\Delta_{l}$ is a diagonal matrix, and by exploiting basic properties of the tensor product, we find $$\begin{aligned}
\lambda_{k}\left(\sum_{j=\underline{0}}^{n[0]-e}C_j^*C_j\right)&=&\overset{(n_{d},\alpha_{d})}
{\underset{l=1}\sum}\overset{n[0]-e}{\underset{j=\underline{0}}\sum}q_{(l-1)n_{d,\alpha}+k}^{(j)}, \quad k=0,1,\ldots,n_{d,\alpha}-1,\\
\lambda_{k}\left(\sum_{j=\underline{0}}^{n[0]-e}C_j^*C_j\right)&=& 0,\qquad k=n_{d,\alpha},\ldots,n_{d}-1.\end{aligned}$$
Now, since $\sum_{j=\underline{0}}^{n[0]-e}C_j^*C_j$ is a positive semidefinite matrix, from $(\ref{bmezzi})$ we finally have $$\begin{aligned}
\sigma_{k}\left(\left(\sum_{j=\underline{0}}^{n[0]-e}C_j^*C_j\right)^{1/2}\right)&=&\sqrt{\overset{(n_{d},\alpha_{d})}
{\underset{l=1}\sum}\overset{n[0]-e}{\underset{j=\underline{0}}\sum}q_{(l-1)n_{d,\alpha}+k}^{(j)}}, \quad k=0,1,\ldots,n_{d,\alpha}-1,\\
\sigma_{k}\left(\left(\sum_{j=\underline{0}}^{n[0]-e}C_j^*C_j\right)^{1/2}\right)&=& 0,\qquad k=n_{d,\alpha},\ldots,n_{d}-1.\end{aligned}$$
Conclusions and future work {#sec:fin}
===========================
In this paper we have studied in detail the singular values of $\alpha$-circulant matrices and we have identified the joint asymptotic distribution of $\alpha$-Toeplitz sequences associated with a given integrable symbol. The generalization to the multilevel block setting has been sketched together with some intriguing relationships with the design of multigrid procedures for structured linear systems. The latter point deserves more attention and will be the subject of future researches. We also would like to study the more involved eigenvalue/eigenvector behavior both for $\alpha$-circulant and $\alpha$-Toeplitz structures.
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[^1]: Dipartimento di Fisica e Matematica, Università dell’Insubria, Via Valleggio 11, 22100 Como (ITALY). Email: {eric.ngondiep,stefano.serrac,debora.sesana}@uninsubria.it; [email protected]
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'Exploiting the deep high-resolution imaging of all 5 CANDELS fields, and accurate redshift information provided by 3D-HST, we investigate the relation between structure and stellar populations for a mass-selected sample of 6764 galaxies above $10^{10}\ M_{\sun}$, spanning the redshift range $0.5<z<2.5$. For the first time, we fit 2-dimensional models comprising a single fit and two-component (i.e., bulge + disk) decompositions not only to the $H$-band light distributions, but also to the stellar mass maps reconstructed from resolved stellar population modeling. We confirm that the increased bulge prominence among quiescent galaxies, as reported previously based on rest-optical observations, remains in place when considering the distributions of stellar mass. Moreover, we observe an increase of the typical index and bulge-to-total ratio (with median $B/T$ reaching 40-50%) among star-forming galaxies above $10^{11}\ M_{\sun}$. Given that quenching for these most massive systems is likely to be imminent, our findings suggest that significant bulge growth precedes a departure from the star-forming main sequence. We demonstrate that the bulge mass (and ideally knowledge of the bulge and total mass) is a more reliable predictor of the star-forming versus quiescent state of a galaxy than the total stellar mass. The same trends are predicted by the state-of-the-art semi-analytic model by Somerville et al. In the latter, bulges and black holes grow hand in hand through merging and/or disk instabilities, and AGN-feedback shuts off star formation. Further observations will be required to pin down star formation quenching mechanisms, but our results imply they must be internal to the galaxies and closely associated with bulge growth.'
author:
- 'Philipp Lang, Stijn Wuyts, Rachel S. Somerville, Natascha M. F[ö]{}rster Schreiber, Reinhard Genzel, Eric F. Bell, Gabe Brammer, Avishai Dekel, Sandra M. Faber, Henry C. Ferguson, Norman A. Grogin, Dale D. Kocevski, Anton M. Koekemoer, Dieter Lutz, Elizabeth J. McGrath, Ivelina Momcheva, Erica J. Nelson, Joel R. Primack, David J. Rosario, Rosalind E. Skelton, Linda J. Tacconi, Pieter G. van Dokkum, Katherine E. Whitaker'
title: 'Bulge Growth and Quenching since z = 2.5 in CANDELS/3D-HST'
---
Introduction {#intro.sec}
============
The mechanisms driving the shutdown of star-forming galaxies (SFGs), ’quenching’, remains one of the least understood puzzles in galaxy formation to date. In the low-redshift universe, galaxies show a bimodal color distribution, accompanied by a bimodality of morphologies [see @kauffmann2003; @strateva2001]. Spiral galaxies have low bulge-to-disk ratios and are commonly the site of active star formation leading to blue colors. Passive galaxies are observed to be mostly spheroid-dominated. The color bimodality has also been observed at higher redshifts [e.g., @brammer2009; @whitaker2011]. These observations have been interpreted as evolutionary paths, in which one or several quenching processes cause the SFG to become red and passive on a short timescale [see @bell2004; @faber2007]. Studies of the shape of the mass function of passive galaxies over cosmic time using large surveys such as SDSS, NMBS, zCOSMOS, UltraVISTA and zFOURGE showed that the probability for a galaxy being quenched increases with its mass (Peng et al. 2010; Brammer et al. 2011; Ilbert et al. 2013; Muzzin et al. 2013; Woo et al. 2013; Tomczak et al. 2014). Further evidence for quenching also comes from studies employing abundance matching techniques, where the cumulative abundance of galaxies is matched to that of haloes using the results of cosmological dark matter simulations alongside with observational constraints on the stellar mass function. Those infer low baryon fractions within dark matter haloes, hinting at an efficient quenching mechanism associated with significant gas mass loss for both low and high mass galaxies [@moster2010; @behroozi2010].
Several quenching mechanisms have been proposed, which act to either remove the gas from the galaxy or prevent the existing/inflowing gas within the galaxy to form stars. For the high-mass regime of galaxies, powerful AGN feedback, which may be induced by galaxy mergers [e.g., @hopkins2006] or internal evolutionary processes triggered by (violent) gravitational disk instabilities [e.g., @bournaud2011], may drive energetic outflows expelling gas out of the galaxy and heating the halo. In addition, a so-called ’radio-mode’ feedback may suppress the cooling of gas onto the galaxy over a longer timespan (Croton et al. 2006).
Moreover, morphological quenching, proposed by [@martig2009] may switch off or reduce the efficiency of star formation in galaxies through the existence of a dominant bulge which stabilizes the gas disk against gravitational instabilities (see Saintonge et al. 2012 and Crocker et al. 2012 for observational hints in the local universe, and Genzel et al. 2014 for galaxies at high redshift). Another process proposed to be responsible for the shut-down of galaxies is halo mass quenching (e.g., Birnboim & Dekel 2003; Kere[š]{} et al. 2005). In this scenario, dark matter haloes exceeding a critical mass of $M \sim 10^{12}M_{\odot}$ are able to stop the flow of incoming cold gas onto their central galaxies via virial shock heating, which leads to a decrease of star formation and/or suppresses star formation over longer timescales. However, at $z>z_{crit}$ (with $ 1 < z_{crit} < 3$), gas is predicted to penetrate to the central galaxy through cold streams even in massive haloes (e.g., Dekel et al. 2009).
The above quenching mechanisms could plausibly leave an imprint on the structure of galaxies, as they may lead to high central concentrations due to internal processes and/or major mergers. This idea triggered a plethora of studies investigating the relation between star formation activity (or absence thereof) and galaxy structure. Local galaxy surveys demonstrated that quiescence is strongly linked with structural and morphological parameters such as the index, stellar mass density and the central velocity dispersion [@kauffmann2003; @kauffmann2006; @schiminovich2007; @bell2008; @fang2013; @cheung2012]. This suggests that a high stellar mass surface density in the center of a galaxy is connected to its quenching.
Recently, high-resolution imaging facilities onboard the Hubble Space Telescope (HST), including critically the NIR Wide-Field Camera 3 (WFC3), have allowed us to trace the relation between structure and stellar populations further back in cosmic time. Early studies on the basis of high-resolution HST-imaging found a correlation between color and bulge fraction for large samples of galaxies at $z\sim 0.7 - 1$ (Bell et al. 2004; Weiner et al. 2005; Koo et al. 2005). Particularly the Cosmic Assembly Near-infrared Deep Extragalactic Legacy Survey [CANDELS, @grogin2011; @koekemoer2011] has played a pivotal role due to its unique combination of multi-wavelength high-resolution imaging, large sample size, and depth. A common theme of several of the early CANDELS studies is that a correlation between galaxy structure and stellar populations (a ‘Hubble sequence’) has already been in place since at least $z \sim 2.5$ (Wuyts et al. 2011; Bell et al. 2012; Wang et al. 2012; Lee et al. 2013; see also Cheung et al. 2012). Additionally, Bruce et al. (2012) show that the majority of star-forming galaxies have disk-dominated light profiles and passive galaxies have bulge-dominated profiles directly from decomposed bulge+disk light-profile fitting of $M > 10^{11}M_{\odot}$ galaxies at $1 < z < 3$. However, it should be noted that they also find a significant fraction ($\sim 30 \%$) of passive disk-dominated galaxies and star-forming bulge-dominated systems in this high-mass regime. These results have set on a firmer footing the pioneering studies by [@franx2008] and [@kriek2009], that used ground-based data and significantly smaller samples, respectively. At all epochs since at least $z \sim 2.5$, SFGs follow a relatively tight relation of $0.2-0.3$ dex scatter between their ongoing star formation rate (SFR) and assembled stellar mass, with a zero point that increases with lookback time. This relation has been dubbed the ’main sequence’ (MS) of star-forming galaxies (Noeske et al. 2007; Elbaz et al. 2007; Daddi et al. 2007; Rodighiero et al. 2011). Typical SFGs that lie on the MS are best approximated by exponential disk profiles, while the quiescent galaxy population below the MS is better described with higher indices [@sw2011b; @bell2012]. In common to all of the above studies, however, is that they were carried out on monochromatic (albeit mostly rest-optical) imaging, furthermore often exploiting only a subset of the entire CANDELS data set. If a galaxy component (e.g., disk) forms significantly more stars than another component (e.g., bulge), their relative weight in light can differ significantly from their contributions in physically more relevant units of stellar mass.
Deep panchromatic high-resolution imaging datasets enable us to go beyond measurements of light profiles, by reconstructing the distribution of stellar mass (see, e.g., Zibetti et al. 2009 for a detailed application to nearby galaxies). While pioneering work on pre-existing high-z datasets using the NICMOS camera onboard HST was limited to small sample sizes [e.g., @elmegreen2009; @fs2011], [@sw2012] made use of resolved 7-band stellar-population synthesis modeling on a pixel-by-pixel basis of over 600 massive ($M > 10^{10}M_{\odot}$) SFGs at $0.5 < z < 2.5$ and found that galaxies are overall smoother and more centrally concentrated in mass than they appear in light. Caused by a combination of internal extinction and star formation history variations , the presence of mass-to-light ratio variations within galaxies emphasizes the importance of making measurements on mass maps when characterizing the stellar structure of high- redshift galaxies. Guo et al. (2011) furthermore demonstrated that also quiescent galaxies at high redshift feature color (and hence mass-to-light ratio) gradients.
In order to further shed light on the connection between galaxy structure and quenching, we reconstruct stellar mass maps of a large mass-selected sample ($> 10^{10}\ M_{\sun}$) of galaxies at $0.5 < z < 2.5$ and subject those to a detailed structural analysis. For this purpose, we exploit the available multi-wavelength data in all 5 CANDELS/3D-HST fields with accurate redshifts from the 3D-HST survey [@brammer2012; @vd2011]. We don’t restrict the structural measurements to one-component () fits, but rather also explore two-component (bulge + disk) decompositions. The latter allows us to carry out a more direct comparison to semi-analytic models (SAMs; specifically those by Somerville et al. 2008, 2012, further developed by Porter et al. 2014), whose prescriptions are formulated in units of disk and bulge components.
The paper is structured as follows. In Section \[obs\_sample.sec\], we give an overview of the observations and sample. The construction of stellar mass maps through resolved stellar population modeling, and the methodology to derive structural parameters, is described in Section \[method.sec\]. We present our observational results in Section \[structure.sec\], followed by a comparison with SAMs in Section \[SAM\_results.sec\]. Finally, we summarize and discuss the implications of our findings in Section \[discussion.sec\].
Throughout this paper, we quote magnitudes in the AB system, assume a Chabrier (2003) initial mass function (IMF), and adopt the cosmological parameters $(\Omega _M, \Omega _{\Lambda}, h) = (0.3, 0.7, 0.7)$.
Data and Sample selection {#obs_sample.sec}
=========================
The core dataset used for this work is the deep space-based HST imaging from the CANDELS multi-cycle treasury program [@grogin2011; @koekemoer2011], complemented with redshift information from the 3D-HST grism survey (van Dokkum et al. 2011; Brammer et al. 2012)[^1].
In order to determine galaxy-integrated masses and SFRs, we make use of additional multi-wavelength data in the CANDELS/3D-HST fields including space-based photometry from [*Spitzer*]{}/IRAC, [*Spitzer*]{}/MIPS and [*Herschel*]{}/PACS and an array of ground-based facilities (see Skelton et al. 2014 for a detailed description of the $H$-selected multi-wavelength catalogs produced by the 3D-HST team).
The galaxy-integrated properties are derived following identical procedures as [@sw2011b]. These include stellar masses, based on $U$- to - $8\mu m$ broad-band SED modeling using population synthesis models from [@bc03] and SFRs derived from a ‘ladder of SFR indicators’. The latter method uses detected emission in either UV + PACS for PACS-detected galaxies (Lutz et al. 2011; Magnelli et al. 2013) or UV + MIPS 24$\mu m$ for MIPS-detected galaxies to compute the sum of the obscured and unobscured SFR. For galaxies lacking an IR detection, the SFR is adopted from the best-fit SED model. We verified that consistent results are obtained when splitting star-forming and quiescent galaxies based on the above measures of star formation activity or when adopting the rest-UVJ diagram (Wuyts et al. 2007; Williams et al. 2009)
HST imaging {#fields_data.sec}
-----------
The HST CANDELS observations used for this study comprise high-resolution imaging in five distinct fields: GOODS-South, GOODS-North, COSMOS, UDS, and EGS, covering a total area of 625 arcmin$^2$. Typical limiting depths in 160 are 27.0 mag for CANDELS/wide and 27.7 mag for CANDELS/deep (the central halves of the GOODS fields), respectively. Additional data used for this work include pre-existing ACS imaging in the GOODS, EGS, and COSMOS fields (Giavalisco et al. 2004; Davis et al. 2007; Koekemoer et al. 2007), plus WFC3 imaging in ERS [@windhorst2011]. The available passbands are $B_{435}$,$V_{606}$,$i_{775}$,$z_{850}$,$J_{125}$,$H_{160}$ for the GOODS fields and $V_{606}$,$I_{814}$,$J_{125}$,$H_{160}$ for the remaining fields. Additionally, $Y_{098}$ imaging was available for ERS as part of GOODS-South and $Y_{105}$ for the regions with CANDELS/Deep coverage.
All imaging used in our analysis was drizzled to a $0\farcs06$ pixel scale. For details on the observations and data reduction, we refer the reader to [@koekemoer2011] and [@grogin2011].
Sample definition {#fields_data.sec}
-----------------
The main focus of this paper is on the structural shape of stellar mass distributions in a mass-selected sample of galaxies in the redshift range $ 0.5 < z < 2.5$. For this purpose, we only apply a mass cut $M > 10^{10}M_{\odot}$ to select our galaxies, well above the mass completeness limit of the $H_{160}$-selected catalogs for all five fields. Our sample consequently spans a wide range of SFRs, from normal SFGs on the main sequence to quiescent galaxies (QGs) below, that already formed the bulk of their stars. For selection, we use the SED modeled galaxy-integrated mass estimates, but note that they match well the masses obtained by summing the resolved stellar mass distributions [see also @sw2012].
In order to determine redshifts for the galaxies in our sample, we use ground-based spectroscopic redshift information whenever available. Otherwise, redshifts are fitted to the combination of 3D-HST grism data and broad-band photometry.
The total sample comprises 6764 galaxies, of which 3839 and 2925 lie within the redshift range $ 0.5 < z < 1.5$ and $ 1.5 < z < 2.5$, respectively. In the following, these two redshift ranges are referred to as $z \sim 1$ and $z \sim 2$, respectively.
Methodology {#method.sec}
===========
Resolved SED Modeling {#fields_data.sec}
---------------------
A detailed description of the resolved SED modeling can be found in [@sw2012]. Here, we review only the key steps involved and additional processing steps.
First, all images in the available wavelength bands are brought to the WFC3 160 resolution ($0\farcs 18$) by using the IRAF PSFMATCH algorithm. Next, a Voronoi-binning scheme from [@cappellari2003] is applied in order to ensure a minimum S/N level of 10 for each bin in the corresponding $H$-band image. The binned multi-wavelength images are then fit with stellar population synthesis models from [@bc03], assuming a Chabrier (2003) IMF, a uniform solar metallicity, and a Calzetti et al. (2000) reddening law with visual extinctions in the range $ 0 < A_v < 4$. The adopted SFHs are exponentially declining, allowing for e-folding timescales down to 300 Myr. This choice is simplistic and may not be representative of the real SFHs (e.g., for SFGs see Maraston et al. 2010). However, we are here interested in the stellar mass, which, among SED-derived properties, is most robust against variations (and uncertainties) in model assumptions (e.g., Papovich et al. 2001; Shapley et al. 2005; Maraston et al. 2010).
In order to conduct further structural analysis using the (binned) output of this SED modeling technique, we constructed pixelized (instead of Voronoi binned) images of the stellar mass surface density distribution in the following way. First, we construct a $M/L$ map which, within the galaxy’s $H_{160}$ segmentation map, has uniform values for pixels belonging to the same Voronoi bins. Pixels outside the segmentation map (i.e., containing sky noise and possibly faint wings of the galaxy extending below the signal-to-noise threshold) are assigned the average $M/L$ of the nearest three Voronoi bins. The resulting expanded $M/L$ map is then combined with the 160 image to construct a final mass map at full $H_{160}$ resolution. This ensures a smooth transition of the galaxy’s mass profile from the brighter central parts to the faintest regions.
Structural Parameters {#fields_data.sec}
---------------------
In order to conduct a structural analysis of our galaxy sample, we employ the GALFIT (Peng et al. 2010) morphology fitting code to fit 2-dimensional parametric models to the stellar mass distribution. For comparison purposes, we model the 2D $H_{160}$ surface brightness distributions as well, following identical procedures.
The models we use comprise on the one hand single models to parametrize the galaxy’s shape with the effective radius ($R_e$) and index $n$. $R_e$ is the galactocentric distance containing half of the total light/mass, and $n$ is a measure of the cuspiness of the overall light/mass profile. We allow $n$ to vary within the range $0.2<n<8$.
While the index $n$ is often used and indeed serves as an approximate measure of the contribution of the bulge, it is important to note that $n$ does not translate one to one to the bulge-to-total ratio $B/T$ (see Appendix A; also Andredakis et al. 1995; de Jong et al. 2004; Cibinel et al. 2013; Bruce et al. 2012, Bruce et al. 2014 in prep.). For a given bulge + disk composite profile, the best-fit index can be increased both by increasing its $B/T$, and by leaving $B/T$ constant while growing the extent of the disk relative to that of the bulge (i.e., lowering $R_{e,B}/R_{e,D}$). For this reason, and to facilitate a more direct comparison to (semi-analytic) models that are parametrized in terms of bulge and disk units, we furthermore decompose the galaxies by fitting 2-component (bulge + disk) models. In addition, we also performed one-component pure disk and pure bulge fits by forcing $n=1$ and $n=4$, respectively.
For the bulge-to-disk decomposition, we adopt a procedure similar to that implemented in Bruce et al. (2012) for the bulge+disk decomposition of $H_{160}$ light profiles, where we fixed the indices to $n=1$ for the disk and to $n=4$ for the bulge.[^2] The centers are left free, but we restrict the relative distance between the bulge and disk centers to be less than 2 pixels. All other parameters defining the two components ($R_{e}$, the axial ratio $b/a$, and the total magnitude/mass of both components) are allowed to vary independently.
In 2-component modeling, the higher number of degrees of freedom increases the odds of the fit being trapped in a local minimum. In order to mitigate this risk, we ran GALFIT using a grid of initial starting values. Our grid was constructed by using a range of size ratios between bulge and disk ($R_{e,B}/R_{e,D}$) ranging from 0.1 to 1, in steps of 0.1. For each initial guess of $R_{e,B}/R_{e,D}$, the corresponding initial guess on $B/T$ was then set such that the index matching this initial configuration matches the one measured in the single-component fit (see Appendix A). Likewise, the initial magnitudes and absolute values of the initial size guesses for bulge and disk were set such that the total magnitude and half-light/mass radius of the composite profile matches the respective values determined from fitting.
In cases where GALFIT yields solutions with implausibly small bulge sizes ($< 0.1$ pixel, corresponding to $\lesssim 1/30$ of the resolution, or $\lesssim$ 50 pc at $z\sim2$) and flags the outcome as potentially not converged and unphysical, we excluded the respective run from the grid. Also, solutions yielding a disk smaller in size than the bulge (i.e., $R_{e,B}/R_{e,D} > 1$) were not included.[^3]
After performing the fits for each point of the grid with initial guesses, we assigned a final $B/T$ ratio for each object as the solution of the fit with the lowest $\chi_{red}^2$. Here, the pure disk and pure bulge fits were also included. Their solutions generally show higher $\chi_{red}^2$ than the 2-component fits, but are occasionally preferred over those in a $\chi_{red}^2$ sense (oftentimes, these are bulgeless systems with $n < 1$). The two-component decompositions are statistically preferred over the single models (as based on both their $\chi_{red}^2$ values and the Akaike information criterion, AIC[^4]) for $\sim2/3$ of the total sample. Those systems for which the single model yields a lower value of $\chi_{red}^2$ and AIC typically feature shallow profiles with $n < 1$.
In both single- and two-component fitting, we use an automated scheme which pre-determines neighboring sources that need to be masked or fitted simultaneously, and passes initial guesses of fitting parameters to GALFIT. Those include estimates on size, total magnitude and center. The initial guess on size is based on the distance from the center to the radius at which the curve of growth reaches 50% of the galaxy’s total flux. The mass-weighted center of the galaxy derived within its segmentation map is adopted as initial estimate for the center. GALFIT takes into account the convolution of the model with the point-spread-function (PSF). Both for fitting light and mass, we use a PSF which is a combination of stacked stars and a Tiny-Tim [@krist1995] model PSF. For a more detailed description of the used PSF, see [@vdw2012].
We emphasize that throughout this paper our working definition adopted for bulge and disk components is based on the above bulge+disk modeling of stellar mass or light maps, as empirical constraints on whether or not stars assigned to a bulge/disk component are dynamically hot/cold are currently lacking. As the SINS survey of $z\sim2$ galaxy kinematics demonstrated that high-redshift SFGs are dynamically (Förster Schreiber et al. 2009) and morphologically (Förster Schreiber et al. 2011) distinct from local spiral galaxies, we caution that bulge fractions as derived by our decompositions may have a somewhat different meaning than they would have in the local universe.
In addition, we estimated a typical measurement uncertainty on $B/T$ by setting up an array of model galaxies with a range of mass, size, $B/T$ ratio, and $R_{e,B}/R_{e,D}$, which is similar to our data. The grid of model galaxies consisted of 5, 3, 11, and 10 grid points in mass, $R_e$, $B/T$ and $R_{e,B}/R_{e,D}$ respectively. These were then inserted in multiple empty sky regions of the CANDELS UDS field to mimic the typical background noise[^5]. We next ran GALFIT on all 8250 mock galaxies using our 2-component fitting scheme. The measurement error on $B/T$ of each galaxy in our sample given its mass, radius, and profile shape is then finally assigned as the scatter among the recovered $B/T$ ratios of the corresponding model galaxy.
Typical measurement errors in $B/T$ for star-forming and quiescent galaxies are on average $\sim0.05$ and $\sim0.06$ at $z\sim 1$ and $\sim0.1$ and $\sim0.13$ at $z\sim 2$, respectively. The distribution of errors peaks below the median ($\lesssim0.05$), and shows a tail towards higher $B/T$ errors. Two alternative methods to estimate the uncertainty in $B/T$, namely the formal random uncertainty reported by GALFIT and re-fitting the observed galaxies after applying additional background and Poisson noise, generally lead to lower estimated uncertainties (by a factor of $ \sim 2$ in the case of GALFIT). In the remainder of the paper, we therefore adopt the most conservative error estimates inferred from our analysis of the inserted mock galaxies.
Results on Galaxy Structure {#structure.sec}
===========================
The Evolving Mass Budget of Disks and Bulges {#budget.sec}
--------------------------------------------
Exploiting the bulge-disk decompositions of the stellar mass maps derived for our sample of massive galaxies, we first evaluate the average mass budget of disks and bulges. Let us consider picking a random star out of our sample of massive galaxies above $10^{10}\ M_{\sun}$. At $1.5<z<2.5$, the probability that this star belongs to a bulge component is $46\%$. Increasing the mass limit to $\log(M) = 10.5$ or 11 yields a higher probability for the star to be associated to the bulge, of $49\%$ and $54\%$, respectively. Perhaps somewhat surprisingly, the fraction of stars residing in a bulge component rises only slightly to $0.5<z<1.5$, to $47\%$, $50\%$, and $56\%$ for galaxies more massive than $\log(M) = 10$, 10.5, and 11 respectively.
The formal uncertainties to the above stated probabilities including sample variance and typical measurement errors on $B/T$ are limited to a few percent. The total error budget is likely dominated by systematics, for example related to the assumptions made in stellar population modeling (see Section 3.1). We note, however, that only $M/L$ uncertainties with a differential impact on bulges and disks will affect the above numbers. Even if the $M/L$ ratio of bulges were systematically under- or overestimated by 0.2 (0.3) dex with respect to those of disks, the change in the above numbers would be limited to $\sim 7 (10)\%$.
As bulges, unlike stellar disks, can be considered sinks in the continuous assembly of a galaxy’s stellar component, the rising mass density of stars in bulges (by a factor of $\sim 1.8$ from the higher to the lower redshift bin) therefore seems to be compensated largely by the continuing assembly of new disks. Splitting our sample in finer redshift bins, we do find the fraction of stellar mass in bulges to increase more significantly, by a factor $\sim 1.5$ over the entire 6 Gyr timespan sampled by our study.
Overall, the bulge mass fractions are higher than what would be inferred from fits to the $H$-band surface brightness profiles, as the median mass-to-light ratio of disk components is 0.2 dex lower than of bulge components. The above numbers address the evolving mass budget of disks and bulges for a mass-limited sample including both star-forming and quiescent galaxies. In the remainder of the paper, we will delve into more depth by breaking down our sample by star formation activity.
{width="\textwidth"}
Profile shape {#profile.sec}
-------------
In recent years, several HST-based studies have investigated the structural differences between star-forming and quiescent galaxies at high redshift. In common to all of these analyses, star-forming systems are found to have significantly larger rest-optical sizes than their quiescent counterparts at the same mass and redshift (e.g., Toft et al. 2009; van der Wel et al. 2014). In addition, their surface brightness profile shapes tend to be shallow ($n \sim 1$), while quiescent galaxies feature cuspier light profiles (e.g., Wuyts et al. 2011; Bell et al. 2012; Cheung et al. 2012, although see also Bruce et al. 2012 for decomposed light profiles at high stellar masses and redshifts).
At the same time, the same multi-wavelength high-resolution lookback surveys have also established that substantial mass-to-light ratio variations in the rest-optical can occur, not only between but also within galaxies (see, e.g., Wuyts et al. 2012; Guo et al. 2012; Boada et al. in prep). Typically, as new stars tend to form from gas settled in a disk configuration, such $M/L$ ratio variations are anticipated to give rise to a composite light profile in which the disk component has a relatively larger weight (per unit mass) than the bulge. It is therefore important to address to which degree the above structural distinction between the two classes of galaxies is intrinsic to their distribution of stellar mass, or, conversely, can be attributed to stellar population effects. The answer to this question is of immediate relevance to our understanding of quenching, as in principle the latter scenario could imply that compact quiescent systems can evolve from the star-forming main sequence by simple fading, without invoking an associated morphological transition. Kriek et al. (2009) investigate this scenario for a spectroscopically confirmed sample of massive $z \sim 2$ galaxies, finding that 3 out of 6 massive star-forming systems have dense cores, and thus may passively evolve into compact galaxies due to fading of the outer star-forming regions. Szomoru et al. (2010), on the other hand, exploit the exquisite depth of the Hubble Ultra Deep Field to probe the surface brightness profile of a massive compact quiescent galaxy at $z = 1.91$, ruling out the existence of a faint extended envelope or disk around the observed galaxy. Another argument against fading comes from Cheung et al. (2012), who derived the stellar masses of bulges for both star-forming and quiescent galaxies at $0.5 \leqslant z < 0.8$. They found that bulges of SFGs are half as massive as those of similar-mass quiescent galaxies, implying they cannot simply fade onto the red sequence without structural evolution.
Using the stellar mass maps reconstructed for our mass-selected sample of 6764 galaxies at $0.5 < z < 2.5$ with $\log(M) > 10$, we are now able to draw statistically significant conclusions on the structural distinction between high-z galaxies prior to and after quenching. In Figure \[struc.fig\], we compare the shape of the stellar mass distributions (i.e., corrected for spatial $M/L$ variations) of star-forming and quiescent galaxies, and study their dependence on the total galaxy stellar mass. We consider profile parameters based on single-component (i.e., ) fits as well as two-component (bulge + disk) decompositions, and show the results for two separate redshift intervals: $0.5 < z < 1.5$ and $1.5 < z < 2.5$. In both cases, we identified a galaxy as quiescent if its specific SFR (sSFR) satisfied $sSFR < \frac{1}{3 t_{Hubble}}$, where $t_{Hubble}$ is the Hubble time at the redshift of the galaxy, and as star-forming otherwise. We tested that a definition of quiescence based on the location of a galaxy in the UVJ diagnostic diagram (Wuyts et al. 2007; Williams et al. 2009) yields effectively identical results.
Figure \[struc.fig\] immediately highlights that the distinct structural appearance of star-forming and quiescent galaxies is intrinsic to its internal distribution of stellar mass, and not just driven by stellar population or obscuration effects. In fact, a comparison to the equivalent plots based on $H$-band surface brightness profiles rather than mass maps (see Appendix B) indicates that stellar population effects (when measuring at rest-optical wavelengths) only induce a modest, albeit non-negligible shift. At all masses, quiescent galaxies feature cuspier stellar mass distributions (i.e., higher $n$) than star-forming systems. Their typical best-fit index is furthermore an increasing function of galaxy mass. Interestingly, also among SFGs the profile shape is not independent of stellar mass. An increase in $n$ is apparent above $10^{11}\ M_{\sun}$, both at $z \sim 1$ and at $z \sim 2$. A similar trend of increasing cuspiness at the tip of the MS was noted by Wuyts et al. (2011, Figure 1; see also Nelson et al. in prep).
Next, it is worthwhile reflecting on what it is that we measure when fitting profiles. Appendix A illustrates that, when considering galaxies as superpositions of bulge and disk components, a given best-fit index does not necessarily correspond one-to-one to a unique $B/T$ value, even though it is often interpreted as such. Given a bulge+disk system with associated best-fit $n$, one can increase its $n$ either by boosting $B/T$, or, alternatively, by growing the extent of the disk with respect to that of the bulge without any change to $B/T$.
Turning to the right-hand panel of Figure \[struc.fig\], we now explore the $B/T$ ratio as a function of galaxy mass, for SFGs and quiescent galaxies separately. Again, we find a clear anti-correlation between star formation activity and bulge prominence. Focussing on the star-forming population, the median $B/T$ is limited to below 30% for intermediate mass SFGs ($10 < \log(M) < 11$), while typical bulge mass fractions rise to 40-50% above $10^{11}\ M_{\sun}$. We note that there is a significant scatter in the distribution of individual $B/T$ values around the median for both quiescent and star-forming galaxies. We investigated the variation in median trends when varying the binning intervals, finding negligible changes at lower masses, while the median $B/T$ of the most massive ($\log(M) \sim 11.3$) SFG bin changes by $\pm 0.1$, depending on the applied binning intervals[^6].
We note that measurements on the $H$-band yield bulge mass fractions among SFGs that are lower by on average $\sim 30\%$, as can be understood from a disk component composed of a younger, lower $M/L$ stellar population than the bulge.
{width="\textwidth"}
From the two-component fits, we infer a typical $R_{e,B}/R_{e,D}$ size ratio of $\sim 0.2$, albeit with significant scatter (see Figure \[n\_bd.fig\]). The median size ratio shows little dependence on star formation activity or mass, over the range probed by our sample. Given the enhanced $B/T$ values in quiescent galaxies, and the fact that bulges have smaller half-mass radii than disks, one may wonder if the difference in total size between SFGs and quiescent galaxies can be accounted for completely by a redistribution of stellar material from the disk to the bulge, without changing the extent of each of the components individually. Our analysis confirms that the change in $B/T$ of SFGs prior or during quenching is to a large extent responsible for the size difference between the quiescent and star-forming population. However, some fraction of the shrinking size is still attributed to the individual components being smaller. In Figure \[Radii.fig\], we show the total sizes as well as the sizes of the individual components for star-forming and quiescent galaxies, as measured on the mass maps. While the total sizes of SFGs and QGs are noticeably different (by a factor $\sim 3$ at $\log(M_*) \sim 10.5$), the difference in size of bulge and disk components between SFGs and QGs respectively is smaller, typically by a factor $\sim 1.5$.
{width="80.00000%"}
Fraction of Quenched galaxies {#f_quench.sec}
-----------------------------
With the morphological parameters of the mass maps for our entire galaxy sample in hand, we now proceed to relate those with galaxy-integrated star formation properties.
The three panels of Figure \[fquench\_all.fig\] show, from left to right, the fraction $f_{quench}$ of quenched galaxies as a function of total stellar mass, bulge mass, and disk mass, respectively. Here, we again define galaxies as quenched/quiescent when $sSFR < 1/3*t_{Hubble}(z)$, and as star-forming otherwise.
The uncertainties in $f_{quench}$ are derived via a bootstrapping method and represent the 68% confidence levels. They include both sample variance and the typical measurement errors on $B/T$, which are derived as described in Section 3.2. For the bootstrapping, we computed $f_{quench}$ for 1000 samples, which are randomly drawn from the original sample, with replacement. For each bootstrap iteration, we displace the $B/T$ values for each galaxy by the typical measurement error in $B/T$, given the galaxy’s magnitude, size and measured profile shape.
The top left panel of Figure \[fquench\_all.fig\] illustrates that the fraction of quenched galaxies increases with increasing mass, from $\sim 0.1$ at around $10^{10} M_{\odot}$ to $\sim 0.5$ at $2 \times 10^{11} M_{\odot}$. A second conclusion is that the fraction of quenched galaxies is overall higher for the lower redshift bin, by a factor of $\sim2$ on average. Both of these results are well established in the literature. The rising mass function of quiescent galaxies over cosmic time has most recently been quantified on a firm statistical footing by Muzzin et al. (2013) and Ilbert et al. (2013), both of which exploit the wide-area UltraVISTA survey. What CANDELS lacks in number statistics compared to UltraVISTA, it adds in depth and high resolution. Exploiting these key strengths, we now turn to the dependence of the quenched fraction on galaxy subcomponents: the mass of their bulge (middle panel) and disk (right-hand panel). Clearly, the dependence of $f_{quench}$ on the bulge mass is much stronger than on the disk mass, which does not show any significant correlation with $f_{quench}$ above $\log(M_{Disk}) \sim 9.5$. Towards lower disk masses, $f_{quench}$ increases rapidly, but we point out that this trend is entirely driven by the (total) stellar mass limit of our sample ($\log(M_*) > 10$; i.e., the galaxies occupying the lowest $M_{Disk}$ bins are necessarily heavily bulge-dominated systems, that tend to form relatively few stars). If lower mass galaxies were to be included, less massive, disk-dominated SFGs would likely outnumber these massive spheroids with small residual disks in the low $M_{Disk}$ bins, producing a flat relation of $f_{quench}$ with $M_{Disk}$ over the full range probed. Above respective masses of $10^{10}\ M_{\sun} $, $f_{quench}$ increases more rapidly with bulge mass than with total stellar mass in both redshift ranges ($\sim 0.35$ per dex of $M_{Bulge}$ compared to $\sim 0.3$ per dex $M_*$, or $\sim 0.1$ per dex of $M_{Disk}$).
With the bulge-to-disk decompositions in hand, we next split the galaxy sample in bins of $B/T$, and explore second parameter dependencies. Considering first the dependence of $f_{quench}$ on the total stellar mass, it is apparent that, at a given total mass, $f_{quench}$ is increasing significantly with increasing $B/T$ ratio. The middle panels of Figure \[fquench\_all.fig\] illustrate that, when considering the dependence of $f_{quench}$ on bulge mass, the different $B/T$ bins align along a much tighter locus. In contrast, a large spread is seen as a function of disk mass (right-hand panels of Figure \[fquench\_all.fig\]). In order to quantify these trends, we compute the Spearman’s rank correlation coefficient ($r_{s}$) for the relations of $f_{quench}$ with $\log(M_*)$, $\log(M_{Disk})$, and $\log(M_{Bulge})$ for respective masses $\log(M) > 10$. We find that $r_s$ is indeed significantly higher for the relation $f_{quench}$ vs. $\log(M_{Bulge})$ ($r_{s} \sim 0.68$) than for both $f_{quench}$ vs. $\log(M_{*})$ ($r_{s} \sim 0.32$) and $f_{quench}$ vs. $\log(M_{Disk})$ ($r_{s} \sim -0.05$), as measured for $z \sim 1$. Consistent results are found for $z \sim 2$.
We investigated the impact of defining quiescence based on a $UVJ$ color-color criterion instead of a sSFR cut. When applying a UVJ-based selection of quiescent galaxies, we find an overall good agreement with the trends presented in Figure \[fquench\_all.fig\]. Quantitatively, small changes occur, with $f_{quench}$ increasing by $ \sim 7$% for the entire $0.5 < z < 2.5$ sample integrated over all masses. The good agreement is not surprising, since the precise threshold in sSFR used to select quiescent galaxies ($sSFR < 1/3*t_{Hubble}(z)$) was chosen to yield maximum overlap with the $UVJ$ selection criterion.
Taken together, this demonstrates that the build-up of a bulge seems to play a critical role in the quenching process of galaxies, whereas the disk does not. The amount of stars in the disk component of a galaxy has little to no predictive power regarding its star-forming or quenched state, unless also $B/T$ (and hence the bulge mass) is known. We find a qualitatively similar behavior at $z \sim 2$ as at $z \sim 1$, but note that the cosmic evolution in the quiescent fraction cannot solely be attributed to continuing bulge growth over time, as galaxies in the same $M_*$ and $B/T$ bin at $z \sim 1$ are more likely to be quenched than those at $z \sim 2$. Appendix B illustrates how the equivalent diagrams composed from fits to the $H$-band surface brightness rather than the stellar mass distribution exhibit a larger spread from low to high $B/T$ bins. This generic behavior can be understood from a physical picture where the disk component has a relatively larger weight in light than in mass.
Our work is in agreement with, and takes the next step beyond previous reports that the inner stellar mass density is better related to the star formation history than the total stellar mass [@franx2008; @bell2012], as inferred from rest-optical imaging of smaller samples of high-redshift galaxies (see Kauffmann et al. 2003 and Fang et al. 2013 for a local universe reference, and Cheung et al. 2012 for intermediate redshifts $z<0.8$)
Importantly, the same behavior explored here over the redshift range $0.5 < z < 2.5$ extends in a strikingly similar fashion all the way to the present day, as demonstrated by Bluck et al. (2014) who exploit the large number statistics of SDSS.
Comparison with SAMs {#SAM_results.sec}
====================
The Somerville model {#somerville.sec}
--------------------
Semi-analytic models (SAMs) have a rich history of trying to reproduce galaxy scaling relations and abundances, with the goal of guiding our interpretation of the observational results. Here, we focus specifically on the SAM developed by Somerville et al. (2008) and further updated by Somerville et al. (2012) and Porter et al. (2014), which is rooted in the Bolshoi cosmological dark matter simulation (Klypin et al. 2011).[^7] As is generic to all SAMs, the model relies on simplified analytic prescriptions for the dynamical and astrophysical processes down from entire galaxy scales, rather than on kiloparsec to parsec scales (the resolution below which state-of-the-art cosmological and zoom-in hydro-simulations resort to subgrid physics, respectively). This limitation, however, yields the enhanced flexibility of a relatively inexpensive runtime, allowing the straightforward generation of statistically significant model galaxy populations, and the tuning of parameters to observational constraints such as mass functions and scaling relations (only empirical constraints from the nearby universe were used in tuning the parameters of the model considered here). The fact that SAMs conceptually are formulated in units of bulge and disk components furthermore makes them suitable for a direct and meaningful comparison to the diagnostics explored in this paper.
A detailed description of the prescriptions for cooling, star formation, feedback, and structural growth is provided by Somerville et al. (2008, 2012), with extensions and applications to the CANDELS data set presented by Porter et al. (2014). The input to and output from the model is further contrasted to that of other SAMs by Lu et al. (2013).
For a detailed discussion of the physical recipes of this SAM and the resulting output in the context of a larger set of SAMs, we refer the reader to Lu et al. (2013). For the sake of the comparison presented here, we emphasize that none of the relations investigated in this paper formed part of the set of observational constraints to which the model parameters were tuned. Model parameters of the SAM were tuned to (approximately) match the global stellar mass function, the stellar mass function of early- and late-type galaxies, the gas fraction as a function of stellar mass for disks, and the mass-metallicity relation for stars.
Also of particular relevance is the fact that, in the model, stars form either in the disk following a Kennicutt-Schmidt law (Kennicutt 1998), where the disk scalelength is set by a similar methodology as Mo, Mao & White (1998), or, in the event of a merger, during a starburst. Bulge formation as well as feeding of the central supermassive black hole can happen through two channels: mergers or disk instabilities (see Porter et al. 2014). The starburst, black hole accretion and morphological transformation induced by mergers depends on the mass ratio and gas fraction, as calibrated using a large suite of binary merger simulations (Somerville et al. 2008; Hopkins et al. 2009; Somerville et al. 2012). Star formation is moderated through heating of gas by supernovae as well as through AGN feedback. In addition to the quasar mode, during which AGN can drive powerful outflows, black holes also grow more gradually over longer timespans through the so-called radio mode (i.e., suppression of cooling via radio jets). No explicit connection between the bulge mass and quenching (as may for example be expected from the Toomre Q stability criterion in a gravitational quenching scenario, see Section 6.2) was built into the model.
![Same as Figure \[fquench\_all.fig\] using galaxies from the Somerville et al. SAM. Galaxies with $sSFR > 1/3*t_{Hubble}$ are assigned as star-forming, the others as quiescent. Uncertainties are derived via bootstrapping to reflect the sample variance. \[SAM\_Somerville1.fig\]](f4.eps){width="49.00000%"}
$f_{quench}$ in the SAM {#fquench_SAM.sec}
-----------------------
In Figure \[SAM\_Somerville1.fig\], we show the model equivalent of Figure \[fquench\_all.fig\], describing how the fraction of galaxies that are quenched depends on the total stellar mass, the mass of stars in the bulge, and the mass of stars in the disk component respectively. As for the observations, we define the threshold for a galaxy to be quenched with a ruler moving with redshift: $sSFR < 1 / (3 t_{Hubble})$. This definition, rather than an application of the UVJ diagnostic, remains closer to the direct output of the SAM, avoiding a translation to mock spectral energy distributions, which would introduce additional assumptions and uncertainties [^8].
At first glance, the SAM features several of the characteristic trends noted earlier for the CANDELS galaxies: $f_{quench}$ rises towards later cosmic times, increases with the total stellar mass, more steeply so with bulge mass, and shows no appreciable correlation with disk mass above $\log(M_{Disk}) = 9.5$. In the interval $9 \lesssim \log(M_{Disk}) \lesssim 9.5$, a sharp drop with increasing $M_{Disk}$ is noted, as also seen in the observations. Given the $M > 10^{10}\ M_{\sun}$ threshold of our sample selection, the latter objects are necessarily heavily bulge-dominated. Without imposing such a mass limit, less massive, disk-dominated SFGs would outnumber massive early-type galaxies with small residual disks in the lower $M_{Disk}$ bins. Despite the qualitative success, quantitative differences in the quenched fractions of model galaxies with respect to those observed are clearly present. The discrepancy is most severe at $z \sim 2$, where modeled $f_{quench}$ values are, on an average over the whole displayed mass range, of order a factor $\sim 2 - 3$ short of observed, hinting at an underestimated quenching rate and/or inefficiency to prevent quiescent systems from rejuvenating[^9].
When splitting the SAM galaxies at each $M_*$, $M_{Bulge}$, and $M_{Disk}$ in bins of $B/T$ (middle and bottom panels of Figure \[SAM\_Somerville1.fig\]), we reproduce a similar behavior as found for the real universe in Section \[f\_quench.sec\]. Namely, the total stellar mass acts as a poorer predictor of the quenched state of a galaxy. This situation can be remedied if in addition to $M_*$ also $B/T$ (and hence $M_{Bulge}$) is known. Quantitatively, the correlation between $f_{quench}$ and $M_{Bulge}$ is measured to be the strongest ($r_s \sim 0.46$ ), whereas $M_*$ and $M_{Disk}$ only show weak correlation with $f_{quench}$ ($r_s \sim 0.21 $ and $r_s \sim -0.1$, respectively). The quoted values of $r_s$ are measured for respective masses of $\log(M) > 10 $ and at $z \sim 1$. The values for $r_{s}$ at $z \sim 2$ are similar, with the correlation between $f_{quench}$ and $\log(M_{*})$ as well as between $f_{quench}$ and $\log(M_{Disk})$ being somewhat stronger. At $0.5 < z < 1.5$, less than 20% of all massive galaxies with $B/T < 0.2$ are classified as quiescent. Conversely, the majority of galaxies in the upper $B/T$ bin (with $B/T > 0.8$) have low sSFR. These inferences are in common between the SAM and the observations. Also in agreement, is the fact that $M_{Bulge}$ serves as a better predictor of $f_{quench}$ than the total stellar mass, with different $B/T$ bins being more (albeit not perfectly) aligned along a single locus in the $f_{quench}$ versus $M_{Bulge}$ diagram.
At $1.5 < z < 2.5$, the model predictions are skewed towards too low $f_{quench}$ values, as noted earlier. However, in relative terms the same generic behavior as a function of bulge prominence is notable.
We note that most of the trends in Figure \[SAM\_Somerville1.fig\] are driven by physical prescriptions in the SAM affecting central galaxies rather than satellites, as centrals account for 80% (90%) of the model galaxy population above $log(M) = 10$ (11). Those massive galaxies classified as satellites are further subjected to additional environmental quenching processes, resulting in a higher $f_{quench}(M_*,M_{Bulge})$ for this particular subpopulation.
The agent of quenching {#agent_SAM.sec}
----------------------
![$f_{quench}$ as a function of halo and black hole mass using galaxies from the Somerville et al. SAM. In the middle and bottom rows, the galaxy sample has been split in bins of $B/T$. Galaxies with $sSFR > 1/3*t_{Hubble}$ are assigned as star-forming, the others as quiescent. Uncertainties are derived via bootstrapping to reflect the sample variance. \[SAM\_Somerville2.fig\]](f5.eps){width="45.00000%"}
Given the qualitative agreement between model and observations, we can now pose the question how, in the context of the Somerville SAM, the relation between structure and stellar populations could be interpreted. To this end, we consider the dependence of $f_{quench}$ on two physical properties of galaxies in the SAM that are not observationally accessible for our CANDELS sample: the halo mass $M_{Halo}$ and the mass of the central supermassive black hole $M_{BH}$. The top panels of Figure \[SAM\_Somerville2.fig\] illustrate that the probability of a galaxy being quenched increases towards high $M_{Halo}$ and high $M_{BH}$. In detail, however, the dependencies on the two look different. The increase in $f_{quench}$ with $M_{Halo}$ is gradual over nearly two orders of magnitude. In contrast, a much sharper upturn of $f_{quench}$ emerges above $\log(M_{BH}) = 7.5$. This behavior is especially notable at $z \sim 1$, but a rise above the same threshold is present in the $z \sim 2$ population as well. Breaking the model galaxy population down by its structural properties, we find a wide spread in $f_{quench}$ for different $B/T$ at a given $M_{Halo}$. Naively, a different, less scattered behavior would be expected if halo mass quenching were the sole and dominant mechanism (Birnboim et al. 2007; Dekel & Birnboim 2008; Dekel et al. 2009). As a function of $M_{BH}$, on the other hand, a similar upturn in $f_{quench}$ is present for all $B/T$ bins equally above $\log(M_{BH}) = 7.5$.
The increased scatter in the relation between $f_{quench}$ and $M_{Bulge}$ compared to the tight correlation of $f_{quench}$ and $M_{BH}$ can be explained by the SAM’s $M_{BH}$ - $M_{Bulge}$ relation. For a given $B/T$ bin, the scatter in the $M_{BH}$ - $M_{Bulge}$ relation is significant compared to the dynamic range in the bulge masses plotted. At a given bulge mass, the scatter stems in part from an anti-correlation between the black hole mass and the level of star formation activity (sSFR).
Our analysis illustrates that in the Somerville model, which includes feedback from both supernovae and AGN, the central supermassive black hole acts as the primary agent of quenching in massive galaxies, and its accumulated mass (i.e., the integral over past accretion activity) is tightly related to the probability of finding a galaxy in a quenched state. Since the physical processes giving rise to bulge and black hole growth are the same in the SAM (mergers and disk instabilities), the stronger relation of $f_{quench}$ with $M_{Bulge}$ than with $M_*$, present in both observations and model predictions, is not surprising.
We stress that in the Somerville model, no direct causal link between the presence of a bulge and quenching is implemented. The bulge is simply the accessible observable that correlates most tightly with the actual agent of quenching in this particular model: the supermassive black hole. Observationally, there is increasing evidence for AGN-driven outflows of massive $z\sim 1-3$ galaxies (Förster Schreiber et al. 2013; Nesvadba et al. 2011; Harrison et al. 2012; Cano-D[í]{}az et al. 2012). Given the shortcomings of the SAM in a quantitative sense, notably its underprediction of the quiescent population at $z \sim 2$, additional or other quenching processes may be at play in the real universe. One such process, that is causally linked to the presence of a bulge, could be morphological quenching (Martig et al. 2009; Genzel et al. 2014). In such a scenario, the high central stellar density provided by a bulge, stabilizes the gas disk and prevents it from forming stars. While plausibly only a temporary measure (as no gas is expelled, nor stopped from accreting through this mechanism), it could potentially contribute to suppressing star formation in $z \sim 2$ galaxies more efficiently and/or preventing them from returning to the star-forming branch in the SAM. In the local universe, star formation efficiencies of bulge-dominated systems are reduced by factors of $\sim 2 -3$ compared to disk-dominated galaxies (Saintonge et al. 2012; Martig et al. 2013).
Discussion {#discussion.sec}
==========
Structural change
-----------------
In order to study the morphological differences between SFGs and quiescent galaxies, and to draw conclusions on the possible structural changes of star-forming galaxies as they move along the MS, we first examined the mass dependence of the profile shape of SFGs and quiescent galaxies as traced by the index and $B/T$ ratio. We have shown that quiescent galaxies are structurally distinct from the star-forming population as seen by overall higher indices and $B/T$ ratios at a given stellar mass. SFGs show rising trends of their median index and $B/T$ ratio with increasing stellar mass, with the latter rising up to $\sim 40-50 \%$ above $10^{11}\ M_{\sun}$. These findings give insights about the link between the structural evolution of SFGs and the quenching process as they move along the MS.
Analyzing the Schechter functional forms of the SFG and QG mass function as a function of redshift, [@peng2010] conclude that the quenching rate of galaxies climbing the MS rises proportionally to the SFR (and given the near-linear MS slope therefore also proportionally to the stellar mass, hence their terminology ’mass quenching’). This corresponds to a survival probability on the MS that drops exponentially with mass, implying that, while nearly all low-mass SFGs are destined to continue growing along the MS, toward the high-mass end the MS becomes progressively more dominated by near-to-be-dead SFGs. In fact, the sub-unity slope of the MS, and possible flattening at the high-mass end (Whitaker et al. 2012), may well be interpreted in this context: the typical SFG above $10^{11}\ M_{\sun}$ is already undergoing some level of quenching, thereby deviating from the projected path along a SFR-Mass relation of slope unity, that could be expected from cosmological accretion rates in the absence of quenching. Tying in our observational results on galaxy structure, the deviation toward high median $n$ and $B/T$ at the massive end reflects the typical structure of soon-to-be-dead star-formers that account for the bulk of SFGs above $10^{11}\ M_{\sun}$. The fact that they are structurally distinct implies that the morphological transition happens first, to be followed later by the departure from the MS. Bulge growth precedes quiescence. Such a morphological change prior to quenching is in line with qualitative predictions based on a toy model by Dekel & Burkert (2013). In the latter study, about half of the star-forming disk galaxies at $z\sim2$ are predicted to evolve into compact star-forming ’blue nuggets’ due to violent disk instabilities before they are quenched into compact quiescent galaxies (’red nuggets’). An observed population of ’blue nuggets’ has been proposed by Barro et al. (2013a,b) to represent an evolutionary link, originating from extended disk galaxies, and evolving into compact quiescent systems.
This does not refute that galaxies also undergo further structural evolution after they are quenched. At least part of the size growth (Cassata et al. 2013; van der Wel et al. 2014) and evolution toward rounder axial ratios (Chang et al. 2013) has been attributed to (minor and/or major) dry mergers, and it is conceivable that similar processes contribute to the observed trend of increasing $B/T$ toward the massive end for the quiescent population.
AGN as the driver of quenching ?
--------------------------------
We have demonstrated that the bulge mass of a system is well correlated with its quenched state and has a stronger predictive power of quiescence than the total stellar mass. The observed trends of $f_{quench}$ with total stellar mass, bulge mass and disk mass as viewed among galaxies in different $B/T$ bins are in good qualitative agreement with predictions from the Somerville et al. SAM. In the context of this model, the growth of the central supermassive black hole, which is the primary quenching agent for massive galaxies in this SAM, is tightly coupled with the growth of bulges through both merging and disk instabilities.
If a black hole - bulge scaling relation is in place during the peak of cosmic star formation as it is in the present-day universe (Häring & Rix 2004), our observational results together with the model comparison could therefore hint at the bulge not being the causal link to quenching, but rather the most accessible observational proxy for the AGN acting as the quenching agent[^10]. In detail, however, there are quantitative differences between the SAM and our observations, most severely in the highest redshift bin ($1.5 < z < 2.5$), where the observed quenched fraction exceeds the value predicted by the SAM by a factor of $\sim 3.5$. The latter difference could hint at a need for more frequent, efficient, or lasting quenching, a possible mechanism we speculate about below. We also note that the same behavior is not necessarily a generic feature to all SAMs (see Appendix C).
It is tempting to draw connections between the emerging bulges in massive MS galaxies out to $z \sim 2.5$ revealed by our analysis, and recent observational results based on deep AO-assisted integral field data sets and grism spectroscopy over the same redshift range. Förster Schreiber et al. (2013) found a high prevalence of powerful nuclear outflows in $\log(M) > 11$ galaxies driven by AGN, which appear to be absent in galaxies at lower masses. Along with star formation driven winds in the outer parts of the galaxies, such outflows could efficiently remove gas out of galaxies and, in this way, contribute to the quenching process.
Meanwhile, the 3D-HST and CANDELS legacy programs have yielded evidence for nuclear depressions in the H$\alpha$ equivalent width in $z \sim 1$ SFGs (Nelson et al. 2012, 2013; Wuyts et al. 2013). At the highest stellar surface mass densities, star formation no longer appears to proceed in lockstep with the assembled stellar mass. Likewise, Genzel et al. (2014) report on ring-shaped H$\alpha$ distributions in $z \sim 2$ SFGs, surrounding a more quiescent center where the dynamically inferred Toomre Q parameter significantly exceeds unity, owing to the emergence of a stellar bulge. As such, the Toomre stability criterion is satisfied in the central galaxy regions, which consequently could prevent the gas reservoir, if present there, from fragmenting and forming stars. While this result suggests that some causal connection between bulge growth and quenching may be at play, it should be noted (as is done also by Genzel et al. 2014) that gravitational quenching by itself does not expel the gas present, neither does it stop the accumulation of a larger gas reservoir by continuing cosmological accretion. Additional maintenance mode might be required for a long-term shut-down of further gas supply.
Conclusions {#conclusions.sec}
===========
We analyzed the structural properties of a sample of 6764 massive ($ > 10^{10}\ M_{\sun}$) galaxies in the redshift range $ 0.5 < z < 2.5$, by exploiting the multi-wavelength CANDELS HST imaging data set in all five CANDELS/3D-HST fields. We carried out single-component () fits and two-component (bulge + disk) decompositions, on stellar mass maps reconstructed from a resolved panchromatic SED modeling technique (Wuyts et al. 2012, 2013), as well as on images of the $H$-band surface brightness distribution. In addition, we compared our findings to predictions by the state-of-the-art semi-analytic model from Somerville et al. (2008, 2012; with extensions including disk instabilities presented by Porter et al. 2014). Our main results are the following:
1. At fixed stellar mass, quiescent galaxies have overall higher indices and $B/T$ ratios than SFGs as measured on their mass maps, in line with previous findings using monochromatic observations. We find an increase of indices among SFGs with increasing total stellar mass, with the median mass profiles increasing from ($n\sim1.3$) at $10^{10}\ M_{\sun}$, to $n \gtrsim 2$ above $10^{11}\ M_{\sun}$. Two-component bulge-disk decompositions confirm that the same rising trend is present when considering the median $B/T$ ratio of SFGs, which is rising up to $\sim 40-50 \%$ above $10^{11}\ M_{\sun}$. The same characteristic behavior is seen at $z \sim 1$ and $z \sim 2$.
2. Quantifying the same trends on the H-band light profiles rather than the mass maps, the indices and $B/T$ fractions are overall lower for SFGs, confirming previous non-parametric measurements for a subset of our sample (Wuyts et al. 2012). The emergence of bulges above $10^{11}\ M_{\sun}$ in SFGs appears to be also slightly less prominent when viewed in light, consistent with the steepest color gradients (blue disks with red central bulges) being found among massive SFGs.
3. The likelihood of a galaxy being quenched, as traced by the fraction of quiescent galaxies, is better correlated with the bulge mass than the total stellar mass and further shows no appreciable correlation with the amount of stellar mass in the disk component. The quenched fraction at redshift 1 is on average higher by a factor $\sim 2$ than at redshift 2.
4. At a given total stellar mass, the quenched fraction exhibits a strong positive correlation with $B/T$, while different $B/T$ bins are confined to a significantly tighter locus in a diagram of $f_{quench}$ versus $M_{Bulge}$. These findings imply that the bulge mass of a system is the single observable parameter with the most predictive power regarding its quenched state, although a somewhat tighter constraint on the probability of quiescence can be obtained if in addition also the total stellar mass is known. The same trend is seen over the full redshift range probed, with the distinction that quenched fractions are lower at higher lookback times.
5. We find a good qualitative agreement between the semi-analytic model by Somerville et al. SAM and our observational findings. Since bulge and black hole growth are tightly coupled in the SAM, the strong dependence of $f_{quench}$ on bulge mass follows rather naturally in this model. Our observational results can [*in the context of this model*]{} therefore be interpreted as the bulge being the closest observable proxy to the underlying agent of quenching: the black hole. Quantitatively, the largest discrepancy between model and observations is found in the highest redshift bin ($1.5 < z < 2.5$), where the observed quenched fraction is larger by a factor of $\sim 3.5$ than predicted by the SAM. We note that the same behavior is not necessarily a generic feature to all SAMs.
{#section .unnumbered}
The authors acknowledge fruitful discussions with Edmond Cheung, David C. Koo, Yu Lu, Casey J. Papovich, Mohammadtaher Safarzadeh, Benjamin J. Weiner and Steven P. Willner. Support for Program number HST-GO-12060 and HST-GO-12177 was provided by NASA through a grant from the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Incorporated, under NASA contract NAS5-26555.
A. The meaning of a Sersic index measurement {#A.app}
============================================
{width="\textwidth"}
The combination of our one and two-component fits on the mass maps enables us to examine empirically how the index relates the amplitude of the bulge (as parametrized by $B/T$) on the one hand, and the relative size-ratio to the bulge and disk component ($R_{e,B}/R_{e,D}$) on the other hand. Figure \[n\_bd.fig\] illustrates that, while $B/T$ shows a clear correlation with $n$, there is no unique one-to-one translation between the two. Instead, the best-fit $n$ to a composite bulge+disk system additionally depends on the size ratio of the two components. In other words, a galaxy’s Sersic index could be increased by placing more material in the bulge, but also by growing the disk at fixed bulge size. The observed CANDELS galaxies occupy a surface in this three-parameter space ($n$, $B/T$, $R_{e,B}/R_{e,D}$) that is in good agreement with what would be anticipated from Sersic fits to idealized, noise-free bulge ($n = 4$) plus disk ($n = 1$) profiles (squares and curves in the left-and right-hand panels of Figure \[n\_bd.fig\], respectively). Evidently, for systems with $B/T$ close to 0 or 1, the size ratio of the two components is ill-constrained as one of them contains barely any mass. The galaxies in our full $0.5 < z < 2.5$ sample span the full range of $B/T$ values, and are located predominantly around bulge-to-disk size ratios of $R_{e,B}/R_{e,D} \sim 0.2$ .
B. Comparison with Measurements on H-band {#B.app}
=========================================
{width="49.00000%"} {width="49.00000%"}
Here, we investigate how the results of our structural analysis change when conducting the measurements on the $H$-band light images rather than on the stellar mass maps. To this end, we present in Figure \[light\_struc.fig\] the same figures as discussed in Section \[structure.sec\], now using the index and $B/T$ values as inferred from the CANDELS $H$-band imaging. Likewise, we compute the bulge and disk mass as $(B/T)_H M_*$ and $(1 - (B/T)_H) M_*$, respectively.
Overall, our analysis reveals a qualitatively similar mass dependence of $n$ and $B/T$, and distinction between SFGs and quiescent galaxies as inferred from the mass maps. In detail, however, modest changes in $n$ and $B/T$ are notable. While the median $z \sim 1$ (2) SFG has $(B/T)_H \lesssim 0.25$ (0.20), the typical bulge fractions increase to above 20%, and reach up to $\sim 40 - 50\%$ at the massive end, once spatial $M/L$ variations are corrected for. Likewise, the corresponding indices measured on $H$-band imaging for SFGs below $\log(M) = 10.8$ are consistent with exponential disk profiles (see also Wuyts et al. 2011), but are slightly cuspier as quantified on mass maps. This is in line with findings based on smaller subsets of CANDELS data by Wuyts et al. (2012) and Guo et al. (2012).
The central 50th percentile intervals marked by the red and blue polygons are somewhat less confined in the plots based on stellar mass maps compared to the $H$-band results. We interpret this to be due to an additional source of random uncertainty introduced by the resolved stellar population modeling. The resolved stellar population modeling itself was motivated by the need to reduce the systematic biases associated with spatial $M/L$ ratio variations.
Focussing on Figure \[light\_struc.fig\], the qualitative trends of $f_{quench}$ with total stellar mass, bulge and disk mass are very similar. However, the lower two rows of Figure \[light\_struc.fig\] compared to Figure \[fquench\_all.fig\] show that the bins of lowest and highest $B/T$ are more separated from each other in $f_{quench}$ in light than in mass. This observation too is in line with the disks of SFGs being dominating by a younger stellar population than that of the bulge, shifting SFGs to lower $B/T$. The middle panels of Figure \[light\_struc.fig\] show a larger scatter than the corresponding panels in Figure \[fquench\_all.fig\].
C. Comparison to Guo et al. (2013) SAM {#C.app}
======================================
{width="58.50000%"} {width="41.00000%"}
In order to evaluate to which extent the characteristic trends of the Somerville et al. SAM, as presented in Section \[SAM\_results.sec\], are generic to all semi-analytic models, we considered an independent semi-analytic model by Guo et al. (2013). The Guo et al. (2013) model is rooted in the dark matter backbone of the Millennium Simulation (Springel et al. 2005), and its output tables with galaxy properties for snapshots of different lookback times are publicly available on the Virgo Millennium Database (G. Lemson & the Virgo Consortium 2006). We present the equivalent plots of Figure \[SAM\_Somerville1.fig\] and \[SAM\_Somerville2.fig\] in Figure \[SAM\_Guo.fig\]. As in the Somerville model, quenched fractions are rising with increasing mass, increasing bulge mass, and decreasing redshift, but broken down in bins of $B/T$, significant differences are notable. Specifically, $f_{quench}$ does not monotonically increase with increasing $B/T$ at a given stellar mass.
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[^1]: The AGHAST G141 grism observations within GOODS-North from the GO-11600 program (PI: B. Weiner) are included in the analysis under ‘3D-HST’ (Brammer et al. 2012).
[^2]: Since the choice of a index value for the bulge profile has long been debated in the literature (see Kormendy & Kennicutt 2004 and references therein), we repeated our two-component fits, with the only change fixing $n=2$ for the bulge component to address the impact on the derived $B/T$ values. We find a median difference of $B_{n=4}/T - B_{n=2}/T = 0.03^{+0.15}_{-0.11}$ for quiescent galaxies and $0.01^{+ 0.09}_{-0.12}$ for SFGs, with the errors marking the 1$\sigma$ scatter. Any systematic trends are small, and our results are therefore robust against the precise value of $n_{bulge}$ adopted.
[^3]: We note that nuclear stellar disks in early-type galaxies do exist (e.g., Jaffe et al. 1994; van den Bosch et al. 1994; Ferrarese et al. 1994), but they are impossible to resolve at the HST resolution for $z \sim 2$ galaxies.
[^4]: In evaluating models with the AIC, the preferred model is the one that minimizes $\chi^2 + 2p + \frac{2p(p+1)}{N-p-1}$, where $p$ is the number of free parameters in the fit and $N$ is the number of data points used in the fit.
[^5]: UDS is part of CANDELS-Wide, which was exposed for two HST orbits divided over F125W and F160W. Part of our data set comes from the CANDELS-Deep regions (the central halves of the GOODS fields), which received 4 orbits per pointing in F125W and F160W each. The inferred uncertainties from our analysis of mock galaxies can therefore be considered as conservative estimates.
[^6]: The binning scheme applied in Figure \[struc.fig\] is such that the most massive bin still contains more than 10 galaxies, allowing a robust estimation of the median.
[^7]: Hereafter, we refer to this model as the Somerville et al. SAM.
[^8]: It should be noted that equivalent assumptions and uncertainties associated with the conversion from light to physical properties enters upon SED modeling of the observed galaxies. The choice of how far to take the models to the observations or visa versa therefore remains somewhat arbitrary.
[^9]: See also Ciambur et al. 2013 for a discussion on the quenched fraction in the Garching semi-analytic models.
[^10]: We note that Rosario et al. (2013) find X-ray signatures of AGN activity at these high redshifts to be most prominent among the star-forming population, most notably at the high-mass end, precisely where we see an upturn in the bulge fraction among SFGs.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We derive an alternative representation of the leading-order contribution to the polarization operator in strong-field quantum electrodynamics with a plane-wave electromagnetic background field, which is manifestly symmetric with respect to the external photon momenta. Our derivation is based on a direct evaluation of the corresponding Feynman diagram, using the Volkov representation of the dressed fermion propagator. Furthermore, the validity of the Ward-Takahashi identity is shown for general loop diagrams in an external plane-wave background field.'
author:
- 'S.'
- 'C. H.'
- 'A.'
title: 'Polarization operator for plane-wave background fields'
---
Introduction
============
The most precise calculations known so far in physics are provided by quantum electrodynamics (QED). The reason for this is the smallness of the fine-structure constant $\alpha = {{e^2}/{(4\pi{\epsilon}_0 \hbar c)}} \approx {{1}/{137}}$, which allows us to use perturbation theory [^1] ($e$ is the electron charge). The most prominent example is probably the electron $g$-factor, for which experimental and theoretical results have been matched on the record accuracy level of parts per billion [@gabrielse_new_2006]. To achieve this outstanding precision, the corresponding theoretical calculation included Feynman diagrams with up to four loops.
A quite different situation is encountered for QED with electromagnetic background fields. A source for strong electromagnetic fields are modern laser systems. If spatial focusing effects are sufficiently small, laser fields can be well approximated by plane-wave fields. For a plane-wave field, we obtain, besides the fine-structure constant, a second gauge and Lorentz-invariant parameter $\xi_0=|e|E_0/(mc\omega_0)$, where $E_0$ and $\omega_0$ are the peak electric field strength and central angular frequency of the plane wave, respectively \[$m$ is the electron mass; see also Sec. \[sec:planewavefields\]\]. If $\xi_0 \gtrsim 1$ the interaction between electron and positrons with the laser field must be taken into account exactly. For optical lasers (photon energy $\hbar\omega_0 \approx \unit[1]{eV}$), this happens already at intensities of the order of $\unitfrac[10^{18}]{W}{cm^2}$. More precisely, we can generally still treat the interaction of the electrons and the positrons with the quantized radiation field perturbatively as in vacuum QED (QED without background fields), but must include the dependence on $\xi_0$ to all orders if the threshold $\xi_0 \approx 1$ is exceeded.
Another important scale is the so-called critical field $E_{cr}=m^2c^3/(\hbar|e|)=1.3\times 10^{16}\;\text{V/cm}$, which corresponds to a peak laser intensity of $I_{\mathrm{cr}}={\epsilon}_0 c E_{\mathrm{cr}}^2 =\unitfrac[4.6\times 10^{29}]{W}{cm^2}$. A constant and uniform electric field of this strength can, in principle, produce electron-positron pairs from the vacuum [@sauter_ueber_1931; @heisenberg_folgerungen_1936; @schwinger_gauge_1951].
The current laser intensity record (in the optical regime) is given by $\unitfrac[2\times 10^{22}]{W}{cm^2}$ [@yanovsky_ultra_2008] and future facilities envisage even intensities of the order of $\unitfrac[10^{24}-10^{25}]{W}{cm^2}$ [@ELI; @HIPER; @XCELS]. Thus, the nonperturbative regime (in $\xi_0$) can be entered with presently available laser systems, and even the critical field can be reached in the rest frame of an ultrarelativistic particle (e.g., a $\sim \unit[1]{GeV}$ electron [@leemans_gev_2006]).
So far only one experiment has been carried out to probe strong-field QED effects using laser fields [@bula_observation_1996; @burke_positron_1997]. However, this is expected to change in the near future and, correspondingly, the experimental progress has stimulated many theoretical investigations during the last years [@di_piazza_quantum_2010; @hu_complete_2010; @king_matterless_2010; @mackenroth_determining_2010; @dumlu_schwinger_2010; @heinzl_beam-shape_2010; @heinzl_finite_2010; @fedotov_limitations_2010; @bulanov_schwinger_2010; @sokolov_pair_2010; @meuren_quantum_2011; @hu_relativistic_2011; @kryuchkyan_bragg_2011; @hartin_high_2011; @boca_thomson_2011; @dumlu_interference_2011; @hebenstreit_pair_2011; @elkina_qed_2011; @nerush_laser_2011; @ilderton_pair_2011; @ilderton_trident_2011; @labun_spectra_2011; @monden_enhancement_2011; @redondo_light_2011; @seipt_nonlinear_2011; @seipt_two-photon_2012; @nousch_pair_2012; @titov_enhanced_2012; @king_photonphoton_2012; @dobrich_magnetically_2012; @dinu_infrared_2012; @harvey_radiation_2012; @krajewska_compton_2012; @king_pair_2012; @mackenroth_nonlinear_2013; @king_photon_2013]. For a more detailed overview, the reader is referred to the review [@di_piazza_extremely_2012].
In contrary to vacuum QED, calculations with a plane-wave background field have not been carried out beyond the one-loop order (for constant-crossed fields higher-order calculations have been performed, see, e.g., [@ritus_radiative_1972; @ritus_vacuum_1972; @ritus_1985]). This can be attributed to the fact that already diagrams with just a few propagators correspond to quite complicated expressions. It is therefore of general interest to investigate new techniques, which have the potential to make also the calculation of complicated diagrams tractable.
Here we present a new derivation of the first-order contribution to the polarization operator given in Fig. \[fig:polarizationoperator\] [@baier_interaction_1975; @becker_vacuum_1975]. In Ref. [@baier_interaction_1975] an operator approach similar to the one introduced by Schwinger [@schwinger_gauge_1951] has been used. We show here how the diagram in Fig. \[fig:polarizationoperator\] can be evaluated directly using the Volkov representation of the dressed propagators. This approach has the appealing feature that it is very similar to the calculation techniques used in vacuum QED. Our final result (which is equivalent to the one in Ref. [@baier_interaction_1975]) has the interesting property that it is manifestly symmetric with respect to the external photon four-momenta $q_1$ and $q_2$ (see Fig. \[fig:polarizationoperator\]). Furthermore, we prove the validity of the Ward-Takahashi identity [@ward_identity_1950; @takahashi_generalized_1957] for general loop diagrams in a plane-wave background field. The calculation techniques employed here are expected to be useful also for other higher-order calculations with strong plane-wave background fields.
The polarization operator itself is of central importance, since it determines the properties of a photon inside the background field via the Schwinger-Dyson equation for the exact photon propagator [@baier_interaction_1975; @landau_quantum_1981]. As a consequence, a plane-wave field acts as an active medium, e.g. the photon obtains a mass and has a nontrivial dispersion relation. Furthermore, due to the unitarity of the $S$-matrix, the imaginary part of the polarization operator is related to the total photo-production probability for an electron-positron pair (see Fig. \[fig:polarizationoperator\]) [@milstein_polarization-operator_2006; @di_piazza_barrier_2009]. The polarization operator is also required for the calculation of radiative corrections to elementary processes like nonlinear Compton scattering or pair production. The significance of the photon polarization tensor can also be inferred from the ongoing effort to calculate it for different field configurations [@gies_vacuum_2011; @karbstein_optical_2012].
![\[fig:polarizationoperator\] The Feynman diagram corresponding to the leading-order contribution to the polarization operator $\mathcal{P}^{\mu\nu}(q_1,q_2)$ in a plane-wave background field. The double lines represent the Volkov propagators for the fermions, which take the external field into account exactly \[see Eq. (\[eqn:dressedpropagator\])\]. The vertical dashed line links the polarization operator to the pair-production diagram due to the unitarity of the $S$-matrix.](Fig1){height="2.1cm"}
The paper is divided into three parts. In Sec. \[sec:strongfieldqed\] the general framework of strong-field QED with plane-wave background fields is described. The actual calculation is then presented in Sec. \[sec:polarizationoperator\]. Finally, we show how our results are related to those obtained in Ref. [@baier_interaction_1975] and discuss the special cases of a constant-crossed field, a linearly polarized field in the quasi-classical approximation, and a circularly polarized, monochromatic field in Sec. \[sec:discussion\].
Strong-field QED {#sec:strongfieldqed}
================
QED is described by the following Lagrangian density [@landau_quantum_1981; @weinberg_quantum_1995]: $$\begin{gathered}
\label{eqn:qedlagrangian}
\mathcal{L}_{\mathrm{QED}}
=
\bar{\psi} {\left(}{i}{}{\slashed{{\partial}}} - m{\right)}\psi - \frac{1}{4} {\mathcal{F}}_{\mu\nu} {\mathcal{F}}^{\mu\nu} - e\bar{\psi} \gamma_\mu \psi {\mathcal{A}}^\mu,\end{gathered}$$ where $\psi$ and ${\mathcal{A}}^\mu$ are the Dirac spinor field and the electromagnetic four-vector potential, respectively, and ${\mathcal{F}}^{\mu\nu} = {\partial}^\mu {\mathcal{A}}^\nu - {\partial}^\nu {\mathcal{A}}^\mu$ is the electromagnetic field tensor \[from now on we will use natural units $\hbar = c = 1$ and Heaviside-Lorentz units for the charge, $\alpha = {{e^2}/{(4\pi)}}$, see Appendix \[sec:notation\] for further details\]. The equation of motion for the spinor field $\psi$, which follows from Eq. (\[eqn:qedlagrangian\]), is the Dirac equation $$\begin{gathered}
\label{eqn:diracequation}
(i{\slashed{{\partial}}} - e{\slashed{{\mathcal{A}}}} - m)\,\psi = 0.\end{gathered}$$ Correspondingly, we obtain for the photon field ${\mathcal{A}}^\mu$ in Lorentz gauge (${\partial}_\mu {\mathcal{A}}^\mu = 0$) the wave equation $$\begin{gathered}
\label{eqn:photonwaveequation}
{\partial}^\rho {\partial}_\rho {\mathcal{A}}^\mu = e J^\mu,
\quad
J^\mu = \bar{\psi} \gamma^\mu \psi.\end{gathered}$$
Vacuum QED
----------
To obtain a quantum theory of electrons, positrons, and photons, both the spinor $\psi$ and the photon field ${\mathcal{A}}^\mu$ are promoted to operators with canonical (anti)commutation relations (alternatively, the functional integral formalism can be employed). Once derived in either way, the $S$-matrix element for a given process can be calculated perturbatively using Feynman rules. In vacuum QED, the starting point for the perturbative expansion is a solution of the free Dirac or the free wave equation \[Eq. (\[eqn:diracequation\]) with ${\mathcal{A}}^\mu=0$ and Eq. (\[eqn:photonwaveequation\]) with $J^\mu=0$, respectively\]. An electron with a given four-momentum $p^\mu=({\epsilon},{\boldsymbol{p}})$ (${\epsilon}> 0, p^2=m^2$) can then be described by the plane-wave solutions [@landau_quantum_1981], $$\begin{gathered}
\label{eqn:freeplanewave}
\psi_{p} = \frac{1}{\sqrt{2{\epsilon}}} e^{-ipx} u_p,
\quad
({\slashed{p}} - m) u_p = 0\end{gathered}$$ and the corresponding propagator is given by $$\begin{gathered}
\label{eqn:freepropagator}
iG(x,y) = i\int \frac{d^4p}{(2\pi)^4} \, e^{-ip(x-y)} \frac{{\slashed{p}} + m}{p^2 - m^2 + i0}.\end{gathered}$$ For a photon with polarization four-vector ${\epsilon}^\mu$ and four-momentum $k^\mu=(\omega,{\boldsymbol{k}})$ ($\omega \geq 0$, $k^2=0$), we obtain the following wave-function: $$\begin{gathered}
{\mathcal{A}}^\mu_{k} = \frac{1}{\sqrt{2\omega}} e^{-ikx} {\epsilon}^\mu\end{gathered}$$ and in the Feynman gauge, the photon propagator is given by $$\begin{gathered}
-iD_{\mu\nu}(x-y) = - i \int \frac{d^4k}{(2\pi)^4} \, e^{-ik(x-y)} \frac{g_{\mu\nu}}{k^2 + i0}.\end{gathered}$$ Finally, the interaction between electrons, positrons, and photons is represented by the interaction vertex $$\begin{gathered}
\label{eqn:freevertex}
-ie \int d^4x \, \cdots \gamma^\mu \cdots,\end{gathered}$$ where the dots indicate that the vertex is always contracted with two fermion and one photon lines. Thus, one can move the exponential functions from the external lines and propagators to the vertex. After the space-time integrals associated with the vertex are taken, momentum-conserving delta functions are obtained. For a more detailed discussion see e.g. Refs. [@landau_quantum_1981; @weinberg_quantum_1995].
QED with background fields {#sec:qedwithbackgroundfields}
--------------------------
Vacuum QED has been tested to a very high precision because, due to the smallness of the fine-structure constant, a perturbative treatment is adequate in most situations. However, for very strong external electromagnetic fields, the (conventional) perturbation series breaks down. Modern laser facilities provide a source of such strong external electromagnetic fields. An intense laser field represents a coherent state of the photon field that can be described by a classical four-potential $A^\mu$. Since only the highly occupied laser modes can be considered as classical, we separate those modes by writing ${\mathcal{A}}^\mu = A^\mu_{\mathrm{rad}} + A^\mu$ in the Lagrangian density. Here, $A^\mu$ is treated as a classical field, whereas all other modes, described by $A^\mu_{\mathrm{rad}}$, are properly quantized [@fradkin_quantum_1991; @glauber_coherent_1963; @harvey_signatures_2009] (see Appendix \[sec:lasercoherentstate\] for further details).
To estimate the laser intensity at which we can start to treat the laser as a classical field, we follow Ref. [@landau_quantum_1981], Sec. 5. The energy density of a laser field with intensity $I_0$ and central angular frequency $\omega_0$ is of the order of $I_0$, and the density of modes is of the order of $\omega_0^3$. Correspondingly, each mode contains $N_\gamma$ photons, where $N_\gamma$ is of the order of ${{I_0}/{\omega_0^4}}$. If $N_\gamma \gg 1$, due to the correspondence principle, we can describe the laser modes by a classical field. Thus, we obtain the following condition for the laser intensity: $$\begin{gathered}
\label{eqn:classicalcondition}
I_0 \gg \omega_0^4 \approx \unitfrac[6 \times 10^5]{W}{cm^2} \, {\left(}\frac{\omega_0}{\unit[1]{eV}} {\right)}^4,\end{gathered}$$ which is well fulfilled at the relativistic intensities ($I_0\gtrsim \unitfrac[10^{18}]{W}{cm^2}$) in the optical regime ($\omega_0 \sim \unit[1]{eV}$) we are interested in here.
Furthermore, we also have to ensure that the depletion of the laser field is sufficiently low, such that the latter can be treated as a given background field. Now, typical available optical petawatt lasers, which are suitable for the investigation of QED processes in a strong laser field, have an energy of the order of 100 J [@di_piazza_extremely_2012], i.e., a total number of about $10^{20}$ photons. In a typical QED process as nonlinear Compton scattering, at an intensity of the order of $\unitfrac[10^{22}]{W}{cm^2}$, about $\xi_0^3\sim 10^6$ photons are absorbed from the laser by each electron [@di_piazza_extremely_2012] (such an intensity is, in principle, reachable with a petawatt laser). By assuming a number of electrons in the beam of the order of $10^9$, we obtain that about $10^{15}$ photons are expected to be absorbed from the laser field, which remains then practically unaffected (this number of electrons is typical for current laser-plasma-based electron accelerators [@leemans_gev_2006]). In conclusion, we can safely assume here that the background laser field can be treated as a given, classical background field (see also Refs. [@fried_scattering_1964; @eberly_electron_1966; @berson_electron_1969; @bergou_nonlinear_1981; @filipowicz_relativistic_1985]).
Working in the so-called Furry picture [@furry51] the classical background field $A^\mu$ is taken into account exactly and only the radiation field is treated perturbatively. The starting point for the perturbative expansion of the $S$-matrix is then the solution of the interacting Dirac equation (\[eqn:diracequation\]) with the replacement ${\mathcal{A}}^\mu\to A^\mu$. Since photons have no self-interactions at tree level, the photon wave functions and propagators are left unchanged by the background field. A more detailed discussion of strong-field QED can be found in Refs. [@battesti_magnetic_2013; @di_piazza_extremely_2012; @ehlotzky_fundamental_2009; @mourou_optics_2006; @marklund_nonlinear_2006; @dittrich_probingquantum_2000; @fradkin_quantum_1991; @ritus_1985] and in the references therein.
Plane-wave fields {#sec:planewavefields}
-----------------
In this paper we will consider only plane-wave external fields, i.e., we require that the field tensor $F^{\mu\nu}$ depends only on the plane-wave phase $\phi = kx$, where $k^\mu$ is a momentum four-vector which characterizes the plane wave ($k^2=0$). In the absence of charges and currents, the field tensor must obey the homogeneous Maxwell equations [@landau_classical_1987]: $$\begin{gathered}
\label{eqn:mwefieldtensor}
{\partial}_\mu F^{\mu\nu} = k_\mu F'^{\mu\nu} = 0,
\quad
{\partial}^\mu F^{*}_{\mu\nu}
=
k^\mu F'^{*}_{\mu\nu} = 0.\end{gathered}$$ Working in a basis where $ka_i = a_1 a_2 = 0$, with $i=1,2$ (see Appendix \[sec:lccappendix\]), we can show that the most general (antisymmetric) field tensor obeying Eq. (\[eqn:mwefieldtensor\]) is given by [@schwinger_gauge_1951] $$\begin{gathered}
\label{eqn:fieldtensor}
F^{\mu\nu}(kx)
=
f_1^{\mu\nu} \psi_1'(kx)
+
f_2^{\mu\nu} \psi_2'(kx),\end{gathered}$$ where $$\begin{gathered}
f_i^{\mu\nu} = k^\mu a_i^\nu - k^\nu a_i^\mu,\\
f^{\mu}_{i\,\rho} f_{j}^{\rho\nu} = -\delta_{ij} a_i^2\, k^\mu k^\nu,
\quad
k_\mu f_i^{\mu\nu} = 0.\end{gathered}$$ The scalar functions $\psi_i(kx)$ are arbitrary, restricted only by the physical requirement that the external field is of finite extent \[i.e., $\psi_i(\pm\infty)=\psi'_i(\pm \infty)=0$, with $\psi_i(kx)$, $\psi'_i(kx)$ vanishing fast enough at infinity\]. Furthermore, we adopt (without restriction) the normalization condition ${\left|\psi_i(kx)\right|},{\left|\psi'_i(kx)\right|} \lesssim 1$. We note that in the Lorentz gauge (${\partial}_\mu A^\mu = 0$) the four-potential corresponding to the field tensor in Eq. (\[eqn:fieldtensor\]) can be chosen in the form $$\begin{gathered}
\label{eqn:planewavefourpotential}
\begin{aligned}
A^\mu(kx) &= a_1^\mu \psi_1(kx) + a_2^\mu \psi_2(kx),
\\
F^{\mu\nu} &= {\partial}^\mu A^\nu - {\partial}^\nu A^\mu.
\end{aligned}\end{gathered}$$
Using the tensors $f_i^{\mu\nu}$ and an arbitrary four-momentum vector $q^\mu$ (most commonly the four-momentum of an incoming particle), we can define the following quantum nonlinearity parameters [@di_piazza_extremely_2012]: $$\begin{gathered}
\chi_{i}
=
-\frac{e\sqrt{qf_i^2q}}{m^3}
=
\eta \, \xi_i,\end{gathered}$$ where $$\begin{gathered}
\eta = \frac{\sqrt{(kq)^2}}{m^2},\end{gathered}$$ and where the quantities $$\begin{gathered}
\label{eqn:xiparameterdef}
\xi_i
=
\frac{1}{m} \sqrt{-a_i^2 e^2}\end{gathered}$$ are the so-called classical intensity parameters. Since both $\eta$ and $\chi_{i}$ are manifestly gauge and Lorentz invariant, also the parameters $\xi_i$ are gauge and Lorentz invariant. Due to the normalization condition for the shape functions $\psi_i(kx)$, the parameters $\xi_i$ characterize the strength of the plane-wave field. It turns out that the plane-wave field must be taken into account exactly in the calculations if $\xi_i \gtrsim 1$ [@di_piazza_extremely_2012]. Modern laser facilities can easily reach this nonperturbative domain, e.g. in Ref. [@yanovsky_ultra_2008] $\xi_i \sim 100$ was obtained \[we point out that the parameter $\xi_0$, mentioned in the introduction, can be related to the parameters $\xi_i$ by noting that, for a linearly-polarized field with $\psi_2(kx)=0$ and with electric-field amplitude $E_0$ and central angular frequency $\omega_0$, the quantity $\sqrt{-a_1^2}$ can be set equal to $E_0/\omega_0$\].
It is also convenient to introduce the integrated field-strength tensor, $$\begin{gathered}
{\mathfrak{F}}^{\mu\nu}(kx)
=
\int^{kx}_{-\infty} d\phi'\, F^{\mu\nu}(\phi'),\end{gathered}$$ which can be written as $$\begin{gathered}
\begin{aligned}
{\mathfrak{F}}^{\mu\nu}(kx)
&=
k^\mu A^\nu(kx) - k^\nu A^\mu(kx)
\\&=
f_1^{\mu\nu} \psi_1(kx)
+
f_2^{\mu\nu} \psi_2(kx)
\end{aligned}\end{gathered}$$ in the Lorentz gauge \[we will use ${\mathfrak{F}}^{\mu\nu}_x = {\mathfrak{F}}^{\mu\nu}(kx)$ interchangeably to denote the argument\]. Both $F^{\mu\nu}$ and ${\mathfrak{F}}^{\mu\nu}$ have the important algebraic property that successive contractions of more than two tensors vanish, and their square is proportional to $k^\mu k^\nu$, e.g., $$\begin{gathered}
{\mathfrak{F}}_x^{\mu\rho} {\mathfrak{F}}_{y\rho\nu}
=
- k^\mu k_\nu \sum_{i=1,2} a_i^2 \psi_i(kx) \psi_i(ky).\end{gathered}$$
If the background field is a plane-wave field, the Dirac equation \[Eq. (\[eqn:diracequation\]) with ${\mathcal{A}}^\mu \to A^\mu$\] can be solved analytically [@volkov_ueber_1935]. The corresponding so-called Volkov solution with the boundary condition $\Psi_p \to \psi_p$ if $kx \to -\infty$ \[see Eq. (\[eqn:freeplanewave\])\] can be written as [@landau_quantum_1981; @ritus_radiative_1972; @mitter_quantum_1975] $$\begin{gathered}
\Psi_p = \frac{1}{\sqrt{2{\epsilon}}} E_{p,x} u_p,
\quad
E_{p,x} = {\left[}{\mathbf{1}}+ \frac{e{\slashed{k}}{\slashed{A}}(kx)}{2\, kp}{\right]}\, e^{iS_p(x)},\end{gathered}$$ where the phase is given by $$\begin{gathered}
S_p(x)
=
- px - \int_{-\infty}^{kx} d\phi' \, {\left[}\frac{e\, pA(\phi')}{\, kp} - \frac{e^2A^2(\phi')}{2\, kp} {\right]}.\end{gathered}$$ Note that Volkov states, although being an exact solution of the Dirac equation and apart from the spin-terms proportional to ${\slashed{k}}{\slashed{A}}(kx)$, have a quasiclassical structure $\sim \exp[iS_p(x)]$, with $S_p(x)$ being the classical action of an electron inside a plane-wave field [@landau_classical_1987].
The dressed propagator (which is the Green’s function of the interacting Dirac equation) is given by $$\begin{gathered}
\label{eqn:dressedpropagator}
iG(x,y)
=
i\int \frac{d^4p}{(2\pi)^4} E_{p,x} \frac{{\slashed{p}} + m}{p^2 - m^2 + i0} \bar{E}_{p,y},\end{gathered}$$ where $$\begin{gathered}
\bar{E}_{p,x}
=
{\left[}{\mathbf{1}}+ \frac{e{\slashed{A}}(kx){\slashed{k}}}{2\, pk}{\right]}\, e^{-iS_p(x)}.\end{gathered}$$ Thus, in comparison with the vacuum case, the plane waves are replaced by the Ritus $E_p$ functions, which depend nontrivially on the plane-wave phase $kx$. However, they also form an orthogonal and complete set [@ritus_radiative_1972]: $$\begin{gathered}
\label{eqn:epcompletenessandorthogonality}
\begin{aligned}
\int \frac{{d}{}^4p}{(2\pi)^4}\, E_{p,x} \bar{E}_{p,x'} &= \delta^4(x-x'),\\
\int {d}{}^4x\, \bar{E}_{p',x} E_{p,x} &= (2\pi)^4 \, \delta^4(p'-p).
\end{aligned}\end{gathered}$$ The $E_p$ functions convert the dressed momentum into the free momentum [@ritus_radiative_1972]: $$\begin{gathered}
\label{eqn:Epprojectionproperty}
\begin{aligned}
\phantom{}[i{\slashed{{\partial}}}_x - e{\slashed{A}}(kx)] E_{p,x} &= E_{p,x} {\slashed{p}},\\
-i {\partial}_x^\mu \bar{E}_{p,x} \gamma_\mu -e \bar{E}_{p,x} {\slashed{A}}(kx) &= {\slashed{p}} \bar{E}_{p,x}
\end{aligned}\end{gathered}$$ (these identities hold only if the derivative acts solely on $E_{p,x}$ and $\bar{E}_{p,x}$, respectively).
Dressed vertex
--------------
To obtain Feynman rules in momentum space, we can proceed analogously as in the vacuum case and move the $E_{p}$-functions to the vertex [@mitter_quantum_1975]. Correspondingly, we define the dressed vertex by $$\begin{gathered}
\label{eqn:sfqed_dressedvertex}
\Gamma^\rho(p',q,p) = -ie \int d^4x \, e^{-iqx}\, \bar{E}_{p',x} \gamma^\rho E_{p,x}.\end{gathered}$$ Working in momentum space, the only difference between vacuum QED and strong-field QED is the vertex we have to use \[i.e., the free vertex in Eq. (\[eqn:freevertex\]) is replaced by the dressed vertex in Eq. (\[eqn:sfqed\_dressedvertex\])\]. Using the relations given in Appendix \[sec:gammamatrixalgebraappendix\], we can write the dressed vertex as $$\begin{gathered}
\label{eqn:dressedvertexfinal}
\Gamma^\rho(p',q,p)
=
-ie \int d^4x \,\big[ \gamma_\mu G^{\mu\rho}(kp',kp;kx)
\\+ i\gamma_\mu \gamma^5 G_5^{\mu\rho}(kp',kp;kx) \big] e^{iS_\Gamma(p',q,p;x)},\end{gathered}$$ where the phase and the coupling tensors are given by $$\begin{gathered}
S_\Gamma(p',q,p;x)
=
-S_{p'}(x) -qx + S_p(x)
\\=
(p'-q-p)x
+
\int_{-\infty}^{kx} d\phi'
\, \Big[ \frac{e p_\mu p'_\nu {\mathfrak{F}}^{\mu\nu}(\phi')}{(kp)(kp')}
\\+ \frac{e^2(kp-kp')}{2(kp)^2(kp')^2} p_\mu p'_\nu {\mathfrak{F}}^{2\mu\nu}(\phi') \Big],\end{gathered}$$ $$\begin{gathered}
\label{eqn:dressedvertexGtensorsdefinition}
\begin{aligned}
G^{\mu\rho}(kp',kp;kx)
&=
g^{\mu\rho}
+
G_1 {\mathfrak{F}}^{\mu\rho}_x
+
G_2 {\mathfrak{F}}^{2\mu\rho}_x,\\
G_{5}^{\mu\rho}(kp',kp;kx)
&=
G_3 {\mathfrak{F}}^{*\mu\rho}_x,
\end{aligned}\end{gathered}$$ $$\begin{gathered}
\label{eqn:Gidefs}
\begin{gathered}
G_1 = -e\, \frac{kp + kp'}{2 kp \, kp'},
\quad
G_2 = \frac{e^2}{2kp\, kp'},\\
G_3 = -e\, \frac{kp - kp'}{2 kp \, kp'}
\end{gathered}\end{gathered}$$ (note that $G_1$ and $G_2$ are even in the permutation $kp \leftrightarrow kp'$ while $G_3$ is odd). We point out that the expression given in Eq. (\[eqn:dressedvertexfinal\]) is manifestly gauge invariant, since it depends on the external field only through the tensor ${\mathfrak{F}}^{\mu\nu}$ [@mitter_quantum_1975].
In position space the dressed propagator in Eq. (\[eqn:dressedpropagator\]) can be interpreted such that the electron (or positron) continuously interacts with the external field during its propagation. Examined in momentum space, we can also visualize the influence of the external field as a modification of the coupling between the photons of the radiation field and the charged particles. From Eq. (\[eqn:dressedvertexfinal\]) we see that, besides the modification of the photon vector current interaction we also obtain a coupling to the axial-vector current inside the plane-wave background. This is possible since the external field provides the pseudotensor ${\mathfrak{F}}^{*\mu\nu}$.
Since the external field depends only on the plane-wave phase $\phi=kx$, it is useful to use light-cone coordinates, which are discussed in Appendix \[sec:lccappendix\]. We can then always take the integrals in $dx^{{{}+}}$ and $dx^{\perp}$ in Eq. (\[eqn:dressedvertexfinal\]) and obtain momentum-conserving delta functions in three of four light-cone components, $$\begin{gathered}
\delta^{({{{}-}},{\perp})}(p'-p-q),\end{gathered}$$ where we used the notation $$\begin{gathered}
\delta^{({{{}-}},{\perp})}(a)
=
\delta(a^{{{}-}}) \delta(a^{{\scalebox{.64}{$\matheuler{I}$}}}) \delta(a^{{\scalebox{.64}{$\matheuler{II}$}}})\end{gathered}$$ for a general four-vector $a^\mu$. Thus, the four-momentum is only conserved up to a four-vector proportional to the plane-wave four-momentum $k^\mu$ at each vertex.
Ward-Takahashi identity {#sec:wardidentity}
-----------------------
The Ward-Takahashi identity [@ward_identity_1950; @takahashi_generalized_1957] is a direct consequence of the gauge invariance of QED, which becomes particularly transparent in the functional integral approach [@collins_renormalization_1984; @weinberg_quantum_1995]. Diagrammatically, it is a functional relation for Feynman diagrams (in momentum space), where the polarization four-vector of an external photon leg is replaced by the corresponding momentum four-vector. In Ref. [@peskin_introduction_2008] a perturbative proof of the Ward-Takahashi identity in vacuum QED is given. We will show now how this proof can be extended to electron-positron loops inside a plane-wave background field.
![\[fig:wardidentity\] Closed electron loop with $n$ dressed vertices and propagators.](Fig2){height="4.1cm"}
The starting point is the following algebraic identity for the dressed vertex [@mitter_quantum_1975] $$\begin{gathered}
\label{eqn:algebraicidentitydressedvertex}
q_\rho \Gamma^\rho(p',q,p)
\\=
({\slashed{p}}' - m) I(p',q,p) - I(p',q,p) ({\slashed{p}} - m),\end{gathered}$$ where $$\begin{gathered}
I(p',q,p)
=
-ie \int d^4x \, e^{-iqx}\, \bar{E}_{p',x} E_{p,x}. \end{gathered}$$ To verify Eq. (\[eqn:algebraicidentitydressedvertex\]), we use Eq. (\[eqn:Epprojectionproperty\]) and note that the identity $$\begin{gathered}
\int d^4x \ i{\partial}_\mu \big[\bar{E}_{p',x} \gamma^\mu e^{-iqx} E_{p,x} \big]
= 0\end{gathered}$$ holds [@mitter_quantum_1975].
Typically, $({\slashed{p}}' - m)$ and $({\slashed{p}} - m)$ in Eq. (\[eqn:algebraicidentitydressedvertex\]) cancel an adjacent propagator, and the associated momentum-integral can be taken using the relations $$\begin{gathered}
\label{eqn:IGammacontraction}
\begin{aligned}
\int \frac{d^4p''}{(2\pi)^4}
I(p,q',p'') \Gamma^\mu(p'',q,p')
&=
-ie \Gamma^\mu (p,q+q',p'),\\
\int \frac{d^4p''}{(2\pi)^4}
\Gamma^\mu(p,q,p'') I(p'',q',p')
&=
-ie \Gamma^\mu (p,q+q',p'),
\end{aligned}\end{gathered}$$ which follow from Eq. (\[eqn:epcompletenessandorthogonality\]). Using Eqs. (\[eqn:algebraicidentitydressedvertex\]) and (\[eqn:IGammacontraction\]), we can simplify diagrams which contain dressed vertices contracted with the corresponding photon four-momenta.
As an example, we consider now a closed electron loop which contains $n$ dressed vertices and electron propagators (see Fig. \[fig:wardidentity\]). The $i$th propagator of such a loop together with its adjacent vertices is given by $$\begin{gathered}
\cdots \Gamma^{\mu_i}(p_{i-1},q_i,p_i) \frac{1}{{\slashed{p}}_i - m} \Gamma^{\mu_{i+1}}(p_i,q_{i+1},p_{i+1}) \cdots\end{gathered}$$ (the electron four-momenta $p_i$ are integrated out). If we insert now a vertex (contracted with its photon four-momentum) at this propagator, we obtain $$\begin{gathered}
\cdots \Gamma^{\mu_i}(p_{i-1},q_i,p_i) \frac{1}{{\slashed{p}}_i - m}
q_\mu \Gamma^{\mu}(p_i,q,p')
\\\times \,
\frac{1}{{\slashed{p}}'-m}
\Gamma^{\mu_{i+1}}(p',q_{i+1},p_{i+1}) \cdots\end{gathered}$$ and, by using Eqs. (\[eqn:algebraicidentitydressedvertex\]) and (\[eqn:IGammacontraction\]), we find that this is equivalent to $$\begin{gathered}
\begin{aligned}
&\phantom{-}\cdots \Gamma^{\mu_i}(p_{i-1},q_i+q,p_i) \frac{1}{{\slashed{p}}_i - m} \Gamma^{\mu_{i+1}}(p_i,q_{i+1},p_{i+1}) \cdots
\\
&-
\cdots \Gamma^{\mu_i}(p_{i-1},q_i,p_i) \frac{1}{{\slashed{p}}_i - m} \Gamma^{\mu_{i+1}}(p_i,q_{i+1}+q,p_{i+1}) \cdots
\end{aligned}\end{gathered}$$ (in the first line, we have changed the name of the integration variable from $p'$ to $p_i$). Thus, the insertion splits the diagram into the sum of twice the original diagram with the additional four-momentum $q$ added once at the adjacent vertex before and after the insertion. If we sum now over all possible insertion points of the loop, we obtain zero since all contributions cancel pairwise (as in the vacuum case [@peskin_introduction_2008]).
We point out that the above discussion is shortened, since possible issues arising due to the renormalization of the theory were not addressed (in general, the validity of the Ward-Takahashi identity may be spoiled by anomalies [@adler_axial-vector_1969]). In this paper, however, we are mainly interested in modifications induced by the background field, which turn out to be finite. Thus, subtleties arising from manipulations of divergent integrals can be addressed as in vacuum QED.
Polarization operator {#sec:polarizationoperator}
=====================
General expression
------------------
The leading-order contribution to the polarization operator $\mathcal{P}^{\mu\nu}(q_1,q_2)$ for plane-wave background fields (see Ref. [@landau_quantum_1981], Sec. 104) is determined by the diagram in Fig. \[fig:polarizationoperator\]. This diagram corresponds to the following expression: $$\begin{gathered}
\label{eqn:polarizationoperator}
{T}^{\mu\nu}(q_1,q_2)
=
\int \frac{d^4p\, d^4p'}{(2\pi)^8}
\operatorname{\mathbf{tr}}\,
\Gamma^\mu(p',q_1,p)
\\ \times \frac{({\slashed{p}} + m)}{p^2-m^2+i0} \,
\Gamma^\nu(p,-q_2,p')
\frac{({\slashed{p}}' + m)}{p'^2-m^2+i0}\end{gathered}$$ and ${T}^{\mu\nu} = i \mathcal{P}^{\mu\nu}$ (see Ref. [@landau_quantum_1981], Sec. 115; [@baier_operator_1975]). We note that ${T}^{\mu\nu}(q_1,q_2)$ is divergent, but if we write $$\begin{gathered}
\label{eqn:regularizationpolarizationoperator}
{T}^{\mu\nu}(q_1,q_2)
=
{\left[}{T}^{\mu\nu}(q_1,q_2) - {T}^{\mu\nu}_{{\mathfrak{F}}=0}(q_1,q_2) {\right]}\\+ {T}^{\mu\nu}_{{\mathfrak{F}}=0}(q_1,q_2),\end{gathered}$$ the contribution in square brackets is finite [@baier_interaction_1975], and the regularization of the vacuum contribution is well known [@landau_quantum_1981; @weinberg_quantum_1995]. In the following, we will focus on the tensor in square brackets which contains only the corrections induced by the external background field.
To determine the expression in Eq. (\[eqn:polarizationoperator\]), we have to insert the dressed vertex given in Eq. (\[eqn:dressedvertexfinal\]) \[we will denote the vertex integrals associated with $\Gamma^\mu(p',q_1,p)$ and $\Gamma^\nu(p,-q_2,p')$ by $d^4x$ and $d^4y$, respectively\]. We then obtain for ${T}^{\mu\nu}(q_1,q_2)$ $$\begin{gathered}
\label{eqn:polarizationoperatorB}
{T}^{\mu\nu}(q_1,q_2)
=
4 \, (-ie)^2 \int \frac{d^4p\, d^4p'}{(2\pi)^8} \int d^4x d^4y\,
\\ \times \,
\frac{\frac14 \operatorname{\mathbf{tr}}\big[ \cdots \big]^{\mu\nu}}{(p^2-m^2+i0)(p'^2-m^2+i0)} e^{iS_{T}}\end{gathered}$$ (the prefactor ${{1}/{4}}$ in front of the trace is included explicitly for later convenience), where the phase reads \[see Eq. (\[eqn:dressedvertexfinal\])\] $$\begin{gathered}
\label{eqn:sfqed_polarizationoperatorphasestructure}
iS_{T}=
i(p'-p-q_1)x
+
i(p-p'+q_2)y
\\+
i\int_{ky}^{kx} d\phi'\, \bigg[ \frac{e p_\mu p'_\nu {\mathfrak{F}}^{\mu\nu}}{(kp)(kp')}
+ \frac{e^2(kp-kp')}{2(kp)^2(kp')^2} p_\mu p'_\nu {\mathfrak{F}}^{2\mu\nu} \bigg]\end{gathered}$$ and $\frac14 \operatorname{\mathbf{tr}}\big[ \cdots \big]^{\mu\nu}$ in Eq. (\[eqn:polarizationoperatorB\]) can be calculated using the relations given in Appendix \[sec:gammamatrixalgebraappendix\]:
$$\begin{gathered}
\label{eqn:sfqed_polarizationoperatortraceA}
\frac14 \operatorname{\mathbf{tr}}\big[ \gamma_\alpha a^{\alpha\mu} + i\gamma_\alpha \gamma^5 b^{\alpha\mu} \big]
({\slashed{p}} + m) \,
\big[ \gamma_\beta c^{\beta\nu} + i\gamma_\beta \gamma^5 d^{\beta\nu} \big]
({\slashed{p}}' + m)
\\
=
m^2[(a^{\alpha\mu} c_\alpha^{\phantom{\alpha}\nu})
+(b^{\alpha\mu}d_\alpha^{\phantom{\alpha}\nu})]
+ (pp')(b^{\alpha\mu}d_\alpha^{\phantom{\alpha}\nu})
- (pp')(a^{\alpha\mu} c_\alpha^{\phantom{\alpha}\nu})
+ (p_\alpha a^{\alpha\mu})(p'_\beta c^{\beta\nu})
+ (p'_\alpha a^{\alpha\mu})(p_\beta c^{\beta\nu})
\\
- (p_\alpha b^{\alpha\mu})(p'_\beta d^{\beta\nu})
- (p'_\alpha b^{\alpha\mu})(p_\beta d^{\beta\nu})
- {\epsilon}_{\rho\sigma\alpha\beta} p^\rho p'^\sigma (a^{\alpha\mu} d^{\beta\nu} + b^{\alpha\mu} c^{\beta\nu}),\end{gathered}$$
where $$\begin{gathered}
\begin{aligned}
a^{\alpha\mu}
&=
G^{\alpha\mu}(kp',kp;kx),&
c^{\beta\nu}
&=
G^{\beta\nu}(kp,kp';ky),\\
b^{\alpha\mu}
&=
G_5^{\alpha\mu}(kp',kp;kx),&
d^{\beta\nu}
&=
G_5^{\beta\nu}(kp,kp';ky).
\end{aligned}\end{gathered}$$
Evaluation of the integrals {#sec:polarizationoperatorintegralevaluation}
---------------------------
Working in light-cone coordinates (see Appendix \[sec:lccappendix\]) we can take all space-time integrals except of those in $dx^{{{}-}}$ and $dy^{{{}-}}$ and obtain the momentum-conserving delta functions $$\begin{gathered}
(2\pi)^6 \delta^{({{{}-}},{\perp})}(p'-p-q) \, \delta^{({{{}-}},{\perp})}(q_1-q_2).\end{gathered}$$ Here and in the following, we write $q^\mu$ if $q^\mu_1$ and $q^\mu_2$ can be used interchangeably due to the above delta function. Successively, we can take the integrals in $dp'^{{{}-}}$ and $dp'^{\perp}$ (for simplicity we will continue writing $p'$ and identify $p'=p+q$ for the components ${{{}-}},{\perp}$).
It is now convenient to introduce the two four-vectors: $$\begin{gathered}
\label{eqn:Lambdavectors}
\Lambda_1^\mu = \frac{f_1^{\mu\nu} q_\nu}{kq \sqrt{-a_1^2}},
\quad
\Lambda_2^\mu = \frac{f_2^{\mu\nu} q_\nu}{kq \sqrt{-a_2^2}},\end{gathered}$$ which obey $\Lambda_i \Lambda_j = -\delta_{ij}$, $k\Lambda_i = q_i\Lambda_j = 0$ and $$\begin{gathered}
f_1^{\mu\nu} \Lambda_{1\nu} = - \frac{m}{e} k^\mu \xi_1,
\quad
f_2^{\mu\nu} \Lambda_{2\nu} = - \frac{m}{e} k^\mu \xi_2.\end{gathered}$$ They allow us to write the remaining phase as $$\begin{gathered}
\label{eqn:sfqed_polarizationoperatorphasestructureB}
iS_{T}=
i(p'-p-q_1)^{{{}+}}x^{{{}-}}\\+
i(p-p'+q_2)^{{{}+}}y^{{{}-}}+
i p\uplambda + i \Uplambda,\end{gathered}$$ where we defined $$\begin{gathered}
\label{eqn:sfqed_polarizationoperatorintegraldefinitions}
\begin{aligned}
\uplambda^\mu
&=
- \frac{m (kq)}{(kp)(kp')} \sum_{i=1,2} \xi_i \Lambda_i^\mu \int_{ky}^{kx} d\phi' \, \psi_i(\phi'),\\
\Uplambda
&=
-\frac{m^2 (kq)}{2(kp)(kp')} \sum_{i=1,2} \xi^2_i \int_{ky}^{kx} d\phi' \, \psi_i^2(\phi').
\end{aligned}\end{gathered}$$ Due to the appearance of $\Lambda_i^\mu$ in $\uplambda^\mu$, it is more convenient to use modified light-cone coordinates from now on \[see Eq. (\[eqn:modifiedlcc\]); the calculation so far is independent of this choice\]. In modified light-cone coordinates, we obtain the convenient relations $$\begin{gathered}
p\uplambda = - p^{\perp}\uplambda^{\perp},
\quad
q^{\perp}= 0,
\quad
p'^{\perp}= p^{\perp},\end{gathered}$$ which simplify the algebra considerably.
If the preexponent would not depend on $p^{{{}+}}$ and $p'^{{{}+}}$, both integrals could now be taken. We therefore introduce the proper-time representation of the scalar propagators [@schwinger_gauge_1951; @dittrich_probingquantum_2000]: $$\begin{gathered}
\label{eqn:sfqed_polarizationoperatorschwingerpropagator}
\frac{1}{p^2-m^2+i0} \frac{1}{p'^2-m^2+i0}
=
(-i)^2 \int_0^\infty ds \, dt
\\ \times \, \exp {\left[}i(p^2-m^2+i0)s + i(p'^2-m^2+i0)t {\right]}.\end{gathered}$$ In the following we will drop the pole prescriptions $i0$ and keep the replacement $m^2 \to m^2 - i0$ in mind. Furthermore, we add the source terms $i p_\mu j^\mu + i p'_\mu j'^\mu$ to the phase, which allows us to make the replacement $$\begin{gathered}
\label{eqn:momentumreplacementtrace}
{\slashed{p}} \longrightarrow (-i){\slashed{{\partial}}}_j,
\quad
{\slashed{p}}' \longrightarrow (-i){\slashed{{\partial}}}_{j'}\end{gathered}$$ in the trace. Now, the preexponent depends only on $p^{{{}-}}$ (through $kp$ and $kp'$). Taking the derivatives with respect to the sources out of the integrals, we can take the integrals in $dp^{{{}+}}$, $dp'^{{{}+}}$, which results in the delta functions, $$\begin{gathered}
\label{eqn:yminuspminusdeltafunctions}
(2\pi) \delta\Big[y^{{{}-}}- x^{{{}-}}-\frac{1}{s+t} (2st q^{{{}-}}- t j^{{{}-}}+ s j'^{{{}-}})\Big]
\\ \times \,
(2\pi) \delta[2p^{{{}-}}(s+t) + 2 q^{{{}-}}t + j^{{{}-}}+ j'^{{{}-}}].\end{gathered}$$ Successively, these delta functions can be used to take also the integrals in $dy^{{{}-}}$ and $dp^{{{}-}}$. To this end we rewrite (since $s+t\geq 0$) $$\begin{gathered}
\delta[2p^{{{}-}}(s+t) + 2 q^{{{}-}}t + j^{{{}-}}+ j'^{{{}-}}]
\\=
\frac{1}{2(s+t)} \delta\Big[p^{{{}-}}+ \frac{1}{2(s+t)} (2q^{{{}-}}t + j^{{{}-}}+ j'^{{{}-}})\Big]\end{gathered}$$ (for simplicity we keep writing $y^{{{}-}}$ and $p^{{{}-}}$). In particular, we obtain the identities $$\begin{gathered}
\label{eqn:kpkpprimekyidentifications}
\begin{aligned}
kp &= -\frac{1}{s+t} \Big[ t kq + \frac{1}{2}(kj + kj') \Big],\\
kp' &= +\frac{1}{s+t} \Big[ s kq - \frac{1}{2}(kj + kj') \Big],\\
ky &= kx + \frac{1}{s+t} (2st kq - t kj + s kj'),
\end{aligned}\end{gathered}$$ which imply for $j=j'=0$ that $$\begin{gathered}
\label{eqn:Givssandt}
\begin{aligned}
G_1 &= \frac{e}{2kq} \frac{(s-t)(s+t)}{st} = \frac{e}{2kq} \frac{v \tau}{\mu},\\
G_2 &= - \frac{e^2}{2(kq)^2} \frac{(s+t)^2}{st} = - \frac{e^2}{2(kq)^2} \frac{\tau}{\mu},\\
G_3 &= - \frac{e}{2kq} \frac{(s+t)^2}{st} = - \frac{e}{2kq} \frac{\tau}{\mu},
\end{aligned}\end{gathered}$$ where we defined [@baier_interaction_1975] $$\begin{gathered}
\label{eqn:tauvmudefinition}
\tau = s+t,
\quad
v = \frac{s-t}{s+t},
\quad
\mu = \frac{st}{s+t}
=
\frac14 \tau (1-v^2)\end{gathered}$$ \[the motivation for these definitions becomes clear in Eq. (\[eqn:sfqed\_polarizationoperatortraceC\])\].
The remaining part of the phase structure (including the part coming from the propagators and the sources) is now given by $$\begin{gathered}
iS'_{T}=
i \Big[ (q_2^{{{}+}}-q_1^{{{}+}}) x^{{{}-}}+
\frac{st}{s+t} q_2^2
-
\frac{1}{s+t} (t \, q_2j - s\, q_2j')
\\
\begin{aligned}
&-
\frac{1}{2(s+t)} (j^{{{}+}}+ j'^{{{}+}}) (j^{{{}-}}+ j'^{{{}-}})
\\
&- (p^{\perp}p^{\perp}+ m^2)(s+t)
\end{aligned}
\\-
(j^{\perp}+ j'^{\perp}+ \uplambda^{\perp}) p^{\perp}+
\Uplambda \Big].\end{gathered}$$ Taking the Gaussian integrals in $p^{{\scalebox{.64}{$\matheuler{I}$}}}$ and $p^{{\scalebox{.64}{$\matheuler{II}$}}}$, we obtain the prefactor $\frac{\pi}{i(s+t)}$, and the final phase is given by $$\begin{gathered}
\label{eqn:sfqed_polarizationoperatorfinalphase}
iS'_{T}=
i \Big[ (q_2^{{{}+}}-q_1^{{{}+}}) x^{{{}-}}-
m^2 (s+t)
+
\frac{st}{s+t} q_2^2
\\-
\frac{1}{s+t} (t \, q_2j - s\, q_2j')
-
\frac{1}{4(s+t)} (j+j')^2
\\-
\frac{1}{2(s+t)} (j+j')\uplambda
-
\frac{1}{4(s+t)} \uplambda^2
+
\Uplambda \Big],\end{gathered}$$ which reads for zero sources ($j=j'=0$) $$\begin{gathered}
\label{eqn:sfqed_polarizationoperatorfinalphasezerosources}
iS'_{T}=
i \Big[ (q_2^{{{}+}}-q_1^{{{}+}}) x^{{{}-}}+
\mu q_2^2
\\-
\tau m^2 + \tau m^2 \sum_{i=1,2} \xi_i^2 (I^2_i-J_i)
\Big],\end{gathered}$$ where we defined $$\begin{gathered}
\label{eqn:defIJintegrals}
\begin{aligned}
I_i
&=
-\frac{1}{2kq \mu} \int_{ky}^{kx} d\phi' \, \psi_i(\phi'),\\
J_i
&=
-\frac{1}{2kq \mu} \int_{ky}^{kx} d\phi' \, \psi^2_i(\phi')
\end{aligned}\end{gathered}$$ (the prefactor is chosen such that, for $j=j'=0$ and $\psi_i= 1$, we obtain $I_i=J_i=1$).
Finally, we can write the tensor ${T}^{\mu\nu}$ as $$\begin{gathered}
\label{eqn:polarizationoperatorafterintegralstaken}
{T}^{\mu\nu}(q_1,q_2)
=
-2i\pi e^2\, \delta^{({{{}-}},{\perp})}(q_1-q_2)
\int_0^\infty ds \, dt
\\ \times\,
\int_{-\infty}^{+\infty} dx^{{{}-}}\,
\frac{1}{(s+t)^2} \frac14 \operatorname{\mathbf{tr}}{\left[}\ldots {\right]}^{\mu\nu} e^{iS'_{T}} \Big|_{j=j'=0},\end{gathered}$$ where the expression for $\frac14 \operatorname{\mathbf{tr}}{\left[}\ldots {\right]}^{\mu\nu}$ is given in Eq. (\[eqn:sfqed\_polarizationoperatortraceA\]) with the replacement in Eq. (\[eqn:momentumreplacementtrace\]) and where the sources are set to zero after the derivatives are taken.
We point out that the two four-momenta $q_1$ and $q_2$ appear asymmetrically in the final expression \[see Eq. (\[eqn:sfqed\_polarizationoperatorfinalphasezerosources\])\]. To remove this asymmetry, we shift the $x^{{{}-}}$ integration by defining $$\begin{gathered}
\label{eqn:symmetricintegralshift}
z^{{{}-}}= x^{{{}-}}+ \mu q^{{{}-}}. \end{gathered}$$ After this shift, the phase contains $q_1q_2$ since $$\begin{gathered}
\label{eqn:symmetricphaseterms}
(q_2^{{{}+}}-q_1^{{{}+}}) x^{{{}-}}+
\mu q_2^2
=
(q_2^{{{}+}}-q_1^{{{}+}}) z^{{{}-}}+
\mu q_1 q_2.\end{gathered}$$ Furthermore, we obtain (for $j=j'=0$) symmetric representations for the functions in Eq. (\[eqn:defIJintegrals\]): $$\begin{gathered}
\label{eqn:IJsymmetricintegrals}
\begin{aligned}
I_i
&=
\frac12 \int_{-1}^{+1} d\lambda \, \psi_i(kz - \lambda \mu kq),\\
J_i
&=
\frac12 \int_{-1}^{+1} d\lambda \, \psi^2_i(kz - \lambda \mu kq),
\end{aligned}\end{gathered}$$ since $$\begin{gathered}
\label{eqn:kxkyversuskz}
\begin{aligned}
kx &= kz - \mu kq,\\
ky &= kz + \mu kq + \frac{1}{s+t}(skj'-tkj).
\end{aligned}\end{gathered}$$
Tensor structure
----------------
In principle, the only remaining task is to evaluate the two derivatives with respect to $j$ and $j'$ and then set $j=j'=0$. Despite being elementary, this is the most tedious part of the calculation, since the sources appear in many places in the final expression. The work is considerably reduced if we expand the polarization operator in a convenient basis [@baier_interaction_1975]. To this end we note that $$\begin{gathered}
q_{1\mu} {T}^{\mu\nu}(q_1,q_2) = 0, \quad {T}^{\mu\nu}(q_1,q_2) q_{2\nu} = 0\end{gathered}$$ due to the Ward-Takahashi identity (see Sec. \[sec:wardidentity\]).
Since the four-vectors $\Lambda_i$ appear in the phase \[see Eq. (\[eqn:sfqed\_polarizationoperatorphasestructureB\])\] and $q_i\Lambda_j=0$, it is natural to introduce the two complete sets $q_1$, ${\mathcal{Q}}_1$, $\Lambda_1$, $\Lambda_2$ and $q_2$, ${\mathcal{Q}}_2$, $\Lambda_1$, $\Lambda_2$, where $$\begin{gathered}
\label{eqn:Qidef}
{\mathcal{Q}}_1^\mu = \frac{k^\mu q_1^2 - q_1^\mu kq}{kq},
\quad
{\mathcal{Q}}_2^\mu = \frac{k^\mu q_2^2 - q_2^\mu kq}{kq}\end{gathered}$$ (${\mathcal{Q}}_1^2 = -q_1^2$, ${\mathcal{Q}}_2^2 = -q_2^2$, ${\mathcal{Q}}_i\Lambda_j=0$, $q_i {\mathcal{Q}}_i=0$). Using the set including $q_1$ for the index $\mu$ and the set including $q_2$ for the index $\nu$, seven of 16 coefficients vanish due to the Ward-Takahashi identity, and we can decompose ${T}^{\mu\nu}(q_1,q_2)$ as [@baier_interaction_1975] $$\begin{gathered}
\label{eqn:polarizationoperatordecomposition}
{T}^{\mu\nu}
=
c_1 \Lambda_1^\mu \Lambda_2^\nu
+
c_2 \Lambda_2^\mu \Lambda_1^\nu
+
c_3 \Lambda_1^\mu \Lambda_1^\nu
\\+
c_4 \Lambda_2^\mu \Lambda_2^\nu
+
c_5 {\mathcal{Q}}_1^\mu {\mathcal{Q}}_2^\nu
+
c_6 {\mathcal{Q}}_1^\mu \Lambda_1^\nu
\\+
c_7 {\mathcal{Q}}_1^\mu \Lambda_2^\nu
+
c_8 \Lambda_1^\mu {\mathcal{Q}}_2^\nu
+
c_9 \Lambda_2^\mu {\mathcal{Q}}_2^\nu.\end{gathered}$$ It turns out that also the coefficients $c_6-c_9$ vanish. If analyzed perturbatively (with respect to the external field coupling) this can be understood from Furry’s theorem [@baier_interaction_1975; @becker_vacuum_1975]. Since a closed fermion loop with an odd number of vertices does not contribute to the final amplitude, only diagrams with an even number of external field couplings ($eA^\mu$) contribute to ${T}^{\mu\nu}$. Due to gauge invariance and the fact that $T^{\mu\nu}$ is a tensor, the external field can enter only as ${\mathfrak{F}}^{\mu\nu}$ (which is linear in $A^\mu$). Since it is not possible to construct a scalar linear in ${\mathfrak{F}}^{\mu\nu}$ using only the four-vectors $q_1^\mu$, $q_2^\mu$ and $k^\mu$, the tensor structure cannot involve an odd number of the tensor ${\mathfrak{F}}^{\mu\nu}$ (note that $q_1{\mathfrak{F}}q_2 = q{\mathfrak{F}}q = 0$). As a consequence, the coefficients $c_6-c_9$ (which are linear in $\Lambda^\mu_i$ and thus in the external field) should vanish. We will later see that this is indeed the case.
The coefficients $c_i$ in Eq. (\[eqn:polarizationoperatordecomposition\]) can be determined by contracting ${T}^{\mu\nu}(q_1,q_2)$ with the appropriate four-vectors. Especially, using again the Ward-Takahashi identity, we obtain $$\begin{gathered}
\label{eqn:Qcontractionrelations}
{\mathcal{Q}}_{1\mu} {T}^{\mu\nu}
=
\frac{q_1^2}{kq} k_\mu {T}^{\mu\nu},
\quad
{T}^{\mu\nu} {\mathcal{Q}}_{2\nu}
=
\frac{q_2^2}{kq} {T}^{\mu\nu} k_\nu.\end{gathered}$$
Thus, effectively, we need to determine the contractions of ${T}^{\mu\nu}(q_1,q_2)$ with the four-vectors $k^\mu$ and $\Lambda_i^\mu$ to determine the coefficients $c_i$, i.e. we need to calculate the $({{{}-}},{\perp})$-components of ${T}^{\mu\nu}(q_1,q_2)$ in modified light-cone coordinates. Since $k^\mu$ has only a ${{{}+}}$-component, the evaluation of the derivatives is now considerably simplified. Leaving the term proportional to $pp'$ aside, we see that all derivatives which act on $kj$ or $kj'$ can be ignored. They would result in the replacement of $p^\mu$ or $p'^\mu$ by $k^\mu$. Since $k_\mu {\mathfrak{F}}^{\mu\nu} = k_\mu {\mathfrak{F}}^{2\mu\nu} = k_\mu {\mathfrak{F}}^{*\mu\nu} = 0$ and $k^2=k\Lambda_i=0$, we do not need to consider those terms. The derivatives acting on $kj$ or $kj'$ are therefore only important to determine the term proportional to $pp'$. However, this is achieved more easily if the calculation presented in Sec. \[sec:polarizationoperatorintegralevaluation\] is repeated with a scalar source term ${\mathcal{J}} pp'$ in the exponent (see Sec. \[sec:scalarterm\]).
To calculate the preexponent of the polarization operator, we must now insert the explicit expressions given in Eq. (\[eqn:dressedvertexGtensorsdefinition\]) into the trace in Eq. (\[eqn:sfqed\_polarizationoperatortraceA\]). Many terms of the trace, e.g., the terms proportional to ${\mathfrak{F}}^{\mu\nu}$, ${\mathfrak{F}}^{2\mu\nu}$, ${\mathfrak{F}}^{2\mu\rho} p_\rho$ vanish, as they are contracted with $k^\mu$ or $\Lambda_i^\mu$ from each side. Using the relations in Appendix \[sec:tensorrelationsappendix\], we can show that Eq. (\[eqn:sfqed\_polarizationoperatortraceA\]) can be substituted by the following expression:
$$\begin{gathered}
\label{eqn:sfqed_polarizationoperatortraceB}
m^2 g^{\mu\nu} + p^\mu p'^\nu + p'^\mu p^\nu
+ g^{\mu\nu} \big[G_3 p{\mathfrak{F}}_{y}p' + G_3 p{\mathfrak{F}}_{x}p' - 2 G_3^2 (p{\mathfrak{F}}^2_{xy} p') -(pp') \big]
\\- G_3 \big[ ({\mathfrak{F}}_y p')^\mu p^\nu - ({\mathfrak{F}}_y p)^\mu p'^\nu + p^\mu ({\mathfrak{F}}_x p')^\nu - p'^\mu ({\mathfrak{F}}_x p)^\nu \big]
- G_1 \big[ p^\mu ({\mathfrak{F}}_y p')^\nu + p'^\mu ({\mathfrak{F}}_y p)^\nu + ({\mathfrak{F}}_x p)^\mu p'^\nu + ({\mathfrak{F}}_x p')^\mu p^\nu \big]
\\+ G_1^2 \big[ ({\mathfrak{F}}_x p)^\mu ({\mathfrak{F}}_y p')^\nu + ({\mathfrak{F}}_x p')^\mu ({\mathfrak{F}}_y p)^\nu \big]
- G^2_3 \big[({\mathfrak{F}}_y p)^\mu ({\mathfrak{F}}_x p')^\nu + ({\mathfrak{F}}_y p')^\mu ({\mathfrak{F}}_x p)^\nu \big],\end{gathered}$$
where ${\mathfrak{F}}^{2\mu\nu}_{xy}={\mathfrak{F}}^{\mu\rho}(kx) {\mathfrak{F}}_{\rho}^{\phantom{\rho}\nu}(ky)={\mathfrak{F}}^{\mu\rho}(ky) {\mathfrak{F}}_{\rho}^{\phantom{\rho}\nu}(kx)$ \[here the replacement $p^\mu \longrightarrow (-i) {\partial}_j^\mu$ and $p'^\mu \longrightarrow (-i) {\partial}_{j'}^\mu$ is understood if the trace is inserted in Eq. (\[eqn:polarizationoperatorafterintegralstaken\]); see Eq. (\[eqn:momentumreplacementtrace\])\]. Since the term proportional to $pp'$ enters as $g^{\mu\nu}$, it modifies only the diagonal coefficients $c_3$ and $c_4$.
Evaluation of the derivatives
-----------------------------
Leaving the term proportional to $pp'$ aside, we can ignore derivatives acting on $kj$ and $kj'$ as discussed above \[this implies that the derivatives do not act on $kp$, $kp'$, and $ky$; see Eq. (\[eqn:kpkpprimekyidentifications\])\]. The remaining nontrivial source-dependent part of the phase is given by \[see Eq. (\[eqn:sfqed\_polarizationoperatorfinalphase\])\] $$\begin{gathered}
-
\frac{i}{s+t} \Big[ t \, q_2j - s\, q_2j'
+
\frac{1}{4} (j+j')^2
+
\frac{1}{2} (j+j')\uplambda \Big].\end{gathered}$$ The squared term contributes only if both derivatives act on it, which results in the replacement
$$\begin{gathered}
\label{eqn:ppprimereplacementA}
p^\alpha p'^\beta
\longrightarrow
(-i)^2 {\partial}^\alpha_j {\partial}^\beta_{j'}
\longrightarrow
\frac{i}{2(s+t)} g^{\alpha\beta}\end{gathered}$$
and the only nonzero contribution arises from the first three terms in Eq. (\[eqn:sfqed\_polarizationoperatortraceB\]). If the derivatives act on the other source terms, we obtain the replacement $$\begin{gathered}
\label{eqn:ppprimereplacementB}
p^\alpha p'^\beta
\longrightarrow
(-i)^2 {\partial}^\alpha_j {\partial}^\beta_{j'}
\\\longrightarrow
-\frac{1}{(s+t)^2}
\Big(tq_2^\alpha + \frac12 \uplambda^\alpha\Big)
\Big(sq_2^\beta - \frac12 \uplambda^\beta\Big).\end{gathered}$$
After these replacements are applied to Eq. (\[eqn:sfqed\_polarizationoperatortraceB\]) and the sources are set to zero, we obtain (effectively) the following expression for Eq. (\[eqn:sfqed\_polarizationoperatortraceB\]): $$\begin{gathered}
\label{eqn:sfqed_polarizationoperatortraceC}
\begin{gathered}
g^{\mu\nu} \Big[ m^2 + \frac{i}{\tau}
- \frac{e}{4kq\, \mu} (q{\mathfrak{F}}_y\uplambda + q{\mathfrak{F}}_x\uplambda) \hspace*{3cm}
\\\hspace*{3cm}+ \frac{e^2}{2(kq)^2} \frac{\tau}{\mu} q{\mathfrak{F}}^2_{xy}q - pp' \Big]\\
\begin{aligned}
&-
2\frac{\mu}{\tau} q_2^\mu q_2^\nu - \frac{v}{2\tau} (q_2^\mu \uplambda^\nu + \uplambda^\mu q_2^\nu) + \frac{1}{2\tau^2} \uplambda^\mu \uplambda^\nu
\\
&+ \frac{e}{kq} v \big[ q_2^\mu ({\mathfrak{F}}_y q)^\nu + ({\mathfrak{F}}_x q)^\mu q_2^\nu \big]
\\
&- \frac{e}{4kq} \frac{1}{\mu}
\big[ ({\mathfrak{F}}_y q)^\mu \uplambda^\nu + \uplambda^\mu ({\mathfrak{F}}_x q)^\nu \big]
\\
&+ \frac{e}{4kq} \frac{v^2}{\mu}
\big[ \uplambda^\mu ({\mathfrak{F}}_y q)^\nu + ({\mathfrak{F}}_x q)^\mu \uplambda^\nu \big]
\\
&+ \frac{e^2}{2(kq)^2} \frac{\tau}{\mu} \big[({\mathfrak{F}}_y q)^\mu ({\mathfrak{F}}_x q)^\nu
-
v^2 ({\mathfrak{F}}_x q)^\mu ({\mathfrak{F}}_y q)^\nu \big]
\end{aligned}
\end{gathered}\end{gathered}$$ \[note that terms proportional to $({\mathfrak{F}}\uplambda)^{\mu}$, $({\mathfrak{F}}\uplambda)^{\nu}$ can be omitted\]. By changing the proper-time integrations from $s$, $t$ to $\tau$, $v$ [@baier_interaction_1975], $$\begin{gathered}
\label{eqn:stintegraltransform}
\int_0^\infty ds \, dt \, f(s,t)
=
\frac12 \int_{-1}^{+1} dv \int_0^\infty d\tau \, \tau \tilde{f}(\tau,v)\end{gathered}$$ we see that the terms linear in $v$ vanish. Those terms determine the coefficients $c_6-c_9$, which are therefore zero (as already anticipated from Furry’s theorem).
Scalar term {#sec:scalarterm}
-----------
To determine the term proportional to $pp'$, we add the scalar source term $i {\mathcal{J}} pp'$ to the phase (instead of $i p_\mu j^\mu + i p'_\mu j'^\mu$) and repeat the calculation presented in Sec. \[sec:polarizationoperatorintegralevaluation\]. The propagators are represented in the same way \[see Eq. (\[eqn:sfqed\_polarizationoperatorschwingerpropagator\])\], and we replace $pp'$ by $-i\frac{{\partial}}{{\partial}{\mathcal{J}}}$. Then, we take the integrals in $dx^{{{}+}}$, $dx^{\perp}$, $dy^{{{}+}}$, $dy^{\perp}$, $dp'^{{{}-}}$, $dp'^{\perp}$, $dp'^{{{}+}}$, and $dp^{{{}+}}$. Instead of Eq. (\[eqn:yminuspminusdeltafunctions\]), we obtain now $$\begin{gathered}
(2\pi) \delta\Big[y^{{{}-}}- x^{{{}-}}-\frac{4st - {\mathcal{J}}^2}{2(s+t+{\mathcal{J}})} q^{{{}-}}\Big]
\\ \times \,
(2\pi) \delta[2(s+t+{\mathcal{J}}) p^{{{}-}}+ (2t+{\mathcal{J}}) q^{{{}-}}].\end{gathered}$$ The remaining part of the phase (including the part from the propagator) can be written as $$\begin{gathered}
iS_{T}' = i\big[ q_2^{{{}+}}y^{{{}-}}- q_1^{{{}+}}x^{{{}-}}- p^{\perp}p^{\perp}{\mathcal{J}} \\+ (-p^{\perp}p^{\perp}- m^2) (s+t) - p^{\perp}\uplambda^{\perp}+ \Uplambda \big].\end{gathered}$$ It is now convenient to shift the proper-time integrations $$\begin{gathered}
\label{eqn:stintegrationshift}
s \longrightarrow s - \frac12 {\mathcal{J}},
\quad
t \longrightarrow t - \frac12 {\mathcal{J}}.\end{gathered}$$ Due to this shift, also the integral boundaries of the proper-time integrations depend on the source. However, if the derivative acts on the integral boundaries, either $s$ or $t$ is set to zero or to infinity. The resulting terms do not depend on the external field since $s=0$ or $t=0$ implies $\mu=0$, $ky=kx$ and thus $\uplambda^\mu = 0$ and $\Uplambda = 0$. On the other hand, the terms at $s\to\infty$ or $t\to\infty$ do not contribute because the field-dependent part of the integral is convergent. Since we want to calculate only the field-dependent part of the polarization operator \[see Eq. (\[eqn:polarizationoperatorfinal\])\], we will ignore the source dependence of the integral boundaries.
After the shift in Eq. (\[eqn:stintegrationshift\]), the delta functions read $$\begin{gathered}
(2\pi) \delta[y^{{{}-}}- x^{{{}-}}- (2\mu-{\mathcal{J}})q^{{{}-}}]
\\ \times \,
(2\pi) \delta[2p^{{{}-}}(s+t) + 2 q^{{{}-}}t]\end{gathered}$$ and the phase is given by $$\begin{gathered}
iS_{T}' = i\Big[ (q_2^{{{}+}}- q_1^{{{}+}}) x^{{{}-}}+ \Big(\mu - \frac12 {\mathcal{J}}\Big) q_2^2 - m^2 (s+t-{\mathcal{J}}) \\- p^{\perp}p^{\perp}(s+t) - p^{\perp}\uplambda^{\perp}+ \Uplambda \Big].\end{gathered}$$ We can now use the delta functions to take the integrals in $dy^{{{}-}}$ and $dp^{{{}-}}$ (we keep writing $y^{{{}-}}$ and $p^{{{}-}}$ for convenience). We then obtain the identities $$\begin{gathered}
\begin{gathered}
kp = - \frac{t}{s+t} kq,
\quad
kp' = \frac{s}{s+t} kq,\\
ky = kx + (2\mu -{\mathcal{J}}) kq
\end{gathered}\end{gathered}$$ \[for ${\mathcal{J}}=0$ this agrees with Eq. (\[eqn:kpkpprimekyidentifications\])\]. The shift in the proper-time integrals has the advantage that $kp$ and $kp'$ are now independent of ${\mathcal{J}}$. We could have proceeded similarly also in the calculation of the other terms. However, since we ignored sources contracted with $k$, this was not necessary.
Taking now the Gaussian integrals in $dp^{{\scalebox{.64}{$\matheuler{I}$}}}$, $dp^{{\scalebox{.64}{$\matheuler{II}$}}}$, we obtain the prefactor $\frac{\pi}{i(s+t)}$, and the final phase is given by $$\begin{gathered}
\label{eqn:sfqed_polarizationoperatorfinalphasezerosourcesscalarterm}
iS_{T}' = i\Big[ (q_2^{{{}+}}- q_1^{{{}+}}) x^{{{}-}}+ \Big(\mu - \frac12 {\mathcal{J}}\Big) q_2^2 - m^2 (\tau-{\mathcal{J}}) \\ + \tau m^2 \sum_{i=1,2} \xi_i^2 (I^2_i-J_i) \Big],\end{gathered}$$ where $I_i$ and $J_i$ are defined in Eq. (\[eqn:defIJintegrals\]) \[for zero sources Eq. (\[eqn:sfqed\_polarizationoperatorfinalphasezerosourcesscalarterm\]) agrees with Eq. (\[eqn:sfqed\_polarizationoperatorfinalphasezerosources\])\]. Since $pp'$ in the preexponent is only multiplied by $g^{\mu\nu}$ \[see Eq. (\[eqn:sfqed\_polarizationoperatortraceC\])\], the evaluation of the derivative is not very cumbersome, and we obtain the replacement $$\begin{gathered}
\label{eqn:ppprimereplacement}
pp'
\longrightarrow
(-i) \frac{{\partial}}{{\partial}{\mathcal{J}}}
\longrightarrow
- \frac12 q_2^2 + m^2 \\+ m^2 \frac{\tau}{2\mu} \sum_{i=1,2} \xi^2_i \big[\psi^2_i(ky) - 2 I_i \psi_i(ky) \big]\end{gathered}$$ after ${\mathcal{J}}$ is set to zero (as explained above, we have ignored the source dependence of the proper-time integral boundaries).
To symmetrize the final expression, we change the $x^{{{}-}}$-integration by defining \[see Eq. (\[eqn:symmetricintegralshift\])\] $$\begin{gathered}
\label{eqn:symmetricintegralshiftscalarterm}
\tilde{z}^{{{}-}}= x^{{{}-}}+ \Big(\mu-\frac12 {\mathcal{J}}\Big) \, q^{{{}-}}\end{gathered}$$ ($\tilde{z}^{{{}-}}=z^{{{}-}}$ for ${\mathcal{J}}=0$), which implies $$\begin{gathered}
\begin{aligned}
kx &= k\tilde{z} - \Big(\mu-\frac12 {\mathcal{J}}\Big) kq,&
ky &= k\tilde{z} + \Big(\mu-\frac12 {\mathcal{J}}\Big) kq
\end{aligned}\end{gathered}$$ and $$\begin{gathered}
(q_2^{{{}+}}- q_1^{{{}+}}) x^{{{}-}}+ \Big(\mu - \frac12 {\mathcal{J}}\Big) q_2^2
\\=
(q_2^{{{}+}}- q_1^{{{}+}}) \tilde{z}^{{{}-}}+ \Big(\mu - \frac12 {\mathcal{J}}\Big) q_1q_2.\end{gathered}$$ Finally, we obtain the symmetric replacement $$\begin{gathered}
\label{eqn:symmetricppprimereplacement}
pp'
\longrightarrow
(-i) \frac{{\partial}}{{\partial}{\mathcal{J}}}
\longrightarrow
- \frac12 q_1q_2+ m^2
+
m^2 \frac{\tau}{2\mu} \sum_{i=1,2} \xi^2_i
\\\times \,
\Big[\frac12\psi^2_i(kx) + \frac12\psi^2_i(ky) - I_i \psi_i(kx) - I_i \psi_i(ky) \Big]\end{gathered}$$ (we assume that at $x^{{{}-}}=\pm\infty$ the external field vanishes, and therefore the derivative does not act on the integral boundaries, which now also depend on the source).
Final result
------------
To determine the nonvanishing coefficients $c_1-c_5$ of the polarization operator \[see Eq. (\[eqn:polarizationoperatordecomposition\])\], we combine now Eqs. (\[eqn:polarizationoperatorafterintegralstaken\]), (\[eqn:symmetricintegralshift\]), (\[eqn:sfqed\_polarizationoperatortraceC\]), and (\[eqn:symmetricppprimereplacement\]). Furthermore, we define the following functions: $$\begin{gathered}
\label{eqn:XZdefinition}
\begin{aligned}
X_{ij}
&=
[I_i-\psi_i(ky)] \, [I_j - \psi_j(kx)],\\
Z_i
&=
\frac12 [\psi_i(kx) - \psi_i(ky)]^2
\end{aligned}\end{gathered}$$ and note that, for $j=j'=0$, $$\begin{aligned}
{\mathfrak{F}}^{\mu\nu}_x \Lambda_{i\nu}
&=
- \frac{m}{e} k^\mu \xi_i \psi_i(kx),
\nonumber\displaybreak[0]\\\nonumber
\uplambda^\mu
&=
- 2m\tau \sum_{i=1,2} \Lambda_i^\mu \xi_i I_i,
\nonumber\displaybreak[0]\\\nonumber
e\Lambda_{i\mu} {\mathfrak{F}}^{\mu\nu}_x q_\nu
&=
m\, kq\, \xi_i\, \psi_i(kx),
\nonumber\displaybreak[0]\\\nonumber
\Lambda_i\uplambda
&=
2m \tau \xi_i I_i,
\nonumber\displaybreak[0]\\\nonumber
e q{\mathfrak{F}}_{x}\uplambda
&=
2 kq \,\tau m^2 \sum_{i=1,2} \xi^2_i \psi_i(kx) I_i,
\displaybreak[0]\nonumber\\
e^2 q{\mathfrak{F}}^2_{x,y}q
&=
m^2 (kq)^2 \sum_{i=1,2} \xi_i^2 \psi_i(kx) \psi_i(ky).\end{aligned}$$
Using these relations, we obtain the following expression for the field-dependent part of the tensor ${T}^{\mu\nu}$: $$\begin{gathered}
\label{eqn:polarizationoperatorfinal}
{T}^{\mu\nu}(q_1,q_2)
-
{T}^{\mu\nu}_{{\mathfrak{F}}=0}(q_1,q_2)
=
-i\pi e^2\, \delta^{({{{}-}},{\perp})}(q_1-q_2)
\\ \times\,
\int_{-1}^{+1} dv \int_0^\infty \frac{d\tau}{\tau} \,
\int_{-\infty}^{+\infty} dz^{{{}-}}\,
\big[
b_1 \Lambda_1^\mu \Lambda_2^\nu
+
b_2 \Lambda_2^\mu \Lambda_1^\nu
\\+
b_3 \Lambda_1^\mu \Lambda_1^\nu
+
b_4 \Lambda_2^\mu \Lambda_2^\nu
+
b_5 {\mathcal{Q}}_1^\mu {\mathcal{Q}}_2^\nu
\big] e^{i\Phi} ,\end{gathered}$$ where the field-independent phase reads \[see Eqs. (\[eqn:sfqed\_polarizationoperatorfinalphasezerosources\]) and (\[eqn:symmetricphaseterms\])\] $$\begin{gathered}
\label{eqn:polarizationoperator_fieldindependentphaseA}
e^{i\Phi}
=
\exp \lcb i {\left[}(q_2^{{{}+}}-q_1^{{{}+}}) z^{{{}-}}+
\mu q_1q_2
-
\tau m^2 {\right]}{\right\}\end{gathered}$$ \[$\mu = \frac14 \tau (1-v^2)$; see Eq. (\[eqn:tauvmudefinition\])\] and $$\begin{aligned}
b_1 &= 2m^2 \xi_1 \xi_2 \Big( \frac{\tau}{4\mu} X_{12} - \frac{\tau v^2}{4\mu} X_{21} \Big) e^{i\tau\beta},\nonumber\displaybreak[0]\\\nonumber
b_2 &= 2m^2 \xi_1 \xi_2 \Big( \frac{\tau}{4\mu} X_{21} - \frac{\tau v^2}{4\mu} X_{12} \Big) e^{i\tau\beta},\nonumber\displaybreak[0]\\\nonumber
b_3 &= - \Big( \frac{i}{\tau} + \frac{q_1q_2}{2} \Big) {\left(}e^{i\tau\beta} -1 {\right)}\\
&\phantom{= }+ 2m^2 \Big[ \frac{\tau}{4\mu} {\left(}\xi_1^2 Z_1 + \xi_2^2 Z_2 {\right)}+ \xi_1^2 X_{11} \Big] e^{i\tau\beta},
\nonumber\displaybreak[0]\\\nonumber
b_4 &= - \Big( \frac{i}{\tau} + \frac{q_1q_2}{2} \Big) {\left(}e^{i\tau\beta} -1 {\right)}\\
&\phantom{= }+ 2m^2 \Big[ \frac{\tau}{4\mu} {\left(}\xi_1^2 Z_1 + \xi_2^2 Z_2 {\right)}+ \xi_2^2 X_{22} \Big] e^{i\tau\beta},
\nonumber\displaybreak[0]\\
b_5 &= - \frac{2\mu}{\tau} {\left(}e^{i\tau\beta} -1 {\right)}.\end{aligned}$$ The field-dependent phase is given by \[see Eq. (\[eqn:sfqed\_polarizationoperatorfinalphasezerosources\])\] $$\begin{gathered}
\label{eqn:polarizationoperator_fielddependentphaseA}
e^{i\tau\beta} = \exp \big[ i \tau m^2 \sum_{i=1,2} \xi_i^2 (I_i^2-J_i) \big],\end{gathered}$$ where \[see Eq. (\[eqn:IJsymmetricintegrals\])\] $$\begin{gathered}
\begin{aligned}
I_i
&=
\frac12 \int_{-1}^{+1} d\lambda \, \psi_i(kz - \lambda \mu kq),\\
J_i
&=
\frac12 \int_{-1}^{+1} d\lambda \, \psi^2_i(kz - \lambda \mu kq)
\end{aligned}\end{gathered}$$ and \[see Eq. (\[eqn:XZdefinition\])\] $$\begin{gathered}
\label{eqn:XYwithkz}
\begin{aligned}
X_{ij}
&=
[I_i-\psi_i(kz + \mu kq)] \, [I_j - \psi_j(kz - \mu kq)],\\
Z_i
&=
\frac12 [\psi_i(kz - \mu kq) - \psi_i(kz + \mu kq)]^2.
\end{aligned}\end{gathered}$$
We note that, using the metric tensor $g^{\mu\nu}$, we can construct the following projection tensor [@becker_vacuum_1975]: $$\begin{gathered}
\label{eqn:projectiontensor}
G^{\mu\nu}
=
q_2^\mu q_1^\nu - q_1q_2 \, g^{\mu\nu}, \end{gathered}$$ which obeys $$\begin{gathered}
q_{1\mu} G^{\mu\nu} = G^{\mu\nu} q_{2\nu} = 0\end{gathered}$$ and can be decomposed as $$\begin{gathered}
\label{eqn:projectiontensordecomposition}
G^{\mu\nu}
=
q_1 q_2 {\left(}\Lambda^\mu_1 \Lambda^\nu_1 + \Lambda^\mu_2 \Lambda^\nu_2 {\right)}+ {\mathcal{Q}}_1^\mu {\mathcal{Q}}^\nu_2.\end{gathered}$$ This shows that the decomposition given in Eq. (\[eqn:polarizationoperatorfinal\]) has the structure claimed in Ref. [@becker_vacuum_1975].
Discussion of the results {#sec:discussion}
=========================
Comparison with the literature {#sec:litcomparison}
------------------------------
The expression we obtained for the field-dependent part of ${T}^{\mu\nu}$ in Eq. (\[eqn:polarizationoperatorfinal\]) is manifestly symmetric in $q_1$ and $q_2$. We will now show how the alternative representation found in Ref. [@baier_interaction_1975] can be derived from our calculation. To this end we do not apply the shift in Eqs. (\[eqn:symmetricintegralshift\]) and (\[eqn:symmetricintegralshiftscalarterm\]), which means that we have to use the replacement given in Eq. (\[eqn:ppprimereplacement\]) \[rather than Eq. (\[eqn:symmetricppprimereplacement\])\] for the $pp'$ term in Eq. (\[eqn:sfqed\_polarizationoperatortraceC\]). This modifies the coefficients $b_3$ and $b_4$. Furthermore, we introduce the variable $$\begin{gathered}
\label{eqn:baierintegralshift}
z'^{{{}-}}= x^{{{}-}}+ 2\mu q^{{{}-}}= z^{{{}-}}+ \mu q^{{{}-}},\end{gathered}$$ which allows us to write \[see Eq. (\[eqn:symmetricphaseterms\])\] $$\begin{gathered}
(q_2^{{{}+}}-q_1^{{{}+}}) x^{{{}-}}+
\mu q_2^2
=
(q_2^{{{}+}}-q_1^{{{}+}}) z^{{{}-}}+
\mu q_1 q_2
\\=
(q_2^{{{}+}}-q_1^{{{}+}}) z'^{{{}-}}+
\mu q_1^2\end{gathered}$$ and \[see Eq. (\[eqn:kpkpprimekyidentifications\])\] $$\begin{gathered}
kx = kz' - 2\mu kq, \quad ky = kz'\end{gathered}$$ (here and in the remaining subsection, we assume that all sources are set to zero). Thus, we obtain the following representation \[see Eq. (\[eqn:defIJintegrals\])\]: $$\begin{gathered}
\begin{aligned}
I_i
&=
\int_{0}^{1} d\lambda\, \psi_i(kz' - 2kq \mu \lambda),\\
J_i
&=
\int_{0}^{1} d\lambda\, \psi^2_i(kz' - 2kq \mu \lambda),
\end{aligned}
\\
I_i^2-J_i
=
\Big[ \int_{0}^{1} d\lambda\, \Delta_i(\mu \lambda) \Big]^2
-
\int_{0}^{1} d\lambda\, \Delta_i^2(\mu \lambda),\end{gathered}$$ where we introduced [@baier_interaction_1975] $$\begin{gathered}
\Delta_i(r)
=
\psi_i(kz'-2kq r) - \psi_i(kz').\end{gathered}$$ Furthermore, it is useful to define \[compare with Eq. (\[eqn:XZdefinition\])\] $$\begin{gathered}
\label{eqn:XYdefinition}
\begin{aligned}
X_{ij}
&=
[I_i-\psi_i(ky)] \, [I_j - \psi_j(kx)],\\
Y_i
&=
[I_i-\psi_i(ky)] \, [\psi_i(kx) - \psi_i(ky)]
\end{aligned}\end{gathered}$$ which can be written as $$\begin{gathered}
\begin{aligned}
X_{ij}
&=
{\left[}\int_0^1 d\lambda \, \Delta_i(\mu\lambda) {\right]}{\left[}\int_0^1 d\lambda \, \Delta_j(\mu\lambda) - \Delta_j(\mu) {\right]},\\
Y_i
&=
{\left[}\int_0^1 d\lambda\, \Delta_i(\mu\lambda) {\right]}\Delta_i(\mu).
\end{aligned}\end{gathered}$$
Finally, we obtain the following alternative representation for the field-dependent part of ${T}^{\mu\nu}$: $$\begin{gathered}
\label{eqn:polarizationoperatorbaier}
{T}^{\mu\nu}(q_1,q_2)
-
{T}^{\mu\nu}_{{\mathfrak{F}}=0}(q_1,q_2)
=
-i\pi e^2\, \delta^{({{{}-}},{\perp})}(q_1-q_2)
\\ \times\,
\int_{-1}^{+1} dv \int_0^\infty \frac{d\tau}{\tau} \,
\int_{-\infty}^{+\infty} dz'^{{{}-}}\,
\big[
b_1 \Lambda_1^\mu \Lambda_2^\nu
+
b_2 \Lambda_2^\mu \Lambda_1^\nu
\\+
b'_3 \Lambda_1^\mu \Lambda_1^\nu
+
b'_4 \Lambda_2^\mu \Lambda_2^\nu
+
b_5 {\mathcal{Q}}_1^\mu {\mathcal{Q}}_2^\nu
\big] e^{i\Phi}\end{gathered}$$ with the coefficients $$\begin{gathered}
\begin{aligned}
b_1 &= 2m^2 \xi_1 \xi_2 \Big( \frac{\tau}{4\mu} X_{12} - \frac{\tau v^2}{4\mu} X_{21} \Big) e^{i\tau\beta},\\
b_2 &= 2m^2 \xi_1 \xi_2 \Big( \frac{\tau}{4\mu} X_{21} - \frac{\tau v^2}{4\mu} X_{12} \Big) e^{i\tau\beta},\\
b'_3 &= - \Big( \frac{i}{\tau} + \frac{q_2^2}{2} \Big) {\left(}e^{i\tau\beta} -1 {\right)}\\
&\phantom{= }+ 2m^2 \Big[ \frac{\tau}{4\mu} {\left(}\xi_1^2 Y_1 + \xi_2^2 Y_2 {\right)}+ \xi_1^2 X_{11} \Big] e^{i\tau\beta},\\
b'_4 &= - \Big( \frac{i}{\tau} + \frac{q_2^2}{2} \Big) {\left(}e^{i\tau\beta} -1 {\right)}\\
&\phantom{= }+ 2m^2 \Big[ \frac{\tau}{4\mu} {\left(}\xi_1^2 Y_1 + \xi_2^2 Y_2 {\right)}+ \xi_2^2 X_{22} \Big] e^{i\tau\beta},\\
b_5 &= - \frac{2\mu}{\tau} {\left(}e^{i\tau\beta} -1 {\right)}\end{aligned}\end{gathered}$$ and phases $$\begin{gathered}
\begin{aligned}
e^{i\Phi}
&=
\exp \lcb i {\left[}(q_2^{{{}+}}-q_1^{{{}+}}) z'^{{{}-}}+
\mu q_1^2
-
\tau m^2 {\right]}{\right\},
\\
e^{i\tau\beta} &= \exp \big[ i \tau m^2 \sum_{i=1,2} \xi_i^2 (I_i^2-J_i) \big].
\end{aligned}\end{gathered}$$ This representation coincides with Eq. (2.27) in Ref. [@baier_interaction_1975].
Constant-crossed field
----------------------
The polarization operator for a constant-crossed field was first obtained in Refs. [@narozhnyi_propagation_1968; @batalin_preprint_1968] (see also Refs. [@batalin_greens_1971; @ritus_radiative_1972; @ritus_1985]). We show now how this result can be obtained from the expression in Eq. (\[eqn:polarizationoperatorfinal\]).
A constant-crossed field is characterized by $$\begin{gathered}
\label{eqn:constantcrossedfield}
\psi_1(\phi) = \phi,
\quad
\psi_2(\phi) = 0\end{gathered}$$ (the latter condition corresponds to $\xi_2=0$, and we will write $\xi = \xi_1$ in this paragraph). The field tensor and its square are then given by \[see Eq. (\[eqn:fieldtensor\])\] $$\begin{gathered}
F^{\mu\nu}
=
f_1^{\mu\nu},
\quad
F^{2\mu\nu}
=
\frac{m^2 \xi^2}{e^2} k^{\mu} k^{\nu}.\end{gathered}$$ For a constant-crossed field, we obtain $$\begin{gathered}
\label{eqn:ccfield_IJXZ}
\begin{gathered}
I_1 = kz,
\quad
J_1 = (kz)^2 + \frac13 (\mu kq)^2,
\quad
I_2 = J_2 = 0,
\\
X_{11} = - (\mu kq)^2,
\quad
Z_{1} = 2 (\mu kq)^2,
\\
Z_{2} = X_{12} = X_{21} = X_{22} = 0.
\end{gathered}\end{gathered}$$ After inserting these expressions into Eq. (\[eqn:polarizationoperatorfinal\]), we can take the integral in $dz^{{{}-}}$ and obtain the remaining delta function $2\pi \, \delta^{({{{}+}})}(q_1-q_2)$, which implies that the polarization tensor for a constant-crossed field is diagonal in the external photon four-momenta. We define therefore the four-vectors \[see Eq. (\[eqn:Qidef\])\] $$\begin{gathered}
q^\mu = q_1^\mu = q_2^\mu,
\quad
{\mathcal{Q}}^\mu = {\mathcal{Q}}_1^\mu = {\mathcal{Q}}_2^\mu = \frac{k^\mu q^2 - q^\mu kq}{kq}.\end{gathered}$$ They obey $$\begin{gathered}
k{\mathcal{Q}}=-kq,
\quad
q{\mathcal{Q}}=0,
\quad
{\mathcal{Q}}^2=-q^2.\end{gathered}$$ The four-vectors $q^\mu$, ${\mathcal{Q}}^\mu$, $\Lambda_1^\mu$, and $\Lambda_2^\mu$ form a complete set, and we obtain the following representation of the metric tensor: $$\begin{gathered}
g^{\mu\nu} = \frac{1}{q^2}{\left(}q^\mu q^\nu - {\mathcal{Q}}^\mu {\mathcal{Q}}^\nu {\right)}- \Lambda_1^\mu \Lambda_1^\nu - \Lambda_2^\mu \Lambda_2^\nu.\end{gathered}$$ From Eq. (\[eqn:polarizationoperatorfinal\]) we obtain now the following representation of the field-dependent part of ${T}^{\mu\nu}$ in a constant-crossed field \[see Eq. (\[eqn:constantcrossedfield\])\]: $$\begin{gathered}
\label{eqn:polarizationoperatorccfieldA}
{T}^{\mu\nu}(q_1,q_2)
-
{T}^{\mu\nu}_{{\mathfrak{F}}=0}(q_1,q_2)
=
-2i\pi^2 e^2\, \delta^{4}(q_1-q_2)
\\ \times\,
\int_{-1}^{+1} dv \int_0^\infty \frac{d\tau}{\tau} \,
\big[
b_{3} \Lambda_1^\mu \Lambda_1^\nu
+
b_{4} \Lambda_2^\mu \Lambda_2^\nu
+
b_{5} {\mathcal{Q}}^\mu {\mathcal{Q}}^\nu
\big] e^{i\Phi_{}},\end{gathered}$$ where $$\begin{gathered}
\begin{aligned}
b_{3} &= - \Big( \frac{i}{\tau} + \frac{q^2}{2} \Big) {\left(}e^{i\tau\beta_{}} -1 {\right)}\\
&\phantom{= }+ m^6 \chi^2 \tau^2 \frac{1}{4}(1-v^2) \Big[ 1 - \frac{1}{2}(1-v^2) \Big] e^{i\tau\beta_{}},\\
b_{4} &= - \Big( \frac{i}{\tau} + \frac{q^2}{2} \Big) {\left(}e^{i\tau\beta_{}} -1 {\right)}+ m^6 \chi^2 \tau^2 \frac{1}{4}(1-v^2) e^{i\tau\beta_{}},\\
b_{5} &= - \frac{1}{2}(1-v^2) {\left(}e^{i\tau\beta_{}} -1 {\right)}\end{aligned}\end{gathered}$$ and the phases are given by $$\begin{gathered}
\begin{aligned}
i\Phi_{} &= -i\tau a,& a &= m^2 \Big[ 1-\frac{1}{4}(1-v^2) \frac{q^2}{m^2} \Big],\\
i\tau\beta_{} &= - \frac{i}{3} \tau^3 b,& b &= m^6 \chi^2 \Big[ \frac14(1-v^2) \Big]^2
\end{aligned}\end{gathered}$$ (in the following, we will make the change of variables $\tau \to t$, where $\tau^3 b = t^3$ and $\rho = {{a}/{\sqrt[3]{b}}}$). Here we have introduced the quantum nonlinearity parameter $$\begin{gathered}
\label{eqn:chidefinition}
\chi
=
-\frac{e\sqrt{qF^2q}}{m^3}
=
\xi \frac{\sqrt{(kq)^2}}{m^2}\end{gathered}$$ ($\kappa$ in Refs. [@narozhnyi_propagation_1968; @ritus_radiative_1972]).
We can rewrite now $$\begin{gathered}
\label{eqn:LambdaiLambdairewritten}
\begin{aligned}
\Lambda_1^\mu \Lambda_1^\nu
&=
-\frac{(Fq)^\mu (Fq)^\nu}{(Fq)^2},\\
\Lambda_2^\mu \Lambda_2^\nu
&= -\frac{(F^*q)^\mu (F^*q)^\nu}{(F^*q)^2},
\end{aligned}\end{gathered}$$ where $$\begin{gathered}
(F^*q)^2 = (Fq)^2 = -\frac{m^2 \xi^2}{e^2} (kq)^2\end{gathered}$$ and obtain \[see Eq. (\[eqn:projectiontensor\])\] $$\begin{gathered}
G^{\mu\nu}
=
q^\mu q^\nu - q^2 \, g^{\mu\nu}
\\=
q^2 {\left(}\Lambda^\mu_1 \Lambda^\nu_1 + \Lambda^\mu_2 \Lambda^\nu_2 {\right)}+ {\mathcal{Q}}^\mu {\mathcal{Q}}^\nu.\end{gathered}$$ We note the following relations: $$\begin{gathered}
\begin{aligned}
q_\rho G^{\rho\nu} &= G^{\mu\rho} q_\rho = 0,
\\
k_\rho G^{\rho\mu} &= G^{\mu\rho} k_\rho = - kq {\mathcal{Q}}^\mu,
\\
G^{\mu\rho} F^{2}_{\rho\sigma} G^{\sigma\nu}
&=
\frac{m^2}{e^2} \xi^2 (kq)^2 {\mathcal{Q}}^\mu {\mathcal{Q}}^\nu.
\end{aligned}\end{gathered}$$ To obtain the representation given in Refs. [@ritus_radiative_1972; @ritus_1985], we pass over to different basis tensors $$\begin{gathered}
b_3 \Lambda_1^\mu \Lambda_1^\nu
+
b_4 \Lambda_2^\mu \Lambda_2^\nu
+
b_5 {\mathcal{Q}}^\mu {\mathcal{Q}}^\nu
=
(q^2 b_5-b_3) \frac{(Fq)^\mu (Fq)^\nu}{(Fq)^2}
\\+ (q^2 b_5-b_4)
\frac{(F^*q)^\mu (F^*q)^\nu}{(F^*q)^2}
+ b_5 G^{\mu\nu}\end{gathered}$$ and define the following functions [@ritus_radiative_1972; @ritus_1985]: $$\begin{aligned}
f(x)
&=
i\int_0^\infty dt \exp\Big[-i\Big(t x + \frac{1}{3}t^3\Big)\Big]
\nonumber\\&\hspace*{3cm}=
\pi \operatorname{Gi}(x) + i\pi \operatorname{Ai}(x),\\
f'(x)
&=
\int_0^\infty t dt \exp\Big[-i\Big(t x + \frac{1}{3}t^3\Big)\Big],\\
f_1(x)
&=
\int_0^\infty \frac{dt}{t} \exp{\left(}-it x {\right)}\Big[ \exp\Big(-\frac{i}{3} t^3 \Big) -1 \Big]
\nonumber\\&\hspace*{3cm}=
\int_x^\infty dt {\left[}f(t) - \frac{1}{t} {\right]}\\
\intertext{and}
f_2(x)
&=
\int_0^\infty \frac{dt}{t^2} \exp{\left(}-it x {\right)}\Big[ \exp\Big(-\frac{i}{3} t^3 \Big) -1\Big]
\nonumber\\&\hspace*{3cm}=
-i{\left[}x f_1(x) + f'(x) {\right]},\end{aligned}$$ where $\operatorname{Ai}$ and $\operatorname{Gi}$ denote the Airy and Scorer functions, respectively [@olver_nist_2010]. These functions obey the following differential equations: $$\begin{gathered}
\begin{aligned}
f''(x) &= x f(x) - 1,
\\
f_1'(x) &= \frac{1}{x} - f(x) = -\frac{1}{x} f''(x).
\end{aligned}\end{gathered}$$ Using the latter, we can replace the function $f_1(x)$ by $f'(x)$ in the following way (if all boundary terms vanish): $$\begin{gathered}
\int_{-1}^{+1} dv \, g(v) f_1[\rho(v)]
=
-\int_{-1}^{+1} dv \, {\left[}\frac{G(v)}{\rho(v)} {\right]}' f'[\rho(v)],\end{gathered}$$ where $G'(v)=g(v)$.
Using the above notation, we can represent the field-dependent part of the tensor ${T}^{\mu\nu}$ for a constant-crossed field given in Eq. (\[eqn:polarizationoperatorccfieldA\]) by $$\begin{gathered}
\label{eqn:polarizationoperatorccfieldB}
{T}^{\mu\nu}(q_1,q_2)
-
{T}^{\mu\nu}_{{\mathfrak{F}}=0}(q_1,q_2)
=
i(2\pi)^4 \delta^{4}(q_1-q_2)
\\ \times\,
\Big[
\pi_1 \frac{(Fq)^\mu (Fq)^\nu}{(Fq)^2}
+
\pi_2 \frac{(F^*q)^\mu (F^*q)^\nu}{(F^*q)^2}
-
\frac{\pi_3}{q^2} G^{\mu\nu}
\Big],\end{gathered}$$ where $$\begin{gathered}
\label{eqn:ccfield_coefficients}
\begin{aligned}
\pi_1
&=
\phantom{-}\alpha\, \frac{m^2}{3\pi} \int_{-1}^{+1} dv \, (w-1) \Big(\frac{\chi}{w}\Big)^{\nicefrac23} f'(\rho),\\
\pi_2
&=
\phantom{-}\alpha\, \frac{m^2}{3\pi} \int_{-1}^{+1} dv \, (w+2) \Big(\frac{\chi}{w}\Big)^{\nicefrac23} f'(\rho),\\
\pi_3
&= - \alpha\, \frac{q^2}{\pi} \int_{-1}^{+1} dv \, \frac{f_1(\rho)}{w}
\end{aligned}\end{gathered}$$ \[$\frac{1}{w} = \frac14(1-v^2)$, $\rho = \big({{w}/{\chi}}\big)^{\nicefrac23}(1- \frac{q^2}{m^2} \frac{1}{w})$\]. Since all nonvanishing functions are even in $v$, we can now apply the following change of variables: $$\begin{gathered}
\int_{-1}^{+1} dv = 2 \int_{0}^{1} dv = \int_{4}^{\infty} dw \, \frac{4}{w\sqrt{w(w-4)}},\end{gathered}$$ which shows that the result in Eq. (\[eqn:polarizationoperatorccfieldB\]) is equivalent to the one given in Refs. [@ritus_radiative_1972; @ritus_1985].
Quasiclassical limit
--------------------
We consider now a linearly polarized plane-wave field $$\begin{gathered}
\label{eqn:linearpolarizationdef}
\psi_1(\phi) = \psi(\phi),
\quad
\psi_2(\phi) = 0 \end{gathered}$$ (we will use $\xi = \xi_1$ and $f^{\mu\nu} = f_1^{\mu\nu}$ in this paragraph) in the quasiclassical limit defined by $\xi\to\infty$ while \[see Eq. (\[eqn:chidefinition\])\] $$\begin{gathered}
\chi
=
-\frac{e\sqrt{qf^2q}}{m^3}
=
\xi \frac{\sqrt{(kq)^2}}{m^2}\end{gathered}$$ is kept constant. In the optical regime (photon energy $\omega_0 \sim \unit[1]{eV}$), $\chi \gtrsim 1$ requires $\xi \gg 1$ (unless the incoming photon energy exceeds the threshold of about 1 TeV), which means that the quasiclassical limit is sufficient to analyze most of the upcoming strong-field experiments with optical lasers.
By employing the identity ${\left|kq\right|} = m^2 {{\chi}/{\xi}}$, we can expand all functions depending on $\mu kq$ $$\begin{gathered}
\label{eqn:qclimit_IJXZ}
\begin{aligned}
I^2_1 - J_1 &= -({{1}/{3}}) (\mu kq)^2 \big[\psi'(kz)\big]^2 + {\mathcal{O}}(\mu kq)^3,\\
Z_1 &= 2 (\mu kq)^2 \big[\psi'(kz)\big]^2 + {\mathcal{O}}(\mu kq)^3,\\
X_{11} &= - (\mu kq)^2 \big[\psi'(kz)\big]^2 + {\mathcal{O}}(\mu kq)^3
\end{aligned}\end{gathered}$$ ($X_{12}=X_{21}=X_{22}=Z_2=I_2=J_2=0$ for linear polarization). Thus, if multiplied by $\xi^2$, only the leading-order terms are independent of $\xi$, and all others are suppressed. In the limit $\xi\to \infty$, the expressions in Eq. (\[eqn:qclimit\_IJXZ\]) correspond to those in Eq. (\[eqn:ccfield\_IJXZ\]) with the replacement $\chi \to \chi(kz) = \chi \psi'(kz)$. The remaining calculation is therefore similar to the one in the constant-crossed field case, and the final result in Eq. (\[eqn:polarizationoperatorlinearpolqc\]) corresponds essentially to Eq. (\[eqn:polarizationoperatorccfieldB\]) with the above replacement. Using \[see Eq. (\[eqn:LambdaiLambdairewritten\])\] $$\begin{gathered}
\begin{aligned}
\Lambda_1^\mu \Lambda_1^\nu
&=
- \frac{(fq)^\mu (fq)^\nu}{(fq)^2},
\\
\Lambda_2^\mu \Lambda_2^\nu
&=
- \frac{(f^*q)^\mu (f^*q)^\nu}{(f^*q)^2}
\end{aligned}\end{gathered}$$ and Eq. (\[eqn:projectiontensordecomposition\]), we obtain for a linearly polarized plane-wave field in the quasiclassical approximation the following representation for the field-dependent part of the tensor ${T}^{\mu\nu}$ \[see Eq. (\[eqn:polarizationoperatorfinal\])\]: $$\begin{gathered}
\label{eqn:polarizationoperatorlinearpolqc}
{T}^{\mu\nu}(q_1,q_2)
-
{T}^{\mu\nu}_{{\mathfrak{F}}=0}(q_1,q_2)
=
i(2\pi)^4\delta^{({{{}-}},{\perp})}(q_1-q_2)
\\ \times\,
\frac{1}{2\pi} \int_{-\infty}^{+\infty} dz^{{{}-}}\, e^{i(q_2^{{{}+}}- q_1^{{{}+}})z^{{{}-}}}
\bigg[\pi'_1\, \frac{(fq)^\mu (fq)^\nu}{(fq)^2}
\\+ \pi'_2\, \frac{(f^*q)^\mu (f^*q)^\nu}{(f^*q)^2} - \frac{\pi'_3}{q_1 q_2}\, G^{\mu\nu} \bigg],\end{gathered}$$ where \[see Eq. (\[eqn:ccfield\_coefficients\])\] $$\begin{gathered}
\begin{aligned}
\pi'_1
&=
\phantom{-}\alpha\, \frac{m^2}{3\pi} \int_{-1}^{+1} dv \, (w-1) \bigg[\frac{{\left|\chi(kz)\right|}}{w}\bigg]^{\nicefrac23} f'(\rho),\\
\pi'_2
&=
\phantom{-}\alpha\, \frac{m^2}{3\pi} \int_{-1}^{+1} dv \, (w+2) \bigg[\frac{{\left|\chi(kz)\right|}}{w}\bigg]^{\nicefrac23} f'(\rho),\\
\pi'_3
&= - \alpha\, \frac{q_1 q_2}{\pi} \int_{-1}^{+1} dv \, \frac{f_1(\rho)}{w}
\end{aligned}\end{gathered}$$ with $\tfrac{1}{w} = \tfrac14(1-v^2)$, $\rho = \big[{{w}/{{\left|\chi(kz)\right|}}}\big]^{\nicefrac23}(1- \tfrac{q_1 q_2}{m^2} \tfrac{1}{w})$ and $G^{\mu\nu} = q_2^\mu q_1^\nu - q_1q_2 \, g^{\mu\nu}$ \[see Eq. (\[eqn:projectiontensor\])\].
Circular polarization
---------------------
The general result in Eq. (\[eqn:polarizationoperatorfinal\]) also simplifies considerably if the plane wave is circularly polarized and monochromatic, $$\begin{gathered}
\psi_1(\phi) = \Re{e^{i\phi}},
\quad
\psi_2(\phi) = \Im{e^{i\phi}},
\quad
\xi_1=\xi_2=\xi.\end{gathered}$$ We then obtain $$\begin{gathered}
\begin{gathered}
I_{1}
=
\operatorname{sinc}(\mu kq) \Re e^{ikz},
\quad
I_{2}
=
\operatorname{sinc}(\mu kq) \Im e^{ikz},\\
J_{1} + J_{2}
= 1,
\quad
Z_1 + Z_2 = 2\sin^2(\mu kq),
\end{gathered}\end{gathered}$$ $$\begin{gathered}
\begin{aligned}
I_{1} - \psi_{1}(kz+\mu kq)
&=
\Re A,\\
I_{2} - \psi_{2}(kz+\mu kq)
&=
\Im A,\\
I_{1} - \psi_{1}(kz-\mu kq)
&=
\Re B,\\
I_{2} - \psi_{2}(kz-\mu kq)
&=
\Im B,
\end{aligned}\end{gathered}$$ where $$\begin{gathered}
\begin{aligned}
A &= e^{ikz} {\left[}\operatorname{sinc}(\mu kq) - \cos(\mu kq) - i \sin(\mu kq) {\right]},\\
B &= e^{ikz} {\left[}\operatorname{sinc}(\mu kq) - \cos(\mu kq) + i \sin(\mu kq) {\right]}\end{aligned}\end{gathered}$$ \[we define $\operatorname{sinc}x = {{(\sin x)}/{x}}$\]. Thus, $$\begin{gathered}
\begin{aligned}
X_{12} - X_{21} &= \Im A^*B,&
X_{11} - X_{22} &= \Re AB,\\
X_{12} + X_{21} &= \Im AB,&
X_{11} + X_{22} &= \Re A^*B,
\end{aligned}\end{gathered}$$ where $$\begin{gathered}
\begin{aligned}
A^*B &= \operatorname{sinc}^2(\mu kq) + \cos(2\mu kq) - 2\operatorname{sinc}(2\mu kq) \\
&\phantom{=}+ i{\left[}-\sin(2\mu kq) + 2 \operatorname{sinc}(\mu kq) \sin(\mu kq) {\right]},\\
AB &= e^{2ikz} {\left[}\operatorname{sinc}^2(\mu kq) - 2\operatorname{sinc}(2\mu kq) + 1 {\right]}.
\end{aligned}\end{gathered}$$
Thus, we can write the field-dependent part of the tensor ${T}^{\mu\nu}$ for a circularly polarized plane wave as \[see Eq. (\[eqn:polarizationoperatorfinal\])\] $$\begin{gathered}
\label{eqn:polarizationoperatorcircularpolarization}
{T}^{\mu\nu}(q_1,q_2)
-
{T}^{\mu\nu}_{{\mathfrak{F}}=0}(q_1,q_2)
=
-i\pi e^2\, \delta^{({{{}-}},{\perp})}(q_1-q_2)
\\
\begin{aligned}
&\times\,
\int_{-1}^{+1} dv \int_0^\infty \frac{d\tau}{\tau} \,
\int_{-\infty}^{+\infty} dz^{{{}-}}\,
\Big[
b_+ \Lambda_+^\mu \Lambda_+^\nu
\\&+
b_- \Lambda_-^\mu \Lambda_-^\nu
+
\frac12(b_1-b_2) (\Lambda^\mu_1 \Lambda^\nu_2 - \Lambda^\mu_2 \Lambda^\nu_1)
\\&+
\frac12(b_3+b_4) (\Lambda^\mu_1 \Lambda^\nu_1 + \Lambda^\mu_2 \Lambda^\nu_2)
+
b_5 {\mathcal{Q}}_1^\mu {\mathcal{Q}}_2^\nu \Big] e^{i\Phi},
\end{aligned}\end{gathered}$$ where we defined $$\begin{gathered}
\Lambda^\mu_{\pm} = \Lambda_1^\mu \pm i \Lambda_2^\mu\end{gathered}$$ and the coefficients are given by $$\begin{gathered}
b_\pm = \frac14{\left[}(b_3-b_4)\mp i(b_1+b_2){\right]}= \frac12 m^2\xi^2
\\\times \,
{\left[}\operatorname{sinc}^2(\mu kq) - 2\operatorname{sinc}(2\mu kq) + 1 {\right]}\, e^{\mp 2ikz +i\tau\beta},\end{gathered}$$ $$\begin{gathered}
\frac12(b_1-b_2)
= m^2\xi^2 \frac{(1+v^2)}{(1-v^2)} \big[ - \sin(2\mu kq)\\
+ 2 \operatorname{sinc}(\mu kq) \sin(\mu kq) \big]\, e^{i\tau\beta}, \end{gathered}$$ $$\begin{gathered}
\frac12(b_3+b_4)
= -\Big( \frac{i}{\tau} + \frac{q_1q_2}{2}\Big) {\left(}e^{i\tau\beta} - 1{\right)}\\
\begin{aligned}
&+ m^2 \xi^2 \Big[ 2 \, \frac{(1+v^2)}{(1-v^2)} \, \sin^2(\mu kq)
\\&+ \operatorname{sinc}^2(\mu kq) - 2\operatorname{sinc}(2\mu kq) + 1 \Big] e^{i\tau\beta},
\end{aligned}\end{gathered}$$ $$\begin{gathered}
b_5 = - \frac{2\mu}{\tau} {\left(}e^{i\tau\beta} -1 {\right)}\end{gathered}$$ and the phases read $$\begin{gathered}
\begin{aligned}
i\tau\beta
&=
i \tau m^2 \xi^2 {\left[}\operatorname{sinc}^2(\mu kq) -1 {\right]},\\
i \Phi
&=
i {\left[}(q_2^{{{}+}}-q_1^{{{}+}}) z^{{{}-}}+ \mu q_1q_2 - \tau m^2 {\right]}\end{aligned}\end{gathered}$$ \[$\mu = \frac14 \tau (1-v^2)$\]. Finally, the integral in $dz^{{{}-}}$ can be taken and we obtain the following expression for the field-dependent part of ${T}^{\mu\nu}(q_1,q_2)$ for a monochromatic, circularly polarized plane-wave laser field $$\begin{gathered}
\label{eqn:polarizationoperatorcircularpolarizationfinal}
{T}^{\mu\nu}(q_1,q_2)
-
{T}^{\mu\nu}_{{\mathfrak{F}}=0}(q_1,q_2)
=
- \frac{i(2\pi)^4 \,e^2}{8\pi^2} \,
\int_{-1}^{+1} dv
\\ \int_0^{\infty} \frac{d\tau}{\tau} \,\, \big[ T^{\mu\nu}_0 \delta^4(q_1-q_2)
+ T^{\mu\nu}_+ \delta^4(q_1-q_2+2k) \\+ T^{\mu\nu}_- \delta^4(q_1-q_2-2k) \big]
e^{i\Phi_{\text{cp}}},\end{gathered}$$ where $$\begin{gathered}
i\Phi_{\text{cp}}
=
- i\tau m^2 \big\{1 + \xi^2 [1 - \operatorname{sinc}^2(\mu kq)] \big\} + i \mu q_1q_2,\end{gathered}$$ $$\begin{gathered}
T^{\mu\nu}_0
=
\tau_1 (\Lambda^\mu_1 \Lambda^\nu_2 - \Lambda^\mu_2 \Lambda^\nu_1)
+
\tau_2 (\Lambda^\mu_1 \Lambda^\nu_1 + \Lambda^\mu_2 \Lambda^\nu_2)
\\+
\tau_3 {\mathcal{Q}}_1^\mu {\mathcal{Q}}_2^\nu, \end{gathered}$$ $$\begin{gathered}
T^{\mu\nu}_\pm = \frac12 m^2 \xi^2 \big[ \operatorname{sinc}^2(\mu kq) - 2\operatorname{sinc}(2\mu kq) + 1 \big]\, \Lambda^\mu_\pm \Lambda^\nu_\pm\end{gathered}$$ and $$\begin{gathered}
\begin{aligned}
\tau_1
&=
m^2\xi^2 \frac{(1+v^2)}{(1-v^2)} \big[2\, {{\sin^2(\mu kq)}/{(\mu kq)}} -\sin(2\mu kq)\big],
\\
\tau_2
&=
2 m^2 \xi^2 \, \frac{(1+v^2)}{(1-v^2)} \, \sin^2(\mu kq)
\\&\phantom{=}+
\Big[ \Big(\frac{\mu}{\tau} - \frac{1}{2}\Big) q_1q_2 - m^2 \Big] {\left(}1 - e^{-i\tau\beta}{\right)},
\\
\tau_3
&=
- \frac{2\mu}{\tau} {\left(}1 - e^{-i\tau\beta} {\right)}.
\end{aligned}\end{gathered}$$ This result agrees with Eq. (2.34) in Ref. [@baier_interaction_1975]. The terms described by $T^{\mu\nu}_\pm$ can be interpreted as describing processes where two photons from the background field are absorbed or emitted, respectively (since the external field is not quantized, this interpretation relies only on the momentum-conserving delta function). In order to obtain Eq. (\[eqn:polarizationoperatorcircularpolarizationfinal\]) from Eq. (\[eqn:polarizationoperatorcircularpolarization\]), we have used the identity $$\begin{gathered}
\int_0^\infty \frac{d\tau}{\tau} e^{i\Phi} \, m^2\xi^2 \big[ \operatorname{sinc}^2(\mu kq) - 2\operatorname{sinc}(2\mu kq) + 1 \big] e^{i\tau\beta}
\\=
\int_0^\infty \frac{d\tau}{\tau} e^{i\Phi} \, \Big[\frac{i}{\tau} + \frac{\mu}{\tau}\, q_1q_2 - m^2 \Big] \big( e^{i\tau\beta} - 1 \big),\end{gathered}$$ which follows from $$\begin{gathered}
i\frac{d}{d\tau} \big( e^{i\tau\beta} - 1 \big)
=
i\frac{d}{d\tau} e^{i\tau\beta}
\\=
m^2\xi^2 \big[ \operatorname{sinc}^2(\mu kq) - 2\operatorname{sinc}(2\mu kq) + 1 \big] e^{i\tau\beta}\end{gathered}$$ via integration by parts.
Conclusion
==========
In the present paper, we have proven for the first time the Ward-Takahashi identity for general loop diagrams in a plane-wave background field (see Sec. \[sec:wardidentity\]). Moreover, we have presented a new derivation of the leading-order contribution to the polarization operator in a plane-wave background field for arbitrary polarization and dependence on the plane-wave phase (see Sec. \[sec:polarizationoperator\]). Our calculation relies on a direct evaluation of the space-time integrals without using Schwinger’s operator method [@schwinger_gauge_1951] that was employed in Ref. [@baier_interaction_1975]. An interesting feature of our final representation is the manifest symmetry with respect to the external photon four-momenta $q_1$ and $q_2$ \[see Eq. (\[eqn:polarizationoperatorfinal\])\].
S.M. is grateful to the Studienstiftung des deutschen Volkes for financial support.
Notation {#sec:notation}
========
In this paper we use natural units $\hbar=c=1$ (in some formulas $\hbar$ and $c$ are restored for clarity) and the charge is measured in Heaviside-Lorentz units (${\epsilon}_0=1$). The electron mass and charge are denoted by $m$ and $e<0$, respectively. Thus, the fine-structure constant is given by $\alpha = {{e^2}/{(4\pi)}} \approx {{1}/{137}}$. In covariant expressions the space-time metric $g_{\mu\nu}$ with signature $(1,-1,-1,-1)$ is used, and ${\partial}_\mu =({\partial}/{\partial}t,{\boldgreek{\nabla}})$ is the four-derivative. This implies ${\partial}_\mu x_\nu = g_{\mu\nu}$, where $x^\mu =(t,{\boldsymbol{x}})$ denotes the position four-vector. The unit tensor is denoted by $\delta^{\mu}_{\nu} = g^{\mu\rho} g_{\rho\nu} = \operatorname{diag}(1,1,1,1)$ ($\delta_\mu^\mu = 4$), and space-time indices (lowercase Greek letters) are raised and lowered using the metric $a_\mu = g_{\mu\nu} a^\nu$ (summation over all types of repeated indices is understood if they do not appear on both sides of an equation). Greek and Latin indices take the values (0,1,2,3) and (1,2,3), respectively. Contractions of four-vectors are denoted by $a^\mu b_\mu = ab$ and scalar products of three-vectors by ${\boldsymbol{a}}^i {\boldsymbol{b}}^i = {\boldsymbol{a}}{\boldsymbol{b}}$. We denote the dual of a second-rank tensor $T^{\mu\nu}$ by $T^{*\mu\nu} = \frac{1}{2} {\epsilon}^{\mu\nu\rho\sigma} T_{\rho\sigma}$, where ${\epsilon}^{\mu\nu\rho\sigma}$ is the totally antisymmetric tensor in four dimensions with ${\epsilon}^{0123}=-{\epsilon}_{0123}=1$. For contractions of second-rank tensors and vectors, a matrix notation is sometimes used, e.g. $aTb= a_\mu T^{\mu\nu} b_\nu$, $(T_1T_2)^{\mu\nu} = T^{\mu}_{1\rho} T_{2}^{\rho\nu}$, $T^{2\mu\nu} = T^{\mu\rho} T_{\rho}^{\phantom{\rho}\nu}$, $(Ta)^\mu = T^{\mu\nu} a_\nu$. All spinors are Dirac spinors (with four components); spinor indices are usual suppressed. The Dirac gamma matrices are denoted by $\gamma^\mu$, ${\slashed{a}}=a_\mu \gamma^\mu$, $\gamma^5 = -i\gamma^0\gamma^1\gamma^2\gamma^3$ and $2\sigma^{\mu\nu} = \gamma^\mu \gamma^\nu - \gamma^\nu \gamma^\mu$ ($\gamma^\mu \gamma^\nu + \gamma^\nu \gamma^\mu = 2g^{\mu\nu}$). For a spinor $u$, we define $\bar{u}=u^\dagger \gamma^0$, and for a matrix in spinor space $M$, correspondingly $\bar{M}=\gamma^0 M^\dagger \gamma^0$. A quantization volume $V=1$ is assumed for the normalization of the single-particle electron, positron and photon states. The total derivative of a function with respect to its argument is denoted by a prime $f'(x) = \frac{d}{dx} f(x)$. Integrals without boundaries range from $-\infty$ to $+\infty$. We use $i0$ as a short notation for $i\upepsilon$ together with the limit $\lim_{\upepsilon \to 0^+}$. In general, our notation therefore follows Ref. [@landau_quantum_1981] with different units for charge.
Laser field as a coherent state of the electromagnetic field {#sec:lasercoherentstate}
============================================================
In this appendix we give a detailed justification why a strong laser can be taken into account by applying the shift ${\mathcal{A}}^\mu \to A^\mu_{\mathrm{rad}}(x) + A^\mu$ in the Lagrangian density, with $A^\mu$ being treated as a classical field and with $A^\mu_{\mathrm{rad}}$ describing all other quantized modes [@fradkin_quantum_1991; @glauber_coherent_1963; @harvey_signatures_2009]. A strong laser field represents a very good experimental realization of a coherent state of the photon field. To be explicit, we consider the four-vector potential operator
\[eqn:photonfieldoperator\] $$\begin{gathered}
\hat{{\mathcal{A}}}^\mu(x)
=
\hat{{\mathcal{A}}}_{+}^\mu(x)
+
\hat{{\mathcal{A}}}_{+}^{\dagger\mu}(x),\end{gathered}$$ $$\begin{gathered}
\hat{{\mathcal{A}}}_{+}^\mu(x)
=
\sum_{\sigma=1,2} \int \frac{d^3q}{(2\pi)^3}
\frac{1}{\sqrt{2\omega_{{\boldsymbol{q}}}}}
\,
\hat{c}_{{\boldsymbol{q}},\sigma}
e^{-{i}qx} {\epsilon}^\mu_{{\boldsymbol{q}},\sigma},\end{gathered}$$
where $\omega_{{\boldsymbol{q}}} = \sqrt{{\boldsymbol{q}}^2}$ and ${\epsilon}^\mu_{{\boldsymbol{q}},\sigma}$ are the orthogonal polarization four-vectors $$\begin{gathered}
({\epsilon}^{*}_{{\boldsymbol{q}},\sigma})^\mu ({\epsilon}_{{\boldsymbol{q}},\tau})_{\mu}
=
- \delta_{\sigma\tau}\end{gathered}$$ \[for simplicity we consider here only the two physical degrees of freedom ($\sigma, \tau = 1,2$), see e.g. [@weinberg_quantum_1995] for further details\]. The photon creation $\hat{c}^\dagger_{{\boldsymbol{q}},\sigma}$ and annihilation operators $\hat{c}_{{\boldsymbol{q}},\sigma}$ obey the canonical commutation relations $$\begin{gathered}
\big[\hat{c}_{{\boldsymbol{p}},\sigma},\hat{c}^\dagger_{{\boldsymbol{q}},\tau}\big]
=
(2\pi)^3 \delta^3({\boldsymbol{p}}-{\boldsymbol{q}}) \delta_{\sigma \tau}.\end{gathered}$$ Using this notation, a coherent state $\ket{A}$ of the photon field can be written as [@glauber_coherent_1963] $$\begin{gathered}
\ket{A} = \hat{D} \, \ket{0},\end{gathered}$$ where $\ket{0}$ is the vacuum state of the photon Fock space ($\hat{c}_{{\boldsymbol{q}},\sigma} \ket{0} = 0$ for all ${\boldsymbol{q}}$ and $\sigma$, and $\braket{0|0} = 1$) and $\hat{D}$ is a unitary displacement operator. If the classical four-potential associated with the coherent state \[compare with Eq. (\[eqn:photonfieldoperator\])\] is
\[eqn:classicallaserfield\] $$\begin{gathered}
A^\mu(x)=A_{+}^\mu(x)+A_{+}^{*\mu}(x),\end{gathered}$$ with $$\begin{gathered}
A_{+}^\mu(x)
=
\sum_{\sigma=1,2} \int \frac{d^3q}{(2\pi)^3}
\frac{1}{\sqrt{2\omega_{{\boldsymbol{q}}}}}
\,
C_{{\boldsymbol{q}},\sigma}
e^{-{i}qx} {\epsilon}^\mu_{{\boldsymbol{q}},\sigma},\end{gathered}$$
the displacement operator $\hat{D}$ has the form $$\begin{gathered}
\label{eqn:displacementoperator}
\hat{D} = \exp \Big[ \sum_{\sigma=1,2} \int \frac{d^3q}{(2\pi)^3} {\left(}C_{{\boldsymbol{q}},\sigma} \hat{c}^\dagger_{{\boldsymbol{q}},\sigma} - C^*_{{\boldsymbol{q}},\sigma} \hat{c}_{{\boldsymbol{q}},\sigma} {\right)}\Big]\end{gathered}$$ and the properties $$\begin{gathered}
\label{eqn:displacementoperatorproperties}
\begin{aligned}
\hat{D}^{-1} \hat{c}_{{\boldsymbol{q}},\sigma} \hat{D} = \hat{c}_{{\boldsymbol{q}},\sigma} + C_{{\boldsymbol{q}},\sigma},\\
\hat{D}^{-1} \hat{c}^\dagger_{{\boldsymbol{q}},\sigma} \hat{D} = \hat{c}^\dagger_{{\boldsymbol{q}},\sigma} + C^*_{{\boldsymbol{q}},\sigma}.
\end{aligned}\end{gathered}$$ By using Eq. (\[eqn:displacementoperatorproperties\]), one can show that $$\begin{gathered}
\hat{{\mathcal{A}}}_{+}^\mu(x) \, \ket{A}
=
A_{+}^\mu(x) \, \ket{A}.\end{gathered}$$ Thus, since $$\begin{gathered}
\braket{A|\hat{{\mathcal{A}}}^\mu(x)|A} = A^\mu(x),\end{gathered}$$ a coherent state can be seen as the most “classical” state of the photon field.
If the coherent part of the photon field is not substantially changed during the interaction, the same coherent state appears on both sides of the $S$-matrix element(s) of an arbitrary QED process, $$\begin{gathered}
\braket{A^\mu| \cdots | A^\mu} = \braket{0|\hat{D}^{-1} \cdots \hat{D}|0}.\end{gathered}$$ Physically, this amounts to the assumption that the laser field is not significantly depleted during the interaction, which can be assumed if the laser is sufficiently intense (see also Sec. \[sec:qedwithbackgroundfields\]). We can then include the coherent part of the photon field nonperturbatively if we adopt the transformation in Eq. (\[eqn:displacementoperatorproperties\]). In particular, we obtain \[see Eqs. (\[eqn:photonfieldoperator\]) and (\[eqn:classicallaserfield\])\] $$\begin{gathered}
\hat{D}^{-1} \hat{{\mathcal{A}}}^\mu(x) \hat{D}
=
\hat{{\mathcal{A}}}^\mu(x) + A^\mu(x).\end{gathered}$$ Thus, instead of calculating $S$-matrix elements between coherent states, we can apply the shift $\hat{{\mathcal{A}}}^\mu(x) \to \hat{A}_{\text{rad}}^\mu(x) + A^\mu(x)$ in the Lagrangian density and consider $S$-matrix elements between vacuum states as usual [@fradkin_quantum_1991; @harvey_signatures_2009].
Light-cone coordinates {#sec:lccappendix}
======================
Calculations involving plane-wave background fields become particular transparent if light-cone coordinates are used [@dirac_forms_1949; @neville_quantum_1971; @mitter_quantum_1975]. Since the nontrivial space-time dependence of the momentum-space vertex in Eq. (\[eqn:dressedvertexfinal\]) is due to the plane-wave phase $\phi = kx$, it is natural to work in a basis where $k^\mu$ is one of the basis four-vectors. However, since $k^2=0$, this will be a light-cone basis. We introduce now a general light-cone basis by adding three four-vectors $\bar{k}^\mu$, $e_i^\mu$ ($i\in 1,2$) to the set and require the following orthogonality relations:
\[eqn:lcc\_orthogonalityrelations\] $$\begin{gathered}
k^2 = \bar{k}^2 = ke_i = \bar{k}e_i = 0,
\,
k\bar{k} = 1,
\,
e_i e_j = - \delta_{ij}\end{gathered}$$ and the orientation $$\begin{gathered}
{\epsilon}_{\mu\nu\rho\sigma} k^\mu \bar{k}^\nu e_1^\rho e_2^\sigma = 1.\end{gathered}$$
To be more specific, we can, in a reference system where the plane wave propagates along the direction ${\boldsymbol{n}}$, take the following four-vectors: $$\begin{gathered}
\label{eqn:canonicallcc}
\begin{gathered}
k^\mu = \omega (1,{\boldsymbol{n}}),
\quad
\bar{k}^\mu = \frac{1}{2\omega} (1,-{\boldsymbol{n}}),
\quad
e_i^\mu=(0,{\boldsymbol{e}}_i) \sim a_i^\mu ,\\
{\boldsymbol{n}}^2 = 1,
\quad
{\boldsymbol{e}}_i {\boldsymbol{e}}_j = \delta_{ij},
\quad
{\boldsymbol{n}} = {\boldsymbol{e}}_1 \times {\boldsymbol{e}}_2
\end{gathered}\end{gathered}$$ (${\boldsymbol{e}}_i$ represent the two polarization directions of the plane-wave field, and $\omega$ has the dimension of a frequency).
Due to the relations given in Eq. (\[eqn:lcc\_orthogonalityrelations\]), we obtain the following decomposition of the metric: $$\begin{gathered}
g_{\mu\nu} = k_\mu \bar{k}_\nu + \bar{k}_\mu k_\nu - e_{1\mu} e_{1\nu} - e_{2\mu} e_{2\nu}.\end{gathered}$$ This allows us to define the transformation to light-cone coordinates (primed indices) by $$a^{\mu'} = \Lambda^{\mu'}_{\phantom{\mu'}\nu} a^\nu\,,
\,
b_{\mu'} = b_\nu {{\Lambda^{\hspace*{-1.8pt}\scalebox{0.65}{$-1$}}}}^{\nu}_{\phantom{\nu}\mu'}\,,
\,
{{\Lambda^{\hspace*{-1.8pt}\scalebox{0.65}{$-1$}}}}^{\rho}_{\phantom{\rho}\mu'} \Lambda^{\mu'}_{\phantom{\mu'}\sigma}
=
\delta^{\rho}_{\sigma},$$ where the components denote the following scalar products: $$\begin{gathered}
\begin{aligned}
\Lambda^{{{{}+}}}_{\phantom{{{{}+}}}\mu} &= \bar{k}_\mu,&
\Lambda^{{{\scalebox{.64}{$\matheuler{I}$}}}}_{\phantom{{{\scalebox{.64}{$\matheuler{I}$}}}}\mu} &= e_{1\mu},\\
\Lambda^{{{{}-}}}_{\phantom{{{{}-}}}\mu} &= k_\mu,&
\Lambda^{{{\scalebox{.64}{$\matheuler{II}$}}}}_{\phantom{{{\scalebox{.64}{$\matheuler{II}$}}}}\mu} &= e_{2\mu}
\end{aligned}\end{gathered}$$ (we label light-cone components by ${{{}+}}$,${{{}-}}$,${{\scalebox{.64}{$\matheuler{I}$}}}$,${{\scalebox{.64}{$\matheuler{II}$}}}$). On the other hand, the inverse transformation is given by $$\begin{gathered}
\begin{aligned}
\label{eqn:transformationmatrixlcc}
{{\Lambda^{\hspace*{-1.8pt}\scalebox{0.65}{$-1$}}}}^{\mu}_{\phantom{\mu}{{{}+}}} &= k^\mu,&
{{\Lambda^{\hspace*{-1.8pt}\scalebox{0.65}{$-1$}}}}^{\mu}_{\phantom{\mu}{{\scalebox{.64}{$\matheuler{I}$}}}} &= -e^\mu_{1},\\
{{\Lambda^{\hspace*{-1.8pt}\scalebox{0.65}{$-1$}}}}^{\mu}_{\phantom{\mu}{{{}-}}} &= \bar{k}^\mu,&
{{\Lambda^{\hspace*{-1.8pt}\scalebox{0.65}{$-1$}}}}^{\mu}_{\phantom{\mu}{{\scalebox{.64}{$\matheuler{II}$}}}} &= -e^\mu_{2}.
\end{aligned}\end{gathered}$$ We point out that $k^\mu$ has dimension of momentum and therefore $\bar{k}^\mu$ must have dimension of inverse momentum ($e^\mu_{i}$ are dimensionless). Hence, the dimensions of $v^{{{}+}}$ and $v^{{{}-}}$ differ from those of $v^\mu$ (here $v^\mu$ is an arbitrary Lorentz four-vector). The different dimensions of the light-cone components can be circumvented by defining $k^\mu = \omega n^\mu$ and using the dimensionless quantity $n^\mu$ in place of $k^\mu$. Then, however, $nv$ is not a Lorentz scalar (contrary to $kv = v^{{{}-}}$), and $\omega$ has to appear explicitly in many places.
In light-cone coordinates, the metric is given by $$\begin{gathered}
g_{\mu'\nu'}
=
g_{\rho\sigma}
{{\Lambda^{\hspace*{-1.8pt}\scalebox{0.65}{$-1$}}}}^{\rho}_{\phantom{\rho}\mu'}
{{\Lambda^{\hspace*{-1.8pt}\scalebox{0.65}{$-1$}}}}^{\sigma}_{\phantom{\sigma}\nu'}
\\=
\delta^{{{}+}}_{\mu'} \delta^{{{}-}}_{\nu'}
+
\delta^{{{}-}}_{\mu'} \delta^{{{}+}}_{\nu'}
-
\delta^{{\scalebox{.64}{$\matheuler{I}$}}}_{\mu'} \delta^{{\scalebox{.64}{$\matheuler{I}$}}}_{\nu'}
-
\delta^{{\scalebox{.64}{$\matheuler{II}$}}}_{\mu'} \delta^{{\scalebox{.64}{$\matheuler{II}$}}}_{\nu'},\end{gathered}$$ which allows us to write the scalar product of two four-vectors as $$\begin{gathered}
a_\mu b^\mu = a^{{{}+}}b^{{{}-}}+ a^{{{}-}}b^{{{}+}}- a^{{\scalebox{.64}{$\matheuler{I}$}}}b^{{\scalebox{.64}{$\matheuler{I}$}}}- a^{{\scalebox{.64}{$\matheuler{II}$}}}b^{{\scalebox{.64}{$\matheuler{II}$}}}\end{gathered}$$ (we also use the short notation $a^{\perp}b^{\perp}= a^{{\scalebox{.64}{$\matheuler{I}$}}}b^{{\scalebox{.64}{$\matheuler{I}$}}}+ a^{{\scalebox{.64}{$\matheuler{II}$}}}b^{{\scalebox{.64}{$\matheuler{II}$}}}$). Due to Eq. (\[eqn:lcc\_orthogonalityrelations\]), we obtain $$\begin{gathered}
{\left|\det \Lambda^{\mu'}_{\phantom{\mu}\nu}\right|}
=
{\left|
\Lambda^{{{{}+}}}_{\phantom{{{{}+}}}\mu} \Lambda^{{{{}-}}}_{\phantom{{{{}-}}}\nu} \Lambda^{{{\scalebox{.64}{$\matheuler{I}$}}}}_{\phantom{{{\scalebox{.64}{$\matheuler{I}$}}}}\rho}\Lambda^{{{\scalebox{.64}{$\matheuler{II}$}}}}_{\phantom{{{\scalebox{.64}{$\matheuler{II}$}}}}\sigma} {\epsilon}^{\mu\nu\rho\sigma}\right|}
=
1.\end{gathered}$$ Thus, the four-dimensional integration measure becomes $$\begin{gathered}
\int d^4a \, = \int da^{{{}+}}da^{{{}-}}da^{\perp},
\quad
da^{\perp}= da^{{\scalebox{.64}{$\matheuler{I}$}}}da^{{\scalebox{.64}{$\matheuler{II}$}}}.\end{gathered}$$
Since all properties of the light-cone coordinates follow from the relations in Eq. (\[eqn:lcc\_orthogonalityrelations\]), we are not forced to use the canonical basis in Eq. (\[eqn:canonicallcc\]). For the calculation of the polarization operator, it is more convenient to use the two four-vectors \[see Eq. (\[eqn:Lambdavectors\])\],
\[eqn:modifiedlcc\] $$\begin{gathered}
e_1'^\mu
=
\Lambda_1^\mu
=
\frac{f^{\mu\nu}_1 q_{\nu}}{kq\, \sqrt{-a_1^2}},
\quad
e_2'^\mu
=
\Lambda_2^\mu
=
\frac{f^{\mu\nu}_2 q_{\nu}}{kq\, \sqrt{-a_2^2}}\end{gathered}$$ together with $k^\mu$ and $$\begin{gathered}
\bar{k}'^\mu = \bar{k}^\mu + \frac{a_1q}{a_1^2\, kq} a_1^\mu + \frac{a_2q}{a_2^2\, kq} a_2^\mu \\- \frac{1}{2(kq)^2} {\left[}\frac{(a_1q)^2}{a_1^2} + \frac{(a_2q)^2}{a_2^2}{\right]}k^\mu.\end{gathered}$$
The set of four-vectors $k^\mu$, $\bar{k}'^\mu$, $e_1'^\mu$, $e_2'^\mu$ also obeys the relations in Eq. (\[eqn:lcc\_orthogonalityrelations\]), and we will call the coordinates, following from this set, modified light-cone coordinates \[the same symbols (${{{}+}}$, ${{{}-}}$, ${{\scalebox{.64}{$\matheuler{I}$}}}$, ${{\scalebox{.64}{$\matheuler{II}$}}}$) are used to denote the corresponding components\].
Gamma matrix algebra {#sec:gammamatrixalgebraappendix}
====================
In this appendix we summarize some general identities, which are useful in calculations involving gamma matrices. The gamma matrices form a complete set in the sense that any matrix in spinor space can be decomposed according to [@leader_spin_2001] $$\begin{gathered}
\label{eqn:decompositionofspinormatrixinfundamentalterms}
\Gamma = c_{\mathbf{1}}{\mathbf{1}}+ c_5 \gamma^5 + c_\mu \gamma^\mu + c_{5\mu} i\gamma^\mu \gamma^5 + c_{\mu\nu} i \sigma^{\mu\nu},\end{gathered}$$ where we assume (without restriction) that $c_{\mu\nu} = - c_{\nu\mu}$ and the coefficients can be calculated using $$\begin{gathered}
\label{eqn:gammamatrixcoefficients}
\begin{gathered}
c_{\mathbf{1}}= \frac{1}{4} \operatorname{\mathbf{tr}}{\mathbf{1}}\Gamma,
\quad
c_5 = \frac{1}{4} \operatorname{\mathbf{tr}}\gamma^5 \Gamma,
\quad
c_\mu = \frac{1}{4} \operatorname{\mathbf{tr}}\gamma_\mu \Gamma,\\
c_{5\mu} = \frac{1}{4} \operatorname{\mathbf{tr}}i \gamma_{\mu} \gamma^5 \Gamma,
\quad
c_{\mu\nu} = \frac{1}{8} \operatorname{\mathbf{tr}}i \sigma_{\mu\nu} \Gamma.
\end{gathered}\end{gathered}$$ Due to the cyclic property of the trace, one can recursively calculate traces of arbitrary length without conceptual difficulties by permuting the first gamma matrix to the last position. For completeness we note the following relations $$\begin{gathered}
\label{eqn:traceformulasuptoordersix}
\begin{aligned}
&\frac14\,\operatorname{\mathbf{tr}}\gamma^\mu \gamma^\nu &
&= g^{\mu\nu},\\
&\frac14\,\operatorname{\mathbf{tr}}\gamma^\mu \gamma^\nu \gamma^\rho \gamma^\sigma &
&=
g^{\mu\sigma}\, g^{\nu\rho}
- \,g^{\mu\rho}\, g^{\nu\sigma}
+\,g^{\mu\nu}\, g^{\rho\sigma},\\
&\frac14\,\operatorname{\mathbf{tr}}\sigma^{\mu\nu} \gamma^\rho \gamma^\sigma &
&=
g^{\mu\sigma}\, g^{\nu\rho}
- \,g^{\mu\rho}\, g^{\nu\sigma},\\
&\frac14\, \operatorname{\mathbf{tr}}\gamma^\mu\gamma^\nu\gamma^\rho\gamma^\sigma \gamma^5&
&=
i{\epsilon}^{\mu\nu\rho\sigma}.\end{aligned}\end{gathered}$$ Thus, any identity involving gamma matrices can be proven by calculating the fundamental terms given in Eq. (\[eqn:decompositionofspinormatrixinfundamentalterms\]) for both sides of the equation. It is in particular possible to map the gamma matrix algebra to a corresponding tensor algebra once the decomposition of the product of two (arbitrary) gamma matrix expressions is known, $$\begin{gathered}
\Gamma_c = \Gamma_a \Gamma_b.\end{gathered}$$ Here $\Gamma_x$ is written as in Eq. (\[eqn:decompositionofspinormatrixinfundamentalterms\]) with the letter $c$ replaced by the letter $x$ appearing in the index. The coefficients of $\Gamma_c$ are then given by $$\begin{gathered}
\label{eqn:gammamatrixalgebra_productdecomposition}
\begin{aligned}
c_{\mathbf{1}}&= a_{\mathbf{1}}b_{\mathbf{1}}+ a_5 b_5 + a^\mu b_\mu + a_5^\mu b_{5\mu} + 2 a_{\mu\nu} b^{\mu\nu},\\[2ex]
c_5 &= {\left(}a_{\mathbf{1}}b_5 + a_5 b_{\mathbf{1}}{\right)}+ {\left(}i a^\mu b_{5\mu} - i a_{5\mu} b^\mu {\right)}\\
&\phantom{=}- i {\epsilon}^{\mu\nu\rho\sigma} a_{\mu\nu} b_{\rho\sigma},\\[2ex]
c_\mu
&=
{\left(}a_{\mathbf{1}}b_\mu + a_\mu b_{\mathbf{1}}{\right)}+ {\left(}i a_{5\mu} b_5 - i a_5 b_{5\mu} {\right)}\\
&\phantom{=}+ 2 {\left(}ia_{\mu\nu} b^\nu - i a^\nu b_{\mu\nu} {\right)}-i {\epsilon}_{\mu\nu\rho\sigma} {\left(}a^\nu_5 b^{\rho\sigma} + a^{\rho\sigma} b_5^{\nu} {\right)},\\[2ex]
c_{5\mu}
&=
{\left(}a_{\mathbf{1}}b_{5\mu} + a_{5\mu} b_{\mathbf{1}}{\right)}+ {\left(}ia_5 b_\mu - i a_\mu b_5 {\right)}\\
&\phantom{=}+ i{\epsilon}_{\mu\nu\rho\sigma} {\left(}a^\nu b^{\rho\sigma} + a^{\rho\sigma} b^\nu {\right)}+ 2{\left(}ia_{\mu\nu} b^\nu_5 -i a^\nu_5 b_{\mu\nu} {\right)},\\[2ex]
c_{\mu\nu}
&=
{\left(}a_{\mathbf{1}}b_{\mu\nu} + a_{\mu\nu} b_{\mathbf{1}}{\right)}-\frac{i}{2} {\epsilon}_{\mu\nu\rho\sigma} {\left(}a^{\rho\sigma} b_5 + a_5 b^{\rho\sigma} {\right)}\\
&\phantom{=}-\frac{i}{2} {\left(}a_\mu b_\nu - a_\nu b_\mu {\right)}-\frac{i}{2} {\epsilon}_{\mu\nu\rho\sigma} {\left(}a^\rho b_5^\sigma + a_5^\sigma b^\rho {\right)}\\
&\phantom{=}- \frac{i}{2} {\left(}a_{5\mu} b_{5\nu} - a_{5\nu} b_{5\mu} {\right)}+
2i {\left(}a_{\mu\rho} b^{\rho}_{\phantom{\rho}\nu} - a_{\nu\rho} b^{\rho}_{\phantom{\rho}\mu} {\right)}.
\end{aligned}\end{gathered}$$ We point out that taking the trace of the gamma matrix expression $\Gamma_c$ projects out the coefficient $c_{\mathbf{1}}$ \[see Eq. (\[eqn:gammamatrixcoefficients\])\]. Therefore, one can also use Eq. (\[eqn:gammamatrixalgebra\_productdecomposition\]) in the calculation of traces.
Tensor relations {#sec:tensorrelationsappendix}
================
If Eq. (\[eqn:gammamatrixalgebra\_productdecomposition\]) is used to simplify large gamma matrix expressions, one typically encounters products or contractions of the totally antisymmetric tensor ${\epsilon}^{\alpha\beta\gamma\delta}$. They can be simplified using well-known identities stated here for completeness [@landau_classical_1987]: $$\begin{gathered}
\label{eqn:epsidentities}
\begin{aligned}
{\epsilon}^{\alpha\beta\gamma\delta}{\epsilon}_{\alpha\beta\gamma\delta} &= -24,\\
{\epsilon}^{\alpha\beta\gamma\mu}{\epsilon}_{\alpha\beta\gamma\nu} &= -6 \delta^{\mu}_{\nu},\\
{\epsilon}^{\alpha\beta\mu\nu}{\epsilon}_{\alpha\beta\rho\sigma} &= -2\big(\delta^{\mu}_{\rho}\delta^{\nu}_{\sigma} - \delta^{\mu}_{\sigma}\delta^{\nu}_{\rho} \big),\\
{\epsilon}^{\mu\nu\rho\sigma} {\epsilon}_{\alpha\beta\gamma\sigma}
& =
-\big(\delta^{\mu}_{\alpha}\delta^{\nu}_{\beta}\delta^{\rho}_{\gamma}
-
\delta^{\mu}_{\alpha}\delta^{\nu}_{\gamma} \delta^{\rho}_{\beta}
+
\delta^{\mu}_{\gamma}\delta^{\nu}_{\alpha}\delta^{\rho}_{\beta}\\
&\phantom{=}- \delta^{\mu}_{\gamma}\delta^{\nu}_{\beta}\delta^{\rho}_{\alpha}
+
\delta^{\mu}_{\beta}\delta^{\nu}_{\gamma}\delta^{\rho}_{\alpha}
-
\delta^{\mu}_{\beta}\delta^{\nu}_{\alpha}\delta^{\rho}_{\gamma}
\big),\\
-{\epsilon}^{\mu\nu\rho\sigma} {\epsilon}_{\alpha\beta\gamma\delta}
&=
\det\left(
\begin{array}{cccc}
\delta^{\mu}_{\alpha} & \delta^{\mu}_{\beta} & \delta^{\mu}_{\gamma} & \delta^{\mu}_{\delta}\\
\delta^{\nu}_{\alpha} & \delta^{\nu}_{\beta} & \delta^{\nu}_{\gamma} & \delta^{\nu}_{\delta}\\
\delta^{\rho}_{\alpha} & \delta^{\rho}_{\beta} & \delta^{\rho}_{\gamma} & \delta^{\rho}_{\delta}\\
\delta^{\sigma}_{\alpha} & \delta^{\sigma}_{\beta} & \delta^{\sigma}_{\gamma} & \delta^{\sigma}_{\delta}
\end{array}\right).
\end{aligned}\end{gathered}$$ In particular, we note the following formulas for antisymmetric tensors $T^{\mu\nu}$, $T_1^{\mu\nu}$, and $T_2^{\alpha\beta}$: $$\begin{gathered}
\begin{aligned}
T_1^{*\mu\nu} T_2^{*\alpha\beta}
&=
\frac12 {\left(}g^{\mu\beta} g^{\nu\alpha} - g^{\mu\alpha} g^{\nu\beta} {\right)}T_{1\rho\sigma} T_2^{\rho\sigma}
- T_1^{\alpha\beta} T_2^{\mu\nu}
\\&+ g^{\nu\alpha} (T_1 T_2)^{\beta\mu} - g^{\mu\alpha} (T_1 T_2)^{\beta\nu}
\\&- g^{\nu\beta} (T_1 T_2)^{\alpha\mu} + g^{\mu\beta} (T_1 T_2)^{\alpha\nu},
\end{aligned}
\\
\begin{aligned}
(T^*_1 T^*_2)^{\mu\nu} &= \frac12 g^{\mu\nu} T_{1\alpha\beta} T_2^{\alpha\beta} + (T_1 T_2)^{\nu\mu},\\
T^*_{1\mu\nu} T_2^{*\mu\nu} &= - T_{1\mu\nu} T_2^{\mu\nu}
\end{aligned}\end{gathered}$$ and $$\begin{gathered}
\begin{aligned}
{\epsilon}^{\mu\nu\rho\sigma} T^*_{\sigma \alpha}
&=
\delta^\mu_\alpha T^{\nu\rho} - \delta^\nu_\alpha T^{\mu\rho} + \delta^{\rho}_{\alpha} T^{\mu\nu},
\\
\frac12 {\epsilon}_{\mu\nu\rho\sigma} T^{*\rho\sigma} &= - T_{\mu\nu}.
\end{aligned}\end{gathered}$$
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[^1]: We briefly mention here that the perturbative approach cannot be applied to processes involving particles with extremely high energies $\varepsilon$ such that $\alpha\log(\varepsilon/mc^2) \sim 1$, with $m$ being the electron mass [@landau_quantum_1981].
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'There are exactly two different types of bi-dimensional improper affine spheres: the non-convex ones can be modeled by the center-chord transform of a pair of planar curves while the convex ones can be modeled by a holomorphic map. In this paper, we show that both constructions can be generalized to arbitrary even dimensions: the former class corresponds to the center-chord transform of a pair of Lagrangian submanifolds while the latter is related to special Kähler manifolds. Furthermore, we show that the improper affine spheres obtained in this way are solutions of certain exterior differential systems. Finally, we also discuss the problem of realization of simple stable Legendrian singularities as singularities of these improper affine spheres.'
address:
- 'Departamento de Matemática - PUC-RioRio de JaneiroBrazil'
- 'Faculty of Mathematics and Information Science Warsaw University of Technology ul. Koszykowa 75, 00-662 WarszawaPoland'
- 'Departamento de Matemática - ICMC Universidade de São Paulo São CarlosBrazil'
author:
- Marcos Craizer
- Wojciech Domitrz
- 'Pedro de M. Rios'
date: 'november 15, 2013'
title: Even Dimensional Improper Affine Spheres
---
[^1]
Introduction
============
A hypersurface whose Blaschke normal vectors are pointing to a center is called an affine sphere. This class of manifolds is quite large and has been studied by various researchers ([@Loftin]). Hypersurfaces with vanishing affine mean curvature are called affine maximal surfaces and have also been extensively studied ([@Aledo09]). Parabolic, or Improper Affine Spheres (IAS) are affine spheres that are also affine maximal. This is equivalent to saying that the Blaschke normal vectors are parallel, i.e., the center of the affine “sphere” is at infinity. There are many articles studying two dimensional IAS ([@Galvez07],[@Craizer11],[@Craizer12],[@Ishikawa06A], [@Martinez05],[@Milan13],[@Milan14]) and there is also some work in dimension three ([@Ishikawa06B]). In this paper we shall consider IAS in arbitrary even dimensions.
Remind that for an immersion $\phi: U\subset {\mathbb{R}}^N\to {\mathbb{R}}^{N+1}$, if $\tilde{\nabla}$ denotes the canonical affine connection on ${\mathbb{R}}^{N+1}$, then any transversal vector field $\xi$ to $\phi(U)$ defines a connection $\nabla$ and symmetric bilinear form $h$ on $TU$ by $$\tilde{\nabla}_{X}\phi_*{Y}=\phi_*(\nabla_{X}{Y})+h(X,Y)\xi,$$ where $X,Y$ are smooth vector fields on $U$. The symmetric bilinear form $h$ defines a volume element on U, denoted $\nu_h$. On the other hand, $\phi^*\Theta_{\xi}$, where $\Theta_{\xi}(\cdot)=\det(\cdot,\xi)$, defines another volume element on $U$. Then, a well-known theorem of Blaschke ([@Blaschke23],[@Nomizu94]) asserts that there exists a unique, up to sign, transversal vector field $\xi$ such that $\nu_h=\phi^*\Theta_{\xi}$ and furthermore $\nabla(\phi^*\Theta_{\xi})=0$. This unique $\xi$ is called the affine normal, or Blaschke normal vector field to the hypersurface $\phi(U)\subset{\mathbb{R}}^{N+1}$.
Let $\xi=(0_N,1)\in{\mathbb{R}}^N\times{\mathbb{R}}$ be a parallel vector field for the canonical connection $\tilde{\nabla}$ on ${\mathbb{R}}^{N+1}$. It is well-known (straightforward computation) that the graph of a function $F:V\subset{\mathbb{R}}^N\to{\mathbb{R}}$ is an improper affine sphere with affine normal $\xi$ if and only if $F$ satisfies the classical Monge-Ampère equation $$\label{eq:MongeAmpere}
\det\left(\frac{\partial^2F}{\partial x^2}\right)=c,$$ for some constant $c$, where the l.h.s. of (\[eq:MongeAmpere\]) denotes the Hessian of $F$. The class of Monge-Ampère equations, in particular the classical one, is an important topic of study in partial differential equations and this highlights the importance of improper affine spheres in geometric analysis (see, e.g., the recent expositions and surveys [@LJSX],[@Loftin]).
Now, for a smooth function $F:V\to{\mathbb{R}}$, where $V$ is an open subset of $\mathbb R^{2n}$, we can translate the Monge-Ampère equation in symplectic terms, as follows. Denote the canonical symplectic form in ${\mathbb{R}}^{2n}$ by $$\label{eq:omega}
\omega=\sum_{i=1}^n dx_i\wedge dx_{i+n}$$ and let $Y_F$ be the Hamiltonian vector field of $F$, i.e., $$\label{eq:sympgradient}
dF=\omega(\cdot ,Y_F).$$ Then $F$ satisfies the classical Monge-Ampère equation (\[eq:MongeAmpere\]) if and only if there is a constant $c$ such that $$\label{eq:symplecticMA}
\det(DY_F)=c,$$ where $DY_F$ denotes the jacobian matrix of the map $x\mapsto Y_F(x)$.
For an open set $U\subset{\mathbb{R}}^{2n}$, consider an immersion $\phi:U\to{\mathbb{R}}^{2n+1}$ transversal to $\xi=(0,1)\in{\mathbb{R}}^{2n}\times{\mathbb{R}}$, where the latter ${\mathbb{R}}^{2n}$ carries the symplectic form $\omega$. We can write $\phi(r)=(x(r), f(r))\in {\mathbb{R}}^{2n}\times{\mathbb{R}}$, where $x(r)\in V\subset{\mathbb{R}}^{2n}$ is locally invertible and $f(r)=F(x(r))$, for some $F:V\to{\mathbb{R}}$. Denote by $Y_F(x)$ the Hamiltonian vector field of $F$ defined by equation and let $y(r)=Y_F(x(r))$. Define $A(r):T_rU\to T_rU$ by $$\label{eq:definea}
Dy(r)=Dx(r)\cdot A(r).$$ It follows from (\[eq:symplecticMA\]) that $\phi$ is an IAS with Blaschke normal $(0,1)$ if and only if $\det A=c$, for some constant $c$.
In dimension two, any non-convex IAS can be parameterized by asymptotic coordinates and modeled by the center-chord transform of a pair of planar curves ([@Craizer08],[@Craizer11],[@Milan13]). In this paper, we show that this construction can be generalized to arbitrary even dimensions, where we consider $x$ as the center and $y$ as the mid-chord of a pair of real Lagrangian submanifolds. In this case, the asymptotic coordinates condition is replaced by the equation $Dy(r)=Dx(r)\cdot K_{2n}$, with $$\label{defk}
K_{2n}=\left[
\begin{array}{cc}
-I_n & 0\\
0& I_n
\end{array}
\right],$$ where $I_n$ denotes the $n\times n$ identity matrix.
Any convex bi-dimensional IAS can be parameterized by isothermal coordinates and modeled by a holomorphic map ([@Galvez07],[@Craizer12],[@Martinez05]). This construction can be generalized to arbitrary even dimensions starting from a holomorphic map $G:{\mathbb{C}}^n\to{\mathbb{C}}$ and the isothermal condition replaced by the relation $Dy(r)=Dx(r)\cdot J_{2n}$, with $$\label{defj}
J_{2n}=\left[
\begin{array}{cc}
0 & I_n\\
-I_n& 0
\end{array}
\right].$$ IAS of this type have already been considered in connection with special Kähler manifolds in [@Cortes00], where they were called [*special*]{}. We shall see that special IAS are naturally related to a rotated center-chord transform of a pair of complex conjugate Lagrangian submanifolds.
Improper affine spheres can also be seen as geometric solutions of a Monge-Ampère system ([@Ishikawa06A]). Consider a contact form $\theta$ in ${\mathbb{R}}^{4n+1}$ given by $$\label{eq:theta}
\theta=dz-\sum_{i=1}^n y_{i+n}dx_i-y_idx_{i+n}$$ and let $$\label{eq:Omega}
\Omega=\sum_{i=1}^n dx_i\wedge dy_{i+n}+dy_i\wedge dx_{i+n}.$$ be the associated canonical symplectic form in ${\mathbb{R}}^{4n}$. For any $F:V\to{\mathbb{R}}$, it follows from (\[eq:sympgradient\]) that the image of a map $L:V\to\mathbb R^{4n}, \ x\mapsto L(x)=(x,Y_F(x))$, is a Lagrangian submanifold of the symplectic space $(\mathbb R^{4n},\Omega)$, i.e. $L^*\Omega=0$, while the image of a map ${\tilde L}:V\to\mathbb R^{4n+1}, \ x\mapsto {\tilde L}(x)=(x,Y_F(x),F(x))$, is a Legendrian submanifold of the contact space $(\mathbb R^{4n+1},\{\theta=0\})$, i.e. ${\tilde L}^*\theta=0$. Then, consider the $2n$-form $\eta$ in ${\mathbb{R}}^{4n}$ given by $$\eta=c\ dx_1\wedge ....\wedge dx_{2n} - dy_1\wedge ....\wedge dy_{2n}.$$ A solution of the Monge-Ampère system $\{\theta,\eta\}$ is a map $F:V\to{\mathbb{R}}$ such that ${\tilde L}^*\theta=0$ and $L^*\eta=0$. Thus $F:V\to{\mathbb{R}}$ is a solution of this Monge-Ampère system if and only if the graph of $F$ is an IAS.
Differently from the case $n=1$, there are other IAS of dimension $2n$ that are neither center-chord nor special, as we show in some examples. On the other hand, as a main result of this paper, we shall prove that center-chord and special IAS can be characterized as solutions of certain exterior differential systems (EDS). Define a symplectic form in ${\mathbb{R}}^{4n}$ by $$\label{eq:Omega1}
\Omega_1=\sum_{i=1}^n dx_i\wedge dx_{i+n}+dy_i\wedge dy_{i+n}$$ and consider the EDS $\mathcal{E}_1=\{\Omega,\Omega_1\}$. We shall verify that the center-chord IAS are exactly the solutions of the EDS $\mathcal{E}_1$. Similarly, let $$\label{eq:Omega2}
\Omega_2=\sum_{i=1}^n dx_i\wedge dx_{i+n}-dy_i\wedge dy_{i+n}.$$ and define the EDS $\mathcal{E}_2=\{\Omega,\Omega_2\}$. We shall prove that special IAS are the solutions of $\mathcal{E}_2$. A natural question that is left out from this paper is whether there are other classes of IAS that are solutions of some EDS.
Therefore, center-chord and special IAS provide two general classes of solutions to the classical Monge-Ampère equation in any even number of variables. But general solutions are known to present singularities. In fact, except for paraboloids, any convex IAS admits singularities, thus singularities appear naturally in the context of improper affine spheres.
Denote by $\pi_1:{\mathbb{R}}^{2n}\times{\mathbb{R}}^{2n}\to{\mathbb{R}}^{2n}$ the projection $\pi_1(x,y)=x$ and by ${\mathcal L}$ the image of the Lagrangian immersion $L$ described above. In the context of IAS, singularities of the Lagrangian map $\pi_1:{\mathcal L}\to{\mathbb{R}}^{2n}$ are the ones which were called admissible in [@Martinez05] and, in that paper, IAS with admissible singularities were called Improper Affine Maps. In dimension $2$, admissible singularities have been well studied ([@Galvez07],[@Craizer11],[@Craizer12],[@Martinez05],[@Milan13],[@Milan14]).
One can also consider singularities of the Legendrian map $\pi_2:{\tilde{\mathcal L}}\to{\mathbb{R}}^{2n+1}$, where $\pi_2:{\mathbb{R}}^{2n}\times{\mathbb{R}}^{2n}\times{\mathbb{R}}\to{\mathbb{R}}^{2n}\times{\mathbb{R}}$ is the projection $\pi_2(x,y,z)=(x,z)$ and ${\tilde {\mathcal L}}$ is the image of the above Legendrian immersion ${\tilde L}$. For $2$-dimension IAS, these singularities were studied in [@Ishikawa06A].
We shall study in this paper the stable singularities of the above Lagrangian and Legendrian maps for general even dimensions. From Theorem 4.1 in [@Domitrz13], we know that any simple stable Lagrangian singularity is realizable as a center-chord IAS. Here we prove that this also holds for special IAS, and our proof extends naturally to the Legendrian setting, showing that any simple stable Lagrangian and Legendrian singularity is realizable as a special IAS. Starting from [@Domitrz13], one can easily verify that the Legendrian statement also holds for center-chord IAS, but here we prove this explicitly in a way that highlights the similarities between the center-chord and the special IAS. In the center-chord case, we also comment on the boundary singularities, or “on-shell” singularities, that appear in the limit of vanishing chords and have a special symmetry, as described in [@DMR].
This paper is organized as follows: In section 2 we establish the notation and describe the symplectic condition for an immersion to be an IAS. In section 3 we describe the models for center-chord and special IAS. In section 4 we prove that the center-chord and special IAS are the solutions of the EDS $\mathcal{E}_1$ and $\mathcal{E}_2$, respectively. In section 5 we discuss the Lagrangian and Legendrian singularities of these maps.
[*Acknowledgements*]{}: The third author benefitted from the hospitality of the Mathematics Department at UC Berkeley during the final stages of the preparation of this manuscript. Special thanks to his host, Alan Weinstein, for comments and suggestions that improved this final version.
Symplectic characterization of IAS
==================================
The symplectic structructure of $TV$ and contact structure of $TV\times {\mathbb{R}}$
-------------------------------------------------------------------------------------
Let $V$ be an open subset of $\mathbb R^{2n}$ and let $\omega$ be the canonical symplectic form on $V$. A map $\flat: TV\ni v\mapsto \omega(v,\cdot) \in T^{\ast}V$ is a isomorphism of the bundles $TV, \ T^{\ast}V$. Let $\alpha$ be the canonical Liouville $1$-form on $T^{\ast}V$. Then $\Omega=\flat^{\ast}d\alpha$ is a symplectic form on $TV$ and $\theta=dz+\flat^{\ast}\alpha$ is a contact form on $TV\times \mathbb R$, where $z$ is a coordinate on $\mathbb R$.
Let $F:V\rightarrow \mathbb R$ be a smooth function. Let $Y_F$ be the Hamiltonian vector field of $F$ e. i. $\omega(Y_F, \cdot) = -dF(\cdot)$.
A map $\tilde{L}:V\ni x\mapsto (x,Y_F(x),F(x))\in TV\times \mathbb R$ is a Legendrian immersion to the contact space $(TV\times \mathbb R, \{ \theta=0 \})$.
It is obvious that $\tilde{L}$ is a immersion. We have $$\tilde{L}^{\ast}\theta=dF+\tilde{L}^{\ast}\flat^{\ast}\alpha=dF+(\flat\circ\tilde{L})^{\ast}\alpha).$$ On the other hand $\flat\circ\tilde{L}=\flat(Y_F)=\omega(Y_F,\cdot)=-dF$. By the tautological property of the Louville $1$-form $\alpha$ we have $(\beta)^{\ast}\alpha=\beta$ for any $1$-form $\beta$ on $V$. Thus we get $\tilde{L}^{\ast}\theta=dF+(-dF)^{\ast}\alpha=dF-dF=0$.
Using the same arguments one can prove the following proposition.
\[LagHam\] A map $L:V\ni x\mapsto (x,Y_F(x))\in TV$ is a Lagrangian immersion to the symplectic space $(TV, \Omega)$.
Let $x=(x_1,\cdots,x_{2n})$ be a coordinate system on $V$ and $$\label{omegacan}
\omega=\sum_{i=1}^{n}dx_i\wedge dx_{i+n}.$$ Let $(x,y)=(x_1,\cdots,x_{2n},y_1,\cdots,y_{2n})$ be the standard coordinate system on $TV$, $(x,y,z)$ be a coordinate system on $TV\times \mathbb R$ and finally let $(x,p)=(x_1,\cdots,x_{2n},p_1,\cdots,p_{2n})$ be the standard coordinate system on $T^{\ast}V$.
The Liouville $1$-form in these coordinates is $\alpha=\sum_{i=1}^n p_i dx_i$ and the isomorphism is $$\flat(x,y)=(x_1,\cdots,x_{2n},-y_{n+1},\cdots,-y_{2n},y_1,\cdots,y_n).$$ Thus the symplectic form and the contact form have the following forms $$\label{Omega}
\Omega=\sum_{i=1}^n dx_i\wedge dy_{i+n} +dy_i\wedge dx_{i+n},$$ $$\label{theta}
\theta=dz-\sum_{i=1}^ny_{i+n}dx_i-y_idx_{i+n}.$$
Center-chord transforms
-----------------------
The form $\Omega$ is also called the tangential lift of $\omega$ and, under the identification $y=\dot x$, it can be formally identified with the “time derivative of $\omega$” and is often denoted by $\dot\omega$. By Proposition \[LagHam\], the “graph” of a Hamiltonian function $F$, i.e. its vector field $Y_F$, is Lagrangian w.r.t. $\Omega$. Such a Hamiltonian vector field is usually seen as the generator of a canonical transformation on $(V,\omega)$.
However, the form $2\Omega$ can also be seen as the pullback of the “difference” symplectic form $\omega\ominus\omega = (\omega,-\omega)$ on $V\times V$ via the linear diffeomorphism $$\label{psi}\Psi:TV\to V\times V , \ (x,y)\mapsto(x+y,x-y)=(x_+,x_-).$$ In this context, the coordinates $(x,y)$ are called the center and mid-chord coordinates and $\Psi^{-1}$ is the [*center-chord transform*]{}. This is globally well-defined when $V=\mathbb R^{2n}$ and thus, in such a case, $$\label{invpsi} \Psi^{-1}:V\times V\to TV , \ (x_+,x_-)\mapsto\left(\frac{x_++x_-}{2},\frac{x_+-x_-}{2}\right)=(x,y),$$ so that $$\label{pullbackOmega} (\Psi^{-1})^*\Omega=\frac{1}{2}(\omega_+-\omega_-),$$ where $\omega_+$ and $\omega_-$ are given as in (\[omegacan\]), for coordinates $x_+=(x^+_i, ..., x^+_{2n})$, $x_-=(x^-_1, ..., x^-_{2n})$ in $V\times V$.
Then, a pair of real Lagrangian submanifolds, $\Lambda_1,\Lambda_2$, of $(V , \omega)$ pulls back to a real Lagrangian submanifold $L=\Psi^{-1}(\Lambda_1\times\Lambda_2)$ of $(TV,\Omega)$ which, when projecting regularly to the center subspace $V\ni x$ can be described as the “graph” of a function $F$ by $L(x)=(x,Y_F(x))$, as above (here the center subspace $V\simeq T^0V$ is seen as the zero section of $TV$).
In this setting, the function $F$ is the Poincaré, or center generating function of the canonical relation $\Lambda_1\times\Lambda_2\subset V\times V$ ([@Poin][@Wein][@RO]). Note that this differs from the usual setting when $Y_F$ is the Hamiltonian vector field that generates an infinitesimal canonical transformation $\Phi:V\to V$ because, in the latter case, although the graph of $\Phi$ is also a real Lagrangian submanifold of $V\times V$, it projects regularly to both copies of $V$.
The above center-chord description can be generalized to study complex Lagrangian submanifolds of a complexified (real) symplectic vector space $(V^{{\mathbb{C}}},\omega)$. In this case, one fixes a complex structure and assigns a pair of holomorphic and anti-holomorphic coordinates, $x=(u,\bar u)\in V^{{\mathbb{C}}}$, so that the symplectic form is given in these complex canonical coordinates by $$\label{complexomega} \omega=\frac{i}{2}du\wedge d\bar u \ ,$$ with index summation subtended, and thus $\omega$ is still a real form, $\bar\omega=\omega$.
The map $\Psi:TV^{{\mathbb{C}}}\to V^{{\mathbb{C}}}\times V^{{\mathbb{C}}}$, given by (\[psi\]), assigns complex canonical coordinates in each copy of $V^{{\mathbb{C}}}$ which are induced from complex canonical coordinates $x=(u,\bar u), y=(w,\bar w)$ in $TV^{{\mathbb{C}}}$, and vice versa, by $$\label{complexpm} x_+=(z_+,\bar z_+) =(u+w,\bar u+\bar w) \ , \ x_-=(z_-,\bar z_-) =(u-w,\bar u-\bar w) \ ,$$ and thus the relevant [*real*]{} symplectic forms in $TV^{{\mathbb{C}}}$ and $V^{{\mathbb{C}}}\times V^{{\mathbb{C}}}$ are written in these complex canonical coordinates as $$\begin{aligned}
& \Psi^*(\omega_+-\omega_-)=2\Omega={i}(dw\wedge d\bar u + du\wedge d\bar w) \ , \\
& (\Psi^{-1})^*(2\Omega)=\omega_+-\omega_-=\frac{i}{2}(dz_+\wedge d\bar z_+ - dz_-\wedge d\bar z_-) \ . \end{aligned}$$
However, for various reasons, some to be made clearer further below, it is also useful to define the [*rotated center-chord transform*]{} as $$\label{complexinvpsi}
\widetilde{\Psi}^{-1}:V^{{\mathbb{C}}}\times V^{{\mathbb{C}}}\to TV^{{\mathbb{C}}}, \ (\tilde x_+,\tilde x_-)\mapsto\left(\frac{\tilde x_++\tilde x_-}{2},\frac{\tilde x_+-\tilde x_-}{2i}\right)=(x,y),$$ with inverse $$\label{complexpsi}
\widetilde{\Psi}: TV^{{\mathbb{C}}}\to V^{{\mathbb{C}}}\times V^{{\mathbb{C}}}, \ (x, y)\mapsto (x+i y, x-i y)=(\tilde x_+,\tilde x_-).$$ Note that the new map $\widetilde{\Psi}$ is obtained from the old one by first multiplying each fiber of $TV^{{\mathbb{C}}}$ by $i$, that is: $$\label{multii} J_x : T_xV^{{\mathbb{C}}}\to T_xV^{{\mathbb{C}}} \ , \ y\mapsto iy \ ,$$ but this is equivalent to performing a $\pi/2$ rotation on each fiber of $TV^{{\mathbb{C}}}$, so that $J_x$ can also be defined using only real coordinates, that is: $$\label{multiir} J_x : T_xV\to T_xV \ , \ J_x^2=-Id_x \ .$$ Now, if $ J$ denotes the map $TV^{{\mathbb{C}}}\to TV^{{\mathbb{C}}}$ (or $TV\to TV$) which is defined by the collection of fiber maps $J_x$ as above, $\forall x\in V$, then $$\label{compJ} \widetilde{\Psi}=\Psi\circ J \ ,$$ so that $\widetilde\Psi$ and its inverse, the rotated center-chord transform $\widetilde\Psi^{-1}$, can also be defined as real maps $TV\to V\times V$ and $V\times V\to TV$, respectively.
Immersions that are transversal to a constant direction
-------------------------------------------------------
In this section we recall some basic facts concerning dual connections. Let $U\subset{\mathbb{R}}^{2n}$ be an open simply connected set and let $V$ be an open set of the symplectic affine space ${\mathbb{R}}^{2n}$ with its canonical symplectic form $\omega$.
Consider an immersion $\phi:U\to V\times{\mathbb{R}}\subset{\mathbb{R}}^{2n}\times{\mathbb{R}}$ transversal to $\xi=(0,1)$. For $r\in U$, write $$\label{eq:graphconnection}
\tilde\nabla_{X}\phi_*{Y}=\phi_*(\nabla_{X}{Y})+h(X,Y)\xi,$$ for any smooth vector fields $X,Y$ on $U$, where $\tilde\nabla$ denotes the canonical connection in ${\mathbb{R}}^{N+1}$.
$\nabla$ is a torsion free affine connection and $h$ is a symmetric bilinear form.
Interchanging the roles of $X$ and $Y$ in and subtracting we obtain $$\nabla_{X}{Y}-\nabla_{Y}X-[X,Y]=0$$ and $$h(X,Y)-h(Y,X)=0.$$
Denote $\phi(r)=(x(r), f(r))$ with $f(r)=F(x(r))$. Denote by $Y_F$ denote the Hamiltonian vector field of $F:V\subset\mathbb R^{2n}\to \mathbb R$ and let $y(r)=Y_F(x(r))$. We have that $$\label{defin1}
df(r)\cdot u=\omega(Dx(r)u,y(r)), \ \ \ \forall u\in T_rU,$$ where the dot $\cdot$ in the l.h.s. of (\[defin1\]) denotes the usual vector-form contraction, seen also as the matrix product of a line $1\times(2n)$ and a column $(2n)\times 1$ matrix, as we shall be using the dot $\cdot$ to denote matrix product in various places below. Fix a basis $\left\{e_i\right\}_{1\leq i\leq 2n}$ of $T_rU$ and write $$x_{r_i}=Dx(r)\cdot e_i;\ \ \ x_{r_ir_j}= Dx_{r_i}\cdot e_j.$$
We have that $$\label{eq:hsym}
h\left(e_i,e_j \right)=\omega(x_{r_i},y_{r_j})=\omega(x_{r_j},y_{r_i})$$ and the $\nabla$-Christoffel symbols $\Gamma_{ij}^k$ are given by the following formula $$\label{eq:Christoffel1}
x_{r_ir_j}=\sum_{k}\Gamma_{ij}^k x_{r_k}.$$
Since $$\phi_{r_i}=\left(x_{r_i},\omega(x_{r_i},y)\right).$$ we obtain $$\phi_{r_ir_j}=\left(x_{r_ir_j},\omega(x_{r_ir_j},y)\right)+\left(0,\omega(x_{r_i},y_{r_j})\right).$$ Now observe that the first parcel in the second member is tangent while the second parcel is a multiple of $\xi$. On the other hand we have $$\left(x_{r_ir_j},\omega(x_{r_ir_j},y)\right)=\phi_{\ast}(\nabla_{e_j}e_i)=\sum_{k}\Gamma_{ij}^k\phi_{r_k}=\left(\sum_{k}\Gamma_{ij}^k x_{r_k}, \omega(\sum_{k}\Gamma_{ij}^k x_{r_k},y)\right)$$ Thus the lemma is proved.
Define $g:U\to{\mathbb{R}}$ by $$dg(r)\cdot u=\omega(Dy(r)u,x(r)), \ \ \ \forall u\in T_rU.$$ Assuming $y(r)$ is locally invertible, the immersion $\psi(r)=(y(r),g(r))$ is called the [*dual immersion*]{} of $\phi$ and the function $G$ such that $g(r)=G(y(r))$, is the [*Legendre transform*]{} of $F$. Equation implies that $g$ is locally well-defined.
Denoting by $\overline{\nabla}$ and $\overline{h}$ the connection and metric of the dual immersion, we have that $\overline{h}=h$ and the $\overline{\nabla}$-Christoffel symbols ${\overline\Gamma_{ij}^k}$ are given by the following formula $$\label{eq:Christoffeldual1}
y_{r_ir_j}=\sum_k {\overline\Gamma_{ij}^k} y_{r_k}.$$
$\overline{\nabla}$ is the $h$-dual, or metric-dual of $\nabla$. In other words, the connection $\hat{\nabla}$ of the metric $h$ is given by $$\hat{\nabla}=\frac{\nabla+{\overline\nabla}}{2}.$$
We must prove that $$\label{eq:dualconnection}
\frac{\partial}{\partial r_k}h\left( e_i, e_j \right) =h\left( \nabla_{e_k} e_i , e_j\right)+h\left(e_i, {\overline \nabla}_{e_k} e_j \right).$$ The first member of is equal to $$\omega\left(x_{r_kr_i}, y_{r_j}\right)+\omega\left(x_{r_i}, y_{r_kr_j}\right)
=\omega\left(\sum_l\Gamma_{ik}^lx_{r_l}, y_{r_j}\right)+\omega\left(x_{r_i}, \sum_l{\overline \Gamma}_{kj}^l y_{r_l}\right)$$ $$=\omega\left(x_*\left(\nabla_{e_k} e_i\right), y_*e_j\right)+\omega\left(x_*e_i, y_*\left({\overline \nabla}_{e_k} e_j \right)\right),$$ which is exactly the second member of .
Denote by $A(r):T_rU\to T_rU$ the invertible linear map satisfying the condition $Dy(r)=Dx(r)\cdot A(r)$. We shall make no distinction between the linear map $A(r)$ and its matrix $A(r)=\left(a_{ij}(r)\right)_{i,j=1,\cdots,2n}$ in the canonical basis $\{e_1,..,e_{2n}\}$.
\[lemma:aconstant\] We have ${\overline\nabla}=A^{-1}\nabla A$.
We must prove that $$A{\overline\nabla}_{e_k}{e_j}=\nabla_{e_k}(Ae_j),$$ for any $1\leq j,k\leq 2n$. This is equivalent to the following formula $$\label{eq:Christoffel2a}
\sum_{l}\overline{\Gamma}_{ik}^{l}a_{sl}=\sum_{j}a_{ji}\Gamma_{jk}^s+\frac{\partial a_{si}}{\partial r_k},$$ for any $1\leq l\leq 2n$. But $$\label{eq:yxaexplicit}
y_{r_i}=\sum_{j}x_{r_j}a_{ji}$$ Differentiating with respect to $r_k$ we obtain $$\label{eq:yxaconstant}
y_{r_ir_k}=\sum_{j}x_{r_jr_k}a_{ji}+\frac{\partial a_{ji}}{\partial r_k}x_{r_j}.$$ Applying and in and using we obtain .
In the next proposition we present a sufficient condition for $A$ to be parallel with respect to the metric connection $\hat{\nabla}$.
\[prop:aparallel\] If $\nabla \left(A^2\right)=0$ then $\hat{\nabla} A=0$.
We have that $$(\hat{\nabla}_XA)Y=\hat{\nabla}_X(AY)-A\hat{\nabla}_XY$$ $$=\frac{1}{2}\left( \nabla_X(AY)+\overline{\nabla}_X(AY) -A\nabla_XY-A\overline{\nabla}_XY \right).$$ Now Proposition \[lemma:aconstant\] implies that $$(\hat{\nabla}_XA)Y=\frac{1}{2}\left( \nabla_X(AY)+A^{-1}{\nabla}_X(A^2Y) -A\nabla_XY-{\nabla}_X(AY) \right)$$ $$=\frac{1}{2}\left( A^{-1} \nabla_X(A^2Y)-A{\nabla}_XY \right)$$ If $\nabla \left(A^2\right)=0$ then $\nabla_X(A^2Y)=A^2\nabla_X Y$ and this last expression vanishes.
In this paper we are specially interested in the cases $A(r)=K_{2n}$ and $A(r)=J_{2n}$. Next result which is a corollary of Proposition \[prop:aparallel\] shows that in this case $A$ is parallel with respect to the metric connection $\hat{\nabla}$.
\[prop:kj-aparallel\] If $A(r)=K_{2n}$ or $A(r)=J_{2n}$ then $\hat{\nabla} A=0$.
Improper affine spheres
-----------------------
An immersion $\phi:U\to{\mathbb{R}}^{2n+1}$ is an improper affine sphere with Blaschke normal vector $\xi=(0,1)$ if the volume determined by the metric $h$ coincides with the volume $\phi^*\Theta_{\xi}$, where $\Theta_{\xi}(\cdot)=\det(\cdot,\xi)$ (see [@Nomizu94]). This is equivalent to $|\det(h)(r)|=det(Dx)^2(r)$, for any $r\in U$.
Let $$\label{defineB}
B(r)=Dx(r)\cdot A(r)\cdot Dx(r)^{-1}.$$
We have that $$\det(h)=\det(Dx)^2\det A.$$
The symmetric matrix $h$ has entries $$h_{ij}=\omega(Dx(r)\cdot e_i, Dx(r)\cdot A(r)\cdot e_j).$$ Since $B(r)=Dx(r)\cdot A(r)\cdot Dx(r)^{-1}$, we have that $\det B=\det A$ and $$h_{ij}=\omega(Dx(r)\cdot e_i,B(r)\cdot Dx(r)\cdot e_j).$$ In terms of matrices, $h=Dx(r)^t \cdot {J_{2n}}\cdot B(r)\cdot Dx(r)$. Hence $$\det(h)=\det(Dx)^2\det B,$$ thus proving the lemma.
The metric $h$ is non-degenerate if and only if $A$ is invertible.
\[prop:Deta\] $\phi$ is an improper affine sphere if and only if $\det A$ is constant.
The immersion $\phi$ is an improper affine sphere if and only if the metric volume in the tangent space is the same as the volume determined by $\xi$. The metric volume is $\sqrt{\det(h)}$, while the volume determined by $\xi$ is $$\det(\phi_{r_1},...,\phi_{r_{2n}},\xi)=\det(x_{r_1},...,x_{r_{2n}})=\det(Dx).$$ Thus $\phi$ is an improper affine sphere if and only if $\sqrt{\det(h)}=c\det(Dx)$, for some constant $c$. Since $$\sqrt{\det(h)}=\det(Dx) \sqrt{\det A},$$ the proposition is proved.
Two distinguished classes of even dimensional improper affine spheres
=====================================================================
In this section we shall describe two classes of even dimensional improper affine spheres. The first one is obtained by taking $x$ as the center and $y$ as the mid-chord of a pair of points of a given pair of real Lagrangian submanifolds. It is a natural generalization of the class of bi-dimensional improper affine spheres with indefinite metric. The second one is a natural generalization of the class of bi-dimensional improper affine spheres with definite metric. IAS in this latter class are called special ([@Cortes00]).
In the center-chord case, the matrix $A$ is similar to $K_{2n}$, while in the special case the matrix $A$ is similar to $J_{2n}$. By proposition \[prop:aparallel\], in both cases the matrix $A$ is parallel with respect to the metric connection. This fact was proved in [@Cortes00] in the special IAS case.
Center-chord improper affine spheres {#sec:Center-chord}
------------------------------------
Let $U_1$, $U_2$ be open subsets of ${\mathbb{R}}^n$ such that $U=U_1\times U_2\subset{\mathbb{R}}^{2n}$ is simply connected. Let $\beta:U_1\to{\mathbb{R}}^{2n}$, $\gamma:U_2\to{\mathbb{R}}^{2n}$ be real Lagrangian embeddings and $\Lambda_1=\beta(U_1)$, $\Lambda_2=\gamma(U_2)$.
Define the center $x:U\to{\mathbb{R}}^{2n}$ by $$x(s,t)=\frac{1}{2}\left(\beta(s)+\gamma(t)\right)$$ and the half-chord $y:U\to{\mathbb{R}}^{2n}$ by $$y(s,t)=\frac{1}{2}\left( \gamma(t)-\beta(s) \right),$$ where $s=(s_1,...,s_n)\in U_1$ and $t=(t_1,...,t_n)\in U_2$. Observe that since $\beta$ and $\gamma$ are Lagrangian, $$\omega(x_{s_i},y_{s_j})=\omega(x_{t_i},y_{t_j})=0.$$ Moreover, $$\omega(x_{s_i},y_{t_j})=\omega(\beta_{s_i},\gamma_{t_j})=\omega(\gamma_{t_j},-\beta_{s_i})=\omega(x_{t_j},y_{s_i}),$$ which implies in the existence of some function $f:U\to{\mathbb{R}}$ satisfying $$f_{s_i}=\omega(x_{s_i},y),\ \ f_{t_i}=\omega(x_{t_i},y), \text{for} \ i=1,\cdots,n .$$
\[ccias\] Assume that the tangent spaces of $\Lambda_1$ at $\beta(s)$ and of $\Lambda_2$ at $\gamma(t)$ are transversal. Then the immersion $\phi(s,t)=(x(s,t),f(s,t))$ is an immersion with $A(r)=A(s,t)=K_{2n}$. As a consequence, $\Sigma^{2n}= \text{Image}(\phi)\subset\mathbb R^{2n+1}$ is an improper affine sphere with Blaschke normal $\xi=(0_{2n},1)$ and Blaschke metric given by $$\label{eq:ccmetric}
h=\frac{1}{4}\left[
\begin{array}{cc}
0 & \omega(\beta_{s_i},\gamma_{t_j}) \\
\omega(\beta_{s_i},\gamma_{t_j}) & 0
\end{array}
\right].$$
The first statement follows from $$y_{s_i}=-\frac{1}{2}\beta_{s_i}=-x_{s_i};\ \ y_{t_j}=\frac{1}{2}\gamma_{t_j}=x_{t_j}.$$ Thus by Proposition \[prop:Deta\], $\phi$ is an improper affine sphere with Blaschke metric given by equation .
The function $f(s,t)$ can be geometrically interpreted as follows: Fix points $\beta(s_0)\in\Lambda_1$ and $\gamma(t_0)\in\gamma$ and take curves $\tilde{\beta}\subset\Lambda_1$ connecting $\beta(s_0)$ with $\beta(t)$ and $\tilde{\gamma}\subset\Lambda_2$ connecting $\gamma(t)$ with $\gamma(t_0)$. Denote by $S$ a $2$-surface whose boundary is the concatenation of the chord $\gamma(t_0)\beta(s_0)$, $\tilde{\beta}$, the chord $\beta(s)\gamma(t)$ and $\tilde{\gamma}$. Then $f(s,t)$ is the area of $S$. Observe that the Lagrangian condition for $\Lambda_1$ and $\Lambda_2$ implies that this area does not depend on the choice of $\tilde{\beta}$ and $\tilde{\gamma}$.
Under the transversality hypothesis of Theorem \[ccias\], the projection $\pi: T\mathbb R^{2n}\to\mathbb R^{2n}$ restricted to $\{(x(s,t),y(s,t)):(s,t)\in U\}=Y_F(V)\subset T\mathbb R^{2n}$ is regular and therefore $ f(s,t)=F(x(s,t))$, where the function $F:V\subset\mathbb R^{2n}\to \mathbb R$ is the center generating function of $\Lambda_1\times \Lambda_2$ and satisfies the classical Monge-Ampère equation, $\det[\partial^2F]=c$, for some constant $c$.
These improper affine spheres that are naturally related to the center-chord transform of a pair of real Lagrangian submanifolds, and its center generating function, shall be called [*center-chord improper affine spheres*]{}. Singularities of these improper affine spheres (or equivalently of this class of solutions to the classical Monge-Ampère equation) occur when the transversality hypothesis fail, and these shall be studied in section \[singularsection\].
Special improper affine spheres
-------------------------------
Let $U$ be open subset of ${\mathbb{C}}^n$. Given a holomorphic map $H:U \to{\mathbb{C}}$, we write $$\label{holom-H}
H(z)=\tilde P(z,\bar z)+i\tilde Q(z, \bar z),$$ with $\tilde P,\tilde Q:U\to{\mathbb{R}}$. Then, for $z=s+it$, $z=(z_1,..,z_n)$, $s=(s_1,..,s_n)$, $t=(t_1,..,t_n)$, we define $$\label{hol2}
P(s,t)=\tilde P(s+it,s-it), \ Q(s,t)=\tilde Q(s+it,s-it).$$ Hence $\frac{\partial Q}{\partial s}=(\frac{\partial Q}{\partial s_1},...,\frac{\partial Q}{\partial s_n} )$ and $\frac{\partial Q}{\partial t}=(\frac{\partial Q}{\partial t_1},...,\frac{\partial Q}{\partial t_n} )$. In this setting, we define $$x(s,t)=(x^{(1)}(s,t),x^{(2)}(s,t))=\left(s, \frac{\partial Q}{\partial t} \right)$$ and $$y(s,t)=(y^{(1)}(s,t),y^{(2)}(s,t))=\left(t, \frac{\partial Q}{\partial s}\right).$$ Let $$f(s,t)= Q(s,t)-\sum_{k=1}^n t_k\cdot \frac{\partial Q}{\partial t_k}(s,t).$$ A straightforward calculation shows that equation holds.
At points $(s,t)$ such that $$\label{transvspecial} \det(\frac{\partial^2 Q}{\partial t^2})\neq 0 \ ,$$ the map $\phi(s,t)=(x(s,t),f(s,t))$ is an immersion with $A(r)=A(s,t)=J_{2n}$. As a consequence, $\Sigma^{2n}= \text{Image}(\phi)\subset\mathbb R^{2n+1}$ is an improper affine sphere with Blaschke normal $\xi=(0_{2n},1)$ and Blaschke metric given by $$\label{eq:specialmetric}
h=\left[
\begin{array}{cc}
\frac{\partial^2 Q}{\partial t^2} & 0 \\
0 & \frac{\partial^2 Q}{\partial t^2}
\end{array}
\right].$$
The first statement follows from $$y_{s_i}=\left( 0, \frac{\partial^2 Q}{\partial s_i\partial s_j}\right)=-\left( 0, \frac{\partial^2 Q}{\partial t_i\partial t_j}\right)=-x_{t_i},$$ where the center equality comes from Cauchy-Riemann equations. Similarly $$y_{t_i}=\left( e_i, \frac{\partial^2 Q}{\partial t_i\partial s_j} \right)=
\left( e_i, \frac{\partial^2 Q}{\partial s_i\partial t_j} \right)=x_{s_i}.$$ Thus by Proposition \[prop:Deta\], $\phi$ is an improper affine sphere with Blaschke metric given by equation .
It is worthwhile to describe this construction in terms of the complex variables $(z,{\overline{z}})$, $z=s+it$, ${\overline{z}}=s-it$. Let $$\label{spcc}
\eta(z,{\overline{z}})=x(z,{\overline{z}})+iy(z,{\overline{z}});\ \ \zeta(z,{\overline{z}})=x(z,{\overline{z}})-iy(z,{\overline{z}}).$$ Then $\eta_{{\overline{z}}}=\zeta_{z}=0$ and so $\eta=\eta(z)$, $\zeta=\zeta({\overline{z}})=\overline{\eta(z)}$. Moreover, the submanifolds defined by $\eta$ and $\zeta$ are Lagrangian. Finally $$\label{specialcc}
x(z,{\overline{z}})=\frac{1}{2}\left( \eta(z)+\zeta({\overline{z}}) \right);\ \ y(z,{\overline{z}})=\frac{i}{2}\left( \zeta({\overline{z}})-\eta(z) \right).$$
We conclude that this immersion has the same algebraic structure as the one in section \[sec:Center-chord\], substituting real variables $(s,t)$ by complex variables $(z,{\overline{z}})$, real immersions $(\beta,\gamma)$ by holomorphic and anti-holomorphic immersions $(\eta,\zeta)$, with the condition $\zeta=\bar\eta$ (and with the $i$ in (\[spcc\])-(\[specialcc\])). In other words, according to (\[complexinvpsi\])-(\[complexpsi\]) and (\[spcc\])-(\[specialcc\]), we have:
Special IAS are naturally related to the rotated center-chord transform of a pair of complex conjugate Lagrangian submanifolds of the complexified (real-$2n$-dimensional) symplectic space.
In spite of this similarity between these two types of IAS, we will reserve the name [*center-chord IAS*]{} for the ones related to the center-chord transform of a pair of real Lagrangian submanifolds, keeping the name [*special IAS*]{} for the complex case. But we stress the fact that, in both cases, center-chord *and* special, each of these IAS is a *real* hypersurface of $\mathbb R^{2n+1}$.
Thus, when condition (\[transvspecial\]) is satisfied, the projection $\pi: T\mathbb R^{2n}\to\mathbb R^{2n}$ restricted to $\{(x(s,t),y(s,t)):(s,t)\in U\}=Y_F(V)\subset T\mathbb R^{2n}$ is regular and therefore $ f(s,t)=F(x(s,t))$, where the function $F:V\subset\mathbb R^{2n}\to \mathbb R$ satisfies the classical Monge-Ampère equation, $\det[\partial^2F]=c$. Singularities of these special IAS occur when condition (\[transvspecial\]) fails. These singularities will be studied in section \[singularsection\].
One parameter families
----------------------
Given a center-chord IAS $\phi$, there exists a one parameter family of center-chord IAS $\phi_{\lambda}$, $\lambda\in{\mathbb{R}},\lambda\neq 0$ with $\phi_{1}=\phi$ such that $\phi_{\lambda}$ has the same Blaschke metric as $\phi_{1}$, for any $\lambda$, and $\phi_{\lambda_1}$ is not affinely equivalent to $\phi_{\lambda_2}$, if $\lambda_1\neq \lambda_2$. In fact, take $\beta_{\lambda}(s)=\lambda\beta(s)$ and $\gamma_{\lambda}(s)=|\lambda|^{-1}\gamma(s)$. For $\lambda=-1$, we get the conjugate IAS. It is not difficult to verify that any center-chord IAS with the same Blaschke metric is affinely equivalent to some IAS of this family.
Similarly, given a special IAS $\phi$, there exists a one parameter family of center-chord IAS $\phi_{\tau}$, $\tau\in[0,2\pi]$, with $\phi_{0}=\phi$ such that $\phi_{\tau}$ has the same Blaschke metric as $\phi_{0}$, for any $\tau$, and $\phi_{\tau_1}$ is not affinely equivalent to $\phi_{\tau_2}$, if $\tau_1\neq \tau_2$. In fact, take $H_{\tau}(z)=e^{2i\tau}H(e^{-i\tau}z)$. For $\tau=\frac{\pi}{2}$, we get the conjugate IAS. It is not difficult to verify that any special IAS with the same Blaschke metric is affinely equivalent to some IAS of this family.
In case $n=1$ these results were proved in [@Simon93A] for any affine sphere, not necessarily improper.
Other examples of even dimensional IAS {#eg}
--------------------------------------
For $n=1$, any IAS is center-chord or special. Next examples show that this is not true for $n>1$.
Consider $Dx=I_{2n}$ and $a\in sp(2n)$. In this case $f$ is quadratic map. For example, take $n=2$ and $$A=\left[
\begin{array}{cccc}
1 & 1 & 0 & 0\\
0 & 1 & 0 & 0\\
0 & 0 & -1 & 0\\
0 & 0 & -1 & -1
\end{array}
\right].$$ Then $f=x_1x_3+x_2x_4+x_2x_3$ and since $A$ is not similar to $K_{4}$ or $J_{4}$, $(x,f)$ is neither center-chord nor special. Observe that, in this example, the Blaschke connection $\nabla$ and its dual $\overline{\nabla}$ are flat and thus $A$ is parallel with respect to $\hat{\nabla}$.
If one considers the product of a center-chord IAS with a special IAS, one obtains a new IAS.
Consider $n=2$ and $f(x_1,x_2,x_3,x_4)=x_1x_3+x_2x_4+h(x_2,x_3)$. Then $$D^2f=\left[
\begin{array}{cccc}
0 & 0 & 1 & 0 \\
0 & h_{x_2x_2} & h_{x_2x_3} & 1 \\
1 & h_{x_2x_3} & h_{x_3x_3} &0 \\
0 & 1 & 0 & 0
\end{array}
\right]$$ and so $\det(D^2f)=1$. The corresponding matrix $A$ is given by $$A=\left[
\begin{array}{cccc}
-1 & -h_{x_2x_3} & -h_{x_3x_3} & 0 \\
0 & -1 & 0 & 0 \\
0 & 0 & 1 &0 \\
0 & h_{x_2x_2} & h_{x_2x_3} & 1
\end{array}
\right]$$ and so this IAS is neither center-chord nor special. If $h(x_2,x_3)$ is not quadratic, the dual connection $\overline{\nabla}$ is not flat and it is not difficult to verify that the matrix $A$ is not parallel with respect to the metric connection $\hat{\nabla}$.
Center-chord and Special IAS as solutions of Exterior Differential Systems
===========================================================================
In this section we shall characterize the center-chord and the special IAS as solutions of certain Exterior Differential Systems (EDS).
Center-chord IAS as solutions of an EDS
---------------------------------------
Define the involution ${\mathcal K}_{4n}:{\mathbb{R}}^{4n}\to{\mathbb{R}}^{4n}$ by $${\mathcal K}_{4n}(v_1,v_2)=\left( v_2, v_1 \right).$$ The symplectic form $\Omega_1$ given by is equivalently defined by $$\Omega_1 \left( v, w \right)=\Omega\left( v, {\mathcal K}_{4n} w \right).$$ Consider the Exterior Differential System ${\mathcal E}_1=\{\Omega,\Omega_1\}$. The main result of this section is the following:
\[thm:EDS1\] The solutions of the EDS ${\mathcal E}_1$ are exactly the center-chord IAS.
We begin with the following lemma:
\[lemma:Center-chordEq\] Consider a $\Omega$-Lagrangian immersion $L$ and denote by $\mathcal{L}$ the image of $U$ by $L$. The following statements are equivalent:
1. $\mathcal{L}$ is $\Omega_1$-Lagrangian, for any $r\in U$.
2. $\mathcal{L}$ is ${\mathcal K}_{4n}$-invariant, for any $r\in U$.
3. $A(r)^2=I_{2n}$, for any $r\in U$.
4. $A(r)$ is equivalent to $K_{2n}$, for any $r\in U$.
[**(1)$\Longleftrightarrow $(2)**]{}. We start with (1)$\Longrightarrow$(2). Fix $w_0\in T_{L(r)}\mathcal{L}$ and take $w_1,w_2\in T_{L(r)}\mathcal{L}$. Then $$\Omega(w_1+\lambda {\mathcal K}_{4n}w_0,w_2+\mu {\mathcal K}_{4n}w_0)=\lambda \Omega_1(w_1,w_0)-\mu \Omega_1(w_2,w_0)=0.$$ Thus $span\{T_{L(r)}\mathcal{L},{\mathcal K}_{4n}w_0\}$ is $\Omega$-Lagrangian and thus ${\mathcal K}_{4n}w_0\in T_{\phi(r)}\mathcal{L}$, which implies (2). The implication (2)$\Longrightarrow$(1) is trivial.
[**(2)$\Longleftrightarrow $(3)**]{}. Any vector $(w,w')\in T_{L(r)}\mathcal{L}$ can be written as $w=Dx(r)\cdot u$, $w'=Dx(r)\cdot A(r)\cdot u$. We have to check when $(w',w)\in T_{L(r)}\mathcal{L}$.
This last condition occurs if and only if $w'=Dx(r)\cdot u_1$, $w'=Dx(r)\cdot A(r)\cdot u_1$, for some $u_1\in T_rU$. But this is equivalent to $u_1=A(r)\cdot u$, $A(r)u_1=u$. We conclude that $(w',w)$ belongs to $T_{L(r)}\mathcal{L}$ if and only if $A^{-1}(r)=A(r)$, which is equivalent to $A(r)^2=I_{2n}$.
[**(3) $\Longleftrightarrow $ (4)**]{}. It is clear that $A(r)$ similar to $K_{2n}$ implies $A(r)^2=I_{2n}$. Assume now that $A(r)^2=I_{2n}$. Then the eigenvalues of $A(r)$ are $\pm 1$. Since $(A(r)-I_{2n})\cdot (A(r)+I_{2n})=0$, the minimal polynomial of $A(r)$ contains only linear factors. Hence $A(r)$, and from equation also $B(r)$, are diagonalizable.
Since $B(r)\in sp(2n)$, $J_{2n}\cdot B(r)=-B^t(r)\cdot J_{2n}$. Take $u$ an eigenvector associated with the eigenvalue $-1$. Then $B(r)^t\cdot J_{2n}u=J_{2n}u$ and so $J_{2n}u$ is an eigenvector associated with the eigenvalue $+1$. We conclude that the dimensions of the $-1$ and $1$ eigenspaces are equal. Hence $B(r)$, and thus also $A(r)$, are equivalent to $K_{2n}$.
The main step in the proof of theorem \[thm:EDS1\] is the following:
\[prop:Center-chordInv\] Consider an immersion $\phi:U\subset{\mathbb{R}}^{2n}\to{\mathbb{R}}^{2n+1}$ transversal to $(0,1)$ such that the matrix $A(r)$ is equivalent to $K_{2n}$, for any $r\in U$. Then we can realize $\phi$ as a center-chord IAS.
The matrix $B(r)$ is similar to $K_{2n}$. Denote by $E_1$ the $-1$-eingenspace and by $E_2$ the $1$-eingenspace. Let $p_1:x(U)\to{\mathbb{R}}^{2n}$ be defined as $p_1(x)=x+y(x)$. Then, for any $v_1\in E_1$, $v_2\in E_2$, $$Dp_1(x)v_1=v_1+Dy(x)v_1=0;\ \, Dp_1(x)v=v_2+Dy(x)v_2=2v_2.$$ Thus $p_1$ has rank $n$ at all points. Denoting $\beta=p_1(U)$, observe that the tangent space to $\beta$ at $p_1(x)$ is exactly $E_1$. For $v_1,w_1\in E_1$, $$\omega(v_1,w_1)=-\omega(v_1,K_{2n}w_1)=-\omega(w_1,K_{2n}v_1)=\omega(w_1,v_1).$$ Thus $\omega(v_1,w_1)=0$ and we conclude that $\beta$ is Lagrangian. Now consider $p_2:x(U)\to{\mathbb{R}}^{2n}$ be defined as $p_2(x)=x-y(x)$. As above, $p_2$ has rank $n$ and the tangent space to $\gamma=p_2(U)$ at $p_2(x)$ equals $E_2$. Moreover, $\gamma$ is Lagrangian.
Now we can prove theorem \[thm:EDS1\]. If $\phi$ is an immersion such that $\Omega^*L=\Omega_1^*L=0$, then lemma \[lemma:Center-chordEq\] implies that $A(r)$ is equivalent to $K_{2n}$, for any $r\in U$. Now proposition \[prop:Center-chordInv\] implies that $\phi$ can be realized an a center-chord IAS.
In case $n=1$, any improper affine sphere $\phi:U\subset{\mathbb{R}}^2\to{\mathbb{R}}^3$ with indefinite metric necessarily satisfies $A^2(r)=I_{2n}$, for any $r\in U$. Thus, by proposition \[prop:Center-chordInv\], $\phi$ can be realized as a center-chord IAS. In this case, the coordinates $(s,t)$ are called asymptotic ([@Craizer11],[@Milan13]).
Special IAS as solutions of an EDS
-----------------------------------
Consider the complex structure $\mathcal{J}_{4n}:{\mathbb{R}}^{4n}\to{\mathbb{R}}^{4n}$ defined by $$\mathcal{J}_{4n}(v_1,v_2)=\left( v_2, -v_1 \right).$$ The symplectic form $\Omega_2$ defined by \[eq:Omega2\] is also given by $$\Omega_2 \left( v, w \right)=\Omega\left( v, \mathcal{J}_{4n} w \right).$$ Consider the Exterior Differential System ${\mathcal E}_2=\{\Omega,\Omega_2\}$. The main result of this section is the following:
\[thm:EDS2\] The solutions of the EDS ${\mathcal E}_2$ are exactly the special IAS.
\[lemma:SpecialEq\] Consider a $\Omega$-Lagrangian immersion $L$. The following statements are equivalent:
1. $\mathcal{L}$ is $\Omega_2$-Lagrangian, for any $r\in U$.
2. $\mathcal{L}$ is $\mathcal{J}_{4n}$-invariant, for any $r\in U$.
3. $A(r)^2=-I_{2n}$, for any $r\in U$.
4. $A(r)$ is equivalent to $J_{2n}$, for any $r\in U$.
Similar to lemma \[lemma:Center-chordEq\].
\[prop:SpecialCS\] Consider an immersion $\phi:U\subset{\mathbb{R}}^{2n}\to{\mathbb{R}}^{2n+1}$ transversal to $(0,1)$ such that $A(r)$ is equivalent to $J_{2n}$, for any $r\in U$. Then $\phi$ can be realized as a special IAS.
Analogous to proposition \[prop:Center-chordInv\] using the complex variables $(z,{\overline{z}})$.
Now we can prove theorem \[thm:EDS2\]. If $\phi$ is an immersion such that $\Omega^*L=\Omega_2^*L=0$, then proposition \[lemma:SpecialEq\] implies that $A(r)$ is equivalent to $K_{2n}$, for any $r\in U$. Now proposition \[prop:SpecialCS\] implies that $\phi$ can be realized an a special IAS.
In case $n=1$, any improper affine sphere $\phi:U\subset{\mathbb{R}}^2\to{\mathbb{R}}^3$ with definite metric necessarily satisfies $A^2(r)=-I_{2n}$, for any $r\in U$. Thus, by proposition \[prop:SpecialCS\], $\phi$ can be realized as a special IAS. In this case, the coordinates $(s,t)$ are called isothermal ([@Galvez07],[@Craizer12],[@Martinez05]).
Lagrangian and Legendrian stable singularities of IAS {#singularsection}
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Consider a Lagrangian immersion $L:\mathbb R^{2n}\rightarrow (T\mathbb R^{2n},\Omega)$ and a Legendrian immersion $\tilde L:\mathbb R^{2n}\rightarrow (T\mathbb R^{2n}\times \mathbb R,\{\theta=0\})$. Denote by $\pi:T\mathbb R^{2n}={\mathbb{R}}^{2n}\times{\mathbb{R}}^{2n}\to{\mathbb{R}}^{2n}$ the projection $\pi(x,y)=x$ and by $\tilde \pi:T{\mathbb{R}}^{2n}\times{\mathbb{R}}={\mathbb{R}}^{2n}\times{\mathbb{R}}^{2n}\times{\mathbb{R}}\to{\mathbb{R}}^{2n}\times{\mathbb{R}}$, the projection $\tilde \pi(x,y,z)=(x,z)$.
In this section we shall consider the singularities of the Lagrangian map $\pi \circ L$ and the Legendrian map $\tilde \pi\circ \tilde L$.
We use the following notation: $x=(x^{(1)},x^{(2)})=({x^{(1)}}, \hat{x}, \check{x})$, $x^{(1)}=(x_1,...,x_n)$, $\hat{x}=(x_{n+1},...,x_{n+m})$, $\check{x}=(x_{n+m+1},...,x_{2n})$ and $y=(y^{(1)},y^{(2)})=(\hat{y},\check{y},{y^{(2)}})$, $\hat{y}=(y_{1},...,y_{m})$, $\check{y}=(y_{m+1},...,y_{n})$, $y^{(2)}=(y_{n+1},...,y_{2n})$ for $0\leq m\leq n-1$.
Let us recall that in this notation $$\Omega=dx^{(1)}\wedge dy^{(2)} +d\hat{y}\wedge d\hat{x}+d\check{y}\wedge d\check{x},$$ $$\theta=dz-y^{(2)}dx^{(1)} +\hat{y}d\hat{x}+\check{y}d\check{x}.$$
Generating functions and generating families
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The main tool used for classifying these singularities are the generating functions and generating families.
Denote by ${\mathcal L}$ the image of the Lagrangian immersion $L$ and by ${\tilde {\mathcal L}}$ the image of the Legendrian immersion $\tilde L$.
A generating function of the Lagrangian submanifold $\mathcal L$ and the Legendrian submanifold $\tilde {\mathcal L}$ is a function $$S:{\mathbb{R}}^{n+m}\times{\mathbb{R}}^{n-m}\ni ({x^{(1)}},\hat{x},\check{y})\mapsto S({x^{(1)}},\hat{x},\check{y})\in{\mathbb{R}},$$ satisfying $$\label{eq:defineGF1}
\mathcal L=\{ (x,y): \frac{\partial S}{\partial {x^{(1)}}}={y^{(2)}}, \frac{\partial S}{\partial \check{y}}=\check{x}, -\frac{\partial S}{\partial \hat{x}}=\hat{y} \}.$$ and $$\label{eq:defineGF2}
\tilde {\mathcal L}=\{ (x,y,z): \frac{\partial S}{\partial {x^{(1)}}}={y^{(2)}}, \frac{\partial S}{\partial \check{y}}=\check{x}, -\frac{\partial S}{\partial \hat{x}}=\hat{y} , z=S({x^{(1)}},\hat{x},\check{y})-\check{y}\cdot\check{x} \}.$$
A generating family of the Lagrangian map $\pi \circ L$ and the Legendrian map $\tilde \pi \circ \tilde L$ is a function $$G:{\mathbb{R}}^{2n}\times{\mathbb{R}}^{n-m}\ni ({x^{(1)}},\hat{x},\check{x},\kappa) \mapsto G({x^{(1)}},\hat{x},\check{x},\kappa)\in {\mathbb{R}},$$ satisfying $$\mathcal L=\{ (x,y): \exists \kappa : \frac{\partial G}{\partial \kappa}=0, \frac{\partial G}{\partial {x^{(1)}}}={y^{(2)}}, -\frac{\partial G}{\partial \check{x}}=\check{y} \}.$$ and $$\tilde{\mathcal L}=\{ (x,y,z): \exists \kappa : \frac{\partial G}{\partial \kappa}=0, \frac{\partial G}{\partial {x^{(1)}}}={y^{(2)}}, -\frac{\partial G}{\partial \check{x}}=\check{y}, z=G({x^{(1)}},\hat{x},\check{x},\kappa) \}.$$
A generating family can be obtained from a generating function by the formula $$\label{gen-fam-gen-fun}
G({x^{(1)}},\hat{x},\check{x},\kappa)=S({x^{(1)}},\hat{x},\kappa)- \check{x}\cdot\kappa.$$
We shall use the following well-known theorem ([@Arnold], Chapter 21):
\[thm-Arnold\] The Lagrangian map-germ $\pi \circ L$ at $0$ generating by $G$ is Lagrangian stable if and only if the classes of function-germs $$1,\frac{\partial G}{\partial x^{(1)}}(0,0,0,\kappa), \frac{\partial G}{\partial \hat{x}}(0,0,0,\kappa),
\frac{\partial G}{\partial \check{x}}(0,0,0,\kappa)$$ generate the linear space $\mathbb R[[\kappa]]/<\frac{\partial G}{\partial \kappa}(0,0,0,\kappa)>$.
The Legendrian map-germ $\tilde \pi \circ \tilde L$ at $0$ generating by $G$ is Legendrian stable if and only if the classes of function-germs $$1,\frac{\partial G}{\partial x^{(1)}}(0,0,0,\kappa), \frac{\partial G}{\partial \hat{x}}(0,0,0,\kappa),
\frac{\partial G}{\partial \check{x}}(0,0,0,\kappa)$$ generate the linear space $\mathbb R[[\kappa]] / <\frac{\partial G}{\partial \kappa}(0,0,0,\kappa), G(0,0,0,\kappa)>$.
Generating functions of center-chord and special IAS
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Consider a center-chord IAS and assume that the Lagrangian submanifolds are given by $(u, dS^{-}(u))$ and $(v,dS^{+}(v))$. Then $$\begin{aligned}
(x^{(1)},x^{(2)})&=&\frac{1}{2}\left( u+v, dS^{+}(v)+dS^{-}(u)\right)\\ (y^{(1)},y^{(2)})&=&\frac{1}{2}\left(v-u,dS^{+}(v)-dS^{-}(u)\right).\end{aligned}$$ straightforward calculations show that $$\label{ccS_0}
S_0({x^{(1)}},{y^{(1)}})=\frac{1}{2} \left( S^{+}({x^{(1)}}+{y^{(1)}})-S^{-}({x^{(1)}}-{y^{(1)}}) \right)$$ satisfies equations - and thus is a generating function for $L$ and ${\tilde L}$. A special IAS is defined by $$(x^{(1)},x^{(2)})=(s, \frac{\partial Q}{\partial t}) ;\ \ (y^{(1)},y^{(2)})=(t, \frac{\partial Q}{\partial s}),$$ where $ Q$ is the imaginary part of a holomorphic function $H$ (see (\[holom-H\])-(\[hol2\])). Thus $$\label{eq:S_0}
S_0(x^{(1)}, y^{(1)})= Q(x^{(1)}, y^{(1)})$$ satisfies equations and and thus is a generating function.
For any $0\leq m\leq n-1$, it follows from equation that the Lagrangian submanifold of the IAS is defined by the equations $$\label{eq:defineSpecialIAS1}
{y^{(2)}}=\frac{\partial S_0}{\partial {x^{(1)}}}, \ \ \hat{x}=\frac{\partial S_0}{\partial \hat{y}}, \ \ \check{x}=\frac{\partial S_0}{\partial \check{y}}.$$ For the Legendrian submanifold, we must consider also $$\label{eq:defineSpecialIAS2}
z=S_0({x^{(1)}}, \hat{y},\check{y})-\hat{y}\cdot \hat{x}- \check{y}\cdot \check{x}.$$ So the generating family has the following form $$\label{G_0}
G_0(x^{(1)},\hat{x},\check{x},\kappa)=S_0({x^{(1)}}, \hat{\kappa},\check{\kappa})-\hat{\kappa}\cdot \hat{x}- \check{\kappa}\cdot \check{x},$$ where $\kappa=(\hat{\kappa},\check{\kappa})$, $\hat{\kappa}=(\kappa_{1},...,\kappa_{m})$, $\check{\kappa}=(\kappa_{m+1},...,\kappa_{n})$.
We can obtain other generating functions when $S_0$ (given by or by ) is quadratic in $\hat{y}$. With this assumption we can write $$\label{eq:defineVquadratic}
S_0({x^{(1)}},\hat{y},\check{y})=\sum_{k=1}^m\left(\frac{1}{2}y_k^2+y_kg_k({x^{(1)}},\check{y})\right) +h({x^{(1)}},\check{y}).$$ Then from $\hat{x}=\frac{\partial S_0}{\partial \hat{y}}$ we obtain $$\label{y_k}
y_k=x_{k+n}-g_k({x^{(1)}},\check{y})$$ for $k=1,\cdots,m$. Let $\hat{g}({x^{(1)}},\check{y})=(g_1({x^{(1)}},\check{y}),\cdots, g_m({x^{(1)}},\check{y}))$. We define a new generating function $$\label{S_m}
S_m({x^{(1)}},\hat{x},\check{y})=S_0({x^{(1)}},\hat{x}-\hat{g}({x^{(1)}},\check{y}),\check{y}) - \sum_{k=1}^m (x_{k+n}-g_k({x^{(1)}},\check{y}))x_{k+n},$$ Thus $$\label{eq:Gfm}
S_m({x^{(1)}},\hat{x},\check{y})=-\frac{1}{2}\sum_{k=1}^m\left(x_{k+n}- g_k({x^{(1)}},\check{y}) \right)^2+h({x^{(1)}},\check{y}).$$ Using equations and , it is straightforward to verify that $S_m$ satisfies and and thus is a generating function for the Lagrangian and Legendrian submanifolds. From we obtain a new generating family $$\label{G_m}
G({x^{(1)}},\hat{x},\check{x},\check{\kappa})=S({x^{(1)}},\hat{x},\kappa)- \check{x}\cdot\check{\kappa}.$$
Realization of simple stable Legendrian singularities of center-chord IAS
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The singular set of a center-chord IAS has a simple geometrical meaning: The tangent planes of the Lagrangian submanifolds $(s, dS^{-}(s))$ and $(t, dS^{+}(t))$ at a singular pair $(s,t)$ must intersect in a $(n-m)$-dimensional vector space, with $n>m$. Thus the image of the singular set by the map $x$ is exactly their Wigner caustic. Moreover, the image of the singular set by the one parameter family $x_{\lambda}$ are the equidistants of the pair of Lagrangian submanifolds.
The singularities of the equidistants and the Wigner caustic of a pair of Lagrangian submanifolds were studied in [@Domitrz13], so the problem of realization of simple singularities for center-chord IAS was solved there, where it is proved that any simple stable Lagrangian singularity can be realized by a center-chord transform and, as it is straightforward to adapt section $4$ of this paper to the Legendrian setting, Theorem $4.1$ in [@Domitrz13] can be restated as follows:
Any germ of a simple stable Legendrian singularity is realizable as a center-chord IAS.
We give below another proof of this theorem, by presenting new generating families that closely resemble the generating families that appear in the proof of Theorem \[specialsing\] for the special IAS, in the next section, thus re-emphasizing the similarities between these two kinds of IAS.
We explicitly describe pairs of functions $(S^{+},S^{-})$ such that the corresponding generating function $S_0$ (see ) or $S_m$ (see ) generates the simple stable Legendrian singularity of the type A-D-E. Using formulas and and Theorem \[thm-Arnold\] one can easily check that for the following pair of functions $(S^{+},S^{-})$ we obtain the following Legendrian singularities:
Denote by $\lfloor a \rfloor$ the greatest integer smaller than or equal to $a$.
For $A_k$ singularity with $k\le n+2$ we take $$S^{+}(v)=\pm (-1)^{ \lfloor \frac{k+1}{2} \rfloor } v_1^{k+1} + \sum_{j=2}^n v_j^2+\sum_{j=1}^{k-2} (-1)^{ \lfloor \frac{k-j+1}{2} \rfloor} v_jv_1^{k-j}$$ $$S^{-}(u)=\pm (-1)^{ \lfloor \frac{k}{2} \rfloor } u_1^{k+1} - \sum_{j=2}^n u_j^2+\sum_{j=1}^{k-2} (-1)^{ \lfloor \frac{k-j+2}{2} \rfloor} u_ju_1^{k-j}$$ Then a generating function is $S_0$ (see ).
For $A_k$ singularity with $n+2<k< 2n+2$ we take $$S^{+}=(-1)^{\lfloor \frac{k+1}{2} \rfloor} v_1^{k+1}+\sum_{j=2}^{k-n-1} (-1)^{\lfloor \frac{k-j}{2} \rfloor} v_jv_1^{k-j+1}+\sum_{j=1}^{n} (-1)^{\lfloor \frac{n-j+2}{2} \rfloor} v_{j}v_1^{n-j+2}$$ $$+\frac{1}{2}\sum_{j=2}^{k-n-1}v_1^{2k-2j+2} +\frac{1}{2}\sum_{j=2}^{k-n-1}v_j^2 +\sum_{j=k-n}^{n}v_j^2.$$ $$S^{-}=(-1)^{\lfloor \frac{k}{2}\rfloor} u_1^{k+1}+\sum_{j=2}^{k-n-1} (-1)^{\lfloor \frac{k-j-1}{2} \rfloor} u_ju_1^{k-j+1}+\sum_{j=1}^{n} (-1)^{\lfloor \frac{n-j+1}{2} \rfloor} u_{j}u_1^{n-j+2}$$ $$-\frac{1}{2}\sum_{j=2}^{k-n-1}u_1^{2k-2j+2} -\frac{1}{2}\sum_{j=2}^{k-n-1}u_j^2 -\sum_{j=k-n}^{n}u_j^2.$$ Then $S_0$ (see ) is quadratic in $y_j=v_j-u_j$ for $j=2,\cdots,k-n-1$. So we can use - to obtain a generating function $S_m$.
For $D_k^{\pm}$ with $4\le k\le n+3$ take $$S^{+}(v)=-v_1v_2^2-v_1^3-v_2v_1^3\pm (-1)^{ \lfloor \frac{k+3}{2} \rfloor}v_1^{k-1}+\sum_{j=3}^{k-3}(-1)^{ \lfloor \frac{j+2}{2} \rfloor}v_jv_1^{j+1}+\sum_{j=3}^{n} v_j^2.$$ $$S^{-}(u)=-u_1u_2^2+u_1^3-u_2u_1^3\pm (-1)^{ \lfloor \frac{k+2}{2} \rfloor}u_1^{k-1}+\sum_{j=3}^{k-3}(-1)^{ \lfloor \frac{j+3}{2} \rfloor}u_ju_1^{j+1}-\sum_{j=3}^{n} u_j^2.$$ Then a generating function is $S_0$ (see ).
For $D_k$ with $n+3<k<2n+2$ take $$S^{+}=(-1)^{\lfloor \frac{k-1}{2} \rfloor}v_1^{k-1}-v_1v_2^2-v_1^3-v_2v_1^3+\sum_{j=3}^{n}(-1)^{\lfloor \frac{j+1}{2} \rfloor}v_jv_1^{j+1}$$ $$+\sum_{j=3}^{k-n-1}(-1)^{\lfloor \frac{j+n}{2} \rfloor}v_{j}v_1^{j+n-1}+\frac{1}{2}\sum_{j=3}^{k-n-1}(-1)^{j+n+1}v_1^{2j+2n-2} -\frac{1}{2}\sum_{j=3}^{k-n-1}v_j^2 -\sum_{j=k-n}^{n}v_j^2.$$ $$S^{-}=(-1)^{\lfloor \frac{k-2}{2}\rfloor}u_1^{k-1}-u_1u_2^2+u_1^3-u_2u_1^3+\sum_{j=3}^{n}(-1)^{\lfloor \frac{j}{2} \rfloor}u_ju_1^{j+1}$$ $$+\sum_{j=3}^{k-n-1}(-1)^{\lfloor \frac{j+n-1}{2}\rfloor}u_{j}u_1^{j+n-1}+\frac{1}{2}\sum_{j=3}^{k-n-1}(-1)^{j+n}u_1^{2j+2n-2} +\frac{1}{2}\sum_{j=3}^{k-n-1}u_j^2 +\sum_{j=k-n}^{n}u_j^2.$$ Then $S_0$ (see ) is quadratic in $y_j=v_j-u_j$ for $j=3,\cdots,k-n-1$. So we can use - to obtain a generating function $S_m$.
For $E_6^{\pm}$ and $n\ge 3$, take $$S^{+}=-v_1^3 \pm v_2^4+v_3^2+v_1^2v_2^2+v_2^2v_1+v_3v_2^2+\sum_{j=4}^n v_j^2$$ $$S^{-}=-u_1^3 \mp u_2^4-u_3^2+u_1^2u_2^2-u_2^2u_1-u_3u_2^2-\sum_{j=4}^n u_j^2$$ and take the generating function $S_0$ (see ).
For $E_7$, $n=3$, take $$S^{+}=-v_1^3+v_1v_2^3-\frac{1}{2}v_3^2+v_3v_2^4+v_2^4+v_1v_2^2+v_3v_1v_2+\frac{1}{2}v_2^8.$$ $$S^{-}=-u_1^3-u_1u_2^3+\frac{1}{2}u_3^2+u_3u_2^4+u_2^4-u_1u_2^2-u_3u_1u_2-\frac{1}{2}u_2^8.$$ Then a generating function $S_0$(see ) is quadratic in $y_3=v_3-u_3$ and we can use - to obtain a generating function $S_m$.
For $E_7$, $n\geq 4$, we take $$S^{+}=-v_1^3+v_1v_2^3-v_3^2-v_4^2+v_2^4v_1+v_2^4+v_3v_2^2+v_4v_1v_2+\sum_{j=5}^n v_j^2$$ $$S^{-}=-u_1^3-u_1u_2^3+u_3^2+u_4^2-u_2^4u_1+u_2^4-u_3u_2^2-u_4u_1u_2-\sum_{j=5}^n u_j^2$$ to obtain a generating function $S_0$ (see ).
For $E_8$, $n=4$, we take $$S^{+}=-v_1^3+v_2^5-v_3^2-\frac{1}{2}v_4^2+v_1v_2^3v_4+v_2^4+v_1^2v_2^2+v_1v_2v_3+v_2^2v_4+\frac{1}{2}v_1^2v_2^6.$$ $$S^{-}=-u_1^3+u_2^5+u_3^2+\frac{1}{2}u_4^2+u_1u_2^3u_4+u_2^4+u_1^2u_2^2-u_1u_2u_3-u_2^2u_4-\frac{1}{2}u_1^2u_2^6.$$ Then a generating function $S_0$ (see ) is quadratic in $y_4=v_4-u_4$ and we can use - to obtain a generating function $S_m$.
For $E_8$, $n\geq 5$, we take $$S^{+}=-v_1^3+v_2^5-v_3^2-v_4^2-v_5^2+v_1^2v_2^3+v_2^4+v_3v_1v_2^2+v_4v_1v_2+v_5v_2^2+\sum_{j=6}^n v_j^2$$ $$S^{-}=-u_1^3+u_2^5+u_3^2+u_4^2+u_5^2-u_1^2u_2^3+u_2^4+u_3u_1u_2^2-u_4u_1u_2-u_5u_2^2-\sum_{j=6}^n u_j^2$$ to obtain a generating function $S_0$ (see ).
We remark that if an IAS is modeled by the center-chord transform of the same Lagrangian submanifold $L\times L\subset V\times V$, then another kind of singularity of the Wigner caustic appears in the limit of vanishing chords, the so-called Wigner caustic on shell. These singularities of the Wigner caustic on shell, which are close to and include the shell $L$, differ from the singularities of the Wigner caustic off shell because, in the former case, their generating families are necessarily [*odd*]{} functions of their variables, so they are symmetric singularities, in this sense. These symmetric singularities that are realized for IAS of this kind (center-chord transform of $L\times L$) have been studied in [@DMR].
Realization of simple stable Legendrian singularities as special IAS
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In this section we show that any simple singularity can be realized as a special IAS, which is a main result of the paper.
\[specialsing\] Any germ of a simple stable Legendrian singularity is realizable as a special IAS.
We explicitly describe holomorphic functions $H$ (see (\[holom-H\])) such that the corresponding generating function $S_0=\text{Im} H$ or $S_m$ (see ) generates the simple stable Legendrian singularity of the type A-D-E. Using formulas and and Theorem \[thm-Arnold\] one can easily check that for the following holomorphic functions $H$ we obtain the following Legendrian singularities:
For $A_k$ singularity with $k\le n+2$ we take $$H(z)=\pm i^{3k}z_1^{k+1} - \sum_{j=2}^n iz_j^2+\sum_{j=1}^{k-2} i^{3k+j+1}z_jz_1^{k-j}.$$ Then a generating function is $S_0=\text{Im}H$.
For $A_k$ singularity with $n+2<k<2n+2$ we take $$H(z)=\pm i^{3k}z_1^{k+1} + \sum_{j=2}^{k-n-1}i^{3k+j+1}z_jz_1^{k-j+1}+ \sum_{j=1}^{n}i^{j-n-1}z_{j}z_1^{n-j+2}$$ $$+\frac{1}{2}\sum_{j=2}^{k-n-1}i^{2k+2j+3}z_1^{2k-2j+2}-\frac{1}{2}\sum_{j=2}^{k-n-1}iz_j^2-\sum_{j=k-n}^{n} iz_j^2$$ Then a generating function $S_0=\text{Im}H$ is quadratic in $y_j=\text{Im}z_j$ for $j=2,\cdots,k-n-1$. So we use - to obtain a generating function $S_m$.
For $D_k^{\pm}$ with $4\le k\le n+3$ take $$H(z)=-z_1z_2^2-iz_1^3-z_2z_1^3\pm i^{3k+2}z_1^{k-1}+\sum_{j=3}^{k-3}i^{3j}z_jz_1^{j+1}-\sum_{j=3}^{n} iz_j^2.$$ Then a generating function is $S_0=\text{Im}H$.
For $D_k^{\pm}$ with $n+3<k<2n+2$ we take $$H(z)=-z_1z_2^2-iz_1^3-z_2z_1^3\pm i^{3k+2}z_1^{k-1}+\sum_{j=3}^{n}i^{3j}z_jz_1^{j+1}+\sum_{j=3}^{k-n-1}i^{3j+3n+1}z_jz_1^{j+n-1}$$ $$+\frac{1}{2}\sum_{j=3}^{k-n-1}i^{2j+2n+3}z_1^{2j+2n-2}-\frac{1}{2}\sum_{j=3}^{k-n-1}iz_j^2-\sum_{j=k-n}^{n} iz_j^2.$$ Then a generating function $S_0=\text{Im}H$ is quadratic in $y_j=\text{Im}z_j$ for $j=3,\cdots,k-n-1$. So we use - to obtain a generating function $S_m$.
For $E_6^{\pm}$ and $n\ge 3$, we take $$H(z)=-z_1^3\pm iz_2^4+iz_3^2+z_1^2z_2^2+iz_2^2z_1+iz_3z_2^2-\sum_{j=4}^n i z_j^2$$ to obtain a generating function $S_0=\text{Im}H$.
For $E_7$, $n=3$, we take $$H(z)=-z_1^3+iz_1z_2^3-\frac{1}{2}iz_3^2+z_3z_2^4+z_2^4+iz_1z_2^2+iz_1z_2z_3+\frac{1}{2}iz_2^8$$ Then a generating function $S_0=\text{Im}H$ is quadratic in $y_3=\text{Im}z_3$ and we can use - to obtain a generating function $S_m$.
For $E_7$, $n\geq 4$, we take $$H(z)=-z_1^3+iz_1z_2^3-iz_3^2-iz_4^2+iz_1z_2^4+z_2^4+iz_2^2z_3+iz_1z_2z_4-\sum_{j=5}^n i z_j^2$$ to obtain a generating function $S_0=\text{Im}H$.
For $E_8$, $n=4$, we take $$H(z)=-z_1^3+z_2^5-iz_3^2-\frac{1}{2}iz_4^2+z_1z_2^3z_4+z_2^4+z_1^2z_2^2+iz_1z_2z_3+iz_2^2z_4+\frac{1}{2}iz_1^2z_2^6$$ Then a generating function $S_0=\text{Im}H$ is quadratic in $y_4=\text{Im}z_4$. So we can use - to obtain a generating function $S_m$.
For $E_8$, $n\geq 5$, we take $$H(z)=-z_1^3+z_2^5-iz_3^2-iz_4^2-iz_5^2+iz_1^2z_2^3+z_2^4+z_1z_2^2z_3+iz_1z_2z_4+iz_2^2z_5-\sum_{j=6}^n i z_j^2$$ to obtain a generating function $S_0=\text{Im}H$.
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Craizer, M.: [*Singularities of Convex Improper Affine Maps*]{}, Journal of Geometry [103(2)]{}, 207-217 (2012).
Domitrz, W., Manoel, M., Rios, P. de M.: [*The Wigner caustic on shell and singularities of odd functions*]{}, J. Geometry and Physics 71, 58-72 (2013).
Domitrz, W., Rios, P. de M.: [*Singularities of equidistants and global centre symmetry sets of Lagrangian submanifolds*]{}, to appear in Geometriae Dedicata (online open access, 2013, DOI 10.1007/s10711-013-9861-2).
Ishikawa, G., Machida, Y.:[*Singularities of improper affine spheres and surfaces of constant gaussian curvature*]{}, Int.J.Math., 17, 269-293, 2006.
Ishikawa, G., Machida, Y.:[*Extra singularities of geometric solutions to Monge-Ampère equation of three variables*]{}, Kyoto University Research Information Repository, 1502: 41-53, 2006.
Li,A.M., Jia,F., Simon,U., Xu,R.:[*Affine Bernstein Problems and Monge-Ampère Equations*]{}. World Scientific, 2010.
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Milán, F.:[*Singularities of improper affine maps and their Hessian equation*]{}. J.Math.Analysis Applications, [405]{}, 183-190 (2013).
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[^1]: The first author thanks CNPq, the second author thanks NCN, and the third author thanks Fapesp, for financial support during the preparation of this manuscript.
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24.2cm 17.0cm -1.0in -42pt plus 2mm minus 1mm
**Nearthreshold Large $Q^2$ Electroproduction**
**off Polarized Deuteron**
L. Frankfurt$^{a,d}$, M. Sargsian$^{a,e}$, M. Strikman$^{b,d}$
> $^{(a)}$ Tel Aviv University, Tel Aviv, Israel\
> $^{(b)}$ Pennsylvania State University, University Park, PA, USA\
> $^{(d)}$ S.Petersburg Nuclear Physics Institute, Russia\
> $^{(e)}$ Yerevan Physics Institute, Yerevan, Armenia
> [**Abstract:**]{} The exclusive and inclusive electroproduction off the polarized deuteron is considered at large $Q^2$ and $x \ge 0.5$. It is shown that the use of a polarized target will allow to emphasize smaller than average internucleon distances in the deuteron. As a result, we expect amplification of all the effects (color transparency, relativistic dynamics, etc.) sensitive to small internucleon distances. Numerical estimates are given for the processes $e+ \vec d \rightarrow e +p+n$ and $e+ \vec d \rightarrow e +X$.
[**1 Motivation**]{}
The theoretical analysis [@FGMSS95] of the intermediate energy $Q^2\sim 1$ GeV$^2$ electrodisintegration of the deuteron at $x\sim 1$ indicates that there is a fast convergence of the higher (large $l$) partial waves of the final $pn$ continuum wave function. As a result, we can substitute the (infinite) sum over the partial waves with the phenomenological amplitude for $pn$ scattering. This simplification allows to implement relativistic kinematics of the final state interaction (FSI) amplitude through the analysis of the corresponding (covariant) Feynman diagrams [@FSS96]. The main theoretical conclusion [@FSS96] is that, at $Q^2\ge 1$ GeV$^2$, there exists a unique scheme of legitimate calculations within the extended eikonal approximation which selfconsistently accounts for relativistic dynamics. This enhances considerably the exploration potential of the electroproduction reaction, especially off a deuteron target, whose wave function is well established at Fermi momenta $\le 400$ MeV/c.
Based on this, we discuss two alternative studies:
$\bullet$ Investigation of the QCD prediction that the absorption of a high momentum virtual photon by a nucleon leads to the production of a small size color singlet state, optimistically called a point-like configuration (PLC). Such a study requires selection of kinematics where small enough Fermi momenta dominate and where the transverse momenta of the spectator nucleons are large enough so that the dominant contribution is given by the reinteraction of the PLC with a spectator nucleon (see Sect. 2).
$\bullet$ Probing relativistic effects in deuteron electrodisintegration at moderate $Q^2\le 4$ GeV$^2$ and rather large longitudinal Fermi momenta. Such a study will provide a critical discrimination between the different approaches to high energy scattering off deeply bound nucleons.
Both these studies would greatly benefit from the use of a polarized target. The reason is that the use of a $\vec d$ allows to enhance the contribution of the $D$-state in the deuteron’s ground state wave function. Due to the diminishing probability of the $D$-state at small Fermi momenta, these reactions would be sensitive to smaller internucleon distances in the deuteron as compared to the unpolarized case, leading to an amplification of all the effects sensitive to small internucleon distances.
[**2 Color Transparency Effects and Vanishing FSI**]{}
In QCD, the absorption of a high $Q^2$ photon by a nucleon produces a PLC, which, at very high energies, would not interact with the nucleons, thus eliminating FSI. This vanishing of
(100,200)(0,0) (-10,255)[[**Figure 1:**]{} [*$p_t$ and $Q^2$ dependence of the ratio*]{}]{} (-10,240)[[*$T^{GA}/T^{CT}$ for $\alpha \equiv (E_s-p_s^z)/m = 1$.*]{}]{} (-10,225)[[*a) quantum diffusion, b) three state model.*]{}]{} (240,255)[[**Figure 2:**]{} [*$Q^2$ dependence of $A_d$ for $\alpha =1$.*]{}]{} (240,240)[[*Solid line - elastic eikonal, dashed - QDM,*]{}]{} (240,225)[[*dashed-dotted - three state model, dotted -*]{}]{} (240,210)[[*PWIA.*]{}]{}
the FSI has been termed color transparency (CT). At high but finite energies, a PLC is actually produced, but it expands as it propagates through the nucleus [@ANN]. To suppress the expansion effects, it is necessary to ensure that the expansion length, $l_h\sim 0.4(p/$GeV), is greater than the characteristic longitudinal distance in the reaction. In the considered $d(e,e'pn)$ and $d(e,e'pN^*)$ reactions, where one nucleon carries almost all the momentum of the photon while the second nucleon (or its resonance) is a spectator, the actual expansion distances are the distances between the nucleons in the deuteron [@FGMSS95]. Thus, suppressing large distance effects through the deuteron’s polarization, one effectively will diminish the PLC’s expansion, leading to an earlier onset of CT.
The scattering amplitude ${\cal M}$, including the $np$ final state interaction, can be written as: $${\cal M} = <p_s^z,\vec p_t|d> -{1\over 4i}\int{d^2k_t\over
(2\pi)^2}<\tilde p_s^z,\vec p_t-\vec k_t|d>
{\bf F^{np}(\vec k)}\left[1-i\beta\right],
\label{eq:amp}$$ where $\tilde p_s^z=p_s^z-(E_s-m){M_d+\nu\over |\vec q|}$ and $E_s=\sqrt{p_s^2+m^2}$. Here, $p_s$ is the spectator momentum and $M_d$ the mass of the deuteron. The difference between $\tilde p_s^z$ and $p_z$ accounts for the longitudinal momentum transfer. Spin indices are suppressed to simplify the notations. The function ${\bf F^{np}}$ represents the FSI between the outgoing baryons and its form depends on the model describing the soft rescattering. Within the elastic eikonal (Glauber) approximation (GA), ${\bf F^{np}(\vec k)}\rightarrow f^{np}({\bf \vec k_t})$, where $f^{pn} =\sigma^{pn}_{tot}(i+a_n)e^{-b_n k_t^2/2}$. At Q$^2>3$ (GeV/c)$^2$, the quantities $\sigma^{pn}_{tot}$, $a_n$ and $b_n$ depend only weakly on the momentum of the knocked-out nucleon, with $\sigma^{pn}_{tot}\approx 40$ mb, $a_n\approx -0.2$ and $b_n\approx~6-8$ GeV$^{-2}$ for the kinematics we use.
The reduced interaction between the PLC and the spectator nucleon can be described in terms of its transverse size and the distance $z$ from the photon absorption point, i.e., in Eq.(\[eq:amp\]) we replace ${\bf F^{pn}}\rightarrow f^{PLC,N}(z,k_t,Q^2)$. For numerical estimates of the reduced FSI $f^{PLC,N}(z,k_t,Q^2)$, we use the quantum diffusion model (QDM) [@FLFS] as well as the three state model [@FGMS93]. Latter is based on the assumption that the hard scattering operator acts on a nucleon and produces a PLC, which is represented as a superposition of three baryonic states, $|PLC\rangle = \sum_{m=N,N^*,N^{**}} F_{m,N}(Q^2) |m\rangle$. In Fig.1, we compare the predictions of the elastic eikonal and the two CT models for the transparency, $T=\sigma^{FSI}_{e,e'p}/
\sigma^{PWIA}_{e,e'p}$, for an unpolarized target. We consider so-called perpendicular kinematics, where the light cone momentum $\alpha={E_s-p^s_s\over m} \approx 1$ and $p_t\le
400~MeV/c$. It was demonstrated in Ref.[@FGMSS95] that these kinematics maximize the contribution from the FSI and minimizes various theoretical uncertainties. One can see from Fig.1 that, optimistically, one may expect $30\%$ effects from CT at $Q^2 \ge 4-6$ GeV$^2$.
Using a polarized target emphasizes the role of the deuteron’s $D$-state, allowing to probe the space-time evolution at smaller space-time intervals. For numerical estimates, we consider the asymmetry $A_d$ measurable in electrodisintegration of a polarized deuteron with helcities of $\pm 1$ and 0: $A_d(Q^2,\vec p_s) = {\sigma(1)+\sigma(-1)-2\cdot\sigma(0)
\over \sigma(1)+\sigma(0)+\sigma(-1)}$, where $\sigma(s_z)\equiv{d\sigma^{\vec s,s_z}\over dE_{e'} d\Omega_{e'}
d^3p}$ and $s_z$ is the deuteron’s helicity. The $Q^2$ dependence of the asymmetry $A_d$ for “perpendicular” kinematics, at $p_t=300$ MeV/c, is presented in Fig.2. One can see from this figure that CT effects can change $A_d$ by as much as factor of two for $Q^2\sim 10$ GeV$^2$.
[**3 Study of the Relativistic Effects**]{}
Let us consider now different kinematics, namely $Q^2\le 4$ GeV$^2$. In this case we expect minimal CT effects and therefore the consequences of the FSI are well under control. The kinematics, where the light-cone momentum $\alpha>1$ and $p_t\approx 0$, are most sensitive to relativistic effects in the deuteron. There are several techniques to treat the deeply bound nucleons as well as relativistic effects in the deuteron. One group of approaches handles the virtuality of the bound nucleon within a description of the deuteron in the lab. frame (we will call them virtual nucleon (VN) approaches) by taking the residue over the energy of the spectator nucleon. One has to deal with negative energy states which arise for non-zero virtualities (see e.g. Ref.[@MST]). Due to the binding, current conservation is not automatic and one has to introduce a prescription to implement e.m. gauge invariance (see e.g. Ref.[@deF]). Another approach is based on the observation that high energy processes evolve along the light-cone. Therefore, it is natural to describe the reaction within the light-cone non-covariant framework [@rep]. Negative energy states do not enter in this case, though one has to take into account so called instantaneous interactions. For this purpose one employs e.m. gauge invariance to express the “bad” electromagnetic current component (containing instantaneous terms) through the “good” component $J^A_+ = -q_+/q_-J^A_-$ [@rep]. In the approximation when non-nucleonic degrees of freedom in the deuteron wave function can be neglected, one can unambiguously relate the light-cone wave functions to those calculated in the lab. frame by introducing the LC $pn$ relative three momentum $k=\sqrt{{m^2+p_t^2\over \alpha(2-\alpha)} - m^2}$.
Turning to numerical estimates, it is worth noting that it is well established that, by using a polarized deuteron target in $(e,e'p)$ reactions, one can decisively disentangle the VN and LC prescriptions (see e.g. [@rep]). Now using the recent advances in the FSI calculation, one can repeat a similar comparison for the tensor asymmetry, $T^{20} = {1\over 3}(\sigma^{1,1} + \sigma^{1,-1}
- 2 \sigma^{1,0})$, accounting also for the FSI diagrams. The result of such a comparison is presented in Fig.3 for backward kinematics ($\theta_{s}=180^o$). One can see that account of the FSI further increases the difference between the predictions of the VN and LC approaches, thus making their experimental investigation more feasible.
The advantage of using a $\vec d$ target to enhance the contribution of small internucleon distances
(100,200)(0,0) (-15,210)[[**Figure 3:**]{} [*$p_s$ dependence of the $(e,e'p)$*]{}]{} (-15,195)[[*tensor polarization at $\theta_{s}=180^0$. Solid*]{}]{} (-15,180)[[*and dashed lines are PWIA predictions*]{}]{} (-15,165)[[*of the LC and VN methods, respective*]{}]{} (-15,150)[[*marked curves include FSI.*]{}]{}
(235,210)[[**Figure 4:**]{} [*$Q^2$ dependence of the unpolarized*]{}]{} (235,195)[[*and tensor polarized cross sections. Solid* ]{}]{} (235,180)[[*line - LC approach with PLC suppression,*]{}]{} (235,165)[[*dashed - LC, and dashed-dotted - VN.*]{}]{} (235,150)[[*Experimental data from Ref.[@Rock].*]{}]{}
holds even for inclusive $\vec d(e,e')$ scattering. In Fig.4, we compare the predictions of the VN and CT approaches for $d(e,e')$ reactions with unpolarized and polarized deuteron targets. Yielding practically the same predictions for a unpolarized target at $x<1$, the two approaches differ by as much as a factor of two in the tensor polarization cross section.
[**3 Conclusions**]{}
We demonstrated that the use of a polarized deuteron target allows to probe effectively smaller internucleon distances in the deuteron ground state wave function for semiexclusive $(e,e'N)$ and inclusive $(e,e')$ reactions. This opportunity can be successfully used to gain a better understanding of the structure of (moderate) high energy, large $Q^2$ $eA$ interactions. In particular, we demonstrated that the use of a $\vec d$ target would allow to observe the onset of Color Transparency at intermediate energies as well as to confront different descriptions of relativistic effects in the deuteron and electromagnetic interactions with deeply bound nucleons.
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---
abstract: 'We present the star formation rate (SFR) and starburst fraction (SBF) for a sample of field galaxies from the ICBS intermediate-redshift cluster survey. We use and Spitzer 4m fluxes to measure SFRs, and 4m fluxes and absorption to measure of SBFs, for both our sample and a present-epoch field sample from the Sloan Digital Sky Survey (SDSS) and Spitzer Wide-area Infrared Extragalactic (SWIRE) survey. We find a precipitous decline in the SFR since $z = 1$, in agreement with other studies, as well as a corresponding rapid decline in the fraction of galaxies undergoing long-duration moderate-amplitude starbursts. We suggest that the change in both the rate and mode of star formation could result from the strong decrease since $z = 1$ of gas available for star formation.'
author:
- 'Alan Dressler, Augustus Oemler, Jr., Michael G. Gladders, Lei Bai, Jane R. Rigby, & Bianca M. Poggianti'
title: 'Evolution of the Rate and Mode of Star Formation in Galaxies since $z = 0.7$'
---
Introduction
============
In recent years, much attention has been focused on the early history of star formation, in particular, the rise of the SFR to its peak around $z \sim 2$ (e.g., Fall, Charlot, & Pei 1996, Madau 1996, Giavalisco 2004, P[é]{}rez-Gonz[á]{}lez 2008, Bouwens 2008). Such a rapid rise is not surprising considering the dynamical timescales of galaxy-sized structures and the lifetimes of stars. More surprising, perhaps, is the precipitous decline in the global SFR since $z \approx 1$ — more than a factor of 10 lower than its peak value, and falling fast (e.g., Lilly 1996, Schiminovich 2005, Hopkins & Beacom2006, Villar 2008). This is remarkable not only for its rapidity but also because it appears to mark our epoch as the beginning of the end of galactic star formation. It is now clear that the unexpected prevalence of starforming galaxies in rich galaxy clusters discovered by Butcher & Oemler (1978), long regarded as a cluster phenomenon, is universal, as starforming galaxies — great and small — surrender youthful vigor and fade towards oblivion in only a few billion years.
Using the wide field of the Inamori-Magellan Areal Camera and Spectrograph (IMACS) on Magellan-Baade, the [*IMACS Cluster Building Survey*]{} (ICBS) is focused on the study of galaxy infall and evolution from $R\sim5$ Mpc into cluster cores. Because the projected density of cluster/supercluster members is low at such large radii, our near-complete samples necessarily include $\sim$1000 “field" galaxies at redshift $0.2 < z < 0.8$ per survey field. This gives us an opportunity to compare galaxy evolution in clusters with the field over this epoch. From these data we report in this [*Letter*]{} on the significant decline in SFR and SBF since $z\sim1$.
Data
====
The data discussed here come from 4 fields that contain rich galaxy clusters at $z = 0.33, 0.38, 0.42,$ $\&\, 0.55$. The IMACS f/2 spectra have an observed-frame resolution of 10Å full-width-half-max with a typical S/N $\sim$ 20-30 in the continuum per resolution element. Spectral coverage varies, but almost all cover and emission, and absorption, our optical diagnostics of star formation; a fraction cover as well. In each 28-diameter IMACS field we have observed 65% of the galaxies that are brighter than $r \sim 22.5$, obtaining adequate spectra of 81% of these. Measurement of spectral features followed procedures described in Dressler (2004, hereinafter D04). For two fields we have confusion-limited Spitzer Multiband Imaging Photometer (MIPS) 4m images (Guest Observer program 40387) that cover nearly the entire IMACS fields. Details of the data, data reduction, and analysis of the field sample are described in Oemler (2009b, hereinafter Oem09b).
In the following analysis we include all galaxies between $0.10 < z < 0.70$ with absolute AB magnitudes at 4400Å brighter than $M_{44}^* +1.0$, except for those with redshifts within $\pm 3 \sigma$ of each targeted cluster. $M_{44}^*$ has been determined to evolve with redshift as $M_{44}^* = -20.00-z$ (Oem09b); the limit of $M_{44}^* +1.0$ is about equivalent to $r = 22.5$ at $z = 0.70$. There are 1144 objects in this sample. Galaxies are given weights proportional to the inverse of the incompleteness of the data at each apparent magnitude. To provide a low–redshift point we use two samples of SDSS galaxies with redshifts $0.04 < z < 0.08$. To compare to infrared-derived properties of the ICBS sample, we use a sample of 385 SDSS galaxies within the SWIRE fields (Lonsdale 2003). For comparison with the optically-derived properties of our sample, we use this SDSS/SWIRE sample and add 690 SDSS galaxies near the North Galactic Pole. We assume a concordance cosmology with $\Omega_{matter} = 0.27$ and $H_o = 71\, \kms\ Mpc^{-1}$.
Measurements of SFR since $z=0.7$
=================================
For galaxies with Spitzer-MIPS coverage we calculate SFRs in a way similar to P[é]{}rez-Gonz[á]{}lez (2006), using the 4m flux to estimate the absorbed UV flux and to estimate the escaped UV flux, but including the k–corrections of Rieke (2008) and a SFR scale based on a Salpeter initial-mass-function (IMF, vid.Oem09b). Donley (2008) report that 10–15% of 4m sources at these flux densities are AGN-dominated. Given the strong redshift evolution of the AGN luminosity function, the contamination in our $z<0.7$ sample should be should be considerably lower. Figure 1 (bottom panel) shows the median SFR per unit $L/L^*$, which we shall refer to as the specific SFR, as a function of redshift. We choose the [*median*]{} specific SFR over the more commonly reported [*mean*]{}, because it is provides a more stable and typical value — particularly at higher redshifts, where the mean is dominated by a small number of very luminous objects. (In the $0.60 < z < 0.70$ interval, only 1.6% of the objects contribute 56% of the total star formation.) We also show, for comparison, values of the median $SFR/(L/L^*)$, derived from the mid-IR data of Damen (2009), and from the P[é]{}rez-Gonz[á]{}lez (2008) study of the buildup of stellar mass (from near-IR measurements), as reinterpreted by Damen and converted by us to a Salpeter IMF scale. To convert the Damen values of SFR per unit mass to $SFR/(L/L^*)$ we take a mean value of $M/L$ at 4400Å of 3, which is the ratio of the cosmic mass density (P[é]{}rez-Gonz[á]{}lez, 2008) to the cosmic luminosity density (Blanton 2003). Although cosmic means, both are dominated by galaxies in the luminosity and mass ranges of our sample, and thus appropriate for our use. With $M/L = 3$, a sample with our absolute magnitude limit has a mean mass of $5x10^{10}\,\Msun$.
Fig. 1. (Bottom) [*Median*]{} Specific SFR, in solar masses per year per L\* luminosity. Black filled circles show values for our sample derived from Spitzer-MIPS 4m fluxes. Red stars are data from Damen (2009), and red open circles are data from P[é]{}rez-Gonz[á]{}lez (2008) as reinterpreted by Damen (2009). The median SFR appears to be declining more rapidly towards the present epoch. (Top) Median values (solid circles) and 90th percentile values (open circles) of for the $z > 0.2$ ICBS field sample, binned to achieve comparable of numbers of -detected galaxies. The single point at $z=0.06$ comes SDSS galaxies with optical spectroscopy (see text).
At $z>0.4$ the [*median*]{} values of SFR increase more slowly than the mean SFR, as reported by Damen (2009), Martin (2007), Zheng (2007), but are otherwise qualitatively similar. However, what has not been apparent from previous observations is that the fall of the median SFR appears to be accelerating in recent epochs.
Optical emission lines , , and are less reliable SFR indicators than 4m flux because of the very large and variable extinction which blankets the HII regions of starforming galaxies. However, they do provide at least a qualitative measure of current star formation and, unlike our Spitzer-MIPS data, are available for all four ICBS fields. Therefore, we plot in the top panel of Figure 1 the 50th and 90th percentiles of the distribution of [*equivalent width*]{} EW() as a function of redshift. The shapes of these distributions are qualitatively similar to the infrared–derived SFR, however, strength increases more slowly with redshift because of the well-known effect that galaxies with higher SFRs also have higher dust extinction.
Were Starbursts the Normal Mode of Star Formation before $z=1$?
===============================================================
It is now well established that starburst and post–starburst galaxies are abundant in intermediate–redshift clusters (Poggianti 1999 (hereinafter P99), Oemler 2009a), however, less is known about the prevalence of starbursts among field galaxies at these epochs. Starburst indicators include exceptionally strong Balmer absorption, intense optical emission, and excess 4m flux. Strong Balmer absorption in the integrated light of galaxies signals a rapid decline in the SFR, because light from A stars persists after the blue continuum from O and B stars, and the emission from HII regions — both of which dilute the Balmer absorption lines — begin to fade or disappear altogether (P99). Although strong Balmer absorption accompanies even the simple truncation of star formation, as O and B stars and their HII regions evolve away, stellar $EW(\Hd) \gs 5$Å occurs only in the aftermath of a significant rise in the SFR above its past average, i.e., a burst (Dressler & Gunn 1983, Couch & Sharples 1987, P99). Using this criterion, P99, D04 and Oemler (2009a) found that the SBF in the field at $z \sim 0.4$ is lower than in clusters, but significantly higher than it is in the field today; however, the data sets used were quite small. More recently, Poggianti (2009) find a large incidence of dusty starburst galaxies for $0.4 < z < 0.8$ in all environments, but particularly in groups. Using an entirely different approach, Bell (2005) find a substantial fraction of massive field galaxies at $z \sim 0.7$ with SFRs much higher than their long–term averages. On the other hand, Noeske (2007) argue that the narrow width of the SFR/mass versus mass relation among field galaxies in the Groth strip of DEEP2 data is inconsistent with a large fraction of strong starbursts.
Determining the starburst frequency — SBF — as a function of cosmic epoch and environment is important both for understanding the nature and cause of starbursts, and their role in galaxy evolution. For example, starbursts provide a quick means of exhausting the gas supply in a galaxy. The dense environment of rich clusters provides a variety of mechanisms for producing starbursts, including ram-pressure from the intergalactic medium, tidal encounters between galaxies, and mergers and accretion. Some of these are also viable in the group environment, to more or less effect, but ram pressure, for example, is likely to be unimportant. Some present-epoch, truly isolated galaxies appear to have experienced starbursts as well, which might point to some instability in the disks of star forming galaxies, or accretion of a small satellite, as other possible causes.
Figure 2 compares results from two different methods of determining the SBF. The top panel shows the fraction of all galaxies in our field sample that have EW() of 2.5Å or greater than expected for a normal, non–bursting galaxy, following the method of D04, but using an improved EW() versus EW() relation (q.v. Oem09). Following Bell (2005), we also calculate the SBF from the Spitzer–derived SFRs, by defining a starburst galaxy as one whose observed SFR is significantly higher than its long-term average rate, which should be, ignoring mass loss during stellar evolution,
$$SFR_{past average} = M_{gal} / t_{gal} = (L_{gal} \times M/L) / t_{gal}$$
where $t_{gal}$ is the length of time that a galaxy has been forming stars, and $M_{gal}$, $L_{gal}$, and $M/L$ are, the galaxy’s [*stellar mass*]{}, luminosity, and stellar mass–to–light ratio. We define $t_{gal}$ to be time elasped from redshift $z=4$ (a reasonable assumption for the start of star formation) to the observed redshift, and assume that $M_{gal}/L_{gal} = 3$, where $L_{gal}$ is calculated at 4400Å. We identify a galaxy as a starburst if $SFR_{obs}/SFR_{past-average} > 3$. We plot in Figure 2 the fraction of galaxies which meet this criterion and perform the same analysis for the observations of Noeske (2007) and Bell (2005). For these we include objects with $M_{gal} \ge 1.6 \times 10^{10}M_{\sun}$, which for $M/L=3$, is equivalent to our luminosity limit of $M_{44}^* +1.0$.
Fig. 2. Two measures of starburst fraction, SBF, versus redshift for galaxies brighter than M\*+1. Top: fraction of galaxies with excess EW() $>$ 2.5Å; bottom: fraction of galaxies with SFR $> 3\times
L_{tot} \times (M/L)$/$\tau$(universe). Black filled circles — ICBS data; open red circles- data from Noeske (2007); red filled circles — data from Bell (2005). Black open circles are ICBS data with SFR $> 10\times L_{tot} \times
(M/L)$/$\tau$(universe). Using this criterion for a starburst, all three samples give consistent results.
Figure 2 shows that such starbursts make up $\sim$20-25% of starforming galaxies by $z\sim0.6$, a much larger fraction than today, Moreover, because the “duty cycle" (fraction of the time these galaxies are identifiable as starbursts) is almost certainly less than 50%, the clear implication is that [*most starforming galaxies at $z > 0.6$ have undergone a starburst of at least moderate strength.*]{} Figure 2 also shows the SBF for a factor-of-ten increase rising in parallel.
Noeske (2007) emphasize the rarity of starbursts over a comparable (but somewhat wider) redshift range, basing their argument on the narrowness of the SFR/M versus M relation. They conclude that no more than one-third of typical starforming galaxies have SFR variations greater than a factor-of -two, while our sample suggests, after correction for the fraction of passive galaxies, that one-third have experienced a rise of a factor of three or more. Put in these terms, the two results appear mildly inconsistent, however, as Figure 2 illustrates, the Noeske data, when analyzed using the same method as we and Bell (2005) have used, is in good agreement with our own.
Finally, we show another way of assessing the SBF(z), using the composite spectra approach developed in D04. Here we add, weighted by luminosity, all ICBS spectra in each of 5 redshift slices $0.20 < z < 0.70$, and for a present-epoch sample of SWIRE galaxies. We plot in Figure 3 EW(, ) measured from these composite spectra, along with expected values for a population of (mostly) continuously-starforming galaxies at the present epoch. Figure 3 includes expected values calculated by D04 from the 2dF & CNOC2 field surveys (see D04) and also new determinations using the SWIRE data, an improvement because we now have individual distributions (on which these predictions rest) from our ICBS sample, for each redshift interval. The Dressler-Shectman low-redshift cluster and field samples follow the “continuous star formation” prediction, but the clusters at $z \sim 0.4$ studied in the Morphs collaboration depart, indicating a higher fraction of starbursts (see D04).
Fig. 3. versus equivalent widths for composite spectra, adapted from D04. The lines ‘2dF’ and ‘CNOC2’ show predictions for continuously starforming populations, and the points ‘Morphs’ and ‘Dressler-Shectman’ cluster and field samples, from studies described in D04. The low-redshift SWIRE field galaxy sample is the blue point. The ICBS field sample measurements are the 5 green points for redshift intervals centered at z = 0.25, 0.35, 0.45, 0.55, 0.65. The 5 joined purple points show the prediction of EW(, ) at these 5 redshift intervals for a population of present-epoch starforming galaxies (the SWIRE sample) with very few starbursts, as described in the text. The observed ICBS measurements at $z = 0.45, 55,
\& 0.65$ far exceed these predicted values, showing the substantial fraction of starburst galaxies in these field populations. The DEEP2 (red) point for field galaxies at $z \sim 0.83$ extends and confirms this result.
For the new field sample, Figure 3 shows that the $z = 0.25, 0.35$ slices are little different from the present-epoch population, but the $z = 0.45, 0.55, 0.65$ slices have much stronger absorption than expected for a population dominated by continuous star formation. We also can compare our EW(, ) values to those from a composite spectrum of $\sim$1000 field galaxies at $z \sim 0.8$ from the DEEP2 Galaxy Redshift Survey (Davis 2009), This coadded spectrum was produced as described in Weiner (2008) by Jeff Newmann and Renbin Yan from galaxies selected with limiting $M_R$ comparable to our own sample. The DEEP2 point continues the trend of yet greater SBF to even higher redshift, for a data set independent of our own, with higher S/N and higher spectral resolution.
In summary, methods based on (1) individual strengths, (2) the SFR-increase-over-past-average, and (3) (EW(, ) from composite spectra, all point to an SBF that increases markedly with redshift.
Discussion and Conclusions
==========================
We have used optical spectra and mid-IR fluxes of field galaxies in our ICBS galaxy cluster survey to measure the rapid decline of the specific star formation rate, SFR, since $z \sim 0.7$. Although qualitatively consistent with other studies, we find that — parameterized as the median specific SFR — this decline appears to be steepening toward the present epoch. This might be evidence for a kind of “downsizing:" elliptical/bulge formation dominates at $2 < z < 6$ and smoothly transitions to the building of massive galaxy disks, but as this wanes for $z < 1$, there is insufficient mass in the still-starforming dwarf galaxy population to prevent what is essentially the end of the era of star formation (Dressler 2004).
We also show that it is not just the SFR, but also its mode — the starburst fraction, SBF — that is evolving rapidly since $z = 0.7$. We use three different methods, from discrete and composite spectra, to show that the SBF also declines dramatically over this period. When accounting for the “duty cycle," the data suggest that the [*majority*]{} of starforming galaxies at $z \gs 0.5$ followed this mode of star formation rather than in the steadier, “continuous" mode that dominates today. We emphasize that, unlike short and often-intense nuclear starbursts, these are typically of moderate strength — the SFR rising by a factor of 3–10. They are likely to be galaxy-wide and of longer duration — consistent with the dynamical timescales of the larger region involved.
It seems possible that these two effects — the rapid decline in specific SFR and the changing mode of star formation — share a common cause. Availability of gas for star formation could be the principal agent. If the Schmidt-Kennicutt (Kennicutt 1998) law holds, higher SFRs of intermediate-redshift galaxies were due to gas fractions higher than the 20-40% typical of today’s luminous spirals (McCaugh & de Blok 1997), although how much greater may depend on the relative importance of $H_{2 }$ versus $HI$ (Robertson & Kravtsov 2009, Krumholz 2009). As discussed by Putnam (2008), this greater gas content could be due to initial supply, accretion of gas-rich satellites, or resupply from surrounding reservoirs of cooling gas, the latter the subject of much recent discussion (see, e.g., Keres 2009). However, direct measurements of gas contents for intermediate-redshift galaxies are not yet possible, and absorption measurements by Prochaska and Wolfe (2009) — albeit of HI alone — support a different picture in which gas fractions do not evolve strongly and high SFRs are instead supported by high rates of gas accretion.
If higher SFRs were the result of higher gas fractions of more than 50% for typical $z > 0.5$ galaxies, then it is reasonable to suppose that the higher SBF might also be a result. Star formation in gas-rich disks may be unstable if high supernovae rates heat disks sufficiently to disrupt conditions favorable for star formation. After a gas-cooling time of several hundred million years, rapid star formation could return; color-magnitude diagrams for some nearby dwarf galaxies show two or more episodes of star formation separated by billions of years (e.g, Gallart 1999; Held 2000). If resupply by cold gas flows is important to the evolution of spiral disks, the higher infall rates of earlier epochs could be subject to significant variability, simply a result of the granularity of the gas density in a complex network of filaments.
Finally, mergers of gas-rich galaxies might contribute to the rising SBF with increasing redshift, although Bell (2005) find only a small fraction of their high SFR galaxies at $z\sim0.7$ in major mergers. Perhaps a higher rate of accretion of small satellites at earlier times might have resulted in more starbursts. Indeed, since these smaller systems were probably gas rich, accretion rather than major mergers might be the dominant starburst trigger.
As yet there are no direct means to measure gas fractions in $z\sim 1$ galaxies, but ALMA and the proposed SKA will in future provide that capability. Until that time, there is an important role for numerical modeling, with its increasing resolution, to explore whether gas fractions, inflow rates, and satellite accretion would affect not just the rate of star formation but its mode as well.
Acknowledgments
===============
Dressler and Oemler acknowledge the support of the NSF grant AST-0407343. All the authors thank NASA for its support through NASA-JPL 1310394. Jane Rigby is supported by a Spitzer Space Telescope Postdoctoral Fellowship. Partial support was also provided through contract 1255094 from JPL/Caltech to the University of Arizona. The authors gratefully acknowledge the Jeff Newman, Renbin Yan, and the DEEP2 team for the composite spectrum used in this paper.
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|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'Let $E/{\mathbb{Q}}$ be an elliptic curve defined over ${\mathbb{Q}}$ with conductor $N$ and ${\operatorname{Gal}(\overline{{\mathbb{Q}}}/{\mathbb{Q}})}$ the absolute Galois group of an algebraic closure $\overline{{\mathbb{Q}}}$ of ${\mathbb{Q}}$. We prove that for every $\sigma\in {\operatorname{Gal}(\overline{{\mathbb{Q}}}/{\mathbb{Q}})}$, the Mordell-Weil group $E({\overline{{\mathbb{Q}}}^{\sigma}})$ of $E$ over the fixed subfield of $\overline{{\mathbb{Q}}}$ under $\sigma$ has infinite rank. Our approach uses the modularity of $E/{\mathbb{Q}}$ and a collection of algebraic points on $E$ – the so-called [*Heegner points*]{} – arising from the theory of complex multiplication. In particular, we show that for some integer $r$, the rank of $E$ over all the ring class fields of conductor of the form $rm$, and of the form $rp^n$ is unbounded, as $m$ goes to infinity, as $m$ and as $n$ goes to infinity respectively, where $m$ is a square-free integer and $p$ is a prime such that $(m, rN)=1$ and $(p, rN)=1$.'
address: 'Department of Mathematics, Indiana University, Bloomington, Indiana 47405'
author:
- 'Bo-Hae Im'
date: 'July 26, 2004'
title: 'Heegner points and Mordell-Weil groups of elliptic curves over large fields'
---
This paper is motivated by the following conjecture of M. Larsen [@larsen]:
[**Conjecture.**]{} Let $K$ be a number field and $E/K$ an elliptic curve over $K$. Then, for every $\sigma\in
\operatorname{Gal}(\overline{K}/K)$, the Mordell-Weil group $E(\overline{K}^{\sigma})$ of $E$ over $\overline{K}^{\sigma}=\{x\in\overline{K}\mid \sigma(x)=x\}$ has infinite rank.
In [@im], we have proved this conjecture in certain cases:
for a number field $K$ and an elliptic curve $E/K$ over $K$,
- if 2-torsion points of $E/K$ are $K$-rational, or
- if $E/K$ has a $K$-rational point $P$ such that $2P\neq O$ and $3P\neq O$,
then for every automorphism $\sigma\in
\operatorname{Gal}(\overline{K}/K)$, the rank of the Mordell-Weil group $E(\overline{K}^{\sigma})$ is infinite.
In this paper, we prove that the conjecture is true for elliptic curves over ${\mathbb{Q}}$ without any hypothesis on rational points of $E/{\mathbb{Q}}$, [*i.e.*]{} if $E/{\mathbb{Q}}$ is an elliptic curve over ${\mathbb{Q}}$, then, for every automorphism $\sigma\in
{\operatorname{Gal}(\overline{{\mathbb{Q}}}/{\mathbb{Q}})}$, the Mordell-Weil group $E({\overline{{\mathbb{Q}}}^{\sigma}})$ over the fixed subfield of $\overline{{\mathbb{Q}}}$ under $\sigma$ has infinite rank.
To prove the conjecture for a given $E/K$, ultimately one must find an infinite supply of rational points of $E$ over finite extensions of $K$ contained in $\overline{K}^{\sigma}$. In [@im], we constructed such points using Diophantine geometry, essentially by searching for sufficiently rational subvarieties of certain quotients of the $n$-fold product $E^n$ of $E$.
Here we use a completely different approach, coming from arithmetic: taking advantage of the modularity of elliptic curves over ${\mathbb{Q}}$, we choose our rational points on $E$ to be algebraic points over ring class fields – the so-called [*Heegner points*]{}. The main strategy is as follows: by using the norm compatibility properties of Heegner points and a generalized dihedral group structure of the Galois groups of ring class fields over ${\mathbb{Q}}$, we show that the rank of $E$ over the ring class fields is unbounded as the ring class fields get larger. And we also show that a given automorphism $\sigma\in {\operatorname{Gal}(\overline{{\mathbb{Q}}}/{\mathbb{Q}})}$ does not fix a quadratic imaginary extension of ${\mathbb{Q}}$ over which all primes dividing the conductor of $E$ are split, then the rank of $E$ over the fixed subfields of the ring class fields under $\sigma$ is unbounded as the ring class fields get larger, which proves that the rank of $E$ over ${\overline{{\mathbb{Q}}}^{\sigma}}$ is infinite. On the other hand, if $\sigma\in {\operatorname{Gal}(\overline{{\mathbb{Q}}}/{\mathbb{Q}})}$ fixes infinitely many quadratic imaginary extensions, then we can construct infinitely many linearly independent points defined over each of them by applying the Hilbert irreducibility theorem to a Weierstrass equation of $E/{\mathbb{Q}}$ directly and this also proves infinite rank of $E$ over ${\overline{{\mathbb{Q}}}^{\sigma}}$.
I would like to thank my thesis advisor, Michael Larsen for suggesting this problem and for valuable discussions and helpful comments on this paper. Also, I wish to thank Henri Darmon for suggesting this approach, and for his guidance and valuable advice and comments on an earlier manuscript of this paper.
the main theorem
================
In this section, we introduce the main theorem. First, we will need the following Hilbert irreducibility and the denseness of Hilbert sets in any open intervals of ${\mathbb{R}}$.
Let $f\in K(t_1,\ldots,t_m)[X_1,\ldots,X_n]$ be a polynomial with coefficients in the quotient field $K(t_1,\ldots,t_m)$ of $K[t_1,\ldots,t_m]$ which is irreducible over $K(t_1,\ldots,t_m)$. We define $$H_K(f)=\{(a_1,\ldots,a_m)\in K^m\mid
f(a_1,\ldots,a_m,X_1,\ldots,X_n) \mbox{ is irreducible over }
K\}$$ to be the Hilbert set of $f$ over $K$. If for every $m \geq
1$, any intersections of a finite number of Hilbert sets with a finite number of nonempty Zariski open subsets in $K^m$ are not empty (in fact, they are infinite), a field $K$ is called *a Hilbertian field*.
\[lem:hilbert\] Let $L$ be a finite separable extension of a Hilbertian field $K$ and let $f$ be a polynomial in $
L(t_1,\ldots,t_m)[X_1,\ldots,X_n]$ which is irreducible over the quotient field $L(t_1,\ldots,t_m)$. Then, there exists a polynomial $p\in K[t_1,\ldots,t_m,X_1,\ldots,X_n]$ such that $p$ is irreducible over $K(t_1,\ldots,t_m)$ and $H_K(p) \subseteq
H_L(f)$.
For a given irreducible polynomial $f \in L(t_1,\ldots,t_m)[X_1,\ldots,X_n]$, by ([@jar], Ch.11, Lemma 11.6), there is an irreducible polynomial $q$ $\in
K(t_1,\ldots,t_m)[X_1,\ldots,X_n]$ such that $H_K(q)\subseteq
H_L(f)$. By ([@jar], Ch.11, Lemma 11.1), there is an irreducible polynomial $p \in K[t_1,\ldots,t_m,X_1,\ldots,X_n]$ which is irreducible over $K(t_1,\ldots,t_m)$ such that $H_K(p)
\subseteq H_K(q)$. Hence the Hilbert set $H_L(f)$ of $f$ over $L$ contains the Hilbert set $H_K(p)$ of $p$ over $K$.
\[lem:dense\] Let $K$ be a number field and $\tau_1,\ldots,\tau_m$ be a family of real embeddings of $K$. For $i=1,2,\ldots,k$, let $f_i(x,y)\in K[x,y]$ be irreducible polynomials over $K(x)$. Let $H_K(f_i)$ $= \{\alpha\in K\mid$ $
f_i(\alpha, y) \in K[y]$ is irreducible over $K\}$ be the Hilbert set of $f_i$ over $K$. Then $$\left(\bigcap\limits _{i=1}^k
H_K(f_i)\right)~\cap~\left(\bigcap\limits
_{j=1}^m\tau_j^{-1}(I)\right)\neq~~ \emptyset,$$ for any open interval $I$ in ${\mathbb{R}}$.
Since $K$ is a finite separable extension of ${\mathbb{Q}}$, by Lemma \[lem:hilbert\], there exist irreducible polynomials $F_i(x,y)\in {\mathbb{Q}}[x,y]$ such that for each $i=1,2,\ldots,k$, the Hilbert set $H_{{\mathbb{Q}}}(F_i)$ of $F_i$ over ${\mathbb{Q}}$ is contained in the Hilbert set $H_{K}(f_i)$ of $f_i$ over $K$.
Let $I$ be an open interval in ${\mathbb{R}}$. Since $\bigcap\limits
_{i=1}^k H_{{\mathbb{Q}}}(F_i)$ is dense in ${\mathbb{Q}}$ by ([@l83], Chapter 9, Corollary 2.5), and ${\mathbb{Q}}$ is dense in ${\mathbb{R}}$, $\left(\bigcap\limits _{i=1}^k H_{{\mathbb{Q}}}(F_i)\right)\cap I$ is not empty. Hence there is $\beta \in \left(\bigcap\limits _{i=1}^k
H_{{\mathbb{Q}}}(F_i)\right)\cap I$. Since $\bigcap\limits _{i=1}^k
H_{{\mathbb{Q}}}(F_i)\subseteq \bigcap\limits _{i=1}^k H_K(f_i)$, we have $\beta\in \bigcap\limits _{i=1}^k H_K(f_i)$. On the other hand, for each real embedding $\tau_j$ of $K$, we have $\tau_j|_{{\mathbb{Q}}}=id_{{\mathbb{Q}}}$. Hence for all $j=1,2,\ldots,m$, $\tau_j(\beta)=\beta\in I$. Hence $\beta\in \bigcap\limits
_{j=1}^m\tau_j^{-1}(I)$. Therefore, $\beta\in \left(\bigcap\limits
_{i=1}^k H_K(f_i)\right)$ $\cap$ $\left(\bigcap\limits
_{j=1}^m\tau_j^{-1}(I)\right)$.
Here is our main theorem.
\[thm:main\] Let $E/{\mathbb{Q}}$ be an elliptic curve over ${\mathbb{Q}}$. Then, for every automorphism $\sigma\in {\operatorname{Gal}(\overline{{\mathbb{Q}}}/{\mathbb{Q}})}$, the rank of $E({\overline{{\mathbb{Q}}}^{\sigma}})$ is infinite.
Let $N$ be the conductor of $E$ and let $y^2=x^3+ax+b$ be a Weierstrass equation of $E/{\mathbb{Q}}$. By the change of variables, we may assume that $a$ and $b$ are integers. Let $M=4p_1p_2\cdot\cdot\cdot p_k$, where $p_k$ are all distinct prime factors of $N$. Consider the polynomial $$f(x)=(1+Mx)^3+aM^4(1+Mx)+bM^6~~~\in{\mathbb{Z}}[x].$$ Then, there exists a real number $r$ such that for all $x<r$, the expression $f(r)$ is strictly negative. Let $I=(-\infty, r)$ be the open interval in ${\mathbb{R}}$ of all real numbers less than $r$. Since ${\mathbb{Q}}$ is Hilbertian, by Lemma \[lem:dense\], there exists an integer $m_1\in I$ such that $y^2-f(m_1)$ is irreducible over ${\mathbb{Q}}$. Let $K_{m_1}={\mathbb{Q}}(\sqrt{f(m_1)})$ be the quadratic imaginary extension of ${\mathbb{Q}}$. By Lemma \[lem:hilbert\], there is a polynomial $p$ over ${\mathbb{Q}}$ such that $H_{{\mathbb{Q}}}(p)\subseteq
H_{K_{m_1}}(y^2-f(x))$. Then, by Lemma \[lem:dense\] again, there exists an integer $m_2\in I\cap H_{{\mathbb{Q}}}(p)$. So $K_{m_2}={\mathbb{Q}}(\sqrt{f(m_2)})$ is a quadratic imaginary extension of ${\mathbb{Q}}$ and $K_{m_1}$ and $K_{m_2}$ are distinct, hence linearly disjoint over ${\mathbb{Q}}$. By repeating this procedure over the composite field of quadratic imaginary extensions obtained from the previous steps inductively, there is an infinite set $S$ of integers such that for all $m\in S$
1. $f(m)<0$, so that $K_m:={\mathbb{Q}}(\sqrt{f(m)})$ is a quadratic imaginary extension of ${\mathbb{Q}}$,
2. the fields in the infinite sequence $\{K_{m}\}_{m\in
S}$ are linearly disjoint over ${\mathbb{Q}}$ and
3. if $E/{\mathbb{Q}}$ has CM, then, $K_m$ is different from $F=\operatorname{End}(E)\otimes {\mathbb{Q}}$.
Note that for each $m\in S$ and for every prime $p_i$ dividing $N$, $$f(m) \equiv \left\{\begin{array} {l@{}l} 1
\mbox{\hspace{.1in }} (\bmod~~p_i), \mbox{\hspace{.1in } if }
p_i\neq 2, \\ 1 \mbox{\hspace{.1in }} (\bmod~~8),
\mbox{\hspace{.1in } if } p_i= 2.
\end{array}\right.$$
Hence, this implies that all primes dividing $N$ are split in $K_m$. On the other hand, the discriminant of $K_m$ is $f(m)$ or $4f(m)$ depending on whether $f(m)\equiv 1$ (mod 4) or not, respectively. And in any case, the discriminant of $K_m$ is prime to $N$.
Let $\sigma \in {\operatorname{Gal}(\overline{{\mathbb{Q}}}/{\mathbb{Q}})}$. We divide into two cases. For the first case, suppose that for all $m\in S$, $\sigma|_{K_m}=id_{K_m}$. Then, for each $m\in S$, consider the number $\displaystyle
\frac{1+Mm}{M^2}\in {\mathbb{Q}}$. By plugging this number into the given Weierstrass equation of $E/{\mathbb{Q}}$, we get $$y^2=\left(\displaystyle\frac{1+Mm}{M^2}\right)^3+a\left(\frac{1+Mm}{M^2}\right)+b
=\frac{f(m)}{M^6}.$$ Hence the point $$P_m=\left(\displaystyle\frac{1+Mm}{M^2},
\frac{\sqrt{f(m)}}{M^3}\right)$$ is in $E(K_m)$ but it is not in $E({\mathbb{Q}})$. And moreover, since $K_m=K_m^{\sigma}$, $P_m$ is fixed under $\sigma$.
So we get an infinite sequence $\{P_m\}_{m\in S}$ of points in $E(\overline{{\mathbb{Q}}}^{\sigma})$ such that each $P_m$ is defined over the quadratic imaginary extension $K_m$ over ${\mathbb{Q}}$. We may assume that these points are not torsion points by ([@sil], Lemma). Now we show the points $P_m$ for $m\in S$ are linearly independent. Suppose that for some integers $a_i$ and $ m_i\in
S$, $$a_1P_{m_1}+a_2P_{m_2}+\cdots+a_kP_{m_k}=O .$$ Since the fields in $\{K_m\}_{m\in S}$ are linearly disjoint over ${\mathbb{Q}}$, for each $i$, there is an automorphism of $\overline{{\mathbb{Q}}}$ which fixes all but one $K_{m_i}$ of $K_{m_1},\ldots,K_{m_k}$. Note that such an automorphism takes $P_{m_i}$ to its inverse, $-P_{m_i}$. Applying this automorphism, we get $$a_1P_{m_1}+\cdots +a_{i-1}P_{m_{i-1}}-a_{i}P_{m_{i}}+\cdots +a_kP_{m_k}=O.$$ By subtracting, we get $2a_iP_{m_i}=O$, which implies $a_i=0$. We conclude that the $P_m$ for $m\in S$ are linearly independent in $E({\overline{{\mathbb{Q}}}^{\sigma}})\otimes {\mathbb{Q}}$. Hence the rank of $E({\overline{{\mathbb{Q}}}^{\sigma}})$ is infinite.
For the second case, suppose that there is an integer $m\in S$ such that $\sigma|_{K_m}\neq id _{K_m}$. Then, fix such a quadratic imaginary extension $K_m$, and call it $K$, and let $K_{ab}$ be the maximal abelian extension of $K$. Then, we complete the proof of this case as a consequence of the following stronger statement:
\[thm:ab\] Under the assumption in the second case above $($i.e. if $K$ is different from $\operatorname{End}(E)\otimes
{\mathbb{Q}}$, all primes dividing $N$ are split in $K$, and $\sigma|_K\neq id_K$$)$, the rank of the Mordell-Weil group of $E$ over the fixed subfield $(K_{ab})^{\sigma}$ of $K_{ab}$ under $\sigma$ is infinite.
The proof of Theorem \[thm:ab\], which lies deeper than the methods used in the first case will be treated in the following section and will be given explicitly in Proposition \[prop:sigma\], as it requires modularity of $E/{\mathbb{Q}}$ and the theory of complex multiplication which give a non-torsion algebraic point of $E$ under the given assumption in the second case.
The rank of $E$ over ring class fields of imaginary quadratic fields
====================================================================
The goal of this section is to prove Theorem \[thm:ab\] — stated at the end of the previous section. By a theorem of Wiles [@Wi] and Taylor-Wiles [@TW] (completed by a later work of Breuil, Conrad, Diamond and Taylor [@BCDT]), the elliptic curve $E/{\mathbb{Q}}$ is known to be [*modular*]{}. Our strategy is to construct algebraic points on $E((K_{ab})^{\sigma})$ using [*Heegner points*]{} over the ring class fields arising from the theory of complex multiplication.
We will need the following lemma later.
\[lem:prime1\] Let $K$ be an imaginary quadratic extension of ${\mathbb{Q}}$ such that $K$ is different from $F=\operatorname{End}(E)\otimes {\mathbb{Q}}$. For a prime $\ell$, let $a_l$ be the coefficient of the Hecke operator $T_{\ell}$ of the modular form of $E$. Then, there is an integer $M$ such that for all $p\geq M$ $($which is inert in $F$, if $E$ has CM$)$, there is a prime $q$ such that
1. $q$ is inert in $K$,
2. $p$ does not divide $a_q$ and
3. $p$ divides $q+1$.
Suppose $E/{\mathbb{Q}}$ has no CM. Then, by ([@se], §4. Theorem 2), there is a large integer $M$ such that for all primes $p\geq M$, the continuous Galois representation $$\rho_p:{\operatorname{Gal}(\overline{{\mathbb{Q}}}/{\mathbb{Q}})}\rightarrow \operatorname{\bf GL}_2({\mathbb{F}}_p)$$ is surjective. In particular, let $M$ be large enough such that every $p\geq M$ is unramified in $K$, since $K$ is ramified at only finitely many primes.
Since $Ker(\rho_p)$ is an open normal subgroup of ${\operatorname{Gal}(\overline{{\mathbb{Q}}}/{\mathbb{Q}})}$, there is a finite extension $L$ over ${\mathbb{Q}}$ such that $Ker(\rho_p)=\operatorname{Gal}(\overline{{\mathbb{Q}}}/L)$.
And since $$\operatorname{Gal}(L/{\mathbb{Q}})\cong{\operatorname{Gal}(\overline{{\mathbb{Q}}}/{\mathbb{Q}})}/Ker(\rho_p)\cong \operatorname{\bf GL}_2({\mathbb{F}}_p)$$ has a unique subgroup of index 2, the kernel of $det^{\frac{p-1}{2}}$, the unique quadratic field $L'$ over ${\mathbb{Q}}$ in $L$ is ramified only at $p$. On the other hand, since $p$ is unramified in $K$, the fields $K$ and $L'$ are linearly disjoint. Let $S$ be a finite set of primes such that the composite field $KL'$ is unramified and $K$ is unramified outside $S$. Then, by the Cebotarev density theorem, each Frobenius automorphism Frob$_q$ comes up infinitely often in $\operatorname{Gal}(KL'/{\mathbb{Q}})$, for $q\notin S$.
Since $K$ and $L'$ are linearly disjoint, we can choose the Frobenius automorphism Frob$_q$ in $\operatorname{Gal}(K/{\mathbb{Q}})$ and $\operatorname{Gal}(L'/{\mathbb{Q}})$, independently. Hence, there is a prime $q\neq 2$ such that $q$ is inert in $K$ and $$\rho_p(\mbox{Frob}_q)=
\begin{pmatrix} a& b\\0 &-a^{-1}\end{pmatrix}$$ where $a$ is in ${\mathbb{F}}_p$ such that the order of $a$ is not $2$ in ${\mathbb{F}}_p$ and $b$ is some element of ${\mathbb{F}}_p$. Then, we have $$a_q=\mbox{Trace}(\rho_p(\mbox{Frob}_q))=a-a^{-1}\not\equiv
0~~~ (\bmod ~ p).$$ This implies that $p$ does not divide $a_q$, which is condition (2). Also, we have $$q=
\mbox{det}(\rho_p(\mbox{Frob}_q))\equiv -1~~~ (\bmod~~ p).$$ This implies that $p$ divides $q+1$ which is condition (3). Hence such a prime $q$ satisfies all conditions (1) though (3).
Suppose $E/{\mathbb{Q}}$ has CM. Let $R=\operatorname{End}(E)$ and $F=R\otimes {\mathbb{Q}}$. Then, $F$ is an imaginary quadratic extension of ${\mathbb{Q}}$ such that $R=\operatorname{End}_F(E)$. By ([@se], §4. Corollary of Theorem 5), there is a large integer $M$ such that for all primes $p\geq M$, the continuous Galois representation $\rho_p$ factors through a surjective homomorphism, $$\sigma_p:\operatorname{Gal}(\overline{F}/F)\rightarrow R_p^*,$$ where $R_p=R\otimes {\mathbb{Z}}_p\cong {\mathbb{Z}}_p^2$ as ${\mathbb{Z}}_p$-modules. Under the canonical embedding $R_p=\operatorname{End}_{R_p}(R_p)\subseteq
\operatorname{End}_{{\mathbb{Z}}_p}({\mathbb{Z}}_p^2)=M_2({\mathbb{Z}}_p)$, the norm map and the trace map from $R_p$ to ${\mathbb{Z}}_p$ are the restrictions of the determinant map and the trace map on $M_2({\mathbb{Z}}_p)$.
Let $\mathcal{O}$ be the maximal order in $F$. Then, $\mathcal{O}/R$ is finite, so if $p\gg 0$, $$R/pR \cong
\mathcal{O}/p\mathcal{O}\cong {\mathbb{F}}_p\times{\mathbb{F}}_p \mbox{ or }
{\mathbb{F}}_{p^2},$$ depending on whether $p$ is split or inert in $F$. In either case, the norm map $N$ is a non-degenerate quadratic form on the underlying 2-dimensional vector space over ${\mathbb{F}}_p$. For any non-zero $k$, $N(\alpha)-kz^2$ is a non-degenerate quadratic form on the 3-dimensional ${\mathbb{F}}_p$-vector space $R/pR\times {\mathbb{F}}_p$. In particular, $N(\alpha)=-z^2$ defines a conic curve. Since any conic has a rational point over a finite field by the Chevalley-Warning theorem ([@se2], Chapter I, Theorem 3), the conic $N(\alpha)=-z^2$ over ${\mathbb{F}}_p$ is isomorphic to ${\mathbb{P}}^1$, hence it has $p+1$ points. Since $N(\alpha)=0$ (when $z=0$) has at most two non-trivial solutions on the line at $\infty$, there are at least $p+1-3$ points of $N(\alpha)=-1$. Therefore, for any prime $p\geq 5$, $N(\alpha)=-1$ has at least three solutions over ${\mathbb{F}}_p$. On the other hand, the trace map $T$ is a linear form on the underlying 2-dimensional vector space. And the system of two equations, $T(\alpha)=0$ and $N(\alpha)=-1$ has at most two solutions. Since $N(\alpha)=-1$ has at least three solutions for any prime $p\geq 5$, we can find $\alpha\in R/pR$ such that $$T(\alpha)\not\equiv 0 \mbox{~~and ~~} N(\alpha)\equiv -1
~(\bmod~p).$$
Since the norm $N(\alpha)$ of $\alpha$ is congruent to $-1$ (mod $p$), we can lift $\alpha$ to a $p$-adic unit $\widetilde{\alpha}\in R_p^*$ by applying the Hensel’s lemma to the norm map.
Let $M\geq 5$ be large enough that for all $p\geq M$, $R/pR\cong\mathcal{O}/p\mathcal{O}$ and let $p\geq M$ be inert in $F$ and unramified in $K$.
Let $d_p$ denote the reduction of $det(\rho_p)$ mod $p$, and let $\beta_p=d_p^{\frac{p-1}{2}}$. Then, $Ker(\beta_p)$ is of index 2 in ${\operatorname{Gal}(\overline{{\mathbb{Q}}}/{\mathbb{Q}})}$ and therefore, corresponds to a quadratic extension $L$ which is ramified only at $p$. Thus, the fields, $F,K$ and $L$ are all linearly disjoint over ${\mathbb{Q}}$. In particular, $\sigma_p(\operatorname{Gal}(\overline{F}/FL))$ is of index 2 in $R_p^*$, which has a unique open subgroup of index 2, since $p$ is inert in $F$. As $FK\neq FL$, $K$ is not contained in the field associated to $Ker(\sigma_p)$, which implies $K$ is linearly disjoint from this field. By the Cebotarev density theorem, there exists $q$ which is inert in $K$ and split in $F$ and such that $$\sigma_p(\mbox{Frob}_q)=\widetilde{\alpha}\in R_p^*.$$ Then, we have $$a_q=T(\sigma_p(\mbox{Frob}_q))=T(\widetilde{\alpha})
\not\equiv 0 ~(\bmod ~p).$$ This implies that $p$ does not divide $a_q$, which is condition (2). Also, we have $$q= \mbox{det}(\sigma_p(\mbox{Frob}_q)) =
N(\widetilde{\alpha}) \equiv -1 ~(\bmod ~p).$$ This implies that $p$ divides $q+1$ which is condition (3). This completes the proof.
For a given elliptic curve $E/{\mathbb{Q}}$, we fix an element $\sigma\in{\operatorname{Gal}(\overline{{\mathbb{Q}}}/{\mathbb{Q}})}$ and a quadratic imaginary extension $K$ of ${\mathbb{Q}}$ which is not fixed under $\sigma$. And we assume that all primes dividing the conductor $N$ of $E/{\mathbb{Q}}$ are split in $K$. Note that this setting corresponds to the second case in the proof of Theorem \[thm:main\].
For each integer $n$ relatively prime to $N$, let $H_n$ denote the ring class field of $K$ of conductor $n$. By ([@dar], Chapter 3, Theorem 3.6), there is an algebraic point $P_n\in E(H_n)$ which is called a [*Heegner point of conductor $n$*]{}. Let $HP(n)\subset E(H_n)$ denote the set of all Heegner points of conductor $n$ in $E(H_n)$. Then, the set $HP(n)$ satisfies the following properties.
First, we recall the norm-compatibility properties of Heegner points.
\[prop:norm\] Let $E/{\mathbb{Q}}$ be a modular elliptic curve over ${\mathbb{Q}}$ and $N$ the conductor of $E/{\mathbb{Q}}$. Let $n$ be an integer and $\ell$ a prime number such that both $n$ and $\ell$ are prime to $N$. Let $P_{n\ell}$ be any point in $HP(n\ell)$ and $a_\ell$ the coefficient of the Hecke operator $T_{\ell}$ of the modular form of $E$. Then, there are points $P_{n}\in HP(n)$ and $($when $\ell|n)$ $P_{n/\ell}\in HP(n/\ell)$ such that $$\mbox{Trace}_{H_{nl}/H_n}(P_{nl})= \left\{\begin{array}
{r@{,\quad}l} a_lP_n & \mbox{if } l\nmid n \mbox{ is inert in } K
\\ (a_l-\sigma_{\lambda}-\sigma_{\lambda}^{-1})P_n &\mbox{if }
l=\lambda\bar{\lambda}\nmid n \mbox{ is split in } K
\\ (a_l-\sigma_{\lambda})P_n & \mbox{if } l=\lambda^2 \mbox{ is ramified in } K
\\ a_lP_n-P_{n/l} & \mbox {if }
l|n. \end{array} \right.$$
See ([@dar], Chapter 3, Proposition 3.10).
\[lem:finite\] Let $H_{\infty}$ be the union of all the ring class fields of conductor prime to $N$. Then, the set $E(H_{\infty})_{tor}$ is finite.
See ([@dar], Chapter 3, Lemma 3.14).
The following lemma describes the structure of the Galois groups of ring class field over an imaginary quadratic extension $K$ of ${\mathbb{Q}}$.
\[lem:degree\] Let $N$ be the conductor of $E/{\mathbb{Q}}$. Let $H_n$ be the ring class field of conductor $n$ over an imaginary quadratic extension $K$ over ${\mathbb{Q}}$.
\(a) If $p$ is a prime not dividing $c\cdot N\cdot [H_c:K]\cdot
disc(H_c)$, then for all integers $n\geq 1$, $$\operatorname{Gal}(H_{cp^{n}}/H_{cp})\cong {\mathbb{Z}}/p^{n-1}{\mathbb{Z}}, \mbox{ and } \operatorname{Gal}(H_{cp^{n+1}}/H_{cp^n})\cong {\mathbb{Z}}/p{\mathbb{Z}}.$$
\(b) If $k=\prod\limits_{j=1}^{m}p_j$ for distinct primes $p_j$ which are relatively prime to $N$ and inert in $K$, then for each $j$, $$\operatorname{Gal}(H_k/H_{\frac{k}{p_j}}) \cong {\mathbb{Z}}/(p_j+1){\mathbb{Z}}.$$
To prove the first part of (a), for $n\geq 1$ and $k\geq 1$,
$\operatorname{Gal}(H_{cp^n}/H_{cp}) \cong ker(\operatorname{Gal}(H_{cp^n}/K) \rightarrow
\operatorname{Gal}(H_{cp}/K))$
$\hspace{1.08in}\cong
ker((\mathcal{O}_K/cp^n\mathcal{O}_K)^*/({\mathbb{Z}}/cp^n{\mathbb{Z}})^*
\rightarrow (\mathcal{O}_K/cp\mathcal{O}_K)^*/({\mathbb{Z}}/cp{\mathbb{Z}})^*)$
$\hspace{1.08in}=
\displaystyle\frac{[(1+cp\mathcal{O}_K)/cp^n\mathcal{O}_K]^*\cdot({\mathbb{Z}}/cp^n{\mathbb{Z}})^*}{({\mathbb{Z}}/cp^n{\mathbb{Z}})^*}$
$\hspace{1.08in}=
\displaystyle\frac{[(1+cp\mathcal{O}_K)/cp^n\mathcal{O}_K]^*}{[(1+cp{\mathbb{Z}})/cp^n{\mathbb{Z}}]^*}$
$\hspace{1.08in}=
\displaystyle\frac{[(1+p\mathcal{O}_K)/p^n\mathcal{O}_K]^*}{[(1+p{\mathbb{Z}})/p^n{\mathbb{Z}}]^*},$ (since $p \nmid c\cdot[H_c:K]\cdot disc(H_c)$.)
$\hspace{1.08in}\cong
\displaystyle\frac{[(1+p\mathcal{O}_K)/p^n\mathcal{O}_K]^*}{({\mathbb{Z}}/p^{n-1}{\mathbb{Z}})},$
(since $[(1+p{\mathbb{Z}})/p^n{\mathbb{Z}}]^*$ $ \cong
ker(({\mathbb{Z}}/p^n{\mathbb{Z}})^* $ $ \rightarrow $ $ ({\mathbb{Z}}/p{\mathbb{Z}})^*)) $ $\cong
{\mathbb{Z}}/p^{n-1}{\mathbb{Z}}.)$
By the logarithmic function from $1+p\widehat{\mathcal{O}}_{K,p} \rightarrow
p\widehat{\mathcal{O}}_{K,p}$ which maps $1+p^n\widehat{\mathcal{O}}_{K,p} \mapsto
p^n\widehat{\mathcal{O}}_{K,p}$, where $\widehat{\mathcal{O}}_{K,p}$ is the completion of $\mathcal{O}_{K}$ at $p$, $$[(1+p\widehat{\mathcal{O}}_{K,p})/p^{n-1}\widehat{\mathcal{O}}_{K,p}]^*
\cong
\widehat{\mathcal{O}}_{K,p}/p^{n-1}\widehat{\mathcal{O}}_{K,p}
\cong \mathcal{O}_{K}/p^{n-1}\mathcal{O}_{K}.$$ Hence, we have $$\operatorname{Gal}(H_{cp^n}/H_{cp}) \cong (\mathcal{O}_{K}/p^{n-1}\mathcal{O}_{K})/({\mathbb{Z}}/p^{n-1}{\mathbb{Z}})\cong {\mathbb{Z}}/p^{n-1}{\mathbb{Z}}.$$
For the second part of (a),
$\operatorname{Gal}(H_{cp^{n+1}}/H_{cp^n})\cong ker( \operatorname{Gal}(H_{cp^{n+1}}/H_{cp})
\rightarrow \operatorname{Gal}(H_{cp^n}/H_{cp}))$
$\hspace{1.28in} \cong ker( ({\mathbb{Z}}/p^n{\mathbb{Z}})^* \rightarrow
({\mathbb{Z}}/p^{n-1}{\mathbb{Z}})^*)$
$\hspace{1.28in} \cong {\mathbb{Z}}/p{\mathbb{Z}}.$
To prove (b), for each $i=1,...,m$,
$\operatorname{Gal}(H_k/H_{\frac{k}{p_i}}) \cong ker(\operatorname{Gal}(H_k/K) \rightarrow
\operatorname{Gal}(H_{\frac{k}{p_i}}/K))$
$\hspace{.96in} \cong
ker((\mathcal{O}_{K}/k\mathcal{O}_{K})^*/({\mathbb{Z}}/k{\mathbb{Z}})^*
\rightarrow
(\mathcal{O}_{K}/\frac{k}{p_i}\mathcal{O}_{K})^*/({\mathbb{Z}}/\frac{k}{p_i}{\mathbb{Z}})^*)$
$\hspace{.96in} \cong ker\left(\prod\limits_{j=1}^{m}
\operatorname{Gal}(H_k/H_{\frac{k}{p_j}}) \rightarrow \prod\limits_{j=1}^{i-1}
\operatorname{Gal}(H_k/H_{\frac{k}{p_j}}) \times \{1\}\times
\prod\limits_{j=i+1}^{m} \operatorname{Gal}(H_k/H_{\frac{k}{p_j}}) \right)$
$\hspace{.96in} \cong \operatorname{Gal}(H_k/H_{\frac{k}{p_i}} ) \cong
({\mathbb{Z}}/\lambda{\mathbb{Z}})^*/({\mathbb{Z}}/p_i{\mathbb{Z}})^*$, where $\lambda$ is a prime factor of $p$.
$\hspace{.96in} \cong {\mathbb{Z}}/(p_i+1){\mathbb{Z}}$.
[*A Heegner system*]{} attached to $(E,K)$ is a collection of points $P_n\in E(H_n)$ indexed by integers $n$ prime to the conductor $N$ of $E$, and satisfying the norm compatibility properties given in ([@dar], Chapter 3, Proposition 3.10 including Proposition \[prop:norm\]) and the behavior under the action of reflections described in ([@dar], Chapter 3, Proposition 3.11).
In our setting, since all primes dividing $N$ are split in $K$ (recall the construction of $K$ in the proof of Theorem \[thm:main\]), there is a Heegner system in which at least one of the points $P_n$ for some $n$ is non-torsion by ([@dar], Chapter 3, Theorem 3.13). We call such a Heegner system a [*non-trivial Heegner system*]{}. We will need the following lemma.
\[lem:bdd\] Suppose there is a nontrivial Heegner system attached to $(E,K)$. Let $n$ be a positive integer prime to the conductor of $E$ such that there are non-torsion points of $E(H_{\infty})$ which are not in $E(H_n)$. Then, there exists a positive integer $M$ such that for all non-torsion points $Q\in E(H_{\infty})$ such that $Q\notin
E(H_n)$, $mQ\notin E(H_n)$ for all integers $m >M$.
Suppose not. Then, for all $M$, there exist an integer $m>M$ and a non-torsion point $Q\in E(H_{\infty})-E(H_n)$ such that $Q, 2Q, \ldots, (m-1)Q \notin E(H_n)$ but $mQ\in
E(H_n)$.
We may assume that $m=p^k$ for some prime $p$ dividing $m$ and for a positive integer $k$ by replacing $Q$ by $\frac{m}{p^k} Q$. Then, either the exponent $k$ or the prime $p$ must go to infinity as $M$ goes to infinity, since $m=p^k >M$.
Since $p^{k-1}<m$, $p^{k-1}Q\notin E(H_n)$. Hence there is an automorphism $\tau\in$ Gal$({H_\infty}/H_n)$ such that $\tau(p^{k-1}Q)\neq p^{k-1}Q$. Hence for the nontrivial point $\tau(p^{k-1}Q)- p^{k-1}Q$, $$p(\tau(p^{k-1}Q)- p^{k-1}Q)=\tau(p^k Q)- p^kQ=p^k Q-p^k Q=O.$$ Since $p$ is a prime, this implies that the point $\tau(p^{k-1}Q)-
p^{k-1}Q$ has order exactly $p$, hence the nontrivial point $\tau(Q)- Q$ has order exactly $m=p^k$. So we have shown that there are torsion points of order exactly $m$ for all integers $m
> M$. Therefore, as $M$ goes to infinity, the order of torsion points in $E(H_{\infty})$ is unbounded, which is a contradiction to the finiteness of the set of torsion points of $E(H_{\infty})$ in Lemma \[lem:finite\]. This completes the proof.
Now, by using Lemma \[lem:prime1\] and the norm-compatibility properties in Proposition \[prop:norm\], we prove that the subgroup generated by all of Heegner points of the given nontrivial Heegner system is not finitely generated as a subgroup of the elliptic curve over the union of all the ring class fields of conductor of the form $rm$ for some $r$, where $m$ is a square-free integer relatively prime to $rN$. Then, by using Lemma \[lem:bdd\], we show the following unboundedness of the rank of Mordell-Weil groups over all of those ring class fields.
\[prop:inf\] Let $N$ be the conductor of the given elliptic curve $E/{\mathbb{Q}}$. If there is a nontrivial Heegner system attached to $(E,K)$, where $K$ is a quadratic imaginary extension of ${\mathbb{Q}}$ which is different from the field $\operatorname{End}(E)\otimes {\mathbb{Q}}$ such that there is a non-torsion Heegner point $P_{r}\in E(H_{r})$ over some ring class field $H_{r}$ of conductor $r$, then the rank of $E(H_{rm})$ is unbounded, as a square-free integer $m$ such that $(m, rN)=1$ goes to infinity.
First, we show that the group generated by all of Heegner points of conductor $rm$ for all square-free integers $m$ such that $(m,rN)=1$ in the given nontrivial Heegner system is not finitely generated. Suppose this group is finitely generated. Then, since there are only finitely many torsion (Heegner) points over all the ring class fields in the system by Lemma \[lem:finite\], for some integer $n$ which is a square-free multiple of $r$ and $(n/r, rN)=1$, there is a fixed ring class field $H_{n}$ over which all Heegner points of conductor $rm$ for all square-free integers $m$ are defined. And we may assume that a Heegner point $P_{n}$ of level $n$ is of infinite order by the assumption that there exists a non-torsion Heegner point $P_{r}\in
E(H_r)$.
Only finitely many primes divide either $P_{n}$ in $E(H_{n})$ or points of $E(H_{\infty})_{tor}$, because $E(H_{n})$ is finitely generated by the Mordell-Weil Theorem and there are only finitely many torsion points over all the ring class fields in the system by Lemma \[lem:finite\]. Let $S$ be the finite set of primes which divide either $P_{n}$ in $E(H_{n})$ or the order of any point of the finite set $E(H_{\infty})_{tor}$. Then, we can choose a large odd prime $p\notin S$ such that $p\geq n+2$ and if $E$ has no CM, then $p$ is unramified in $K$ and if $E$ has CM, then $p$ is inert in the imaginary quadratic extension, $\operatorname{End}(E)\otimes {\mathbb{Q}}$ of ${\mathbb{Q}}$.
Then by Lemma \[lem:prime1\], there is a prime $q$ such that
1. $q$ is inert in $K$,
2. $p$ divides $q+1$ and
3. $p$ does not divide $a_q$.
And since $q+1\geq p$ by (2) and $p\geq n+2$, $q$ is strictly greater than $n$. This implies $q$ does not divide $n$. Therefore, the ring class field $H_{nq}$ is a proper finite extension of $H_{n}$ and $nq$ is again a square-free multiple of $r$ and $(nq/r, rN)=1$.
By the norm-compatibility property given in Proposition \[prop:norm\], when $q\nmid n$ is inert in $K$, we have $$\mbox{Trace}_{H_{nq}/H_{n}}(P_{nq})=a_qP_{n}.$$
On the other hand, $E(H_{nq})=E(H_{n})$, by assumption. Hence the trace of $P_{nq}$ from $H_{nq}$ to $H_{n}$ is divisible by the degree of $H_{nq}$ over $H_{n}$ which is $q+1$ by (b) of Lemma \[lem:degree\].
Hence, $\mbox{Trace}_{H_{nq}/H_{n}}(P_{nq})$ is divisible by $p$ by the property (3). But by the property (4) and since $p\notin
S$, $p$ divides neither $a_q$ nor the point $P_{n}$, which is a contradiction. So we have shown that the group generated by the Heegner points of conductor $rm$ for all square-free integers $m$ is not finitely generated. In particular, this shows that there is a non-torsion point $P_{nq}\in E(H_{nq})$ but not in $E(H_n)$.
By Lemma \[lem:bdd\], for such a non-torsion point $P_{nq}\notin
E(H_n)$, there exists an integer $M$ such that $mP_{nq}\notin
E(H_n)$ for all $m > M$. In other words, the point $P_{nq}$ is independent of any points in $E(H_n)$. Hence, $E(H_n)\otimes {\mathbb{Q}}\neq E(H_{nq})\otimes {\mathbb{Q}}$. Therefore, we conclude that the rank of $E(H_{rm})$ cannot be bounded, as a square-free integer $m$ such that $(m,N)=1$ goes to infinity.
Now we prove the unboundedness of the rank of Mordell-Weil groups over all the ring class fields of conductor $cp^n$ as $n$ goes to infinity and this will be an important role in proving the main theorem. To prove this, we need the following simple lemma.
\[lem:seq\] For an elliptic curve $E/{\mathbb{Q}}$ and for a prime $p$ not dividing the conductor of $E$, let $a_p=p+1-\#E({\mathbb{F}}_p)$. Then, there is no infinite sequence $\{c_n\}_{n=0}^{\infty}$ of integers with $c_0\neq 0$ and satisfying the following linear recurrence, $$pc_{n+1}=a_pc_n-c_{n-1}, \mbox{ for } n\geq 1,$$ and for every $N$, there exists $n >N $ such that $c_n\neq 0$.
Suppose there is such an infinite sequence $\{c_n\}_{n=0}^{\infty}$ of integers satisfying the above conditions. Then, the linear recurrence implies that $$(*) \mbox{\hspace{5 cm}} c_n =\alpha^nb_0
+\beta^nb_1, \mbox{ for
all } n\geq 1,\mbox{\hspace{6 cm}}$$ where $\alpha$ and $\beta$ are two solutions of the quadratic equation $x^2-\frac{a_p}{p}x+\frac{1}{p}=0$ and $$b_0=\left(\frac{-\beta}{\alpha-\beta}c_0
\frac{1}{\alpha-\beta}c_1\right) \mbox{ and }
b_1=\left(\frac{\alpha}{\alpha-\beta}c_0
-\frac{1}{\alpha-\beta}c_1\right).$$ Note that this quadratic equation has no rational solutions, because if there was, the only possible pairs of rational solutions are $1$ and $\frac{1}{p}$ or $-1$ and $-\frac{1}{p}$ and in either case, we get $a_p=\pm (p+1)$ which is impossible since $|a_p|<2\sqrt{p}$ by Hesse’s inequality ([@sil1], Chapter V, Theorem 1.1). And $b_0\neq 0$ and $b_1\neq 0$ since $c_0$ is a nonzero integer and $\alpha$ and $\beta$ are not rational numbers.
Let $F$ be a quadratic extension of ${\mathbb{Q}}$ containing $\alpha$ and $\beta$ and choose an embedding of $F$ into $\overline {\mathbb{Q}}_p$ with the valuation $v_p$ such that $v_p(\alpha)<0$ and $v_p(\beta)=0$, where $\overline{\mathbb{Q}}_p$ is an algebraic closure of the $p$-adic field ${\mathbb{Q}}_p$. This is possible because $\alpha\beta=\frac{1}{p}$ and we can take an automorphism of $F$ and take an embedding of $F$ into $\overline{\mathbb{Q}}_p$ such that $v_p(\alpha)<0$ and $v_p(\beta)=0$.
Since $b_0\neq 0$ and $v_p(\beta)=0$, the recurrence relation $(*)$ implies that for all large integers $n$, $v_p(c_n)$ is dominated by $v_p(\alpha^{n-1})$ which is negative. But for each $N$, there exists $n>N$ such that $c_n\neq 0$. Since a nonzero integer $c_n$ has a nonnegative valuation, we get a contradiction. Hence, there is no such a sequence.
\[prop:inf2\] Let $N$ be the conductor of the given elliptic curve $E/{\mathbb{Q}}$. If there is a nontrivial Heegner system attached to $(E,K)$, where $K$ is a quadratic imaginary extension of ${\mathbb{Q}}$ which is different from the field $\operatorname{End}(E)\otimes {\mathbb{Q}}$, then for a prime $p$ such that $p\nmid r\cdot N \cdot [H_r:K]\cdot
disc(H_r)$, there exist an integer $r$ such that the rank of $E(H_{rp^n})$ is unbounded, as $n$ goes to infinity.
Let $p$ be a prime $p\nmid r\cdot N \cdot [H_r:K]\cdot
disc(H_r)$. As we have shown in Proposition \[prop:inf\], we show that for some integer $r$, the group generated by all of Heegner points of conductor $rp^n$ for all integer $n\geq 1$ in the given nontrivial Heegner system is not finitely generated. Suppose for any integer $r$, this group is finitely generated. Then, since there are only finitely many torsion (Heegner) points over all the ring class fields in the system by Lemma \[lem:finite\], for some integer $k$, there is a fixed ring class field $H_{rp^k}$ over which all Heegner points of conductor $rp^n$ for all $n \geq 1$ are defined. Let $n_0=rp^k$. Since the given Heegner system is nontrivial, we may assume that a Heegner point $P_0$ of conductor $n_0=rp^k$ is of infinite order.
By the norm-compatibility property given in Proposition \[prop:norm\], we have that for all $n\geq 1$, $$\mbox{Trace}_{H_{n_0p^{n+1}}/H_{n_0p^n}}(P_{n+1})=a_pP_{n}-P_{n-1},$$ where $P_i$ are Heegner points of conductor $n_0p^i$.
Since $P_{n+1}$ is defined over $H_{n_0}$, hence over $H_{n_0p^n}$ by assumption, the trace of $P_{n+1}$ from $H_{n_0p^{n+1}}$ to $H_{n_0p^n}$ is the degree of $H_{n_0p^{n+1}}$ over $H_{n_0p^n}$ which is $p$ by (a) of Lemma \[lem:degree\]. Hence, the infinite sequence of Heegner points of conductor of the form $n_0p^n$ satisfies the linear recurrence relation, $$(**)
\mbox{ \hspace{4 cm} } pP_{n+1}=a_pP_n-P_{n-1}, \mbox{ for all }
n\geq 1.\mbox{ \hspace{6 cm} }$$
By the Mordell-Weil Theorem, $E(H_{n_0})$ is finitely generated, so by dividing by its torsion subgroup $E(H_{n_0})_{tor}$, all points $P_n \mod
E(H_{n_0})_{tor}$ lie in ${\mathbb{Z}}^k$ for some $k$. Suppose that $E(H_{n_0})\cong
{\mathbb{Z}}Q_1+\cdots +{\mathbb{Z}}Q_k+ E(H_{n_0})_{tor}$. Now we consider all points $P_n \mod
E(H_{n_0})_{tor}$ and denote it by $P_n$ again by abuse of natation. Let $P_n=\sum\limits_{i=1}^k c_{n,i}Q_i$ for integers $c_{n,i}$. Since $P_0$ is not a torsion point by the assumption, without loss of generality we may assume that $c_{0,1}\neq 0$. Then, either $c_{2,1}$ or $c_{3,1}$ is nonzero, since otherwise, the relation $(**)$ implies that $Q_1$ is a linear combination of points $Q_2, Q_3, \ldots,Q_k$ over ${\mathbb{Z}}$ which contradicts the linear independence of points $Q_i$. By using $(**)$ and linear dependence of points $Q_i$ inductively, we can show that $pc_{n+1}=a_pc_n-c_{n-1}$ for all $n\geq 1$, and if $c_{n,1}\neq 0$, then either $c_{n+1,1}\neq 0$ or $c_{n+2,1}\neq
0$. Hence, by letting $c_n=c_{n,1}$, we get an infinite sequence $\{c_n\}_{n=0}^{\infty}$ of integers satisfying the $pc_{n+1}=a_pc_n-c_{n-1}$ for all $n\geq 1$ with $c_0\neq 0$ and for each $N$, there exists $n>N$ such that $c_n\neq 0$. This is impossible by Lemma \[lem:seq\]. Therefore, we conclude that the rank of $E(H_{rp^n})$ is not finitely generated, as $n \rightarrow
\infty$. In particular, by Lemma \[lem:bdd\], we conclude that the rank of $E(H_{rp^n})$ cannot be bounded, as $n$ goes to infinity.
Finally, the following proposition proves Theorem \[thm:ab\] hence, completes Theorem \[thm:main\].
\[prop:sigma\] Let $\sigma\in {\operatorname{Gal}(\overline{{\mathbb{Q}}}/{\mathbb{Q}})}$. Let $K$ be a quadratic imaginary extension of ${\mathbb{Q}}$ such that $\sigma|_K\neq id_K$ and $K$ is different from $\operatorname{End}(E)\otimes {\mathbb{Q}}$. Suppose all primes dividing the conductor $N$ of $E$ are split in $K$. Then, the rank of the Mordell-Weil group $E(H_n^{\sigma})$ over the fixed subfield of $H_n$ under $\sigma$ is unbounded, as $n$ goes to $\infty$. Hence, the rank of $E((K_{ab})^{\sigma})$ is infinite, where $K_{ab}$ is the maximal abelian extension of $K$.
Since all primes dividing the conductor $N$ of $E$ are split in $K$, there is a nontrivial Heegner system attached to $(E,K)$ by ([@dar], Chapter 3, Theorem 3.13). For a given $\sigma\in {\operatorname{Gal}(\overline{{\mathbb{Q}}}/{\mathbb{Q}})}$, since $\sigma|_K\neq id_K$, the restriction of $\sigma$ to each ring class field $H_n$ in the given Heegner system can be lifted as an involution of $H_n$. Let $\sigma_n=\sigma|_{H_n}$ be the restriction of $\sigma$ to $H_n$. Then, each ring class field $H_n$ has a generalized dihedral group structure as its Galois group over ${\mathbb{Q}}$ with an involution $\sigma_n$ such that for any $\tau\in \operatorname{Gal}(H_n/K)$, $\sigma_n\tau\sigma_n=\tau^{-1}$.
By Proposition \[prop:inf2\], we fix a prime $p$ and an integer $r$ such that $p\nmid r\cdot N \cdot [H_r:K]\cdot disc(H_r)$ and the rank of $E(H_{rp^n})$ is unbounded, as $n$ goes to infinity. We prove that for an odd prime $p$ not dividing $rN[H_r:K]disc(H_r)$, the rank of $E(H_{rp^n}^{\sigma})$ is unbounded as $n$ goes to infinity. Suppose not. Then since the restriction $\sigma_n$ of $\sigma$ to $H_{rp^n}$ acts by an involution of each ring class field $H_{rp^n}$, there exists a fixed integer $n_0=rp^k$ for some $k\geq 1$ such that $\sigma$ acts by $-1$ on any nontrivial quotient $(E(H_{n_0p^n})\otimes {\mathbb{Q}})/(E(H_{n_0})\otimes{\mathbb{Q}})$, for all $n\geq 1$.
By (a) of Lemma \[lem:degree\], $\operatorname{Gal}(H_{n_0p^n}/H_{rp}) =
\operatorname{Gal}(H_{rp^{k+n}}/H_{rp})$ is a cyclic group of order $p^{k+n-1}$. And $\operatorname{Gal}(H_{n_0p^n}/H_{n_0})$ is a subgroup of $\operatorname{Gal}(H_{n_0p^n}/H_{rp})$. Hence, it is a cyclic subgroup of order $p^m$ for some $m < k+n-1$.
Let $\tau_n$ be a generator of $\operatorname{Gal}(H_{n_0p^n}/H_{n_0})$. Consider $E(H_{n_0p^n})\otimes {\mathbb{Q}}$ as a representation of $\operatorname{Gal}(H_{n_0p^n}/{\mathbb{Q}})$. And for each n, let $$M_n=(E(H_{n_0p^n})\otimes {\mathbb{Q}})/(E(H_{n_0})\otimes{\mathbb{Q}}).$$
For every element $\alpha\in \operatorname{Gal}(H_{n_0p^n}/{\mathbb{Q}})$, let $\alpha|_{H_{n_0}}$ be the restriction of $\alpha$ to $H_{n_0}$. Then, $\alpha|_{H_{n_0}}$ is an element of $\operatorname{Gal}(H_{n_0}/K)$, since $H_{n_0}$ is Galois over ${\mathbb{Q}}$. Therefore, $\operatorname{Gal}(H_{n_0p^n}/{\mathbb{Q}})$ acts on $E(H_{n_0})\otimes {\mathbb{Q}}$ as well. So we can consider the quotient $M_n$ as a representation of $\operatorname{Gal}(H_{n_0p^n}/{\mathbb{Q}})$.
Let $$\rho : \operatorname{Gal}(H_{n_0p^n}/{\mathbb{Q}}) \rightarrow \mbox{GL}(M_n)$$ be the representation of $\operatorname{Gal}(H_{n_0p^n}/{\mathbb{Q}})$. Then, by the hypothesis, $\sigma_n$ acts by $-1$ on $M_n$. Hence, $\rho(\sigma_n)=-id$ on $M_n$. On the other hand, by the dihedral group structure of $\operatorname{Gal}(H_{n_0p^n}/{\mathbb{Q}})$, $\sigma_n\tau_n\sigma_n=\tau_n^{-1}$. Therefore, $$\rho(\tau_n^2)=\rho(\tau_n)\rho(\tau_n)=(-id)\rho(\tau_n)(-id)\rho(\tau_n)=\rho(\sigma_n\tau_n\sigma_n\tau_n)=\rho(1)=id.$$
Hence, the restriction of $\rho$ to the cyclic subgroup $\operatorname{Gal}(H_{n_0p^n}/H_{n_0})$ of $Gal(H_{n_0p^n}/{\mathbb{Q}})$ generated by $\tau_n^2$ is a trivial representation of $M_n$. Since the order of $\tau_n$ is an odd integer $p^m$, $$\langle \tau_n^2 \rangle
=\langle \tau_n \rangle = \operatorname{Gal}(H_{n_0p^n}/H_{n_0}).$$ Therefore, we have $$M_n^{\operatorname{Gal}(H_{n_0p^n}/H_{n_0})} = M_n^{\langle \tau_n^2 \rangle}=M_n.$$ This implies that $$E(H_{n_0p^n}^{\operatorname{Gal}(H_{n_0p^n}/H_{n_0})})\otimes{\mathbb{Q}}+E(H_{n_0})\otimes{\mathbb{Q}}=E(H_{n_0p^n})\otimes{\mathbb{Q}}.$$ Since $H_{n_0p^n}^{\operatorname{Gal}(H_{n_0p^n}/H_{n_0})}=H_{n_0}$, $$E(H_{n_0})\otimes{\mathbb{Q}}=E(H_{n_0p^n}) \otimes{\mathbb{Q}}, ~~~\mbox{~~for all }
n \geq 1,$$ which is a contradiction to Proposition \[prop:inf2\].
Hence, the rank of $E(H_{n_0p^n}^{\sigma})$ is unbounded, as $n\rightarrow\infty$. Since all ring class fields $H_{n_0p^n}$ are abelian over $K$, this implies that the rank of $E((K_{ab})^{\sigma})$ is infinite.
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H. Darmon, *Rational points on modular elliptic curves*, CBMS Regional Conference Series in Mathematics, 101. Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 2004, (or see *http://www.math.mcgill.ca/darmon/pub/pub.html*).
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M. Larsen, *Rank of elliptic curves over almost algebraically closed fields*, *Bull. London Math. Soc.* [**35**]{} (2003), 817–820.
J.-P. Serre, *A course in Arithmetic*, Springer-Verlag, GTM [**7**]{}, 1973.
J.-P. Serre, *Propriétés Galoisiennes des points d’ordre fini des courbes elliptiques*, *Invent. math.* [**15**]{} (1972), 259-331.
J. H. Silverman, *Integer points on curves of genus $1$*, *J. London Math. Soc. [**(2), 28**]{} (1983)*, 1-7.
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|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We study the one-dimensional quarter-filled extended Hubbard model with an alternating transfer integral. In the strong-dimerization limit the charge part is described by the quantum Ising model which shows the two-dimensional Ising criticality at the self-dual point, and it is naturally connected to the double-frequency sine-Gordon theory in the weak dimerization. Treating low-lying excitations in finite-size systems, we numerically determine a phase boundary between two types of $4k_{\rm F}$ density-wave states and clarify the ground-state phase diagram. Further, we refer to its relevances to the charge-ordered phase observed in the charge-transfer organic salts.'
address: |
$^1$Department of Physics, Tokyo Metropolitan University, Tokyo 192-0397 Japan\
$^2$Department of Applied Physics, Faculty of Science, Tokyo University of Science, Tokyo 162-8601 Japan
author:
- 'Hiromi Otsuka$^{1}$ and Masaaki Nakamura$^{2}$'
title: |
Quarter-filled extended Hubbard model with alternating transfer integral:\
Two-dimensional Ising transition in the ground state
---
The organic materials described by the chemical formula (TMTSF)$_2$X (X=PF$_6$, ClO$_4$, etc) and (TMTTF)$_2$X (X=PF$_6$, AsF$_6$, etc) form a class of the quasi one-dimensional (1D) conductors; a large number of investigations on these materials have been accumulated in the literature.[@Bour99] While the various types of electronic phases, e.g., spin/charge-density-wave (SDW/CDW), the spin-Peierls and the superconducting states have been observed in the low-temperature region, newly discovered charge-ordered (CO) phase in (TMTTF)$_2$X exhibiting an anomaly in the low-frequency dielectric constant [@Nad_00] and the charge disproportionation in the NMR studies [@Chow00] has received intensive current interest. Although the stabilizations of these phases at finite temperature resort to interchain couplings, it is believed that the intrachain interaction effects play a leading role to describe them.
For the study of the CO phase, the 1D quarter-filled extended Hubbard model (EHM) with an alternating transfer integral has been employed:[@Mila95; @Seo_97; @Mazu99; @Tsuc01; @Tsuc02; @Nish00; @Shib01] $H=H_1+H_2$ with $$\begin{aligned}
&&H_1=
\sum_{j,s}
-t\left[1-\delta(-1)^j\right]
\left(
c^\dagger_{j,s}c^{}_{j+1,s}+{\rm H.c.}
\right),\\
&&H_2=
\sum_j \left(U n_{j,\uparrow} n_{j,\downarrow}+ V n_{j} n_{j+1}\right),
\end{aligned}$$ where $c^{}_{j,s}$ annihilates an $s$-spin electron ($s=\uparrow$ or $\downarrow$) on the $j$th site and satisfies the periodic boundary condition $c^{}_{L+1,s}=c^{}_{1,s}$ ($j\in[1,L]$; $L$ is an even number). The number operators are defined as $n_{j,s}=c^\dagger_{j,s}c^{}_{j,s}$ and $n_{j}=n_{j,\uparrow}+n_{j,\downarrow}$. The parameters $U$ and $V$ taking positive values stand for the onsite and nearest-neighbor Coulomb repulsion. The dimerization parameter $\delta$ shows the alternation in the transfer integral of the molecular chains (we set $t=1$ in the following). For the theoretical descriptions of the 1D electrons, the Tomonaga-Luttinger liquid (TLL) picture has been widely adopted.[@Hald81] Since TLL consists of the massless charge and spin parts both controlled by the Gaussian fixed point \[the conformal field theory (CFT) with the central charge $c=1$\], it is important to understand its instabilities. In particular, the CO transition may be related to the crossover of the criticality embedded in the renormalization group (RG) flow,[@Tsuc02] which is one of the typical instability of the $c=$1 CFT.
In this paper, we present the numerical calculation results of the ground-state phase diagrams of $H$. Our method being deeply connected to the instability will be explained briefly. Further, in the strong-dimerization limit ($\delta=1$), we show that the charge part can be described by the so-called quantum Ising chain, which is complementary to the bosonization argument and gives us an exact limiting condition of the phase boundary line. Since the region with sufficiently large Coulomb repulsions is relevant to the CO transition, an occurrence of the phase separation or the transition to the superconducting phases is outside of our research scope.
Let us start with the description of the low-energy physics in the weak-coupling region, where the bosonization method provides a reliable approach, i.e., linearizing the dispersion at Fermi points $\pm k_{\rm F}=\pm\pi
n/2a$ (electron density $n=N/L=1/2$) and applying the method, we can obtain an effective Hamiltonian. For the present case, according to the recent research results,[@Bour99; @Tsuc01; @Yosh00] we can use the following expression: $H\rightarrow{\cal H}={\cal H}_\rho+{\cal H}_\sigma$ with $$\begin{aligned}
{\cal H}_\rho
\!\!&=&\!\!
\int {d}x \frac{v_\rho}{2\pi}
\left[
{ K_\rho}\left(\partial_x \theta_\rho\right)^2+
{1\over K_\rho}\left(\partial_x \phi_\rho\right)^2
\right]\nonumber\\
\!\!&+&\!\!
\int {d}x \frac{2}{(2\pi\alpha)^2}\!
\left(
-g_\rho{\sin\sqrt8\phi_\rho}+g_{1/4}{\cos2\sqrt{8}\phi_\rho}
\right),
\label{eq_chag}
\end{aligned}$$ where the operator $\theta_\rho$ is the dual field of $\phi_\rho$ satisfying the commutation relation $\left[\phi_\rho(x),\partial_y\theta_{\rho}(y)/\pi\right]={i}\delta(x-y)$ and parameters $K_\rho$ and $v_\rho$ are the Gaussian coupling and the velocity of the charge excitation, respectively.[@GNT] A benefit to use the bosonized expression is now clear, i.e., since the spin-charge separation occurs in ${\cal H}$ and ${\cal H}_\sigma$ is the SU(2) critical Gaussian model in the present case,[@Tsuc01; @Tsuc02] we can concentrate on the charge part ${\cal
H}_\rho$ which takes a form of the so-called double-frequency sine-Gordon (DSG) model. In uniform case ($\delta=0$), the $8k_{\rm F}$-Umklapp scattering with $g_{1/4}\propto U^2(U-4V)$ (Refs. ) brings about the Berezinskii-Kosterlitz-Thouless (BKT) transition, and then the charge part becomes massive for large values of the Coulomb interactions. For the BKT transition point, values in the strong coupling limit are known as $V^*(U\to\infty)=2$ and $U^*(V\to\infty)=4$.[@Ovch73; @Lin-G-C-F-G] Further the estimations for the intermediate region are available.[@Yosh00; @Mila93; @NakaEX] In the case of nonzero dimerization ($\delta\ne0$), the scaling dimension of the “half-filled Umklapp scattering” term with $g_{\rho}\propto U\delta[1-A(U-2V)]$ ($A$ is a constant) [@Tsuc01; @Giam97] on the Gaussian fixed point is small ($x_{\rm
4B}=2K_\rho$) enough to bring about the second-order phase transition for $V\le V^*(U)$, which is accompanied by the divergent correlation length of the form $\xi\propto
\delta^{-1/(2-2K_\rho)}$.[@Bour99; @Penc94] For $V>V^*(U)$, since the charge gap may survive in a weak-dimerization region, the transition point $\delta_\rho(U,V)$ takes nonzero values depending on $U$ and $V$, and more importantly, the universality of the transition is changed. Recently, Tsuchiizu and Orignac,[@Tsuc02] on the basis of the DSG theory,[@Delf98] argued that the charge part on $\delta_\rho(U,V)$ \[$V>V^*(U)$\] is renormalized to the 2D-Ising fixed point with $c=\frac12$ (i.e., the fixed point with lower symmetry), which is in accord with Zamolodchikov’s $c$-theorem [@Zamo86] (see also Refs. and ). Then, the critical line corresponds to the phase boundary and satisfies a condition $\delta_\rho(U,V\to V^*(U))\searrow 0$ in the weak-dimerization region. To characterize the phases, we shall use the CDW and the bond-order-wave (BOW) order parameters with the $4k_{\rm F}$ wave vector:[@Yosh00] $${\cal O}_{\rm 4C} \propto \cos\sqrt8\phi_\rho,~~~
{\cal O}_{\rm 4B} \propto \sin\sqrt8\phi_\rho.$$ Here, note that the expectation value of the $4k_{\rm F}$-BOW order parameter is finite, $\langle{\cal O}_{\rm 4B}\rangle\ne0$ and $\langle{\cal O}_{\rm 4C}\rangle =0$ in the upper region of the boundary, but both of these are finite in the lower region ($\delta\ne0$). While this “mixed” state is basically the $4k_{\rm F}$-CDW phase, we shall use the double quotation marks “$4k_{\rm F}$-CDW” to express this situation.[@Fabr00]
On the other hand, another condition of the boundary can be found in the strong-dimerization limit ($\delta=1$). To derive an effective Hamiltonian, it is convenient to work with the orbital operators defined by $ d^{}_{m,\pm,s}
\equiv \left(c^{}_{2m-1,s}\pm c^{}_{2m,s}\right)/{\sqrt2} $, where $d^{}_{m,l,s}$ annihilates an $s$-spin electron in the $l$-orbital ($l=\pm$) on the $m$th unit cell ($m\in[1,L/2]$). In this limit, $H_1$ consists of a sum of the intracell electron hopping, which is diagonalized by using the operators as $H_1=\sum_{m,l,s}-2ld^{\dagger}_{m,l,s}d^{}_{m,l,s}$. For sufficiently large $U$ and $V$, since the one-electron states $|l,s\rangle_m=d^\dagger_{m,l,s}|0\rangle$ have a principal role to describe the $m$th unit cell in the quarter-filled ground state, and the Hamiltonian does not change the electron number in each cell, we shall introduce the pseudospin operators, $${{\mbox{\boldmath$T$}}}_m
\equiv
\sum_{l,l',s}
\frac12
d^\dagger_{m,l,s }
\left[\mbox{\boldmath$\tau$}\right]^{}_{l,l'}
d^{ }_{m,l',s},$$ acting on the orbital space as, for instance, $T^3_m |\pm,s\rangle_m =
\pm\frac12 |\pm,s\rangle_m$ \[$\mbox{\boldmath$\tau$}=(\tau^1,\tau^2,\tau^3)$; $\tau^i$ is the Pauli matrix\]. Using these, $H_1=\sum_{m}-4T^3_m$. For $H_2$, since the intracell Coulomb interactions are absent and the intercell Coulomb repulsion only remains in the restricted Hilbert space spanned by the direct product of one-particle states $\{\otimes_m|l,s\rangle_m\}$, a straightforward calculation brings about the expression $H_2=\sum_{m} \left(-V T^1_mT^1_{m+1}+{\rm const}\right)$. Now, since the Hamiltonian acts only on the orbital space, its eigenstate takes a form of the direct product of vectors in the spin and the orbital spaces as $|\Phi\rangle=|{\rm spin}\rangle\otimes|{\rm
orbital}\rangle$. Thus, assuming a certain spin configuration belonging to the 2$^{L/2}$-dimensional space for spins and restricting ourselves to the orbital (or charge) part, we see that the Hamiltonian $H$ with $\delta=1$ is reduced to the quantum Ising chain [@Pfeu70] $$\begin{aligned}
H_{\rho,\delta=1}=\sum_m
\left(-\Gamma T^3_m-JT^1_mT^1_{m+1}\right)
\label{Eq_qim}
\end{aligned}$$ ($\Gamma=4$, $J=V$). Note that this possibility was mentioned qualitatively in Ref. . Then, the ground state of Eq. (\[Eq\_qim\]) is known to show the 2D-Ising criticality at its self-dual point $\Gamma=J/2$ ($V=8)$, which separates ordered ($\langle T^1_m\rangle\ne0$) and disordered ($\langle T^1_m\rangle =0$) phases. The ordered state is realized via the breaking of the Z$_2$ symmetry $(\tau^1\!\to-\tau^1)$, and it is doubly degenerated, e.g., $$\begin{aligned}
|\pm\tau^1\rangle
=
\prod_m\frac{1}{\sqrt2}(d^{\dag}_{m,+}\pm d^{\dag}_{m,-})|0\rangle
=
\prod_m c^{\dagger}_{2m-1} (c^{\dagger}_{2m})|0\rangle
\label{4C}
\end{aligned}$$ (we dropped the spin index). This expresses the 4$k_{\rm F}$-CDW state with the perfect microscopic polarization, $
\langle\pm\tau^1|T^1_m|\pm\tau^1\rangle
=
\langle\pm\tau^1|\frac12(n_{2m-1}-n_{2m})|\pm\tau^1\rangle
=
\pm\frac12
$. On one hand, a disordered state is supported by the external field in $\tau^3$-direction, and an ideal one is given by $$\begin{aligned}
|+\tau^3\rangle
=
\prod_m d^{\dag}_{m,+}|0\rangle
=\prod_m\frac{1}{\sqrt2}(c^{\dagger}_{2m-1}+c^{\dagger}_{2m})|0\rangle,
\label{4B}
\end{aligned}$$ which expresses the 4$k_{\rm F}$-BOW state as expected. Here it is worthy of noticing that these states can be distinguished by the expectation value of the twist operator[@Naka02] $$z_\rho\equiv\Bigl\langle{\rm exp}
\Bigl(\frac{4\pi i}{L}\sum_j jn_j\Bigr)\Bigr\rangle.$$ This quantity takes values $z_\rho=1$ for Eq. (\[4C\]) and $z_\rho=-[\cos(2\pi/L)]^{L/2}$ for Eq. (\[4B\]), so the sign of $z_\rho$ characterizes these two density-wave states (see below). Consequently, in the strong-dimerization limit, the orbital degrees of freedom show the 2D-Ising type transition between the “4$k_{\rm
F}$-CDW” and the 4$k_{\rm F}$-BOW phases at $V=8$, where $U$ is irrelevant. Since this pseudospin representation is naturally connected to the bosonization picture in the weak couplings,[@Tsuc02] the phase boundary belongs to the 2D-Ising universality and satisfies the limiting condition $\delta_\rho(U,V\to 8)\nearrow 1$, which provides a solid guide to investigations in the strong-dimerization region.
Here, note that the qualitative estimation of the phase boundary might be possible in the weak- and strong-dimerization region.[@Tsuc01] To evaluate the entire phase diagram precisely, however, a numerical treatment of the 1D electron model is required. For this issue, recently the present authors have numerically treated the same instability observed in the quantum-spin chain and interacting electron systems.[@Otsu02] Therefore, we shall employ the same approach to the present system (see also Ref. ). Since there are two critical fixed points connected by the RG flow, a relationship between lower-energy excitations on these fixed points — the ultraviolet-infrared (UV-IR) operator correspondence — has essential significance in the investigations.[@Tsuc02; @Fabr00] To see this, let us rescale phase fields and the Gaussian coupling as $2\phi_\rho\to\phi$, $\theta_\rho/2\to\theta$, and $4K_\rho\to K\simeq 1$, which makes it possible to directly adopt our previous research.[@Otsu02] With respect to $\phi$, the nonlinear potential density is given as $-g_\rho\sin\sqrt2\phi+g_{1/4}\cos\sqrt{8}\phi$, and the order parameters as ${\cal O}_{\rm 4C}\propto\cos\sqrt2\phi$ $(x_{\rm 4C}=K/2)$ and ${\cal O}_{\rm 4B}\propto\sin\sqrt2\phi$ $(x_{\rm 4B}=K/2)$. Along the RG flow these operators on the Gaussian fixed point (UV) are transmuted to those on the 2D-Ising fixed point (IR) as $${\cal O}_{\rm 4C} \to \mu,~~~
{\cal O}_{\rm 4B} \to I+\epsilon,$$ where $\mu$ is the disorder field (Z$_2$ odd), and $\epsilon$ is the energy density operator (Z$_2$ even) with scaling dimensions $x_\mu=\frac18$ and $x_\epsilon=1$, respectively. Since the dimerization $\delta$ couples with ${\cal O}_{\rm 4B}$ in the Hamiltonian (\[eq\_chag\]), a deviation from the transition point $\delta-\delta_\rho(U,V)$ plays a role of the “thermal scaling variable” and brings about $\xi\propto [\delta-\delta_\rho(U,V)]^{-\nu}$ with $1/\nu=2-x_\epsilon=1$. On one hand, the operator $\mu$ corresponding to ${\cal O}_{\rm 4C}$ provides a most divergent fluctuation.
Now, we shall explain our numerical procedure to determine the transition point. We shall focus our attention on the level $\Delta E$ in finite-size systems which corresponds to the operator ${\cal O}_{\rm 4C}$ (taking the ground-state energy as zero). According to the finite-size-scaling argument based on CFT, $\Delta
E\simeq 2\pi x_{\rm 4C}/L$ on the UV fixed point;[@Card84] we can numerically obtain the level by using discrete symmetries of the lattice Hamiltonian in the diagonalization calculations. Various excitations observed in TLL are characterized by a set of quantum numbers for symmetry operations. With respect to ${\cal O}_{\rm 4C}$, it can be found in the subspace of the total spin $S^z_{\rm T}=0$ and the space inversion $P=-1$ (the boundary condition is the same as that for the ground state).[@NakaEX] Suppose that $\Delta E(U,V,\delta,L)$ is a level corresponding to ${\cal O}_{\rm 4C}$ in the $L$-site system. Then, we numerically solve the phenomenological renormalization-group (PRG) equation $(L+2)\Delta E(U,V,\delta,L+2)=L\Delta E(U,V,\delta,L)$ with respect to $\delta$ for given values of $U$ and $V$, where the gap behaves as $\Delta E\propto 1/L$ \[i.e., an $L$-dependent transition point $\delta_\rho(U,V,L+1)$ (see Fig. \[FIG1\])\].[@Otsu02] After evaluating $\delta_\rho(U,V,L+1)$, we extrapolate them to the limit $L\to\infty$ using the formula $\delta_\rho(U,V,L)=\delta_\rho(U,V)+a L^{-3},$[@Itzy89] where $\delta_\rho(U,V)$ and $a$ are determined by the least-square-fitting condition.
![ The ground-state phase diagram of the quarter-filled EHM with an alternating transfer integral. The correspondence between marks and system sizes is given in the figure. The double circles show the limiting values, i.e., $(V^*(U),0)$ at which criticality changes from the Gaussian to the 2D-Ising type, and $(8,1)$ the self-dual point. []{data-label="FIG1"}](fig1.eps){width="3.29in"}
![ The $\delta$ dependence of $z_\rho(L)$ at $U=16$ and $V=4$. The vertical dotted line indicates the transition point (i.e., the PRG result) $\delta_\rho(16,4)\simeq 0.12$. Inset plots $L$ dependences of $\delta'_\rho(U,V,L)$ (crosses) and the PRG data $\delta_\rho(U,V,L)$ (circles) with the fitting line. []{data-label="FIG2"}](fig2.eps){width="3.29in"}
From the data of $L=12$-20, we obtain the phase boundary $\delta_\rho(U,V)$ as shown in Fig. \[FIG1\]. We can check that, for all values of $U$ used here, the phase boundary lines converge to the point $(V,\delta)=(8,1)$ with the 2D-Ising criticality. On the other hand, while the finite-size corrections to the boundary may be large in weak-dimerization region, the boundaries also show convergences to the BKT-transition points $(V^*(U),0)$. Next we demonstrate the $\delta$ dependence of $z_\rho(L)$ in Fig. \[FIG2\]. With the increase of $\delta$, $z_\rho(L)$ decreases and becomes negative \[we denote the zero point of $z_\rho(L)$ as $\delta'_\rho(U,V,L)$\]; this corresponds to the change of the center of mass as demonstrated in the above.[@Naka02] However, unlike, for instance, the Gaussian transition, $z_\rho(\infty)$ can take a finite value on the Ising transition point, so $\delta'_\rho(U,V,L)$ may not give an estimation of the transition point. In fact, the inset of Fig. \[FIG2\] exhibits that $\delta'_\rho(U,V,L)$ may be extrapolated to a value different from the PRG result. On the other hand, Fig. \[FIG2\] also shows that there is a point $\delta\simeq 0.12$ at which $z_\rho(L)$ is almost independent of $L$. This crossing point is expected to be a good estimator for the Ising transition point because this is quite close to the PRG result even for small $L$. However, this issue remains as a future problem.
Lastly, we shall refer to some implications of our study to the real materials. Besides the quantum-chemistry calculations,[@Duca86] the numerical estimations of the model parameters have been performed based upon the experimental data.[@Mila95; @Nish00] For example, the realistic values of the dimerization parameter and the onsite Coulomb repulsion of (TMTTF)$_2$PF$_6$ have been estimated as $t_2/t_1\simeq 0.7$, $U/t_1\simeq 7.0$ ($t_{1,2}=1\pm \delta$), but the value of $V$ is still controversial (an uncertainty exists also in the value of $t_2/t_1$[@Nad_00; @Frit91]). Our numerical estimation of the transition point using these values is $V_{\rm c}/t_1\simeq$ 4.0, while generally the mean-field-type calculations tend to predict somewhat smaller values due to an overestimation of $V$ effects.[@Seo_97; @Shib01] On the other hand, several values have been reported for this material, e.g., $V/t_1\simeq$ 2.8 (1.4) in Ref. (Ref. ), which is much smaller than the critical value, and thus predicts a uniform charge distribution (this conclusion may not be changed even in smaller dimerization cases). However, (TMTTF)$_2$PF$_6$ has the CO phase in the region above the lower-temperature spin-Peierls phase, and further it was theoretically suggested that a huge anomaly in the dielectric constant may reflect a nature of systems in the critical region.[@Tsuc02] This discrepancy may be attributed to many other interaction effects not included in the Hamiltonian. However, we think that since experimental findings seem to support the spin-charge separation with respect to the CO transition,[@Nad_00; @Chow00] the 1D electron models are to provide a primary description of real materials.
To summarize, we investigated the ground-state phase diagram of the 1D quarter-filled extended Hubbard model with alternating transfer integral. Especially, the criticality on the phase boundary and the implication to the CO transition observed in the charge-transfer organic salts were mainly argued.
One of the authors (H.O.) would like to thank K. Mizoguchi and Y. Okabe for stimulating discussions. M.N. is partly supported by the Ministry of Education, Culture, Sports, Science and Technology of Japan through Grants-in-Aid No. 14740241. Main computations were performed using the facilities of Yukawa Institute for Theoretical Physics, and the Supercomputer Center, Institute for Solid State Physics, University of Tokyo.
For review, C. Bourbonnais and D. Jérome, in [*Advances in Synthetic Metals, Twenty Years of Progress in Science and Technology*]{}, edited by P. Bernier, S. Lefrant, and G. Bidan (Elsevier, New York, 1999).
F. Nad, [[*et al.*]{}]{}, .
D.S. Chow, [[*et al.*]{}]{}, .
F. Mila, .
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S. Mazumdar, [[*et al.*]{}]{}, .
M. Tsuchiizu, H. Yoshioka, and Y. Suzumura, .
M. Tsuchiizu and E. Orignac, .
S. Nishimoto, M. Takahashi, and Y. Ohta, .
Y. Shibata, S. Nishimoto, and Y. Ohta, .
F.D.M. Haldane, .
H. Yoshioka, M. Tsuchiizu, and Y. Suzumura, .
For a review, A.O. Gogolin, A.A. Nersesyan, and A.M. Tsvelik, [*Bosonization and Strongly Correlated Systems*]{} (Cambridge U.P., Cambridge, 1998).
H. Yoshioka, M. Tsuchiizu, and Y. Suzumura, , and references therein.
T. Giamarchi, , and the references therein.
M. Nakamura, .
A.A. Ovchinnikov, \[\].
H.Q. Lin, E.R. Gagliano, D.K. Campbell, E.H. Fradkin, and J.E. Gubernatis, in [*The Hubbard Model*]{}, edited by D. Baeriswyl, D.K. Campbell, J.M.P. Carmelo, F. Guinea, and E. Louis (Plenum P., New York, 1995), p. 315.
F. Mila and X. Zotos, ; K. Sano and Y. $\rm \bar O$no, ; K. Penc and F. Mila, .
K. Penc and F. Mila, .
G. Delfino and G. Mussardo, .
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M. Fabrizio, A.O. Gogolin, and A.A. Nersesyan, .
M. Fabrizio, A.O. Gogolin, and A.A. Nersesyan, .
For example, P. Pfeuty, .
R. Resta and S. Sorella, ; A. Aligia and G. Ortiz, ; M. Nakamura and J. Voit, .
H. Otsuka, ; H. Otsuka and M. Nakamura, condmat/0403630 (unpublished).
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|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We consider a $C^\infty$ Anosov diffeomorphism $T$ with a $C^\infty$ stable dynamical foliation. We show upper bounds on the essential spectral radius of its transfer operator acting on anisotropic Sobolev spaces. (Such bounds are related to the essential decorrelation rate for the SRB measure.) We compare our results to the estimates of Kitaev on the domain of holomorphy of dynamical determinants for differentiable dynamics.'
address: 'CNRS-UMR 7586, Institut de Mathématiques Jussieu, Paris, France'
author:
- Viviane Baladi
date: December 2004
title: 'Anisotropic Sobolev spaces and dynamical transfer operators: $C^\infty$ foliations'
---
[^1]
Introduction
============
Let $T$ be an Anosov diffeomorphism on a $d$-dimensional compact connected $C^\infty$ Riemann manifold $\XX$ (i.e., $T\XX =E^u \oplus E^s$ and there are $C>0$, $\gamma >1$, with $|DT ^n|E^s|\le C \gamma^{-n}$, $|DT ^{-n}|E^u|\le C \gamma^{-n}$ for all $n\ge 1$). Denote the Jacobian of $T$ with respect to Lebesgue by $|\det DT|$. To construct SRB measures and to analyse their speed of mixing, it is natural to consider the following operators, defined initially on $C^\infty$ functions: $$\label{xfer}
\MM \varphi=\frac{ \varphi\circ T^{-1}}{|\det DT|\circ T^{-1}}\, ,
\qquad
\LL \varphi = \varphi \circ T \, .$$ The operator $\LL$ fixes the constant functions, while $\MM$ fixes the constant functions if and only if $\det DT$ is constant (i.e., if $T$ is volume preserving). The dual of $\MM$ restricted to elements of the dual of $\CC^\infty$ which are finite complex measures, absolutely continuous with respect to Lebesgue with a $\CC^\infty$ density, coincides with $\LL$ viewed as acting on the density and vice-versa. Alternatively, the dual of $\MM$ acting on $L^1(\XX, \Leb)$ is $\LL$ acting on $L^\infty(\XX, \Leb)$.
For $w \in \XX$, and $\widetilde T$ an Anosov diffeomorphism on $\XX$, introduce local hyperbolicity exponents ($|\cdot|$ denotes euclidean norm) $$\nonumber
\begin{split}
\lambda_{w}(\widetilde T)^{-1}
&=\sup_{v\in E^u(\widetilde T(w)), |v|=1} |D_{T(w)} \widetilde T^{-1} (v)|\, ,\cr
\nu_{w}(\widetilde T)&=\sup_{v\in E^s(w), |v|=1} |D_w\widetilde T (v)|\, .
\end{split}$$ Assume $T$ is $C^{r+1}$ for some $r>0$. Kitaev [@Ki] proved that the following “dynamical Fredholm determinant” $$\nonumber
\nonumber
d(z)=\exp -\sum_{n=1}^\infty
\frac {z^n} {n} \sum_{T^n(x)=x} \frac {1}{ |\det (DT^n(x)-\Id)]}$$ extends to a holomorphic function in each disc $\{z\mid |z| \cdot \rho^{(p,s)}_1(T)<1\}$, where $p\in (-r,0)$, $s=r+p$, and $$\nonumber
\rho^{(p,s)}_1(T)=\lim_{n\to \infty}
\biggl ( \int_\XX
\max\bigl ( (\lambda_{w} (T^n))^{p}, (\nu_{w} (T^n))^{s}
\bigr ) \, dLeb(w) \biggr )^{1/n} <1 \, .$$ One may take $s=-p=r/2$: Kitaev’s result is then reminiscent of the “loss of one half of the Hölder exponent” which occurs when going from two-sided subshifts to one-sided subshifts in symbolic dynamics [@Bow], since one easily sees that $\rho^{(-r/2,r/2)}_1(T)\le \gamma^{-r/2}$.
In view of the results of Ruelle [@Ru] for smooth expanding maps, it is natural to look for Banach spaces $\BB_{p,s,\LL}$, respectively $\BB_{p,s,\MM}$, on which the essential spectral radius of $\LL$, respectively $\MM$, is $\le \rho^{(p,s)}_1$. Set $$\nonumber
\rho^{(p,s)}_\infty(T)=\lim_{n\to \infty}
\biggl ( \sup_\XX
\max\bigl ( (\lambda_{w} (T^n))^{p}, (\nu_{w} (T^n))^{s}
\bigr ) \biggr )^{1/n} <1 \, .$$ Clearly $\rho^{(p,s)}_\infty(T)\ge \rho^{(p,s)}_1(T)$ and, e.g., $\rho^{(-r/2,r/2)}_\infty(T)\le \gamma^{-r/2}$.
We shall assume that $T$ is $C^\infty$ and the stable foliation of $T$ (or its unstable foliation) is $C^\infty$. (This is a very strong assumption, and the corresponding case should be viewed as a “toy model" in which the features of our symbolic calculus approach are completely transparent: The heart of the proof is contained in a half page, between (\[heart2\]) and (\[heartn\]) below.) We introduce in Subsection \[space\], for $p$, $s$ in $\real$ and $1<t< \infty$, a Banach space $W^{p,s-p,t}(\XX)=W^{p,s-p,t}(\XX, T)$ of distributions, based on $L^t(\Leb)$. [^2]
Our main result (Theorem \[mainthm\]) when the stable foliation is $C^\infty$ is that, if $T$ is volume preserving, the essential spectral radius $\rho_{ess}$ of $\LL$ on $W^{p,s-p,t}(\XX)$ is at most $\rho^{(p,s)}_\infty(T)$ for all $p <0$, $s >0$ and $t \in (1, \infty)$; while if $T$ does not preserve volume thn $\limsup_{t\to \infty}\rho_{ess}(\LL|_{W^{p,s-p,t}(\XX)})
\le \rho^{(p,s)}_\infty(T)$. If the unstable foliation is $C^\infty$, the essential spectral radius of $\MM$ on $W^{p,s-p,t}(\XX, T^{-1})$ is at most $\lim_{n \to \infty} \sup_\XX |\det DT^n|^{-(t-1)/tn} \cdot \rho^{(-s,-p)}_\infty(T)$ for all $p <0$, $s>0$ and $t\in (1, \infty)$ (Theorem \[mainthm2\]).
Propositions \[L1\] and \[L1,2\] give upper bounds related to $\rho^{(p,s)}_1(T)$: They imply $$\nonumber
\begin{split}
\limsup_{t\to \infty}&\rho_{ess} (\LL|_{W^{p,s-p,t}})\le
\lim_{n\to \infty} \| \det DT^n|_{E^u} \|_{L^\infty(\Leb)}^{1/n} \rho^{(p,s)}_1(T)\, ,\cr
\limsup_{t\to 1}
&\rho_{ess} (\LL|_{W^{p,s-p,t}})\le \lim_{n\to \infty} \| (\det DT^n|_{E^s})^{-1} \|_{L^\infty(\Leb)}^{1/n} \rho^{(p,s)}_1(T)
\, .
\end{split}$$
Finally, we study in the appendix the essential spectral radii of $$\LL_t \varphi =|\det DT|^{1/t} \cdot (\varphi \circ T) \, ,
\qquad \MM_t\varphi =\frac {\varphi \circ T^{-1} } {|\det DT|^{1-1/t} \circ T^{-1}}\, .$$
The case when $T$ is $C^{r+1}$ (for some $r>0$) and neither of the dynamical foliations is $C^\infty$, but at least one of them is $C^{1+\epsilon}$ (for $\epsilon>0$) will be treated in a forthcoming work [@Ba], using spaces due to Alinhac [@A2]. We hope that the (general) $C^\alpha$ foliation case will be amenable to the present approach. Gouëzel and Liverani [@GL] have independently obtained non trivial, but weaker, bounds for the essential spectral radius of $\MM$, on a different Banach space, in this general case.
We end this introduction with three open problems:
With the techniques of Blank–Keller–Liverani [@BKL], one should obtain that the spectral radius of $\LL$ on each $W^{p,q,t}(\XX)$ is one, that $1$ is is a semi-simple eigenvalue, and that the corresponding eigenvector (in the dual of $W^{p,q,t}(\XX)$) for the dual of $\LL$ is an invariant probability measure $\mu$ with ergodic basin of full Lebesgue measure. Furthermore, the multiplicity of the eigenvalue $1$ is equal to the number of ergodic components of $\mu$, and each ergodic component is an SRB measure. Also, if $1$ is a simple eigenvalue then it is the only eigenvalue on the unit circle: this corresponds to exponential decay of correlations for smooth observables. If the unstable foliation is $C^\infty$, the SRB measure(s) of $T$ can alternatively be constructed with the fixed point of $\MM$ in $W^{p,q,t}(\XX, T^{-1})$. [^3]
The perturbation techniques of [@BKL], should imply stability of the spectrum of $\LL$ (including eigenprojectors) outside a disc of radius $\rho$, under stochastic and $C^{r+1}$ deterministic perturbations of $T$, perhaps up to taking $\rho >\rho^{(p,s)}_\infty$. For deterministic perturbations $\widetilde T$ of $T$, the Banach spaces $W^{p,q,t}(\XX,T)$ and $\widetilde W^{p,q,t}=W^{p,q,t}(\XX,\widetilde T)$ are different. “Stability of the eigenprojector” $\Pi$ of $T$ associated to an eigenvalue $\tau$ of large enough modulus means the following (assume $\tau$ is simple): Let $\widetilde \LL$ denote the transfer operator of $\widetilde T$; then, if $\widetilde T$ is close enough to $T$, there are a Banach space $ W_\epsilon$ contained in the intersection of $\widetilde W^{p,q,t}$ and $W^{p,q,t}$, a rank-one projector $ \Pi_\epsilon$ (on $W_\epsilon$, $\widetilde W^{p,q,t}$, and $W^{p,q,t}$), and a simple eigenvalue $(\tilde \tau,\widetilde \Pi)$ for $\widetilde \LL$ on $\widetilde W^{p,q,t}$, so that both $\|\Pi_\epsilon-\Pi\|_W$ and $\| \Pi_\epsilon-\widetilde \Pi\|_{\widetilde W}$ are small.
For $C^{r+1}$ expanding circle endomorphisms $F$, the essential spectral radius $\rho_{ess}(\MM_F|_{C^r})$ of $\MM_F \varphi(x)=\sum_{F(y)=x} \varphi(y)/|\det DF(y)|$ acting on $C^r$ functions (see [@CE] and references therein) is equal to $$\lim_{n\to \infty} \bigl (\int |(F^n)'(x)|^{-r} \, dLeb(x) \bigr )^{1/n}=
\lim_{n\to \infty} \bigl (\int |(F^n)'(x)|^{-r} \, d\mu_{SRB}(x)\bigr )^{1/n} \, .$$ However, for $C^{r+1}$ expanding maps in arbitrary dimension [@GLa] $$\label{GuL}
\begin{split}
\rho_{ess}(\MM_F|_{C^r})=&\exp (\sup_{\mu}
\{ h_\mu - \int \log |\det DF | \, d\mu - r \cdot \chi_\mu\}) \cr
&\qquad \qquad\le
\lim_{n\to \infty} \bigl (\int \sup_{|v|=1} |D_x(F^n)(v)|^{-r} \, dLeb(x)\bigr )^{1/n} \, ,
\end{split}$$ where $\mu$ ranges over ergodic $F$-invariant probability measures, $h_\mu$ is the entropy of $\mu$, and $\chi_\mu$ denotes the smallest (positive) Lyapunov exponent of $DF$. The inequality in (\[GuL\]) can be strict. In the other direction, note that $\rho_{ess}(\MM_F|_{C^r})\ge \exp(-r \chi_{\mu_{SRB}})$, and the inequality can be strict [@CE], even in dimension one. The results of Avila et al. [@AGT], indicate that in dimension at least two there may be Banach spaces containing all $C^r$ functions on which the essential spectral radius of $\MM_F$ is strictly smaller than $\rho_{ess}(\MM_F|_{C^r})$. (This would imply [@CE] that $\rho_{ess}(\MM_F|_{C^r})$ may be strictly larger than the essential decorrelation radius of $F$ for $C^r$ observables and thus $\rho_{point-ess}(\MM_F|_{C^r})<\rho_{ess}(\MM_F|_{C^r})$.)
Let $T$ be a transitive $C^{\infty}$ Anosov diffeomorphism with both foliations $C^\infty$. Let $\rho_{ess}^+(p,s,t)$ and $\rho_{ess}^-(p,s,t)$ be the essential spectral radii of $\LL$ acting on $W^{p,s-p,t}(\XX)$ and $W^{-p,-s+p,t}(\XX, T^{-1})$, respectively, and set $$\rho(r):=
\min\bigl (\inf_{\stackrel{t, p \in(-r,0)}{ s\in (0,r+p)}} \rho^+_{ess} (p,s,t),
\inf_{\stackrel{t, p \in(-r,0)} { s\in (0,r+p) }} \rho^-_{ess} (p,s,t) \bigr )
\, .$$ We expect that $\inf_{\BB_r} \rho_{ess}(\LL|_{\BB_r})$, where $\BB_r$ spans all Banach spaces of distributions of order $\le r$, containing all $C^r$ functions, and on which $\LL$ acts boundedly, coincides with the essential decorrelation radius $\hat \rho(r)$ of $T$ for $C^r$ functions, and that $\hat \rho(r) < \rho(r)$ can occur.
Bounding the essential spectral radius
======================================
Preliminaries {#prel}
-------------
From now on and until the end of Subsection \[proofLY\], $T$ is Anosov and $C^\infty$, with a $C^\infty$ stable foliation $\FF^s$. Write $I=(-1,1)$, and let $d_s$ be the dimension of $\FF^s$. We work with $C^\infty$ foliated charts $\kappa$, $V$: let $\cup_{i \in \II} V_i$ be a finite covering of $\XX$ by small open sets, and let $U_i=I^d=I^{d_s}\times I^{d-d_s}$ be $\#\II$ disjoint copies of $I^d$, viewed as subsets of disjoint copies $\real^d_i$ of $\real^d$. Let $\kappa_i : V_i \to U_i$ be $C^\infty$ diffeomorphisms so that $\kappa_i^{-1}$ of each horizontal segment is the intersection of a leaf of $\FF^s$ with $V_i$. In addition, we require that $\kappa_i^{-1}(\{(0,y)\})$ is the unstable leaf of $\kappa_i^{-1}(0,0)$ intersected with $V_i$ (this is a way to require closeness of the vertical foliation in $I^d$ and the image of leaves of the unstable foliation $\FF^u$).
Choose a $C^\infty$ partition of the unity $\{\psi_i\}$ on $\XX$, compatible with $V=\{V_i\}$, i.e., each $\psi_i$ is supported in $V_i$. Then, for each $n\ge 1$ $$\label{maintransfer}
\LL ^n\varphi (w)=\sum_{i, j} \psi_j(T^n (w)){\psi}_{i}(w)\cdot \varphi (T^n(w)) \, .$$
If $V_{ij}=V_{ij,n}:=T^{-n}(V_j)\cap V_i \ne \emptyset$, setting $U_{ij,n}:=\kappa_i(T^{-n}(V_j)\cap V_i)\subset U_i$, the map $T^n_{ij}:U_{ij,n}\to U_j$ has a derivative in block form: $$\nonumber
\left ( \begin{array}{cc}
A^{tr}_{ij,n}(x,y) & B^{tr}_{ij,n}(x,y) \\
0 & D^{tr}_{ij,n}(x,y) \\
\end{array} \right ) \, , \, (x,y)\in (I^{d_s}, I^{d-d_s}) \, ,$$ with $A_{ij,n}$ a $d_s\times d_s$ matrix, $D_{ij,n}$ a $(d-d_s)\times (d-d_s)$ matrix, and $$\label{rem0}
\begin{split}
&| A_{ij,n}(x,y)|\le\nu_{ij}(T^n) := \sup_{w\in V_{ij}} \nu_w(T^n)<1\, ,\cr
& | D_{ij,n}(x,y) ^{-1}|\le\lambda_{ij}(T^n)^{-1}:=
\sup_{w\in V_{ij}} (\lambda_{T^n(w)}(T^n) )^{-1}<1\ . \cr
\end{split}$$ Furthermore, for each $\epsilon$, there exists $\delta$ so that if ${\rm diam}\, V <\delta$ then $$\label{rem}
|B_{ij, n}(x,y) v |\le \epsilon | D_{ij, n}(x,y) v | \, , \forall n\ge 1\, , \,
\forall v \in \real^{d-d_s} \, .$$
Elementary spaces $W^{p,q,t}(\real^d)$
--------------------------------------
Let $p$ and $q$ be real numbers. We introduce the “symbol” $a_{p,q}(\xi,\eta)$, for $(\xi, \eta) \in \real^{d_s}\times \real^{d-d_s}$: $$a_{p,q}(\xi,\eta)=
(1+|\xi|^2 +|\eta|^2)^{p/2} (1+|\xi|^2 )^ {q /2}\, .$$ The corresponding linear operator $a^{Op}_{p,q}$ maps the space $\SS$ of rapidly decaying $C^\infty$ functions on $\real^d$ into itself via $$a^{Op}_{p,q} (f) (x,y)
=(2\pi)^ {-d}\int \int e^{ix\xi} e^{iy\eta} a_{p,q}(\xi,\eta) \hat f(\xi,\eta)\, d\xi \, d\eta \, ,$$ where the Fourier transform of $f$ is $
\hat f(\xi,\eta)=\int \int e^{-ix\xi} e^{-iy\eta} f(x,y) \, dx \, dy
$.
For $1\le t\le\infty$, let $W^{p,q,t}(\real^d)$ be the closure of $\{ f \in \SS(\real^d)\}$ for the $L^t(\real^d, \Leb)$ norm of $a^{Op}_{p,q} (f)$, with induced norm, denoted $\|\cdot\|_{p,q,t,\real^d}$.
By construction, $a^{Op}_{p,q}$ extends to a bounded invertible operator from $W^{p,q,t}(\real^d)$ to $L^t(\real^d)$. Clearly, $H^{p,q}(\real^d)=W^{p,q,2}(\real^d)$ is a Hilbert space.
\[compact\] Assume that $1<t<\infty$. Denote by $W^{p',q',t}_C(\real^d)$ those $f\in W^{p',q',t}(\real^d)$ supported in a compact subset of $\real^d$. If $q' \ge q$ and $p'\ge p$ then the natural injection $W^{p',q',t}_C(\real^d) \subset W^{p,q,t}(\real^d)$ is bounded. This injection is compact if $q' \ge q$ and $p'>p $ .
If $t=2$, the proofs of Theorems 2.5.2 and 2.5.3 in [@H0] adapt readily. The general case is an easy exercise.
More generally, we may introduce classes of (symbols) of pseudodifferential operators: Let $p$ and $q$ be real numbers. We say that $b\in C^{\infty}(I^d \times \real^d, \real)$, belongs to $S^{p, q}$ if for any multi-indices[^4] $\alpha=(\alpha',\alpha'')$ and $\beta=(\beta',\beta'')$ in $\integer^{d_s+(d-d_s)}_+$, there exists $C_{\alpha, \beta}$ so that $$\begin{split}
\sup \left |
\partial^{\alpha'}_\xi \partial^{\alpha''}_\eta \partial^{\beta'}_x
\partial^{\beta''} _y b(x,y,\xi,\eta)\right |
&\le C_{\alpha, \beta} (1+|\xi|+|\eta|)^{p-|\alpha''|} (1+|\xi|)^{q-|\alpha'|} \, .
\end{split}$$ The spaces $S^{p,q}$ and $H^{p,q}$ were studied by Kordyukov [@Ko]. The 1963 edition of Hörmander’s book [@H0 II.2.5] contains a treatment of a special case of the spaces $S^{p,q}$ when $d_s=1$. See also Sablé-Tougeron [@ST] for applications of these special cases.
Banach spaces $W^{p,q,t}(\XX)$ and Leibniz formula {#space}
--------------------------------------------------
Let $\kappa$, $V$ be a chart and $\psi$ be a compatible partition of unity as in Subsection \[prel\].
\[defnorm\] Let $p$, $q$ be real numbers, and let $1\le t\le \infty$. $W^{p,q,t}(\XX, \kappa,V,\psi)$ is $\{ \varphi \in \DD'(\XX)
\mid (\psi_i \cdot \varphi)\circ \kappa_i^{-1} \in W^{p,q,t}(\real^d_i)\, ,
\forall i \in \II \}$, normed by $$\|\varphi\|_{p,q,t}=\sum_{i\in \II} \|(\psi_i \cdot \varphi)\circ \kappa_i^{-1}\|_{p,q,t,\real^d_i} \, .$$
If $1<t< \infty$, the Banach spaces $W^{p,q,t}(\XX,\kappa,V,\psi)$ are independent of the charts $(\kappa,V)$ and of the partition of unity $\psi$: A version of the change of variables theorem for pseudodifferential operators, see e.g. [@AG I.7.1], shows that the norms corresponding to different $(\kappa, V, \psi)$ are equivalent. (See Lemmas \[Leibniz\] and \[comp\] below.) We may thus write $W^{p,q,t}(\XX)$. $H^{p,q}(\XX)=W^{p,q,2}(\XX)$ is a Hilbert space.
$W^{p,q,t}(\XX)$ is the Banach space of distributions $f$ on $\XX$ so that $(1+\Delta_s)^{q/2} (1+\Delta)^{p/2} f
\in L^t(\XX)$, with the induced $L^t(\XX)$ norm, where $\Delta$ is the Laplacian and $\Delta_s$ is the stable foliated Laplacian. In particular, if $p\le 0$ and $0\le q \le r$, it contains all $C^r$ functions.
We start with a useful remark:
\[proper\] For every compact subsets $K_1$, $K_2$ of $I^d$, with $K_1$ included in the interior of $K_2$, and for every $1<t<\infty$, $p$, $q$, there are $C>0$ and a $C^\infty$ function $\Psi_K:\real^d\to [0,1]$, supported in $K_2$, so that for each $f \in W^{p,q,t}(\real^d)$ supported in $K_1$, $$\|\Psi_K \cdot a_{p,q}^{Op}(f)-a_{p,q}^{Op}(f)\|_{L^t} \le C
\| f\|_{p-1,q,t} \, .$$
Using that the kernel of a pseudo-differential operator is $C^\infty$ outside of the diagonal, a standard construction allows to write $\Psi\cdot a-a$ (acting on compactly supported distributions) as an operator with a $C^\infty$ kernel (see e.g. [@AG Prop 6.3]). Integrate by parts to conclude.
\[Leibniz\] Let $1<t<\infty$, let $p$, $q$ be real numbers and let $h$ be a compactly supported $C^\infty$ function on $I^d$. Then there is $C(h)>0$, and there exists $C_x(h)$, depending only on $$\label{opM}
\sup_{|\beta'|\in \{1, 2\}, (x,y)\in I^d} |\partial^{\beta'}_x h(x,y)|\, ,$$ so that for every $f \in W^{p,q,t}(\real^d)$ $$\label{Leibnizz}
a^{Op}_{p,q}( h \cdot f) =h \cdot a^{Op}_{p,q}( f)+
g_1 + g_2 \, ,$$ with $\|g_1\|_{L^t} \le C_x(h) \| f\|_{p,q-1,t}$ and $\|g_2\|_{L^t} \le C(h) \| f\|_{p-1,q,t}$.
Multiplication by $h$ is a pseudodifferential operator. Composing it with $a^{Op}_{p,q}$, we get a new operator $b^{Op}$. Using a Taylor series of order one (see e.g. [@AG Théorème I.4.1 and §I.8.2]), we find $b (x,y,\xi,\eta)=$ $$\nonumber
\begin{split}
& a_{p,q}(\xi,\eta) \cdot h(x,y)+ \frac {2}{ (2\pi)^{d} }\sum_{|\alpha|+|\beta|=2}
\frac {(-1)^{|\alpha|+|\beta|} }{ \alpha ! \beta !} \int_0^1 (1-s) \cdot \cr
&\qquad\, \cdot \int e^{-i (u,v)(\omega,\theta)}
\omega^{\beta'} \theta^{\beta''} u^{\alpha'} v^{\alpha''}
\cdot \partial^{\beta'}_\omega \partial^{\beta''}_\theta
a_{p,q}(\xi-s\omega, \eta-s\theta) \, d\omega\, d\theta\cr
&\qquad\qquad\qquad \qquad \qquad\qquad \cdot
\partial^{\alpha'}_u\partial^{\alpha''}_v h(x-s u , y-s v)
\, du \, dv \, ds \, .\cr
\end{split}$$ The symbol $a_{p,q}(\xi,\eta) \cdot h(x,y)$ gives rise to the first term in the right-hand-side of (\[Leibnizz\]). For the remainder term, the usual integrations by parts [@AG §I.8.2, p.56] yield a linear combination of terms $$\nonumber
\begin{split}
b ^{\gamma, j}(x,y,\xi,\eta)&:= \int_0^1 (1-s) s^j \cdot \int e^{-i (u,v)(\omega,\theta)} \cdot
\partial^{\gamma'}_\omega\partial^{\gamma''}_\theta
a_{p,q}(\xi-s\omega, \eta-s\theta) \cr
&\qquad \qquad\quad \cdot
\partial^{\gamma'}_u \partial^{\gamma''}_v h(x-s u , y-s v)
\, d\omega d\theta \, du dv \, ds \, ,\cr
\end{split}$$ where $j\in \{0,1,2\}$ and $|\gamma'|+|\gamma''|\in\{ 1,2\}$ (the number of terms and the coefficients in the linear combination are independent of $h$ and $a_{p,q}$). If $|\gamma''|=0$ then $|\gamma'|\in\{1,2\}$, and this gives $g_1$, as we explain next. Define a symbol $\tilde b=b ^{\gamma, j} (a_{p,q-1})^{-1}$. By [@CM Théorème 9] it suffices to show that there is $C_x(h)$ so that $\sup | \partial^{ \alpha}_{x,y}
\partial^{\beta}_{\xi,\eta} \tilde b(x,y,\xi,\eta)|\le C_x(h)(1+|\xi|+|\eta|)^{-|\beta|}$, for all $| \alpha|\le 1$ and all $ \beta$. This can be shown by a straightforward (although tedious) implementation of the standard oscillatory integral argument [@AG §I.8.2, p.56]. Finally, if $|\gamma''|\ge 1$ then the term corresponding to $b ^{\gamma, j}$ may be included in $g_2$, working with $b ^{\gamma, j} ( a_{p-1,q})^{-1}$.
Bounding the essential spectral radius of $\LL$
-----------------------------------------------
\[mainthm\] Let $T$ be a $C^\infty$ Anosov diffeomorphism on a compact manifold, with a $C^\infty$ stable foliation. For any $p<0$, $s>0$, and $t\in(1,\infty)$, the essential spectral radius of $\LL$ on $W^{p,s-p,t}(\XX)$ is not larger than $\lim_{n \to \infty} \sup_\XX |\det DT^n|^{-1/tn} \cdot \rho^{(p,s)}_\infty(T)$.
Note that the essential spectral radius of the dual of $\LL$ i.e. (an extension of) $\MM$ acting on the dual of $W^{p,s-p,t}(\XX)$ coincides with the essential spectral radius of $\LL$ on $W^{p,s-p,t}(\XX)$. Also, if the unstable foliation is $C^\infty$, then $\varphi \mapsto \varphi \circ T^{-1}$ on $W^{p,s-p,t}(\XX,T^{-1})$ has essential spectral radius $\le
\lim_{n \to \infty} \sup_\XX |\det DT^{-n}|^{-1/tn} \cdot\rho^{(p,s)}_\infty(T^{-1})$ (note that $\rho^{(p,s)}_\infty(T^{-1})=\rho^{(-s,-p)}_\infty(T)$).
It is convenient to extend each $T^n_{ij}=
\kappa_j\circ T^n \circ\kappa_i^{-1}:U_{ij,n} \to U_j$ to a $C^\infty$ diffeomorphism from $\real^d_i$ onto its image in $\real^d_j$ in such a way that the intersection of $U_j$ with the image of $U_i$ by the extended map coincides with $T^n_{ij}(U_j)$, and so that the extended map is the identity outside of a large compact set. (The extension is still noted $T^n_{ij}$.) The theorem will be a consequence of the following lemma, proved in §\[proofLY\]:
\[comp\] There exist $\delta_0 >0$ and $C_0$, so that for each cover with ${\rm diam} \, V <\delta_0$, and for each $n\ge 1$ there exists $C(n)>1$, so that for every $f \in W^{p,q,t}(\real^d_j)$, compactly supported in $U_j$, and each $C^\infty$ function $\Psi_{ij}:\real_i^d
\to [0,1]$ compactly supported in $U_{ij,n}$ $$\nonumber
\begin{split}
\| & \Psi_{ij} \cdot a_{p,q}^{Op}(f \circ T_{ij}^n)\|_{L^t}
\cdot \inf_{V_{ij}} |\det DT^n|^{1/t}\cr
&\qquad \le C_0 \cdot
\max((\lambda_{ij}(T^n))^{p},(\nu_{ij}(T^n))^{q+p})
\| f \|_{p,q,t,\real^d_j}\cr
&\qquad\qquad\quad+ C(n) \| f \|_{p-1/2,q,t,\real^d_j} \, , \, \forall
p\le 0\, , \, q\ge -p\, , \, 1<t<\infty \, .
\end{split}$$
Let $\delta_0$ be as in Lemma \[comp\]. For $\delta <\delta_0$, let $(\kappa, V)$ be a foliated chart of diameter at most $\delta$, and let $\psi$ be an adapted partition of unity. Set $f_j|_{U_j}=(\psi_j \cdot \varphi) \circ \kappa_j^{-1}$, extending by zero on $\real^d_j$. By definition, for all $n\ge 1$, $$\| \LL^n \varphi \|_{p,q,t}\le
\sum_i \sum_{j :T^n(V_i)\cap V_j \ne \emptyset}
\| (\psi_i \circ \kappa_i^{-1})
\cdot ( f_j \circ T^n_{ij}) \|_{p,q,t,\real^d_i}\, .$$ By Lemma \[proper\], there is a $C^\infty$ function $\Psi_{ij}:\real^d_i\to [0,1]$, supported in a compact subset of $U_{ij,n}$, so that $\| (\psi_i \circ \kappa_i^{-1})
\cdot ( f_j \circ T^n_{ij})\|_{p,q,t,\real^d_i}\le$ $$\nonumber
\begin{split}
& \| \Psi_{ij} a_{p,q}^{Op}( (\psi_i \circ \kappa_i^{-1})
f_j \circ T^n_{ij} )\|_{L^t}
+C \|(\psi_i \circ \kappa_i^{-1})
(f_j \circ T^n_{ij}) \|_{p-1,q,t} \, .
\end{split}$$ By Lemma \[Leibniz\], the first term in the above sum is bounded by $$\nonumber
\begin{split}
& C(\psi) \cdot \| \Psi_{ij} a_{p,q}^{Op}( f_j \circ T^n_{ij} )\|_{L^t}
+ C(\psi) \cdot \| ( f_j \circ T^n_{ij})\|_{p-1,q,t} \, .
\end{split}$$ Set $\rho(p,s,n)=\max_{i,j} \max(( \lambda_{ij}(T^n))^p,(\nu_{ij}(T^n))^s))$. By Lemma \[comp\] $$\nonumber
\begin{split}
&\| \LL^n \varphi \|_{p,s-p,t}\cdot\inf_{V_{ij}} |\det DT^n|^{1/t}
\cr
&\qquad\qquad\le C_0 C(\psi) \rho(p,s,n)
\cdot\sum_{i,j}
\| a_{p,s-p}^{Op}( (\psi_j \cdot \varphi)\circ \kappa_j^{-1})\|_{L^t}\cr
&\qquad\qquad\qquad+ C(n, \psi) \sum_{i,j}
\|(\psi_j \cdot \varphi)\circ \kappa_j^{-1}\|_{p-1/2,s-p,t}\cr
&\qquad\qquad\le\# \II \cdot C_0 C(\psi) \rho(p,s,n)
\cdot
\sum_j \|(\psi_j \cdot \varphi) \circ \kappa_j^{-1}\|_{p,s-p,t} \cr
&\qquad\qquad
\qquad+ \# \II \cdot C(n,\psi)
\sum_j \| (\psi_j \cdot \varphi)\circ \kappa_j^{-1}\|_{p-1/2,s-p,t}\cr
&\qquad \qquad\le C_1
\rho(p,s,n) \cdot
\| \varphi\|_{p,s-p,t} + C_2(n)\| \varphi\|_{p-1/2,s-p,t}
\,.\cr
\end{split}$$ By Lemma \[compact\], we can apply Hennion’s theorem [@He].
Bounds involving averaged hyperbolicity exponents
-------------------------------------------------
\[L1\] Let $T$ be a $C^\infty$ Anosov diffeomorphism on a compact manifold, with a $C^\infty$ stable foliation. For any $p<0$, $s>0$, and $1<t< \infty$, the essential spectral radius of $\LL$ on $W^{p,s-p,t}(\XX)$ is $$\begin{split}\nonumber
&\le
\lim_{n\to \infty}
\biggl ( \int_\XX
\max\bigl ( (\lambda_{w} (T^n))^{p}, (\nu_{w} (T^n))^{s}
\bigr ) \cr
&\qquad\qquad\qquad\qquad \cdot | \det DT^n|_{E^u} | \cdot
|\det DT^n |^{-1/t} \, dLeb(w) \biggr )^{1/n} \cr
&
=
\lim_{n\to \infty}
\biggl ( \int_\XX
\max\bigl ( (\lambda_{w} (T^{n}))^{p}, (\nu_{w} (T^{n}))^{s}
\bigr ) \cr
&\qquad\qquad\qquad\qquad \cdot | \det DT^n|_{E^s} |^{-1}\cdot
|\det DT^n |^{1-1/t} \, dLeb(w) \biggr )^{1/n}
\, .
\end{split}$$
The reader is invited to check that there is $C_3>1$ (depending on $T$) so that for all $n\ge 1$, each cover $V$, all $i$, $j$, all $p\le 0$ $$\max_{w\in \overline V_{ij,n}} (\lambda_{w} (T^n))^{p}
-\min_{w\in \overline V_{ij,n}}(\lambda_{w} (T^n))^{p}
\le n C_3 (\lambda_{ij}(T^n))^p {\rm diam}\, V \, ,$$ and similarly for the $\nu_w$ (this is a bounded distortion argument). If $\ell_{ij}(n):=\max((\lambda_{ij}(T^n))^p, (\nu_{ij}(T^n))^s)=(\lambda_{ij}(T^n))^p
$ (the other case is similar) then $$\nonumber
\begin{split}
\max_{w\in \overline V_{ij}}(\ell_{ij}(n)& -\max( (\lambda_{w} (T^n))^{p}, (\nu_{w} (T^n))^{s}))
\le \ell_{ij} (n)-\min_{w\in \overline V_{ij}}(\lambda_{w} (T^n))^{p}\cr
&\quad\qquad\le \max_{w\in \overline V_{ij}}(\lambda_{w} (T^n))^{p}
-\min_{w\in \overline V_{ij}}(\lambda_{w} (T^n))^{p} \, .
\end{split}$$ Choose a partition $\XX=\cup_{i \in \II} W_i$ with $W_i\subset V_i$, and write $W_{ij}=W_i \cap T^{-n}W_j$. Then $$\nonumber
\begin{split}
&\sum_{i,j} \Leb(W_{ij}) \ell_{ij}(n) - \int_\XX
\max( (\lambda_{w} (T^n))^{p}, (\nu_{w} (T^n))^{s} ) \, d Leb\cr
&\qquad \le
\sum_{i,j} \Leb(W_{ij})\biggl ( \ell_{ij}(n) - \min_{W_{ij}}
\max( (\lambda_{w} (T^n))^{p}, (\nu_{w} (T^n))^{s} )\biggr ) \cr
&\qquad \le
C_3 n \cdot {\rm diam}\, V \sum_{i, j} \Leb(W_{ij}) \ell_{ij}(n) \, . \cr
\end{split}$$ Therefore, fixing $\delta \in (0,1)$, if $V(n)$ satisfies ${\rm diam}\, V (n)= \delta/ (C_3n)$, $$\sum_{i,j} \Leb(V_{ij,n}) \ell_{ij}(n)
\le \frac{ \# V(n) }{(1-\delta)} \int_\XX
\max( (\lambda_{w} (T^n))^{p}, (\nu_{w} (T^n))^{s} ) \, d Leb \, .$$ Choose $V(n)$ and $\psi(n)$ with $\# V=O(n)$ and $(\min_i \Leb (V_i) )^{-1}=O (n^{d})$, ensuring that the derivatives of the $\psi_i$ from Lemma \[Leibniz\] satisfy $O(n^Q)$ bounds; for some $Q\ge 1$. Finally, there is $C_4\ge 1$ so that $$\frac{1}{\Leb(V_{ij,n})} \le \frac{ C_4 }{\min_i \Leb (V_i))} \inf_{V_{ij,n}}
|\det DT^n|_{E^u}|\, .$$ for all $n$ and all covers $V$. Lemma \[comp\] allows to conclude, by a straightforward adaptation of the proof of Theorem \[mainthm\].
Proof of the Lasota-Yorke inequality {#proofLY}
------------------------------------
We replace $T$ by $T^n$ (the reader should keep in mind that $A_{ij}$, $B_{ij}$, $D_{ij}$, $\lambda_{ij}$, and $\nu_{ij}$ depend on $n$) and drop the indices $i$, $j$. We study the action of the composition by $T$ on our symbol $a_{p,q}(\xi,\eta)$ (see e.g. [@AG Chapter I.7, Proposition 7.1, and Chapter I.8, Théorème 3]).
Taking a Taylor series of order $0$ (i.e., $k=1$ in the proof of [@AG I.8, Lemme 4]), we find that $(\Psi_{ij} a_{p,q}(\xi,\eta)^{Op} (f\circ T )) \circ T^{-1}$ decomposes as $$\label{heart1}
\begin{split}
&( \Psi_{ij}\circ T^{-1})\cdot \bigl (
( a_{p,q}((DT)^{tr}_{T^{-1}(x,y)}(\xi,\eta))^{Op}( f)\cr
&\qquad \qquad\qquad+ r_1(x,y,\xi,\eta)^{Op} (f)
+r_2(x,y,\xi,\eta)^{Op} (f) \bigr )\, ,
\end{split}$$ where $r_1$ and $r_2$ are described next. The symbol $r_1(T(x,y),\xi,\eta)$ is a universal finite linear combination of $$\label{r1}
\begin{split}
&
\int_{\real^d} du dv\,
\int_{\real^{d}} d\omega d\theta \, e^{-i(u,v)(\omega, \theta)}
\int_0^1 ds\,
(1-s) s^j \cr
&\cdot
\partial_{u_\ell} \bigl (e^{i(R_{(x,y)} (x+su,y+sv) )^{tr} (\xi,\eta)}
\bigr )\cr
&\cdot
(1+|s\omega+A_{(x,y)}\xi|^2+|s\theta+B_{(x,y)}\xi+D_{(x,y)}\eta|^2)^{p/2} \cr
&\cdot
\partial_{\omega_\ell}
\bigl (1+|s\omega+ A_{(x,y)}\xi|^2 )^ {q /2}
\chi \biggl ( \frac{(s\omega+ A_{(x,y)} \xi,
s\theta+B_{(x,y)}\xi+D_{(x,y)}\eta)}
{1+|(A_{(0,0)}\xi,B_{(0,0)}\xi+D_{(0,0)}\eta)|} \biggr ) \, ,
\end{split}$$ where $j\in \{0, 1\}$, $1\le \ell \le d_s$, the function $\chi:\real^{d}\to [0,1]$ is $C^\infty$ and compactly supported in a suitable annulus, and $$R_{(x,y)}(u,v)=T(u,v)-T(x,y) -DT_{(x,y)} (u-x, v-y)\, .$$ To describe $r_2$, set $\tilde r_2=r_2\cdot (a_{p-1/2,q})^{-1}$, so that $r_2^{Op}= \tilde r_2^{Op} a_{p-1/2,q}^{Op}$. (In the proof we shall use the notation $\tilde r=r\cdot (a_{p-1/2,q})^{-1}$ several times.) We claim that $\tilde r_2^{Op}$ is a bounded operator on $L^t( \real^d,\Leb)$, so that $$\nonumber
\begin{split}
\|\Psi_{ij}\cdot \bigl ((r_2^{Op} f) \circ T\bigr) \|_{L^t}
&\le C \sup_{V_{ijj}} |\det DT|^{-1/t} \cdot
\biggl (\int | r_2^{Op}(f)|^t \, dLeb\biggr ) ^{1/t}\cr
&\le C_2(n) \| f\|_{p-1/2, q, t,\real^d_j}\, .
\end{split}$$ By [@CM Théorème 9] it suffices to show that for all $| \alpha|\le 1$ and all $ \beta$ we have $\sup |
(1+|\xi|+|\eta|)^{|\beta| } \partial^{ \alpha}_{x,y}
\partial^{\beta}_{\xi,\eta} \tilde r_2(x,y,\xi,\eta)|<\infty$. This can be seen by observing that $r_2$ is made on the one hand with contributions due to $1-\chi$, which have rapid decay in $1+|(\omega,\theta)|+
|(A_{(0,0)}\xi,B_{(0,0)}\xi+D_{(0,0)}\eta)|$ (by a small modification of [@AG p.58], using bounded distortion for $DT^n$). The other terms forming $r_2$ correspond to a $\partial_{\theta_\ell}$ derivative, or to a $\partial_{\omega_\ell}$, but acting on a factor $
(1+|s\omega+A_{(x,y)}\xi|^2+|s\theta+B_{(x,y)}\xi+D_{(x,y)}\eta|^2)^{p/2}
$. (Details are left to the reader, see [@AG p.60].)
We may thus concentrate on the first two terms in (\[heart1\]). The first one is called the [*principal symbol.*]{}
We get $a_{p,q}((DT_{(x,y)})^{tr}(\xi,\eta))^{Op}=b^{Op}\circ a^{Op}_{p,q}$ by setting $b(x,y,\xi,\eta)=a_{p,q}((DT_{(x,y)})^{tr}(\xi,\eta))/a_{p,q}(\xi,\eta)$. Again by [@CM Théorème 9] it suffices to show that, up to replacing $b$ by $b-r_3$, with $\tilde r_3^{Op}$ bounded on each $L^t$, we have $\sup | (1+|\xi|+|\eta|)^{|\beta|} \partial^{ \alpha}_{x,y}
\partial^{\beta}_{\xi,\eta} b|\le( C_\delta/2)
\max(\lambda^{p},\nu^{q+p})$, for all $| \alpha|\le 1$ and all $ \beta$. Of course, we must also prove the same bounds for $r_1 \cdot (a_{p,q})^{-1}$ (modulo $r_4+r_5$, with $\tilde r_4^{Op}$ and $\tilde r_5^{Op}$ bounded on each $L^t(\real^d)$).
Consider first $ \alpha= \beta=0$ and the principal symbol, i.e., the bound for $\sup|b|$. For $\xi\ne 0$ write $\nu_\xi= \sup_{x,y}|A_{(x,y)}\xi|/|\xi|$. Then, setting $\Lambda_1=\max_\xi ((1-2 \nu_\xi)/\nu_\xi)$, we get for any $|\xi|\ge \Lambda_1$ $$\label{heart2}
(1+|A_{(x,y)}\xi|^2)^{q/2}
\le (1+\nu_\xi^2|\xi|^2)^{q/2} \le 2 \nu_{\xi/|\xi|}^{q} (1+|\xi|^2)^{q/2}\, .$$ For $|\xi|<\Lambda_1$ we always have $
(1+|A_{(x,y)}\xi|^2)^{q /2} \le (1+|\xi|^2)^{q/2}
$.
If $|\xi|\ge\max( \Lambda_1,|\eta|)$ then [^5] $$(1+|A_{(x,y)}\xi|^2+|B_{(x,y)}\xi+D_{(x,y)}\eta|^2)^{p/2}
\le 2 \nu_{\xi/|\xi|}^{p} (1+|\xi|^2+ |\eta|^2)^{p/2}\, .$$
For $\eta\ne 0$ write $\lambda_\eta= \inf_{x,y} |D_{(x,y)}\eta|/|\eta|$. Fix $\Lambda_2=\max_\eta \lambda_\eta$. If $\epsilon <1/4$ and $|\eta|\ge \max(\Lambda_2, |\xi|)$ then [^6] $$(1+|A_{(x,y)}\xi|^2+|B_{(x,y)}\xi+D_{(x,y)}\eta|^2)^{p/2}
\le 3 \lambda^{p} (1+|\xi|^2+|\eta|^2)^{p/2}\, .$$
Finally, if $|\xi|\le \Lambda_1$, and $|\eta|\le \Lambda_2$, there is $C_{\Lambda_1, \Lambda_2}(n)$ with $$\label{heartn}
(1+|A_{(x,y)}\xi|^2+|B_{(x,y)}\xi+D_{(x,y)}\eta|^2)^{p/2} \le C_{\Lambda_1, \Lambda_2}
(1+|\xi|^2+|\eta|^2)^{p/2-1}\, .$$ We include the above contribution in $r_3$.
To bound $\sup_{x,y,\xi,\eta}|r_1\cdot (a_{p,q})^{-1}|$, multiply the integrand of (\[r1\]) by $$\biggl (1- \tilde \chi \bigl (\frac{ s\omega+ A_{(x,y)} \xi }{1+|A_{(0,0)}\xi|}
\bigr ) \biggr ) +
\tilde \chi \bigl ( \frac{s\omega+ A_{(x,y)} \xi}{1+|A_{(0,0)}\xi|} \bigr )\, ,$$ where $\tilde \chi:
\real^{d_s} \to [0,1]$ is $C^\infty$ and compactly supported in an annulus. We consider separately the two terms in this decomposition:
The term containing $\chi \cdot (1-\tilde \chi)$ enjoys $C_k(n)(1+|A_{(0,0)}\xi|+|\omega|)^{-k}$ rapid decay (adapting [@AG p.58]). By choosing first $k$ and then $\Lambda_3$ we get a bound $(C_\delta/4)
\max(\lambda^{p},\nu^{q+p})$ for $|\xi|\ge \Lambda_3$. If $|\xi|\le \Lambda_3$, we use that if $|\eta|>\max( |\xi|,\Lambda_4)$ then $$\label{suivantes}
\begin{split}
\sup_{s,(\omega,\theta),(u,v),(x,y)}&
(1+|s\omega+A_{(x,y)}\xi|^2+|s\theta+B_{(x,y)}\xi+D_{(x,y)}\eta|^2)^{p/2}\cr
&\quad \cdot \chi \biggl ( \frac{(s\omega+ A_{(x,y)} \xi,
s\theta+B_{(x,y)}\xi+D_{(x,y)}\eta)}
{1+|(A_{(0,0)}\xi,B_{(0,0)}\xi+D_{(0,0)}\eta)|} \biggr )\cr
&\qquad \le 2 \lambda^p (1+|\xi|^2+|\eta|^2)^{p/2} \, .
\end{split}$$ The compact set $\{ |\xi|\le \Lambda_3, |\eta|\le \Lambda_4\}$ gives rise to a term $r_4$.
For the $\chi \cdot \tilde \chi$ term, use the ideas exploited for the principal symbol (see also – again – [@AG p.60]) to get a bound $(C_\delta/4)
\max(\lambda^{p},\nu^{q+p})$, up to a perturbation $r_5$. In particular, if $|\xi|$ is large with respect to $|\eta|$ then $$\label{term}
\begin{split}
\sup_{s,(\omega,\theta),(x,y)} &(1+|s\omega+A_{(x,y)}\xi|^2+|s\theta+B_{(x,y)}\xi+D_{(x,y)}\eta|^2)^{p/2}\cr
&\qquad \cdot \partial_{\omega_\ell}
\bigl ( (1+|s\omega+ A_{(x,y)}\xi|^2 )^ {q /2}
\chi (\cdots) \cdot \tilde \chi (\cdots) \bigr )
\end{split}$$ is bounded by $C (1+|\xi| +|\eta|)^{p-1} (1+|\xi| )^ q$ (giving a contribution $r_5$); while if $|\eta|$ is large then (\[term\]) is bounded by $2 \lambda^{p} (1+|\xi|^2 +|\eta|^2)^{p/2} (1+|\xi|^2 )^{q/2}$.
The control of the derivatives of $b$ and $r_1\cdot( a_{p,q})^{-1}$, i.e., the case of nonzero $| \alpha|+| \beta|$, is straightforward although rather tedious.
The essential spectral radius of $\MM$ {#last}
--------------------------------------
Let $T$ be a $C^\infty$ Anosov diffeomorphism on a compact manifold, with a $C^\infty$ unstable foliation. Let $W^{p,q,t}(\XX,T^{-1})$ denote the Banach space in Definition \[defnorm\]. (We use now stable foliation of $T^{-1}$, i.e. the unstable foliation of $T$, in other words, $W^{p,q,t}(\XX,T^{-1})=(1+\Delta_u)^{-q/2} (1+\Delta)^{-p/2} ( L^t(\XX))$.)
\[mainthm2\] For any $p<0$ and $s>0$, the essential spectral radius of $\MM$ on $W^{p,s-p,t}(\XX,T^{-1})$ is not larger than $\lim_{n \to \infty} \sup_\XX |\det DT^n|^{-(t-1)/tn} \cdot \rho^{(-s,-p)}_\infty(T)$ for $t \in(1,\infty)$.
\[L1,2\] For any $p<0$, $s>0$, and $1< t< \infty$, the essential spectral radius of $\MM$ on $W^{p,s-p,t}(\XX,T^{ -1})$ is $$\begin{split}
&\le
\lim_{n\to \infty}
\biggl ( \int_\XX
\max\bigl ( (\lambda_{w} (T^n))^{-s}, (\nu_{w} (T^n))^{-p}
\bigr )\cr
& \qquad\qquad\qquad\qquad \cdot | \det DT^{n}|_{E^u} |\cdot
|\det DT^{n} |^{-(t-1)/t} \, dLeb(w) \biggr )^{1/n}
\, .
\end{split}$$
Adapt the proofs of Theorem \[mainthm\] and Proposition \[L1\], using distortion estimates to bound (\[opM\]) when exploiting Lemma \[Leibniz\] for a weight $(1/|\det DT^n |)\circ \kappa_i^{-1}$ (see also the comments before Corollary \[compint\]).
Operators $\MM_t$ and $\LL_t$
=============================
\[mainthm3\]
\(1) If the stable foliation of $T$ is $C^\infty$ then for any $p<0$ and $s>0$ with $q=s-p$ integer, there exists $t_1(q)>1$ so that the essential spectral radius of $\LL_t$ on $W^{p,s-p,t}(\XX)$ is $\le \rho^{(p,s)}_\infty(T)$, for each $1<t<t_1$.
\(2) If the unstable foliation of $T$ is $C^\infty$ then for any $p<0$ and $s>0$ with $q=s-p$ integer there exists $t_2(q)\ge 1$ so that the essential spectral radius of $\MM_t$ on $W^{p,s-p,t}(\XX,T^{-1})$ is $\le \rho^{(-s,-p)}_\infty(T)$, for each $t_2 < t <\infty$.
For any integer $q\ge 1$, there exist $C\ge 1$ and $t_1(q) >1$ so that for every $\gamma'$ with $1\le |\gamma'|\le q$, all $1\le t < t_1$ and all $n\ge 1$, setting $h(x,y)=|\det DT^{n}| ^{1/t} \circ \kappa_i^{-1}(x,y)$, then $|\partial^{\gamma'}_x h(x,y)|\le C h(x,y)$. (If $q=1$ we may take $t_1=\infty$.) Theorem \[mainthm3\] is therefore a consequence of the following corollary of the proof of Lemma \[comp\] combined with a refinement of the Leibniz formula for $a_{p,q}$ if $q\in \integer_+$, Lemma \[Leibnizint\].
\[compint\] There exist $\delta_0 >0$ and $C_0$ so that, for all $V$ with ${\rm diam} \, V <\delta_0$ and $n\ge 1$, there exists $C(n)>1$ so that for any multi-index $\gamma'$ with $|\gamma'|\le q$, any $f \in W^{p,q,t}$, compactly supported in $U_j$, and each $C^\infty$ function $\Psi_{ij}:\real^d_i\to [0,1]$ compactly supported in $U_{ij,n}$ $$\nonumber
\begin{split}
\| \Psi_{ij}\cdot & |\det DT^n \circ \kappa^{-1}_i|^{1/t} \cdot a_{p,0}\, \partial_x^{\gamma'} (f \circ T_{ij}^n)\|_{L^t} \cr
&\qquad\le C_0
\max((\lambda_{ij}(T^n))^{p},(\nu_{ij}(T^n))^{q+p})
\| f \|_{p,q,t,\real^d_j}\cr
&\qquad\qquad\quad+ C(n) \| f \|_{p-1/2,q,t,\real^d_j} \, , \, \forall
p\le 0\, , \, q\ge -p\, , \, 1<t<\infty \, .
\end{split}$$
\[Leibnizint\] Let $1<t<\infty$. Let $q \in \integer_+$ and let $p \in \real$. There exists $C\ge 1$, and for every compactly supported $h\in C^\infty(I^d)$ there exists $C(h)>0$ so that for each $f \in W^{p,q,t}(\real^d)$, we have $a^{Op}_{p,q}( h \cdot f) =h \cdot a^{Op}_{p,q}(f)+ g_1 + g_2$ with $$\|g_1\|_{L^t} \le C
\sum_{\stackrel{|\gamma'| =q-1}{ \gamma'_1+\gamma_2'=\gamma'}}
\| \partial^{\gamma'_1}_x h \cdot a_{p, 0}^{Op}( \partial^{\gamma'_2}_x f)\|_{L^t}\, , \,\,
\|g_2\|_{L^t} \le C(h) \| f\|_{p-1,q,t} \, .$$
Decompose $a_{p,q}^{Op}=a_{p,0}^{Op} \circ a_{0,q}^{Op}$. The proof of Lemma \[Leibniz\] gives that if $\tilde h \in
C^\infty(I^d)$ is compactly supported, then there is $C(\tilde h)\ge $ so that for all $\tilde f \in W^{p,0,t}(\real^d)$, we have $
a^{Op}_{p,0}( \tilde h \cdot \tilde f) =\tilde h \cdot a^{Op}_{p,0}(\tilde f)+
\tilde g
$ with $\| \tilde g \|_{L^t}\le C(\tilde h) \|\tilde f \|_{p-1, 0, t}$.
Since $q$ is an integer, $a^{Op}_{0,q}(h \cdot f)$ decomposes as: $$\nonumber
a^{Op}_{0,q}(h \cdot f)=
\begin{cases} h\cdot f+
\sum_{j=1}^\ell
\binom {\ell}{j} \bigl (\partial^2_{x_1}+ \cdots +\partial^2_{x_{d_s}} \bigr )^j (h \cdot f)&\cr
\qquad \qquad\qquad\qquad \qquad\qquad\qquad\text{if } q=2\ell \text{ is even,}\cr
\mu_1 * a^{Op}_{0,q-1}(h \cdot f) +
\mu_2 * \bigl ( \sum_{j=1}^{d_s} R_{x_j} ( \partial_{x_j} a_{0, q-1}^{Op}(h\cdot f )\bigr )\cr
\qquad\qquad\qquad\qquad \qquad\qquad\qquad \text{if } q=2\ell+1 \text{ is odd,}\cr
\end{cases}$$ where $\mu_1$ and $\mu_2$ are finite measures (which do not depend on $h$ or $f$) and $*$ denotes convolution. Indeed, $q=2\ell$ is even, just recall that $a_{0,q}^{Op}=(1+\Delta_s)^{q/2}= (1+\sum_{j=1}^{d_s} \partial^2_{x_j})^{\ell}$. If $q=2\ell+1$, recall [@St V.3.2–V.3.4] that $
(1+\Delta_s)^{1/2} (\varphi)
= \mu_{1} * \varphi + \mu_2 *
\bigl ( \sum_{j=1}^{d_s} R_{x_j} ( \partial_{x_j} \varphi ) \bigr )
$, where $R_{x_j}$ is the Riesz transform [@St III.1]. Finally, use the ordinary Leibniz formula for partial derivatives, that $a^{Op}_{p,0}$ commutes with each $R_{x_j}$, that $R_{x_j}$ is bounded on $L^t$, and that if $\mu$ is a measure, with total mass $|\mu|$, then $a_{p,0} (\mu * f)= \mu * a_{p,0} (f)$ and $\|\mu * f \|_{L^t}\le |\mu| \cdot \|f\|_{L^t}$.
[10]{}
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A. Avila, S. Gouëzel, and M. Tsujii, *Smoothness of fat solenoidal attractors,* in preparation (2004).
V. Baladi, *Anisotropic Sobolev spaces and dynamical transfer operators: $C^{1+\alpha}$ foliations,* in preparation.
V. Baladi and M. Baillif, *Kneading determinants and spectra of transfer operators in higher dimensions, the isotropic case,* Preprint (2003).
M. Blank, G. Keller, and C. Liverani, *Ruelle-Perron-Frobenius spectrum for Anosov maps,* Nonlinearity 15 (2002) 1905–1973.
R. Bowen, [*Equilibrium states and the ergodic theory of Anosov diffeomorphisms,*]{} Springer Lecture Notes in Mathematics Vol 470 (1975).
R.R. Coifman and Y. Meyer, *Au-delà des opérateurs pseudo-différentiels,* Astérisque 57 (1978).
P. Collet and J.-P. Eckmann, *Liapunov multipliers and decay of correlations in dynamical systems,* J. Stat. Phys. 115 (2004) 217–254.
S. Gouëzel and C. Liverani, *Banach spaces adapted to Anosov systems,* preprint (2004).
M. Gundlach and Y. Latushkin, *A sharp formula for the essential spectral radius of the Ruelle transfer operator on smooth and Hölder spaces,* Ergodic Theory Dynam. Systems 23 (2003) 175-191.
H. Hennion, *Sur un théorème spectral et son application aux noyaux lipschitziens,* Proc. Amer. Math. Soc. 118 (1993) 627–634.
L. Hörmander, [*Linear Partial Differential Operators,*]{} Springer (1963).
A.Yu. Kitaev, *Fredholm determinants for hyperbolic diffeomorphisms of finite smoothness,* Nonlinearity 12 (1999) 141–179. *Corrigendum:* Nonlinearity 12 (1999) 1717–1719.
Y.A. Kordyukov, *Functional calculus for tangentially elliptic operators on foliated manifolds,* in Analysis and Geometry in Foliated Manifolds: international conference on differ. geom., Santiago de Compostela, 1994, World Scientific.
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[^1]: Thanks to A. Avila, L. Boutet de Monvel, G. David, P. Gérard, S. Gouëzel, F. Ledrappier, C. Liverani, and M. Tsujii for useful conversations.
[^2]: Controlling the spectrum on a scale of Sobolev spaces may be useful: see [@BB].
[^3]: The operators $\LL_t$, $\MM_t$ “interpolate” between the SRB measures of $T$, $T^{-1}$.
[^4]: We decompose multi-indices $\gamma=(\gamma', \gamma'')$ in this way tacitly from now on.
[^5]: Here we pay the price of $p< 0$.
[^6]: If $|\xi|$ is small but $|\eta|$ is large we need $p< 0$ to get a contraction here.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'Predictive modelling relies on the assumption that observations used for training are representative of the data that will be encountered in future samples. In a variety of applications, this assumption is severely violated, since observational training data are often collected under sampling processes which are systematically biased with respect to group membership. Without explicit adjustment, machine learning algorithms can produce predictions that have poor generalization error with performance that varies widely by group. We propose a method to pre-process the training data, producing an adjusted dataset that is independent of the group variable with minimum information loss. We develop a conceptually simple approach for creating such a set of features in high dimensional settings based on a constrained form of principal components analysis. The resulting dataset can then be used in any predictive algorithm with the guarantee that predictions will be independent of the group variable. We develop a scalable algorithm for implementing the method, along with theory support in the form of independence guarantees and optimality. The method is illustrated on some simulation examples and applied to two real examples: removing machine-specific correlations from brain scan data, and removing race and ethnicity information from a dataset used to predict recidivism.'
author:
- Emanuele Aliverti
- Kristian Lum
- 'James E. Johndrow'
- 'David B. Dunson'
bibliography:
- 'short.bib'
title: 'Removing the influence of a group variable in high-dimensional predictive modelling'
---
[[**Keywords:**]{} Constrained optimisation; Criminal justice; Neuroscience; Orthogonal predictions; Predictive modelling; Singular value decomposition; $\ell$-1 norm.]{}
Introduction {#sec:1}
============
Modern statistical and machine learning applications often rely on large datasets constructed by automated systems; for example, web scraping results from a search term or aggregating publicly accessible information, such as images and online articles. Alternatively, data may be collected within observational studies for a convenience sample of individuals, and interest is on detecting relationships between features and outcome variables; for example, disease or behavior. Such processes create datasets in which the sampling mechanism is complex, unknown, and often subject to some form of systematic bias [@dunson2018statistics].
When selection bias exists in the sampling mechanism, inferences or predictions produced using the data often encode spurious associations. There is growing recognition that machine learning algorithms will reproduce and often amplify sampling bias in the data upon which they were trained [e.g. @angwin2016machine; @zech2018confounding; @dunson2018statistics]. For example, racial bias in police records can be propagated to predictive algorithms trained on these data [@lum2016predict]. Often in such settings there is concern about sampling bias with respect to a key group membership variable, such as ethnicity, gender or a blocking factor in scientific studies (e.g. the clinical facility in multisite study designs, or the machine on which an assay was performed). Motivated by this problem, we propose a method to adjust high-dimensional datasets in order to remove associations between covariates of interest and group membership, and provide predictions that are free from the effect of sampling bias.
A popular and important application for automated predictions is disease recognition from medical imaging data, such as radiography, ultrasonography and brain scans. Modern machine learning can obtain impressive predictive performance, similar to or better than expert practitioners in the field [e.g. @obermeyer2016predicting; @wang2012machine]. Unfortunately, it is virtually impossible to determine and describe the sampling mechanism of such complex data, which contain millions of records collected across different regions, clinical facilities, and equipment. @zech2018confounding recently showed that radiographic image data encoded information on the specific hospital system from which the data were collected, likely because different systems tended to use different imaging equipment. When these data were used to train a model for pneumonia screening, the model learned to associate these hospital-specific characteristics to the outcome of interest. The model’s heavy reliance on such associations jeopardizes the generalizability of the results. This issue is also detrimental for in-sample evaluation, since regarding such associations as risk factors for the outcome of interest is clearly misleading.
Another area in which unwanted associations arise is in criminal justice data. For example, there has been much recent attention on the use of criminal risk assessment models, many of which use demographic, criminal history, and other information to predict whether someone who has been arrested will be re-arrested in the future. These predictions then inform decisions on pre-trial detention, sentencing, and parole. In practice, the data used to train the models are based on arrest records, which are well known to be subject to racial bias [@simoiu2017problem; @rudovsky2001law; @bridges1988law; @lum2017limitations]. When risk assessment models are trained using these data, the end result is that racial minority groups tend to be systematically assigned to higher risk categories on average [@lum2016statistical; @angwin2016machine].
In this article we focus on developing data pre-processing methods to remove the influence of group membership variables, such as hospital id or racial group. There are several application areas in which datasets are pre-processed in order to remove or obscure specific information. For example, in data confidentiality, there is strong interest in releasing synthetic datasets which minimise the probability to disclose respondents’ identities [@reiter2005releasing; @raghunathan2003multiple]. Our work is also closely related to recent methods on data pre-processing in the “algorithmic fairness” literature, where one of the crucial aims is to create datasets which produce predictions that are independent of sensitive variables, such as race or ethnicity or gender [e.g. @feldman2015certifying; @kamiran2012data; @lum2016statistical]
The main advantage of our contribution is its simplicity and scalability to a large number of covariates ($p$), particularly when $p$ is greater than the number of observations $n$. In this sense, it is particularly well-suited to applications like brain imaging, in which the observed covariates are high-dimensional. It also has significant advantages in the case of highly collinear predictors, which is very common in applications. Moreover, we will show that the solution we propose has theoretical guarantees, both in terms of optimal dimension reduction and statistical parity under a linearity condition.
Our method is motivated by the reliance on very high-dimensional data, such as medical imaging and “omic" data, for an increasing number of prediction tasks. As previously described, such type of data may encode unnecessary or detrimental correlations that threaten the generalizability of models trained with the data. Removing these associations will be important to ensure that complex prediction models are relying on reliable features that will generalize across a wide variety of patient populations, not just artifacts of the data collection process.
Generating data orthogonal to groups
====================================
Notation and setup
------------------
Let $X$ denote an $n \times p$ data matrix of $p$ features measured over $n$ subjects, and let $Z$ denote an additional group membership variable; for example ethnicity, gender or clinical facility. We focus for simplicity on a scalar $Z$. We seek to estimate $\widetilde {X}$, an $n \times p$ reconstructed version of the data matrix that is orthogonal to $Z$ with minimal information loss. In our setting, the reconstructed version is used to produce a prediction rule $\hat y(\widetilde x)$ that returns a prediction of $y$ for any input $\tilde x$. Our aim is to guarantee that $\hat y(\widetilde x)$ is uncorrelated with the group variable. In the sequel, we refer to this condition generically as “independent”.
We will focus on statistical models linear in the covariates, such as generalised linear models, support vector machines with linear kernels, and many others. It is easy to check that when $\hat y(\tilde x)$ is a linear function of $\tilde x$, $\operatorname{cov}(\hat Y, Z ) = 0$ naturally follows from $\operatorname{cov}( \tilde X, Z ) = 0$. A natural procedure to transform the original data, then, consists in imposing orthogonality between $\widetilde {X}$ and $Z$, while attempting to preserve as much of the information in ${X}$ as possible. The former requirement is geometrically analogous to requiring that the projection of $z$ onto the range of the reconstructed covariates $\widetilde {X}$ is null.
In terms of the geometry of least squares estimation, the orthogonality condition guarantees that the columns of $\widetilde {X}$ provide no information about the variable $Z$ [@hastiefriedman]. This assumption implies that it is not possible to predict the group membership using the transformed variables as covariates in a statistical model which is linear in the covariates. Potentially, non-linear dependencies could still be present in the transformed matrix $\widetilde X$, and hence affect predictions of non-linear models. Our method can be motivated as a second-order adjustment, which attempts to accommodate for effects that are simple to measure and generally more relevant. As we will illustrate, in our experience higher order dependencies are often relatively modest in concrete applications, and our procedure also performs well when non-linear models are employed.
In high-dimensional settings, it is often assumed that large collections of variables have approximately a low-rank representation, meaning that observations lie close to a lower-dimensional subspace that captures the most salient properties of the data. We express the reduced rank approximation as $\widetilde{X}=SU^T$, where ${U}$ is a $p \times k$ matrix of $k$ linear orthonormal basis vectors and ${S}$ is the $n \times k$ matrix of associated scores.
The problem of preprocessing the data to ensure Orthogonality to Groups (henceforth OG) can be expressed as a minimization of the Frobenius distance between the original data and the approximated version, $\|{X}- \widetilde {X}\|^2_F$, under the constraint $\langle\widetilde{X},Z\rangle=0$. Given the particular structure assumed for $\widetilde {X}$, this leads to the following optimization problem: $$\label{eq:min}
\operatorname*{arg\,min}_{{S},{U}} \;\|{X}- {S}{U}^T\|^2_F, \quad
\text{subject to}\; \langle {S}{U}^T,Z \rangle = 0, \quad {U}\in \mathcal{G}_{p,k}$$ where $\mathcal{G}_{p,k}$ is the Grassman manifold of orthonormal matrices.
Since the constraints are separable, it is possible to reformulate as $p$ distinct constraints, one over each column of $\widetilde {X}$. Moreover, since any column of $\widetilde {X}$ is a linear combination of the $k$ columns of ${S}$, and ${U}$ is orthonormal, the $p$ constraints over $\widetilde {X}$ can be equivalently expressed as $k$ constraints over the columns of the score matrix ${S}$. The matrix ${U}$ is forced to lie on the Grassman manifold to prevent degenerate conditions, such as basis vectors being identically zero or solutions with double multiplicity. The optimisation problem admits an equivalent formulation in terms of Lagrange multipliers, $$\label{eq:prob}
\operatorname*{arg\,min}_{{S},{U}}\bigg\{ \frac {1}{n}\sum_{i=1}^n \|x_i - \sum_{j=1}^k s_{ij} u_j^T\|^2 + \frac{2}{n}\sum_{j=1}^k\lambda_j \sum_{i=1}^n s_{ij}z_i\bigg\},
$$ with the introduction of the factor $2/n$ for ease of computation.
Theoretical support {#sec:fpl}
-------------------
The following Lemma characterizes the solution of the Orthogonal to Groups (OG) optimization problem, which can be interpreted as the residual of a multivariate regression among left singular values and a group variable. Let $V_k\Sigma_k{U}_k^T$ denote the rank-$k$ singular values decomposition of $X$.
\[lemma1\] The problem stated in \[eq:min\] can be solved exactly, and admits an explicit solution in terms of singular values. The solution is equal to $\widetilde X = (I_n - P_z)V_k\Sigma_k{U}_k^T$, with $P_Z = Z(Z^TZ)^{-1}Z^T$.
All proofs are given in \[appendix\]. The computational cost of the overall procedure is dominated by the cost of the partial singular value decomposition. This can computed with modern methods in $O(nk^2)$ [@golub2012matrix]. The procedure outlined in \[lemma1\] is simple and only involves matrix decomposition and least squares theory; hence we can fully characterise the solution and its properties. The following Lemma characterises the proposed solution within the class of constrained low-rank representations.
\[lemma2\] The solution $\widetilde X$ of the orthogonal to group algorithm is the best rank-$k$ approximation, in Frobenius norm, of the data matrix ${X}$ under the OG constraint.
The singular value decomposition achieves the minimum error in Frobenius distance among all matrices of rank-$k$ [e.g., @golub2012matrix]. Naturally, the introduction of additional constraints reduces the accuracy of the approximation, with respect to the optimal one. The following result allows us to bound the additional error analytically.
\[lemma3\] Let $\widetilde X_k = V_k D_kU_k^T$ denote the best rank-$k$ approximation of the matrix $X$ obtained from the partial singular value decomposition of rank $k$. The reconstruction error of the OG algorithm is lower bounded by the optimal error rate of $\widetilde X_k$, and the amount of additional error is equal to $||P_zV_kD_k||_F^2$, where $P_Z = Z(Z^TZ)^{-1}Z^T$.
The additional reconstruction error can be interpreted as a measure of the collinearity between the subspace spanned by $Z$ and the left singular vectors of the data $X$. The more collinear the singular vectors are with the group variable, the greater is the amount of additional error with respect to the solution without the OG constraint. When these quantities are already orthogonal, then the solution is identical to the truncated singular value decomposition and the reconstruction achieves the minimum error.
Sparse OG procedure
-------------------
In order to reduce the solution to a more interpretable structure, common methods in multivariate analysis impose constraints over the elements of a matrix decomposition, usually through an $\ell_1$-norm penalty to favour sparsity [e.g. @zou2006sparse; @jolliffe2003modified; @witten2009]. Besides improving the interpretability of the results, constraints improve the numerical estimation of the eigenvectors, which can be problematic in very high-dimensional applications [@johnstone2001].
To make the OG problem tractable and stable when the number of features is very large – potentially larger than the number of observations – we introduce additional constraints in the algorithm. We will build our method on a standard procedure to perform sparse matrix decomposition [e.g. @hastie2015statistical Chapter 8], and adapt the computations to introduce the orthogonality constraint. We define the sparse orthogonal to group (SOG) optimization problem as follows. $$\label{eq:sfpl}
\begin{split}
\operatorname*{arg\,min}_{S,U} &\left\|X - SU^T \right\|_F^2 \\
\text{subject to}\quad ||u_j||_2 \leq 1, ||u_j||_1 \leq t,& \;||s_j||_2 \leq 1, s_j^Ts_l=0, \;s_j^T{Z}=0,
\end{split}$$ for $j=1,\dots,k$ and $l\neq j$. The problem in \[eq:sfpl\] includes sparsity constraints over the vectors $u_j$ and imposes orthogonality constraints among the score vectors $s_j$ and the group variable, since the reconstructed version of $\widetilde {X}= {S}{U}^T$ is a linear combination of the vectors $s_j$.
Focus now on the case of rank-$1$ approximation. As outlined in @witten2009, it is possible to show that the solutions in $s$ and $u$ for \[eq:sfpl\] when $k=1$ also solve $$\label{eq:equivalent}
\operatorname*{arg\,max}_{s,u}\; s^T X u\quad \text{subject to}\quad ||u||_2 \leq 1, ||u||_1 \leq t, \;||s||_2 \leq 1, \;s^T{Z}=0.
$$ Although the minimisation in \[eq:equivalent\] is not jointly convex in $s$ and $u$, it can be solved with an iterative algorithm. Since the additional constraints do not involve the vector $u$, when $s$ is fixed the minimisation step is mathematically equivalent to a sparse matrix decomposition with constraints on the right singular vectors, and takes the following form. $$\label{eq:s}
\operatorname*{arg\,max}_{u}\; b u\quad \text{subject to}\quad ||u||_2 \leq 1, ||u||_1 \leq t,
$$ with $b=s^TX$ and solution equal to $$u=g(b,\theta)=\frac{\mathcal{S}_\theta (b)}{||\mathcal{S}_\theta(b)||_2},$$ where $\mathcal{S}_\theta$ is the soft threshold operator, defined as $\mathcal{S}_\theta(x) = \text{sign}(x)(|x| -\theta)\mathbb{I}(|x|\geq \theta)$ and applied over every element separately. The value of $\theta$ is $0$ if $||b||_1\leq t$, and otherwise $\theta > 0$ is selected such that $||g(b,\theta)||_1= t$ [@hastie2015statistical; @witten2009].
When $u$ is fixed, the problem is similar to the solution described in \[sec:fpl\], after arranging as follows. $$\label{eq:u}
\operatorname*{arg\,max}_{S}\; s^T a \quad \text{subject to}\quad||s||_2 \leq 1, \;s^T{Z}=0,$$ with $a=Xu$. The solution to is directly related to the method outlined in \[sec:fpl\], and is given by the following expression. $$s=\frac{a-\beta Z}{||a-\beta Z||_2},$$ with $\beta = (Z^TZ)^{-1}Z^Ta$.
Solutions with rank greater than $1$ are obtained by consecutive univariate optimisation. For the $j$-th pair $(u_j,s_j)$, $j=2,\dots,k$, the vector $a$ of the partial problem outlined in \[eq:equivalent\] is replaced with $P_{k-1}Xu_j^T$, where $P_{k-1} = I_{n\times n} - \sum_{l=1}^{k-1} s_l s_l^T$ projects $Xu_j^T$ onto the complement of the orthogonal subspace $\text{span}( s_l,\dots, s_{k-1})$, thus guaranteeing orthogonality among the vectors $s_j$, $j=1,\dots,k$. A detailed description of the algorithm outlined above is given in the \[appendix\].
Simulation Study
================
We conduct a simulation study to evaluate the empirical performance of the proposed algorithms. The focus of the simulation is on assessing the fidelity in recovering a high-dimensional data matrix and success in removing the influence of the group variable from predictions for future subjects. We set $n = 5000, p = 200, k = 10$, and construct a loading matrix $S$, with size $(n,k)$, and a score matrix $W$ with size $(k,p)$, with entries sampled from independent normal distributions. A group variable $Z$ of length $n$ is sampled from independent Bernoulli distributions with probability equal to $0.5$. Each $p$-dimensional row of the $n\times p$ data matrix $X$ is drawn from a $p$-variate standard normal distribution with mean vector $\mu_i = (s_i - 2 z_i1_k^T)W$, $i=1, \dots, n$. Lastly, a synthetic continuous response $y_i$, $i=1,\dots,n$ is sampled from independent Normal random variables with mean $(s_i - 2z_i1_k^T)\beta$ and elements of $\beta$ sampled uniformly in $(-5, 5)$. We evaluate two aspects of performance: accuracy of the approximation of $X$ by $\widetilde X$ and success in achieving $\langle \hat Y,Z \rangle \approx 0$, where $\hat Y$ is the prediction for $Y$ estimated from $\widetilde X$.
The left-panel (a) of illustrates the relative error between the reconstructed and original data, measured in terms of Frobenius distance for increasing values of the approximation rank $k$, and normalised with the Frobenius norm of $X$. As expected, for every value of $k$, the truncated SVD provides the reconstruction with minimum error. The approximation provided by the OG algorithm is competitive with the optimal reconstruction, and the relative improvement of the latter increases with the rank until the true value $k=10$ is reached, and remains constant afterwards. This behaviour is not surprising, since the simulation setting for $X$ suggests that the singular values $\sigma_k$ for $k >10$ are nearly $0$, not affecting the quantity in \[lemma3\]. Moreover, although there are no theoretical guarantees on the error achieved with the SOG algorithm, in this simulation study its performance is numerically comparable to the performance of the OG algorithm.
Panel (b) in , compares the empirical distribution of the out of sample predictions for $\hat Y \mid Z=0$ and $\hat Y \mid Z=1$, under different approaches. Data have been divided into a training and a test set, and a standard linear regression estimated on the training data is used to predict the outcome $\hat Y$ on the remaining part. The dimensionality of the matrix $X$ was reduced using the OG and SOG algorithms, using an approximation rank $k=10$.
In the first panel, the predictive model was estimated over the ordinary SVD decomposition. Without adjustment, the distributions of the predicted values differ markedly as a function of $Z$, with the predictions for $Z=1$ more concentrated at low values and predictions with $Z=0$ at high values. The second and third panel illustrate the effect of removing the influence of the group $Z$ with the SO algorithm and SOG algorithm, respectively. After removing the effect of the group variable $Z$ from the transformed features $\widetilde X$, predictions across different values of $Z$ are similar in terms of their empirical distribution, with the conditional empirical cumulative distribution function matching almost perfectly.
\(a) (b)
Application
===========
Human connectome project {#sec:brains}
------------------------
Our motivating application is drawn from a study of the Human Connectome Project (HCP) on $n=1056$ adult subjects [@glasser2016human; @glasser2013minimal]. The study provides, for each individual, information on the structural interconnections among the $68$ brain regions characterizing the Desikan atlas [@desikan:2006], measured through a combination of diffusion and structural magnetic resonance imaging [@zhang2018relationships]. Many different features are also available, covering a wide range of biographical, physiological and behavioural information at the individual level and technical information related to the specific session in which brain scans were collected. For an extended description of the Humane Connectome Project, the tools involved in the collection process and the aims of the study see @zhang2018relationships [@glasser2016human; @glasser2013minimal]. For our purposes, it is enough to characterize the outcomes of interest as physiological and behavioural traits and the covariates as data on the presence and strength of connections between the 68 brain regions.
Recent developments in neuroscience and novel preprocessing pipelines have stimulated considerable interest in analysing the relation among brain structure/activity and subject-specific traits, with the main focus on detecting if variations in the brain structure are associated with variation in phenotypes [e.g. @genovese2002thresholding; @zhang2018relationships; @Durante:2018]. Our specific focus is on investigating the relation between brain structural connectivity patterns and “hard” drug consumption, in order to establish whether the latter can be predicted on the basis of information recorded in the brain scans. There is evidence for the existence of brain differences across subjects with severe drug addictions, both in terms of functional connectivity [@wilcox2011enhanced; @kelly2011reduced] and volumes [@beck2012effect; @goldstein2009neurocircuitry]. As discussed in \[sec:1\], a fundamental problem with observational medical data is the presence of spurious or nuisance associations. In neuroscience, harmful factors that can negatively impact on predictive modelling include subject motion, eye movements, different protocols and hardware-specific features, among many others [@basser2002diffusion; @sandrini2011use]. In the motivating application, a binary variable $y_i$ indicates a positive result for subject $i$ to a drug test for at least one among Cocaine, Opiates, Amphetamines, MethAmphetamine and Oxycontin. We regard the machine on which the data were gathered as the group variable whose influence we want to remove. For every observation, a binary variable $z_i$ indicates which scanner had recorded the $i$-th observation. In order to apply our proposed method, the brains scans were vectorised into a $n \times p$ matrix ${X}$, with $p=2278$ corresponding to the vectorised numerical data on the strength of connection among all pairs of brain regions.
Predictive performance for the different approaches is evaluated over an independent test set, randomly sampled from the original data. In order to reduce the randomness from a single split, the results are averaged over $300$ different splits into training and test. As a preliminary approach, analyses are conducted with two different naive approaches: using the original covariates, and including the scanner id variable $z$ as a covariate. The first and second columns of \[table:brains\] represent, respectively, results for a logistic regression using standard sparse SVD with $30$ components (SVD) as covariates, and the same covariates including scanner as an additional covariate (SVD,Z). The third and fourth columns compare predictive performance for Lasso (Lasso(u)) and random forest (RF(u)), using all the unadjusted available covariates. Results suggest that predictions differ markedly across the two different scanners, and this issue equally affects both linear and non-linear models. This problem is exacerbated when the scanner id is used as a covariate. For example, classification error for observations in the second scanner is roughly two times greater than for the first one, suggesting that predictions are strongly related with the scanner used to collect data.
The right half of \[table:brains\] shows results for the adjusted procedures. The fifth and sixth columns refer to predictions for a logistic regression estimated over $30$ covariates extracted from the OG algorithm and its sparse version SOG, respectively. Predictions are now more similar across different scanners. For example, false negative rates are almost identical, and the global performance is comparable to the unadjusted setting. In addition, our methodology allows estimation of any baseline model on our pre-processed data. In the seventh column, a Lasso model has been estimated using data from the OG algorithm, although the results are worse than for the other baseline models. Although the proposed methods provide strong guarantees only for models which are linear in the covariates, the last column of \[table:brains\] suggests that also a highly non-linear model (such as random forest) can obtain strong benefit from our methods. In this case, predictions across scanners are more similar, although less precise.
As a concluding check, we conducted separate Mann-Whitney tests to evaluate if the distribution of $\hat Y$ is statistically different across different scanners. The null hypothesis of independence was not rejected for all the adjusted methods, and rejected for the unadjusted. We also conducted sensitivity analysis for different values of the approximation rank ranging in $\{10,50,100\}$, and results were consistent with the main empirical findings.
\[table:brains\] 0.15in
----- ---- ------- -------- ---------- ------- ------- ------- ------- -------
SVD SVD, Z LASSO(u) RF(u) OG SOG LASSO RF
CE S1 0.302 0.266 0.161 0.454 0.293 0.312 0.403 0.451
S2 0.352 0.463 0.181 0.430 0.271 0.259 0.409 0.472
AUC S1 0.564 0.573 0.546 0.601 0.547 0.534 0.529 0.552
S2 0.647 0.631 0.644 0.714 0.547 0.552 0.552 0.621
ACC S1 0.608 0.644 0.749 0.456 0.617 0.598 0.507 0.459
S2 0.558 0.447 0.729 0.480 0.639 0.651 0.501 0.438
FNR S1 0.286 0.250 0.144 0.442 0.279 0.298 0.437 0.441
S2 0.313 0.430 0.154 0.392 0.241 0.222 0.427 0.441
FPR S1 0.015 0.016 0.017 0.012 0.014 0.014 0.010 0.010
S2 0.039 0.033 0.035 0.037 0.030 0.037 0.024 0.031
----- ---- ------- -------- ---------- ------- ------- ------- ------- -------
: Predictive performance on the HCP dataset for the approaches described in \[sec:brains\]. S1 and S2 denote first and second scanner, respectively. Metrics: Classification Error, AUC, Accuracy, False negative rates, False positive rates.
-0.1in
COMPAS recidivism data {#sec:COMPAS}
----------------------
As outlined in \[sec:1\], another important area in which it is fundamental to remove unwanted association is criminal risk assessment. We will focus here on the COMPAS dataset, a standard dataset in the fairness literature which includes detailed information on criminal history for more than $6000$ defendants over a time range of two years [@angwin2016machine]. For each defendant, several features of criminal history are available, such as the number of past felonies, misdemeanors, and juvenile offenses. The defendant’s sex, age and race are also available. The focus of this example is on predicting two-year recidivism, with particular interest on providing predictions that are independent of race/ethnicity.
The design matrix $X$ was constructed including the available features and all the interaction terms, for a total of $64$ variables. Although the number of features is only moderately large, and considerably smaller than in the previous example, the use of approaches that requires manual intervention, such as @lum2016statistical, is burdensome. Moreover, the inclusion of every interaction term induces collinearity in the design matrix, thus motivating an approximation through a lower-dimensional structure.
Predictive performance for different competitors is evaluated over an independent test set, randomly sampled from the original data. compares the out-of-sample predictive distribution for four logistic regressions, trained on different data. The first two panels show unadjusted predictions, the first excluding racial information and the second using it as a covariate. Results show that Caucasian individuals (light continuous line) are systematically assigned lower probabilities of recidivism, at any level of the predicted risk. The third and fourth panels correspond, respectively, to predictions obtained from the OG and SOG algorithms, with $k=10$. The gap between the two curves is notably reduced, both with the standard OG and the sparse implementation, leading to predictions which are more similar across different racial groups.
\[table:COMPAS\] reports results for the model previously described and other competitor approaches. The first and second columns of \[table:COMPAS\] represent, respectively, results for a logistic regression using all the available original covariates (OR) and all the variables and race (OR,Z). The third column (OR $Z_0$) corresponds to one approach discussed in @pope2011implementing, which consists in the estimation of a complete model with all the variables and race, and then setting to $0$ the coefficient associated with group membership when predictions are performed. An alternative approach suggested in @pope2011implementing consists in a full model estimation, and predictions obtained replacing $z$ with its mean value $\bar z$ in the test data. In this application, such approach leads to metrics identical to column (OR Z), and has not been reported.
The fourth and fifth columns compare predictive performance for Lasso (Lasso (u)) and random forest (RF(u)) respectively, using all the unadjusted available covariates. Numerical results suggest that predictions are strongly different across ethnic groups, and this issue affects both models linear and non-linear in the covariates. For example, the proportion of reoffenders correctly classified (TPR, true positive rates) is roughly $1.5$ times higher for white than for non-white subjects, and this issue holds for all the unadjusted approaches. Conversely, the True Negative Rate (TNR) is significantly lower for whites than for non-white individuals, suggesting that models estimated on the unadjusted features disproportionately assign low probabilities of recidivism to white subjects.
The second part of \[table:COMPAS\] illustrates results for the adjusted procedures. The sixth and seventh columns show results for a logistic regression estimated over $10$ covariates extracted from the OG algorithm and its sparse version SOG, respectively, also reported in the third and fourth panel of . The discrepancy in prediction metrics across racial group is less severe under these approaches. For example the True Positive Rate is increased and more similar across groups. Similarly to the brain scan application, results seems satisfactory also when a highly non-linear model is employed, such as a random forest in column (RF); predictive performance using the adjusted data is also on par with other articles that have analysed the same dataset [e.g. @dieterich2016compas; @lum2016statistical].
These empirical findings also highlight a compelling argument for using of the proposed method in risk assessment. Considering predictive performance of models with and without OG pre-processing, the values for Area Under the Roc curve and classification error before and after adjustment are very similar, in particular for the logistic regressions reported in \[fig:COMPAS\] and in the columns OR and OG, SOG in \[table:COMPAS\]. This result suggests that, when interest is on predicting two-year recidivism, predictions from our methods achieve orthogonality from groups without large effects on the global accuracy.
0.15in
----- ---- ------- ------- ---------- ---------- ------- ------- ------- ------- -------
OR OR Z OR $Z_0$ LASSO(u) RF(u) OG SOG LASSO RF
CE W 0.281 0.285 0.281 0.303 0.285 0.329 0.329 0.344 0.354
NW 0.318 0.316 0.316 0.321 0.325 0.333 0.333 0.332 0.395
AUC W 0.734 0.734 0.734 0.715 0.725 0.722 0.722 0.717 0.655
NW 0.729 0.729 0.729 0.720 0.733 0.731 0.731 0.732 0.651
TPR W 0.569 0.562 0.569 0.534 0.577 0.453 0.453 0.434 0.444
NW 0.401 0.408 0.419 0.389 0.428 0.434 0.434 0.447 0.325
TNR W 0.150 0.153 0.150 0.163 0.138 0.218 0.218 0.221 0.201
NW 0.282 0.276 0.266 0.290 0.247 0.233 0.233 0.221 0.280
FNR W 0.045 0.052 0.045 0.080 0.037 0.161 0.161 0.180 0.170
NW 0.123 0.116 0.105 0.135 0.096 0.089 0.089 0.077 0.199
FPR W 0.236 0.233 0.236 0.223 0.248 0.168 0.168 0.165 0.185
NW 0.194 0.200 0.211 0.186 0.229 0.244 0.244 0.255 0.196
----- ---- ------- ------- ---------- ---------- ------- ------- ------- ------- -------
: Predictive performance on the COMPAS dataset for the approaches described in \[sec:COMPAS\]. W stands for White ethnicity, NW for the remaining ethnic groups. Metrics: Classification Error, AUC, Total positive Rate, Total Negative Rate, False negative rates, False positive rates. []{data-label="table:COMPAS"}
-0.1in
Discussion {#discussion .unnumbered}
==========
In this article we proposed an efficient method to pre-process high-dimensional datasets to remove the influence of groups in predictive modelling. Our approach is simple, scalable, and has theoretical guarantees regarding approximation error and orthogonality from the group variables. Although the theoretical properties of the methods hold on models linear in the covariates, the empirical findings suggest that also when non-linear models are employed, the methods still work well, provide reasonable global predictive performance and predictions that are more similar across groups. A promising future extension for the proposed method involves generalisation to more complex multidimensional data structures. For example, in the neuroscience application considered in \[sec:brains\], brain scans for multiple subjects can be represented as a 3-dimensional array, where every slice corresponds to a brain network for a single subject [@zhang2018relationships]. The inclusion of constraints on orthogonality to groups can be accomplished adapting the contribution proposed in this article to some methods in tensor decompositions [e.g. @kolda2009tensor].
Appendix
========
Focus on the case $k=1$, where the approximation of the original set of data consists of finding the closest rank-$1$ matrix (vector). is reformulated as
$$\begin{aligned}
\label{eq:1d}
\operatorname*{arg\,min}_{s_1,u_1}&\bigg\{ \frac {1}{n}\sum_{i=1}^n ||x_i - s_{i1} u_1^T||^2 + \frac{2}{n}\lambda_1 \sum_{i=1}^n s_{i1}z_i\bigg\},\end{aligned}$$
and some algebra and the orthonormal condition on $u_1$ allows us to express the loss function to be minimized as
$$\begin{aligned}
L(s_1,u_1) &=\frac {1}{n}\sum_{i=1}^n (x_i - s_{i1} u_1^T)^T (x_i - s_{i1} u_1^T) + \frac{2}{n}\lambda_1 \sum_{i=1}^n s_{i1}z_i \\
&=\frac {1}{n}\sum_{i=1}^n (x_i^Tx_i - 2s_{i1}x_iu_1^T+s_{i1}^2)+\frac{2}{n}\lambda_1 \sum_{i=1}^n s_{i1}z_i.\end{aligned}$$
The function is quadratic, and the partial derivative in $s_{i1}$ leads to $$\begin{aligned}
\frac{\partial}{\partial s_{i1}}L(s_1,u_1) = \frac{1}{n}(- 2x_iu_1^T+2s_{i1}) +\frac{2}{n}\lambda_1 z_i,\end{aligned}$$ with stationary point given by $s_{i1} = x_iu_1^T-\lambda_1 z_i$. The optimal score for the $i$-th subject is obtained by projecting the observed data onto the first basis, and then subtracting $\lambda_1$-times $z$. The constraint does not involve the orthonormal basis $u_1$, hence the solution of for $u_1$ is equivalent to the unconstrained scenario. A standard result of linear algebra states that the optimal $u_1$ for without constraints equivalent to the first right singular vector of ${X}$, or equivalently to the first eigenvector of the matrix ${X}^T{X}$ [e.g., @hastiefriedman]. Plugging in the solution for $u_1$ and setting the derivative with respect to $\lambda_1$ equal to $0$ leads to
$$\sum_{i=1}^n (x_i u_1^T-\lambda_1 z_i) ^T z_i = 0 \qquad \lambda_1 = \frac{\sum_{i=1}^n x_i u_1^T z_i}{\sum_{i=1}^n z_i^2} = \frac{\langle Xu_1^T,z \rangle}{\langle z,z\rangle},$$
a least squares estimate of ${X}{u}_1^T$ over $z$.
Consider the complete problem formulated in . The derivatives with respect to the generic element $s_{ij}$ can be calculated easily due to the constraint ${U}\in G_{k,p}$, which simplifies the mixed products among the $u_j$s. The optimal solution for the generic score $s_{ij}$ is given by $$\label{eq:derk}
s_{ij} = x_iu_j^T-\lambda_j z_i,$$ since $u_i^Tu_j=0$ for all $i \neq j$ and $u_j^Tu_j = 1$ for $j=1,\dots,k$. The solution has an intuitive interpretation, since it implies that the optimal scores for the $j$-th dimension are obtained projecting the original data over the $j$-th basis, and then subtracting $\lambda_j$-times the observed value of $z$. Moreover, since the OG constraints do not involve any vector $u_j$, the optimization with respect to the basis can be derived from known results in linear algebra. The optimal value for the vector $u_j$, with $j=1,\dots,k$, is equal to the first $k$ right singular values of ${X}$, sorted accordingly to the associated singular values [e.g., @bishop2006pattern; @hastie2015statistical].
The global solution for $\lambda = (\lambda_1, \dots, \lambda_k)$ can be derived from least squares theory, since we can interpret as a multivariate linear regression where the $k$ columns of the projected matrix ${X}{U}^T$ are response variables and $z$ a covariate. The general optimal value for $\lambda_k$ is then equal to the multiple least squares solution $$\lambda_k = \frac{\langle{X}u_k^T,z\rangle}{\langle z,z\rangle}$$.
Since the optimization problem of is quadratic with a linear constraint, any local minima is also a global minima. The solution performed via the singular value decomposition and the least squares constitute a stationary point, that is also global minimum.
Let $V_kD_kU_k^T$ define the rank-$k$ SVD decomposition of the matrix ${X}$, using the first $k$ left and right singular vectors, and the first $k$ singular vales. Let $\widetilde X_{OG}$ define the approximated reconstruction obtained by the OG algorithm. The reconstruction error between the original data matrix $X$ and its low-rank approximation $\widetilde X_{OG}$ can be decomposed as follow. $$\begin{aligned}
||{X}- \widetilde X_{OG}||^2_F =&||{X}- (V_k D_k-Z\lambda )U_k^T||^2_F \\
=&||{X}- V_kD_kU_k^T+Z\lambda U_k^T||^2_F\\
=&||{X}- V_kD_kU_k^T||_F^2 + ||Z\lambda U_k^T||^2_F
+&2\langle{X}- V_kD_kU_k^T,Z\lambda U_k^T \rangle_F.\end{aligned}$$
The Frobenius-inner product term vanishes due to the orthogonality of the singular vectors, and rearranging terms the following expression is obtained. $$\begin{aligned}
||{X}- \widetilde{X}_{OR}||^2_F - ||{X}- V_kD_kU_k^T||_F^2 = &||Z\lambda U^T||^2_F=||Z\lambda ||_F^2,\end{aligned}$$
Since the optimal value for $\lambda$ is equal to the least squares solution of $Z$ over $V_kD_k$, it follows that $||Z\lambda ||_F^2=||Z(Z^TZ)^{-1}Z^TV_kD_k||_F^2$, and the proof is complete.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'Two approaches to nonperturbative renormalization are discussed for theories quantized on the light cone. One is tailored specifically to a calculation of the dressed-electron state in quantum electrodynamics, where an invariant-mass cutoff is used as a regulator and a Tamm–Dancoff truncation is made to include no more than two photons. The other approach is based on Pauli–Villars regulators and is applied to Yukawa theory and a related soluble model. In both cases discretized light-cone quantization is used to obtain a finite matrix problem that can be solved nonperturbatively.'
address: 'Department of Physics, University of Minnesota, Duluth, Minnesota 55812'
author:
- 'John R. Hiller'
date: 'March 20, 1997'
title: 'Nonperturbative Renormalization in Light-Cone Quantization[^1] '
---
INTRODUCTION
============
Light-cone quantization[@Dirac] has attracted some interest as a means to perform nonperturbative analyses of quantum field theories[@Reviews]. There are good reasons to hope that this technique will provide the leverage needed to obtain a qualitative, and perhaps quantitative, connection between quantum chromodynamics (QCD) and the constituent quark model[@Wilson]. Given the complexity of QCD, it is useful to first study simpler theories such as quantum electrodynamics (QED) and even models in $1+1$ spacetime dimensions rather than $3+1$ dimensions.
Bound-state calculations in QCD$_{3+1}$ and QED$_{3+1}$ require nonperturbative renormalization. Most attempts at such calculations have used Tamm–Dancoff truncations[@TammDancoff] and cutoff-type regularization, which require counterterms that depend on Fock sector[@SectorDependent]. An example of such a calculation is given here for the electron’s anomalous moment[@Zakopane]. We then explore the practicality of Pauli–Villars regularization[@PauliVillars] as an alternative. In particular, we consider a simple heavy-fermion model abstracted from the Yukawa model.
We define light-cone coordinates by $$\label{eq:coordinates}
x^\pm=t\pm z\,,\;\;{\bf x}_\perp=(x,y)\,.$$ Momentum variables are similarly constructed as $$\label{eq:momentum}
p^\pm=E\pm p_z\,,\;\;{\mathbf p}_\perp=(p_x,p_y).$$ The dot product is written $$p\cdot x=\frac{1}{2}(p^+x^-+p^-x^+)
-{\bf p}_\perp\cdot{\bf x}_\perp\,.$$ The time variable is taken to be $x^+$, and time evolution of a system is then determined by the conjugate operator ${\cal P}^-$. The energy $E$ is replaced by the light-cone energy $p^-$, and the momentum ${\bf p}$ by the light-cone momentum $\underline{p}\equiv (p^+,{\bf p}_\perp)$. The light-cone Hamiltonian is $$\label{eq:HLC}
H_{\rm LC}={\cal P}^+{\cal P}^- - {\cal P}^2_\perp\,,$$ where ${\cal P}^+$ and ${\bf \cal P}_\perp$ are momentum operators conjugate to $x^-$ and ${\bf x}_\perp$. The eigenvalue problem is $$\label{eq:EigenProb}
H_{\rm LC}\Psi=M^2\Psi\,,\;\;
\underline{\cal P}\Psi=\underline{P}\Psi\,,$$ where $M$ is the mass of the state.
Some of the advantages of light-cone coordinates are the following: They admit the largest possible set of nondynamical generators. In particular, boosts are kinematical. For many theories of massive particles, the perturbative vacuum is the physical vacuum, because $p_i^+=\sqrt{p^2+m^2}+p_z>0$ implies that no particle state can contribute to the $P^+=0$ vacuum. Thus there is no need to compute the vacuum state before computing massive states. Also, well-defined Fock-state expansions exist, with no disconnected vacuum pieces.
Such expansions are written as $$\label{eq:Psi}
\Psi=\sum_n\int [dx]_n\,[d^2k_\perp]_n\,
\psi_n(x,{\bf k}_\perp)
|n:xP^+,x{\bf P}_\perp+{\bf k}_\perp\rangle\,,$$ with $n$ the number of particles, $i$ ranging between 1 and $n$, $(P^+,{\bf P}_\perp)$ the total light-cone momentum, and $$\label{eq:dx}
[dx]_n=4\pi\delta(1-\sum_{i=1}^nx_i)
\prod_{i=1}^n\frac{dx_i}{4\pi\sqrt{x_i}}\,,\;\;\;
[d^2k_\perp]_n=4\pi^2\delta(\sum_{i=1}^n{\bf k}_{\perp i})
\prod_{i=1}^n\frac{d^2k_{\perp i}}{4\pi^2}\,.$$ In the Fock basis $\{|n:p_i^+,{\bf p}_{\perp i}\rangle\}$, ${\cal P}^+$ and ${\bf \cal P}_\perp$ are diagonal. The amplitude $\psi_n$ is interpreted as the wave function of the contribution from states with $n$ particles.
A common numerical technique is discretized light-cone quantization (DLCQ)[@PauliBrodsky], in which periodic boundary conditions are assigned to bosons and antiperiodic to fermions in a light-cone box $-L<x^-<L$, $-L_\perp<x,y<L_\perp$. Integrals are replaced by trapezoidal approximations on a grid: $p^+\rightarrow\frac{\pi}{L}n$, ${\bf p}_\perp\rightarrow
(\frac{\pi}{L_\perp}n_x,\frac{\pi}{L_\perp}n_y)$, with $n$ even for bosons and odd for fermions. The limit $L\rightarrow\infty$ can be exchanged for a limit in terms of the integer [*resolution*]{} $K\equiv\frac{L}{\pi}P^+$. The longitudinal momentum fraction $x=p^+/P^+$ becomes $n/K$. $H_{\rm LC}$ is independent of $L$.
Because the $n_i$ are all positive, DLCQ automatically limits the number of particles to no more than $\sim\!\!K/2$. The integers $n_x$ and $n_y$ range between limits associated with some maximum integer $N_\perp$ fixed by $L_\perp$ and a cutoff that limits transverse momentum.
To reduce the size of the discrete matrix problem, a Tamm–Dancoff truncation[@TammDancoff] in the number of particles can be applied. This has serious implications for renormalization. These include severe sector dependence of counterterms[@SectorDependent], and, for QED, violation of the Ward identity.
Regularization via cutoffs typically involves limits on the invariant mass. A limit can be placed on the total invariant mass of each Fock state $$\label{eq:cutoff1}
\sum_i \frac{m_i^2+k_{\perp i}^2}{x_i}\leq\Lambda^2$$ or on the invariant mass of each particle $$\label{eq:cutoff2}
\frac{m_i^2+k_{\perp i}^2}{x_i}\leq\Lambda^2\,.$$ There can also be a limit on the change in invariant mass across each matrix element of $H_{\rm LC}$[@Lepage] $$\label{eq:cutoff3}
\left|\sum_i^n \frac{m_i^2+k_{\perp i}^2}{x_i}
-\sum_j^m \frac{m_j^2+k_{\perp j}^2}{x_j}\right|
\leq\Lambda^2\,.$$
THE ANOMALOUS MOMENT
====================
The anomalous moment $a_e=F_2(0)$ can be computed from a spin-flip matrix element of the electromagnetic current $$\label{eq:current}
-\frac{q_1}{2m_e}F_2(q^2)=
\frac{1}{2P^+}\langle P+q,\uparrow|J^+(0)|P,\downarrow\rangle$$ in the standard light-cone frame $q=(0,q_\perp^2/P^+,{\bf q}_\perp=q_1\hat{x})$. Brodsky and Drell[@BrodskyDrell] have given a useful reduction of this matrix element to the form $$\label{eq:aeLC}
a_e=-2m_e\sum_je_j\sum_n\int\,[dx]_n\,[d^2k_\perp]_n\,
\psi_{n\uparrow}^*(x,{\bf k}_\perp)
\sum_{i\neq j}x_i\frac{\partial}{\partial k_{1i}}
\psi_{n\downarrow}(x,{\bf k}_\perp)\,,$$ where $e_j$ is the fractional charge of the struck constituent and $x_i=p_i^+/P^+$. The wave functions $\psi_n$ satisfy coupled integral equations obtained from $H_{\rm LC}\Psi=M^2\Psi$. The QED light-cone Hamiltonian has been given by Tang [*et al.*]{}[@Tang]. However, the bare masses and couplings must be computed from sector dependent renormalization conditions.
Consider the case where there are at most two photons and only one electron. The Fock-state expansion can be written schematically as $$\label{eq:SchematicExpansion}
\Psi=\psi_0|e\rangle+\vec{\psi}_1|e\gamma\rangle
+\vec{\psi}_2|e\gamma\gamma\rangle\,.$$ Here $\vec{\psi}_1$ and $\vec{\psi}_2$ are column vectors that contain the amplitudes for individual Fock states with one and two photons, respectively. The eigenvalue problem becomes a coupled set of three integral equations $$\begin{aligned}
\label{eq:IntegralEqns}
m_0^2\psi_0 + {\bf b}_1^\dagger\cdot\vec{\psi}_1
+ {\bf b}_2^\dagger\cdot\vec{\psi}_2 & = & M^2\psi_0\,,
\nonumber \\
{\bf b}_1\psi_0 + A_{11}\vec{\psi}_1
+ A_{12}\vec{\psi}_2 & = & M^2\vec{\psi}_1\,, \\
{\bf b}_2\psi_0 + A_{12}^\dagger\vec{\psi}_1
+ A_{22}\vec{\psi}_2 & = & M^2\vec{\psi}_2\,, \nonumber\end{aligned}$$ where $m_0$ is the bare electron mass and ${\bf b}_i^\dagger$ and $A_{ij}$ are integral operators obtained from matrix elements of $H_{\rm LC}$.
The bare electron mass in the one-photon sector is computed from the one-loop self energy allowed by the two-photon states. We then require that $m_0$ be such that $M^2=m_e^2$ is an eigenvalue. The second and third equations can be solved for $\vec{\psi}_1/\psi_0$ and $\vec{\psi}_2/\psi_0$. Then the first equation yields $m_0$. Normalization of $\Psi$ fixes the value of $\psi_0$.
The bare coupling for the electron-photon three-point vertex depends on the initial and final momenta of the electron and on the sectors between which the coupling acts. The momentum dependence is present because the amount of momentum available constrains the extent to which higher order corrections can contribute. Similarly, the sector dependence makes itself felt when the number of additional particles in higher-order corrections is restricted. The coupling is fixed by the ratio of the e$\gamma\rightarrow$e transition matrix element to the bare vertex at zero photon momentum.
In the present calculation we use a Tamm–Dancoff truncation to {e, e$\gamma$, e$\gamma\gamma$}, a nonzero photon mass $m_\gamma=m_e/10$, and a moderate coupling $\alpha=1/10$. Some results are given elsewhere[@Zakopane]. When only states with at most one photon and no pairs are retained, one can show that $a_e$ reduces to $$a_e=\frac{\alpha m_e^2}{\pi^2}\int\,\frac{dx\,d^2k_\perp}{x}
\frac{\theta(\Lambda^2-(m_e^2+k_\perp^2)/x
-(m_\gamma^2+k_\perp^2)/(1-x))}
{[m_e^2-(m_e^2+k_\perp^2)/x
-(m_\gamma^2+k_\perp^2)/(1-x)]^2}\,,$$ which in the limit of $\Lambda\longrightarrow\infty$ becomes[@BrodskyDrell] $$a_e=\frac{\alpha}{2\pi}\int_0^1
\frac{2x^2(1-x)dx}{x^2+(1-x)(m_\gamma/m_e)^2}\,.$$ For $m_\gamma=0$, this yields the standard Schwinger contribution[@Schwinger] of $\alpha/2\pi$.
YUKAWA THEORY AT ONE LOOP
=========================
As an alternative approach to regularization, we consider Pauli–Villars[@PauliVillars] regularization of the $3+1$ Yukawa model[@Yukawa; @BurkardtLangnau]. The one-loop fermion self-energy is proportional to $$I(\mu^2,M^2)\equiv-\frac{1}{\mu^2}\int
\frac{dl^+d^2l_\perp}{l^+(q^+-l^+)^2}
\frac{(q^+)^2{\bf l}_\perp^2+(2q^+-l^+)^2M^2}
{M^2-D_1}\theta(\Lambda^2-D_1)\,,$$ where $\underline{q}$ is the fermion momentum, $\mu$ is the boson mass, $M$ is the fermion mass, and $$D_1=\frac{\mu^2+{\bf l}_\perp^2}{l^+/q^+}
+\frac{M^2+{\bf l}_\perp^2}{(q^+-l^+)/q^+}\,.$$ The boson mass $\mu$ sets the energy scale. When $M^2=0$ we obtain $$I(\mu^2,0)=\frac{\pi}{\mu^2}\left[\frac{\Lambda^2}{2}-
\frac{\mu^4}{2\Lambda^2}-
\mu^2\ln\left(\frac{\Lambda^2}{\mu^2}\right)\right]\,.$$ In order to maintain $I(\mu^2,M^2)\propto M^2$, three Pauli-Villars bosons are needed:[@ChangYan] $$\label{eq:Isub}
I_{\rm sub}(\mu^2,M^2,\mu_i^2)=
I(\mu^2,M^2)+\sum_{i=1}^3 C_i I(\mu_i^2,M^2)\,.$$ The $C_i$ are chosen to satisfy $$1+\sum_{i=1}^3 C_i=0\,, \;\;
\mu^2+\sum_{i=1}^3 C_i\mu_i^2=0\,, \;\;
\sum_{i=1}^3 C_i\mu_i^2\ln(\mu_i^2/\mu^2)=0\,.$$ A DLCQ calculation of $I_{\rm sub}$ has been done[@BHM], with values of 20, 22, and 24 for $K$ and 25 through 30 for $N_\perp$. Modification of the trapezoidal rule, with introduction of unequal weights, is necessary to obtain sufficient accuracy. Each integral in (\[eq:Isub\]) was separately extrapolated to infinite $K$ and $N_\perp$ via fits to either $c_0+a_1/K^3+b_1/N_\perp^2$ or $c_0+a_1/K^3+a_2/K^4+b_1/N_\perp^2+b_2/N_\perp^3$. The latter was used for the $\mu_1$ integral. Extrapolation after subtraction is not as accurate. The resulting values of $I_{\rm sub}$ were extrapolated to infinite cutoff by fits to $a+b/\Lambda^2$. These fully extrapolated values are given in Table \[tab:Linfinity\].
$M^2$ 0 $0.05\mu^2$ $0.1\mu^2$ $0.2\mu^2$
--------------- -------- ------------- ------------ ------------
$I_{\rm sub}$ -0.064 0.70 1.37 2.70
: Values of the subtracted integral $I_{\rm sub}(M^2/\mu^2,\mu_i^2/\mu^2)$ in the limit of infinite cutoff. The Pauli–Villars masses are $\mu_1^2=10\mu^2$, $\mu_2^2=50\mu^2$ and $\mu_3^2=100\mu^2$.[]{data-label="tab:Linfinity"}
The magnitude of the error in each extrapolated integral was found to be $\leq0.02$ when compared to the analytic result for $M^2=0$. This implies an error of $\pm0.04$ in the $I_{\rm sub}$ values. The extrapolation in $\Lambda^2$ induces additional uncertainty reflected in the miss of zero by 0.06 for $M^2=0$. The values in Table \[tab:Linfinity\] are consistent with $I_{\rm sub}\propto M^2$ to within this amount of error.
The number of Fock states required for Pauli–Villars particles is approximately 1.5 times the number for physical states. A listing of counts for two cases is given in Table \[tab:FockStates\]. Making $\mu_1$ larger does decrease the number of Pauli-Villars states but this increases the coefficients $C_i$ and thereby amplifies errors in the integrals. Also, with fewer states, the integrals themselves are approximated less accurately.
------------------- ----- ----------- -------------- ------------------- ------------------- -------------------- -------
physical
$\Lambda^2/\mu^2$ $K$ $N_\perp$ boson states $\mu_1^2=10\mu^2$ $\mu_2^2=50\mu^2$ $\mu_3^2=100\mu^2$ total
200 20 25 25975 22602 11142 3305 37049
200 24 30 44943 39162 19293 5695 64150
------------------- ----- ----------- -------------- ------------------- ------------------- -------------------- -------
: Number of Fock states used in two typical cases.[]{data-label="tab:FockStates"}
We could also consider the boson self energy. To lowest order there is a fermion loop contribution $$\int\frac{dl^+d^2l_\perp}{4LL_\perp^2}
\frac{q^+(l_\perp^2+M^2)}{l^{+2}(q^+-l^+)^2}
\frac{\theta\left(\Lambda^2-D_2\right)}{\mu^2-D_2}\,,$$ where $$D_2\equiv q^{+2}(M^2+l_\perp^2)/[l^+(q^+-l^+)]\,,$$ and a $\phi^4$ contribution $$\int\frac{dl^+d^2l_\perp dk^+d^2k_\perp}{q^+l^+k^+(q^+-l^+-k^+)}
\frac{\theta\left(\Lambda^2-D_4\right)}{\mu^2-D_4}\,,$$ where $$D_4\equiv\frac{\mu^2+l_\perp^2}{l^+/q^+}
+\frac{\mu^2+k_\perp^2}{k^+/q^+}
+\frac{\mu^2+(l_\perp+k_\perp)^2}{(q^+-l^+-k^+)/q^+}\,.$$ A Pauli–Villars fermion may be needed.
A HEAVY-FERMION MODEL
=====================
By some severe modifications of the Yukawa Hamiltonian[@McCartorRobertson] we obtain the following model Hamiltonian: $$\begin{aligned}
H_{\rm LC}^{\rm eff}
&=&M_0^2\int\frac{dp^+d^2p_\perp}{16\pi^3p^+}
\sum_\sigma b_{\underline{p}\sigma}^\dagger
b_{\underline{p}\sigma}
+P^+\int\frac{dq^+d^2q_\perp}{16\pi^3q^+}
\left[\frac{\mu^2+q_\perp^2}{q^+}
a_{\underline{q}}^\dagger a_{\underline{q}}
+ \frac{\mu_1^2+q_\perp^2}{q^+}
a_{1\underline{q}}^\dagger a_{1\underline{q}}
\right] \nonumber \\
& & +g\int\frac{dp_1^+d^2p_{\perp1}}{\sqrt{16\pi^3p_1^+}}
\int\frac{dp_2^+d^2p_{\perp2}}{\sqrt{16\pi^3p_2^+}}
\int\frac{dq^+d^2q_\perp}{16\pi^3q^+}
\sum_\sigma b_{\underline{p}_1\sigma}^\dagger
b_{\underline{p}_2\sigma} \\
& & \rule{0.5in}{0mm}\times \left[
a_{\underline{q}}^\dagger
\delta(\underline{p}_1-\underline{p}_2+\underline{q})
+a_{\underline{q}}
\delta(\underline{p}_1-\underline{p}_2-\underline{q})
\right. \nonumber \\
& & \rule{0.75in}{0mm} \left.
+ia_{1\underline{q}}^\dagger
\delta(\underline{p}_1-\underline{p}_2+\underline{q})
+ia_{1\underline{q}}
\delta(\underline{p}_1-\underline{p}_2-\underline{q})
\right]\,. \nonumber\end{aligned}$$ The kinetic energy of the fermion is no longer momentum dependent and only a modified no-flip three-point vertex remains as an interaction. The fermion then acts as a “static” source for the boson. We include one Pauli–Villars field, which will prove sufficient in this case. Similar Hamiltonians, without the Pauli–Villars field, have been considered in equal-time[@SchweberGreenberg] and light-cone coordinates[@GlazekPerry].
We write the eigenvector as a Fock-state expansion $$\begin{aligned}
\Phi_\sigma&=&\sqrt{16\pi^3P^+}\sum_{n,n_1}
\int\frac{dp^+d^2p_\perp}{\sqrt{16\pi^3p^+}}
\prod_{i=1}^n\int\frac{dq_i^+d^2q_{\perp i}}{\sqrt{16\pi^3q_i^+}}
\prod_{j=1}^{n_1}
\int\frac{dr_j^+d^2r_{\perp j}}{\sqrt{16\pi^3r_j^+}} \\
& & \times \delta(\underline{P}-\underline{p}
-\sum_i^n\underline{q}_i-\sum_j^{n_1}\underline{r}_j)
\phi^{(n,n_1)}(\underline{q}_i,\underline{r}_j;\underline{p})
\frac{1}{\sqrt{n!n_1!}}b_{\underline{p}\sigma}^\dagger
\prod_i^n a_{\underline{q}_i}^\dagger
\prod_j^{n_1} a_{1\underline{r}_j}^\dagger |0\rangle \,,
\nonumber\end{aligned}$$ normalized according to $\Phi_\sigma^{\prime\dagger}\cdot\Phi_\sigma
=16\pi^3P^+\delta(\underline{P}'-\underline{P})$, which yields $$\label{eq:NormCondition}
1=\sum_{n,n_1}\prod_i^n\int\,dq_i^+d^2q_{\perp i}
\prod_j^{n_1}\int\,dr_j^+d^2r_{\perp j}
\left|\phi^{(n,n_1)}(\underline{q}_i,\underline{r}_j;
\underline{P}-\sum_i\underline{q}_i
-\sum_j\underline{r}_j)\right|^2\,.$$ For $\Phi_\sigma$ to satisfy the Schrödinger equation (\[eq:EigenProb\]), the amplitudes must satisfy $$\begin{aligned}
\left[M^2-M_0^2\rule{0mm}{0.35in}\right.
&&\left.\rule{0mm}{0.35in}-\sum_i\frac{\mu^2+q_{\perp i}^2}{y_i}
-\sum_j\frac{\mu_1^2+r_{\perp j}^2}{z_j}
\right]\phi^{(n,n_1)} \\
& & =g\left\{\sqrt{n+1}\int\frac{dq^+d^2q_\perp}{\sqrt{16\pi^3q^+}}
\phi^{(n+1,n_1)}(\underline{q}_i,\underline{q},
\underline{r}_j,\underline{p})\right.
\nonumber \\
& & \rule{0.75in}{0mm}
+\frac{1}{\sqrt{n}}\sum_i\frac{1}{\sqrt{16\pi^3q_i^+}}
\phi^{(n-1,n_1)}(\underline{q}_1,\ldots,\underline{q}_{i-1},
\underline{q}_{i+1},\ldots,\underline{q}_n,
\underline{r}_j,\underline{p})
\nonumber \\
& &\rule{0.75in}{0mm}+i\sqrt{n_1+1}
\int\frac{dr^+d^2r_\perp}{\sqrt{16\pi^3r^+}}
\phi^{(n,n_1+1)}(\underline{q}_i,\underline{r}_j,
\underline{r},\underline{p})
\nonumber \\
& & \rule{0.75in}{0mm}
+\left.\frac{i}{\sqrt{n}}\sum_j\frac{1}{\sqrt{16\pi^3r_j^+}}
\phi^{(n,n_1-1)}(\underline{q}_i,
\underline{r}_1,\ldots,\underline{r}_{j-1},
\underline{r}_{j+1},\ldots,\underline{r}_{n_1},
\underline{p}) \right\}\,. \nonumber\end{aligned}$$ The structure of this coupled set of integral equations is deliberately identical in basic form to the equations considered by Greenberg and Schweber[@SchweberGreenberg]. Therefore, we transcribe their [*ansatz*]{} for a solution to light-cone form $$\phi^{(n,n_1)}=\sqrt{Z}\frac{(-g)^n(-ig)^{n_1}}{\sqrt{n!n_1!}}
\prod_i\frac{q_i^+}{\sqrt{16\pi^3q_i^+}(\mu^2+q_{\perp i}^2)}
\prod_j\frac{r_j^+}{\sqrt{16\pi^3r_j^+}(\mu_1^2+r_{\perp j}^2)}\,.$$ This does work as a solution if $M_0^2(\mu_1)$ is chosen to satisfy $$M^2-M_0^2=-\frac{g^2}{16\pi^3}
\left\{\int\frac{dyd^2q_\perp}{\mu^2+q_\perp^2}
-\int\frac{dzd^2r_\perp}{\mu_1^2+r_\perp^2}\right\}\,.$$ From the normalization condition (\[eq:NormCondition\]) we obtain $$\frac{1}{Z}=\exp\left\{\frac{g^2}{16\pi^3}
\left[\int\frac{ydyd^2q_\perp}{(\mu^2+q_\perp^2)^2}
+\int\frac{zdzd^2r_\perp}{(\mu_1^2+r_\perp^2)^2}
\right]\right\}\,.$$ The bare mass and wave function renormalization are thus determined as functions of the Pauli–Villars mass.
To fix the coupling we could use the slope of the fermion no-flip form factor, which is related to the transverse size of the dressed fermion. The form factor is most easily evaluated from[@BrodskyDrell] $$\begin{aligned}
F(Q^2)&=&\frac{1}{2P^+}
\langle P+p_\gamma\uparrow |J^+(0)|P\uparrow\rangle \\
&=&\sum_j e_j\int 16\pi^3\delta(1-\sum_i x_i)
\delta(\sum_i {\bf k}_{\perp i})
\prod_i \frac{dx_id^2p_{\perp i}}{16\pi^3} \nonumber \\
& & \rule{0.75in}{0mm}
\times \psi_{P+p_\gamma\uparrow}^*(x_i,{\bf p}'_{\perp i})
\psi_{P\uparrow}(x_i,{\bf p}_{\perp i})\,, \nonumber\end{aligned}$$ where the matrix element has been evaluated in the frame with $$P=(P^+,P^-=\frac{M^2}{P^+},{\bf 0}_\perp)\,,\;\;
p_\gamma=(0,p_\gamma^-=2p_\gamma\cdot P/P^+,
{\bf p}_{\gamma\perp})\,,\;\;
Q^2\equiv p_{\gamma\perp}^2\,,$$ $e_j$ is the charge of the jth constituent, and $${\bf p}'_{\perp i}=
\left\{\begin{array}{cc}
{\bf p}_{\perp i}-x_i{\bf p}_{\gamma\perp} & i\neq j \\
{\bf p}_{\perp i}+(1-x_i){\bf p}_{\gamma\perp} & i=j\,.
\end{array}\right.$$ A sum over Fock states is understood.
When the fermion is assigned a charge of 1, and the bosons remain neutral, the analytic solution for the amplitudes yields $$F(Q^2)=Z\exp\left\{g^2\int\frac{dy d^2q_\perp}{16\pi^3}
\frac{\sqrt{y}}{\mu^2+q_\perp^{\prime 2}}
\frac{\sqrt{y}}{\mu^2+q_\perp^2}
+\mbox{P-V term}\right\}\,,$$ with $${\bf q}'_\perp={\bf q}_\perp-y{\bf p}_{\gamma\perp}\,.$$ From this we find $$F'(0)=-g^2\int\frac{dyd^2q_\perp}{16\pi^3}
\frac{y^3}{(\mu^2+q_\perp^2)^3}
\left[\frac{2\mu^2}{\mu^2+q_\perp^2}-1\right]
+\mbox{P-V term}\,.$$ Numerically, the slope is computed from a finite-difference approximation to $$\begin{aligned}
F'(0)&=&\sum_{n,n_1}\prod_i^n\int\,dq_i^+d^2q_{\perp i}
\prod_j^{n_1}\int\,dr_j^+d^2r_{\perp j} \\
& & \times
\left[\left(\sum_i \frac{y_i^2}{4}\nabla_{\perp i}^2+
\sum_j \frac{z_j^2}{4}\nabla_{\perp j}^2\right)
\phi^{(n,n_1)}(\underline{q}_i,\underline{r}_j;
\underline{P}-\sum_i\underline{q}_i-\sum_j\underline{r}_j)
\right]^*
\nonumber \\
& & \rule{1in}{0mm} \times
\phi^{(n,n_1)}(\underline{q}_i,\underline{r}_j;
\underline{P}-\sum_i\underline{q}_i-\sum_j\underline{r}_j)\,.
\nonumber\end{aligned}$$ With the bare parameters determined, we “predict” a value for $\langle :\!\!\phi^2(0)\!\!:\rangle$. For the analytic solution, this expectation value reduces to $$\langle :\!\!\phi^2(0)\!\!:\rangle=
\frac{g^2}{8\pi^2\mu^2}\left[1-\frac{\mu^2}{\Lambda^2}
-\frac{\mu^2}{\Lambda^2}\ln\frac{\mu^2}{\Lambda^2}
\right]\,.$$ From a numerical solution it can be computed from a sum similar to the normalization sum $$\begin{aligned}
\langle :\!\!\phi^2(0)\!\!:\rangle
&=&\sum_{n=1,n_1=0}\prod_i^n\int\,dq_i^+d^2q_{\perp i}
\prod_j^{n_1}\int\,dr_j^+d^2r_{\perp j}
\left(\sum_{k=1}^n \frac{2}{q_k^+/P^+}
\right) \\
& & \rule{0.75in}{0mm}
\times \left|\phi^{(n,n_1)}(\underline{q}_i,\underline{r}_j;
\underline{P}-\sum_i\underline{q}_i-\sum_j\underline{r}_j)
\right|^2\,.
\nonumber\end{aligned}$$
SUMMARY
=======
For the anomalous moment calculation there remain several hurdles. The Tamm–Dancoff truncation results in logarithmically divergent four-point graphs. To deal with these will probably require use of scattering processes, such as Compton scattering[@MustakiPinsky], to obtain renormalization conditions. Verification of the removal of all logarithms and restoration of symmetries can then be undertaken. Also neglected up to this point have been zero modes, photon modes of zero longitudinal momentum[@ZeroModes]. How they might be included has been indicated by Kalloniatis and Robertson[@KalloniatisRobertson].
Additional physics could be included in the calculation by introducing an effective interaction from Z graphs or even putting eee$^+$ states in the basis. In the latter case, photon mass renormalization must be done.
In the Yukawa-model calculations we have learned that Pauli–Villars Fock states increase the basis size by only 150%, which may not be prohibitive. To perform such calculations accurately with a minimal basis size, improvement of ordinary DLCQ, by inclusion of weighting factors, is critical.
We have found a simple $3+1$ model, related to Yukawa theory, which can be solved analytically. Here we will attempt a nonperturbative numerical solution to further test the use of Pauli–Villars regularization in DLCQ. If successful, we can begin to increase the complexity of the model, eventually reaching the full Yukawa theory.
ACKNOWLEDGMENTS {#acknowledgments .unnumbered}
===============
This work was supported in part by the Minnesota Supercomputer Institute through grants of computing time. It was done in collaboration with S.J. Brodsky and G. McCartor.
[^1]: To appear in the proceedings of Orbis Scientiae 1997: [*Twenty-Five Coral Gables Conferences and their Impact on High Energy Physics and Cosmology*]{}, B.N. Kursunoglu, ed., (Plenum, New York, 1997).
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We report the discovery of a faint L6$\pm$1 companion to the previously known M9 dwarf, [[2MASS J01303563$-$4445411]{}]{}, based on our near-infrared imaging and spectroscopic observations with the 3m Infrared Telescope Facility SpeX imager/spectrometer. The visual binary is separated by 3$\farcs$28$\pm$0$\farcs$05 on the sky at a spectrophotometric distance of 40$\pm$14 pc. The projected physical separation is 130$\pm$50 AU, making it one of the widest VLM field multiples containing a brown dwarf companion. [[2MASS J0130$-$4445]{}]{} is only one of ten wide VLM pairs and only one of six in the field. The secondary is considerably fainter ($\Delta K \approx$ 2.35 mag) and redder ($\Delta$ ([[$J-K_s$]{}]{}) $\approx$ 0.81 dex), consistent with component near-infrared types of M9.0$\pm$0.5 and L6$\pm$1 based on our resolved spectroscopy. The component types suggest a secondary mass below the hydrogen-burning limit and an age-dependent mass ratio of 0.6–0.9. The system’s space motion and spectroscopic indicators suggest an age of 2–4 Gyr while the model-dependent masses and binding energies suggest that this system is unlikely to have formed via dynamical ejection. The age, composition, and separation of the [[2MASS J01303563$-$4445411]{}]{} system make it useful for tests of VLM formation theories and of condensate cloud formation in L dwarfs.'
author:
- 'Saurav Dhital , Adam J. Burgasser , Dagny L. Looper , Keivan G. Stassun'
bibliography:
- 'ads.bib'
title: |
Resolved Spectroscopy of M Dwarf/L Dwarf Binaries.\
IV. Discovery of an M9 + L6 Binary Separated by Over 100 AU
---
Introduction {#Sec: intro}
============
The processes by which very low-mass (VLM; M $\lesssim$0.1 [[M$_{\sun}$]{}]{}, @Burgasser2007b) stars and brown dwarfs (BD) form, and whether these processes are similar to those of higher-mass stars, is an open question. The VLMs/BDs exhibit significant differences in the distribution of binary/multiple systems when compared to their more massive brethren. The resolved binary fraction of $\sim$20–30% in VLMs/BDs [@Basri2006; @Joergens2008] is significantly lower than in F and G dwarfs [$\sim$60%; @Duquennoy1991] and modestly lower than M dwarfs [$\sim$27–42%; @Fischer1992; @Reid1997]. The typical orbital separation of $\sim$4–5 AU in VLMs/BDs is much smaller compared to $\sim$30 AU for F, G, and M dwarf binaries [@Duquennoy1991; @Fischer1992]. In addition, while stellar binaries are known to have separations in excess of $\sim$1 pc [e.g., @Lepine2007a; @Dhital2010], no VLM system has a separation greater than 6700 AU. Indeed, only 15 of the known 99 VLM systems have projected physical separations larger than 20 AU and only nine systems are wider than 100 AU[^1]. Energetically, the VLM binaries seem to stand apart as well: based on empirical data, @Close2003 suggested minimum binding energies of $10^{42.5}$ erg for field VLM systems, $\sim$300 times higher than the $10^{40}$ erg limit for stellar binary systems. Lastly, most VLM binaries are close to equal-mass. All of these differences indicate that the same formation process(es) may not be responsible for the two populations.
It is now generally believed that most stars form in multiple systems via fragmentation of the protostellar cloud, with single stars being the result of decay of unstable multiples [e.g., @Kroupa1995a]. The most favored process is gravoturbulence where the fragmentation is the result of a combination of turbulent gas flows and gravity. Hydrodynamical simulations have shown that when turbulent gas flows in protostellar clouds collide, they form clumps that are gravitationally unstable and, hence, collapse forming multiple stellar embryos [e.g., @Caselli2002; @Goodwin2004a; @Goodwin2004b; @Bate2009]. Within a few freefall times, most of these embryos are ejected due to mutual dynamical interactions, preferentially the ones with lower masses.
To then explain the observed distributions of VLM binaries separations, two explanations have been proffered. The first so-called “ejection hypothesis” suggests that most VLM binaries, unlike the more-massive stellar systems, are the result of the ejected embryos [@Reipurth2001]. The wider systems get disrupted, explaining the overall rarity of VLM and BD binaries. The second is preferential accretion within the first 0.1 Myr ($\sim$1 freefall time), making VLM systems tighter and more equal-mass. As a result, even VLM distributions that initially may have looked similar to that of higher mass stars are transformed and look like the observed VLM distributions [@Bate2009]. However, neither hypothesis explains why $\sim$10% of observed VLM binaries are wider than 100 AU. Two other theories on VLM/BD formation, disk fragmentation [e.g., @Watkins1998a; @Watkins1998b] and photoablation [@Whitworth2004], require massive stars to trigger the process and cannot explain the existence of VLM binaries in the field. To resolve the differences between observational and numerical results and to distinguish between the various formation scenarios, a larger sample of VLM binaries—especially very wide systems that are most susceptible to dynamical effects—is needed.
In this paper, we report the discovery of a wide VLM binary [[2MASS J01303563$-$4445411]{}]{} (hereafter [[2MASS J0130$-$4445]{}]{}) separated by 130 AU, 3$\farcs$3. The brighter primary component of [[2MASS J0130$-$4445]{}]{} was identified by @Reid2008 in the Two Micron All Sky Survey [2MASS; @Skrutskie2006] and classified as an M9 dwarf on the @Kirkpatrick1999 red optical scheme, indicating a spectrophotometric distance of 33.1$\pm$2.2 pc. Neither [[H$\alpha$]{}]{} nor , activity and age indicators, respectively, were evident in the optical spectrum. The primary has a proper motion of (120$\pm$14, -25$\pm$20) [mas yr$^{-1}$]{}and a tangential velocity of 19$\pm$3 [km s$^{-1}$]{}[@Faherty2009]. The system is unresolved in 2MASS, and there have been no reports of a faint companion to this source in either optical survey data or follow-up observations [@Reid2008; @Faherty2009].
In our own follow-up observations of [[2MASS J0130$-$4445]{}]{}, we have identified a well-separated, faint L dwarf companion, indicating that this is a wide VLM binary system with a probable BD component. In Sections \[Sec: NIRimaging\] and \[Sec: NIRspectra\], we describe our imaging and spectroscopic observations, respectively, and discuss the properties of the components of the resolved binary system. We discuss the physical association, mass, and age of the binary [[2MASS J0130$-$4445]{}]{}AB in Section \[Sec: analysis\] and its implications on VLM formation and evolution scenarios in Section \[Sec: discussion\]. The conclusions are presented in Section \[Sec: summary\].
Near-Infrared Imaging {#Sec: NIRimaging}
=====================
Observations and Data Reduction
-------------------------------
[[2MASS J0130$-$4445]{}]{} was imaged with the 3m NASA Infrared Telescope Facility (IRTF) SpeX spectrograph [@Rayner2003] on December 7, 2009 (UT), as part of a program to identify unresolved M/L dwarf plus T dwarf spectral binaries [e.g., @Burgasser2008a]. Conditions were clear but with poor seeing, 1$\farcs$2 at $K$-band, due in part to the large airmass of the observation (2.34–2.37). These images revealed a faint point source due east of the primary target at a separation of roughly 3$\arcsec$. Four dithered exposures were obtained of the pair in each of the MKO[^2] $J$, $H$, and $K$ filters, with individual exposure times of 45s, 30s, and 30s, respectively. The field rotator was aligned at a position angle of 0$\degr$; i.e., north up and east to the left.
Imaging data were reduced in a standard manner using custom IDL routines. Raw images were mirror-flipped about the y-axis to reproduce the sky orientation and pair-wise subtracted to remove sky contributions. The difference images were divided by normalized flat field frames, constructed by median-combining the imaging data for each filter after masking out the sources. Subsections of each image, 10$\arcsec$ (83 pixels) on a side and centered on the target source, were extracted from these calibrated frames. A final image for each filter/target pair (Figure \[Fig: image\]) was produced by averaging the registered subframes together, rejecting 5$\sigma$ pixel outliers. The two sources of [[2MASS J0130$-$4445]{}]{} are well resolved along a nearly east-west axis. The brighter western component is hereafter referred to as [[2MASS J0130$-$4445]{}]{}A and the eastern component as [[2MASS J0130$-$4445]{}]{}B.
![Combined SpeX images of [[2MASS J0130$-$4445]{}]{}AB in the $J$-, $H$- and $K$-bands (left to right), showing the 10$\arcsec\times$10$\arcsec$ (83$\times$83 pixel) region around both sources. Images are aligned with north up and east to the left.[]{data-label="Fig: image"}](fig1.eps){width="1\linewidth"}
Analysis
--------
Component magnitudes and the angular separation of the [[2MASS J0130$-$4445]{}]{} pair were determined through point spread function (PSF) fits to the reduced imaging data, following the prescription described in @McElwain2006. The PSF models were derived from Gaussian fits to the primary component in the individual subimage frames. For each filter, four distinct PSF models were produced, each of which were fit to the individual images, resulting in a total of 16 independent measures of the relative component magnitudes and 48 independent measures of the separation and orientation of the pair, in each of the $JHK$ filters. However, as the secondary was undetected in one of the four $J$-band images, four measures of the relative $J$-band flux and separation were discarded before computing mean values and standard deviations. Separation measurements were converted from pixels to arcseconds assuming a plate scale of 0$\farcs$120$\pm$0$\farcs$002 pixel$^{-1}$ (J. Rayner, 2005, private communication) and no distortion. The position angle (set at 0$\degr$) was assumed to be accurate to within 0$\fdg$25 (ibid.).
Results are listed in Table \[Tab: psf\]. The angular separation of the pair was measured to be 3$\farcs$282$\pm$0$\farcs$047 at a position angle of 87$\fdg$3$\pm$0$\fdg$9; i.e., along an east-west line. The secondary is both considerably fainter and significantly redder than the primary. We derived relative magnitudes of $\Delta{J}=$ 3.11$\pm$0.06 and $\Delta{K}=$ 2.34$\pm$0.04. Using the combined-light 2MASS photometry for the system[^3], this translates into [[$J-K_s$]{}]{} colors of 1.13$\pm$0.04 and 1.94$\pm$0.08 for the primary and secondary, respectively.
[lc]{} $\Delta{\alpha}\cos{\delta}$ ($\arcsec$) & 3.28$\pm$0.05\
$\Delta{\delta}$ ($\arcsec$) & 0.15$\pm$0.06\
Apparent Separation ($\arcsec$) & 3.28$\pm$0.05\
Position Angle ($\degr$) & 87.3$\pm$0.9\
$\Delta{J}$ (mag) & 3.11$\pm$0.06\
$\Delta{H}$ (mag) & 2.68$\pm$0.11\
$\Delta{K}$ (mag) & 2.34$\pm$0.04\
\[Tab: psf\]
Near-Infrared Spectroscopy {#Sec: NIRspectra}
==========================
Observations and Data Reduction
-------------------------------
The two components of [[2MASS J0130$-$4445]{}]{} were observed on separate nights with the prism-dispersed mode of SpeX, the primary on December 7, 2009 (the same night as the imaging observations) and the secondary on December 28, 2009 (UT). Conditions on the latter night were clear with a stable seeing of 0$\farcs$6 at $K$-band. The SpeX prism mode provides 0.75–2.5 $\micron$ continuous spectroscopy with resolution [[$\lambda/{\Delta}{\lambda}$]{}]{} $\approx 120$ for the 0$\farcs$5 slit employed (dispersion across the chip is 20–30 [Å]{} pixel$^{-1}$). Both components were observed separately, with the slit oriented north-south, roughly aligned with the parallactic angle and perpendicular to the separation axis. For the primary, eight exposures of 90s each were obtained at an average airmass of 2.33, while guiding on spillover light from the slit. For the secondary, eight exposures of 150s each were obtained at an average airmass of 2.34, while guiding on the primary off-slit. For both sources, the A0 V star HD 8977 was observed immediately before the target for telluric and flux calibration while the quartz and Ar arc lamps were observed for flat field and wavelength calibration, respectively. Data were reduced using the SpeXtool package, version 3.4 [@Vacca2003; @Cushing2004] using standard settings; see @Burgasser2007d for details.
Analysis {#Sec: NIRanalysis}
--------
Figure \[Fig: nirspec\] shows the spectra of the two components of [[2MASS J0130$-$4445]{}]{}AB; signal-to-noise at the $JHK$ flux peaks was 100–150 and 25–35 for the A and B components, respectively. Both spectra show the characteristic near-infrared (NIR) features of late-type M and L dwarfs [e.g., @Reid2001a; @McLean2003; @Cushing2005]: steep red optical slopes (0.8–1.0 $\micron$) from the pressure broadened wing of the 0.77 $\micron$ doublet; molecular absorption bands arising from [[H$_{2}$O]{}]{} (1.4 and 1.9 $\micron$), CO (2.3 $\micron$), and FeH (0.99 $\micron$); and an overall red spectral energy distribution (SED), consistent with the photometric colors. [[2MASS J0130$-$4445]{}]{}A also exhibits additional absorption features in the 0.8–1.2 $\micron$ region arising from TiO, VO, FeH, and unresolved and lines, all typical for a late-type M dwarf. The corresponding region in the spectrum of [[2MASS J0130$-$4445]{}]{}B is considerably smoother, albeit more noisy, suggesting that many of these gaseous species have condensed out [e.g., @Tsuji1998; @Ackerman2001]. The appearance of weak [[H$_{2}$O]{}]{} absorption at 1.15 $\micron$ and the very red SED of the NIR spectrum all indicate that [[2MASS J0130$-$4445]{}]{}B is a mid- to late-type L dwarf with relatively thick condensate clouds at the photosphere.
![NIR spectra of [[2MASS J0130$-$4445]{}]{}A (top) and [[2MASS J0130$-$4445]{}]{}B (bottom) obtained with IRTF/SpeX. Data are normalized at the peak of each spectra, and the spectrum for [[2MASS J0130$-$4445]{}]{}A is vertically offset by 0.5 dex for clarity (dotted lines). NIR spectral features are labeled.[]{data-label="Fig: nirspec"}](fig2.eps){width="0.8\linewidth"}
To determine spectral types we compared our NIR spectra of [[2MASS J0130$-$4445]{}]{}AB with 463 spectra of 439 M7 or later dwarfs from the SpeX Prism Spectral Libraries[^4]. All templates wre chosen to have median S/N $>$ 10 and could not be binaries, giants, subdwarfs, or spectral classifications that were peculiar or uncertain. Best matches were determined by finding the minimum [[$\chi^2$]{}]{} deviation between component spectra and templates in the 0.95–1.35, 1.45–1.80, and 2.00–2.35 $\micron$ regions (i.e., avoiding telluric bands), following the procedure of @Cushing2008 with no pixel weighting. The two best matching templates to both components of [[2MASS J0130$-$4445]{}]{}AB are shown in Figure \[Fig: nirspec\_temp\]. For [[2MASS J0130$-$4445]{}]{}A, these are the optically classified M8 dwarf 2MASS J05173729$-$3348593 [@Cruz2003] and L0 dwarf DENIS–P J0652197$-$253450 [@Phan-Bao2008]; for [[2MASS J0130$-$4445]{}]{}B these are the optically classified L5 dwarf 2MASSW J1326201$-$272937 [@Gizis2002] and L7 dwarf 2MASS J03185403$-$3421292 [@Kirkpatrick2008]. Note that despite the differences in optical type, these spectra provide equivalently good fits—the two fits were different only by 1.7$\sigma$ for the primary and 1.1$\sigma$ for the secondary based on the F-test which gauges whether two different fits to data are significantly distinct based on the ratio of [[$\chi^2$]{}]{} values and degrees of freedom [@Burgasser2010a]—to the components of [[2MASS J0130$-$4445]{}]{}AB, a reflection of the discrepancies between optical and NIR spectral morphologies for late-type M and L dwarfs [e.g., @Geballe2002; @Kirkpatrick2005]. A $\chi^2$ weighted mean of all the SpeX templates [e.g., @Burgasser2010a] indicates component types of M9.0$\pm$0.5 for [[2MASS J0130$-$4445]{}]{}A and L6$\pm$1 for [[2MASS J0130$-$4445]{}]{}B.
We also derived classifications using a suite of spectral indices and spectral index/spectral type relations from @Tokunaga1999, @Reid2001a, @Geballe2002, @Burgasser2006b, and @Burgasser2007c. Table \[Tab: indices\] shows the measured values and the inferred spectral subtypes of [[2MASS J0130$-$4445]{}]{}A and [[2MASS J0130$-$4445]{}]{}B for each of the indices. The mean and scatter from these indices yields classifications of L0.5$\pm$1.0 for [[2MASS J0130$-$4445]{}]{}A and L7.0$\pm$1.5 for [[2MASS J0130$-$4445]{}]{}B. These are consistent with, but less precise than, the types inferred from spectral template matching, so we adopt the latter for our subsequent analysis.
![The spectral types for [[2MASS J0130$-$4445]{}]{}AB as determined by matching their spectra with templates from the SpeX Prism Spectral Libraries. The best matches for [[2MASS J0130$-$4445]{}]{}A were 2MASS J05173729$-$3348593 [optically classified M8; @Cruz2003; @Schmidt2007] and DENIS$-$P J0652197$-$253450 [optically classified L0; @Phan-Bao2008] while 2MASSW J1326201$-$272937 [optically classified L5; @Gizis2002] and 2MASS J03185403$-$3421292 [optically classified L7; @Kirkpatrick2008] were the best matches for [[2MASS J0130$-$4445]{}]{}B. SpeX data for the templates are from @Burgasser2010a and A. J. Burgasser et al. (in preparation). A $\chi^2$ weighted mean of all the best fit templates gives spectral types of M9.0$\pm$0.5 and L6$\pm$1 for the two components, respectively. The residuals of the comparison (target $-$ spectral type templates) are shown at the bottom of each panel.[]{data-label="Fig: nirspec_temp"}](fig3.eps){width="1\linewidth"}
[lcccccl]{} -J & 0.954 & L0.3 && 0.706 & L7.1 & 1, 2\
[[H$_{2}$O]{}]{}-H & 0.876 & M9.6 && 0.649 & L8.3 & 1, 2\
[[H$_{2}$O]{}]{}-A & 0.686 & L1.4 && 0.602 & L4.1 & 3\
[[H$_{2}$O]{}]{}-B & 0.818 & L0.3 && 0.517 & L7.8 & 3\
[[H$_{2}$O]{}]{}-1.5 $\micron$ & 1.216 & M9.8 && 1.882 & L9.0 & 4\
[[CH$_{4}$]{}]{}-K & 1.056 & L2.1 && 0.968 & L5.5 & 1, 2\
[[CH$_{4}$]{}]{}-2.2 $\micron$ & & && 1.033 & L5.7 & 4\
K1 & 0.069 & M8.7 && & & 3, 5\
Average SpT & &L0.5$\pm$1.0 && &L7.0$\pm$1.5 &\
\[Tab: indices\]
System Properties {#Sec: analysis}
=================
Is [[2MASS J0130$-$4445]{}]{} A Physical Binary?
------------------------------------------------
To assess whether two stars comprise a physical binary or are just a chance alignment of random stars, the most reliable method used is to check for a common systemic velocity. However, [[2MASS J0130$-$4445]{}]{}B is very faint, even in the infrared, and has not been detected in any earlier epoch; hence, we do not have proper motions for the secondary nor radial velocities for either component. In the absence of kinematic information, we employed two other tests to examine whether [[2MASS J0130$-$4445]{}]{}AB is a physical pair: (1) the heliocentric distances of the two components and (2) the probability that the sources are a chance alignment based on the surface distribution of stars on the sky.
The spectrophotometric distances to each component of [[2MASS J0130$-$4445]{}]{}AB were derived using the $M_J$/spectral type relations from @Cruz2003 based on the combined-light 2MASS photometry and our relative SpeX photometry (Table \[Tab: properties\]). The derived distances are 34.5$\pm$3.2 pc for the primary and 45.8$\pm$13.6 pc for the secondary, where the errors are from the uncertainties in the NIR spectral types (see Sec. \[Sec: NIRanalysis\]). These distances are consistent with each other within their associated errors.
[lccl]{} Optical Spectral Type & M9 & & 1\
NIR Spectral Type & M9.0$\pm$0.5 & L6$\pm$1 & 2\
$J$ (mag) & 14.12$\pm$0.03 & 17.28$\pm$0.06 & 2,3\
$H$ (mag) & 13.48$\pm$0.03 & 16.13$\pm$0.10 & 2,3\
$K_s$ (mag) & 12.99$\pm$0.03 & 15.34$\pm$0.05 & 2,3\
[[$J-K_s$]{}]{} & 1.13$\pm$0.04 & 1.94$\pm$0.08 & 2,3\
Est. [[T$_{\rm eff}$]{}]{} (K) & 2400$\pm$110 & 1450$\pm$100 & 4\
Est. Distance (pc) & 35$\pm$3 & 46$\pm$14 & 2,5\
Projected Separation (AU) & & 1,2\
[[V$_{\rm tan}$]{}]{} ([[km s$^{-1}$]{}]{}) & & 2,6\
\[Tab: properties\]
Next, we calculated the probability that [[2MASS J0130$-$4445]{}]{}AB is a random chance alignment along our line-of-sight based on its three-dimensional position in the Galaxy. We followed @Dhital2010, constructing a three-component Galactic model with the thin disk, thick disk, and halo, constrained with empirical stellar density profiles [@Juric2008; @Bochanski2010]. The model recreates a 30$\arcmin\times$30$\arcmin$ region in the sky, centered around the coordinates of the given binary system, and out to heliocentric distances of 2500 pc. As all the simulated stars are single and non-associated, any visual binary is a random chance alignment. In $10^7$ Monte Carlo realizations, we found, on average, 0.0285 chance alignments per realization on the sky, within the 3$\farcs$3 angular separation of [[2MASS J0130$-$4445]{}]{}AB. More importantly, none of these chance alignments were within the range of spectrophotometric distances (40$\pm$14 pc) estimated for [[2MASS J0130$-$4445]{}]{}AB. As such, we conservatively infer a $\lesssim 10^{-7}$ probability of positional coincidence. We therefore conclude that [[2MASS J0130$-$4445]{}]{}AB is a physically-bound binary and not a chance alignment of two unassociated stars.
Age & Mass Estimates for [[2MASS J0130$-$4445]{}]{}AB
-----------------------------------------------------
The NIR spectral types of [[2MASS J0130$-$4445]{}]{}A and [[2MASS J0130$-$4445]{}]{}B correspond to effective temperatures, [[T$_{\rm eff}$]{}]{}, of 2400$\pm$110 K and 1450$\pm$100 K, respectively, based on the [[T$_{\rm eff}$]{}]{}/spectral type relation of @Stephens2009. Uncertainties include scatter in the [[T$_{\rm eff}$]{}]{} relation and uncertainties in the classifications (Sec. \[Sec: NIRanalysis\]) added in quadrature. Figure \[Fig: evol\_track\] shows the @Burrows1993 [@Burrows1997] evolutionary models, displaying [[T$_{\rm eff}$]{}]{} as a function of mass and age; the Burrows mass tracks and the observed [[T$_{\rm eff}$]{}]{} ranges are shown as dotted and dashed lines and gray boxes, respectively. As the masses of the two components vary quite a bit with assumed age, it is imperative to constrain the age of [[2MASS J0130$-$4445]{}]{}AB.
![[[T$_{\rm eff}$]{}]{} versus age for VLM stars and BDs based on the evolutionary models of @Burrows1997. Tracks for masses of 0.02–0.09 [[M$_{\sun}$]{}]{} in 0.01 [[M$_{\sun}$]{}]{} steps are shown as dotted lines while the LDB limit of 0.06 [[M$_{\sun}$]{}]{} is shown as a dashed line. The locus defined by this lower limit on age and [[T$_{\rm eff}$]{}]{} derived from their spectral types is shown in gray for both components. The solid rectangle shows the age estimate of 2–4 Gyr based on the kinematics of @Faherty2009. Note that the broad range of possible ages for this system allow a wide range of mass ratios, from 0.6 to 0.9.[]{data-label="Fig: evol_track"}](fig4.eps){width="1\linewidth"}
The absence of in [[2MASS J0130$-$4445]{}]{}A indicates that it is more massive than the predicted lithium depletion boundary (LDB) mass, $\sim$0.06 [[M$_{\sun}$]{}]{} [@Chabrier1996; @Burrows2001] for field stars of solar metallicity. Using the [[T$_{\rm eff}$]{}]{} for the primary based on its spectral type (including uncertainties) and assuming a mass $\gtrsim$0.06 [[M$_{\sun}$]{}]{}, the evolutionary models of @Burrows1997 indicate an age $\gtrsim$250 Myr (Figure \[Fig: evol\_track\]). We note that recent work by @Baraffe2010 has suggested that episodic accretion during the pre-main sequence stages causes central temperature of a star to increase up to 1 dex, with a sharp dependence on the frequency and magnitude of the episodic accretion. This serves to deplete earlier than in non-accreting stars of the same mass and effectively reduces the inferred LDB mass and, thus, the minimum allowable age. Here, we have not taken episodic accretion into account. While we do not have an optical spectrum of the secondary, the presence (absence) of in the spectrum would set a upper (lower) limit on the mass and, hence, the age of the system, in this case $\lesssim$1.8 Gyr ($\gtrsim$1.1 Gyr). The likely proximity of the mass of the secondary to the LDB is motivation to obtain an optical spectrum of this component.
The absence of [[H$\alpha$]{}]{} emission in the optical spectrum of [[2MASS J0130$-$4445]{}]{}A and lack of UV or X-ray flux—the system is not detected in the GALEX [@Martin2005b] or the ROSAT [@Voges1999] All-Sky Surveys—indicates that the system is not particularly active and, hence, not likely to be a very young system. This absence indicates that [[2MASS J0130$-$4445]{}]{}AB is probably older than 1–100 Myr, as such emission has been detected in brown dwarfs in the Orion Nebula Cluster [isochronal age $\sim$1 Myr; @Peterson2008], Taurus [$\sim$3 Myr; @Guieu2006], $\sigma$ Orionis [2–7 Myr; @Zapatero-Osorio2002], $\alpha$ Persei [$\sim$80 Myr; @Stauffer1999], Pleiades [$\sim$100 Myr; @Stauffer1998; @Martin2000], and Blanco 1 ($\sim$100 Myr; P. A. Cargile et al., in prep.). However, activity signatures might not be reliable age indicators in the VLM regime. Both [[H$\alpha$]{}]{} and X-ray emission drop precipitously across the M dwarf/L dwarf transition [e.g., @Kirkpatrick2000; @Gizis2000; @West2004; @Stelzer2006], likely the result of reduced magnetic field coupling with increasingly neutral photospheres [e.g., @Gelino2002; @Mohanty2002].
Extreme youth can also be ruled out based on the the NIR spectra of these sources, which do not exhibit the triangular H-band peaks seen in $\sim$100 Myr Pleiades M and L dwarfs [@Bihain2010] and young field L dwarfs [e.g., @Kirkpatrick2006]. It is notable that [[2MASS J0130$-$4445]{}]{}B is somewhat red compared to typical L6 dwarfs [$\langle{J-K_s}\rangle =$ 1.82$\pm$0.07; @Schmidt2010], as red sources have been shown to exhibit smaller velocity dispersions and, hence, younger ages [@Faherty2009; @Schmidt2010]. However, [[2MASS J0130$-$4445]{}]{}A is not unusually red for its spectral type ($\langle{J-K_s}\rangle =$ 1.12$\pm$0.10); and the red color of the secondary may reflect an unusually dusty atmosphere [e.g., @Looper2008a]. Also, neither NIR spectra nor the optical spectrum of [[2MASS J0130$-$4445]{}]{}A show high gravity signatures, i.e., unusually blue colors form enhanced [[H$_{2}$]{}]{}, or evidence of the system being metal-poor, making it unlikely that the system is as old as $\sim$10 Gyr [@Burgasser2003b; @Reid2007].
Considering the kinematics of the system, the tangential velocity of [[2MASS J0130$-$4445]{}]{}A, 19$\pm$3 [[km s$^{-1}$]{}]{} is similar to the median velocities of the L dwarfs in the SDSS sample [28$\pm$25 [[km s$^{-1}$]{}]{}; @Schmidt2010], the M9 dwarfs in the BDKP sample [23$\pm$23 [[km s$^{-1}$]{}]{}; @Faherty2009], and the M7–L8 dwarfs in the 2MASS sample [25$\pm$21 [[km s$^{-1}$]{}]{}; @Schmidt2007], with the quoted errors being the 1$\sigma$ dispersions. The low tangential velocity suggests that [[2MASS J0130$-$4445]{}]{}AB is part of the thin disk, although we note that we cannot rule out a higher space velocity for the binary system. Kinematic studies have found that late-M and L dwarfs with average kinematics are typically $\sim$2–4 Gyr old [@Wielen1977; @Faherty2009].
In conclusion, based on the absence of in the primary, we can place a (model-dependent) hard limit on the minimum age of [[2MASS J0130$-$4445]{}]{}AB to be $\sim$250 Myr while its kinematics indicate a preferred age of $\sim$2–4 Gyr. Spectral features for both components are in agreement with these ages. For the ages of 0.25–10 Gyr, based on the @Burrows1993 and @Burrows1997 models, the estimated masses for [[2MASS J0130$-$4445]{}]{}A and [[2MASS J0130$-$4445]{}]{}B are 0.055–0.083 and 0.032–0.076 [[M$_{\sun}$]{}]{}, respectively, and the mass ratio is 0.57–0.92. For kinematics-based age limits of 2–4 Gyr, the estimated masses for [[2MASS J0130$-$4445]{}]{}A and [[2MASS J0130$-$4445]{}]{}B are 0.082–0.083 and 0.066–0.073 [[M$_{\sun}$]{}]{}, respectively, and the mass ratio is 0.81–0.89 (Table \[Tab: model\_props\]). Hence, the components straddle the hydrogen-burning mass limit; and this system is likely composed of a very low mass star and (massive) brown dwarf pair.
[lllll]{} Primary Mass ([[M$_{\sun}$]{}]{}) & 0.055 & 0.082 & 0.083 & 0.083\
Secondary Mass ([[M$_{\sun}$]{}]{}) & 0.032 & 0.066 & 0.073 & 0.076\
Mass Ratio & 0.57 & 0.81 & 0.89 & 0.92\
Log Binding Energy (erg) & 41.61 & 41.96 & 41.97 & 41.97\
Period (yr) & 5030 & 3860 & 3760 & 3720\
\[Tab: model\_props\]
Discussion {#Sec: discussion}
==========
Formation of Wide VLM Binaries in the Field
-------------------------------------------
With a projected separation of 130$\pm$50 AU, [[2MASS J0130$-$4445]{}]{}AB is one of only ten VLM systems wider than 100 AU, with six of them in the field. All of these systems have been identified relatively recently; prior to their discovery, it was believed that VLM field systems were nearly all tight, a possible consequence of dynamic ejection early on. Based on this idea and the VLM binary population known at the time, two relations to define the largest possible separation of VLM binaries were proposed. First, @Burgasser2003a suggested that the maximum separation of a system was dependent on its mass: $a_{\rm max}$ (AU) $=$ 1400 ([[M$_{\rm tot}$]{}]{}/[[M$_{\sun}$]{}]{})$^2$. Second, @Close2003 proposed that the stability of binary systems was contingent on their binding energy—a criterion based on the product rather than the sum of component masses; thus, only systems with binding energy $\geq
10^{42.5}$ erg would exist in the field[^5]. For the (age-dependent) estimated mass of [[2MASS J0130$-$4445]{}]{}AB, the @Burgasser2003a relation equates to maximum physical separations of only 9.2 AU and 35.4 AU for ages of 0.25 and 10 Gyr, respectively, which are both much smaller than the physical separation we have measured for [[2MASS J0130$-$4445]{}]{}AB. Similarly, the binding energies for the system are $10^{41.61}$ and $10^{41.97}$ erg for the same ages (see Table \[Tab: model\_props\]). Both the @Burgasser2003a and @Close2003 relations are definitively violated by [[2MASS J0130$-$4445]{}]{}AB, for all ages and mass ratios. Assuming these limits emerge from dynamical scattering processes, this binary seems unlikely to have formed via the ejection of protostellar embryos.
More recently, @Zuckerman2009 have argued that fragmentation, rather than dynamical, processes are more likely to describe the boundary for the lowest binding energy systems. A protostellar cloud can only fragment if it is more massive than the minimum Jeans mass [$\sim$7 M$_{\rm J}$; @Low1976]. Assuming fiducial separations of 300 AU for the fragments, they derived a cut-off for binding energy as a function of total systemic mass. Finding that this disfavors the formation of very wide and/or high mass ratio binaries, @Faherty2010 used the Jeans length, instead of the fiducial separation, and mass ratio of the system. For [[2MASS J0130$-$4445]{}]{}AB at 0.25 Gyr ([[M$_{\rm tot}$]{}]{}$\approx$0.1 [[M$_{\sun}$]{}]{}), @Zuckerman2009 and @Faherty2010 relations suggest minimum binding energies of 10$^{40.5}$ and 10$^{39}$ erg, respectively; if the system were older, they would be even more stable due to the higher masses. [[2MASS J0130$-$4445]{}]{}AB is well within the bounds of both @Zuckerman2009 and @Faherty2010 formation criteria for all ages and mass ratios. Hence, the observed wide, low binding energy VLM binaries could have formed from small protostellar clouds, with masess close to the local Jeans mass.
Current numerical simulations have suggested an alternative mechanism to form wide binaries: N-body dynamics in small clusters disrupt the VLM pairs wider than $\sim$60 AU, and very wide systems ($>10^4-10^5$ AU) can then be formed when stars are ejected into the field in the same direction [@Moeckel2010; @Kouwenhoven2010]. While this provides a mechanism to create the most fragile VLM pairs identified to date, it does not aid in the formation of 100–1000 AU pairs like [[2MASS J0130$-$4445]{}]{}AB. Two other VLM systems in this separation range are known: 2MASSJ 1623361$-$240221 [220 AU; @Billeres2005] and SDSS J141623.94$+$134826.3 [100 AU; @Burningham2010; @Scholz2010].
[[2MASS J0130$-$4445]{}]{}AB as a Probe of the M Dwarf/L Dwarf Transition {#Sec: context}
-------------------------------------------------------------------------
The components of [[2MASS J0130$-$4445]{}]{}AB straddle a spectral type range that is particularly interesting for three reasons. First, condensate clouds become an important source of photospheric opacity and thermal evolution starting at the end of the M dwarf sequence and peaking in influence in the middle of the L dwarf sequence [@Ackerman2001; @Kirkpatrick2008; @Saumon2008]. With a component on either end of this regime, [[2MASS J0130$-$4445]{}]{}AB is a particularly useful coeval laboratory for studying the emergence and dispersal of these clouds. Second, both components straddle the hydrogen-burning mass limit; and the secondary is very near the LDB. Detection of in the optical spectrum of the secondary could provide a relatively precise constraint on the age of this system (0.25–1.8 Gyr) and thereby make it a useful benchmark for studies of brown dwarf thermal evolution and atmospheric models [e.g., @Pinfield2006]. Third, the M dwarf/L dwarf transition exhibits a steep decline in magnetic activity metrics, including [[H$\alpha$]{}]{}, UV, and X-ray emission [e.g., @Gizis2000; @West2004] but notably not radio emission [e.g., @Berger2006; @Berger2010]. This is believed to be due to the decoupling of magnetic fields from an increasingly neutral photosphere [@Gelino2002; @Mohanty2002] but does not rule out the presence of significant magnetic fields [@Reiners2007]. While [[H$\alpha$]{}]{} is absent in the spectrum of [[2MASS J0130$-$4445]{}]{}A, examination of field strengths and radio emission in this coeval pair may facilitate understanding of how magnetic fields evolve across the stellar/brown dwarf transition. As one of only three binaries spanning the M dwarf/L dwarf transition whose components are easily resolvable from ground-based facilities , [[2MASS J0130$-$4445]{}]{}AB is an important laboratory for studying how condensate clouds, lithium burning, and magnetic activity trends vary across this transition.
Summary {#Sec: summary}
=======
We have identified a binary companion to [[2MASS J01303563$-$4445411]{}]{}A based on NIR imaging and spectroscopic observations. The secondary is well-separated ($\Delta\theta=3\farcs2$) and much fainter ($\Delta K \approx$ 2.35 mags). Based on template matching and spectral indices, we have calculated the NIR spectral types to be M9.0$\pm$0.5 and L6$\pm$1 for the two components. The optical spectrum of [[2MASS J0130$-$4445]{}]{}A shows no evidence of either [[H$\alpha$]{}]{} or indicating a minimum age of 0.25 Gyr, while the kinematics suggest an age of 2–4 Gyr. However, we would like to stress that 0.25–10 Gyr, with the lower bound set by LDB and activity, is the more secure age for the system. [[2MASS J0130$-$4445]{}]{}AB is likely a “grown-up” wide binary that has survived ejections and/or dynamical interactions. More importantly, the system definitively violates the binary stability limits based on the ejection hypothesis [@Burgasser2003a; @Close2003] and satisfies the limits based on the idea that wide VLM binaries are formed from approximately Jeans mass-sized protostellar clouds [@Zuckerman2009; @Faherty2010]. This suggests that observed wide VLM binaries may have formed differently than single VLMs and/or tighter binaries. As one of ten VLM systems with separations $\gtrsim$100 AU, [[2MASS J0130$-$4445]{}]{}AB provides a stringent test for theoretical studies of VLM binary formation, as well as a well-resolved, coeval laboratory for studying empirical trends across the M dwarf/L dwarf and stellar/brown dwarf transitions.
The authors would like to thank the anonymous referee for his/her comments on the manuscript, telescope operator Paul Sears and instrument specialist John Rayner for their assistance during the IRTF observations, and Kelle Cruz for providing an electronic version of the optical spectrum of [[2MASS J0130$-$4445]{}]{}A. SD and KGS acknowledge funding support through NSF grant AST-0909463. This publication makes use of data from the Two Micron All Sky Survey, which is a joint project of the University of Massachusetts and the Infrared Processing and Analysis Center, and funded by the National Aeronautics and Space Administration and the National Science Foundation. 2MASS data were obtained from the NASA/IPAC Infrared Science Archive, which is operated by the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and Space Administration. This research has made use of the VLM Binaries Archive, maintained by Nick Siegler at <http://www.vlmbinaries.org>; the SpeX Prism Spectral Libraries, maintained by Adam Burgasser at <http://www.browndwarfs.org/spexprism>; and the M, L, and T dwarf compendium housed at DwarfArchives.org and maintained by Chris Gelino, Davy Kirkpatrick, and Adam Burgasser. The authors wish to recognize and acknowledge the very significant cultural role and reverence that the summit of Mauna Kea has always had within the indigenous Hawaiian community. We are most fortunate to have the opportunity to conduct observations from this mountain.
Facilities:
[^1]: VLM Binaries Archive (<http://vlmbinaries.org/>) and references therein.
[^2]: Mauna Kea Observatory filter system; see @Tokunaga2002 and @Simons2002.
[^3]: We included small corrections to the relative magnitudes in converting from the MKO to 2MASS photometric systems: 0.009, -0.006, and -0.003 mag in the $J$, $H$, and $K/K_s$ bands, respectively, calculated directly from the spectral data.
[^4]: <http://browndwarfs.org/spexprism/>
[^5]: We note that for the small separations and a mass ratio highly skewed toward one, which was the case for the VLM binaries known at the time, the @Burgasser2003a and @Close2003 limits are essentially equivalent.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We examined the effects of the weak magnetic field on the properties of heavy quarkonia immersed in a thermal medium of quarks and gluons and studied how the magnetic field affects the quasi-free dissociation of quarkonia in the aforementioned medium. For that purpose, we have revisited the general structure of gluon self-energy tensor in the presence of a weak magnetic field in thermal medium and obtained the relevant structure functions using the imaginary-time formalism. The structure functions give rise to the real and imaginary parts of the resummed gluon propagator, which further give the real and imaginary parts of the dielectric permittivity. The real and imaginary parts of the dielectric permittivity will be used to evaluate the real and imaginary parts of the complex heavy quark potential. We have observed that the real-part of the potential is found to be more screened, whereas the magnitude of the imaginary-part of the potential gets increased on increasing the value of both temperature and magnetic field. In addition to this, we have observed that the real-part gets slightly more screened while the imaginary part gets increased in the presence of a weak magnetic field as compared to their counterparts in the absence of a magnetic field (pure thermal). The increase in the screening of the real-part of the potential leads to the decrease of binding energies of $J/\Psi$ and $\Upsilon$, whereas the increase in the magnitude of the imaginary part leads to the increase of thermal width with the temperature and magnetic field both. Also the binding energy and thermal width in the presence of weak magnetic field become smaller and larger, respectively, as compared to that in the pure thermal case. With the observations of binding energy and thermal width in hands, we have finally obtained the dissociation temperatures for $J/\Psi$ and $\Upsilon$, which become slightly lower in the presence of weak magnetic field. [*For example*]{}, with $eB = 0m_\pi^2$ the $J/\psi$ and $\Upsilon$ are dissociated at $1.17T_c$ and $3.98T_c$, respectively, whereas with $eB = 0.5m_\pi^2$ they dissociated at slightly lower value $1.13T_c$ and $3.94T_c$, respectively. This observation leads to the slightly early dissociation of quarkonia because of the presence of a weak magnetic field.'
---
0.1in
0.1in
Mujeeb Hasan$^\dag$[^1] and Binoy Krishna Patra$^\dag$ [^2] 0.02in
0.01in
PACS: 12.39.-x,11.10.St,12.38.Mh,12.39.Pn 12.75.N, 12.38.G\
[**Keywords**]{}: Thermal QCD; Weak magnetic field; Resummed propagator; Dielectric permittivity; Heavy quark potential;\
Introduction
============
Lattice QCD predicted that at sufficiently high temperatures and/or densities the quarks and gluons confined inside hadrons get deconfined into a medium of quarks and gluons coined as quark-gluon Plasma. In the last few decades a large number of experiment has been involved in identifying this new state of matter in ultrarelativistic heavy-ion collisions (URHICs) at RHIC and LHC. However, for the noncentral events in URHICs, a strong magnetic field is generated at the very early stages of the collisions due to very high relative velocities of the spectator quarks with respect to the fireball. Depending on the centralities of the collisions, the strength of the magnetic fields may vary from $m_{\pi}^2$ ($\sim 10^{18}$ Gauss) at RHIC to 10 $m_{\pi}^2$ at LHC [@Skokov:IJMPA24'2009; @Voronyuk:PRC83'2011]. Motivated by this, in the recent past many theoretical works have started emerging to explore the effects of this strong magnetic field on the various QCD phenomena [@Fukushima:PRD78'2008; @Braguta:PRD89'2014; @Kharzeev:PRL106'2011; @Gusynin:PRL73'1994]. Earlier the nascent strong magnetic field was thought to decay very fast with time, resulting the magnetic field of weaker strength. However, it was later found that the realistic estimates of electrical conductivity of the medium may elongate the life-time of the magnetic field [@Tuchin:AHEP2013'2013; @Mclerran:NPA929'2014; @Rath:PRD100'2019]. It thus becomes imperative to investigate the effects of both strong and weak magnetic field on the signature of the novel matter produced in URHICs.
The heavy quarkonia is one of the probe to study the properties of nuclear matter under extreme condition of temperature and magnetic field, because the quarkonia are formed in URHICs on a very short time-scale $\sim 1/2m_Q$ (where $m_Q$ is the mass of the charm or bottom quark), which is similar to the time-scale at which the magnetic field is generated. Therefore the study of the effects of magnetic field on the properties of heavy quarkonia is worth of investigation. We have recently studied the properties of quarkonia in strong magnetic field. However, as we know the quarkonia, the physical resonances of $Q \bar Q$ states, are formed in the plasma frame at a time, $t_F$ (=$\gamma \tau_F$), which is order of 1-2fm depending on the resonances and their momenta. By the time elapsed, the magnetic field may become weak, so in our present study, we aim to understand theoretically the properties of heavy quarkonia and their dissociation in the presence of weak magnetic field ($T^2>|q_fB|$, $T^2>m_f^2$, where $|q_f|$ ($m_f$) is the absolute electric charge (mass) of the $f$-th quark flavour). As we know that, in order to study the dissociation of quarkonia the perturbative computation of heavy quarkonium potential is needed.
The perturbative computations of the potential at high temperatures show that the potential of $Q \bar Q$ is complex [@Laine:JHEP03'2007], where the real part is screened due to the existence of deconfined color charges [@Matsui:PLB178'1986] and the imaginary part [@Beraudo:NPA806'2008] assigns the thermal width to the resonance. Therefore the physics of quarkonium dissociation in a medium has been refined in the last two decades, where the resonances were initially thought to be dissociated when the screening is strong enough, [*i.e.*]{} the real-part of the potential is too weak to keep the $Q\bar Q$ pair together. Nowadays, the dissociation is thought to be primarily because of the widening of the resonance width arising either from the inelastic parton scattering mechanism mediated by the spacelike gluons, known as Landau damping [@Laine:JHEP03'2007] or from the gluo-dissociation process during which color singlet state undergoes into a color octet state by a hard thermal gluon [@Brambilla:JHEP1305'2013]. The latter processes take precedence when the medium temperature is lower than the binding energy of the particular resonance. This dissociates the quarkonium even at lower temperatures where the probability of color screening is negligible. Recently one of us estimated the imaginary-part of the potential perturbatively, where the inclusion of a confining string term makes the (magnitude) imaginary component smaller [@Lata:PRD89'2014; @Lata:PRD88'2013], compared to the medium modification of the perturbative term alone [@Adiran:PRD79'2009]. Gauge-gravity duality also indicates that in strong coupling limit the potential also develops an imaginary component beyond a critical separation of $Q \bar Q$ pair [@Binoy:PRD92'2015; @Binoy:PRD91'2015]. Moreover lattice studies have also shown that the potential may have a sizable imaginary part [@Rothkopf:PRL'2012]. There are, however, other processes which may cause the depopulation of the resonance states either through the transition from ground state to the excited states during the non adiabatic evolution of quarkonia [@Bagchi:MPLA30'2015] or through the swelling or shrinking of states due to the Brownian motion of $Q \bar Q$ states in the parton plasma [@Binoy:NPA708'2002]. Very recently the change in the properties of heavy quarkonia immersed in a weakly-coupled thermal QCD medium has been described by HTL permittivity [@Lafferty:arxiv:1906.00035]. They used the generalized Gauss law in conjunction to linear response theory to obtain the real and imaginary parts of the heavy quark potential, where a logarithmic divergence in imaginary part is found due to string contribution at large $r$. They have circumvented by regularizing weak infrared diverging ($1/p$) term in the resummed gluon propagator by choosing the regulation scale in terms of Debye mass. There is another recent work [@Guo:PRD100'2019], where a nonperturbative term induced by the dimension two gluon condensate besides the usual HTL resummed contribution is included in the resummed gluon propagator to obtain the string contribution in the potential, in addition to the KMS potential [@KMS].
The abovementioned studies are attributed for a thermal medium in the absence of a magnetic field. However, as mentioned earlier that a magnetic field is also generated in the heavy ion collisions, thus the influence of a homogeneous and constant external magnetic field on the heavy meson spectroscopy has been investigated quantum mechanically subjected to a three-dimensional harmonic potential and Cornell potential plus spin-spin interaction term [@Alford:PRD88'2013; @Bonati:PRD92'2015]. Further, the effect of a constant uniform magnetic field on the static quarkonium potential at zero and finite temperature [@Bonati:PRD94'2016] and on the screening masses [@Bonati:PRD95'2017] have been investigated. The momentum diffusion coefficients of heavy quarks in a strong magnetic field along the directions parallel and perpendicular to the magnetic field at the leading order in QCD coupling constant has been studied [@Fukushima:PRD93'2016]. Recently some of us have explored the effects of strong magnetic field on the properties of the heavy-quarkonium in finite temperature by computing the real part of the $Q \bar Q$ potential [@Mujeeb:EPJC77'2017] in the framework of perturbative thermal QCD and studied the dissociation of heavy quarkonia due to the color screening. Successivley, we made an attempt to study the dissociation of heavy quarkonia due to Landau damping in presence of strong magnetic field by calculating the real and imaginary parts of the heavy quark potential in presence of strong magnetic field [@Mujeeb:NPA995'2020]. The complex heavy quark potential in presence of strong magnetic field has also been obtained in [@Balbeer:PRD97'2018]. We have also explored the effects of strong magnetic field on the wakes in the induced charge density and in the potential due to the passage of highly energetic partons through a thermal QCD medium [@Mujeeb:1901.03497]. Recently, the dispersion spectra of a gluon in hot QCD medium in presence of strong as well as weak magnetic field limit is studied [@karmakar:EPJC79'2019].
In the present study, we aim to obtain the complex heavy quark anti-quark potential in an environment of temperature and weak magnetic field. For that purpose, we first start with the evaluation of gluon self energy in the similar environment using the imaginary-time formalism. As the quark-loop is only affected with the magnetic field thus, the quark-loop in the said environment is now dictated by both the scales namely the magnetic field as well as the temperature, whereas for the gluon-loop, the temperature is the only available scale in the medium as the gluon-loop is not affected with the magnetic field. Furthermore, we have revisited the general structure of gluon self energy tensor in presence of weak magnetic field in thermal medium and obtained the relevant structure functions. Hence the real and imaginary parts of the resummed gluon propagator have been obtained, which give the real and imaginary parts of the dielectric permittivity. The real and imaginary parts of the dielectric permittivity will inturn give the real and imaginary parts of the complex heavy quark potential. The real part of the potential is used in the Schrödinger equation to obtain the binding energy of heavy quarkonia whereas the imaginary part is used to calculate the thermal width. Finally, we have obtained the dissociation temperatures of heavy quarkonia and studied how the dissociation temperatures get affected in presence of magnetic field.
Thus, our work proceeds as follows. In section 2, we will calculate the gluon self energy in a weak magnetic field wherein, we will discuss the general structure of gluon self energy and resuumed gluon propagator at finite temperature in presence of weak magnetic field and will calculate the relevant form factors in subsection 2.1 and subsection 2.2, respectively. Thus, the real and imaginary parts of the resummed gluon propagator will give the real and imaginary parts of the dielectric permittivity in subsection 3.1, which gives the real and imaginary parts of complex heavy quark potential in subsection 3.2. We will use the real and imaginary parts of the potential to obtain the binding energy and thermal width in subsection 4.1 and 4.2, respectively, which will then give the dissociation temperatures of heavy quarkonia in subsection 4.3. Finally, we will conclude our findings in section 5.
Gluon self energy in a weak magnetic field
==========================================
In this section we will evaluate the gluon self energy in a weak magnetic field. As we know that for the evaluation of gluon self energy, we need to evaluate both the quark loop and gluon loop contributions in presence of weak magnetic field. Because of weak magnetic field, only the quark loop will get affected whereas the gluon loop remain as such. Now, we will first start with the quark-loop contribution to gluon self energy $$\begin{aligned}
i\Pi^{\mu\nu}_{ab}(Q)&=&-\int\frac{d^4K}{(2\pi)^4}Tr
\left[ i g t_b \gamma^\nu i S(K) i g t_a \gamma^\mu i S(P)
\right],
\nonumber\\
&=&\sum_f\frac{g^2\delta_{ab}}{2}\int\frac{d^4K}{(2\pi)^4}Tr
\left[ \gamma^\nu i S(K) \gamma^\mu i S(P)\right],
\label{self_energy}\end{aligned}$$ where $P=(K-Q)$ and $Tr(t_a t_b)=\frac{\delta_{ab}}{2}$. The $S(k)$ is the quark propagator in a weak magnetic field which can be written upto order of $O(q_fB)^2$ as [@ayala:1805.07344] $$\begin{aligned}
iS(K)=i\frac{(\slashed{K}+m_f)}{K^2-m^2_f}-q_fB
\frac{\gamma_1\gamma_2(\slashed{K}+m_f)}{(K^2-m_f^2)^2}
-2i(q_f B)^2\frac{[K_\perp^2(\slashed{K}_\parallel+m_f)
+\slashed{K}_\perp(m_f^2-K_\parallel^2)]}
{(K^2-m_f^2)^4},
\label{weak_propagator}\end{aligned}$$ where $m_f$ and $q_f$ are the mass and charge of the $f^{th}$ flavor quark. According to the following choice of metric tensors, $$\begin{aligned}
g^{\mu\nu}_\parallel&=& {\rm diag} (1,0,0-1),\\
~g^{\mu\nu}_\perp&=&{\rm diag} (0,-1,-1,0),\end{aligned}$$ the four-momentum suitable in a magnetic field directed along the $z$ axis, $n^\mu =(0,0,0,-1)$, is given by $$\begin{aligned}
K^{\mu}_\parallel&=&(k_0,0,0,k_z),\label{momentum_parallel}\\
K^{\mu}_\perp&=&(0,k_x,k_y,0),\label{momentum_perpendicular}\\
K_{\parallel}^2&=&k_{0}^2-k_{z}^2,\\
K_{\perp}^2&=&k_{x}^2+k_{y}^2.\end{aligned}$$ The above Eq. can be recast in the following form $$\begin{aligned}
iS(K)=S_0(K)+S_1(K)+S_2(K),
\label{weak_propagator1}\end{aligned}$$ where $S_0(K)$ is the contribution of the order $O[(q_fB)^0]$, $S_1(K)$ is the contribution of the order $O[(q_fB)^1]$ and $S_2(K)$ is the contribution of the order $O[(q_fB)^2]$. Using Eq., the Eq. can be written as $$\begin{aligned}
\Pi^{\mu\nu}(Q)=-\sum_f\frac{ig^2}{2}\int\frac{d^4K}
{(2\pi)^4}Tr\left[\gamma^\nu \lbrace S_0(K)+S_1(K)+S_2(K)
\rbrace \gamma^\mu \lbrace S_0(P)+S_1(P)+S_2(P)\rbrace
\right].
\label{self_energy1}\end{aligned}$$ After simplifying, the above gluon self energy given by Eq. can be expressed as follows $$\begin{aligned}
\Pi^{\mu\nu}(Q)=\Pi^{\mu\nu}_{(0,0)}(Q)+\Pi^{\mu\nu}_{(1,1)}(Q)
+2\Pi^{\mu\nu}_{(2,0)}(Q)+O[(q_fB)^3],
\label{self_energy2}\end{aligned}$$ where $$\begin{aligned}
\Pi^{\mu\nu}_{(0,0)}(Q)&=&-\sum_f\frac{ig^2}{2}\int\frac{d^4K}
{(2\pi)^4}Tr[\gamma^\nu S_0(K)\gamma^\mu S_0(P)],\label{pi_00}\\
\Pi^{\mu\nu}_{(1,1)}(Q)&=&-\sum_f\frac{ig^2}{2}\int\frac{d^4K}
{(2\pi)^4}Tr\lbrace \gamma^\nu S_1(K) \gamma^\mu S_1(P)
\rbrace,\label{pi_11}\\
\Pi^{\mu\nu}_{(2,0)}(Q)&=&-\sum_f\frac{ig^2}{2}\int\frac{d^4K}
{(2\pi)^4}Tr\left[\gamma^\nu S_2(K) \gamma^\mu S_0(P)\right].
\label{pi_20}\end{aligned}$$ The term $\Pi^{\mu\nu}_{(0,0)}$ is of the order $O[(q_fB)^0]$, where $\Pi^{\mu\nu}_{(1,1)}$ and $\Pi^{\mu\nu}_{(2,0)}$ both are of the order $O[(q_fB)^2]$. The term which is of the order $O[(q_fB)^1]$ vanishes. Substituting the values of $S_0$, $S_1$ and $S_2$ in Eq., Eq. and Eq. by comparing Eq. with Eq., we get
$$\begin{aligned}
\Pi^{\mu\nu}_{(0,0)}(Q)&=&\sum_f\frac{ig^2}{2}\int\frac{d^4K}{(2\pi)^4}
\frac{Tr[\gamma^\nu(\slashed{K}+m_f)\gamma^\mu(\slashed{P}+m_f)]}
{(K^2-m^2_f)(P^2-m_f^2)},\nonumber\\
&=&\sum_f i2g^2\int\frac{d^4K}{(2\pi)^4}\frac{\left[P^\mu K^\nu+
K^\mu P^\nu-g^{\mu\nu}(K.P-m_f^2)\right]}
{(K^2-m^2_f)(P^2-m_f^2)},\\
\Pi^{\mu\nu}_{(1,1)}(Q)&=&-\sum_f\frac{ig^2(q_fB)^2}{2}\int\frac{d^4K}
{(2\pi)^4}\frac{Tr[\gamma^\nu\gamma_1\gamma_2
(\slashed{K}_\parallel+m_f)\gamma^\mu\gamma_1\gamma_2
(\slashed{P}_\parallel+m_f)]}
{(K^2-m^2_f)^2(P^2-m_f^2)^2},\nonumber\\
&=&\sum_f 2ig^2(q_fB)^2\int\frac{d^4K}{(2\pi)^4}
\frac{\left[P_\parallel^\mu K_\parallel^\nu +K_\parallel^\mu
P_\parallel^\nu +(g_\parallel^{\mu\nu}-g_\perp^{\mu\nu})
(m_f^2-K_\parallel .P_\parallel)\right]}
{(K^2-m^2_f)^2(P^2-m_f^2)^2},\\
\Pi^{\mu\nu}_{(2,0)}(Q)&=&-\sum_f\frac{2ig^2(q_fB)^2}{2}\int
\frac{d^4K}{(2\pi)^4}\frac{Tr\left[\gamma^\nu(\slashed{K}+m_f)
\gamma^\mu\lbrace K_\perp^2
(\slashed{K}_\parallel+m_f)+\slashed{K}_\perp(m_f^2-K_{\parallel}^2)
\rbrace \right]}{(K^2-m_f^2)(K^2-m_f^2)^4}\nonumber\\
&&-\sum_f\frac{2ig^2(q_fB)^2}{2}\int\frac{d^4K}{(2\pi)^4}
\frac{Tr\left[\gamma^\nu\lbrace K_\perp^2
(\slashed{K}_\parallel+m_f)+\slashed{K}_\perp
(m_f^2-K_{\parallel}^2)\rbrace\gamma^\mu(\slashed{K}+m_f)\right]}
{(K^2-m_f^2)^4(P^2-m_f^2)},
\nonumber\\
&=&-\sum_f 4ig^2(q_fB)^2\int\frac{d^4K}{(2\pi)^4}
\left[\frac{M^{\mu\nu}}{(K^2-m_f^2)^4(P^2-m_f^2)}\right],\end{aligned}$$
where $$\begin{aligned}
M^{\mu\nu}&=&K_\perp^2\left[P^\mu K_\parallel^\nu+
K_\parallel^\mu P^\nu-g^{\mu\nu}(K_\parallel .P-m_f^2)\right]
+(m_f^2-K_\parallel^2)\left[P^\mu K_\perp^\nu +K_\perp^\mu
P^\nu-g^{\mu\nu}(K_\perp .P)\right].\end{aligned}$$ Here the strong coupling $g$ runs with the magnetic field and temperature both, which is recently obtained in [@ayala:PRD98'2018] $$\begin{aligned}
\alpha_s(\Lambda^2,eB)=\frac{g^2}{4\pi}=\frac{\alpha_s(\Lambda^2)}{1+
b_1\alpha_s(\Lambda^2)\ln\left(\frac{\Lambda^2}
{\Lambda^2+eB}\right)},\end{aligned}$$ with $$\begin{aligned}
\alpha_s(\Lambda^2)=\frac{1}{
b_1\ln\left(\frac{\Lambda^2}
{\Lambda_{\overline{MS}}^2}\right)},\end{aligned}$$ where $\Lambda$ is set at $2\pi T$, $b_1=\frac{11N_c-2N_f}{12\pi}$ and $\Lambda_{\overline{MS}}=0.176GeV$.
Before evaluating further, we will first discuss the structure of gluon self energy in thermal medium in presence of weak magnetic field in the next subsection.
Structure of Gluon self energy and resummed gluon propagator for thermal medium in the presence of weak magnetic field
----------------------------------------------------------------------------------------------------------------------
In this subsection, we will briefly discuss the general structure of gluon self energy tensor and resummed gluon propagator for thermal medium in the presence of weak magnetic field. The general structure of gluon self energy in a thermal medium defined by the heat bath in local rest frame, $u^\mu=(1,0,0,0)$ and in the presence of magnetic field directed along the $z$-direction, $n_\mu=(0,0,0,-1)$ is recently obtained as follows [@karmakar:EPJC79'2019] $$\begin{aligned}
\Pi^{\mu\nu}(Q)=b(Q)B^{\mu\nu}(Q)+c(Q)R^{\mu\nu}(Q)+d(Q)M^{\mu\nu}(Q)
+a(Q)N^{\mu\nu}(Q),
\label{self_decomposition}\end{aligned}$$ where $$\begin{aligned}
B^{\mu\nu}(Q)&=&\frac{{\bar{u}}^\mu{\bar{u}}^\nu}{{\bar{u}}^2},\\
R^{\mu\nu}(Q)&=&g_{\perp}^{\mu\nu}-\frac{Q_{\perp}^{\mu}Q_{\perp}^{\nu}}
{Q_{\perp}^2},\\
M^{\mu\nu}(Q)&=&\frac{{\bar{n}}^\mu{\bar{n}}^\nu}{{\bar{n}}^2},\\
N^{\mu\nu}(Q)&=&\frac{{\bar{u}}^\mu{\bar{n}}^\nu+{\bar{u}}^\nu{\bar{n}}^\mu}
{\sqrt{{\bar{u}}^2}\sqrt{{\bar{n}}^2}},\end{aligned}$$ the four vectors ${\bar{u}}^\mu$ and ${\bar{n}}^\mu$ used in the construction of above tensors are defined as follows $$\begin{aligned}
\bar{u}^\mu &=&\left(g^{\mu\nu}-\frac{Q^\mu Q^\nu}{Q^2}\right)u_\nu,\\
\bar{n}^\mu &=&\left(\tilde{g}^{\mu\nu}-\frac{\tilde{Q}^\mu\tilde{Q}^\nu}
{\tilde{Q}^2}\right)n_\nu,\end{aligned}$$ where ${\tilde{g}}^{\mu\nu}=g^{\mu\nu}-u^\mu u^\nu$ and $\tilde{Q}^\mu=Q^\mu-(Q.u)u^\mu$. Using the properties of projection tensors, the form factors appear in can be obtained as $$\begin{aligned}
b(Q)&=&B^{\mu\nu}(Q)\Pi_{\mu\nu}(Q),
\label{form_b}\\
c(Q)&=&R^{\mu\nu}(Q)\Pi_{\mu\nu}(Q),
\label{form_c}\\
d(Q)&=&M^{\mu\nu}(Q)\Pi_{\mu\nu}(Q),
\label{form_d}\\
a(Q)&=&\frac{1}{2}N^{\mu\nu}(Q)\Pi_{\mu\nu}(Q)
\label{form_a}.\end{aligned}$$ Now we can obtained the resummed gluon propagator in thermal medium in presence of weak magnetic field. The general form of the resummed gluon propagator in Landau gauge can be written as [@karmakar:EPJC79'2019] $$\begin{aligned}
D^{\mu\nu}(Q)=\frac{(Q^2-d)B^{\mu\nu}}{(Q^2-b)(Q^2-d)-a^2}
+\frac{R^{\mu\nu}}{Q^2-c}+\frac{(Q^2-b)M^{\mu\nu}}{(Q^2-b)(Q^2-d)-a^2}
+\frac{aN^{\mu\nu}}{(Q^2-b)(Q^2-d)-a^2}.\end{aligned}$$ The point to be noted here is that, we required only the “00”-component of resummed gluon propagator for deriving the heavy quark potential. Hence the “00”-component of the propagator can be obtained as $$\begin{aligned}
D^{00}(Q)=\frac{(Q^2-d)\bar{u}^2}{(Q^2-b)(Q^2-d)-a^2},
\label{propagator_00}\end{aligned}$$ where $R^{00}=M^{00}=N^{00}=0$. Now we will obtained the form factors appear in the above propagator . We will first start with the form factor $a$, which can be obtained using Eq. with Eq. as $$\begin{aligned}
a(Q)=a_0(Q)+a_2(Q),\end{aligned}$$ where $a_0$ is of the order of $O(q_fB)^0$ and $a_2$ is of the order of $O(q_fB)^2$. An important point to be noted here is that the zero magnetic field contribution of form factor $a$ vanishes, that is $a_0=0$, whereas $a_2$ gives the contribution of order $O(q_fB)^2$. However the contribution of form factor $a$ in the denominator of the propagator appear as $a^2$, which becomes of the order of $O(q_fB)^4$. Since in the current theoretical calculation we are considering contribution upto $O(q_fB)^2$, so we can neglect the contribution appear from the form factor $a$. Thus, the “00”-component of resummed gluon propagator upto $O(q_fB)^2$ can be written as $$\begin{aligned}
D^{00}(Q)=\frac{\bar{u}^2}{(Q^2-b)},
\label{propagator_final}\end{aligned}$$ so we end up with only one form factor $b$, which we will evaluate in the next subsection.
Real and Imaginary parts of the form factor $b(Q)$
--------------------------------------------------
In this subsection, we will calculate the real and imaginary parts of the form factor $b$. Using Eq., the form factor $b$ can be evaluated as follows $$\begin{aligned}
b(Q)&=&B_{\mu\nu}(Q)\Pi^{\mu\nu}(Q),\nonumber\\
b(Q)&=&\frac{u^\mu u^\nu}{\bar{u}^2}\Pi^{\mu\nu}(Q),\end{aligned}$$ thus using Eq., the form factor b can be written upto $O[(q_fB)^2]$ as $$\begin{aligned}
b(Q)=b_0(Q)+b_2(Q),
\label{formfactor_b}\end{aligned}$$ where the form factors $b_0$ and $b_2$ are defined as follows $$\begin{aligned}
b_0(Q)&=&\frac{u^\mu u^\nu}{\bar{u}^2}\Pi^{\mu\nu}_{(0,0)}(Q),
\label{formfactor_b0}\\
b_2(Q)&=&\frac{u^\mu u^\nu}{\bar{u}^2}[\Pi^{\mu\nu}_{(1,1)}(Q)+
2\Pi^{\mu\nu}_{(2,0)}(Q)].
\label{formfactor_b2}\end{aligned}$$ ****:\
\
Here we will solve the form factor $b_0$. Using Eq., the form factor can be written as $$\begin{aligned}
b_0(Q)&=&\frac{u^\mu u^\nu}{\bar{u}^2}\Pi^{\mu\nu}_{(0,0)}(Q),
\nonumber\\
&=&\sum_f \frac{i2g^2}{\bar{u}^2}\int\frac{d^4K}{(2\pi)^4}\frac
{\left[2k_0^2-K^2+m_f^2)\right]}
{(K^2-m^2_f)(P^2-m_f^2)}.\end{aligned}$$ Now we will solve the form factor $b_0$ using the imaginary-time formalism, the detailed calculation for which has been shown in appendix \[b\_0\]. Thus, the real and imaginary parts of the form factor $b_0$ in the static limit are given as follows $$\begin{aligned}
{\rm Re}~b_0(q_0=0)&=&g^2T^2\frac{N_f}{6},\\
\left[\frac{{\rm Im}~b_0(q_0,q)}{q_0}\right]_
{q_0=0}&=&\frac{g^2T^2N_f}{6}\frac{\pi}{2q}.\end{aligned}$$ Now we will evaluate the gluonic contribution. The temporal component of gluon self energy due to the gluon-loop contribution can be calculated as [@Weldon:PRD26'1982; @Pisarski:PRL63'1989], $$\begin{aligned}
\Pi^{00}(q_0,q)=-g^2 T^2 \frac{N_c}{3}\left(\frac{q_0}
{2q}\ln\frac
{q_0+q+i\epsilon}{q_0-q+i\epsilon}-1\right)~,\end{aligned}$$ which gives the real and imaginary parts of form factor $b_0$ due to the gluonic contribution in the static limit $$\begin{aligned}
{\rm Re}~b_0(q_0=0)&=&g^2T^2\left(\frac{N_c}{3}\right),\\
\left[\frac{{\rm Im}~b_0(q_0,q)}{q_0}\right]_{q_0=0}
&=&g^2T^2\left(\frac{N_c}{3}\right)\frac{\pi}{2q}.
\label{img_b0}\end{aligned}$$ Now we add the quark and gluon-loop contributions together to obtain the real and imaginary parts of form factor $b_0$ in the static limit as follows $$\begin{aligned}
{\rm Re}~b_0(q_0=0)&=&g^2T^2\left(\frac{N_c}{3}+\frac{N_f}{6}\right),
\label{real_b0}\\
\left[\frac{{\rm Im}~b_0(q_0,q)}{q_0}\right]_{q_0=0}
&=&g^2T^2\left(\frac{N_c}{3}+\frac{N_f}{6}\right)\frac{\pi}{2q}.
\label{img_b0}\end{aligned}$$ Thus we can see that the form factor $b_0$ is independent of the magnetic field as it is $O[(q_fB)^0]$ and depends only on the temperature of the medium. This form factor $b_0$ coincides with the HTL form factor $\Pi_L$ in absence of the magnetic field [@Weldon:PRD26'1982; @Pisarski:PRL63'1989].
****:\
\
Here we will discuss the form factor $b_2$, which is of the order of $O[(q_fB)^2]$. Using Eq., the form factor is given by $$\begin{aligned}
b_2(Q)&=&\frac{u^\mu u^\nu}{\bar{u}^2}[\Pi^{\mu\nu}_{(1,1)}(Q)+
2\Pi^{\mu\nu}_{(2,0)}(Q)],
\nonumber\\
&=&\sum_f \frac{i2g^2(q_fB)^2}{\bar{u}^2}\left[\int\frac{d^4K}{(2\pi)
^4}\left\lbrace\frac{\left(2k_0^2-K_\parallel^2+m_f^2\right)}
{(K^2-m^2_f)^2(P^2-m_f^2)^2}
-\frac{\left(8k_0^2K_\perp^2\right)}{(K^2-m^2_f)^4(P^2-m_f^2)}\right
\rbrace\right].\end{aligned}$$ We have calculated the real and imaginary parts of the form factor $b_2$ in the appendix \[b\_2\], which gives the real and imaginary parts of the form factor $b_2$ in the static limit as follows $$\begin{aligned}
{\rm Re}~b_2(q_0=0)=\sum_f\frac{g^2}{12\pi^2 T^2}(q_fB)^2
\sum_{l=1}^{\infty}(-1)^{l+1}l^2K_0(\frac{m_fl}{T}).
\label{real_b2}\end{aligned}$$ $$\begin{aligned}
\left[\frac{{\rm Im}~b_2(q_0,q)}{q_0}\right]_
{q_0=0}&=&\frac{1}{q}\left[\sum_f\frac{g^2(q_fB)^2}{16\pi T^2}
\sum_{l=1}^\infty(-1)^{l+1}l^2K_0\left(\frac{m_f l}{T}\right)
\right. \nonumber\\ &&\left.
-\sum_f\frac{g^2(q_fB)^2}{96\pi T^2}
\sum_{l=1}^\infty(-1)^{l+1}l^2K_2\left(\frac{m_f l}{T}\right)
\right. \nonumber\\ &&\left.
+\sum_f\frac{g^2(q_fB)^2}{768\pi}\frac{(8T-7\pi m_f)}{m_f^2 T}
\right],
\label{img_b2}\end{aligned}$$ where $K_0$ and $K_2$ are the modified Bessel functions of second kind.
After obtaining the real and imaginary parts of the form factor $b_0$ and $b_2$, we can write the real and imaginary parts of form factor $b$ using Eq. as follows $$\begin{aligned}
{\rm Re}~b(q_0=0)&=&
g^2T^2\left(\frac{N_c}{3}+\frac{N_f}{6}\right)+
\sum_f\frac{g^2}{12\pi^2 T^2}(q_fB)^2
\sum_{l=1}^{\infty}(-1)^{l+1}l^2K_0(\frac{m_fl}{T}),
\label{real_b}\\
\left[\frac{{\rm Im}~b(q_0,q)}{q_0}\right]_
{q_0=0}&=&g^2T^2\left(\frac{N_c}{3}+\frac{N_f}{6}\right)\frac{\pi}{2q}
+\frac{1}{q}\left[\sum_f\frac{g^2(q_fB)^2}{16\pi T^2}
\sum_{l=1}^\infty(-1)^{l+1}l^2K_0\left(\frac{m_f l}{T}\right)
\right. \nonumber\\ &-&\left.
\sum_f\frac{g^2(q_fB)^2}{96\pi T^2}
\sum_{l=1}^\infty(-1)^{l+1}l^2K_2\left(\frac{m_f l}{T}\right)
+\sum_f\frac{g^2(q_fB)^2}{768\pi}\frac{(8T-7\pi m_f)}{m_f^2 T}
\right],
~~\label{img_b}\end{aligned}$$ where the Eq. is the real-part of the form factor in the static limit which gives the Debye screening mass in the presence of weak magnetic field as follows $$\begin{aligned}
M_D^2=g^2T^2\left(\frac{N_c}{3}+\frac{N_f}{6}\right)+
\sum_f\frac{g^2}{12\pi^2 T^2}(q_fB)^2
\sum_{l=1}^{\infty}(-1)^{l+1}l^2K_0(\frac{m_fl}{T}).
\label{debyemass}\end{aligned}$$ Thus, it is observed that Debye screening mass of the thermal medium in the presence of weak magnetic field is affected by both the temperature and magnetic field. Now in order to see how the Debye mass is changed in the presence of weak magnetic field we have mentioned the leading order result of Debye mass for thermal medium in absence of magnetic (termed as “Pure Thermal”) [@Shuryak:ZETF'1978]. $$\begin{aligned}
M^2_D({\rm Pure~Thermal})=g^2 T^2
\left(\frac{N_c}{3} +\frac{N_f}{6}\right).\end{aligned}$$ In the left panel of Fig.\[debye\], we have quantitatively studied the variation of Debye mass with varying strength of weak magnetic field for a fixed value of temperature. We have observed that the debye mass is found to increase with the varying strength of magnetic field. On the other hand, in the right panel of Fig.\[debye\], we have studied the variation with the temperature for a fixed value of magnetic field and observed that the Debye mass is also found to increase with increasing temperature, but the increase of Debye mass with temperature is fast as compared to the slow increase with magnetic field. In addition to this, we have also made a comparison of Debye mass in presence of magnetic field with the one in absence of magnetic field and observed that the Debye mass in presence of weak magnetic field is found to be slightly higher as compared to the one in pure thermal case.
--------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
![Variation of Debye mass with magnetic field (left panel) and with temperature (right panel).[]{data-label="debye"}](debye_mag.eps "fig:"){width="7.6cm" height="8.4cm"} ![Variation of Debye mass with magnetic field (left panel) and with temperature (right panel).[]{data-label="debye"}](debye_temp.eps "fig:"){width="7.5cm" height="7.6cm"}
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Medium modified heavy quark potential
=====================================
In this section we will discuss the medium modification to the potential between a heavy quark $Q$ and its anti-quark $\bar{Q}$ in the presence of weak magnetic field. Since the mass of the heavy quark ($m_Q$) is very large, so the requirements - $m_Q \gg T \gg \Lambda_{QCD}$ and $m_Q \gg \sqrt{eB}$ are satisfied for the description of the interactions between a pair of heavy quark and anti-quark at finite temperature in a weak magnetic field in terms of quantum mechanical potential, that leads to the validity of taking the static heavy quark potential. Thus we can obtain the medium-modification to the vacuum potential in the presence of magnetic field by correcting both its short and long-distance part with a dielectric function $\epsilon(q)$ as $$V(r;T,B)=\int\frac{d^3q}{(2\pi)^{3/2}}
({e^{ik.r}-1})\frac{V(q)}{\epsilon(q)},
\label{pot_defn}$$ where the $r$-independent term has subtracted to re normalize the heavy quark free energy, which is the perturbation free energy of quarkonium at infinite separation. The Fourier transform, $V(q)$ of the perturbative part of the Cornell potential ($V(r)=-\frac{4\alpha_s}{3r}$) is given by $${V}(q)=-\frac{4}{3}\sqrt{\frac{2}{\pi}}
\frac{\alpha_s}{q^2},
\label{ft_pot}$$ and the dielectric permittivity, $\epsilon(q)$ embodies the effects of confined medium in the presence of magnetic field and is to be calculated next. The important point to be noted here is that we have taken the Fourier transform of the perturbative part of the vacuum potential only, the reason for this is that we can not use the same screening scale for both Coulomb and string terms because of the non-perturbative nature of the string term. To include the non-perturbative part of the potential, we will use the method of dimension two gluon condensate.
The complex permittivity for a hot QCD medium in a weak magnetic field
----------------------------------------------------------------------
The complex dielectric permittivity, $\epsilon (q)$ is defined by the static limit of “00”-component of resummed gluon propagator from the linear response theory $$\frac{1}{\epsilon (q)}=-\displaystyle
{\lim_{q_0 \rightarrow 0}}{q}^{2}D^{00}(q_{0},
q).
\label{dielectric}$$ Now we will evaluate the “00”-component of resummed gluon propagator. The real-part of the resummed gluon propagator in the static limit can be evaluated by using Eq. and Eq. $$\begin{aligned}
{\rm Re}~D^{00}(q_0=0)=\frac{-1}{q^2+M_D^2}.
\label{real_resummed}\end{aligned}$$ The imaginary part of resummed gluon propagator can be written in terms of the real and imaginary parts of the form factor by using the following formula [@Weldon:PRD42'1990] $$\begin{aligned}
{\rm Im}~D^{00}(q_0,q)=\frac{2T}{q_0}\frac{{\rm Im}~b(q_0,q)}
{(Q^2-{\rm Re}~b(q_0,q))^2+({\rm Im}~b(q_0,q))^2},\end{aligned}$$ which can be recast into the following form $$\begin{aligned}
{\rm Im}~D^{00}(q_0,q)=2T\frac{\left[\frac{{\rm Im}~b(q_0,q)}
{q_0}\right]}
{(Q^2-{\rm Re}~b(q_0,q))^2+(q_0\left[\frac{{\rm Im}~b(q_0,q)}
{q_0}\right])^2},\end{aligned}$$ in the static limit the above equation reduces to the simplified form $$\begin{aligned}
{\rm Im}~D^{00}(q_0=0)=2T\frac{\left[\frac{{\rm Im}~
b(q_0,q)}{q_0}\right]_{q_0=0}}{(q^2+M_D^2)^2},
\label{resummed}\end{aligned}$$ where we have substituted ${\rm Re}~b(q_0=0)=M_D^2$. Using Eq. and the above Eq., the imaginary part of “00”-component of resummed gluon propagator can be written as follows $$\begin{aligned}
{\rm Im}~D^{00}(q_0=0,q)=\frac{\pi T M^2_{(T,B)}}{q(q^2+M_D^2)},
\label{img_resummed}\end{aligned}$$ where we have defined the quantity $M^2_{(T,B)}$ as follows $$\begin{aligned}
M_{(T,B)}^2&=&g^2T^2\left(\frac{N_c}{3}+\frac{N_f}{6}\right)
+\left[\sum_f\frac{g^2(q_fB)^2}{8\pi^2 T^2}
\sum_{l=1}^\infty(-1)^{l+1}l^2K_0\left(\frac{m_f l}{T}\right)
\right. \nonumber\\ &-&\left.
\sum_f\frac{g^2(q_fB)^2}{48\pi^2 T^2}
\sum_{l=1}^\infty(-1)^{l+1}l^2K_2\left(\frac{m_f l}{T}\right)
+\sum_f\frac{g^2(q_fB)^2}{384\pi^2}\frac{(8T-7\pi m_f)}{m_f^2 T}
\right].\end{aligned}$$ Now we will obtain the real and imaginary parts of dielectric permittivity, before evaluating them we will discuss the procedure to handle the nonperturbative part of the heavy quark potential. The nonperturbative part of the potential is recently been discussed in [@Guo:PRD100'2019]. The procedure is to include a nonperturbative term in the real and imaginary parts of the “00”-component of resummed gluon propagator along with the usual Hard Thermal Loop (HTL) propagator which we have obtained earlier. The real and imaginary parts of the nonperturbative (NP) term by using the dimension two gluon condensate are given as follows $$\begin{aligned}
{\rm Re}~D^{00}_{NP}(q_0=0,q)=-\frac{m_G^2}{(q^2+M_D^2)^2},\\
{\rm Im}~D^{00}_{NP}(q_0=0,q)=\frac{2\pi TM^2_{(T,B)}m_G^2}
{q(q^2+M_D^2)^3},\end{aligned}$$ where $m_G^2$ is a dimensional constant, which can be related to the string tension through the relation $\sigma=\alpha m_G^2/2$. Thus, the real and imaginary parts of the “00”-component of the resummed gluon propagator that consists of both the HTL and the NP contributions can be written as follows $$\begin{aligned}
{\rm Re}~D^{00}(q_0=0,q)&=&-\frac{1}
{q^2+M_D^2}-\frac{m_G^2}
{(q^2+M_D^2)^2}
\label{real_propagator},\\
{\rm Im}~D^{00}(q_0=0,q)&=&\frac{2\pi T M^2_{(T,B)}}
{q(q^2+M_D^2)^2}+\frac{2q\pi T M^2_{(T,B)}m_G^2}
{q(q^2+M_D^2)^3}.
\label{imaginary_propagator}\end{aligned}$$ Now substituting Eq. and Eq. in Eq. gives the real and imaginary parts of the dielectric permittivity, respectively $$\begin{aligned}
\frac{1}{{\rm Re}~\epsilon (q)}&=&\frac{q^2}
{q^2+M_D^2}+\frac{q^2 m_G^2}
{(q^2+M_D^2)^2}
\label{real_dielectric},\\
\frac{1}{{\rm Im}~\epsilon (q)}&=&-\frac{2q\pi T M^2_{(T,B)}}
{(q^2+M_D^2)^2}-\frac{2q\pi T M^2_{(T,B)}m_G^2}
{(q^2+M_D^2)^3}.
\label{imaginary_dielectric}\end{aligned}$$ We are now going to derive the real and imaginary parts of the complex potential from the real and imaginary parts of dielectric permittivities, respectively in the next subsection. The important point to be noted here is that the non perturbative terms in the real and imaginary parts of the dielectric permittivity will lead to the string contribution in the real and imaginary parts of the potential.
Real and Imaginary parts of the potential
-----------------------------------------
Here we will calculate the real and imaginary parts of the heavy quark potential in presence of weak magnetic field. The real-part of the dielectric permittivity in Eq. is substituted into the definition of potential in Eq. to obtain the real-part of $Q \bar Q$ potential in the presence of weak magnetic field (with $\hat{r}=rM_{D}$) $$\begin{aligned}
\rm{Re} V(r;T,B)&=&-\frac{4}{3}\alpha_s\left(\frac{e^{-\hat{r}}}
{r}+M_D\right)+\frac{4}{3}\frac{\sigma}{M_D}
\left(1-e^{-\hat{r}}\right),
\label{real_potential}\end{aligned}$$ where the temperature and magnetic field dependence in the potential enters through the Debye mass. While plotting the real-part of the potential we have excluded the non-local terms which are however, required to reduce the potential in the medium $V(r;T,B)$ to the vacuum potential in $(T, B) \rightarrow 0 $ limit. In Fig.\[realb\], we have plotted the real-part of the potential as a function of interquark distance ($r$). In the left panel of Fig.\[realb\] we have plotted the real-part of the potential for different strengths of weak magnetic field like $eB=0.5m_\pi^2$ and $2m_\pi^2$ for a fixed value of temperature $T=2T_c$. We observed that on increasing the value of magnetic field the real-part become more screened. Whereas in the right panel of Fig.\[realb\], the real-part is plotted for different strengths of temperature like $T=1.5T_c$ and $T=2T_c$ and found to be more screened on increasing the value of temperature. Thus, the real-part of the potential is found to be more screened on increasing the value of both temperature and magnetic field. This observation of the real-part of the potential can be understood in terms of the observation of the Debye mass which is found to be increased both with temperature and magnetic field as shown earlier in Fig.\[debye\].
------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
![Real-part of the potential for different strengths of magnetic field (left panel) and for different strengths of temperature (right panel).[]{data-label="realb"}](rpot_mag.eps "fig:"){width="7.5cm" height="7.5cm"} ![Real-part of the potential for different strengths of magnetic field (left panel) and for different strengths of temperature (right panel).[]{data-label="realb"}](rpot_temp.eps "fig:"){width="7.5cm" height="7.5cm"}
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![Real-part of the potential in the absence and presence of weak magnetic field.[]{data-label="real_comp"}](rpot_mag_comp.eps){width="7.5cm" height="7.5cm"}
We have made a comparison in Fig.\[real\_comp\] to see how the magnetic field will affect the real-part of the potential, for that we have plotted the real-part of the potential in presence of magnetic field with the one for pure thermal case. As we have seen in the right panel of Fig.\[debye\] that the Debye mass in presence of magnetic field is slightly higher as compared to the Debye mass in pure thermal medium, that leads to the slightly more screening of the real-part of the potential in presence of weak magnetic field as compared to the same in the pure thermal case.
We will now evaluate the imaginary-part of the potential in presence of weak magnetic field. The imaginary-part of the potential is obtained by substituting the imaginary part of dielectric permittivity from Eq. into the definition of the potential Eq. $$\begin{aligned}
\rm{Im} V_C(r;T,B)&=&-\frac{4}{3}\frac{\alpha_s T M^2_{(T,B)}}{M_D^2}\
\phi_2(\hat{r}),\\
\rm{Im} V_S(r;T,B)&=&-\frac{4 \sigma T M^2_{(T,B)}}{M_D^4}\
\phi_3(\hat{r}),
\label{imaginary_potential}\end{aligned}$$ where the function $\phi_2(\hat{r})$ and $\phi_3(\hat{r})$ are given in [@Guo:PRD100'2019] $$\begin{aligned}
\phi_2(\hat{r})&=&2\int_0^{\infty}\frac{zdz}{(z^2+1)^2}
\left[1-\frac{\sin z\hat{r}}{z\hat{r}}\right],\\
\phi_3(\hat{r})&=&2\int_0^{\infty}\frac{zdz}{(z^2+1)^3}
\left[1-\frac{\sin z\hat{r}}{z\hat{r}}\right],\end{aligned}$$ in the small $\hat{r}$ limit $(\hat{r}\ll 1)$, the above functions become $$\begin{aligned}
\phi_2(\hat{r})&\approx &-\frac{1}{9}{\hat{r}}^2\left(3\ln \hat{r}-
4+3\gamma_E\right),\\
\phi_3(\hat{r})&\approx&-\frac{{\hat{r}}^2}{12}+\frac{{\hat{r}}^4}{900}
\left(15\ln\hat{r}-23+15\gamma_E\right).\end{aligned}$$ Thus, the imaginary part of the potential in the small $\hat{r}$ limit becomes, keeping only the leading-log term $$\begin{aligned}
\rm{Im} V_C(r;T,B)&=&\frac{4}{9}\frac{\alpha_s T M^2_{(T,B)}}{M_D^2}
{\hat{r}}^2\ln \hat{r},
\label{imaginary_coulomb}\\
\rm{Im} V_S(r;T,B)&=&-\frac{\sigma T M^2_{(T,B)}}{15 M_D^4}\
{\hat{r}}^4\ln \hat{r}.
\label{imaginary_string}\end{aligned}$$ Similar to the real-part of the potential we have plotted the imaginary-part of the potential as a function of interquark distance ($r$) in Fig.\[imgb\]. We have calculated the imaginary-part of the potential for different strengths of weak magnetic field like $eB=0.5m_\pi^2$ and $2m_\pi^2$ in the left panel of Fig.\[imgb\]. We found that on increasing the value of magnetic field the magnitude of imaginary-part gets increased. On the other hand, in the right panel of Fig.\[realb\], the imaginary-part is calculated for different strengths of temperature like $T=1.5T_c$ and $T=2T_c$ here also the imaginary-part is found to increase with the temperature. Hence the magnitude of the imaginary-part of the potential gets increased with the value of temperature and magnetic field both. This observation also attributed to the fact that the Debye mass is found to be increased with temperature and magnetic field both.
----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- -----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
![Imaginary-part of the potential for different strengths of magnetic field (left panel) and for different strengths of temperature (right panel).[]{data-label="imgb"}](ipot_mag.eps "fig:"){width="7.5cm" height="7.5cm"} ![Imaginary-part of the potential for different strengths of magnetic field (left panel) and for different strengths of temperature (right panel).[]{data-label="imgb"}](ipot_temp.eps "fig:"){width="7.5cm" height="7.5cm"}
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Here also we have calculated the imaginary-part of the potential in presence of magnetic field with the one for pure thermal case in Fig.\[img\_comp\]. We have observed that the imaginary-part of the potential in presence of magnetic field is increased slightly as compared to the one in pure thermal case.
![Imaginary-part of the potential in the absence and presence of weak magnetic field.[]{data-label="img_comp"}](ipot_mag_comp.eps){width="7.5cm" height="7.5cm"}
Properties of quarkonia
=======================
In this section we first explore the effects of weak magnetic field on the properties of heavy quarkonia. The obtained real and imaginary parts of the heavy quark potential will be used to evaluate the binding energy and thermal width of the heavy quarkonia, respectively.
Binding energy
--------------
In this subsection, we have obtained the binding energy of $J/\psi$ and $\Upsilon$. In order to calculate the binding energy, the real part of the potential Eq. is put into the radial part of the Schrödinger equation, which is then solved numerically to obtain the energy eigenvalues that inturns give the binding energies of quarkonia. To see how the presence of weak magnetic field affects the binding of quarkonia, we have plotted the binding energies of $J/\psi$ as a function of $T/T_c$ for different strengths of magnetic field in the left panel of Fig.\[psi\]. We observed that the binding energy is found to decrease with the temperature and magnetic field both, we can attribute this finding in terms of the increasing of screening with the temperature and magnetic field that we have observed in the real-part of the potential. The point to be noted here is that the difference between the values of binding energies plotted for the magnetic field $eB=0.5m_\pi^2$ and $eB=2m_\pi^2$ is pronounced at higher temperature, this is in accordance with the validity of our work in the weak field limit $(T^2>>|q_fB|)$.
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![The binding energy of $J/\psi$ as a function of temperature.[]{data-label="psi"}](binding_psi.eps "fig:"){width="7.5cm" height="7cm"} ![The binding energy of $J/\psi$ as a function of temperature.[]{data-label="psi"}](binding_psi_comp.eps "fig:"){width="7.5cm" height="7cm"}
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![The binding energy of $\Upsilon$ as a function of temperature.[]{data-label="upsilon"}](binding_upsilon.eps "fig:"){width="7.5cm" height="7cm"} ![The binding energy of $\Upsilon$ as a function of temperature.[]{data-label="upsilon"}](binding_upsilon_comp.eps "fig:"){width="7.5cm" height="7cm"}
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In the right panel of Fig.\[psi\], we have also compared the binding energy of $J/\psi$ in presence of weak magnetic field with the pure thermal case. We found that the binding energy in presence of magnetic field is smaller as compared to the one in pure thermal case, this is because the real-part of the potential in presence of magnetic becomes more screened as compared to pure thermal case. The similar observation has also been observed for $\Upsilon$, except that the value of binding energy for $\Upsilon$ is higher as compared to the value for $J/\Psi$. The variation of binding energy for $\Upsilon$ is studied in the left and right panel of Fig.\[upsilon\].
Thermal width
-------------
We will now use the imaginary part of the potential obtained in presence of weak magnetic field to estimate the broadening of the resonance states in a thermal medium. So using the first-order time-independent perturbation theory, the width ($\Gamma$) has been evaluated by folding with ($\Phi(r)$), $$\begin{aligned}
\Gamma({\rm T,B})=-2\int_0^\infty \rm{Im}~V(r;T,B) |\Phi(r)|^2 d\tau,
\label{gammaT}\end{aligned}$$ the wave function $\Phi(r)$ is taken to be the Coloumbic wave function for the ground state $$\begin{aligned}
\Phi(r)=\frac{1}{\sqrt{\pi a_0^3}}e^{-r/a_0},\end{aligned}$$ where $a_0=\frac{3}{2m_Q\alpha_s}$ is the Bohr radius of the heavy quarkonium system. The imaginary-part of the potential contain the Coulombic and string contributions, thus we can write the thermal width as follows $$\begin{aligned}
\Gamma({\rm T,B})=\Gamma_C({\rm T,B})+\Gamma_S({\rm T,B}).\end{aligned}$$ Here we have used the imaginary-part of the potential as a perturbative to obtain the thermal width, for that purpose we have obtained the imaginary-part of the potential in the small distance limit. Thus, the Coulombic and string contributions to the thermal width are calculated analytically by substituting the imaginary parts of the potential due to the coulombic Eq. and string terms Eq. in the small distance limit, respectively $$\begin{aligned}
\Gamma_C({\rm T,B})&=&\frac{6TM^2_{(T,B)}}{m_Q^2\alpha_s}\left[\ln\left(\frac
{3M_D}{4m_Q\alpha_s}\right)+1.5061\right],\\
\Gamma_S({\rm T,B})&=&-\frac{243\sigma TM^2_{(T,B)}}{16m_Q^4{\alpha_s}^4}
\left[\ln\left(\frac
{3M_D}{4m_Q\alpha_s}\right)+1.872\right].\end{aligned}$$
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![Variation of the thermal widths with the temperature for $J/\psi$.[]{data-label="decay_psi"}](width_psi.eps "fig:"){width="7.5cm" height="7.5cm"} ![Variation of the thermal widths with the temperature for $J/\psi$.[]{data-label="decay_psi"}](width_psi_comp.eps "fig:"){width="7.5cm" height="7.5cm"}
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![Variation of the thermal widths with the temperature for $\Upsilon$.[]{data-label="decay_upsilon"}](width_upsilon.eps "fig:"){width="7.5cm" height="7.5cm"} ![Variation of the thermal widths with the temperature for $\Upsilon$.[]{data-label="decay_upsilon"}](width_upsilon_comp.eps "fig:"){width="7.5cm" height="7.5cm"}
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We have obtained the thermal width analytically and observed that it depend on the temperature as well as the weak magnetic field. To explore the effects of the weak magnetic field on the thermal width of heavy quarkonia, we have plotted the thermal width of $J/\psi$ and $\Upsilon$ as a function of $T/T_c$ for different strengths of magnetic field in Fig.\[decay\_psi\] and Fig.\[decay\_upsilon\], respectively. We observed that the thermal widths for $J/\psi$ and $\Upsilon$ get increased both with the temperature and magnetic field as depicted in the left panels of Fig.\[decay\_psi\] and Fig.\[decay\_upsilon\]. We can understood this finding in terms of the increase of the imaginary-part of the potential, the magnitude of which gets enhanced both with temperature and magnetic field. We also made a comparison of thermal width in presence of weak magnetic field with its counter part in absence of magnetic field in the right panels of Fig.\[decay\_psi\] and Fig.\[decay\_upsilon\], where we found that the decay widths for $J/\Psi$ and $\Upsilon$ get increased in the presence of magnetic field as compared to the pure thermal case.
Dissociation of quarkonia
-------------------------
In the previous subsections, we have obtained the binding energies and thermal widths of heavy quarkonia, $J/\psi$ and $\Upsilon$. Now we will study the quasi-free dissociation of heavy quarkonia in a thermal QCD medium and see how the dissociation temperatures of quarkonia are affected in the presence of weak magnetic field. For that purpose we use the criterion on the width of the resonance ($\Gamma$): $\Gamma \ge 2 ~{\rm{BE}}$ [@Mocsy:PRL99'2007] (where ${\rm{BE}}$ is the binding energy of the heavy quarkonia) to estimate the dissociation temperature for $J/\psi$ and $\Upsilon$.
----------------------- ---------- ------------
State $J/\psi$ $\Upsilon$
Pure Thermal ($eB=0$) 1.17 3.98
$eB=0.5m_\pi^2$ 1.13 3.94
----------------------- ---------- ------------
: Dissociation temperatures in absence and presence of weak magnetic field.[]{data-label="table_diss"}
We have obtained the dissociation temperatures of $J/\Psi$ and $\Upsilon$ in the absence and presence of weak magnetic field in Table.\[table\_diss\], and observed that the dissociation temperatures become slightly lower in the presence of weak magnetic field. [*For example*]{}, with $eB = 0m_\pi^2$ the $J/\psi$ and $\Upsilon$ are dissociated at $1.17T_c$ and $3.98T_c$, respectively whereas with $eB = 0.5m_\pi^2$ the $J/\psi$ and $\Upsilon$ are dissociated at $1.13T_c$ and $3.94T_c$. This observation leads to the slightly early dissociation of heavy quarkonia in the presence of the weak magnetic field.
Conclusions
===========
In the present theoretical study, we have explored the effects of weak magnetic field on the dissociation of quarkonia in a thermal QCD by calculating the complex heavy quark potential perturbatively in the aforesaid medium. For that purpose, we first evaluate the gluon self-energy in a similar environment using the imaginary-time formalism. Furthermore, we have revisited the general structure of gluon self-energy tensor in the presence of weak magnetic field in thermal medium and obtained the relevant structure functions, that inturns give rise to the real and imaginary parts of the resummed gluon propagator, which give the real and imaginary parts of the dielectric permittivity. To include the medium modification to the non-perturbative part of the vacuum heavy quark potential, we have included a non-perturbative term in the resummed gluon propagator induced by the dimension two gluon condensate besides the usual hard thermal loop resummed contribution. Thus, the real and imaginary parts of the dielectric permittivity will be used to evaluate the real and imaginary parts of the complex heavy quark potential. We have studied the effects of weak magnetic field on the real and imaginary parts of the potential. We have found that the real-part of the potential is found to be more screened on increasing the value of temperature and magnetic field both. In addition to this, we have observed that the real-part gets slightly more screened in the presence of weak magnetic field as compared to its counter part in the absence of magnetic field. On the other hand, the magnitude of the imaginary-part of the potential gets increased with the value of both temperature and magnetic field, and its magnitude also gets increased in the presence of weak magnetic field as compared to pure thermal case. The real part of the potential is used in the Schrödinger equation to obtain the binding energy of heavy quarkonia, whereas the imaginary part is used to calculate the thermal width. We observed that the binding energies of $J/\Psi$ and $\Upsilon$ are found to decrease with the temperature and magnetic field both, we can attribute this findings in terms of the increasing of screening of the real-part of the potential with the temperature and magnetic field both. We also observed that the binding energy of $J/\Psi$ and $\Upsilon$ in the presence of magnetic field are smaller as compared to the one in the pure thermal case. The increase in the magnitude of the imaginary-part of the potential will leads to the increase of decay width with temperature and magnetic field both. The thermal width for $J/\Psi$ and $\Upsilon$ get increased in presence of magnetic field as compared to pure thermal case. With the observations of binding energy and decay width in hands, we have finally studied the dissociation of quarkonia in the presence of weak magnetic field. The dissociation temperatures for $J/\Psi$ and $\Upsilon$ become slightly lower in the the presence of weak magnetic field. [*For example*]{}, with $eB = 0m_\pi^2$ the $J/\psi$ and $\Upsilon$ are dissociated at $1.17T_c$ and $3.98T_c$, respectively whereas with $eB = 0.5m_\pi^2$ the $J/\psi$ and $\Upsilon$ are dissociated at $1.13T_c$ and $3.94T_c$. This observation leads to the slightly early dissociation of quarkonia because of the presence of a weak magnetic field.
Acknowledgements {#acknowledgements .unnumbered}
================
One of the author BKP is thankful to the CSIR (Grant No.03 (1407)/17/EMR-II), Government of India for the financial assistance. In the following appendices we have shown the explicit calculations of form factors $b_0(Q)$ and $b_2(Q)$.
Calculation of the form factor $b_0(Q)$ {#b_0}
=======================================
In this appendix, we will use the imaginary-time formalism to calculate the form factor $b_0$, which is given by $$\begin{aligned}
b_0(Q)&=&\sum_f \frac{i2g^2}{\bar{u}^2}\int\frac{d^4K}{(2\pi)^4}\frac
{\left[2k_0^2-K^2+m_f^2)\right]}
{(K^2-m^2_f)(P^2-m_f^2)},\nonumber\\
&=&-N_f \frac{2g^2}{\bar{u}^2}\int\frac{d^3k}{(2\pi)^3}T\sum_n\frac
{\left[K^2+2k^2\right]}
{(K^2-m^2_f)(P^2-m_f^2)},\nonumber\\
&=&-N_f \frac{2g^2}{\bar{u}^2}[I_1(Q)+I_2(Q)],
\label{formfactor1_b0}\end{aligned}$$ where we have neglected $m_f$ in numerator in the Hard Thermal Loop (HTL) approximation and $\int\frac{d^4K}{(2\pi)^4}
\rightarrow iT\int\frac{d^3k}{(2\pi)^3}\sum_n$, the $I_1$ and $I_2$ are given as $$\begin{aligned}
I_1(Q)&=&\int\frac{d^3k}{(2\pi)^3}T\sum_n\frac
{K^2}{(K^2-m^2_f)(P^2-m_f^2)},\\
I_2(Q)&=&\int\frac{d^3k}{(2\pi)^3}T\sum_n\frac
{2k^2}{(K^2-m^2_f)(P^2-m_f^2)}.\end{aligned}$$ Now we substitute $k_0=i\omega_n$, $q_0=i\omega$, $E_1=\sqrt{k^2+m^2_f}$ and $E_2=\sqrt{(k-q)^2+m^2_f}$, and then perform the frequency sum, which gives $I_1$ as $$\begin{aligned}
I_1(Q)&=&-\int\frac{d^3k}{(2\pi)^3}T\sum_n\frac
{1}{(\omega_n^2+E_1^2)},\nonumber\\
&=&-\int\frac{d^3k}{(2\pi)^3}\frac{1}{2E_1}[1-2n_F(E_1)],\end{aligned}$$ where the first term is the non-leading term in $T$, thus retaining only the leading term in $T$, the $I_1$ becomes $$\begin{aligned}
I_1(Q)&=&\int\frac{d^3k}{(2\pi)^3}\frac{n_F(E_1)}{E_1},\end{aligned}$$ now taking $I_2$, which becomes $$\begin{aligned}
I_2(Q)&=&2\int\frac{d^3k}{(2\pi)^3}k^2~T\sum_n\frac
{1}{(\omega_n^2+E_1^2)[{(\omega_n-\omega)}^2+E_2^2]},\nonumber\\
&=&-\int\frac{d^3k}{(2\pi)^3}\left[\frac{n_F(E_1)}{E_1}
+q\cos\theta\frac{dn_F(E_1)}{dk}\frac{1}{i\omega-
q\cos\theta}\right].\end{aligned}$$ Substituting $I_1$ and $I_2$ in Eq., the form factor $b_0$ becomes $$\begin{aligned}
b_0(q_0,q)=-N_f \frac{2g^2}{\bar{u}^2}\int\frac{d^3k}{(2\pi)^3}
\frac{dn_F(E_1)}{dk}\left(1-\frac{q_0}{q_0-q\cos\theta}\right),\end{aligned}$$ where we have again resubstituted $q_0=i\omega$. Now we will evaluate the real and imaginary parts of the form factor $b_0$.
The real-part of $b_0$ in the static limit is given by $$\begin{aligned}
\rm{Re}~b_0(q_0=0)&=&-N_f \frac{g^2}{\pi^2}\int k^2 dk
\frac{dn_F(E_1)}{dk},\nonumber\\
&=&N_f\frac{g^2T^2}{6}.\end{aligned}$$ On the other hand, for the evaluation of the imaginary part of $b_0$ we will us the following identity $$\begin{aligned}
{\rm Im}~b_0(q_0,q)=\frac{1}{2i}\lim_{\eta\rightarrow 0}
\left[b(q_0+i\eta,q)-b(q_0-i\eta,q)\right],
\label{identity1}\end{aligned}$$ along with the following expression $$\begin{aligned}
\frac{1}{2i}\left(\frac{1}{q_0+\sum_j E_j+i\eta}-\frac{1}
{q_0+\sum_j E_j-i\eta}\right)=-\pi\delta(q_0+\sum_j E_j),
\label{identity2}\end{aligned}$$ thus using the above identities Eq. and Eq., the imaginary-part of $b_0$ becomes $$\begin{aligned}
\rm {Im}~b_0(q_0,q)&=&N_f\frac{2g^2}{\bar{u}^2}
\frac{1}{2i}\lim_{\eta\rightarrow 0}\int \frac{d^3k}{(2\pi)^3}
\frac{dn_F(k)}{dk}
\left(\frac{q\cos\theta}{q_0-q\cos\theta+i\eta}
-\frac{q\cos\theta}{q_0-q\cos\theta-i\eta}\right),\nonumber\\
&=&-N_f\frac{\pi g^2}{2\pi^2\bar{u}^2}
\frac{q_0}{q}\int k^2~dk\frac{dn_F(k)}{dk},\end{aligned}$$ which in the static limit takes the simplified form $$\begin{aligned}
\left[\frac{{\rm Im}~b_0(q_0,q)}{q_0}\right]_
{q_0=0}=\frac{g^2T^2N_f}{6}\frac{\pi}{2q}.\end{aligned}$$
Calculation of the form factor $b_2(Q)$ {#b_2}
=======================================
Similar to the form factor $b_0$, here we will solve the form factor $b_2$, which is given by $$\begin{aligned}
b_2(Q)&=&\sum_f \frac{i2g^2(q_fB)^2}{\bar{u}^2}\left[\int\frac
{d^4K}{(2\pi)
^4}\left\lbrace\frac{\left(2k_0^2-K_\parallel^2+m_f^2\right)}
{(K^2-m^2_f)^2(P^2-m_f^2)^2}
-\frac{\left(8k_0^2K_\perp^2\right)}{(K^2-m^2_f)^4(P^2-m_f^2)}\right
\rbrace\right],\nonumber\\
&=&-\sum_f \frac{2g^2(q_fB)^2}{\bar{u}^2}\int\frac{d^3k}{(2\pi)^3}
T\sum_n\left\lbrace\frac{K^2+k^2(1+\cos^2\theta)+m_f^2)}
{(K^2-m^2_f)^2(P^2-m_f^2)^2}\right.\nonumber\\&&\left.-
\frac{8(k^4+k^2K^2)(1-\cos^2\theta)}
{(K^2-m^2_f)^4(P^2-m_f^2)}\right\rbrace,\end{aligned}$$ where we have used the spherical polar coordinate system for $k=(k\sin\theta\sin\phi,k\sin\theta
\cos\phi,k\cos\theta)$. In order to solve the form factor $b_2$, we will use the method as shown in [@karmakar:EPJC79'2019] , which gives $$\begin{aligned}
b_2(Q)&=&-\sum_f\frac{2g^2q_f^2B^2}{\bar{u}^2}\left[
\left\lbrace\frac{\partial}{\partial(m_f^2)}+\frac{5}{6}
m_f^2\frac
{\partial^2}{\partial^2(m_f^2)}\right\rbrace
\int\frac{d^3k}{(2\pi)^3}T\sum_n\frac{1}{(K^2-m_f^2)
(P^2-m_f^2)}\right.\nonumber\\&&\left.-\left\lbrace\frac
{\partial}{\partial(m_f^2)}+\frac{m_f^2}{2}\frac
{\partial^2}{\partial^2(m_f^2)}\right\rbrace
\int\frac{d^3k}{(2\pi)^3}T\sum_n\frac{\cos^2\theta}
{(K^2-m_f^2)(P^2-m_f^2)}
\right],\end{aligned}$$ now we will perform the following frequency sum $$\begin{aligned}
T\sum_n\frac{1}{(\omega_n^2+E_1^2)[(\omega_n-\omega)^2+E_2^2}
&=&\frac{[1-n_F(E_1)-n_F(E_2)]}{4E_1E_2}\left\lbrace\frac{1}
{i\omega+E_1+E_2}-\frac{1}{i\omega-E_1-E_2}\right\rbrace\nonumber\\
&&+\frac{[n_F(E_1)-n_F(E_2)]}{4E_1E_2}\left\lbrace\frac{1}
{i\omega+E_1-E_2}-\frac{1}{i\omega-E_1+E_2}\right\rbrace.\end{aligned}$$ Thus, after simplification the form factor $b_2$ becomes $$\begin{aligned}
b_2(q_0,q)&=&\sum_f\frac{2g^2q_f^2B^2}{\bar{u}^2}\left\lbrace\
\left(\frac{\partial^2}{\partial^2(m_f^2)}+\frac{5}{6}
m_f^2\frac{\partial^3}{\partial^3(m_f^2)}\right)
\int\frac{d^3k}{(2\pi)^3}\frac{n_F(E_1)}{E_1}\left
(\frac{q_0}{q_0-q\cos\theta}-1\right)\right.\nonumber\\&&+\left.
\left(\frac{\partial}{\partial(m_f^2)}+\frac{5}{6}
m_f^2\frac{\partial^2}{\partial^2(m_f^2)}\right)
\int\frac{d^3k}{(2\pi)^3}\frac{n_F(E_1)}{2E_1^3}
\left(\frac{q_0}{q_0-q\cos\theta}\right)
\right.\nonumber\\&&-\left.
\left(\frac{\partial^2}{\partial^2(m_f^2)}+\frac{
m_f^2}{2}\frac{\partial^3}{\partial^3(m_f^2)}\right)
\int\frac{d^3k}{(2\pi)^3}\frac{n_F(E_1)}{E_1}\cos^2\theta
\left(\frac{q_0}{q_0-q\cos\theta}-1\right)
\right.\nonumber\\&&-\left.
\left(\frac{\partial}{\partial(m_f^2)}+\frac{
m_f^2}{2}\frac{\partial^2}{\partial^2(m_f^2)}\right)
\int\frac{d^3k}{(2\pi)^3}\frac{n_F(E_1)}{2E_1^3}\cos^2\theta
\left(\frac{q_0}{q_0-q\cos\theta}\right)
\right\rbrace.\end{aligned}$$ Thus, the real part of $b_2$ in the static limit is obtained as [@karmakar:EPJC79'2019] $$\begin{aligned}
{\rm Re}~b_2(q_0=0)=\sum_f\frac{g^2}{12\pi^2 T^2}(q_fB)^2
\sum_{l=1}^{\infty}(-1)^{l+1}l^2K_0(\frac{m_fl}{T}).\end{aligned}$$ Now we will evaluate the imaginary part of form factor $b_2$, for that we write $b_2$ as $$\begin{aligned}
b_2(q_0,q)=\sum_f\frac{2g^2q_f^2B^2}{\bar{u}^2}[I_3(q_0,q)+
I_4(q_0,q)+I_5(q_0,q)+I_6(q_0,q)],
\label{sum}\end{aligned}$$ where we have defined the following functions $$\begin{aligned}
I_3(q_0,q)&=&
\left(\frac{\partial^2}{\partial^2(m_f^2)}+\frac{5}{6}
m_f^2\frac{\partial^3}{\partial^3(m_f^2)}\right)
\int\frac{d^3k}{(2\pi)^3}\frac{n_F(E_1)}{E_1}\left
(\frac{q\cos\theta}{q_0-q\cos\theta}\right),\\
I_4(q_0,q)&=&
\left(\frac{\partial}{\partial(m_f^2)}+\frac{5}{6}
m_f^2\frac{\partial^2}{\partial^2(m_f^2)}\right)
\int\frac{d^3k}{(2\pi)^3}\frac{n_F(E_1)}{2E_1^3}
\left(\frac{q_0}{q_0-q\cos\theta}\right),\\
I_5(q_0,q)&=&
\left(\frac{\partial^2}{\partial^2(m_f^2)}+\frac{
m_f^2}{2}\frac{\partial^3}{\partial^3(m_f^2)}\right)
\int\frac{d^3k}{(2\pi)^3}\frac{n_F(E_1)}{E_1}\cos^2\theta
\left(\frac{q\cos\theta}{q_0-q\cos\theta}\right),\\
I_6(q_0,q)&=&
\left(\frac{\partial}{\partial(m_f^2)}+\frac{
m_f^2}{2}\frac{\partial^2}{\partial^2(m_f^2)}\right)
\int\frac{d^3k}{(2\pi)^3}\frac{n_F(E_1)}{2E_1^3}\cos^2\theta
\left(\frac{q_0}{q_0-q\cos\theta}\right),\end{aligned}$$ Now we will evaluate the imaginary parts of all the above four terms one by one using the identities Eq. and Eq., first we start with $I_3(q_0,q)$ $$\begin{aligned}
{\rm Im} I_3(q_0,q)&=&X_3(m_f)
\frac{1}{2i}\lim_{\eta\rightarrow 0}\left[
\int\frac{d^3k}{(2\pi)^3}
\frac{n_F(E_1)}{E_1}
\left(\frac{q\cos\theta}{q_0-q\cos\theta+i\eta}
-\frac{q\cos\theta}{q_0-q\cos\theta-i\eta}\right)\right],
\nonumber\\\label{function_3}\end{aligned}$$ where $X_3(m_f)=\left(\frac{\partial^2}{\partial^2
(m_f^2)}+\frac{5}{6}
m_f^2\frac{\partial^3}{\partial^3(m_f^2)}\right)$, now the Eq. in the static limit becomes $$\begin{aligned}
\left[\frac{{\rm Im}~I_3(q_0,q)}{q_0}\right]_{q_0=0}&=
&-\frac{1}{4\pi q}X_3(m_f)
\int k^2~dk\frac{n_F(E_1)}{E_1},\nonumber\\
&=&-\frac{1}{4\pi q}X_3(m_f)
\sum_{l=1}^{\infty}\frac{m_f^2}{2}\left[K_2
(\frac{m_fl}{T})-K_0(\frac{m_fl}{T})\right],\nonumber\\
&=&\frac{1}{32\pi q T^2}
\sum_{l=1}^{\infty}(-1)^{l+1}l^2 K_0(\frac{m_fl}{T})
-\frac{1}{192\pi q T^2}
\sum_{l=1}^{\infty}(-1)^{l+1}l^2 K_2(\frac{m_fl}{T}),
\nonumber\\\label{i3}\end{aligned}$$ where $K_0$ and $K_2$ are the modified Bessel functions of second kind. Now we take $I_4(q_0,q)$ $$\begin{aligned}
{\rm Im} I_4(q_0,q)&=&X_4(m_f)
\frac{1}{2i}\lim_{\eta\rightarrow 0}\left[
\int\frac{d^3k}{(2\pi)^3}
\frac{n_F(E_1)}{2E_1^3}
\left(\frac{q_0}{q_0-q\cos\theta+i\eta}
-\frac{q_0}{q_0-q\cos\theta-i\eta}\right)\right],
\nonumber\\\label{function_4}\end{aligned}$$ where $X_4(m_f)=\left(\frac{\partial}{\partial
(m_f^2)}+\frac{5}{6}
m_f^2\frac{\partial^2}{\partial^2(m_f^2)}\right)$, the Eq., takes the following form in the static limit $$\begin{aligned}
\left[\frac{{\rm Im}~I_4(q_0,q)}{q_0}\right]_{q_0=0}&=
&-\frac{1}{8\pi q}X_4(m_f)
\int k^2~dk\frac{n_F(E_1)}{E_1^3},\nonumber\\
&=&\frac{1}{16\pi q}X_4(m_f)\left[1+\gamma_E-\frac
{\pi m_f}{4T}+\log\frac{m_f}{\pi T}\right],\nonumber\\
&=&\frac{1}{1536\pi q}\frac{(8T-7\pi m_f)}{m_f^2T}.
\nonumber\\\label{i4}\end{aligned}$$ Similarly the imaginary part of $I_5(q_0,q)$ and $I_6(q_0,q)$ $$\begin{aligned}
{\rm Im} I_5(q_0,q)&=&-X_5(m_f)
\frac{1}{2i}\lim_{\eta\rightarrow 0}\left[
\int\frac{d^3k}{(2\pi)^3}
\frac{n_F(E_1)}{E_1}
\left(\frac{q\cos^3\theta}{q_0-q\cos\theta+i\eta}
-\frac{q\cos^3\theta}{q_0-q\cos\theta-i\eta}\right)
\right],
\nonumber\label{function_5}\end{aligned}$$ where $X_5(m_f)=\left(\frac{\partial^2}{\partial^2
(m_f^2)}+\frac{
m_f^2}{2}\frac{\partial^3}{\partial^3(m_f^2)}\right)$, the Eq. vanishes in the static limit $$\begin{aligned}
\left[\frac{{\rm Im}~I_5(q_0,q)}{q_0}\right]_{q_0=0}&=
&0,
\label{i5}\end{aligned}$$ $$\begin{aligned}
{\rm Im} I_6(q_0,q)&=&-X_6(m_f)
\frac{1}{2i}\lim_{\eta\rightarrow 0}\left[
\int\frac{d^3k}{(2\pi)^3}
\frac{n_F(E_1)}{2E_1^3}
\left(\frac{q_0\cos^2\theta}{q_0-q\cos\theta+i\eta}
-\frac{q_0\cos^2\theta}{q_0-q\cos\theta-i\eta}\right)
\right],
\nonumber\\\label{function_6}\end{aligned}$$ where $X_6(m_f)=\left(\frac{\partial}{\partial
(m_f^2)}+\frac{
m_f^2}{2}\frac{\partial^2}{\partial^2(m_f^2)}\right)$, the Eq. also vanishes in the static limit $$\begin{aligned}
\left[\frac{{\rm Im}~I_6(q_0,q)}{q_0}\right]_{q_0=0}&=
&0.
\label{i6}\end{aligned}$$ Finally, we substitute Eq., Eq., Eq. and Eq. in Eq., to evaluate the imaginary part of $b_2(q_0,q)$, which in the static limit can be written as $$\begin{aligned}
\left[\frac{{\rm Im}~b_2(q_0,q)}{q_0}\right]_
{q_0=0}&=&\frac{1}{q}\left[\sum_f\frac{g^2(q_fB)^2}{16\pi T^2}
\sum_{l=1}^\infty(-1)^{l+1}l^2K_0\left(\frac{m_f l}{T}\right)
\right. \nonumber\\ &&\left.
-\sum_f\frac{g^2(q_fB)^2}{96\pi T^2}
\sum_{l=1}^\infty(-1)^{l+1}l^2K_2\left(\frac{m_f l}{T}\right)
\right. \nonumber\\ &&\left.
+\sum_f\frac{g^2(q_fB)^2}{768\pi}\frac{(8T-7\pi m_f)}{m_f^2 T}
\right].\end{aligned}$$
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[^1]: [email protected]
[^2]: [email protected]
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'The order in which plane-filling curves visit points in the plane can be exploited to design efficient algorithms. Typically, the curves are useful because they preserve locality: points that are close to each other along the curve tend to be close to each other in the plane, and vice versa. However, sketches of plane-filling curves do not show this well: they are hard to read on different levels of detail and it is hard to see how far apart points are along the curve. This paper presents a software tool to produce compelling visualisations that may give more insight in the structure of the curves.'
author:
- Herman Haverkort
bibliography:
- 'plane-filling-trails.bib'
title: 'Plane-filling trails'
---
##### Plane-filling curves
A plane-filling curve is a continuous surjective mapping $f$ from the unit interval to a subset of the plane that has positive area, that is, Jordan content. Although such a mapping cannot be one-to-one, an unambiguous inverse can be defined with a tie-breaking rule. Thus, the mapping provides an order in which to process points in the plane. Famous examples include Pólya’s triangle-filling curve [@Pol1913] and square-filling curves by Peano [@Pea1890] and Hilbert [@Hil1891]. Continuity of the mapping is not always required: if we drop this requirement, we speak of plane-filling *traversals*. Z-Order [@Mor66] is an example that is often applied in practice.
Plane-filling traversals and their inverses have been used to design efficient solutions for various applications, including indexing of points in the plane, geometric algorithms and data structures, finite element methods, load balancing in parallel computing, improving cache utilization in computations on large matrices or images, combinatorial optimization, image compression, information visualization, and sonification [@Bad13; @Hav17]. It is therefore interesting to see the differences between the various plane-filling traversals that have been proposed.
##### Defining a plane-filling curve
Plane-filling traversals are usually visualised in a way that follows their definition. Consider Pólya’s curve. To define it, we start with a single line segment (Figure \[fig:polyadefinition\]a). We refine this simple drawing as follows. Let $p$ and $r$ be the end points of the original line segment. Imagine a circle with centre line $pr$ and draw another point $q$ halfway on the circle as we follow it clockwise from $p$ to $r$. Erase the original line segment $pr$ and replace it by two smaller segments $pq$ and $qr$ (Figure \[fig:polyadefinition\]b). Next, refine the drawing again by applying the same refinement procedure to each segment, but this time changing the orientation: to find the new intermediate points, we now follow the circles in counterclockwise direction. To indicate this change in orientation, we add an arrow head to $pr$ on the left side, and put the arrow heads for $pq$ and $qr$ on the right side. Thus, two line segments become four segments (Figure \[fig:polyadefinition\]c). Note that the middle two segments lie on top of each other, but they have different directions. If we repeat this refinement process six more times, alternating clockwise and counterclockwise, and move all points slightly so that the curve does not back up on itself, we get Figure \[fig:polyadefinition\]d. If we continue ad infinitum, the curve fills a right isosceles triangle.
![Pólya’s triangle-filling curve.[]{data-label="fig:polyadefinition"}](polyadevelopment-paper.pdf)
##### The challenge of visualising plane-filling curves
Figures \[fig:polyadefinition\]b and \[fig:polyadefinition\]d are typical of the way in which plane-filling curves are usually sketched. Neither figure makes it clear in an instant in what order the curve fills what parts of the plane—not to mention showing the curve’s locality-preserving properties and violations thereof. Try comparing, for example, Hilbert’s curve in Figure \[fig:hilbertsketch\]a to the $\beta\Omega$-curve in Figure \[fig:hilbertsketch\]b (a promising alternative [@YL06]). Given a pair of points in the plane, these drawings do not allow us to see at a glance how long the path from one point to another along the curve is, and what regions of the plane are visited on the way. Furthermore, the impression one gets of the curve depends heavily on how one chooses to define it and on the details of how it is sketched. Figure \[fig:confusingsketches\]a shows three sketches that all sketch the same curve, and Figure \[fig:confusingsketches\]b shows a sketch of a trapezoid-filling curve that is nothing else than the first three quarters of Pólya’s curve: none of this is visually obvious from the drawings.
![Sketches of a) the Hilbert curve [@Hil1891] and b) an $\Omega$ section of the $\beta\Omega$-curve [@Wie02].[]{data-label="fig:hilbertsketch"}](hilbertsketch.pdf)
![a) Three traditional-style sketches of Peano’s curve, mapped to a $\sqrt{3} :1$ rectangle. The left sketch connects the end points of the defining segments; the other two sketches connect their centre points, on different levels of refinement. Note how these different sketches give conflicting impressions of the curve, in particular of the main axes of movement. b) A sketch of a trapezoid-filling curve.[]{data-label="fig:confusingsketches"}](balanced-peano-sketches.pdf "fig:")![a) Three traditional-style sketches of Peano’s curve, mapped to a $\sqrt{3} :1$ rectangle. The left sketch connects the end points of the defining segments; the other two sketches connect their centre points, on different levels of refinement. Note how these different sketches give conflicting impressions of the curve, in particular of the main axes of movement. b) A sketch of a trapezoid-filling curve.[]{data-label="fig:confusingsketches"}](polyatrapezoid-paper.pdf "fig:")
##### Visualisation as three-dimensional landscapes
To visualise plane-filling curves and traversals more clearly, I developed a tool . The tool reads a definition of a plane-filling curve and produces a *plane-filling trail*, a model of the curve on a three-dimensional landscape, in which each point $f(t) = (x,y)$ of the curve is rendered as a point $(x,y,t)$. Thus the curve becomes a steadily ascending path in the landscape, see Figure \[fig:polyatrail\]. At a low resolution, the concept can be seen in action in a Hilbert curve marble run design by Ortiz [@Ort18]. At higher resolutions, we obtain a clear visualisation of the locality-preserving and locality-violating properties of the curve that can be studied at different levels of detail. High, steep slopes reveal pairs of points that are close in the plane but far apart along the curve. Narrow corridors reveal sections between points that are relatively close to each other along the curve, but far apart in the plane. Wide corridors show sections of the curve that have good locality-preserving properties in both directions. The global course of the curve is easy to follow, but the image also facilitates studying the curve in more detail. Moreover, the visualisation is independent of what definition of the curve is used, out of multiple equivalent definitions. For example, the fact that the trapezoid-filling curve is simply the first three quarters of the Pólya curve is now obvious, see Figure \[fig:polyatrail\]. The visualisation gives the user the possibility to study the curve without any bias towards an arbitrary underlying tessellation.
to
Note that in these renderings, the space under the plane-filling trail is filled, thus creating the vertical walls in the landscape. Without these vertical walls, the visualisation would be rather ambiguous: it would be difficult or impossible to see if a gap between two points $p$ and $q$ in the three-dimensional view is strictly vertical, or if the gap also exists in the projection on a horizontal plane. The first would imply that $p$ and $q$ are the same point in two-dimensional space that is visited by the curve twice; the second could hint at gaps in the two-dimensional image of the curve. Filling up the space under each point of the trail eliminates the ambiguity that results from projecting the three-dimensional model onto a two-dimensional viewing plane. In many of the examples in Figure \[fig:examples\], an additional visual frame of reference is provided by a background that consists of a low plane in the front, a cliff, and a high half-plane in the back.
##### Alternative visualisations
Alternative visualisation methods that come closest to meeting the same goals render the $t$-coordinate as values on a grey or colour scale instead of elevation. Indeed, such renderings are quite common, and they truly show the curve, not merely its definition, colouring the entire image according to the order in which points are visited. Figure \[fig:colourgradient\] shows two examples.
to
In comparison to the plane-filling trail method, the following differences can be observed. The colour progressions show the complete curve without distortion, whereas the plane-filling trail methods are affected by perspective distortion and occlusion. However, the colour progressions have less discernible detail. One complicating factor is that the perception of colour depends very much on the local context. As a result, it is sometimes hard to see where the curve goes: in the grey-scale image, the Pólya curve sometimes seems to alternate between becoming brighter and darker, and may even seem to cross itself. The colour image shows more detail than the grey-scale image, but has other problems. Colour gradients do not expose similarities between different parts of the curve well and they do not communicate distances along the curve well, because colour difference establishes, at best, an ordinal scale, not an interval scale. Our visual system may not perceive translations and scalings on the colour scale as similarity transformations. This is where the plane-filling trails excel, since elevation gives us an effective interval scale on which we can read distances along the curve.
Another interesting alternative are the three-dimensional models by Irving and Segerman [@SI13] that stack different refinement levels according to a definition of the curve. However, these models are hard to “read” when presented as a two-dimensional printed image, and they are inherently dependent on the chosen definition of the curve. That is fine if one wants to illustrate the definition, but it is a shortcoming if one wants to be able to reveal the equivalence of different definitions by producing the same image in such cases.
##### The tool
The tool reads the definition of a plane-filling traversal in the format from Ventrella [@Ven12], extended to support discontinuities and multiple refinement rules (known as *generators*). Thus, the various traversals that have been proposed in the computer science literature [@BuH16; @Sam06; @Wie02] can all be rendered and it is easy to explore new designs. Traversals are not confined to an integer grid, so we can also visualise interesting traversals related to, for example, the Rauzy fractal [@Rau82] (see Figure \[fig:examples\]). For rendering, the traversal is sampled and drawn on a grid of hexagonal cells; thus operates without any knowledge of the shape that is filled by the traversal (which can be a complicated fractal). The tool offers various options to control parameters such as camera position, resolution of the rendering grid, visualisation style, colour scheme, and what to do with sample points of different elevations in the same cell. Small “parapets” can be added at the top of high cliffs, to make the edge of the cliff more visible when seen from the “mountain” side, thus enhancing the perception of depth. The output is a [<span style="font-variant:small-caps;">collada</span>]{} file that can be rendered with, for example, Blender; if the resolution is not too high, it can also be moved around in Blender in real time.
##### Sampling density
The sampling algorithm in adapts automatically to the grid size. To make use of the resolution of the grid and to avoid spurious holes in the image, the sample points must be sufficiently dense, so that, at least, the following conditions are satisfied: (1) if a grid cell is partially covered by the curve, then there is at least one sample point in that grid cell or one of its neighbours; (2) if a grid cell is entirely covered by the curve, then there is at least one sample point inside that grid cell; (3) if an edge between two grid cells is entirely covered by the curve, then there is a sample point in at least one of the two cells. The implementation exploits the curve’s self-similar structure to compute a good upper bound on the radius $r$ such that any section of the curve whose endpoints are a distance $d$ apart, can only visit points that are within distance $dr$ from the closest end point. To fulfil all requirements, it then suffices to make sure that the distance between sample points along the curve is at most half of the grid edge length divided by $r$.
With the current implementation, I found grid sizes of at least $2000 \times 2000$ hexagonal cells to be feasible on my laptop; the model is then generated within a few minutes. Blender could generate a poster-size image ($5000 \times 5000$ pixels) from such a model in a few hours, using the Cycles ray-tracer. For the figures in this paper I used a grid of at least $500 \times 500$ cells, from which Blender generated an image in a few minutes.
##### Polynomial close-up
Special features of the software include “polynomial” close-up: given a focus point $p$ and a zoom parameter $\zeta$, any point $q$ at distance $r$ from $p$ is moved to a point at distance $r^{1/\zeta}$ from $p$. More precisely, if we take $p$ as the origin of the $x,y,t$-coordinate system, then any point $q$ with coordinates $(x,y,t) = (r \cos\phi, r \sin\phi, t)$ is mapped to $(r^{1/\zeta} \cos\phi, r^{1/\zeta} \sin\phi, \mathrm{sign}(t) \cdot |t|^{1/(1.5\zeta-0.5)})$. This allows us to zoom in on features that remain invisible in normal close-up views.
![Definitions of a) the Gosper curve [@Gar76] and b) the Inner-flip [@Ven12] or Alternating [@Sch12] Gosper curve.[]{data-label="fig:gosperdefinition"}](gosper-definition.pdf)
For example, consider the Gosper curve, which is defined in Figure \[fig:gosperdefinition\]a and shown as a plane-filling trail in Figure \[fig:gosperzoom\], top left. The curve follows a tessellation with tiles arranged in a hexagonal grid pattern. In particular, $f(2/7)$ is the vertex where the second, third, and seventh tile meet. At such vertices, the tiles wind around each other like spirals. From geometric descriptions of the tiling [@Hav11] one can derive that the spirals are logarithmic spirals with a growth rate of a factor $\sqrt{7}$ per rotation of $\arctan\sqrt{3/25}$, which is approximately $9 \cdot 10^7$ per revolution. Thus, at such a vertex, we cannot say in what direction to travel to enter a particular one of the three adjacent tiles: any ray from $f(2/7)$ with positive length, no matter the direction and no matter how short, crosses all three tiles.
In contrast, we may consider a variation of the Gosper curve in which all the defining line segments are reflected such that the arrowheads move to the other end (see Figure \[fig:gosperdefinition\]b). Ventrella [@Ven12] calls this variation *Inner-flip Gosper*. The Inner-flip Gosper does not exhibit this logarithmic spiral behaviour at the vertices of the tessellation. The boundaries between the tiles are still fractals, but they are confined to 30 degrees’ wedges that meet at the vertices of the tessellation. Between these wedges, there are 90 degree’s wedges that each lie entirely inside a single tile.
No normal close-up views could show this remarkable difference in character between these two plane-filling curves, since no normal close-up view could show logarithmic spirals that shrink as fast as a factor $9 \cdot 10^7$ per revolution. But with polynomial close-up views, we can reveal these spirals and wedges clearly. Figure \[fig:gosperzoom\] shows the result of zooming in, with increasing values of $\zeta$, on the point $f(2/7)$ in the Gosper and Inner-flip Gosper curves.
to 0.01to 0.01to
to 0.01to 0.01to 0.01to
##### Limitations of path-filling trails
The attentive viewer may notice a particular aspect of the plane-filling trails: the visualisation is not scale-independent. A section of the curve that is similar to the curve as a whole will look like the curve as a whole, but considerably flatter. If the curve is scaled down by a factor $s$ in the plane, its area shrinks by a factor $s^2$, and so does the one-dimensional pre-image, that is, the projection on the vertical axis. Relative to the horizontal dimensions, the vertical dimension is therefore scaled down by an additional factor $s$. This has advantages and disadvantages: on the one hand, it means that we have to make the model quite high to be able to see details; on the other hand, it may make it easier to focus on the coarser levels of detail. Despite the flattening, I believe the plane-filling trails still tend to show a wider range of levels of detail clearly than traditional drawings achieve with line work and colours, but one should be aware that the flattening effect can be a limiting factor. From a certain resolution onwards, the amount of detail that can be shown along a path-filling trail will scale linearly with the height, not the area, of the drawing.
Another aspect that one should be aware of, is that similarity by reversal, that is, by reflection of the pre-image of the curve, may not be as easy to recognize as similarity by rotation, reflection, translation and scaling of the image. It may not be visually obvious that one part of the terrain is the same as another part of the terrain turned upside down; recognizing such similarities may require that the viewer makes a conscious effort to look for them.
##### Remaining challenges
There are several ways in which could be improved. The following issues are high on the priority list.
By design, currently operates without being given any knowledge of the image of the plane-filling traversal. Thus it is flexible, and the images that are produced depend mostly on the traversal itself, and hardly on the particular iterated function system that is used to define it. Of course the iterated function system determines the sample points that are used, but due to the conditions on the sampling density as explained above, this effect is small. However, there are circumstances in which the current approach does not suffice to produce the perfect figure, and in which it would be useful to be able to give some hints.
In particular, the following circumstances can cause suboptimal rendering. First, in some traversals, jumps create bridges and tunnels that start or end on gentle slopes. These bridges and tunnels interfere with the relief on that slope, so that relief, tunnel entrances, or bridgeheads are distorted or disappear entirely. Although currently provides several options to control what happens in such cases, it seems that in some cases, the best solution would be a context-dependent asymmetric rendering of the bridge or tunnel. Second, in many curves, there are narrow passages where the trail connects two regions that only meet in a point. For clarity, it would often be good to make such passages wider—but to be able to do so, would need to know where the narrow passages are. Third, interference between the rendering grid pattern and the sampling pattern may lead to small but obtrusive visual artefacts. Oversampling, which is an option in already, may reduce or eliminate these artefacts, but this solution can be quite inefficient. Given a simple description of the image of each curve section, it would be possible to guarantee that sample points are placed in *all* grid cells intersected by the traversal. This would eliminate artefacts that stem from the sampling pattern.
Finally, to enable the production of 3D-printable models, a solution must be implemented that prevents or eliminates (near-)degeneracies around bridges and saddle points, without interfering with the semantics of the model.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'Two recently proposed concepts to improve the perturbative calculation of exclusive amplitudes, gluonic radiative corrections (Sudakov factor) and confinement size effects (intrinsic transverse momentum) are combined to study the neutron magnetic form factor in the space-like region. We find that nucleon distribution amplitudes modelled on the basis of current QCD sum rules indicate overlap with the existing data at the highest measured values of momentum transfer. However, sizeable higher-order perturbative corrections (K-factor) and/or higher-twist contributions cannot be excluded, although they may be weaker than in the proton case.'
address:
- |
Fachbereich Physik\
Universität Wuppertal\
D-42097 Wuppertal, Germany
- |
Institut für Theoretische Physik II\
Ruhr-Universität Bochum\
D-44780 Bochum, Germany\
author:
- 'J. BOLZ, R. JAKOB and P. KROLL[^1]'
- 'M. BERGMANN and N. G. STEFANIS[^2]'
title: 'NEUTRON FORM FACTOR: SUDAKOV SUPPRESSION AND INTRINSIC TRANSVERSE SIZE EFFECT '
---
=0
In a recent paper [@BJKBS94] we have studied the space-like proton form factor within a theoretical scheme proposed by Li and Sterman [@LS92], which takes into account gluonic radiative corrections in the form of a Sudakov factor [@BS89]. This scheme naturally generalizes the standard hard scattering picture (HSP) of Brodsky and Lepage [@LB80]—commonly used to calculate exclusive reactions within perturbative QCD—by taking into account the transverse momentum of the partons.
A major point in our proton form-factor analysis was to show that proper treatment of the $\alpha _{s}$-singularities demands the imposition of an appropriate infrared (IR) cut-off to render the form-factor calculation both finite and insensitive to the inclusion of the soft region of phase space. This is in contrast to the pion case [@LS92], where a “natural” IR cut-off appears in the form of the interquark separation. Considering in detail optional IR cut-off prescriptions [@Li93; @Hye93; @SS94], we found that maximum IR protection is provided by introducing as a common IR cut-off in the Sudakov (suppression) factor the maximum interquark separation (“MAX” prescription [@BJKBS94]). The underlying physical idea is the following: One expects that because of the color neutrality of a hadron, its quark distribution cannot be resolved by gluons with a wavelength much larger than a characteristic interquark separation scale $\tilde{b}_{l}$. Thus, gluons with wavelengths large compared to the (transverse) hadron size probe the hadron as a whole, i.e., in a color-singlet state and decouple. As a result, quarks in such configurations act coherently and therefore (soft) gluon radiation is dynamically inhibited.
The “MAX” presription not only suffices to suppress the $\alpha _{s}$-singularities, but also preserves the finiteness of the integrand of the expression for the form factor, even when renormalization-group (RG) evolution of the wave function is included. An additional bonus of the “MAX” prescription is that the proton form factor saturates, i.e., becomes insensitive to the contributions from large transverse separations. Admittedly, little confidence is put in perturbative treatments of the large-distance region, so that saturation of the form factor at transverse distances as low as possible is a prerequisite for a self-consistent perturbative calculation.
In [@BJKBS94] we have pointed out that the recent numerical analysis by Li [@Li93] of the proton form factor has serious drawbacks: (i) The cancellation of the $\alpha _{s}$-singularities in the region $\tilde{b}_{l}\Lambda _{\text{QCD}} \simeq 1$ ($x_{l}$ fixed) is incomplete for different $\tilde{b}_{l}$, amounting to uncompensated singularities of the form $$\sim \ln \left(
\frac{1}{\tilde{b}_{l}\Lambda _{\text{QCD}}}
\right)^{\kappa}.
\label{eq:sing}$$ (ii) There is no saturation, meaning that the main form-factor contributions are accumulated in the “forbidden” soft region.
A second element of our approach in [@BJKBS94] was the incorporation of the intrinsic transverse momentum in the proton wave function, following a previous work by two of us [@JK93] on the pion form factor. The intrinsic transverse momentum reflects confinement-size effects [@Kro93] and improves the saturation behavior of the form factor. As a consequence of these effects (Sudakov factor and intrinsic transverse momentum), the self-consistent perturbative contribution to the proton form factor turns out to be reduced by at least a factor of two compared to the existing experimental data (see, e.g., [@Arn86; @Sil93]). This is true for a variety of nucleon distribution amplitudes (DA), recently determined by two of us [@BS93; @BS94] on the theoretical basis of QCD sum rules [@COZ89; @KS87].
In the present work we extend this type of analysis to the neutron magnetic form factor. While in our previous work the focus was on the self-consistent implementation of the Sudakov factor and the proper identification and matching of the scales involved, the subject of the current effort will be on the phenomenological side. In particular, we consider observables involving the ratio of the neutron to the proton form factor $G_{M}^{n}/G_{M}^{p}$. It has been discussed in [@Ste89] (and previous references cited therein) and more recently in [@Ste93a] that proposed model distribution amplitudes for the nucleon can be classified according to this ratio (an observable quantity) and the theoretical parameters $B_{4}$ (projection coefficient on the corresponding eigenfunction of the nucleon evolution equation) and the “hybridity” angle $\vartheta$, as detailed in [@BS94]. One of the crucial questions posed in the present work is whether the emerging pattern of solutions to the sum rules, found within the standard HSP [@LB80; @COZ89] to constitute a smooth and finite “orbit” in the $(B_{4},-G_{M}^{n}/G_{M}^{p})$ plane, pertains to the inclusion of transverse-momentum contributions.
The starting point of our analysis is to consider the neutron magnetic form factor within the modified HSP: $$G_{M}^{n}(Q^{2})
=
\frac{16}{3}
\int_{0}^{1}[dx][dx']
\int_{}^{}\frac{d{}^{2}b_{1}}{(4\pi )^{2}}
\frac{d{}^{2}b_{2}}{(4\pi )^{2}}
\sum_{j=1}^{2}\, \hat{T}_{j}(x,x',\vec{b},Q,\mu )
\hat{Y}_{j}^{n}(x,x',\vec{b},\mu _{F}) \,
{\rm e}^{-S_{j}},
\label{eq:G_M^n(b)}$$ with $[dx]=dx_{1}dx_{2}dx_{3}\delta (1-\sum_{}^{}x_{i})$; $x_{i}$ being the momentum fractions carried by the valence quarks. The Fourier-transformed hard scattering amplitudes are given by $$\hat{T}_{1}
=
\frac{8}{3}\,C_{\text{F}}\,\alpha _{s}(t_{11}) \alpha _{s}(t_{12})
K_{0}
\left(
\sqrt{(1-x_{1})(1-x_{1}')}\,Qb_{1}
\right)
K_{0}\left(
\sqrt{x_{2}x_{2}'}\,Qb_{2}
\right),
\label{eq:FourierT_1}$$ $$\hat{T}_{2}
=
\frac{8}{3}\,C_{\text{F}}\,\alpha _{s}(t_{21}) \alpha _{s}(t_{22})
K_{0}
\left(
\sqrt{x_{1}x_{1}'}\,Qb_{1}
\right)
K_{0}
\left(
\sqrt{x_{2}x_{2}'}\,Qb_{2}
\right),
\label{eq:FourierT_2}$$ where the $K_{0}$ are modified Bessel functions of order $0$ and $b_{l}$ denotes the length of the corresponding transverse-distance vector.
The renormalization scale is chosen in such a way that each hard gluon refers to its own individual momentum scale $t_{ji}$ to be used in the argument of the corresponding $\alpha _{s}$. The $t_{ji}$ is defined as the maximum scale of either the longitudinal momentum or the inverse transverse separation, associated with each of the gluons: viz. $$\begin{aligned}
& t_{11} &
=
{\rm max} \left[
\sqrt{(1-x_{1})(1-x_{1}^{\prime})}\,Q, 1/b_{1}
\right],
\nonumber \\
& t_{21} &
=
{\rm max} \left[
\sqrt{x_{1}x_{1}^{\prime}}\,Q, 1/b_{1}
\right],
\nonumber \\
& t_{12} &
=
t_{22}
=
{\rm max} \left[
\sqrt{x_{2}x_{2}^{\prime}}\,Q, 1/b_{2}
\right].
\label{eq:t_ij}\end{aligned}$$ Since the hard scattering amplitudes depend only on the differences of initial and final state transverse momenta, there are only two transverse separation vectors, namely those between quarks 1 and 3 and between quarks 2 and 3: $\vec{b}_{1}\;(=\vec{b}_{1}')$, $\vec{b}_{2}\;(=\vec{b}_{2}')$. Accordingly, the transverse separation between quark 1 and quark 2 is $
\vec{b}_{3}
=
\vec{b}_{2} - \vec{b}_{1}.
$
The soft part of the form factor is given by the following expressions which contain linear combinations of products of the initial and final state wave functions in the transverse configuration space, weighted by $x_{i}$-dependent factors arising from the fermion propagators: $$\begin{aligned}
\hat{Y}_{1}^{n}
=
\frac{1}{(1-x_{1})(1-x^{\prime}_{1})}
\Bigl\{ & \phantom{} &
\!\!\!\!\!\!
-2\hat{\Psi} ^{\star\prime}_{123}\hat{\Psi} _{123}
-2\hat{\Psi} ^{\star\prime}_{132}\hat{\Psi} _{132}
+ \hat{\Psi} ^{\star\prime}_{231}\hat{\Psi} _{231}
+ \hat{\Psi} ^{\star\prime}_{321}\hat{\Psi} _{321}
\nonumber \\
& - & \hat{\Psi} ^{\star\prime}_{231}\hat{\Psi} _{132}
-\hat{\Psi} ^{\star\prime}_{132}\hat{\Psi} _{231}
-\hat{\Psi} ^{\star\prime}_{321}\hat{\Psi} _{123}
-\hat{\Psi} ^{\star\prime}_{123}\hat{\Psi} _{321}
\Bigr\}
\label{eq:Y_1^n}\end{aligned}$$ $$\begin{aligned}
\hat{Y}_{2}^{n}
= & \phantom{} &
\!\!\!\!\!\!
\frac{1}{(1-x_{2})(1-x^{\prime}_{1})}
\left\{
\hat{\Psi} ^{\star\prime}_{231}\hat{\Psi} _{231}
+ \hat{\Psi} ^{\star\prime}_{231}\hat{\Psi} _{132}
+ \hat{\Psi} ^{\star\prime}_{132}\hat{\Psi} _{231}
\right\}
\nonumber \\
& + & \frac{1}{(1-x_{3})(1-x^{\prime}_{1})}
\left\{
2\hat{\Psi} ^{\star\prime}_{321}\hat{\Psi} _{321}
- \hat{\Psi} ^{\star\prime}_{123}\hat{\Psi} _{123}
+ \hat{\Psi} ^{\star\prime}_{321}\hat{\Psi} _{123}
+ \hat{\Psi} ^{\star\prime}_{123}\hat{\Psi} _{321}
\right\}.
\label{eq:Y_2^n}\end{aligned}$$ The Fourier transform of the wave function reads $$\hat{\Psi}_{123}(x,\vec{b},\mu _{F})
=
\frac{1}{8\sqrt{N_{\text{c}}!}}
f_{\text{N}}(\mu _{F})
\Phi _{123}(x,\mu _{F})
\hat{\Omega}_{123}(x,\vec{b}),
\label{eq:FourierPsi}$$ wherein its intrinsic $k_{\perp}$-dependence is parametrized according to the Gaussian $$\hat{\Omega}_{123}(x,\vec{b})
=
(4\pi )^{2}
{\rm exp}
\left\{
- \frac{1}{4a^{2}}
\Bigl[
x_{1}x_{3}b_{1}^{2} + x_{2}x_{3}b_{2}^{2}
+ x_{1}x_{2}b_{3}^{2}
\Bigr]
\right\}.
\label{eq:FourierOmega}$$ We have used the convenient short-hand notation $
\hat{\Psi} _{123}(x,\vec{b})
=
\hat{\Psi} (x_{1},\vec{b};
x_{2},\vec{b};
x_{3},\vec{b})
$ denoting in $
\Phi _{123}(x,\mu _{F})
$ the factorization scale of short-and large-distance contributions by $\mu _{F}$.
The exponentials ${\rm e}^{-S_{j}}$ in (\[eq:G\_M\^n(b)\]) are the Sudakov factors responsible for the effects of gluonic radiative corrections. They have been calculated by Botts and Sterman [@BS89] using resummation techniques in the context of the renormalization group (RG) and having recourse to previous extensive work by Collins, Soper, and Sterman [@CS81]. The explicit expressions for the Sudakov exponents are given in [@LS92; @Li93].
The analytical and numerical evaluation of the neutron form factor is performed under the imposition of the “MAX” IR-prescription [@BJKBS94] on the Sudakov factor, i.e., setting $$\tilde{b}\equiv {\rm max}\{b_{1},b_{2},b_{3}\}
=
\tilde{b}_{1}=\tilde{b}_{2}=\tilde{b}_{3}.
\label{eq:MAX}$$ The results of this calculation are typified by the curves shown in Fig. \[fig:G\_M\^n(Q\^2)\] for the COZ DA [@COZ89] (solid line), including also the intrinsic transverse momentum in two different ways: (i) by normalizing the probability $P_{3q}$ for finding three valence quarks in the neutron to unity (dashed line), which results to $\langle k^{2}_{\perp}\rangle ^{1/2}=271$ MeV; and (ii) by setting the value of the r.m.s. transverse momentum equal to $600$ MeV (dotted line), which implies $P_{3q}=0.042$. Here and below we throughout use the values $\Lambda _{\text{QCD}}=180$ MeV and $|f_{N}|=(5.0\pm 0.3)\times 10^{-3}$ GeV${}^{2}$ [@COZ89], the latter being the value of the nucleon DA at the origin. The momentum evolution of the DA is RG-controlled—provided the model DA is satisfying the nucleon evolution equation [@LB80], which is true for all DAs we consider in this work. Then the nucleon DA can be expanded in terms of the eigenfunctions of the one-gluon exchange kernel to read $$\Phi _{123}(x,\mu )
=
\Phi _{123}^{\text{as}}(x)
\sum_{n}^{}B_{n}
\left(\frac{\alpha _{s}(\mu )}{\alpha _{s}(\mu _{0})}
\right)^{\tilde{\gamma} _{n}/\beta_{0}}
\tilde{\Phi}_{123}^{n}(x),
\label{eq:Phi}$$ where the notations of [@Ste89] are adopted and $
\Phi _{123}^{as}(x) = 120 x_{1}x_{2}x_{3}
$ is the asymptotic DA. The exponents $\tilde{\gamma}_{n}$ are related to the anomalous dimensions of trilinear quark operators with isospin $1/2$ (see [@Pes79]) and resemble the $b_{n}$ in the Brodsky-Lepage notation [@LB80]. Because they are positive fractional numbers increasing with n, higher-order terms in (\[eq:Phi\]) are gradually suppressed. The constants $\tilde{\gamma}_{n}$ are given in [@BJKBS94; @Li93]; $\beta_{0}=11-2n_{f}/3=9$ for three flavors. Under these conditions we may use QCD sum-rule results on the moments $$\Phi ^{(n_{1}n_{2}n_{3})}(\mu _{0})
=
\int_{0}^{1}[dx]x_{1}^{n_{1}}x_{2}^{n_{2}}x_{3}^{n_{3}}
\Phi _{123}(x,\mu _{0})$$ \[eq:moments\] to constrain the first few expansion coefficients $B_{n}$ (for more details see [@BS94; @Ste89; @Ste93a]).
As can be seen from Fig. \[fig:G\_M\^n(Q\^2)\] and comparison with [@Ste93a], the magnitude of the neutron magnetic form factor with Sudakov correction is reduced by more than a factor of 2.5 with respect to the standard HSP. This reduction is enhanced when the intrinsic transverse momentum is included and is pertinent to all nucleon DAs [@BS93; @BS94] modelled on the basis of existing QCD sum-rules [@COZ89; @KS87] (shaded band in Fig. \[fig:stripn\]). The upper region of the band is characterized by COZ-like [@COZ89] DAs, whereas its lower part is associated with the recently proposed “heterotic” DA [@SB93]. We note that the perturbative contribution becomes self-consistent for momentum transfers larger than 8 GeV${}^{2}$ (using $\langle k_{\perp}^{2}\rangle ^{1/2}=271$ MeV) in the sense that at least 50 % of the result is accumulated in regions where $\alpha _{s}^2$ is smaller than $0.5$.
It is important to emphasize that the neutron magnetic form factor is the first process calculated within the modified HSP that yields predictions which indicate overlap with the existing data [@Bos92], as can be seen from Fig. \[fig:stripn\]. This tentative agreement occurs at data points corresponding to the largest momentum transfers measured, where, incidentally, our theoretical calculations become self-consistent. Therefore, measurements of the neutron magnetic form factor beyond 10 GeV${}^{2}$ are extremely important in order to check the validity of the theoretical predictions in a more quantitative way.
One place to test these results is in the data for the differential cross sections for elastic electron-proton and electron-neutron scattering $\sigma_{\text{p}}$ and $\sigma_{\text{n}}$, respectively. For small scattering angles, where the terms $\propto \tan^{2}(\theta/2)$ can be neglected, and for large $Q^{2}$, the ratio $
\sigma_{\text{n}}/ \sigma_{\text{p}}
$ becomes in a slightly model-dependent way proportional to the square of the ratio of the neutron to the proton magnetic form factor.
Combining our calculations for the proton [@BJKBS94] with those presented here for the neutron, we can extract theoretical predictions for $
\sigma_{\text{n}}/ \sigma_{\text{p}}
$ by inputting the same set of model DAs for the nucleon [@BS93; @BS94] as before. The results are shown in Fig. \[fig:stripsigma\] (shaded area) in comparison with available data [@Roc92]. From this figure we see that the measured values of $\sigma_{\text{n}}/\sigma_{\text{p}}$ enter the estimated range already at $Q^{2}\approx 8$ GeV${}^{2}$. \[The corresponding values of the ratio $-G_{M}^{n}/G_{M}^{p}$, allowed by our analysis, range between -0.2 and 0.5.\] The fair agreement between theoretical predictions and data is partly deceptive owing to the fact that the self-consistent calculation of the leading-order perturbative contribution to the proton magnetic form factor within the modified HSP yields a rather small value [@BJKBS94]. This missing part of the proton form factor can arise from many sources, e.g, from a large K-factor and/or higher-twist contributions and would certainly affect the width of the strip. Such contributions are also conceivable for the neutron form factor.
In any case it is remarkable that the collective pattern of solutions to the QCD sum rules [@COZ89; @KS87], found within the standard HSP [@BS93; @BS94], pertains to the inclusion of transverse-momentum contributions comprising the Sudakov factor and those due to the intrinsic transverse momentum (see Fig. \[fig:orbit\]). Indeed, the solutions arrange themselves across an “orbit” in the ($B_{4},-G_{M}^{n}/G_{M}^{p}$) plane which is somewhat shifted compared to the original one. In contrast to the standard HSP version, within the present context, the “orbit” is slightly $Q^{2}$-dependent, as shown in Fig. \[fig:orbit\]. The new “orbit” at $Q^{2}=30$ GeV${}^{2}$ can be characterized by the empirical relation $
-G_{M}^{n}/G_{M}^{p}
=
0.426 - 9.91 \times 10^{-3} B_{4} - 4.27 \times 10^{-4} B_{4}^{2}
+ 4.59 \times 10^{-6} B_{4}^{3} ,
$ which complies with that found in [@BS93]. The dashed line in Fig. \[fig:orbit\] represents a similar fit for $Q^{2}=10^{3}$ GeV${}^{2}$. We observe that with increasing momentum transfer, the “orbit” within the modified HSP transmutes into that of the standard HSP. We note that the coefficient $B_{4}$ projects onto the eigenfunction $\tilde{\Phi}_{4}(x_{i})$ and hence provides an effective measure to account for the antisymmetric content of the nucleon DA, since the other antisymmetric eigenfunctions are offset by this term [@BS93].
In summary, in this letter the modified HSP has been applied to the neutron magnetic form factor for the first time. In contrast to other cases (e.g., pion and proton electromagnetic form factors), the band of predictions obtained with the set of model DAs for the nucleon indicates overlap with the experimental data at the largest measured values of momentum transfer, where the theoretical predictions become self-consistent. Nevertheless, it is likely that in order to improve agreement with the data, several additional contributions have to be included: Perturbative higher-order corrections may give rise to a rather large K-factor of the order of 2 multiplying the leading-order contribution, as found for other large-momentum transfer processes [@Ant83]. However, for the case of the pion form factor, already existing calculations [@Fie81; @DR81] of the K-factor to one-loop order indicate that with an appropriate choice of the renormalization point, the actual value of the K-factor is rather small, i.e., of order unity. On the other hand, still unestimated contributions due to higher twists are presumably sizeable in the experimentally accessible region and may also be important.
This work was supported in part by the Deutsche Forschungsgemeinschaft and the Bundesministerium für Forschung und Technologie FRG under contract 06WU737.
[99]{} J. Bolz, R. Jakob, P. Kroll, M. Bergmann, and N.G. Stefanis, Wuppertal preprint\
WU-B-94-06 and Bochum preprint RUB-TPII-94-01 (May 1994), submitted to\
Z. Phys. C - Particles and Fields. H.-N. Li and G. Sterman, Nucl. Phys. B 381 (1992) 129. J. Botts and G. Sterman, Nucl. Phys. B 325 (1989) 62. G.P. Lepage and S.J. Brodsky, Phys. Rev. D 22 (1980) 2157. H.-N. Li, Phys. Rev. D 48 (1993) 4243. T. Hyer, Phys. Rev. D 47 (1993) 3875. M.G. Sotiropoulos and G. Sterman, Stony Brook Report ITP-SB-93-83 (January 1994). R. Jakob and P. Kroll, Phys. Lett. B 315 (1993) 463; B 319 (1993) 545(E). P. Kroll, Proceedings of the Workshop on Hadron Structure ’93, S. Dubnička and\
A.Z. Dubničková (Eds.), Banská Štiavnica, Slovakia, Sept. 5-10, 1993. R.G. Arnold et al., Phys. Rev. Lett. 57 (1986) 174. A.F. Sill et al., Phys. Rev. D 48 (1993) 29. M. Bergmann and N.G. Stefanis, Phys. Rev. D 48 (1993) R2990. M. Bergmann and N.G. Stefanis, Phys. Lett. B 325 (1994) 183. V.L. Chernyak, A.A. Ogloblin, and I.R. Zhitnitsky, Z. Phys. C 42 (1989) 569. I.D. King and C.T. Sachrajda, Nucl. Phys. B 279 (1987) 785. N.G. Stefanis, Phys. Rev. D 40 (1989) 2305; D 44 (1991) 1616(E) N.G. Stefanis and M. Bergmann, Proceedings of the Workshop on Exclusive Reactions at High Momentum Transfer, C.E. Carlson, P. Stoler, and M. Taiuti (Eds.), Elba, Italy, 24-26 June, 1993, World Scientific, Singapore, 1994; Proceedings of the Workshop on Hadron Structure ’93, S. Dubnička and A.Z. Dubničková (Eds.), Banská Štiavnica, Slovakia, Sept. 5-10, 1993; Proceedings of the Workshop on Quantum Field Theoretical Aspects of High Energy Physics, B. Geyer and E.-M. Ilgenfritz (Eds.), Kyffhäuser, Germany, Sept. 20-24, 1993. J.C. Collins and D.E. Soper, Nucl. Phys. B 193 (1981) 381; B 194 (1982) 445; J.C. Collins, D.E. Soper, and G. Sterman, Nucl. Phys. B 261 (1985) 104. M. Peskin, Phys. Lett. 88 B (1979) 128. N.G. Stefanis and M. Bergmann, Phys. Rev. D 47 (1993) R3685. P. Bosted et al., Phys. Rev. Lett. 68 (1992) 3841; S. Platchkov et al., Nucl. Phys. A 510 (1990) 740. S. Rock et al., Phys. Rev. D 46 (1992) 24. N.G. Antoniou et al., Phys. Lett. B 128 (1983) 257. R.D. Field et al., Nucl. Phys. B 186 (1981) 429. F.M. Dittes and A.V. Radyushkin, Yad. Fiz. 34 (1981) 529 \[Sov. J. Nucl. Phys. 34 (1981) 293\].
[^1]: E-mail address: [email protected]
[^2]: E-mail address: [email protected]
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{
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abstract: 'Differential production cross sections of $K^-$ and $K^+$ mesons have been measured in Ni+Ni and Au+Au collisions at a beam energy of 1.5 $A\cdot$GeV. The $K^-/K^+$ ratio is found to be nearly constant as a function of the collision centrality and system size. The spectral slopes and the polar emission pattern differ for $K^-$ and $K^+$ mesons. These observations indicate that $K^+$ mesons decouple earlier from the fireball than $K^-$ mesons.'
author:
- |
A. Förster$^b$, F. Uhlig$^b$, I. Böttcher$^d$, M. Dȩbowski$^{e,f}$, F. Dohrmann$^f$, E. Grosse$^{f,g}$, P. Koczoń$^a$, B. Kohlmeyer$^d$, F. Laue$^{a,*}$, M. Menzel$^d$, L. Naumann$^f$, H. Oeschler$^b$, W. Scheinast$^f$, E. Schwab$^a$, P. Senger$^a$, Y. Shin$^c$, H. Ströbele$^c$, C. Sturm$^{b,a}$, G. Surówka$^{a,e}$, A. Wagner$^f$, W. Waluś$^e$\
KaoS Collaboration\
$^a$ Gesellschaft für Schwerionenforschung, D-64220 Darmstadt, Germany\
$^b$ Technische Universität Darmstadt, D-64289 Darmstadt, Germany\
$^c$ Johann Wolfgang Goethe-Universität, D-60325 Frankfurt am Main, Germany\
$^d$ Phillips Universität, D-35037 Marburg, Germany\
$^e$ Jagiellonian University, PL-30059 Kraków, Poland\
$^f$ Forschungszentrum Rossendorf, D-01314 Dresden, Germany\
$^g$ Technische Universität Dresden, D-01062 Dresden, Germany\
$^*$ Present address: Brookhaven National Laboratory, USA
title: 'First evidence for different freeze-out conditions for kaons and antikaons observed in heavy-ion collisions'
---
Heavy-ion collisions provide the unique possibility to study baryonic matter well above saturation density. The conditions inside the dense reaction zone and the in-medium properties of hadrons can be explored by measuring the particles created in such collisions [@aich; @brown]. In particular, strange mesons produced at beam energies below or close to the $NN$ threshold are well suited for these studies. The yield of $K^+$ mesons as measured in Au+Au collisions at SIS/GSI [@sturm] constrains the nuclear matter equation-of-state [@fuchs]. The pronounced patterns of the elliptic and directed flow of kaons provide evidence for the existence of a repulsive kaon-nucleon in-medium potential [@shin; @crochet]. The $K^-$/$K^+$ ratio is enhanced in heavy-ion collisions as compared to proton-proton collisions [@barth; @laue]. In order to reproduce the measured yields, transport model calculations have to take into account density-dependent $KN$ potentials corresponding to an in-medium modification of the $K$ meson mass [@cass_brat; @li_brown]. On the other hand, the measured ratios of strange particles can be explained within statistical models without any in-medium modification of the masses. These models reproduce the measured ratios by choosing an appropriate pair of values for the temperature and the baryon chemical potential assuming thereby a simultaneous chemical freeze-out [@cley00].
Quantitative information on the production mechanisms and the properties of strange mesons in dense baryonic matter can be extracted from the phase-space distributions of $K^+$ and $K^-$ mesons observed in heavy-ion collisions. In central Au+Au collisions at beam energies above 4 $A\cdot$GeV the spectral slopes were found to be similar for $K^+$ and $K^-$ mesons [@Ahle]. The rapidity density distributions of strange mesons from central Au+Au collisions at 10.7 $A\cdot$GeV and Pb+Pb collisions at 158 $A\cdot$GeV were found to be wider for $K^+$ than for $K^-$ mesons [@AGS; @NA49]. In both cases, the $K^-/K^+$ ratio is constant as a function of the collision centrality. At beam energies below the $NN$ thresholds for strangeness production ($NN \to K^+\Lambda N$ at E = 1.6 GeV, $NN
\to K^+ K^-NN$ at E = 2.5 GeV), where in-medium effects are expected to influence the kaon production significantly, a systematic comparison of $K^+$ and $K^-$ phase-space distributions has not yet been published.
In this Letter we present results of experiments on $K^+$ and $K^-$ production in Ni+Ni and Au+Au collisions studied at a beam energy of 1.5 $A\cdot$GeV. This is the lowest beam energy where antikaons have been observed so far in collisions between heavy nuclei. We have measured the spectral and angular distributions of strange mesons as function of the collision centrality and have found significant differences between kaons and antikaons.
The experiments were performed with the Kaon Spectrometer (KaoS) at the heavy-ion synchrotron (SIS) at GSI in Darmstadt [@senger]. Due to the energy loss in the Au target (thickness 0.5 mm) the average energy of the Au beam is 1.48 $A\cdot$GeV. The energy loss of the Ni ions in the Ni target is negligible. In order to reach an energy of 1.5 $A\cdot$GeV for Au beams, an exceptional operation of the GSI accelerator facility was required: acceleration of the $^{197}$Au$^{63+}$ ions with the synchrotron up to an energy of 0.3 $A\cdot$GeV, then extraction and full stripping, then injection into the Experimental Storage Ring (ESR) where the beam was cooled by electron cooling, then re-injection into the synchrotron and acceleration up to 1.5 $A\cdot$GeV.
In order to study the centrality dependence we grouped the data measured close to midrapidity ($\theta_{lab} = 40^{\circ}$) into five centrality bins both for Ni+Ni and Au+Au collisions. The centrality of the collision is derived from the multiplicity of charged particles measured in the interval 12$^{\circ} <
\theta_{lab} < 48^{\circ}$. The most central collisions correspond to 5% of the total reaction cross-section $\sigma_R$, the subsequent centrality bins correspond to 15%, 15% and 25% of $\sigma_R$. The most peripheral collisions correspond to 40% of $\sigma_R$. The total reaction cross-section has been derived from a measurement with a minimum bias trigger and was found to be $\sigma_R$ = 6.0$\pm$0.5 barn for Au+Au and $\sigma_R$ = 2.9$\pm$0.3 barn for Ni+Ni collisions. The corresponding number of participating nucleons $A_{part}$ has been calculated from the measured reaction cross-section fractions using a geometrical model assuming a sharp nuclear surface.
Figure \[spectra\] shows the production cross sections for $K^+$ and $K^-$ mesons measured close to midrapidity as a function of the kinetic energy in the center-of-momentum system for the five centrality bins in Au+Au collisions. The uppermost spectra correspond to the most central reactions. The error bars represent the statistical uncertainties of the kaon and the background events. An overall systematic error of 10% due to efficiency corrections and beam normalization has to be added. The solid lines represent the function $ E \cdot d^3\sigma/dp^3 = C
\cdot E \cdot exp(-E/T)$ fitted to the data. $C$ is a normalization constant and the exponential describes the energy distribution with $T$ as the inverse slope parameter.
The spectra presented in Fig. \[spectra\] exhibit a distinct difference between $K^-$ and $K^+$: The slopes of the $K^-$ spectra are steeper than those of the $K^+$ spectra. The inverse slope parameters $T$ are displayed in the upper panel of Figure \[ratio\] as a function of the number of participating nucleons $A_{part}$. $T$ increases with increasing centrality and is found to be significantly lower for antikaons than for kaons, even for the most central collisions. When interpreting spectral slopes one should keep in mind that they are influenced by both the random and the collective motion of the particles (temperature and flow). The radial-flow contribution to the slope depends on the particle mass and hence cannot cause a difference between the $K^+$ and the $K^-$ spectra. The temperature contribution to the slope is determined at kinetic freeze-out, i.e. at the time when the particles cease to interact.
The kaon multiplicity is defined for each centrality bin as $M =
\sigma_K/\sigma_r$ with $\sigma_K$ the kaon production cross section and $\sigma_r$ the reaction cross-section of the particular event class. Figure \[ratio\] presents $M/A_{part}$ for $K^+$ (second panel) and for $K^-$ (third panel) as a function of $A_{part}$. Both for $K^+$ and $K^-$ mesons the multiplicities exhibit a similar rise with $A_{part}$. Moreover, $M/A_{part}$ is found to be almost identical in Ni+Ni and Au+Au collisions. The $K^-/K^+$ ratio is about 0.02 below $A_{part} = 100$ and decreases slightly to about 0.015 for the most central collisions (Figure \[ratio\], lowest panel).
Another observable sensitive to the production mechanism is the polar angle emission pattern. The deviation from isotropy of the $K^+$ and the $K^-$ emission can be studied by the ratio $\sigma_{inv}(E_{CM},\theta_{CM})$/$\sigma_{inv}(E_{CM},
90^\circ)$ as a function of $cos(\theta_{CM})$. Here, $\sigma_{inv}(E_{CM},\theta_{CM})$ is the invariant kaon production cross-section measured at the polar angle $\theta_{CM}$ in the center-of-momentum frame and $\sigma_{inv}(E_{CM},
90^\circ)$ is the one measured at $\theta_{CM} = 90^\circ$. Due to limited statistics we considered only Au+Au collisions grouped into two centrality bins: near-central (impact parameter $b<$6 fm) and non-central collisions ($b>$6 fm). Figure \[polar\] displays the anisotropy ratio for $K^+$ (upper panel) and $K^-$ (lower panel) and for near-central (right) and non-central collisions (left). For an isotropic distribution this ratio would be constant and identical to 1.
The solid lines in Fig. \[polar\] represent the function $1 +
a_2 \cdot \cos^2(\theta_{CM})$ which is fitted to the experimental distributions with the values of $a_2$ given in the figure. In near-central collisions the $K^-$ mesons exhibit an isotropic emission pattern whereas the emission of $K^+$ mesons is forward-backward peaked. The angular distributions observed for $K^+$ and $K^-$ in Ni+Ni collisions at 1.93 $A\cdot$GeV are similar to the ones presented in Fig. \[polar\] [@menzel]. The measured emission patterns indicate that the antikaons - in contrast to the kaons - have lost the memory of the beam direction for central heavy-ion collisions.
In the following we compare our data to the results of theoretical calculations. Statistical models using a canonical formulation of strangeness conservation predict a constant $K^-/K^+$ ratio as a function of system size for heavy-ion collisions at SIS beam energies [@cley00]. The result of such a calculation is shown in the lowest panel of Figure \[ratio\] as a dashed line [@cley00]. In this case a baryochemical potential of $\mu$ = 770 MeV and a chemical freeze-out temperature of T = 63 MeV was assumed. Measured inverse slope parameters refer to thermal freeze-out and are substantially larger: T($K^+$) = 103$\pm$6 MeV and T($K^-$) = 93$\pm$6 MeV for near-central Ni+Ni collisions, and T($K^+$) = 116$\pm$7 MeV and T($K^-$) = 90$\pm$8 MeV for near-central Au+Au collisions (corresponding to the average value of the two most central bins in figure 2, upper panel).
The observation of different spectral slopes or mean energy for $K^+$ and $K^-$ mesons is at variance with a scenario in which both kaons and antikaons have the same flow velocity and thermal energy at chemical freeze-out. In consequence, the statistical model does not offer a consistent explanation for both the yields and spectral slopes of $K^+$ and $K^-$ mesons. The difference in spectral slopes rather indicates that $K^+$ and $K^-$ mesons decouple from the fireball sequentially due to their very different KN inelastic cross sections.
Microscopic transport models predict that the kaons and the hyperons are produced via processes like $NN\to K^+ YN$ or $\pi
N\to K^+ Y$ with $Y=\Lambda,\Sigma$ in the early phase of a heavy-ion collision [@fuchs; @aichelin; @cass_brat]. The $K^+$ mesons leave the reaction volume with little rescattering because of their long mean free path. Therefore, the $K^+$ mesons probe the early, dense and hot phase of the collision and have been used to obtain information on the nuclear equation-of-state [@sturm; @fuchs]. Within the transport calculations the production of antikaons proceeds predominantly via strangeness-exchange reactions $\pi Y \to K^-N$ [@Ko; @cass_brat; @Hart02]. The mean free path of the $K^-$ mesons is about 1.5 fm in nuclear matter due to absorption via reactions like $K^-N\to Y\pi$. However, via the inverse reaction ($\pi Y \to
K^- N$) the antikaons may reappear again thus propagating to the surface of the fireball. Consequently, the yields of $K^+$ and $K^-$ mesons are both related to the hyperon yield, but the observed $K^-$ mesons in average are produced later than the $K^+$ mesons [@Hart02].
The various transport calculations do not yet provide a consistent picture concerning the in-medium properties of antikaons. Recent QMD model calculations predict a rather weak sensitivity of the $K^-$ yield on the $K^-N$ potential [@Hart02]. The calculations result in a $K^-/K^+$ ratio which systematically underestimates the experimental data (see the hatched area in the lowest panel of Figure \[ratio\])[@Hart02]. This result is based on the assumption of in-medium $K^+$ and $K^-$ masses. A very similar result is obtained for free masses. On the other hand, BUU calculations need to take into account an attractive in-medium $K^-N$ potential in order to explain the $K^-$ yields [@cass_brat; @li_brown]. Predictions of a BUU model calculation [@li_brown] for the $K^-/K^+$ ratio as a function of transverse mass for near-central Au+Au collisions at 1.5 $A\cdot$GeV are shown in Figure \[BUU\] together with our experimental results. A similar result was found for Ni+Ni collisions at 1.93 $A\cdot$GeV [@wisniewski]. As demonstrated in Figure \[BUU\] the calculations assuming free K meson masses (dashed line) and in-medium masses (solid line) clearly disagree. In this case the differences in spectral slope are caused by the opposite mean-field potentials of kaons and antikaons (see also [@brat]). However, both the QMD [@Hart02] and the BUU models [@li_brown; @brat] use a rather simple parametrization of the effective mass of $K^+$ and $K^-$ mesons in nuclear matter. New theoretical concepts are required to improve the interpretation of experimental data. This is expected from the next generation of transport calculations which take into account off-shell effects like in-medium spectral functions and in-medium cross sections [@cass; @lutz].
In summary, we have presented differential cross sections and phase-space distributions of kaons and antikaons produced in heavy-ion collisions at 1.5 $A\cdot$GeV. We observed the following features: (i) The $K^-/K^+$ yield ratio is quite independent of $A_{part}$ both for Ni+Ni and Au+Au collisions, (ii) in near-central collisions $K^-$ mesons are emitted almost isotropically whereas $K^+$ mesons exhibit a forward-backward enhanced emission pattern, and (iii) the inverse slope parameters are significantly smaller for $K^-$ than for $K^+$ mesons even for the most central Au+Au collisions. These findings indicate that (i) the production mechanisms of $K^+$ and $K^-$ mesons are correlated by strangeness-exchange reactions, (ii) $K^-$ mesons undergo many collisions before leaving the fireball and, as a consequence, (iii) $K^-$ and $K^+$ mesons experience different freeze-out conditions.
We thank the GSI accelerator crew for an exceptional operation of the GSI accelerator facilities resulting in a high-energy gold beam of excellent quality. This work was supported by the German Federal Government (BMBF), by the Polish Committee of Scientific Research (No. 2P3B11515) and by the GSI fund for Universities.
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abstract: 'By means of density functional theory (DFT) calculations (with and without inclusion of spin-orbit (SO) coupling) we present a detailed study of the electronic structure and corresponding microscopic Hamiltonian parameters of [[Na$_2$IrO$_3$]{}]{}. In particular, we address the following aspects: (i) We investigate the role of the various structural distortions and show that the electronic structure of [[Na$_2$IrO$_3$]{}]{} is exceptionally sensitive to structural details. (ii) We discuss both limiting descriptions for [[Na$_2$IrO$_3$]{}]{}; quasi-molecular orbitals (small SO limit, itinerant) versus relativistic orbitals (large SO limit, localized) and show that the description of [[Na$_2$IrO$_3$]{}]{} lies in an intermediate regime. (iii) We investigate whether the nearest neighbor Kitaev-Heisenberg model is sufficient to describe the electronic structure and magnetism in [[Na$_2$IrO$_3$]{}]{}. In particular, we verify the recent suggestion of an antiferromagnetic Kitaev interaction and show that it is not consistent with actual or even plausible electronic parameters. Finally, (iv) we discuss correlation effects in [[Na$_2$IrO$_3$]{}]{}. We conclude that while the Kitaev-Heisenberg Hamiltonian is the most general expression of the quadratic spin-spin interaction in the presence of spin-orbit coupling (neglecting single-site anisotropy), the itinerant character of the electrons in [[Na$_2$IrO$_3$]{}]{} makes other terms beyond this model (including, but not limited to 2nd and 3rd neighbor interactions) essential.'
author:
- Kateryna Foyevtsova
- 'Harald O. Jeschke'
- 'I. I. Mazin'
- 'D. I. Khomskii'
- Roser Valentí
title: '*Ab initio* analysis of the tight-binding parameters and magnetic interactions in [Na$_{2}$IrO$_{3}$]{}'
---
Introduction
============
The electronic and magnetic behavior of layered $5d$ transition metal oxides [@Kim_2009] has been a subject of intensive discussion in the last years. Particularly exciting has been the suggestion by the authors of Ref. that hexagonal iridates such as [[Na$_2$IrO$_3$]{}]{} are a realization of the nearest neighbor Kitaev-Heisenberg (nnKH) model: $$H_{ij}^{(\gamma )}=2KS_{i}^{\gamma }S_{j}^{\gamma }+J{\bf S}_{i}\cdot
{\bf S}_{j} \label{nnKHmodel}$$ This proposal is based on the premise that spin-orbit (SO) coupling is the most important energy scale for the description of these systems so that Ir $5d$ $t_{2g}$ orbitals are written in terms of $j_{\text{eff}}=1/2$ and $j_{\text{eff}}=3/2$ relativistic orbitals, with the Kramers doublet $j_{\text{eff}}=1/2$ represented by the operator $S=1/2$. The combination of Kitaev and Heisenberg terms leads to a complex phase diagram with various magnetic and spin-liquid phases [@Chaloupka2010; @trebst1; @Chaloupka2012]. Obviously, some of these properties can only manifest themselves when the Kitaev term dominates or is at least comparable to the Heisenberg term. Also, other possible contributions, such as magnetic anisotropy, ring exchange or biquadratic exchange, to mention a few, may alter the phase diagram and the properties of the model considerably. Most importantly, while the Kitaev-Heisenberg expression is the most general fully-symmetric expression for anisotropic pairwise magnetic interactions in the second order in spin in the presence of SO coupling (just as the Heisenberg exchange represents the same in the isotropic non-relativistic case), it is not necessarily short ranged in the presence of considerable itinerancy.
So far, essentially all analyses of the nnKH model for [[Na$_2$IrO$_3$]{}]{} have been performed in the localized limit, where an assembly of weakly interacting relativistic atomic orbitals is assumed to be a good starting approximation. On the other hand, first principles calculations suggest considerable delocalization of electrons over individual Ir hexagons building quasi-molecular orbitals (QMOs) [@we]. The associated itinerant energy scale (the band width) is $\approx1.5$ eV, to be compared to the single-site spin-orbit splitting scale [@comment_split] $(3/2)\lambda\approx0.7$ eV and the Hubbard and Hund’s rule correlation energy scale of $U-J_{\mathrm{H}}\approx0.5-1$ eV. This makes the entire premise of the nnKH model questionable. At the same time, it has also been pointed out [@trebst; @Radu] that the nnKH model with the addition of the 2nd and 3rd neighbors Heisenberg interaction is easier to reconcile with the experimental data. Such relatively long-range exchange interaction is another hallmark of considerable itinerancy (here and below, when we speak of itinerancy, we imply mostly delocalization over Ir$_6$ rings, but not necessarily over the entire crystal).
In the present work we revisit and discuss the validity of both limiting descriptions for [[Na$_2$IrO$_3$]{}]{}; itinerant (QMO picture) versus localized ($j_{\text{eff}}=1/2$ Kramers doublet). To this end, we perform a thorough analysis of the electronic properties of [[Na$_2$IrO$_3$]{}]{} within [*non-relativistic*]{} and [*relativistic*]{} density functional theory (DFT) and derive, using projection on Wannier functions, the relevant hopping parameters and show that QMOs are naturally obtained as linear combinations of Ir $t_{2g}$ Wannier functions. We discuss the relation between the quasi-molecular orbital and the relativistic orbital, $j_{\text{eff}}$, representations and show that the behavior of [[Na$_2$IrO$_3$]{}]{} lies in between a fully localized and fully itinerant description. Finally, the parametrization of the electronic bandstructure allows us to provide realistic estimates for the model parameters in the localized nnKH model. We thus investigate whether we are close to a regime where the Kitaev interaction plays a decisive role or not.
Quite unexpectedly, we find that [[Na$_2$IrO$_3$]{}]{} is an example of a material where minor details of the crystal structure can dramatically affect the electronic structure, and simple guessing of the band structure parameters, or estimating them from simplified crystallographic models (so far all model calculations for this compound were utilizing one or the other approach) can be exceptionally misleading. In fact some of the models energetically discussed in the community, while of undeniable theoretical appeal, are not even qualitatively close to the actual parameter range in [[Na$_2$IrO$_3$]{}]{}.
While this particular compound is very intriguing and has been enjoying extraordinary popularity lately, we want to emphasize that this strong dependence of the electronic properties on details of the crystal structure is an important result, whose relevance goes beyond specifically [[Na$_2$IrO$_3$]{}]{} and is likely true for many other materials based on honeycomb transition-metal layers.
The paper is organized as follows. In Section \[sec:two\] we review the crystal structure and magnetic properties of [[Na$_2$IrO$_3$]{}]{}. In Section \[sec:three\] we provide details of the DFT calculations and the projector method. In Section \[sec:four\] we present the results of the electronic structure analysis without inclusion of spin-orbit coupling and analyze the role of the structural distortions in [[Na$_2$IrO$_3$]{}]{}. In Section \[sec:five\] we investigate the role of spin-orbit coupling and discuss the relation between the QMOs and the relativistic orbitals ($j_{\text{eff}}$). In this context, we discuss whether the existing experimental situation can distinguish between the DFT description (with the resulting itinerancy) and localized ($j_{\mathrm{eff}}=1/2)$ models. We proceed with an analysis of the single-site magnetic anisotropy in [[Na$_2$IrO$_3$]{}]{} and find it to be relevant (pure $j_{\mathrm{eff}}=1/2$ states do not have any single-site anisotropy). In Section \[sec:six\] we provide *ab initio*-derived estimates for the parameters appearing in the Kitaev and Heisenberg terms in [[Na$_2$IrO$_3$]{}]{} and discuss the validity of the nnKH model by considering the experimentally observed magnetic order and attempts to explain it from a local point of view. Finally in Section \[sec:seven\] we present our conclusions.
![ Crystal structure of [[Na$_2$IrO$_3$]{}]{}. (a) Projection on the $ac$ plane and (b) projection on the $ab$ plane. []{data-label="crystal"}](fig1){width="0.95\columnwidth"}
Crystal structure and magnetic properties of [N$_2$IO$_3$]{} {#sec:two}
============================================================
[[Na$_2$IrO$_3$]{}]{} crystallizes in the monoclinic space group $C\,2/m$ (No. 12) [@Radu] (see Fig. \[crystal\]) and consists of Ir honeycomb layers (Fig. \[crystal\] (b)) stacked along the monoclinic [**c**]{} axis (Fig. \[crystal\] (a)) with an in-plane off-set along [**a**]{}. Na ions occupy both the interlayer positions and 1/3 of the in-plane positions at the centers of Ir hexagons. This structure can be visualized as proceeding from NaIrO$_{2}$ with a CdI$_{2}$ structure with triangular IrO$_{2}$ layers. In these layers 1/3 of the in-plane iridium atoms are substituted by extra Na, [[*[i. e.]{}*]{}]{}, its formula can be written as Na(Na$_{1/3}$Ir$_{2/3})$O$_{2}$, which, multiplied by 3/2, gives the usual formula of [Na$_{2}$IrO$_{3}
$]{} [@comment_cava].
An idealized crystal structure of this kind corresponds to having all nearest neighbor (NN) Ir-Ir and NN Ir-O distances equal and Ir-O-Ir angles of 90 degrees. The experimental structure of [[Na$_2$IrO$_3$]{}]{} departs from the idealized case and shows a few distortions: (i) orthorhombic distortion that introduces inequality among NN Ir-Ir distances and among NN Ir-O distances, (ii) IrO$_{6}$ octahedra rotations that place O atoms on the faces of a cube containing an Ir hexagon (see Fig. 2 of Ref. ) and (iii) trigonal distortion which is a compression of the IrO$_{6}$ octahedra in the [**c**]{}-direction that induces a departure from 90 degrees of the Ir-O-Ir angles. In Section \[sec:four\] we will discuss the effect of these distortions on the electronic structure of [[Na$_2$IrO$_3$]{}]{}.
As shown by transport, optical and high-energy spectroscopy studies [@Singh10; @Comin12], [[Na$_2$IrO$_3$]{}]{} is an insulator with an energy gap $E_{g}$ of 340 meV. Magnetic susceptibility measurements indicate a Curie-Weiss behavior at high temperatures with a Curie-Weiss temperature $\Theta _{CW}=-116$ K and an effective Ir moment $\mu_{\text{eff}}=1.82\mu_{{\text{B}}}$. [[Na$_2$IrO$_3$]{}]{} orders antiferromagnetically below $T_{N}=15$ K with an ordered magnetic moment $\mu_{ord} \sim 0.2\mu_{\text{B}}$. The fact that $T_{N}$ is much smaller than $\Theta _{CW}$ may be a signature of frustration, but it may be also caused by the itinerancy of Ir $5d$ electrons [@we] as will be discussed in Section \[sec:six\].
![Possible antiferromagnetic patterns in a honeycomb lattice[]{data-label="patterns"}](fig2){width="50.00000%"}
The magnetic pattern observed experimentally [@Radu] corresponds to a zigzag ordering, in contrast to the prediction of a stripe order by the nnKH model [@Chaloupka2010] (see Fig. \[patterns\]). Recently, Chaloupka *et al.* [@Chaloupka2012] argued that such a zigzag ordering can be also obtained by the nnKH model, when one correctly includes all the terms contributing to NN Ir-Ir exchange. In Section \[sec:five\], we will discuss this proposition in more detail.
Method {#sec:three}
======
In this work we perform DFT calculations using the linearized augmented plane wave (LAPW) method as implemented in the full-potential code WIEN2k [@w2k]. We employ the Perdew-Burke-Ernzerhof generalized gradient approximation [@PBE] to the DFT exchange-correlation functional and set the basis-size controlling parameter $RK_{\text{max}}$ [@Cottenier] to 7. We consider a mesh of 500 ${\bf k}$-points in the first Brillouin zone. Relativistic effects are treated within the second variational approach. Convergence with respect to relevant parameters (the ${\bf k}$-point mesh, the $RK_{\max }$ and the second variational energy cutoff, *etc.*) has been carefully checked.
Calculation of hopping integrals
--------------------------------
In order to be able to discuss various Ir-Ir $5d$ processes, we parameterize our non-relativistic DFT results in terms of a tight-binding (TB) model where the TB Ir $5d$ hopping parameters are obtained through the Wannier function projection formalism proposed in Ref. and generalized to molecular Wannier functions in Ref. . We first construct Wannier function projectors $P^{\alpha}_{m\nu}({\bf k})$ for the three $t_{2g}$ Ir $5d$ orbitals and calculate the TB Hamiltonian $H^{\text{TB}}({\bf k})$ (in matrix form) via $$H^{\text{TB}}({\bf k}) = P({\bf k}) D({\bf k}) P^{\dagger}({\bf k}),$$ where $D({\bf k})$ is a diagonal matrix of Ir $5d$ $t_{2g}$ Bloch eigenvalues and the matrix $P({\bf k})$ is formed by the projectors $P^{\alpha}_{m\nu}({\bf k}) $. Here, indices $\alpha$, $m$, and $\nu$ run over equivalent Ir atoms in the unit cell, Ir $t_{2g}$ orbitals, and Bloch bands, respectively. [[Na$_2$IrO$_3$]{}]{} has two Ir per unit cell and only the six Ir $t_{2g}$ bands near the Fermi level $E_{\text{F}}$ are considered in the construction of projectors.
We calculate the hopping integral $t^{mm^{\prime}}_{\alpha- {\bf R},
\alpha^{\prime}- {\bf R}^{\prime}}$ between orbital $m$ on Ir atom $\alpha$ in the unit cell at a distance ${\bf R}$ from a reference unit cell and orbital $m^{\prime}$ on Ir atom $\alpha^{\prime}$ in the unit cell at a distance ${\bf R}^{\prime}$ from a reference unit cell by integrating $H^{\text{TB}}({\bf k})$ over $N_{{\bf k}}$ ${\bf k}$-vectors in the first Brillouin zone: $$t^{mm^{\prime}}_{\alpha- {\bf R}, \alpha^{\prime}- {\bf R}^{\prime}} = \frac{1}{N_{{\bf k}}} \sum_{{\bf k}} H^{\text{TB}}_{\alpha m, \alpha^{\prime
}m^{\prime}}({\bf k}) e^{-i{\bf k}({\bf R}-{\bf R}^{\prime})}.
\label{TB_param}$$ where $H^{\text{TB}}_{\alpha m, \alpha^{\prime}m^{\prime}}({\bf k})$ are the matrix elements of $H^{\text{TB}}({\bf k})$. Correspondingly, the diagonal matrix elements $t^{mm}_{\alpha\alpha}$ give the on-site energies.
Construction of quasi-molecular projectors
------------------------------------------
As was argued in Ref. , the most natural description of the electronic structure of [[Na$_2$IrO$_3$]{}]{} is in terms of quasi-molecular (QMO) orbitals localized on a hexagon. The strongest Ir-Ir hopping is between $5d$ $t_{2g}$ orbitals of neighboring iridium ions via common oxygens. In this case, an electron on a given Ir $t_{2g}$ orbital propagates around an Ir$_{6}$ hexagon with the peculiarity than only a certain $t_{2g}$ orbital at each Ir participates in the hopping [@axes], e.g. Ir1(${xy}$)-Ir2(${xz}$)-Ir3(${yz}$)-Ir4(${xy}$)-Ir5(${xz}$)- Ir6(${yz}$) (see Fig. 2 of Ref. ). These QMOs are analogous to the molecular orbitals of the benzene molecule C$_{6}$H$_{6}$ except for the fact that in benzene the same $p$-orbital on each carbon ion participates in the formation of the molecular orbital while in [[Na$_2$IrO$_3$]{}]{}, as described above, different $t_{2g}$ orbitals are involved in one QMO and the three $t_{2g}$ orbitals on one Ir ion contribute to three different neighboring QMOs. We elaborate the details of the construction of the QMOs in what follows.
QMO projectors $P_{\mathcal{M}\nu }({\bf k})$ are obtained as linear combinations of Ir $t_{2g}$ projectors $P_{M\nu }({\bf k})$: $$P_{\mathcal{M}\nu }({\bf k})=\sum_{M}U_{\mathcal{M},M}T_{M}({\bf k})P_{M\nu
}({\bf k}).\label{E.QMOproj}$$ where in the Ir $t_{2g}$ projectors $P_{M\nu }({\bf k})$, the index $M$ combines now the atomic index $\alpha $ and orbital index $m$, [*i.e.*]{} $M$ runs over all $t_{2g}$ orbitals of all equivalent Ir atoms. With QMOs ordered as $\mathcal{M}=A_{1g},E_{2u},E_{1g},B_{1u},E_{1g},E_{2u}$ and Ir $t_{2g}$ orbitals ordered as $M={xy}^{1},{xz}^{1},{yz}^{1},{xy}^{2},{xz}^{2},{yz}^{2}$ (the upper index labels Ir atoms), $U$ is given by \[$\omega =\exp (i\pi /3)$\] $$U=
\begin{pmatrix}
1 & 1 & 1 & 1 & 1 & 1 \\
1 & \omega ^{4} & \omega ^{2} & -1 & \omega & \omega ^{5} \\
1 & \omega ^{2} & \omega ^{4} & 1 & \omega ^{2} & \omega ^{4} \\
1 & 1 & 1 & -1 & -1 & -1 \\
1 & \omega ^{4} & \omega ^{2} & 1 & \omega ^{4} & \omega ^{2} \\
1 & \omega ^{2} & \omega ^{4} & -1 & \omega ^{5} & \omega
\end{pmatrix},
\label{E.unitary}$$ and $T_{M}({\bf k})$ are the Bloch factors, accounting for the fact that the 6 sites forming a QMO belong to several different unit cells. Actual values for these factors depend on the manner in which a particular band structure code selects the unit cell (see the Appendix for the WIEN2k settings).
Non-Relativistic electronic structure {#sec:four}
=====================================
In this Section we analyze and discuss the Ir-Ir $5d$ $t_{2g}$ tight-binding parameters for [[Na$_2$IrO$_3$]{}]{} [*up to second nearest neighbors*]{}. As mentioned in Section \[sec:two\], three structural distortions are present in [[Na$_2$IrO$_3$]{}]{}: orthorhombic distortion, IrO$_6$ octahedra rotation and trigonal distortion. Besides, the stacking of the honeycomb planes inherently violates the rhombohedral symmetry even if each plane is ideal. The formation of QMOs relies on the dominance of *intra*hexagon hopping [@we] and therefore is sensitive to structural details. Therefore it is important to understand the role of structural distortions in establishing electron hopping paths. This motivates us to study electronic properties of a number of artificially idealized [[Na$_2$IrO$_3$]{}]{} unit cells where structural distortions of different types are systematically eliminated [@Radu_comment]. Such a procedure has proven very useful [@Martins11] in understanding the behavior of Sr$_2$IrO$_4$.
We consider four different crystal structures: (i) the experimental crystal structure [@Radu], $S_{\rm exp}$, (ii) an artificially idealized [[Na$_2$IrO$_3$]{}]{} unit cell, $S_1$, where the orthorhombic distortion has been removed from the experimental crystal structure, (iii) an artificially idealized [[Na$_2$IrO$_3$]{}]{} unit cell, $S_2$, where the IrO$_6$ octahedra rotations have been removed from $S_1$, and (iv) an artificially idealized [[Na$_2$IrO$_3$]{}]{} unit cell, $S_3$, where the trigonal distortion has been removed from $S_2$. Table \[total\_energies\] shows a comparison of total (non-magnetic) DFT energies for the various structures.
Structure $S_{\rm exp}$ $S_1$ $S_2$ $S_3$
---------------------------------- --------------- ------- ------- --------
$E_{S_i}-E_{S_{\rm exp}}$ (mRyd) 0 0.95 78.90 180.01
: Non-relativistic total energies obtained within DFT for the experimental, $E_{S_{\rm exp}}$, and the three idealized, $S_i$ ($i=1,2,3$), [[Na$_2$IrO$_3$]{}]{} crystal structures. Energy is given per unit cell containing two formula units.[]{data-label="total_energies"}
As it is to be expected, the experimental structure is the energetically most stable case. Tight-binding hopping parameters between Ir $t_{2g}$ orbitals up to second nearest neighbors calculated for the four structures are given in Table \[T.AOhoppings\] and schematically represented in Fig. \[hoppings\]. We consider the following rationale for labeling of the hopping parameters Eq. \[TB\_param\]. In the experimental structure of [[Na$_2$IrO$_3$]{}]{} there are two first NN Ir-Ir distances and two second NN Ir-Ir distances due to the fact that the Ir$_6$ hexagons are not perfect. We denote the corresponding Ir $t_{2g}$ - Ir $t_{2g}$ hopping parameters as $t_1$ and $t_{\bar{1}}$ for the first NN and, respectively, $t_2$ and $t_{\bar{2}}$ for the second NN hoppings. Further, we have various possible hoppings between equal and different $t_{2g}$ orbitals. Regarding first NN, we denote $t_{1\,{\rm O}}$ and $t_{{\bar{1}}\,{\rm
O}}$ the hoppings between unlike $t_{2g}$ orbitals via O $p$ states (Fig. \[hoppings\] (a)). $t_{1\sigma}$ and $t_{\bar{1}\sigma}$ denote NN direct hoppings of $\sigma$-type. $t_{1\parallel}$ and $t_{\bar{1}\parallel}$ denote NN hoppings between like orbitals lying in parallel planes. In the ideal structure such hoppings consist of linear combinations with equal weight of $dd\pi$ and $dd\delta$ bonds. $t_{1\perp}$ and $t_{\bar{1}\perp}$ denote NN hoppings between unlike orbitals lying in perpendicular planes (see Figs. \[hoppings\] (b) and (c)).
Regarding the second NN hopping parameters, $t_{2\,{\rm O}}$ and $t_{\bar{2}\,{\rm O}}$ denote hoppings between unlike orbitals via O $p$ and Na $s$ states (Fig. \[hoppings\] (e)). $t_{2a}$ and $t_{2b}$ ($t_{\bar{2}a}$ and $t_{\bar{2}b}$) denote hoppings between like orbitals as shown in Fig. \[hoppings\] (d) and $t_{2c}$, $t_{2d}$ and $t_{2e}$ ($t_{\bar{2}c}$, $t_{\bar{2}d}$ and $t_{\bar{2}e}$) denote hoppings between unlike orbitals (Fig. \[hoppings\] (e)).
Experimental crystal structure
------------------------------
Previous electronic structure calculations [@we] have identified the dominant hopping integrals for [[Na$_2$IrO$_3$]{}]{} to be $t_{1\,{\rm O}}$ and $t_{2\,{\rm O}}$ \[as well as $t_{\bar{1}\,{\rm O}}$ and $t_{\bar{2}\,{\rm O}}$; further on, if not explicitly stated otherwise, we refer to both equivalent $t_{1}$ ($t_{2}$) and $t_{\bar{1}}$ ($t_{\bar{2}}$) when writing $t_{1}$ ($t_{2}$)\]. In Table \[T.AOhoppings\] column $S_{exp}$ we present the complete list of hopping parameters up to the second nearest neighbors. A TB model based only on these hopping integrals provides already a reasonable description of [[Na$_2$IrO$_3$]{}]{} Ir $t_{2g}$ states near the Fermi level $E_{\text{F}}$ \[Fig. \[F.AObands\] (a)\].
{width="80.00000%"}
We first note a very good agreement between the $t_{1\,{\rm O}}$ ($\sim 270$ meV) and $t_{2\,{\rm O}}$ ($\sim -75$ meV) values obtained with our WIEN2k-based projection method and with the FPLO code [@FPLO] as was used in Ref. . These leading Ir $t_{2g}$ hoppings strongly tend to confine the electron’s motion to a single Ir hexagon and, as a result, the electronic structure of [[Na$_2$IrO$_3$]{}]{} near the Fermi level is dominated by the formation of well separated and relatively weakly dispersive QMOs [@we]. On an Ir hexagon, as shown above, each Ir atom participates with one of its $t_{2g}$ orbitals (see Fig. 2 of Ref. ). These orbitals combine to form six QMOs according to the unitary transformation Eq. (\[E.unitary\]). In support of this picture, Fig. \[F.QMOdos\] (a) shows the density of states of [[Na$_2$IrO$_3$]{}]{} projected onto the six QMOs (singlets $A_{1g}$ and $B_{1u}$ and doublets $E_{2u}$ and $E_{1g}$), where states with certain predominant QMO character are clearly separated in energy from one another. The near-degeneracy of $A_{1g}$ and $E_{2u}$ states around $E_{\text{F}}$ is rather accidental resulting from the $t_{1\,{\rm O}}/t_{2\,{\rm
O}}\sim-3.6$ ratio (see Table \[T.AOhoppings\] and Ref. ). The real-space representations of the QMO Wannier functions onto which the [[Na$_2$IrO$_3$]{}]{} DOS is being projected are shown in Fig. \[F.Wannier\]. The QMO Wannier functions were constructed as described in Section \[sec:two\] by explicitly accounting for the location of each Ir $t_{2g}$ orbital in the crystal [@note1].
Other NN and second NN hopping processes involving intraorbital and interorbital hoppings (see Table \[T.AOhoppings\]) allow an electron to jump from one QMO to another and hence are responsible for the band dispersion. Many of those hoppings are of the same order of magnitude (although mostly by at least an order of magnitude smaller) than $t_{2\,{\rm O}}$, like, for example, $t_{1\parallel}$ and $t_{\bar{1}\parallel}$. For the z bond such hoppings will be between $xz$ and $xz$ or $yz$ and $yz$ orbitals (see Fig. \[hoppings\] (b)). These hoppings are equal to 47.7, 30.0, and 33.1 meV, depending on the NN bond (see Table \[T.AOhoppings\]). In fact, such appreciable variations in magnitude, which violate the $D_{6h}$ symmetry of an ideal Ir hexagon, are ubiquitous among the hoppings that connect neighboring QMOs. Some of them even change sign, as, for instance $t_{1\sigma}$ and $t_{\bar{1}\sigma}$. This feature results from the orthorhombic stacking, distortions within the Ir$_{2}$Na planes, and rotations of IrO$_{6}$ octahedra.
NN $S_{exp}$ $S_1$ $S_2$ $S_3$
----------- ------------------------------------------------- ----------- -------- -------- --------
0 $xy\to xy$ -448.8 -422.9 -422.8 -601.1
$xz\to xz$ -421.5 -421.8 -421.2 -601.1
$yz\to yz$ -421.5 -421.8 -421.2 -601.1
$xy\to xz$, $xy\to yz$ -27.8 -26.4 -21.2 -13.5
$xz\to yz$ -23.1 -25.2 -18.8 -14.7
1 $xy\to xy$ ($t_{1\parallel}$) 47.7 34.1 27.8 120.8
$xy\to xz$, $xy\to yz$ ($t_{1\,{\rm O}}$) 269.6 268.5 231.7 209.7
$xy\to xz$, $xy\to yz$ ($t_{1\perp}$ ) -25.6 -16.6 43.7 -5.3
$xz\to xz$, $yz\to yz$ ($t_{1\parallel}$) 30.0 33.2 17.2 118.9
$xz\to xz$, $yz\to yz$ ($t_{1\sigma}$) -20.7 3.5 -66.5 -381.6
$xz\to yz$ ($t_{1\perp}$) -21.4 -16.4 41.7 -4.9
$\bar{1}$ $xy\to xy$ ($t_{\bar{1}\sigma}$) 25.4 0.2 -65.5 -382.8
$xy\to xz$, $xy\to yz$ ($t_{\bar{1}\perp}$) -11.9 -17.6 46.9 -5.3
$xz\to xz$, $yz\to yz$ ($t_{\bar{1}\parallel}$) 33.1 33.9 21.2 120.5
$xz\to yz$ ($t_{\bar{1}\,{\rm O}}$) 264.4 264.8 228.7 211.7
2 $xy\to xy$ ($t_{2a}$) -3.5 -2.6 -18.9 2.0
$xy\to xz$, $xy\to yz$ ($t_{2\,{\rm O}}$) -75.8 -77.4 -94.7 -82.1
$xy\to xz$, $xy\to yz$ ($t_{2c}$) -36.5 -35.3 -52.1 -38.5
$xy\to xz$, $xy\to yz$ ( $t_{2d}$) 12.5 10.1 1.7 6.9
$xy\to xz$, $xy\to yz$ ($t_{2e}$) -21.4 -19.2 -7.3 1.9
$xz\to xz$, $yz\to yz$ ( $t_{2a}$) -0.6 -3.1 -16.6 1.4
$xz\to xz$, $yz\to yz$ ($t_{2b}$) -1.5 -1.6 -1.0 5.7
$xz\to yz$ ($t_{2e}$) -18.6 -19.0 -7.1 2.4
$xz\to yz$ ($t_{2d}$) 10.2 10.2 2.4 6.6
$\bar{2}$ $xy\to xy$ ($t_{\bar{2}b}$) -1.4 -1.4 -1.2 5.7
$xy\to xz$, $xy\to yz$ ($t_{\bar{2}e}$) -19.0 -19.2 -8.4 2.1
$xy\to xz$, $xy\to yz$ ($t_{\bar{2}d}$) 9.3 10.2 0.7 7.5
$xz\to xz$, $yz\to yz$ ($t_{\bar{2}a}$) -1.4 -3.0 -17.7 1.5
$xz\to yz$ ($t_{\bar{2}\,{\rm O}}$) -77.0 -78.0 -95.2 -81.9
$xz\to yz$ ($t_{\bar{2}c}$) -30.4 -35.1 -51.6 -38.9
: Nearest neighbor (NN) and second NN hopping integrals in meV between Ir $t_{2g}$ orbitals for the experimental structure and three idealized structures $S_1$, $S_2$, $S_3$ of [[Na$_2$IrO$_3$]{}]{} (see text and Appendix for a description of the structures and parameter labeling). The $\text{NN}=0$ data are Ir $t_{2g}$ on-site energies and *inter*orbital hoppings; the $\text{NN}=1$ and $\text{NN}=\bar{1}$ ($\text{NN}=2$ and $\text{NN}=\bar{2}$) data are hoppings over nonequivalent (due to orthorhombic distortion) NN (second NN) Ir bonds.[]{data-label="T.AOhoppings"}
![[[Na$_2$IrO$_3$]{}]{} bandstructures near the Fermi level $E_{\text{F}}=0$ calculated using DFT (black solid lines) and a TB model that considers [*only*]{} up to NNN hopping processes between Ir $t_{2g}$ orbitals (dashed lines). The data are obtained with (a) experimental crystal structure and idealized structures (b) $S_1$, (c) $S_2$, and (d) $S_3$. []{data-label="F.AObands"}](fig4){width="45.00000%"}
![[[Na$_2$IrO$_3$]{}]{} DOS projected onto QMOs for (a) experimental crystal structure and idealized structures (b) $S_1$, (c) $S_2$, and (d) $S_3$. The Fermi level is set to zero.[]{data-label="F.QMOdos"}](fig5){width="40.00000%"}
![Real-space representation of the QMOs in [[Na$_2$IrO$_3$]{}]{} obtained by the Wannier projector method.[]{data-label="F.Wannier"}](fig6){width="50.00000%"}
Structure $S_1$ obtained by removing the orthorhombic distortion
----------------------------------------------------------------
We now consider an idealized [[Na$_2$IrO$_3$]{}]{} structure without the orthorhombic distortion of Ir hexagons; this structure, which we call $S_1$, as well as other structures in this Section, is tabulated in the Appendix. In the structure $S_1$: (i) all *intra*layer Ir-Ir bonds are of the same length, [*i.e.*]{}, the $D_{6h}$ symmetry of an Ir hexagon is restored, (ii) all NN Ir-O bonds are of the same length, (iii) all Ir-O-Ir bond angles are equal to 98.7$^\circ$, and (iv) the oxygens lie on the faces of a cube drawn around an Ir hexagon (see Fig. 2 of Ref. ). The 3D crystal structure, though, remains orthorhombic in this approximation, due to the presence of multiple Ir layers. This explains small residual variations among the nominally equivalent TB model parameters (Table \[T.AOhoppings\], column $S_1$): [*E.g.*]{}, comparing parameters labeled with and without overbar; also, onsite energies like the $xy$ on-site energy is slightly lower than the $xz/yz$ on-site energy. However, these variations of $t_{2g}$ orbital on-site energies, as well as of equivalent hopping integrals, are now noticeably smaller than in the experimental [[Na$_2$IrO$_3$]{}]{} structure.
We conclude that removal of the orthorhombic distortion restores (to a certain degree) the degeneracy of the Ir $t_{2g}$ orbitals, but does not change the hierarchy of hopping integrals. In the structure $S_1$, the $t_{1\,{\rm O}}$ and $t_{2\,{\rm O}}$ values are close to the respective values in the experimental [[Na$_2$IrO$_3$]{}]{} structure and, as a consequence, the overall structure of the $t_{2g}$ bands is only slightly changed \[Figs. \[F.AObands\] (b) and \[F.QMOdos\] (b)\].
Structure $S_2$ obtained by removing the IrO$_6$ octahedra rotations
--------------------------------------------------------------------
In the structure $S_1$ that we designed in the previous Section, two types of distortions are still present: (i) trigonal squeezing of IrO$_6$ octahedra along the (111) direction perpendicular to Ir hexagon planes and (ii) IrO$_{6}$ octahedra rotations that place O atoms on the cube’s faces. We now consider structure $S_2$, where the IrO$_{6}$ octahedra rotations are removed from $S_1$. In this structure, the Na-O and Ir-O bond lengths are the same (in the experimental structure, the former is considerably longer). This feature enhances the second NN hopping processes through Na $s$ states, such as $t_{2\,{\rm O}}$, $t_{2a}$, $t_{2c}$ (and the equivalent overbar hoppings) as shown in Table \[T.AOhoppings\], column $S_2$. At the same time, the NN O-assisted hopping $t_{1\,{\rm
O}}$ gets reduced and the $t_{1\,{\rm O}}/t_{2\,{\rm O}}$ ratio decreases to $\sim -2.4$, resulting in a larger separation of the lowest ($B_{1u}$) band from the rest of $t_{2g}$ bands \[Fig. \[F.AObands\] (c)\]. Formation of QMOs still takes place in structure $S_2$ \[Fig. \[F.QMOdos\] (c)\], but the QMO bands are more dispersive compared to the experimental or $S_1$ structures, due to increased *inter*hexagon NN hopping integrals $t_{1\sigma}$, $t_{1\perp}$ (and equivalent $t_{\bar{1}\sigma}$, $t_{\bar{1}\perp}$): thus, one observes broadening of the $A_{1g}$ band and redistribution of weight away from the $E_{2u}$ doublet.
Structure $S_3$ obtained by removing the trigonal distortion
------------------------------------------------------------
We finally consider a most idealized [[Na$_2$IrO$_3$]{}]{} structure $S_3$ without the trigonal distortion, [*i.e.*]{}, with 90$^\circ$ Ir-O-Ir bond angles. Importantly, one can only remove this distortion, while keeping the Ir-O bond length the same, if the Ir-Ir bonds are shortened. Because of that, the hierarchy of hopping integrals changes drastically (Table \[T.AOhoppings\], column $S_3$). The dominant hopping is now the direct NN hopping between like orbitals $t_{1\sigma}$ (and the equivalent $t_{\bar{1}\sigma}$) reaching $\sim
-380$ meV, while the O-assisted hopping $t_{1\,{\rm O}}$ ($t_{\bar{1}\,{\rm O}}$) has been reduced to $\sim 210$ meV. Accordingly, the large *inter*hexagon interaction destroys the QMO picture, as illustrated by the strongly dispersive $t_{2g}$ manifold in Fig. \[F.AObands\] (d) and the delocalization of individual QMO characters over the whole DOS range in Fig. \[F.QMOdos\] (d). We also observe that the main reason for the trigonal squeeze is the geometrical effect of optimizing simultaneously the Ir-Ir and Ir-O bonds. As a result, even though the on-site $t_{2g}$ orbitals split into an $a_{1g}$ singlet and an $e_{g}$ doublet, this is not a strong effect and not the driving force for the squeeze, as it is often assumed in the spirit of localized limit and the Jahn-Teller effect.
Summarizing these results, in the $S_3$ structure, the NN direct hopping increases by an order of magnitude compared to the experimental [[Na$_2$IrO$_3$]{}]{} structure and the NN O-assisted hoppings get suppressed. Therefore we conclude that structural distortions of all types in [[Na$_2$IrO$_3$]{}]{} act constructively to enhance the *intra*hexagon effective hopping parameters (such as $t_{1\,{\rm O}}$ and $t_{2\,{\rm O}}$ ) and suppress the *inter*hexagon ones (such as NN direct hopping) favoring the formation of QMOs.
Spin-orbit coupling {#sec:five}
===================
We proceed now with the analysis of the electronic structure of [[Na$_2$IrO$_3$]{}]{} in the presence of spin-orbit (SO) coupling. Previous relativistic DFT calculations [@Shitade2009] showed that [[Na$_2$IrO$_3$]{}]{} states near the Fermi level experience strong relativistic splitting with pronounced concentration of $j_{\text{eff}}=\frac{1}{2}$ character in the upper two bands. However, the [[Na$_2$IrO$_3$]{}]{} relativistic states seem to preserve their QMO identity as well \[see Fig. S6 (b) of Ref. \]. In order to understand such duality, we set up a TB model for the Ir $t_{2g}$ orbitals that includes also local SO interaction terms. With this TB+SO model, we are able not only to confirm the relativistic DFT results by calculating DOS but also to access the composition of individual states and trace their evolution as a function of the spin-orbit coupling $\lambda$.
TB+SO model {#sec:fiveA}
-----------
We start with a TB model that perfectly describes the non-relativistic DFT Ir $t_{2g}$ bands of [[Na$_2$IrO$_3$]{}]{}. It includes three hundred and twenty one hopping integrals between up to 50 nearest neighbors. We then double the dimension of the TB Hamiltonian matrix to introduce spin dependence and add local SO coupling terms $\langle \lambda
{\bf L}\cdot {\bf S}\rangle $ that mix spin-$\uparrow $ and spin-$\downarrow $ subspaces: $$\begin{array}{c|cccccc}
& {xy}\uparrow & {xz}\uparrow & {yz}\uparrow & {xy}\downarrow &
{xz}\downarrow & {yz}\downarrow \\ \hline
{xy}\uparrow & 0 & 0 & 0 & 0 & \frac{\lambda }{2} & -\frac{i\lambda }{2}
\\
{xz}\uparrow & 0 & 0 & \frac{i\lambda }{2} & -\frac{\lambda }{2} & 0 & 0
\\
{yz}\uparrow & 0 & -\frac{i\lambda }{2} & 0 & \frac{i\lambda }{2} & 0 & 0
\\
{xy}\downarrow & 0 & -\frac{\lambda }{2} & -\frac{i\lambda }{2} & 0 & 0 &
0 \\
{xz}\downarrow & \frac{\lambda }{2} & 0 & 0 & 0 & 0 & -\frac{i\lambda }{2}
\\
{yz}\downarrow & \frac{i\lambda }{2} & 0 & 0 & 0 & \frac{i\lambda }{2} & 0
\end{array}\label{matrix_SO}$$ Importantly, even though SO coupling is a local on-site interaction, it couples neighboring quasi-molecular orbitals and therefore is [${\bf k}$-vector]{} dependent in the QMO basis.
Having thus set up the TB model, we vary the SO coupling strength $\lambda$ until the best matching with the DFT relativistic bands is achieved, which is found to correspond to $\lambda=0.44$ eV (Fig. \[F.RELbands\]).
![WIEN2k relativistic bandstructure (black solid lines) versus TB+SO model relativistic bandstructure (red dashed lines) of [[Na$_2$IrO$_3$]{}]{} as described in the text. In model calculations, $\protect\lambda=0.44$ eV was used.[]{data-label="F.RELbands"}](fig7){width="45.00000%"}
Since our purpose is to reconcile the QMO and relativistic orbital (RO) pictures, we analyze the $\lambda{\bf L}\cdot{\bf S}$ matrix elements between spin-$\uparrow$ and spin-$\downarrow$ QMOs to see how SO coupling mixes QMO characters. They can be easily obtained by applying the unitary transformation $UT({\bf k})$ \[Eq. (\[E.unitary\]) \] to the $\lambda{\bf L}\cdot{\bf S}$ matrix in the $t_{2g}$ basis: $$H^{\text{SO}}_{\text{QMO}}({\bf k}) = U T({\bf k}) H^{\text{SO}}_{t_{2g}}
T^{H}({\bf k}) U^{H}. \label{E.SOtrans}$$ This equation explicitly illustrates how [${\bf k}$-vector]{} dependence enters the SO matrix elements in the QMO basis. Concise expressions can be derived if one notes that QMOs can be represented by their “winding number” $n$ which defines a phase change $\Delta\phi=\frac{n\pi}{3}$ of $t_{2g}$ orbitals around a hexagon. In this notation, QMOs $A_{1g}, E_{2u}, E_{1g}, B_{1u}, E_{1g}, E_{2u}$ correspond to, respectively, $n=0,1,2,3,4,5$ winding numbers. The $\lambda{\bf L}\cdot{\bf S}$ matrix elements in the QMO basis are given then by $$\begin{aligned}
H^{\text{SO}}_{n\uparrow n^{\prime}\uparrow} & = \frac{\lambda}{2} ie^{\frac{(n^{\prime}-n)\pi i}{2}} \cos\frac{(n^{\prime}-n)\pi}{2}\cos (k_{x}+k_{y})
\notag \\
& \times\left( e^{\frac{2(2n^{\prime}-n)\pi i}{3}}-e^{-\frac{2(2n-n^{\prime
})\pi i}{3}}\right) \notag \\
& + \frac{\lambda}{2} ie^{\frac{(n^{\prime}-n)\pi i}{2}} \sin\frac{(n^{\prime }-n)\pi}{2}\sin(k_{x}+k_{y}) \notag \\
& \times\left( e^{\frac{2(2n^{\prime}-n)\pi i}{3}}+e^{-\frac{2(2n-n^{\prime
})\pi i}{3}}\right)\end{aligned}$$ and $$\begin{aligned}
H^{\text{SO}}_{n\uparrow n^{\prime}\downarrow} & = 2\,e^{\frac{(n^{\prime
}-n)\pi i}{2}} \left( e^{\frac{4n^{\prime}\pi i}{3}}\cos(-\frac{(n^{\prime
}-n)\pi}{2} + k_{y})\right. \notag \\
& -e^{-\frac{4n\pi i}{3}}\cos(-\frac{(n^{\prime}-n)\pi}{2} - k_{y}) \notag
\\
& + ie^{\frac{2n^{\prime}\pi i}{3}}\cos(-\frac{(n^{\prime}-n)\pi}{2} - k_{x})
\notag \\
& \left. - ie^{-\frac{2n\pi i}{3}}\cos(-\frac{(n^{\prime}-n)\pi}{2} + k_{x})
\right) .\end{aligned}$$ We list numerical values of the matrix elements for two representative [${\bf k}$-vector]{}s: ${\bf k}=(0,0,0)$ (point $\Gamma$) (first two tables) and ${\bf k}=(\frac{\pi}{2},0,0)$ (last two tables).
$$\label{E.Guu}
\begin{array}{c|ccc|ccc}
& n=5 & n=0 & n=1 & n=2 & n=3 & n=4 \\
& E_{2u}\uparrow & A_{1g}\uparrow & E_{2u}\uparrow & E_{1g}\uparrow &
B_{1u}\uparrow & E_{1g}\uparrow \\ \hline
E_{2u}\uparrow & C_{1} & 0 & 0 & 0 & -C_{1} & 0 \\
A_{1g}\uparrow & 0 & 0 & 0 & -C_{1} & 0 & C_{1} \\
E_{2u}\uparrow & 0 & 0 & -C_{1} & 0 & C_{1} & 0 \\ \hline
E_{1g}\uparrow & 0 & -C_{1} & 0 & C_{1} & 0 & 0 \\
B_{1u}\uparrow & -C_{1} & 0 & C_{1} & 0 & 0 & 0 \\
E_{1g}\uparrow & 0 & C_{1} & 0 & 0 & 0 & -C_{1}
\end{array}$$
$$\label{E.Gud}
\begin{array}{c|ccc|ccc}
& n=5 & n=0 & n=1 & n=2 & n=3 & n=4 \\
& E_{2u}\downarrow & A_{1g}\downarrow & E_{2u}\downarrow & E_{1g}\downarrow
& B_{1u}\downarrow & E_{1g}\downarrow \\ \hline
E_{2u}\uparrow & C_{2} & 0 & 0 & 0 & C_{4} & 0 \\
A_{1g}\uparrow & 0 & 0 & 0 & -C_{3} & 0 & -C_{4} \\
E_{2u}\uparrow & 0 & 0 & -C_{2} & 0 & C_{3} & 0 \\ \hline
E_{1g}\uparrow & 0 & C_{4} & 0 & C_{2} & 0 & 0 \\
B_{1u}\uparrow & -C_{3} & 0 & -C_{4} & 0 & 0 & 0 \\
E_{1g}\uparrow & 0 & C_{3} & 0 & 0 & 0 & -C_{2}
\end{array}$$
$$\begin{array}{c|ccc|ccc}
& n=5 & n=0 & n=1 & n=2 & n=3 & n=4 \\
& E_{2u}\uparrow & A_{1g}\uparrow & E_{2u}\uparrow & E_{1g}\uparrow &
B_{1u}\uparrow & E_{1g}\uparrow \\ \hline
E_{2u}\uparrow & \frac{\lambda}{6} & 0 & \frac{\lambda}{6} & 0 & -\frac{\lambda}{3} & 0 \\
A_{1g}\uparrow & 0 & \frac{\lambda}{6} & 0 & -\frac{\lambda}{3} & 0 & \frac{\lambda}{6} \\
E_{2u}\uparrow & \frac{\lambda}{6} & 0 & -\frac{\lambda}{3} & 0 & \frac{\lambda}{6} & 0 \\ \hline
E_{1g}\uparrow & 0 & -\frac{\lambda}{3} & 0 & \frac{\lambda}{6} & 0 & \frac{\lambda}{6} \\
B_{1u}\uparrow & -\frac{\lambda}{3} & 0 & \frac{\lambda}{6} & 0 & \frac{\lambda}{6} & 0 \\
E_{1g}\uparrow & 0 & \frac{\lambda}{6} & 0 & \frac{\lambda}{6} & 0 & -\frac{\lambda}{3}
\end{array}$$
$$\begin{array}{c|ccc|ccc}
& n=5 & n=0 & n=1 & n=2 & n=3 & n=4 \\
& E_{2u}\downarrow & A_{1g}\downarrow & E_{2u}\downarrow & E_{1g}\downarrow
& B_{1u}\downarrow & E_{1g}\downarrow \\ \hline
E_{2u}\uparrow & C_{1} i & C_{5} & 0 & -\frac{\lambda}{6} & C_{6} & C_{7} \\
A_{1g}\uparrow & C^{*}_{5} & 0 & C_{5} & -C^{*}_{6} & \frac{\lambda}{3} &
-C_{6} \\
E_{2u}\uparrow & 0 & C^{*}_{5} & -C_{1} i & C^{*}_{7} & C^{*}_{6} & -\frac{\lambda}{6} \\ \hline
E_{1g}\uparrow & -\frac{\lambda}{6} & C_{6} & C_{7} & C_{1} i & C_{5} & 0 \\
B_{1u}\uparrow & -C^{*}_{6} & \frac{\lambda}{3} & -C_{6} & C^{*}_{5} & 0 &
C_{5} \\
E_{1g}\uparrow & C^{*}_{7} & C^{*}_{6} & -\frac{\lambda}{6} & 0 & C^{*}_{5}
& -C_{1} i\end{array}$$
with $C_{1} = \frac{\lambda}{\sqrt{12}}$, $C_{2} = \frac{\lambda}{\sqrt{12}}(1+i)$, $C_{3} = 0.105663\lambda(1+i)$, $C_{4} = 0.394337\lambda(1+i)$, $C_{5} = \frac{\lambda}{12} + \frac{\lambda}{2\sqrt{12}}i$, $C_{6} = \frac{\lambda}{4} + \frac{\lambda}{2\sqrt{12}}i$, $C_{7} = -\frac{\lambda}{6} +
\frac{\lambda}{\sqrt{12}}i$.
Several comments are in place here. First, spin-orbit coupling mixes QMOs at all [${\bf k}$-vector]{}s. Even at the $\Gamma $ point, [*i. e.*]{}, on the same hexagon, the three upper QMOs ($A_{1g}$ and two $E_{2u}$) are SO coupled to the three lower QMOs ($B_{1u}$ and two $E_{1g}$), which explains sizable shifts of the relativistic bands compared to the non-relativistic ones at this [${\bf k}$-vector]{}. Additionally, SO coupling induces splitting of the degenerate $E_{2u}$ and $E_{1g}$ states at all [${\bf k}$-vector]{}s. Another striking feature of the calculated $\lambda {\bf L}\cdot
{\bf S}$ matrix is that its $A_{1g}$, $E_{2u}$ (upper triplet) and $B_{1u}$, $E_{1g}$ (lower triplet) blocks are identical. This means that if not for the accidental near-degeneracy of the $A_{1g}$ and $E_{2u}$ states (which magnifies the SO induced energy shifts) the upper and the lower triplets would have been equally affected by the SO coupling.
Quasimolecular orbital basis versus relativistic basis
------------------------------------------------------
The main difficulty in describing the [[Na$_2$IrO$_3$]{}]{} bandstructure is that it interpolates between eigenstates of two Hamiltonians: the *itinerant* TB Hamiltonian of (primarily) *intra*hexagon electron hopping that preserves the $s_z$ spin subspace and the *local* spin-orbit (SO) interaction $\lambda{\bf L}\cdot{\bf S}$ Hamiltonian that couples *different* spin subspaces. The eigenstates of the TB Hamiltonian are quasi-molecular orbitals (QMOs), while the eigenstates of the SO interaction (in the $t_{2g}$ subspace) are *relativistic orbitals* (ROs) $|j_{\text{eff}},j^{z}_{\text{eff}}\rangle$ characterized by an effective total angular momentum $j_{\text{eff}}$ and its $z$-projection $j^{z}_{\text{eff}}$: $$\begin{aligned}
|\tfrac{1}{2},\tfrac{1}{2}\rangle & = \frac{1}{\sqrt{3}}|{xy}\uparrow
\rangle+\frac{i}{\sqrt{3}}|{xz}\downarrow\rangle+ \frac{1}{\sqrt{3}}|{yz}\downarrow\rangle, \notag \\
|\tfrac{1}{2},-\tfrac{1}{2}\rangle & = \frac{i}{\sqrt{3}}|{xz}\uparrow\rangle-\frac{1}{\sqrt{3}}|{yz}\uparrow\rangle+ \frac{1}{\sqrt{3}}|{xy}\downarrow\rangle, \notag \\
|\tfrac{3}{2},\tfrac{3}{2}\rangle & = \frac{i}{\sqrt{2}}|{xz}\uparrow
\rangle+\frac{1}{\sqrt{2}}|{yz}\uparrow\rangle, \notag \\
|\tfrac{3}{2},\tfrac{1}{2}\rangle & = -\sqrt{\frac{2}{3}}|{xy}\uparrow\rangle+\frac{i}{\sqrt{6}}|{xz}\downarrow\rangle+ \frac{1}{\sqrt{6}}|{yz}\downarrow\rangle, \notag \\
|\tfrac{3}{2},-\tfrac{1}{2}\rangle & = \frac{i}{\sqrt{6}}|{xz}\uparrow\rangle-\frac{1}{\sqrt{6}}|{yz}\uparrow\rangle-\sqrt{\frac{2}{3}}|{xy}\downarrow\rangle, \notag \\
|\tfrac{3}{2},-\tfrac{3}{2}\rangle & = -\frac{i}{\sqrt{2}}|{xz}\downarrow\rangle+\frac{1}{\sqrt{2}}|{yz}\downarrow\rangle.\end{aligned}$$
This basis [@comment_basis] can be explained as follows; three $t_{2g}$ orbitals (total degeneracy, including spins, is 6) are split into a lower-lying quartet $j_{\text{eff}} =3/2$ and an upper lying $j_{\text{eff}}=1/2$ doublet, and the $5d$-electrons of Ir$^{4+}$ fully occupy the lower quartet leaving the upper $j_{\text{eff}} =1/2$ doublet half-filled. This makes this situation similar to a non-degenerate Hubbard model (S=1/2 doublet on a site), with the important difference that in the Hubbard model the hopping matrix elements preserve the $s_z$ spin subspace, while here the states of the $j_{\text{eff}}=1/2$ doublet are spin-orbit mixed states, leading to a strong anisotropy of hoppings and their dependence on spin (or rather total moment) direction. This may bring about anisotropic exchange, [*e. g.*]{}, the Kitaev exchange on a honeycomb lattice [@Chaloupka2010].
By gradually increasing an effective spin-orbit coupling strength $\lambda_{\text{eff}}$, $$\lambda_{\text{eff}}=\frac{\lambda^{2}}{(
t_{1\,{\rm O}})^{2}+\lambda^{2}},\quad t_{1\,{\rm O}} =0.270~\mbox{eV}, \label{leff}$$ from 0 to 1, one can trace a smooth evolution of the TB+SO model eigenvalues from, respectively, the non-relativistic (QMO) limit to the fully relativistic (RO) limit (see Fig. \[F.evolution\] (a) for the data at the $\Gamma$ point). An SO coupling parameter of $\lambda=0.44$ eV for [[Na$_2$IrO$_3$]{}]{} corresponds to $\lambda_{\text{eff}}=0.73$, which is marked by a vertical dotted line in Fig. \[F.evolution\].
![Properties of the TB+SO model of [[Na$_2$IrO$_3$]{}]{} at the $\Gamma$ point as a function of effective SO coupling $\protect\lambda_{\text{eff}}$ defined in Eq. (\[leff\]). The vertical dotted line marks the realistic $\protect\lambda_{\text{eff}}=0.73$ value for [[Na$_2$IrO$_3$]{}]{}. (a) The eigenvalues of the TB+SO model at $\Gamma$. Eigenenergies have been scaled by $\sqrt{1-\lambda_{\rm eff}}$ to keep them within the $[-1.5,0.2]$ eV range. (b) The $j_{\text{eff}}=\frac{3}{2}$ (solid line) and $B_{1u}$ (dashed line) weights on the lowest state. (c) The $j_{\text{eff}}=\frac{1}{2}$ (solid line) and total $E_{2u}$ (dashed line) weights on the uppermost state. Inset shows individual contributions from the two $E_{2u}$ QMOs. []{data-label="F.evolution"}](fig8){width="45.00000%"}
The RO basis is an attractive starting point to describe the low-energy physics of [[Na$_2$IrO$_3$]{}]{} as it allows to truncate the Hamiltonian to only $j_{\text{eff}}=\frac{1}{2}$ states that dominate near the Fermi energy and map [[Na$_2$IrO$_3$]{}]{} onto the Kitaev-Heisenberg model. Although this approach might seem reasonable given the noticeable separation of the $j_{\text{eff}}=\frac{1}{2}$ and $j_{\text{eff}}=\frac{3}{2}$ characters in the DOS of [[Na$_2$IrO$_3$]{}]{} \[[*cf.*]{} Fig. 2 (b) of Ref. \], we argue that the itinerant terms are too strong to be neglected (which should not be surprising since $\lambda =0.44~\mbox{eV}<W\approx 4t_{1\,{\rm O}} =1~\mbox{eV}$) and that, consequently, the QMO basis is as well (or as poorly) justified to work with as the RO basis.
To support this statement, let us concentrate on the TB+SO model states at the $\Gamma$ point. Fig. \[F.evolution\] (a) shows the evolution of the model eigenvalues as a function of $\lambda_{\text{eff}}$ (Eq. \[leff\]). In the non-relativistic limit ($\lambda_{\text{eff}}=0$), the states are almost purely (with slight deviation due to orthorhombic distortion) QMOs, ordered as $B_{1u},E_{1g},A_{1g},E_{2u}$ with increasing energy [@note2]. At the same time, at each state the $j_{\text{eff}}=\frac{1}{2}$ contribution is $1/3$ and the $j_{\text{eff}}=\frac{3}{2}$ contribution is, correspondingly, $2/3$ (for one of the two Ir atoms). Note that, since the model distinguishes spin-$\uparrow$ and spin-$\downarrow$ states each level is doubly degenerate.
With the QMO splitting obviously prevailing for zero SO coupling, we now want to quantify the QMO character rectification upon increasing $\lambda_{\text{eff}}$ by calculating the QMO and RO weights on two selected states: the lowest ($B_{1u}$) and the uppermost ($E_{2u}$). The $B_{1u}$ state \[Fig. \[F.evolution\] (b)\] is a simpler case as it is non-degenerate (apart from spin) and quite well separated from the rest of the QMOs so that the SO effects here should be less important. Changing $\lambda_{\text{eff}}$ from 0 to 0.73 ([[Na$_2$IrO$_3$]{}]{} value), the $j_{\text{eff}}=\frac{3}{2}$ weight on this state increases from 0.6667 to 0.8320, whereas the $B_{1u}$ weight is only slightly reduced from 0.9932 to 0.9567. This indicates that the lowest relativistic state at the $\Gamma$ point in [[Na$_2$IrO$_3$]{}]{} is better described by a QMO $B_{1u}$ than by one of the $j_{\text{eff}}=\frac{3}{2}$ ROs. In fact, this turns out to hold for the whole lowest relativistic band \[[*cf.*]{} the $j_{\text{eff}}$- and QMO-projected [[Na$_2$IrO$_3$]{}]{} DOS in, respectively, Fig. 2 (b) of Ref. and Fig. S6 (b) of Ref. \].
The uppermost state is one of the $E_{2u}$ doublet states. It is near-degenerate with $A_{1g}$ and the other $E_{2u}$ and, therefore, the SO effects are here particularly strong. At the $\Gamma$ point, though, it can only couple to itself or to the other $E_{2u}$ \[see Eqs. (\[E.Guu\]) and (\[E.Gud\])\], depending on which linear combination of these degenerate states is considered. Upon switching $\lambda_{\text{eff}}$ on, the $j_{\text{eff}}=\frac{1}{2}$ weight on this upper states rapidly grows from 0.3333 to $\sim0.6$ in the range $0<\lambda_{\text{eff}}<0.05$, and then gradually increases to 0.8295 at $\lambda_{\text{eff}}=0.73$ \[Fig. \[F.evolution\] (c)\]. At the same time, the weight of one of the $E_{2u}$ states (we may call it $E_{2u}^{\prime}$) is reduced from 1.0 to 0.53730 \[see inset of Fig. \[F.evolution\] (c)\]. However, the total weight of two $E_{2u}$ states is barely changed: at $\lambda_{\text{eff}}=0.73$ it equals 0.9617. This means that the uppermost relativistic state at the $\Gamma$ point in [[Na$_2$IrO$_3$]{}]{} is very well described by a linear combination of two $E_{2u}$ states (which is also a QMO) with, in general, $\lambda_{\text{eff}}$-dependent individual contributions.
The $B_{1u}$ and $E_{2u}$ states (at $\lambda =0$) seem to simultaneously bear both RO and QMO features up to very strong SO coupling, with the QMO character dominating for $\lambda_{\text{eff}}<0.9$. This can also be illustrated by inspecting the composition of, [*e. g.*]{}, the lowest energy band state as shown in Table \[T.states\]. At zero SO coupling, the doubly degenerate lowest state corresponds to (almost) pure $B_{1u}\uparrow$ and $B_{1u}\downarrow$ QMOs [@note3]. At $\lambda_{\text{eff}}=0.73$, the structure of this state is strikingly similar to the $B_{1u}$ states, with only slight admixtures of the ${xz}$ and ${yz}$ orbitals of opposite spin. Even at some very high $\lambda _{\text{eff}}$, when the RO $j_{\text{eff}}=\frac{3}{2}$ weight is close to 1, the states retain the $B_{1u}\uparrow$ and $B_{1u}\downarrow$ QMO features.
)
------------------------------------- -- --------- --------- -- ------------------- ------------------- -- ------------------- -------------------
${xy}^{1} \uparrow$ -0.454 [0.0]{} -0.453 [0.0]{} -0.444 [0.0]{}
${xz}^{1} \uparrow$ -0.383 [0.0]{} -0.363 + 0.056$i$ -0.053 – 0.100$i$ -0.263 + 0.150$i$ -0.142 – 0.200$i$
${yz}^{1} \uparrow$ -0.383 [0.0]{} -0.363 – 0.056$i$ -0.100 – 0.053$i$ -0.263 – 0.150$i$ -0.200 – 0.142$i$
${xy}^{2} \uparrow$ 0.454 [0.0]{} 0.453 [0.0]{} 0.444 [0.0]{}
${xz}^{2} \uparrow$ 0.383 [0.0]{} 0.363 – 0.056$i$ 0.053 + 0.100$i$ 0.263 – 0.150$i$ 0.142 + 0.200$i$
${yz}^{2} \uparrow$ 0.383 [0.0]{} 0.363 + 0.056$i$ 0.100 + 0.053$i$ 0.263 + 0.150$i$ 0.200 + 0.142$i$
${xy}^{1} \downarrow$ [0.0]{} -0.454 [0.0]{} -0.453 [0.0]{} -0.444
${xz}^{1} \downarrow$ [0.0]{} -0.383 0.053 – 0.100$i$ -0.363 – 0.056$i$ 0.142 – 0.200$i$ -0.263 – 0.150$i$
${yz}^{1} \downarrow$ [0.0]{} -0.383 0.100 – 0.053$i$ -0.363 + 0.056$i$ 0.200 – 0.142$i$ -0.263 + 0.150$i$
${xy}^{2} \downarrow$ [0.0]{} 0.454 [0.0]{} 0.453 [0.0]{} 0.444
${xz}^{2} \downarrow$ [0.0]{} 0.383 -0.053 + 0.100$i$ 0.363 + 0.056$i$ -0.142 + 0.200$i$ 0.263 + 0.150$i$
${yz}^{2} \downarrow$ [0.0]{} 0.383 -0.100 + 0.053$i$ 0.363 – 0.056$i$ -0.200 + 0.142$i$ 0.263 – 0.150$i$
$j_{\text{eff}}=\frac{3}{2}$ weight
$B_{1u}$ weight
The features shown in this Section, not unexpectedly, characterize [[Na$_2$IrO$_3$]{}]{} as intermediate between the non-relativistic (pure quasi-molecular orbital) and fully relativistic (pure RO) cases.
Moreover, these results show that, in the RO representation, the upper band states are not pure $j_{\text{eff}}=1/2$ states but there is some significant mixing of $j_{\text{eff}}=3/2$ states. In fact, for the upper band states, the projections onto $j_{\mathrm{eff}}=1/2$ and $j_{\mathrm{eff}}=3/2$ are, respectively, 0.64 and 0.21 with 2(0.64$^{2}+2\times 0.21^{2})=1$, while in the non-relativistic case these projections are both equal to $\sqrt{1/6}=0.41$. Note that looking at the weights may be misleading. Indeed this state appears to be 2$\times 0.64^{2}=82\%$ pure $j_{\text{eff}}=1/2$ state \[Fig. \[F.evolution\] (bottom)\], but its $projection$ on the $j_{\mathrm{eff}}=3/2$ state is only twice smaller than in the non-relativistic case. In other words, the hopping between the upper Kramers doublets, initially not considered in Ref. , is only reduced by about a factor of two compared to the non-relativistic case. One but possibly not the only consequence of this fact is that the contribution of the Kitaev term in the analysis below may be overestimated, probably by as much as a factor of two.
Comparison with experiment: branching ratio
-------------------------------------------
An argument frequently used to justify the assumption of pure ROs in [[Na$_2$IrO$_3$]{}]{} is that it is experimentally supported. However, the experimental evidence is inconclusive. It is first assumed that the electronic states are pure ROs and then it is shown that this assumption does not contradict the experiment, yet the experiments, upon a closer look, do not falsify the DFT picture, either. A typical and, by far, the most often used quantity to discuss the nature of the states in iridates is the branching ratio (BR) extracted from X-ray absorption spectroscopy (XAS) experiments. In XAS, essentially, $\langle{\bf L}\cdot{\bf S}\rangle$ is measured. This expectation value is of course zero without spin-orbit coupling. A detailed and very insightful analysis can be found, for instance, in Refs. . In particular, it is shown that, for a related iridate, the main contribution to $\langle{\bf L}\cdot{\bf S}\rangle$ $(1.4$ out of $2.1$) doesn’t come from the $t_{2g}$ orbitals, which define the $j_{\mathrm{eff}}=1/2$ states, but from the admixture of the $e_{g}$ orbitals. In our calculations –shown below– we observe the same behavior.
We apply our TB+SO model to calculate $\langle{\bf L}\cdot{\bf
S}\rangle$ for [[Na$_2$IrO$_3$]{}]{} where ${\bf L}$ and ${\bf S}$ are, respectively, the *total* orbital and spin angular momenta of Ir $5d$ electrons. $\langle{\bf L}\cdot{\bf S}\rangle$ is related to the experimentally accessible branching ratio as $$\mbox{BR}=\frac{(2-r)}{(1+r)},\quad r=\frac{\langle{\bf L}\cdot{\bf S}\rangle }{n_{\text{h}}},$$ with $n_{\text{h}}=5$ being the average number of $5d$ Ir holes [@Laan88; @Thole88]. In recent XAS measurements [@Clancy], $\mbox{BR}=5.5-5.7$, translating to $\langle {\bf L}\cdot{\bf
S}\rangle=-2.7\hbar^{2}$, was obtained for [[Na$_2$IrO$_3$]{}]{} and interpreted as a sign of strong spin-orbit coupling.
When applying the TB+SO model that we constructed for [[Na$_2$IrO$_3$]{}]{} in Section \[sec:fiveA\] the calculated $\langle{\bf L}\cdot{\bf S}\rangle=-0.73\hbar^{2}$ (as compared to $-1\hbar^{2}$ in the limit $\lambda_{\rm eff}=1$). This value is several times smaller than the experimental value. This is, however, not unexpected given the significant contribution of the Ir $e_{g}$ empty states to $\langle{\bf L}\cdot{\bf S}\rangle$ ([*cf.*]{} Ref. ), which are not considered in the TB+SO model discussed in the previous Section. In order to make a meaningful comparison with experiment, we extend our TB+SO model to include (in the same spirit) also the Ir $e_{g}$ states. $\langle{\bf L}\cdot{\bf S}\rangle$ within such a model is $-1.91\hbar^{2}$. This is about 30% less than the experimental value reported by Clancy [*et al.*]{}[@Clancy]. This result is indeed in good agreement with experiment, given the large fluctuations in experimental values. For instance, Ref. reported $\langle{\bf L}\cdot{\bf S}\rangle$ = $-3.1\hbar^{2}$ for Sr$_{2}$IrO$_{4}$ while Ref. reported $-2.1\hbar^{2}$ (about 30% difference) for the same compound. This example gives a sense of possible fluctuations between results of different experimental groups, and therefore our theoretical $\langle{\bf L}\cdot{\bf S}\rangle$ value for [[Na$_2$IrO$_3$]{}]{} might be even closer to the true result.
The main conclusion from these calculations is that with the TB+SO model based on all [*five*]{} Ir $5d$ orbitals we are able to reasonably reproduce the large experimentally measured $\langle{\bf L}\cdot{\bf S}\rangle$ value in [[Na$_2$IrO$_3$]{}]{}, which validates our approach. As our analysis shows, the large $\langle{\bf L}\cdot{\bf S}\rangle$ does not necessarily mean an ideal separation of $j_{\text{eff}}=\frac{3}{2}$ and $j_{\text{eff}}=\frac{1}{2}$ RO states, but rather the effect of $e_g$ states also contributing in the process. Due to the peculiar electron hopping hierarchy in [[Na$_2$IrO$_3$]{}]{}, QMOs might be a better basis.
In conclusion, the XAS experiments only tell us that the upper Kramers doublet has a considerable contribution coming from $j_{\mathrm{eff}}=1/2$, but not that it is a pure RO state.
Comparison with experiment: RIXS
--------------------------------
Another experiment sometimes quoted as supporting the fully relativistic $j_{\text{eff}}=\frac{1}{2}$ picture is resonant inelastic x-ray scattering (RIXS) [@RIXS]. In this experiment a joint density of electronic states (JDOS) is probed, somewhat similar to that in the infrared absorption but with different matrix elements. The authors of Ref. observed several peaks in JDOS, of which the lowest peak at $\sim0.42$ eV was interpreted as transitions across the Mott-Hubbard gap, consistent with a 30% smaller optical absorption threshold. The next two peaks are close to each other at 0.72 and 0.83 eV and were ascribed to transitions from the $j_{\text{eff}}=3/2$ quartet into the upper $j_{\text{eff}}=1/2$ doublet. The splitting of 110 meV was ascribed to the trigonal splitting. Altogether, this interpretation suggests an SO coupling $\lambda\sim\frac{2}{3}(\frac{0.72+0.83}{2}-\frac{0.42}{2})\,\text{eV}\approx0.39\,\text{eV}$, a very reasonable number, if slightly too small.
This analysis, even though it looks reasonable on the first glance, has serious shortcomings. First, the deduced trigonal splitting is nearly twice as large as the actual trigonal splitting. In fact, the trigonal splitting is decided by the electrostatic field of the ligands, and in addition one-electron hoppings; both are very well accounted for by the DFT calculations, which give $\Delta_T$ = 75 meV. Second, even a $\Delta_T= 110$ meV cannot produce well separated peaks in JDOS, given that the Ir-Ir hopping is $t_{1\,{\rm O}}$ = 270 meV. Third, even if one completely neglects the Ir-Ir hoppings [@RIXS], in order to extract $\lambda $ and $\Delta _{T}$ one has to diagonalize the full Hamiltonian including both factors and then fit the resulting eigenvalues to the observed peaks. After doing that, one gets $\lambda
=0.5$ eV and $\Delta _{T}=180$ meV. Although the previous numbers are a rough estimate since they depend on the direction of the Ir spins as well as on $U$ (here we considered $U=0$), the latter number is more than twice the actual trigonal splitting. This argument shows that an interpretation of RIXS in terms of infinitely narrow bands split by the trigonal field may not be completely correct.
We find, on the other hand, that this experiment is consistent with DFT band structure. To demonstrate that, we have performed DFT calculations for the magnetic zigzag phase. We note that the results do not depend qualitatively on the choice of the pattern and the magnetization direction. In order to account for the missing correlation effects and adjust the direct gap to be consistent with infrared measurements[@Comin12], we applied a rigid shift of 200 meV between the occupied and empty bands (scissor operator). This exercise gives a JDOS which has a broad feature, consisting of (i) a peak at 0.42 and a shoulder 0.48 eV (compared to 0.42 eV in the experiment) corresponding to the transition between the top QMOs and (ii) a peak at 0.77 eV and a shoulder at 0.81 eV corresponding to transitions from the lower QMOs. While the experiment finds two peaks at 0.72 and 0.83 eV, one should keep in mind that the matrix elements, omitted in our calculation, can easily suppress or enhance a shoulder, making it disappear (at 0.48 eV) or become a separate peak (at 0.81 eV). Therefore we conclude that the agreement between experiment and our calculations, simplified as they are, is reasonably good.
Magnetism {#sec:six}
=========
We proceed now with the discussion of the magnetic behavior of [[Na$_2$IrO$_3$]{}]{}. Neutron diffraction experiments reported long-range antiferromagnetic order at low temperatures in a zigzag pattern [@Radu]. This ordering was confirmed by relativistic spin-polarized DFT calculations [@we] where we showed that it is the itinerancy of the system that stabilizes the zigzag configuration. Such a pattern was also predicted from the localized nnKH model [@Chaloupka2010; @Chaloupka2012] (Eq. \[nnKHmodel\]). In the following we will provide [*ab initio*]{}-derived estimates for the Kitaev and Heisenberg terms and will show that in the physically reasonable parameter range this model unfortunately fails to reproduce the experimentally observed magnetic order.
Nearest neighbor Kitaev-Heisenberg model
----------------------------------------
One term neglected in the conventional Kitaev-Heisenberg model treatment is the single-site magnetocrystalline anisotropy. Localized electrons with the spin $1/2$ do not have any anisotropy, no matter how strong the spin-orbit coupling is. However, if hopping is considered, electrons can have a preferred spin direction, which in the language of the nnKH Hamiltonian would be reflected in a single-site term proportional, in the lowest order, to $({\bf A}\cdot{\bf S})^2$ where ${\bf A}$ is a vector. Such terms are usually neglected when dealing with the nnKH model. Our calculations [@we] without including $U$ show a magnetic anisotropy as large as 3 meV per Ir (in order to address the single-site anisotropy, we compared ferromagnetic calculations). This energy should be compared to the total magnetic stabilization energy (i.e. the energy difference between magnetic and non-magnetic solutions) of maximally 5 meV. When the DFT calculations are performed including a $U=2$ eV, the magnetic anisotropy is as large as 8 meV out of a total energy of 28 meV. This substantial anisotropy suggests that a single site term should be added to the Kitaev-Heisenberg Hamiltonian, probably resulting in a rather different phase diagram.
With all these caveats, it is still instructive to analyze where [[Na$_2$IrO$_3$]{}]{} is to be found in the parametric space of the nnKH model. We make the following assumptions: (i) that the atomic orbitals are fully localized and the appropriate basis is given by pure $j_{\mathrm{eff}}=1/2$ orbitals; (ii) that the only hoppings relevant for magnetic interactions are $pd$ hoppings, so that the only oxygen assisted Ir-Ir hoppings are specific $t_{2g}-t_{2g}$ hoppings between unlike orbitals, as outlined in Refs. , and the $t_{2g}-e_{g}$ hoppings given in Ref. ; and (iii) that the only processes contributing to magnetic interactions are those listed in Ref. .
Indeed, the fact that the experimentally observed magnetic order is zigzag suggests that either the Heisenberg terms are exceptionally long ranged (the 3rd neighbor exchange is comparable to the 1st one) [@Radu; @trebst], or that the Kitaev term is strong and antiferromagnetic [@Chaloupka2012; @chinese]. The former suggestion is seemingly in contradiction with the fact that the calculated 3rd neighbor hoppings are substantially smaller than the 1st neighbor ones. This makes it impossible to explain the large 3rd neighbor exchange integral in terms of superexchange. However, there is a possibility, suggested in Ref. , that the Ir electrons are itinerant over individual hexagons, which makes magnetic interactions naturally long ranged, and not directly related to the hopping integrals.
The second suggestion, which is the one we will focus on in what follows, was proposed in Ref. , namely that of an antiferromagnetic Kitaev term. If strong enough, this could explain the observed magnetic order. Below we consider the expressions presented in Ref. and substitute the unknown variables with *ab initio*-derived parameters.
Chaloupka *et al.* [@Chaloupka2012] discuss four relevant processes contributing to the exchange interactions in [[Na$_2$IrO$_3$]{}]{}: (1) Direct hopping $t_{1\sigma}$ between nearest neighbor Ir $t_{2g}$ orbitals contributing with a term $I_{1}=\left(
\frac{2}{3}t_{1\sigma}\right) ^{2}/U$ to the Heisenberg term, where $U$ is the Coulomb repulsion between $t_{2g}$ electrons.
\(2) Interorbital nearest neighbor Ir $t_{2g}$-$e_{g}$ hopping via intermediate oxygens $\tilde{t}_{1}$, with $\tilde{t}_{1}=t_{pd\sigma
}t_{pd\pi }/\Delta $, where $\Delta $ is the charge-transfer energy (the difference between the O $p$ and Ir $d$ levels) contributing with a term $I_{2}=\frac{4}{9}\frac{\tilde{t}_{1}^{2}}{\tilde{U}}\,\frac{\tilde{J}_{\mathrm{H}}}{\tilde{U}}$ both to the Kitaev and Heisenberg terms, but with the opposite signs. Here $\tilde{U}$ is the excitation energy associated with the $t_{2g}$-$e_{g}$ hopping $i.e$. it also includes crystal field splitting, $\tilde{U}$ = $U+10Dq$. $\tilde{J}_{\mathrm{H}}$ is the Hund’s rule coupling between $t_{2g}$ and $e_{g}$ electrons.
\(3) Oxygen-assisted hopping between two nearest neighbor Ir $t_{2g}$ orbitals $t_{1\,{\rm O}}$ contributing with a term $I_{3}=\frac{8}{3}\frac{ t_{1\,{\rm
O}}^{2}}{U}\,\frac{J_{\mathrm{H}}}{U}$ to the Kitaev term, where $J_{\mathrm{H}}$ is the Hund’s rule coupling between $t_{2g}$ electrons, and, we remind, $ t_{1\,{\rm O}} =t_{pd\pi }^{2}/\Delta $.
\(4) Oxygen-$2p$ – Iridium-$5d$ charge transfer contributing with a term $I_{4}=\frac{8 t_{1\,{\rm O}}^{2}}{9}[\frac{1}{2\Delta
+U_{p}-3J_{p}}+\frac{1}{3(2\Delta +U_{p}-J_{p})}+\frac{2}{3(2\Delta
+U_{p}+2J_{p})}-\frac{1}{\Delta }],$ where $U_{p}$ and $J_{p}$ are, respectively, the Hubbard repulsion and the Hund’s rule parameter for oxygen. This expression was derived by G. Khaliullin [@Khaliullin] and is worth some additional discussion. The first three terms correspond to processes where two holes of the same or of opposite spins meet at an oxygen atom. Neglecting $J_{p},$ one gets simply $\frac{8t_{1\,{\rm O}}^{2}}{9}\frac{1}{\Delta +U_{p}/2},$ which reflects the fact that if the Ir atoms have opposite spins one can create an intermediate state with two holes on the same oxygen orbital, which lowers the total energy. The last term appears due to ring exchange, with an intermediate state where two holes are located on different oxygens. This process is only allowed when the ground state is FM, and only if the ground state hole is in an $a_{1g}$ or $j_{\rm eff}=1/2$ state, but not for pure $t_{2g}$ orbitals. However, contrary to a common misconception, $J_{p}$ is large, between 1.2 and 1.6 eV. We have estimated $U_{p}$ and $J_{p}$, using the technique described in Ref. , and obtained $U_{p}= 2.7$ and $J_{p}= 1.6$ eV, consistent with earlier DFT estimates [@Mazin97]. For non-relativistic orbitals it is comparatively straightforward to account for the Hund’s rule coupling on O, but for relativistic orbitals it becomes more tedious.
If we expand $I_4$ in both $U_{p}$ and $J_{p}$, then $I_{4}\approx \frac{8 t_{1\,{\rm O}}^{2}}{9}
\frac{U_{p}-J_{p}}{2\Delta ^{2}}.$ This expression shows that $U_{p}$ alone contributes ferromagnetically to the Heisenberg term and antiferromagnetically to the Kitaev term and may shift the various phases in the nnKH model. Together with $J_{p}$ though, for the values suggested above the effect of $U_{p}$ and $J_{p}$ largely cancels and $I_4$ appears to be unimportant (note though that if $J_{p}$ is entirely neglected, as in Ref. , this proposition becomes more questionable).
Summarizing the above terms into a single expression, Eq. (\[nnKHmodel\]) can be written as: $$H_{ij}^{(\gamma)}=\underbrace{(2I_{2}-I_{3}+2I_{4})}_{\displaystyle2K}S_{i}^{\gamma
}S_{j}^{\gamma }+\underbrace{(I_{1}-I_{2}-I_{4})}_{\displaystyle J}{\bf S}_{i}\cdot {\bf S}_{j}. \label{eqKJ}$$
This model has a zigzag magnetic ground state [@Chaloupka2012] if the Kitaev term is antiferromagnetic (AFM) and the Heisenberg term is ferromagnetic (FM), with $K>0$, $J<0$ and $-26\lesssim J/K \lesssim -0.3$.
In Table \[tab:parameters\] we provide the parameter values relevant for [[Na$_2$IrO$_3$]{}]{}, as obtained from our DFT results. Note that the $\tilde{t}_{1}$ parameter was assumed to be $2t_{1\,{\rm O}}$ in Ref. , while in the calculations (DFT calculations are usually very reliable in this respect) $\tilde{t}_{1}/t_{1\,{\rm O}}$ is 1.4. However, using the ratio of 2 hardly changes any conclusions.
parameter value (eV) meaning
-------------------------- ------------- ----------------------------------------
$t_{1\sigma}$ 0.03 direct Ir-Ir hopping
$t_{1\,{\rm O}}$ 0.27 O assisted Ir-Ir hopping
$\tilde{t}_1$ 0.38 Ir $t_{2g}-e_{g}$ hopping
$t_{pd\pi}$ 0.57$^{*}$ Ir-O $\pi$ hopping
$t_{pd\sigma}$ 1.6$^{*}$ Ir-O $\sigma$ hopping
$\Delta$ 2.4 charge transfer energy
between the O $p$ and Ir $d$ levels
$J_{\mathrm{H}}$ 0.5 Ir $t_{2g}$ Hund’s rule coupling
$\tilde{J}_{\mathrm{H}}$ 0.5 Ir $t_{2g}-e_{g}$ Hund’s rule coupling
: DFT-calculated values of transfer integrals and charge transfer energy for [[Na$_2$IrO$_3$]{}]{} and estimates of Hund’s rule coupling strength as described in the text. The values marked with $*$ were obtained from $\tilde{t}_{1}$, $t_{1 O}$ and $\Delta$.[]{data-label="tab:parameters"}
We present our results in Figures \[KJ\] and \[all\]. In Fig. \[KJ\] we show the calculated values of $K$ and $J$ as a function of two variables: the $x$ axis is the Hubbard $U$ associated with the upper Kramers doublet, and the $y$ axis is the energy $\tilde{U}$, associated with exciting an individual electron from the upper $t_{2g}$ to an average $e_{g}$ state. The Hubbard $U$ for $5d$ electrons is, generally speaking, 1.5 to 2 eV. However, in this case it is additionally screened by the $e_{g}$ electrons, and also reduced by hybridization (*cf*. Na$_{x}$CoO$_{2}$ [@Liebsch] and Fe pnictides [@Georges]). Experimental estimates of the Hubbard $U$ defined as the energy cost for exciting electrons across the insulating gap (which is the definition relevant to superexchange) yield 0.3-0.5 eV [@Comin12; @RIXS]. Additionally, LDA+U calculations with $U\sim2 $ eV yield an excitation gap of the same order. We conclude that the realistic range of this parameter is 0.5–2 eV, with the smaller values more likely.
For the second parameter, $\tilde{U}$, DFT calculations give $\sim2.5$ eV. This should be considered as a lower bound since DFT tends to slightly overestimate the orbital overlap and crystal fields, and misses the effects of the $t_{2g}-e_{g}$ Hubbard interaction. One can thus limit the physically admissible range in the region 2.5 eV $\lesssim\tilde{U}\lesssim3$ eV.
In Figure \[all\] we show the phase diagram in the space of the two parameters above. Several observations are in place: (1) While there is a zigzag phase in this diagram, it is very far removed from the range of the parameters that can be called physical, 0.5 eV $\lesssim
U\lesssim2$ eV, 2 eV$\lesssim$ $\tilde{U}\lesssim3$ eV (even though in the above estimate we have liberally stretched the admissible range in favor of a zigzag phase). In fact, the zigzag regime appears only when $\tilde{U}<0.6U$, i.e. when the Hubbard gap is larger than the $e_{g}-t_{2g}$ splitting, a rather unlikely proposition. (2) In the physical range of parameters, the ground state is either ferromagnetic or the spin liquid phase. It is rather curious that the very narrow slivers of the phase space in the $J,K$ coordinates [@Chaloupka2012] are transformed into a very large range in the $U,\tilde{U}$ space.
It is also worth mentioning that in order to explain the experimental data of Refs. one needs not only the ground state to be zigzag, but also that $K$ be several times larger than $|J|$; Chaloupka *et al.* [@Chaloupka2012] used $K=10.44$ and $J=-4.01$ meV. This solution cannot be obtained for a given set of $U$ and $\tilde{U}$ (see Fig. \[KJ\]). Moreover, a closer look at the expressions in their work reveals that $K+J=I_{1}-I_{3}/2$, which does not depend on $\tilde{U}$ and is always negative. Thus the two equalities above cannot be satisfied simultaneously for any choice of parameters, be they physical or not. Moreover, the values of $J$ and $K$ used in Ref. can only be obtained if $\tilde{U}<0.2$ eV, which is clearly an impossible regime.
{width="80.00000%"}
![(Color online) Phase diagram of the Kitaev-Heisenberg model for [[Na$_2$IrO$_3$]{}]{} with parameters determined following Ref. . The calculated exchange integrals are functions of the Mott-Hubbard gap $U$ and the cubic crystal field splitting $\tilde{U}$. The contours mark isolines of the ratio $K/J$.[]{data-label="all"}](fig10){width="\columnwidth"}
Long-range exchange
-------------------
As mentioned above, an alternative interpretation of the experimental results, given in Refs. , is in terms of sizable 2nd and 3rd neighbor exchange constants, comparable to the nearest neighbor exchange. In this picture the Kitaev term may or may not play a role, but this role is not decisive in establishing the observed magnetic order. Given that the calculated hopping amplitudes (Table I) are clearly dominated by the nearest neighbor terms, standard superexchange cannot explain such long range interactions.
However, it is important to remember that in the opposite, itinerant limit every electron is fully delocalized over a hexagon and, as such, is equally sensitive to the mean field magnetization pattern on the 1st, 2nd or 3rd nearest neighbors. As discussed in our earlier work [@we], the zigzag order, as compared to the stripy one, results in a sizable pseudogap at the Fermi level even without a Hubbard $U$. This creates an energy gain that cannot be cast in a form of nearest neighbor interaction, as it depends on the magnetization pattern over an entire hexagon.
We are far from stating that the superexchange Hamiltonian outlined in Ref. is irrelevant and an itinerant description will give the final answer to all questions regarding the magnetism in [[Na$_2$IrO$_3$]{}]{}. However, relying solely on the localized picture and, correspondingly, on the nnKH model, is, apparently, inadequate.
Conclusions
===========
In summary, we have performed an extensive investigation of the electronic properties of [[Na$_2$IrO$_3$]{}]{} in the framework of non-relativistic and relativistic density functional theory calculations and derived by means of the Wannier function projector method, the corresponding microscopic parameters. We resolved the following open questions: (1) By considering various idealized crystal structures for [[Na$_2$IrO$_3$]{}]{} we could disentangle the effect of each of the structural distortions present in this system and concluded that it is the joint effect of these distortions that constructively enhances the intrahexagon effective hopping parameters and suppresses the interhexagon ones favoring the formation of quasi-molecular orbitals. (2) We modelled the relativistic DFT results in terms of a tight-binding model including the spin-orbit coupling term and analyzed the electronic properties of [[Na$_2$IrO$_3$]{}]{} in terms of two complementary descriptions, the (itinerant) quasi-molecular basis and the (localized) relativistic $j_{\rm eff}$ basis. We observed that the behavior of [[Na$_2$IrO$_3$]{}]{} lies in between the fully itinerant and the fully localized description and that a quasi-molecular orbital description keeps its character even at large values of the spin-orbit coupling strength. (3) We showed that XAS and RIXS observations can be well understood within an itinerant description of [[Na$_2$IrO$_3$]{}]{} in contrast to other iridates like Sr$_3$CuIrO$_6$ where localization is imposed by the crystallographic arrangement of the IrO$_6$ octahedra [@Liu2012]. (4) Finally, we provided [*ab initio*]{}-derived estimates for the parameters appearing in the Kitaev and Heisenberg terms in [[Na$_2$IrO$_3$]{}]{} and found that the recently proposed nnKH model (see Section \[sec:six\]), even though it is a very interesting model [*per se*]{}, is unfortunately not realistic for [[Na$_2$IrO$_3$]{}]{}. In conclusion, in order to obtain a full understanding of the behavior of [[Na$_2$IrO$_3$]{}]{} all three features; spin-orbit, Coulomb correlations and delocalization of valence electrons over Ir$_6$ hexagons are essential.
H.O.J., D.Kh. and R.V. acknowledge support by the Deutsche Forschungsgemeinschaft through grants SFB/TR 49 and FOR 1346 (H.O.J. and R.V.) and SFB 608 and FOR 1346 (D.Kh.). \[sec:seven\]
Appendix. Idealized [N$_{2}$IO$_{3}$]{} crystal structures as used in WIEN2
===========================================================================
Experimental structure from Ref.
---------------------------------
[cp[1cm]{}cp[1.0cm]{}cp[0.6cm]{}c]{}\
\
Atom & & $x$ & & $y$ & & $z$\
Na1 & & 0.0 & & 0.0 & & 0.5\
Na2 & & 0.5 & & 0.5 & & 0.0\
Na3 & & 0.5 & & 0.5 & & 0.3400\
Ir & & 0.0 & & 0.0 & & 0.1670\
O1 & & 0.4590 & & 0.2110 & & 0.1780\
O2 & & 0.0070 & & 0.7960 & & 0.0\
Idealized structure $S_1$
-------------------------
[cp[1cm]{}cp[1.0cm]{}cp[0.6cm]{}c]{}\
\
Atom & & $x$ & & $y$ & & $z$\
Na1 & & 0.0 & & 0.0 & & 0.5\
Na2 & & 0.5 & & 0.5 & & 0.0\
Na3 & & 0.5 & & 0.5 & & 0.3333\
Ir & & 0.0 & & 0.0 & & 0.1667\
O1 & & 0.4646 & & 0.2097 & & 0.1785\
O2 & & 0.0000 & & 0.7903 & & 0.0\
Idealized structure $S_2$
-------------------------
[cp[1cm]{}cp[1.0cm]{}cp[0.6cm]{}c]{}\
\
Atom & & $x$ & & $y$ & & $z$\
Na1 & & 0.0 & & 0.0 & & 0.5\
Na2 & & 0.5 & & 0.5 & & 0.0\
Na3 & & 0.5 & & 0.5 & & 0.3333\
Ir & & 0.0 & & 0.0 & & 0.1667\
O1 & & 0.4606 & & 0.1909 & & 0.1667\
O2 & & 0.0000 & & 0.8091 & & 0.0\
Idealized structure $S_3$
-------------------------
[cp[1cm]{}cp[1.0cm]{}cp[0.6cm]{}c]{}\
\
Atom & & $x$ & & $y$ & & $z$\
Na1 & & 0.0 & & 0.0 & & 0.5\
Na2 & & 0.5 & & 0.5 & & 0.0\
Na3 & & 0.5 & & 0.5 & & 0.3333\
Ir & & 0.0 & & 0.0 & & 0.1667\
O1 & & 0.5000 & & 0.2443 & & 0.1667\
O2 & & 0.0000 & & 0.7557 & & 0.0\
Note that due to the necessity of using a monoclinic angle $\gamma$ in WIEN2k, the Ir honeycomb layers in the [[Na$_2$IrO$_3$]{}]{} unit cells presented above are parallel to the $ac$ plane. Accordingly, within this convention the vector of the Bloch factors in Eqs. (\[E.QMOproj\]) and (\[E.SOtrans\]) is given by $$T_{M=1\ldots6}({\bf k})=
(1, e^{-ik_{x}\tilde{a}}, e^{ik_{z}\tilde{c}},
e^{i(k_{z}\tilde{c}-k_{x}\tilde{a})}, e^{ik_{z}\tilde{c}}, e^{-ik_{x}\tilde{a}}),$$ where $\tilde{a}$ and $\tilde{c}$ are the lengths of the two primitive lattice vectors lying in the $ac$ plane. Here, one explicitly accounts for the choice of WIEN2k of the actual positions of the two Ir atoms in the primitive unit cell, which are, [[*[e. g.]{}*]{}]{}, (-0.167,0,0.167) and (-0.833,0,0.833) in the experimental [[Na$_2$IrO$_3$]{}]{} structure.
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The spin-orbit splitting is given by the difference of the eigenvalues $\lambda /2$ and $-\lambda$ in the $j_{\rm eff}$ space.
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|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'For a Legendrian link $\Lambda \subset J^1M$ with $M = {\mathbb {R}}$ or $S^1$, immersed exact Lagrangian fillings $L \subset \mbox{Symp}(J^1M) \cong T^*({\mathbb {R}}_{>0} \times M)$ of $\Lambda$ can be lifted to conical Legendrian fillings $\Sigma \subset J^1({\mathbb {R}}_{>0} \times M)$ of $\Lambda$. When $\Sigma$ is embedded, using the version of functoriality for Legendrian contact homology (LCH) from [@PanRu1], for each augmentation $\alpha: \mathcal{A}(\Sigma) \rightarrow {\mathbb {Z}}/2$ of the LCH algebra of $\Sigma$, there is an induced augmentation $\epsilon_{(\Sigma,\alpha)}: \mathcal{A}(\Lambda) \rightarrow {\mathbb {Z}}/2$. With $\Sigma$ fixed, the set of homotopy classes of all such induced augmentations, $I_\Sigma \subset \mathit{Aug}(\Lambda)/{\sim}$, is a Legendrian isotopy invariant of $\Sigma$. We establish methods to compute $I_\Sigma$ based on the correspondence between Morse complex families and augmentations. This includes developing a functoriality for the cellular DGA from [@RuSu1] with respect to Legendrian cobordisms, and proving its equivalence to the functoriality for LCH. For arbitrary $n \geq 1$, we give examples of Legendrian torus knots with $2n$ distinct conical Legendrian fillings distinguished by their induced augmentation sets. We prove that when $\rho \neq 1$ and $\Lambda \subset J^1{\mathbb {R}}$ [*every*]{} $\rho$-graded augmentation of $\Lambda$ can be induced in this manner by an immersed Lagrangian filling. Alternatively, this is viewed as a computation of cobordism classes for an appropriate notion of $\rho$-graded augmented Legendrian cobordism.'
address:
- Massachusetts Institute of Technology
- Ball State University
author:
- Yu Pan
- Dan Rutherford
bibliography:
- 'PR.bib'
title: Augmentations and immersed Lagrangian fillings
---
|
{
"pile_set_name": "ArXiv"
}
|
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